tag:blogger.com,1999:blog-57035688076152638512018-03-24T11:53:34.971-07:00How to Prepare for the Gifted And Talented TestGAT tests measure cognitive skills but schools don't teach them. This site helps parents identify the material, teach the skills, and not only gain GAT entry but succeed in the program.Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.comBlogger229125tag:blogger.com,1999:blog-5703568807615263851.post-23046390086372947412018-03-24T11:53:00.002-07:002018-03-24T11:53:35.085-07:00Really Bad AdviceMy google news feed has delivered articles from the SAT experts. I've been reading how to solve the hardest math problems on the SAT to see if I can improve my super secret strategy for the TTWBN test. I'm learning that the experts don't know anything useful.<br /><br />The goal of an expert is to dissect the problem down to steps that lead to the solution. This teaches nothing, of course, except for a memorized list of steps for a problem that will never be seen again. The solution advice has an element of time management, as in narrow down the problem to the work that has to be done in the shortest time possible. This approach will backfire, because once you short circuit the analysis with time pressure, it's much harder to find the right path. Dead ends will be stress inducing. Unless you are an expert and already at the 1600 level, in which case it's easy. <br /><br />The TTWBN test has no time limit, and we're going to take full advantage of that fact, like 4 hours per topic. The difference between preparing for the TTWBN test now and the SAT in high school is that we'll spend one or two sessions under time pressure before the SAT . The prep process is going to be identical, including spending 10 minutes per question.<br /><br />I've rarely mentioned one of George Poyla's strategies for solving geometry problems. Rarely mentioned it, but we do it all of the time and it's behind 'Read The Question' for little people preparing for the COGAT. He warns readers that geometry proofs will need to use prior results, maybe from the last proof or from last week, to solve the current problem.<br /><br />The version that I use for grade school is that if you see a geometry, solve everything before you read the question. I want every line labelled with a length and every angle with degrees. If it's an algebra problem, be prepared to rearrange and transform. I've written before about this in the context of verbal analogies. Here's what inevitably follows:<br /><br /><ul><li>We get stuck because someone forgot that a + b + c = 180 or adjacent angles sum to 180 or something else that we didn't cover yet. So we cover it.</li><li>During this process, the characteristics of the problem at hand become clear.</li><li>The solution strategy presents itself and the answer usually becomes known before the pick list is surveyed.</li></ul><div>This is a much better approach than "What I am supposed to do?" followed by me explaining solution steps. I might as well talk to the wall.</div><div><br /></div><div>Before 4th grade, this skill is called 'Read The Question' and involves me asking lots of what if questions about a figure matrix or verbal analogy for 20 minutes before we actually pick an answer. I originally did this because really challenging COGAT test prep questions take me a long time to create and aren't found on practice tests so I wanted to get the most out of each question.</div><div><br /></div><div>I'm currently experimenting with similar approaches to Reading Comp. When I get to the end of a boring passage, I remember very little about the passage, maybe 2 nouns like bridge and engineer. Then I get a list of questions that ask who the author is, what type of writing is this, how are they feeling, how many arguments are in the passage. Then I go back and reread the passage to find out. 3 years into this, it dawned on me that I'm going to be asked this stuff anyway, so I might as well look for it.</div><div><br /></div><div>A parent might be fooled by the engaging quality of most reading comp passages. Don't be fooled. You're an adult now. Everything is interesting to you. Your child is totally bored beyond comprehension. So I announced that after the passage is read, and before we begin work on the questions, I want be told a lot about the passage, like who's writing it, what type of writing is it, what's the point of each paragraph, when did it happen? I'm inching our way toward not having to read the passage a second time thoroughly (thinking ahead to a timed test). I'm the same way about the questions. Was line 32 about eclectic dissension? Exuberant facilitation? Ascetic abnormalism? If we're luck, the answers will have about 20 words that need definitions analysis in the context of the narrative. Unless it isn't a narrative.</div><div><br /></div><div>Will the child take the hint and adopt this approach to reading or math? Certainly not in my presence, out of spite, but probably in the classroom and when it counts on the test. I've caught them both doing things properly when they thought I wasn't looking.</div><div><br /></div><div>So here's my bad advice. If you follow my approach properly, your child will get through very little material, probably do it wrong 5 times, forget the next day what was learned, and not have any academic knowledge to show for it. All the while, the important skills will be forming. Then one day they will magically know everything and things will be really easy. The first few months are a struggle.</div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-14247862306411663862018-03-17T09:57:00.000-07:002018-03-17T10:35:38.320-07:00The Test That Won't Be NamedIn this article, I'm going to jam 7 articles into one because I'm really pressed for time on the weekends.<br /><br /><b>Review of Home Schooling Literature</b><br />I've been reviewing home schooling guides lately to see if there's anything that I can add to my At Home School curriculum. "At Home Schooling" means doing a little extra work weeknights and weekends to make up for the slow pace of learning at school.<br /><br />Home school curriculum guides are pretty disappointing. If I were full time home schooling my child, I would be planning to send the child to Stanford at age 14 because home schooling is so easy. The curriculum guides shoot for something more average.<br /><br />As most curriculum guides point out, trying to teach anything to your child is really hard. What they don't point out is that your child will learn at an accelerated pace once you stop teaching. The impossible becomes the easy. The secret is in the approach, which I will describe in the next article (below).<br /><br /><b>The Secret to Learning</b><br />Almost every week, I have to remind my kids that they have to slow down. I had to tell the younger one this story again.<br /><br /><i>There were two equally bright, equally capable children. One was dumb and one was a genius. The dumb one looked at a hard problem, became frustrated because he didn't know it, and started guessing. He got the wrong answer. The genius looked at the same problem, became frustrated because he didn't know it, and started to work on it slowly one step at a time. He tried 3 times to do it, and finally chose the answer, which was also wrong.</i><br /><i><br /></i><i>A third child who was equally bright and capable also struggled with this problem. He was smart. He also took a long time to work through this frustrating problem, and after his fifth try, he bothered to check his answer, found a mistake, and fixed it. The smart child got the correct answer. </i><br /><i><br /></i>The smart child is getting 99% on the Test That Won't Be Named, but the genius is stuck between 85% and 95%. Both are learning about the same amount. Maybe the smart child is getting a bit more out of the learning process because he's checking his work. What's the problem here? The problem is that the smart child is fixated on the goal of a solution, especially the correct solution, and the genius is more interested in the learning process. Eventually, the smart child is going to be in an advanced accelerated course (or maybe pre Algebra) and the work is going to be really hard. Both the genius and the smart kid will make lots of mistakes, and this will bother the smart kid so much that he drops out. But the genius, who doesn't care about the answer in the first place, will just plod on as usual until he has a PhD in a joint Law Medicine Chemical Engineering Medieval Slovakian Literature.<br /><br />I've warned the genius that he better start checking answers because if he doesn't get a perfect score on the TTWBN test he can forget about AP courses because he won't get into a good school.<br /><br /><b>The Secret For Parents</b><br />Among equally capable parents, we find dumb, smart, and genius parents. The problem that parents need to solve is that you have a child doing a problem - whether it's a cognitive skills exam, or one of the 2 main sections on the TTWBN test - and your child is totally not getting it. Dumb parents expect their child to get it, smart parents expect their child to get it after a long struggle, and genius parents really don't care.<br /><br />Once you see a child go through this process, you get it as a parent, and work and frustration is replaced by work and learning. For this reason, the 2nd child should always end up twice as smart as the oldest sibling, given a fraction of the learning time.<br /><br />When I was a dumb parent, I came up with the parent skill set in order to survive the first few rounds of my ridiculous At Home School curriculum goals. The very first goal was to skip first grade math and do 2nd grade math starting on winter break in Kindergarten. This was the worst and best idea I ever came up with. (Tip - if you do hard core COGAT test prep at age 4, 2nd grade math at age 5 isn't all that challenging).<br /><br />As a reminder, my survival steps include start every problem by acknowledging that you are totally baffled, take a long, long time reading the question, going so far as to do a workbook on the topic before you get to the answer, make a lot of mistakes and go out for ice cream any time the child gets 100% wrong, and if a test is coming up, check the #%$!!!! answer. The parent will encourage these steps. For the parent, I'd like to add 1) set your expectations at zero, 2) I really mean zero, not .0001 but zero, and 3) stop looking at the solutions.<br /><br />You can't practice learning skills (see prior paragraph) if your child is doing a 30 question timed worksheet or knows the material or doesn't make mistakes. That's why we have a pace of 1 to 5 super hard problems in Math House.<br /><br /><b>Reading</b><br />I always considered reading to be a filler activity. I'm beginning to think differently. Competition for GAT seats is between kids who read 6 hours a day, and those of us who will just become really good problem solvers (aka shapes, math and logic) and cheat our way into the program. Cheating is much more satisfying and is the basis for higher order math.<br /><br />To be on the safe side, we did lots of vocab (vocabularyworkshop.com) and 2nd grade phonics starting on day 1 (Pre-K Phonics Conceptual Vocabulary and Thinking). But it was always primarily silly and fun. Why discourage a life of reading by putting pressure on the first year?<br /><br />I think my casual approach to reading is the reason language arts thrived in Math House.<br /><br />Yes, I grilled the kids at the Word Board (<i>How would a commander on the battle field use the word 'dispersion' in a sentence?</i>), but they didn't actually have give me a proper response and I didn't want to take the words down because I was going to quiz them on the synonyms in a few days anyway.<br /><br />But mainly we went slowly and had fun. When I say slowly, I mean when you try to slow down to nothing the child learns at a highly accelerated rate. That doesn't make sense until you see it happen, but it always does.<br /><br /><b>The Magic of Slow</b><br />I've decided that I'm no longer teaching math in Math House. Once again, I want to teach How To Figure Out A Problem. We lost that last summer trying to tackle high school math. Figuring out a concept is a much more useful skill than getting a correct answer on known material. That was the whole point of TPM. If your child masters Figuring Out A Concept, then At Home Schooling is more productive.<br /><br />In order to prepare for TTWBN, we've been working with an SAT test prep book. This doesn't mean that we're tackling high school material at a high school level. The SAT is more like grade school material for an advanced child in really convoluted problems. This characterization of the SAT motivated Test Prep Math and it's been paying dividends ever since, until we started doing high school math last summer and started to focus on knowing match concepts.<br /><br />Here's a problem that demonstrates the full range of skills, those listed above, and the skill of Seeing (aka take time to look at every element of the problem and see the things that other kids miss for lack of vocabulary or patience). No matter how old your child is or how long he's received this training, he still forgets to practice the basic skills because he's in a hurry to finish math and get on to something more enjoyable, like going to the dentist.<br /><br /><div style="text-align: center;"><a href="http://3.bp.blogspot.com/-zGTNqFB1cNs/Wq04dffOShI/AAAAAAAABvs/ubA0SSJcPswof4SaRcfoo7wN5WQZwQJaQCK4BGAYYCw/s1600/Untitled%2Bdrawing%2B%25288%2529.jpg" imageanchor="1"><img border="0" height="188" src="https://3.bp.blogspot.com/-zGTNqFB1cNs/Wq04dffOShI/AAAAAAAABvs/ubA0SSJcPswof4SaRcfoo7wN5WQZwQJaQCK4BGAYYCw/s200/Untitled%2Bdrawing%2B%25288%2529.jpg" width="200" /></a></div><br />The triangle above is isosceles and AB > AC. Which of the following is false?<br /><br />I'm going to omit the answers because of an important technique Poyla's How To Solve it (1945). When I translated Poyla for 5 year old's preparing for a gifted exam, it becomes 'Read the Question'. The translation applicable to preparing for the TTWBN is 'if you see a geometry problem, solve everything before you look at the actual question.' (If this were age appropriate SAT test prep, then I'd take Poyla at face value because the topic of the book is geometry proofs for high school students and we'd be working under time limits.) The version for algebra is 'you're going to transform the equation so stop trying to solve it in your brain' and for trig, 'get out the basic formulas and be prepared to do geometry or algebra on top of that'.<br /><br />Anyway, we reviewed the definition of an isosceles triangle (totally forgotten since age 5 training), the sum of the angles in a triangle, and a hint where the base of the triangle is. There are the 3 steps that require Working Memory. Love this problem. Don't care about the solution.<br /><br />Initially, this problem resulted in guessing so I had to jump in and 'help' by asking questions. When I work with other people's children, they are more than happy to work thoroughly and patiently, but when I work with my own children they get frustrated and guess. Am I exaggerating? No. This is why it's so much work for a parent. Other kids just assume that I'm a teacher and therefore this will be a doable problem or else I wouldn't teach it, and things go well, but my own kids assume I'm an Insane Tyrannical Cruel Math Despot and am torturing them. You will face the same problem with your own children, which is why the survival skills above are so important.<br /><br />We've been working consistently at a pace of about 5 problems per day, and over time the child might do 3 problems on his own (incorrectly) and only need help on 2, and before you know it, he's back to needing help on all 5 problems because I had to switched to much harder material.<br /><br />Anyway, it was this problem where we ran into guessing and I decided I would much rather have him just work the question than try to solve it until he substitutes his subpar approach with '15 minutes of reading the question and 1 minute of getting it right'.<br /><br /><b>Reading</b><br />I've been happy to ignore reading until now, just doing the minimum lots of vocab and a couple hours of reading a day, an approach that paid dividends, but this year the older one has to take TTWBN for real and the younger one would rather do the verbal sections than the math sections to spite me. So it's time to get serious.<br /><br />When I bought the SAT books a few years ago (2nd dumbest and smartest idea ever), we had a lot of success but my 5th grader and I failed at the reading comp. We never made it past baffled.<br /><br />I knew a high school English teacher named Yoda who taught SAT test prep classes and begged the little green guy for advice. He said, 'Ask why you got the question wrong, you must'. I'm not kidding, aside from the Yodese accent; this is the only thing he said because we were sitting in a Boy Scout meeting whispering and then got shushed, and I haven't seen him since. For a year, we kept coming up with the answer 'Because neither of us know what the heck we're doing trying to do with SAT reading comp questions in 5th grade' and then gave up.<br /><br />Now I've got a 4th grader and a 7th grader with identical books (each have a copy) and I'm starting to get it. If you've got a 99.6% GRE level in vocabulary (because on the pre-test you got a 50% so you did some serious test prep back in the day) or a good dictionary, the reading comp section boils down to...but first I should point out that given the age difference, it's a totally different experience with each and the 4th grader finds those small passages that ask about sentence structure - saving the long passages for 6th or 7th grade.<br /><br />By the way, to overcome the vocab deficit, I've found that about half the time if you just add a 'y' to a word it's good enough. Decisive becomes Decisiony and we can move on. The rest of the time its a longer discussion.<br /><br />Anyway, it once again boils down to Math. It boils down to math. It's all just logic, one word at a time, counting sentences, iterating. If Math is 100% language based (I've said that before) it's only fair that reading becomes 100% math based. The left-brain-right-brain theory turned out to be totally wrong.<br /><br />Or, if you don't like that answer, it boils down to math in the sense of be baffled, spend a lot of time on the question (including the pick list), go slow, make mistakes and try again, and check your work.<br /><br />It's also patterns. By the time we're done, I'll know every technique, aspect, variation, and trick of the SAT. For example, when an answer choice is 'the author reluctantly agrees partially', you need to find concrete evidence in lines 30-33 of reluctant, agreement, and partial not whole. Applying Poyla to this material, you better be able to tell me the author's life story after you read the passage and before you start answering questions. It took me a year to figure that out, but now it seems obvious.<br /><br /><b>The Danger of Test Prep Classes</b><br />The problem of a classroom of any type is that to serve all 20 or 30 students, you have to TELL them the material. All kids are paying the same amount, and they'll all come out KNOWING the material and performing well on a test if you just tell them. This will work on a standardized test or even some gifted tests for some kids with specific learning styles. I worry about the longer term impact (jury is deliberating).<br /><br />The problem of TTWBN is that there isn't enough time to teach all of the material that the test covers at the level we need to be each year, and this is the big year. So I'm back to focusing on figuring things out.<br /><br />How important is At Home Schooling? Is it important enough for me to set aside a few hours a week, maybe a few more for research and preparation? Is it important enough for me to go through the frustration and headaches?<br /><br />What will the child think if I say 'This is not important at all to me to spend any time on it, but I'm going to make you go to this totally unimportant class'. The child cannot visualize money and he doesn't visualize you sitting in traffic. If you are not physically there going through the same pain, a bright child will conclude you do not value this activity at all that you are making him do. You won't see an impact with little kids, but you will see it later.Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-79926587295775160362018-03-11T09:21:00.003-07:002018-03-11T09:21:36.094-07:00The Kindergarten ChallengeHere's a challenge I received from a reader. <br /><br />The 1st grade child scores 99% on the NNAT one year than falls to 80% the next. Reading and math scores also fall. All scores have to be near the 100% mark in three months for GAT entry.<br /><br />The child is going to be home schooled. I'm very excited about this. It only takes a few hours a day to give the child 8 hours of education, and the child can sleep in every day which is critical for intense instruction. This leaves about 50 minutes for test prep and 2 hours for art, crafts and projects every day and 3 hours of reading. I consider science to fall under crafts and projects at this age. Think sorting rocks, vinegar and baking soda. First grade will take about 4 months under these conditions, and second grade another 4 months.<br /><br />The parent needs to find out the times of day when math works. Is math first thing in the morning, or is it morning painting and Read To? Test prep needs 2 times, one in early, late or mid morning, and one sometime in the afternoon.<br /><br />There are a few reasons why the scores fell year-over-year. I could write a whole article just on that topic. For now, the things I care about are a) anything score 50% is not a bad starting point, b) three months of prep is better than eight weeks, and c) we need to slow down the pace of learning, probably by about 90% and ramp up the complexity of the material. If the child did not do well on the test because the parent teaching methods and attitude are a total disaster (been there) then we need to fix this, which will be a separate article.<br /><br />Yes, I said slow down the pace of learning. This is probably the biggest factor in GAT preparation. My pace when I coach is 1 problem in 20 to 30 minutes (depending on the child's age) and 5 or 6 problems when the child works alone. We're just as slow in math, and I've managed to get two kids into high school math at age 9 or 10 on 5 problems a day. Not that they're especially talented in math.<br /><br />The premise of "slow" is slightly counter intuitive under a deadline. Here's the explanation. When you build an academic culture where a little work goes a long way, you're using the skills measured by the GAT tests, skills that are also critical to standardized tests like the MAP. Unless it's a timed test, but we can account for that after the learning takes place. When you have a culture where problems are easy, correct answers are expected, and worksheets are long and fast, the child is going to totally bomb on a test like the COGAT and NNAT.<br /><br />I would make time for Vocabulary Workshop because it's so much fun and children learn how to eliminate answer choices as they quickly progress toward harder material. I would have a Word Board for something because it's where adult discussions take place and where the child has to stand up and deliver. Or fail. There's always the next day.<br /><br />For math and test prep, let's teach this child so that he or she gets to 99%. I've been going back through my articles thinking about my teaching methods. I don't think articles are clear on my preferred approach:<br />1. Give the child super advanced material and let them flounder. Eventually they will pick up the skills to work with super hard advanced material.<br />2. Give them advanced material and let them do all the work before you don't grade it. (No typo, read that again.)<br />3. Walk through the super hard material together, one question at a time after they do it.<br />4. Do it with them, one question at a time, mostly just asking questions.<br />5. Give them simple material on a super advanced topic so that they can learn one step at a time on their own.<br />6. Give them last year's workbook (last year may actually be next year depending on the circumstances) so that they can catch up on material they need to know in order to keep up with 1 to 3 above. They can do this on their own, or with some starter help.<br />7. Lay 5 skittles on the table, one of each color, and provide a skittle each time your child gets a correct answer.<br />8. Give them a skittle just for making an attempt.<br />9. Do the problems yourself while they watch.<br /><br />Lately I've been doing 4b, which is to break down a problem entirely and a class or rules, but I didn't do this in first grade. I did say Shape Size Color Count over and over when they were stuck to remind them not to look at a problem for 15 seconds and announce that they were stuck, because that's called 'The Beginning of the Work'. <br /><br />Which approach do you use? I used them all.<br /><br />I used a variety of material, not because of the Spaghetti approach, but because sampling is the best way to find out what works, a child needs to learn from all materials, and a child needs to learn all learning styles and accommodate all teaching styles. It's not a matter of what the child likes best (aka the easiest), but what works best on which day to meet our goals.<br /><br />Finally, both cognitive kills tests and the upper levels of standardized tests in math and reading require deep, careful thinking over an extended period of time, mistrust of answers, tackling something unknown, surprising, new, with subtle, hidden complexity. How to you train a child to have these skills? #1 through #5 on the list above. It works the best with 1 super hard long 25 minute mind numbing problem, but in practice, this is a total disaster with crying and yelling, so I've settled on 5 medium really hard problems in 25 minutes. After that, brain exhausted.<br /><br />I almost forgot. We also did music starting in Kindergarten. I gave my child an electric piano and the Piano Adventure series, and no help what so ever except for tempo. <br /><br />Remind yourself that the child will be sitting in some advanced class someday without your help. The child will be taking a test without your help. This is what you are preparing them for. So many people get hung up on them having to know math because they have to get above 95% on the math section. It's so much easier to train them to think and then math comes really easily after that.<br /><br />What would this take? I think a few reading comp books, about 10 to 15 each, maybe 3 math workbooks, judicious use of the web, 2 vocab workshop books (current followed by current + 1 for starters), maybe one reading comp book, but lots of reading of all kinds. I would go to Michaels and buy lots of cheap crafts and things like that bead thing, concentrating and creativity activities, painting, and then whatever test prep books you want. <br /><br />Origami. Almost forgot. Origami is really good for visual spacial and fun, and the test we're challenged with in this case is the NNAT after all. You can create all sorts of animals. Do not let you're child do an activity that requires you to do it. It's kind of the opposite of test prep and how I do math. <br /><br />Totally excited about this. The thing I got out of this time period is a) I learned how not to be impatient or expect anything or care about correct answers and b) I ended up with a much closer relationship with my children and some credibility with them. a) led to b). a) also leads to a boatload of learning in a short period of time.<br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com1tag:blogger.com,1999:blog-5703568807615263851.post-42592442784695003442018-03-03T12:09:00.001-08:002018-03-03T12:09:20.341-08:00Putting Skills To The TestI've been working with a few hypotheses since about 2011 based on my research into the COGAT. Six years later, it's time to see how these hypotheses faired over time.<br /><br />Since I am the only person in the universe who a) believes skills exist, b) believes that skills are learnable and c) isn't making you buy a product or service to learn about skills, this website is pretty much your only resource. It would be nice if it were accurate as well.<br /><br />Here are my hypotheses.<br /><ol><li>The COGAT measures the skills that predict academic success.</li></ol><div>This hypothesis is based on the simple observation that school districts pay a lot of money for the COGAT in order to populate their gifted and talented programs. I read the research of the current test author and determined that he stands apart form cognitive skills researchers - all skills and no genetic intelligence.</div><div><br /></div><div>Any parent who forgoes COGAT test prep (or a similar cognitive abilities test) has no interest in a child with cognitive abilities. </div><div><br /></div><div>Unfortunately, future academic success is dependent on a child who has continued interest in academic pursuits. If the child lives in a house that devalues academics or goes to a school that devalues learning (aka most schools governed by No Child Left Behind) then hypothesis #1 may be undone. What started as an assumption is now ongoing research. So far, so good.</div><div><ol start="2"><li>The skills are age independent.</li></ol><div>Another way of stating this hypothesis is that once the skills are learned, the child has them forever. A child could pick these skills up at age 3, or age 15. Everything I've seen in the last 6 years supports this hypothesis. A corollary to this hypothesis is that the probability that the child will pick up these skills decreases every year after 1st grade, probably because of NCWLB, with the exception of age 15 (which I haven't personally researched yet.)<br /><br /></div></div><div>I first came up with this hypothesis while reading a description of the classical education in the Well Trained Mind. The classical education has a breakpoint every 4 years and is based on the development of the child, brain or otherwise.<br /><br /></div><div>I've noticed leap in skills around 5th/6th grade academic material, certainly by middle school, which I've had some fun with recently and describe below.</div><div><ol start="3"><li>The list of skills is boring and unremarkable.</li></ol></div><div>I'm not going to restate my skill inventory here, but if you read the list in prior articles, it's not really earth shattering. I think I would have more readership if I could come up with clever sounding names for the skills or write articles like '10 Things You Didn't Know About Skills', but there are only 4 or 5 things you didn't know, and those are the skills. </div><div><br /></div><div>What I find more interesting is watching a child go through the transition from not using the skills to overcoming very difficult material by applying the skills. Take Mistakes, for example. A child doesn't need this skill, and is not incented to use it because it requires some effort and controlling emotions. The reason the child doesn't need this skill is because parents and teachers are willing to explain the mistake, show the solution, explain the solution. There is a high price on making mistakes in the first place. Once the support structure and penalties are removed, the child has to go through the process of proving to himself the value of mistakes, as in make one, learn something, try again and again, and achieve the solution with no help. It's like military boot camp. Not fun when you're there, but it pays off.</div><div><br /></div><div>In practice, I observe the emergence or application of about a dozen sub-skills during this process. The sub-skills are germane to the subject and child specific. I've never seen a reason to discuss most of these (except the big 5) because we'd end up just replacing spoon-feeding-training subject matter with spoon-feeding-training sub-skills and be back to a helpless child who's not getting it. Right now I'm tackling middle school reading comprehension with a vengeance and we are heads down on the sub-skills, but that article will wait until we get past the high school entrance requirements.</div><div><br /></div><div>Recently, a 4th grade buddy came over to play Minecraft. In Math House, the rule is no math, no computer. In this case, 'math' meant learning algebra from scratch in 25 minutes or less. This child is solidly at the top of the gifted spectrum. I don't know why his parents didn't bother to teach their 9 year old algebra yet - probably because they are not insane - but it qualified him for my research.<br /><br />During this experiment, I noted that there is a leap in skills required of algebra. I'm not talking about- abstract thinking or a new language in the form of different syntax or seeing pre-algebra for the first time. Because of this leap, the child went from 99% in skills to 0% in skills before working his way back. Also, note that parentheses alone work a magic spell on children that makes them forget everything they've ever learned. <br /><br />Here's a transcript of the experiment.<br /><br />Me: Solve this equation: 3 + 5 = ? (He responded 8, then looked at me like I was a moron.)<br /><br />Me: Solve this equation: 3 + 5 = ___ Does it matter that I changed the question mark with a blank? (He responded no.)<br /><br />Me: Now solve this equation: 3 + ___ = 8. Is it totally confusing that the blank has moved? (He answered no.)<br /><br />Me: Not solve this equation: 3 + x = 8. I am replacing the blank with an x. Instead of telling me what goes in the blank, tell me what x is. Is this to confusing for you? (He answered no.)<br /><br />Me: Now I want you to use algebra. Instead of just solving for x, you have to transform the equation one step at a time. You can either add a number to both sides, subtract a number from both sides, multiply both sides by a number, or divide each side by a number. (There are a few more transformations, and I didn't mention expressions, but we're keeping it simple because we only have 25 minutes for this experiment.)<br /><br />Me: Here is everything you need to know about algebra. Look at these 2 equations and tell me what is wrong with the second one:<br /><div><ul><li>x = 2</li><li>3 + x = 8 - 5x</li></ul><div>Me (after a brief discussion): The first one is perfect. I know the answer immediately. The second one is broken because it doesn't have a letter on the left side and a number on the right side. Fix it. You can only use 1 of the four transformations, and you can only do one transformation at a time.</div></div><br />Rules: a) apply one of the 4 transformations to both sides, b) only apply one transformation at a time.<br /><br />We took a break at this point to remember the scale problems from 2nd or 3rd grade math (which he forgot) and assure ourselves that the 2 sides stay equal when these transformations take place. Then he had to tackle these 2 problems:<br /><br /><ul><li>3 + x = 8 - 5x</li><li>7x - 15x = x(x + 5)</li></ul><div>It's really fun to watch what happens next. First of all, rules a) and b) from above are both violated repeatedly. "Both sides" is forgotten. Gifted kids are gifted in part because they can solve complicated expressions in one shot. In practice, they combine steps. Doing only one step and writing it out is like eating broccoli. When I teach algebra to young kids, I'm always battling them trying to figure out the answer in their head, which they can do. I'm asking them to stop doing things in the way that they are good at, and start doing things in a way that they are not good at and will likely lead to an incorrect answer. It's more than Baffling for this reason. </div><div><br /></div><div>Next, they forget how to add and subtract single digit numbers.</div><div><br /></div><div>Any pre-algebra kids learned up to this point is also forgotten. This includes parenthesis and not adding x to 5, because you can't and x to 5 and get 5x or 6.</div><div><br /></div><div>I made some really cool observations during this experiment. The subject wondered what 5x means, and then realized why dot means multiplication - because writing 7 x x to mean 7 times x doesn't make sense. His skills of analyzing the question were strong. Analyzing the question in algebra, at least initially, means learning quite a bit on the spot that was not previously known (which I minimized in the problem above). It's a leap in this skill. Once we get beyond simple one variable equations, the question analysis takes a leap.</div><div><br /></div><div>It's hard to make the leap to 5x + x. What does this mean? It means that you have 5 x's, and I give you another x, how many x's do you have now? It's like working with a 3 year old on addition. Did you forget to add? Do you want to do it on your fingers, butter bean? Do I need to invite the 3 year old down the street here to teach you how to count on your fingers? I really need a control group where I don't antagonize the subject.</div><div><br /></div><div>The most remarkable observation for this experiment is that the child typically (100% of the cases) get's stuck on what to do even though according the rules, the only thing to do is apply one of the four transformations to the equation. Maybe they can clean up the expression by making 7x - 15x equal -8x, but that's not what they are stuck on. Without doing enough of these problems, it is not clear which arithmetic operation to apply to each side. Addition? Multiplication? Subtraction? Division? These kids break the transformation down to a simple question & answer, and they don't know the answer. The correct approach is to try all 4 and see if the resulting equation is getting fixed (aka easier) or more broken. Algebra has the skill of Mistakes build right into the process.</div><div><br /></div><div>That is the biggest leap in skills.</div><div><br /></div><div>At age 5, a gifted child will make a mistake, not be bothered, and try again until the solution is correct. Really gifted children (on standardized tests, anyway), check their answers to verify that they didn't make a mistake.</div><div><br /></div><div>With algebra, initially, on each step there is a 75% chance that you will make a mistake, and you may have to try all 4 to see where the equation is going. That is a 25% error rate built in to each and every step. Sometimes you might even have to do 2 or 3 steps, trying a series of transformations, before you know you are on track, and you've ended up with a score in the single digits before you get past the first problem.</div><div><br /></div><div>I've occasionally mentioned that I think drawing is a valid way to teach a child to be gifted in math. Hand your child a 2 inch stack of paper and a dozen pencils, and ask them to draw a realistic looking horse. All of the cognitive skills are used to their extreme in this exercise. Children who draw for a living should become math powerhouses*. </div><div><br /></div><div>*It depends on what they draw. Horses aren't good enough. Needs something with lines and circles in it.</div><div><br /></div><div>I prefer crafts for math training to prepare for algebra.</div><div><br /></div><div>Anyway, the subject passed the 25 minute algebra lesson and his parents didn't complain yet about any signs of psychological damage.</div></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com3tag:blogger.com,1999:blog-5703568807615263851.post-8603296718896808752018-02-28T21:11:00.000-08:002018-02-28T21:11:02.230-08:00Skills In PerspectiveIn the last article, I went a little overboard on the technical detail with some middle school competitive math. I tried my best to lay out problem solving so that you can see it is consistent with little children and consistent with the high school, college, graduate school and post doc experience.<br /><br /><hr />Let me explain this bluntly.<br /><br />You want your child to have problem solving skills. This is much better than having to help with math or hire a tutor to spoon feed your child steps from question to answer.<br /><br />But if you try to teach your child problem solving skills in the hopes that these spur cognitive growth, you will fail. It's as bad as having your child memorize formulas and rote practice applying them.<br /><br /><hr /><br />Here's a brief history of skills. In 1945, a researcher at Stanford named George Poyla took 3,000 years of research into how mathematicians solve problems from philosophers, ancient Greeks, and mathematicians themselves, and wrote a book called How To Solve it to help high school teachers mentor their students on solving geometry proofs. The emphasis of How To Solve it is 'mentoring', not doing any work for the student or teaching problem solving algorithms or heuristics.<br /><br />By the 1970's problem solving was turned into a pre-packaged, spoon feeding program to help students apply problem solving methods to pre-algebra and more advanced maths without the need to understand anything that they are doing, let alone math.<br /><br />The #1 problem in problem solving is that the defective learning approach that emphasizes a speedy, correct answer that has been memorized and practiced has evolved into a defective learning approach that emphasizes a speedy, correct answer using a problem solving technique that has been memorized and practiced.<br /><br />When I finished translating How To Solve it into a method suitable for parents of 4 year olds, I was stunned to find a solid approach that also works for graduate school. I added a step that researchers at Berkeley identified as the #1 success factor for surviving their first year calculus courses. The first experimentee of the program is now 9 year's old, and needs about 10 minutes to get a score of 50% on SAT reading comprehension tests. Obviously, we have a way to go, but the method is so general that if pretty much works everywhere, including assembling Ikea furniture and fixing plumbing issues. I would recommend it simply for the benefit of not having to call a plumber.<br /><br />Here is the short version of the problem solving method:<br /><ol><li>Be Baffled (thanks Berkeley math department)</li><li>Spend a lot of time thinking about and exploring the problem</li><li>Make mistakes and try again</li><li>Check your work (I added this because it raises test scores)</li></ol><div>In between #2 and #3 sit the process of problem solving. In the last article, I demonstrated the most powerful problem solving techniques from the standpoint of a baffled parent trying to help their child learn some new material that is way beyond the child's skill level. Think figure matrices, multiplication, fractions, exponents, algebra, trig or whatever. I'm going to continue the numbering from the above list and explain why shortly.</div><div><ol start="5"><li>Start with a much, much easier version of the problem, like 1 x 2 = 2 and just keep adding to it and iterating until you are back to the original problem. This can take weeks if you're trying to teach multiplication to a 5 year old. In some cases, the child is missing something fundamental from material we skipped, so we just backtrack to an easier math book to practice the prior material and then come back to the problem. Backtracking happens a lot in Math House. Ironically, I can teach basic Trig in about 30 minutes, but it takes months to teach basic alegra.</li><li>Translate the hard problem into 2 easier problems and solve the easier problems instead. This approach usually involves decomposition or regrouping in the early years, and gets trickier in high school math.</li></ol><div>There are other good approaches for more advanced topics outlined in Poyla, like solving the problem backwards, applying some theorem or proof that you just learned in the prior problem (which works for both Geometry and the COGAT), filling in the missing word or shape. If you give the child enough space to explore the problem and make mistakes, the child will learn these methods on their own, or even better, make up their own methods however inefficient.<br /><br />When I combine the two lists, which is why they are numbered contiguously, I end up with 90% of my teaching method for math until we get to Algebra and Geometry.<br /><br />There is a great deal of contemporary discussion on the topic of why students are struggling in Physics. The consensus of physics teachers is that students are more interested in getting to the solution (using the internet to find the method) and less interested in learning physics. You can find many, many books written to demonstrate the step-by-step approach to solving every class, subclass, and subsubclass of algebra problem if you wish to be an algebra expert without knowing what you are doing. If a parent would just take a step back from Teach To The Test, you'd find that it takes a fraction of the time to get a 99.9% based on thinking and learning than a 90% based on practice and memorization. To emphasize this point, we tend to do 2 to 5 problems a day and make much more progress more quickly than children who do 30 or 40 easy problems a day.<br /><br />Learning happens from the start of the first problem until the student realizes that there is a formula or method that can be used to solve problems of this type. When the child struggles with 2/3+ 5/7, lots of learning is happening. But once the child realizes that each fraction has to be transformed to share common denominators, we're done with learning. Learning also stops when the solution is checked as well, right or wrong.<br /><br />The biggest complaint I receive from parents who start down the path that I recommend is that it doesn't work. By 'doesn't work', it means that their child is frustrated, lost, and getting nowhere. To me, this is a description of the initial stages of the process a not a defect or shortcoming in the approach. Some stubborn kids need about 6 weeks to undo the programming from school, programming that you must know what you are doing, do it quickly, and obtain the correct answer without effort or challenge. It takes a while for the child to realize that expectations have changed. <br /><br />Sometimes it takes 2 or 3 weeks on a half dozen problems to teach the child that we are going to go slow, think a lot, be confused, hit dead ends, have to backtrack, and get things wrong a lot. To accelerate this process (meaning show the student that the rules have changed), I'm usually confused, get the wrong answer, and don't check the solutions. Once the child gets past this hurdle, the pace begins to go very quickly, and if you stick with this approach, the child will in a few years teach themselves entire subjects very quickly, or if you insist on teaching your 9 year old algebra, not very quickly but adequately.</div></div><div><br /></div><div><br /></div><div><br /></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-414815082458240142018-02-24T16:14:00.002-08:002018-02-24T16:14:31.860-08:00Skills in ActionA reader challenged me to explain how to do competitive math. I'm excited about these problems because they demonstrate a fundamental skill set that is developed during learning to read at age 4. It's very similar to the skills that cognitive skill sets like the COGAT teaches. It's 100% learnable.<br /><br />What I like about this problem set is that a parent can work through these and have the exact same experience that your child goes through. I'm hoping that parents who take the time to work through this material will have better training when your child shows you a figure matrix that is baffling. It's also a good opportunity for the parent who discovers this website 6 or 7 years too late so that I can show that it's never too late.<br /><br />I've gotten numerous questions about what follows Test Prep Math 3. I like competitive math as a warm up for SAT test prep. We dabble with pre-algebra, but usually only in a Algebra 1 setting. Sounds hard, but this is the skill set.<br /><br />Here we go. Picture a competitive math worksheet with 40 problems on it, that has a 45 minute time limit. I suppose if we were serious about competition, we'd train for learned strategies to address the time limit, but we're not serious about competition, just doing a bit of daily math. I think 5 problems is asking a lot of an 11 year old.<br /><br /><b>Question 1:</b> F - T - L - T - ? - ? Find the last 2 letters in this series.<br /><ol><li>I have no clue how to do this. Anyone you has seen this question type probably doesn't have any clue because it has unlimited subjects. But I have the most important skill of all, which is the proper way to be Baffled, which is to not care that I'm clueless.</li><li>I think for a minute about adult IQ tests. Friday, Tuesday, Something That Begins with L, Thursday. Fail.</li><li>F = 6, T = 20, a difference of 14. L = 12, T = 20, - 8 + 8? Fail. Skill 2 - don't care how many incorrect answers I get.</li><li>I stop and think about the question a bit. Kids only know arithmetic, language, geometry, and a tiny bit of algebra. Pre-Algebra is fair game. In the real world, I should have used Skill #3 which is to spend more time thinking about the question and less time getting incorrect answers, but in competitive math with no time limit, lots of learning can happen in dead ends.</li><li>Going the geometry route, all the letters have a single vertical line. F has 2 vertical lines and T has 1. That's 3. L has 1 vertical line and T has 1. That's 2. I forgot to look at the answer set. My skills are rusty because the answer set is part of the question.</li><li>The answer set is:</li><ol type="A"><li>L - T</li><li>L - B ( I think this B has no vertical lines.)</li><li>L - M</li><li>T - P</li></ol><li>If B is the answer, using counting horizontal lines in the series, we get 2 - 1 - 1 - 1 - 0, but if we take pairs of vertical lines, we get 3 - 2 -1. B is the answer, I accidentally stumbled on it, and I have no clue why I am correct. But it was the best of a bunch of confusing bad answers.</li></ol><div>This problem took me about 15 minutes. It's very similar to the type of work I do on a daily basis. I wonder if competitive math tests are structured so that super duper problem solving kids prioritize the questions and their time before answering and just skip this one. Probably. </div><div><br /></div><div>When you work with your child and do a problem that is really hard for their age and skill set, just like the one above, here's the benefit you both gain:</div><div><ol><li>You get used to working with baffling things and don't get put off.</li><li>You make a lot of mistakes and don't get put off. In fact, in my failed attempts (attempts not include above), I learned a lot of interesting things and picked up a few mini-skills on the way to dead ends.</li><li>The solver is forced to think creatively and view the problem from different angels. It will take a lot more problems to learn creativity, but since I am making a habit of baffled and mistakes as skills by force feeding my child these problems over and over again, we'll get their eventually.</li><li>I never looked at the clock or the solutions. This problem is kind of tricky and fun. The solution will end the learning process and reinforce the Rule #1 that it's not about learning or getting better at something, it's about being right or wrong. Rule #1 will destroy your child's ability to learn. Rule #1 is an anti-skill.</li></ol><div>When I work with kids, a team will really help, and I'm the only one available for the other team member, so in practice I ask a lot of questions (as needed) and make suggestions for the next attempt (as needed). I'm always baffled. In practice, I'm suggesting skills and approaches from my toolset of exactly 5 approaches to any math problems. </div><div><br /></div><div>Why is it that when your child comes to you and asks what 'dispersed' means, you're more than happy to tell him, in fact you're so happy your bright little child has an interested in vocabulary and is not skipping over unknown words when reading, but when your child gets a math problem wrong, you're disappointed? What a horrible destructive way to teach children to hate math. Adding a time limit makes it even worse, because then a teacher can mark of a series of unanswered questions. This is why schools can completely eliminate tests through Junior year in high school and produce kids who blow away college entrance exams.</div></div><div><br /></div><div>OK, let's see what we get out of more baffling problems.</div><div><br /></div><div>What is the remainder when the 15-digit number 444444444444444 is divided by 9?</div><div><ol><li>Are you kidding - this is too big to fit in the calculator. Curse you competitive math test author. The answer pick list is irrelevant. Again, I have no clue.</li><li>Too hard of a problem. So I fall back to how we tackled any math - starting at age 4, when it's too hard. We start with the easiest version of the problem and work our way back to the harder problem:</li><ol><li>4/9 ~ r 4</li><li>44/9 ~ r 8</li><li>444/9 ~ r 3</li><li>4444/9 ~ r 7 this is good practice for division but a fail in solving the problem.</li></ol><li>Then I remembered that when I teach division, I always make the student turn 36 ÷ 9 into 3*3*2*2/3*3. Now were trying to turn this problem into a more solvable, easier version of this problem. Here's goes:</li><ol><li>4*111111111111111/9 = ? Still hard. Fortunately, I can look back on the first fail and continue.</li><li>1/9 ~ r 1</li><li>11/9 ~ r 2</li><li>111/9 ~ r 3. Get it? Light bulb. </li><li>Continuing, I get to r 0 at 1111111111 which puts 111111111111111 (15 digits) at r 5.</li></ol><li>Unfortunately, I'm stuck having to multiply the whole thing by the remainder. This stinks, I stink, and your child stinks, so we're going to have to take baby steps.</li><ol><li>Since 1/9 = 0 + 1/9, 4*(0 + 1/9) = (0 + 4/9) ~ r 4, which is what I got in the first fail. Notice I'm checking the answer, which is skill #4 at the base of the cognitive skills pyramid. I suppose this requires some pre-algebra.</li><li>11/9 = (1 + 2/9), so 4*(1 + 2/9) ~ r 8, again, just like above.</li><li>111/9 = (12 + 3/9) but 4(12 + 3/9) is going to give us 48 + 12/9, slightly confusing, and I have to go read the question yet again. Oh yea, we're dividing by 9, and trying to find the remainder, so I can write 48 + 1 + 3/9 ~ r 3 just like expected.</li><li>At some point, the lightbulb goes off, and I can just jump to 15 ones's/9 = (something big + 5/9), and I multiply by 4 and get 4*something big + 20/9 ~ r 2, which is not even on the answer list. The choices are 4, 5, 6, and 7. </li><li>So starting over, which I'm totally used to because we do it all the time, I note that the 9 digit number 111,111,111/9 = 12,345,679 r 0, duh, should have thought this though. This makes 111,111,111,111,111/9 = something big 6/9 (since 15 digits is 6 more than 9 digits), and 4*(something big + 6/9) = 4*big + 24/9 = 4*big + 2 + 6/9, giving me the correct answer of 6.</li></ol></ol><div>We've got 3 big solutions approaches that we start using when the child is about 3 years old. </div><div><br /></div><div>At some point, your child is looking at * * * * * * of something and you ask her to count. She answers 12 or 5 or gives up, so you start small, like *, then * *, then * * *. I teach addition, fractions, and multiplication this way. It works in graduate school and it was by experimenting that I found it works really well at the youngest ages. It works on pre-algebra. It works on all forms of high school math. It's required for competitive math. Math books do this from chapter 1 through chapter 15, but we do it in 5 minute increments and don't really need a math book.</div></div><div><br /></div><div>Next, when a problem is too hard, turn it into an easier problem. This is the foundation of algebra. You might as well start now.</div><div><br /></div><div>Finally, notice that there are 3 steps to this problem. If you've seen TPM, you know why I think 3 is so important. It builds working memory. For the age group for the problem above, we're probably beyond working memory, and if not, doing these problems will bring it back. But the working part in 3 steps is where the little brain turns itself into a big brain by defining relationships and patterns and working abstractions into algorithms from one part of the problem to the second to the third. You see all three in the solution above. A genius can do it in one step only under one condition: the genius worked through enough of these problems to get really good at devising and applying algorithms. Don't be fooled into thinking it's genetic. The rest of us are happy doing the 3 steps one step at a time. One step at a time is good for 99%.</div><div><br /></div><div>Moving on, how about this problem. What is the value of 1 - 2 + 3 - 4 + 5 - 6 + ... + 81 - 82?</div><div><br /></div><div>This problem not only demonstrates the value of spending way more time exploring the question than trying to answer the question, it also demonstrates the value of what I call "Seeing". I learned it from the COGAT. It involves looking at the problem from different perspectives. </div><div><br /></div><div>I checked to see that there were an even number of elements to this equation, all equaling negative one when paired, and came up with -41. Eight minutes of thinking about the equation and 4 seconds deriving the answer. With 40 questions and a 45 minute time limit, I would have come in last on the competitive math exam. Can you picture me sitting with a bunch of 6th and 7th graders? </div><div><br /></div><div>This next question is my favorite and a really great exercise on it's own to teach exponents. I love this question. This differs in an important way from the math I would give a younger child but is identical in nature to the non-verbal section in TPM. It involves doing a lot of work, organizing and thinking about it, and then answering. </div><div><br /></div><div>If a and b can take on the values in [0,9] (meaning that they can each be 0, 1, 2, ... 9), then the expression a<sup>b</sup> can take on how many different odd number values?<br /><ol><li>To start, I just created a grid with 0-9 on the rows and 0-9 in the columns and started calculating the expression based on inputs. In a competitive math situation, this is a waste of time and requires thinking, but with most kids (and 9 year olds), I make them use the brute force approach because they usually have never seen a<sup>b </sup>outside of 4<sup>2</sup>. I've got a whole exponent crash course (including negative and fraction exponents), but this seems to be a good starter exercise. </li><li>The rows are a and the columns are b. I didn't calculate the *'s but I could have. <table><tbody><tr><td>*</td><td><b>0</b></td><td><b>1</b></td><td><b>2</b></td><td><b>3</b></td><td><b>4</b></td><td><b>5</b></td><td><b>6</b></td><td><b>7</b></td><td><b>8</b></td><td><b>9</b></td></tr><tr><td><b>0</b></td><td>?</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td><b>1</b></td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td><b>2</b></td><td>1</td><td>2</td><td>4</td><td>8</td><td>16</td><td>32</td><td>64</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>3</b></td><td>1</td><td>3</td><td>9</td><td>81</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>4</b></td><td>1</td><td>4</td><td>16</td><td>64</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>5</b></td><td>1</td><td>5</td><td>25</td><td>125</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>6</b></td><td>1</td><td>6</td><td>36</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>7</b></td><td>1</td><td>7</td><td>49</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>8</b></td><td>1</td><td>8</td><td>64</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td><b>9</b></td><td>1</td><td>9</td><td>81</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr></tbody></table></li><li>This seems to be a fail. Too hard. I did notice that only one zero in the top row and one from row 2 and column 2 are going to be included. What is zero raised to zero? It's either one, zero, or undefined, but if you read the question again (and you should because it's a skill), it doesn't matter to the answer.</li><li>After rereading the question yet again, I noticed that I only have to deal with ODD numbers. With the exception of '1', the rows with 5, 7, and 9 qualify, and since 3*3*3*3 = 9*9, the row of threes where the exponent is odd also qualifies but not when the exponent is even. And we can add 1 only once and ignore zero. And that gives the correct answer of 27 (the whole row of 5,7,9) + 1 (from the one) + 5 (from 3 row where it doesn't repeat a value from the 9 row) = 33.</li><li>It's possible to jump to step 4 as a competitive math coach, but not a regular bright kid doing competitive math coach.</li></ol><div>I'm guessing the question needs about 5 readings before this work can begin. I've watched little mathematicians create charts to answer questions and it's very gratifying.</div></div><div><br />Finally, the last question is this. If x and y are integers and 360x = y<sup>3</sup>, what is the minimum possible value for x + y? At this point, we left all kids under 4th grade behind and we're just looking at algebra. Or are we? Yes, I'm running out of steam and have already covered all the really great problem solving techniques. <br /><br /><ol><li>After 30 minutes with the question, I decided that x is just a function of y, so forget about x. Just find the smallest possible value of y. Or do algebra. It's late, I've exceeded the maximum good thinking time of a grade school child of 25 minutes, and the Olympics are on.</li><li>But I don't like 360, so I wrote 2*2*3*3*2*5x = y<sup>3</sup>. Then I rewrote it to be 2*2*2*3*3*5x = y<sup>3</sup>. You can see that if y is an integer, x has to be 3*5*5, making y = 2*3*5. So x = 75, y = 30, and the answer is 105. </li></ol>I got the entire solution correct by following these steps:<br /><br /><ol><li>I had no clue what to do.</li><li>I went off in the wrong direction by trying to use algebra, which I can, but doesn't solve the problem for a kid who doesn't know algebra. Fail.</li><li>I tried again.</li><li>I spent more time looking at the question and eventually started to rearrange it in the hopes of finding an easier problem. (I.e., I used one of the big five 5 math problem solving techniques.)</li><li>I looked at it, specifically looking at the root primes against the exponent on the other side of the equation. I used my power of seeing things differently.</li><li>The answer emerged with no effort.</li></ol>This is why studying for the COGAT is so critically important. It's the easiest way to get the skills. If you missed this opportunity, there are other opportunities including competitive math. It seems harder and more complicated because your child is older and the math more obscure, but it's about the same. If you did this when your child was younger, you would have blocked out all of the tears and frustration by now and just remember how it all worked out. Same with bed wetting in the middle of the night. Remember that? Of course not.<br /><br />Is there anything different between a child who does this problem successfully and one who gives up? Not mathematically. It's all in these base skills which are 100% learnable and needed for high school math. If you want a strong competitor in a math contest, you'll need interest and a lot more practice, but if you just want a five on the BC Calculus without having to nag your child or hire a tutor, do a few problems and focus on the skills.<br /><br /></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-24285755285232039592018-02-17T08:14:00.003-08:002018-02-17T08:22:46.174-08:00Totally Doable If Done RightIn the last few weeks, I've stumbled across a whole new group of people who are suddenly concerned about their child's education either because they decided it would be nice to have an actual child in the next few years, or they have an actual child and just found out about the COGAT, or they are getting COGAT scores back and deciding that it's time to get serious.<br /><div><br /></div><div>My Power Mom's Group, or PMG, from last year is officially demoted to Last Year's Power Mom's Group because your kids all met their ridiculously high cutoff goals (and are solidly on their way to additional goals). There's one more item on the todo list for the next few months and then I'll declare a 100% success rate based on selection criteria that includes a) great parents and b) capable kids. The new members of LYPMG are going to get heavy doses of my super secret program to crush the MAP test in the coming years. How similar are the COGAT and the MAP? COGAT skills are a prerequisite of the MAP, but the COGAT type math isn't what people generally consider to be math and the MAP has way more math than anyone realizes. If you are not in LYPMG, then you'll read about my super secret MAP program but you won't realize that you're reading about it until I can get everyone in the house past the 7th grade MAP.</div><div><br /></div><div>For newbies, I've been working on a less insane sounding description of my math approach, with a nice sounding title like Easy Fun Math*. (*Also known as Ridiculously Hard Insane Math until you get it, and then it's just Ridiculously Hard Math.)</div><div><br /></div><div>Here goes.</div><div><br /></div><div>First, read read read read. If your child only has 60 minutes per day of at home schooling, devote 40 minutes to reading. If your child has 6 hours a day because it's Saturday, devote 5 hours and 40 minutes a day to reading.</div><div><br /></div><div>Secondly, do not, under any conditions, every teach math. The skills your child needs to excel in math are organizing, seeing patterns, trying again, iterating, comparing, trying out different options, defining, extending, explaining, rethinking, simplifying (ie organizing), decomposing (ie organizing), and not being put off by mistakes, lack of information and clarity, and total confusion because if your child isn't working in on a math problem that starts with mistakes, lack of information and clarity, and total confusion then they are not working on a math problem that will develop the skillset. The super advanced skill set for math includes good executive skills and a lot of Grit. If your child develops these skills under your guidance, your child will excel in math. If you teach math, your child won't need any of these skills, won't develop them, and then someday will fail at math.</div><div><br /></div><div>Look at 'First' and 'Second' again. Higher order math skills are developed by reading. This really matters when your child is 2 and 3. By 4th grade, it will be assumed but not a major factor in the program.</div><div><br /></div><div>Third, at the 99.8% level, which is totally doable if done right (Totally Doable If Done Right, my new motto, and this just replaced the original title for this article which was Advice for Newby Math Parents), there are a lot of parent skills involved. While the child is learning each new skill, you will be learning a new skill. Your child will see math in a different way, and you will see coaching math in a different way.</div><div><br /></div><div>Forth, your child's math score is going to be constrained by working memory. I can't stress this enough. School math needs one or zero working memory buckets in the brain. Think 'Ann has 2 apples and Bob has 5 apples. How many apples do they have together.' Test Prep Math starts with 2 and ends up with 3 working memory buckets - or more - on every problem. I've settled on 3 since it appears after 3 a pencil is needed. </div><div><br /></div><div>Test Prep Math emphasizes messy, sometimes unanswerable problems (in clumps of 3, all mixed up and interspersed with vague words and ridiculous plots). Now you know why.</div><div><br /></div><div>There is an ongoing debate on whether or not children should memorize their math facts. Teachers who need to get all 30 kids in the class past arithmetic errors in 2nd or 3rd grade are generally stuck with memorization exercises - even in GAT classes. Researchers who are figuring out how to get kids to the upper levels of math excellence can explain why memorization is counter productive. </div><div>If you search 'Boaler Memorize Math Facts' you should find a few really good articles explanating why memorization is a bad idea by the leading researcher in this field. You may also come across an <a href="https://gregashman.wordpress.com/2015/09/19/jo-boaler-is-wrong-about-multiplication-tables/" target="_blank">counter argument from Greg Ashman</a> that totally misses the point, but get's so close with this diagram that he's one sentence away from solving his own problem. Look at this diagram:</div><div><br /></div><div style="text-align: center;"><a href="http://2.bp.blogspot.com/-4er-FaL1jsk/WohLlqGxh_I/AAAAAAAABP0/GiT1FVB1xIY6Xru5RRFHo-IhQMnFdhxVACK4BGAYYCw/s1600/Ashman.jpg" imageanchor="1"><img border="0" height="234" src="https://2.bp.blogspot.com/-4er-FaL1jsk/WohLlqGxh_I/AAAAAAAABP0/GiT1FVB1xIY6Xru5RRFHo-IhQMnFdhxVACK4BGAYYCw/s320/Ashman.jpg" width="320" /></a></div><div>Note to Ashman, the goal here is not to use long term memory to help with the math facts but to triple working memory. Also note that this diagram makes me want to sneeze.</div><div><br /></div><div>Boaler attributes number sense to strong math skills. Number sense and math fact memorization are two exclusive roads to math, and memorization falls short. In my ground breaking research I found that use of working memory isn't just a tool for math, it's a math skills generation factory. The child learns the next level of math skills while working arithmetic in working memory. When people see the term 'Working Memory', they see 'working MEMORY'. It's more accurate to view this as 'WORKING memory AND MATH SKILLS GENERATION FACTORY'. Please note that Boaler's research concerns making math accessible to everyone, but my research concerns a child who just blew away the COGAT and is looking for the next big leap in skills. </div><div><br /></div><div>Maybe groundbreaking doesn't cover it. Here's what we got out of the workings of working memory in action: an 8 year old who is solving problems off of middle school competitive math tests.</div><div><br /></div><div>When I wrote, and rewrote, and refactored and added to Test Prep Math, I met my goals to tackle working memory, base skills, and no math if it can be helped. I failed on the no math part because I couldn't help sneaking in math. A little geometry, a little algebra, and if you look closely, you'll see the makings of other maths, but I generally avoided division, and avoided decimals and anything else that is on a Common Core list. This approach doesn't work for everyone. Some people are short sighted and think of math as topics from a math book. Others already taught their kids math and the horse already left the barn.</div><div><br /></div><div>One of my favorite exercises is to do Every Day Math Grade 2 in K. For those that missed the opportunity in K, it's simply known as Current+2. I think of this as an exercise in Grit and not math, kind of a warm up to the challenge that will follow. Last week, a reader shared her child's current math situation which sounds so dire, what with mistakes, frustration, and not getting it. Once again, my children are even worse in comparison, but we manage to score consistently in the high 90's (like 99, which is what I expect) and do almost no work at all. One year ahead in math for us and maybe 40 to 60 minutes during the week. That leaves plenty of time for reading, crafts, and projects. My secret isn't smarter kids but kids who don't quit. And we do things totally different, like work smarter and not harder.</div><div><br /></div><div>After successfully avoiding the memorization of math facts, I've extended the counter cultural approach with not really ever learning math or being remotely competent in any one math topic. Focusing on underlying skills for years at the expense of math has really paid off in a big way.</div><div><br /></div><div>You'd think the next step after Test Prep Math would be learning actual math, maybe tackling Pre Algebra. Instead, we took a detour into competitive math, not really like school math at all, and then I've opened up 7th through 12th grade math topics for any given weekend. I think we have about 3 20 to 30 minute sessions each week, and the topic could be a first look at derivatives, exponents, polynomial zeros, 'What is sine and why am I making you go through this pain?' or anything else. One week it was exponents, and the next week my older kid saw logs for the first time and had to invent and derive formulas for logs that corresponded to the exponential formulas that we worked in the prior week. When this child sees logs again in a month, he will have remember exactly zero of it, but he's got the tools to make short work of it.</div><div><br /></div><div>After 4th grade, the little one will spend the next year or two working through SAT books. Other parents will try this and find that it's a disaster. Our experience will be even worse, but we'll plod on come out with 2 completed books, about 18 practice tests in all, and then move on to the reading comp sections. I've recently summarize the parent coaching skills needed to get through this approach successfully. When the 9 year old gets through the first page, 3 problems attempted, 3 wrong answers, and a lot of complaining and tears, I'll wonder why the heck I'm doing this. Then I'll remember that I've done this type of thing many times before, and it will magically work out in the end.</div><div><br /></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com4tag:blogger.com,1999:blog-5703568807615263851.post-73186882225368901442018-02-10T19:47:00.000-08:002018-02-10T19:47:31.197-08:00Visual Math Et CeteraFor years, I have been asked for a recommendation for 4th grade math. I now have one, and one for 5th grade as well. It's called <a href="https://www.amazon.com/Mindset-Mathematics-Visualizing-Investigating-Ideas/dp/111935871X/ref=sr_1_1?ie=UTF8&qid=1517965885&sr=8-1&keywords=mindset+mathematics+grade+5&utm_source=Youcubed+Updates&utm_campaign=dffbcc2fb8-EMAIL_CAMPAIGN_2018_02_04&utm_medium=email&utm_term=0_230e567c40-dffbcc2fb8-133942873" target="_blank">Visual Math</a>. These are not expensive books. The authors are from a ground breaking group of researchers that I've been following since the beginning of getyourchildintogat.com. Back in January, I wrote an article where I said that our current math curriculum needs to be flushed as an artifact of the Industrial Revolution. There is equally challenging, more engaging, more pertinent math to the information age. Visual Math.<br /><br />Except that I'm stuck on fractions, polynomials, mononomials, exponents, algebra, trig and calculous because darn it, they show up everywhere in math and all fields whether you're doing machine learning, number theory, or Hollywood CGI. I guess I'm always one rebellion ahead of the next trend.<br /><br />I don't face the same broad classroom education challenges that the authors of Visual Math face. I face the challenge of a single kid. My idea of visual math starts with COGAT test prep, Building Thinking Skills, and the rest starting ASAP, like age 4. See my curriculum page. In a house enriched with crafts followed by Minecraft, visual skills are overdeveloped.<br /><br />But the genius of Visual Math isn't just a much better more appropriate visual (and thus more timely) curriculum, it's the approach outlined by Jo Boaler years ago that is question heavy and solution light. In other words, spending time understanding and defining the problem, whatever that may be, in the process really learning math, and as an after thought deriving a solution. You've heard it before from me, and this is where I got it. There is much more to the approach beyond this.<br /><br />I'm a big fan of a single problem that is hard, multi-step (working memory intense) and requires a lot of time to solve, preferably something goofy or non-sensical, if that's what it takes to turn a predictable answer into an argument. I don't want a child to come out of this having mastered 3 x 5, which is useless, but having mastered getting there from the unknown, or better yet, an unknown mess.<br /><br />And that brings us to 1/2 and 2/3. A few months ago, a reader asked what to do about struggling with fractions. I offered to get on Skype, but since I'm insane, and can turn any 30 second problem into a 30 minute challenge, the reader declined. Too bad.<br /><br />There are 2 parts to a good fraction problem. <br /><br />The first part is 1/2 takes about 3 brain clicks to understand. I think 98% of the problem with fractions is that kids expect 1 click, they don't get it on one click, and they are frustrated or worse. I watch this with the brightest children trying to tackle fractions at a totally inappropriate age. The second part is the fraction in a more complicated setting of a pre-algebra problem. Too hard for younger kids, but doable at a pace 10 times slower than a 5th or 6th grader. Solving a fraction problem is multi-step. When I work with fractions and children, or algebra, or exponents, I expect a few weeks to get them to admit that they have to work the problem step by step. They are determined to do one single step, because it's one problem after all, and if they have to do 3 steps, then it becomes three times the work.<br /><br />Kids who are trained in math hit a wall with fractions. Kids where are 99.9% wizzes hit a wall for the opposite reason. Both groups underestimate the problem.<br /><br />Lately I've been working on the next challenge. How quickly can I get kids to be adept with pre-algebra, exponents/logs, functions, geometry proofs, algebra, trig and calculus? By quickly, I mean a small number of problems and weeks per topic. My group is 4th to 7th.<br /><br />In each case, a few problems can be used to explore the basics. During this time, there is wonder involved with the new syntax and the concepts that it articulates. Like the first time a child stumbles on negative numbers or square roots. A few problems get the job done. To take the next step requires a special problem solving approach for each field. We avoid the complicated applications that fill 90% of a decent text book and just stick with the basics. <br /><br />I've come up with a one session introduction to trig that addresses many of the questions (about 25%) on a good trig final. One session for a 9 year old. I remember struggling with this exact same material for about a month in high school, trying to remember formulas. I'm really disappointed about how bad the course was and how unprepared I was (not having studied math between 1st grade and trig). But I'm mainly disappointed with the approach to math from the 1920's which I used in high school. <br /><br />The last thing I'm going to do is explore the other 75% or so of each of these topics. I think this will be an 8th grade exercise. Is it possible to send a child to high school prepared to be bored with A/B calculus or chemistry? Can this be done with almost no work whatsoever? I'm starting to think so. <br /><br />I enjoy getting articles from readers that include an age and a topic and a description of how much they are struggling. I think, wow, we struggled much worse. I can tell them that and actually solve a problem. I can also state, if needed, 2 or 3 ways to get past it and how long it will take (longer than you think.) In some ways, this is just like potty training. Some parents wring their hands over every trip to the potty, and others let their kids poop all over the place until the problem takes care of itself. The only thing I did differently was discuss plumbing while cleaning the poop off so that I'd have someone I could count on someday to clear clogs.<br /><br />Someday is almost here in math. In plumbing, my 13 year old routed the pipes right before his birthday.Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-87379727663213174052018-02-03T09:45:00.001-08:002018-02-03T09:45:17.168-08:00Innovations in Math EducationI promise as soon as I complete this article I'm going to start populating the reading list in the prior article on reading. But first...<br /><br />My 8 year math program is about to come to fruition. In April, math Experimentee #1 (a newly minted teenager who started K with a bunch of my newby missteps) is going to take the MAP test, and after a long interval of not caring, this score counts and it has to be a 99%. In this article, I'm going to summarize where we are, demonstrate the leap in math skills that happens in 4th grade, demonstrate how my math program is dramatically different than regular programs, and present it in such a way that I lose most readers before I get to the end because that 99% is competing against about 10,000 other kids in Chicago who's parents are all googling 'How To Get 99% on the MAP So My Child Gets Into A Decent High School'. Also, I'm going to discuss my approach in purely in 4th grade terms to help parents of younger children plan ahead, and explain why Test Prep Math is the way it is. <br /><br />Let's start at the beginning. My first goal back in K was to conquer Every Day Math. We didn't have to pick everything up at once, just a lot of hard work to show 'You Can Do This'. My goal was simple. For Experimentee #1, the goals focused on entering a GAT program in 1st grade, given that we were totally behind because we did nothing to prepare for it, not even phonics or learning to read, but at least the math would be familiar (it would be EDM Grade 2 - a complete repeat) and he would have some confidence.<br /><br />After crying, forgetting, getting them all wrong, spending a week or two on a single 6 question worksheet page, having to find 1st grade books to practice concepts and skills we never saw before, I transformed the following survival skills into Academic Coaching Skills that we would use for the rest of our lives and pass down many generations (hopefully) of bright descendants until one actually wants to study math in graduate school. Here they are:<br /><br /><ol><li>Set Your Expectations To Zero. Don't expect your child to get anything correct, understand it, remember it, work on their own, or anything. Even if you do the same problem every day for a week and it's 7 + 6 = ? This is the parent skill. The child-parent team skill is to enjoy 'Being Baffled' on totally hard work that has never been encountered before that will take a lot of time to sort out.</li><li>Make Mistakes. Mistakes are the key. After a while I stopped looking at solutions because I expected mistakes. </li><li>Take A Long Time. When we slowed down to 30 minutes per problem, we started making progress. This is also known as 'Read The Question' where we spent more time thinking about 7 + 6 and what it could mean and how to work it before solving it. </li><li>Other tips I put in the blog over the years, but the top 3 were the key.</li></ol><div>So here's what we got. At one point, we sat down and looked at Student Journal #1, with every single problem answered. Every single problem. No child anywhere does every single problem in a math book, or every page or even every chapter. This is a rare and invaluable life lesson. Experimentee #1 has an extremely high tolerance for work, chores, painful work, hard chores, ridiculously hard chores. Even better, Experimentee #1 is not put off in the slightest by being totally confused on material that is way beyond his abilities.</div><div><br /></div><div>Somewhere during this process, the speed of learning and work accelerated to match the challenge, and by about 1/2 through Student Math Journal #2, we quit because the challenge was gone.</div><div><br /></div><div>Experimentee #2 experienced hard core phonics (Pre K Phonics Conceptual Vocabulary and Thinking age 4.0) and hard core math (Shape Size Color Count age 3.9) because I wanted to address any gaps I found in GAT preparation and more importantly COGAT prep, and did it with a sledge hammer the size of an SUV. Experimentee #2 has math skills that Experimentee #1 will never have, like a child who learns to play the violin from birth will always outplay a child who picks it up at age 6, but Experimentee #2 has a completely different work ethic. Experimentee #2 will sit down with something quietly for hours and master it, but not without a lot of complaining about the fact that he can't pick it up immediately. Experimentee #1 never complains.</div><div><br /></div><div>There is a completely different path for K and 1st grade that will produce almost identical short term results. Many parents enroll their children in an after school math program. In a good program, the child learns problem solving skills and solution strategies as well as practices math daily. This is not a bad approach, but it is not consistent with the goals I mentioned above and a few I am going to add shortly.</div><div><br /></div><div>After 1st grade, we stopped learning math and went more hard core into Test Prep Math. This series is not about becoming adept at advanced math topics, but becoming adept at navigating convoluted questions, staying in the 'math game' because the questions are somewhat on the goofy side and don't include boring, manufactured math book type questions, and building working memory. This book is not designed for children already at the 99% level for math, it's designed to get them there shortly thereafter. I've had a few parents who's kids finish 2 years of after school math (and are at 99% already) complain that the beginning of the book is too easy. Kind of a 'duh' moment for me, but one I need to mention for those kids, Test Prep Math Level 3 in 2nd grade is preferred. The purpose of this book is to lay the groundwork for 99% thereafter, not to put a 99% kid at 99.9%, except by accident (which is what we experienced, by the way). </div><div><br /></div><div>Instead of more math, we went directly from Test Prep Math into reading comp questions. This should be obvious from the problems in Section 1. Section 2 takes us directly into competitive math questions (because I need something to fill the gaps before ramping up real math in 4th grade). But the MAP score is only half math; the rest is reading comp.</div><div><br /></div><div>From 1st through 4th grade, we only stayed a year ahead in math while I put together the basic skill set that we need. This basic skill set is very similar to the skill set that kids would use to survive an advanced engineering or abstract math course in college but it's missing formal solution strategies. College is the other goal, and I'm thinking ahead as usual.</div><div><br /></div><div>At inappropriately young ages, while we were biding our time putzing around with current + 1, I started introducing advanced topics, just for fun, just to exercise thinking and start to explore the wonder of math. It was enormously enjoyable to surprise a kid with these types of questions:</div><div><ol><li>What is 5 minus 3?</li><li>What is the square root of 4? 9?</li><li>What is 1 divided by 2?</li><li>If 1/3 of my donuts are chocolate, what percent of these are not chocolate?</li></ol><div>If your child sees any of these questions for the first time in school, I guarantee the wonder, fun, learning and enjoyment of math will be totally crushed out of the experience because your child will be presented with definitions, comprehensive examples, and a long list of routine problems that have nothing to do with anything. It then just becomes a pattern matching and lookup referencing exercise. The child will 'learn' math, but not know how to learn.</div></div><div><br /></div><div>Sometimes we would resort to backtracking, which is finding a workbook or online resource to practice the material during the learning process. If we got '1/4' kind of but not really, a worksheet might fill in the gaps. If a concept (fractions in this case) requires an understanding of division that is not there, we would certainly backtrack to a division worksheet and then come back to fractions.</div><div><br /></div><div>Over time, however, I discovered the power of bucketing, which I subsequently labeled 'Power Bucketing'. This is very similar to what I witnessed with Experimentee #1 going into 1st grade and being handed the same EDM Grade 2 workbooks that were completed the previous year. Math is much easier to understand the 2nd or 3rd time than the first time, and quick mastery is the likely result.</div><div><br /></div><div>With '3 - 5', I would just leave the question out there and not answer it. Or maybe I would answer it, but then a month later I would ask it again and watch the same process starting over again from the start, but going a bit faster and progressing a bit farther. When this come up again out of nowhere the third month, we might end up with mastery with almost no work and exactly zero practice. Even better if the child sees negative numbers on his own in a book, he dives right in and the result is not only self-mastery, but he owns it.</div><div><br /></div><div>SQRT(4) and also 5x - 13 = 2 will demonstrate the leap in skills that takes place around 4th grade. Kids coming off work like Section 2 of TPM can calculate both of these without understanding how they do it. Good little mathematicians iterate through possible solution values until they arrive at the answer, and great little mathematicians add weighing with high-low bands that narrow to the solution strategy to arrive at the answer more efficiently.</div><div><br /></div><div>After 4th grade, when the brain is capable of higher order thinking, these two exercises gain new meaning. The definition of SQRT(4) is the number when squared that equals 4. In other words x^(1/2) is solved backwards. Square roots present the opposite syntax of squares, and the solution is to back into the answer. This is critical for topics that are going to come later. 5x - 13 = 3 is a simple introduction of y = mx + b, which is an important framework for characterizing more complicated problems, and the elements of y = mx + b have additional meaning besides finding a number.</div><div><br /></div><div>There are also new skills that come with these math concepts. A 3rd grader will jump in and solve either problem to get a number. It's all one step. A 4th or 5th grader will decompose the problem, spend more time analyzing the question, and learn more during the problem. I've introduced younger children to the next level of math skills, like problem decomposition and making a hard problem easier; this exercise can take 20 minutes and is really good for thinking. It requires a lot of working memory which is why in 2nd and 3rd grade working memory is most of the focus. But older children do this intentionally, quickly, and know why they are doing it.</div><div><br /></div><div>Let's look at some pre-algebra concepts that have been a real struggle for me to teach. </div><div><br /></div><div>First, x^2x^3 versus (x^2)^3. Per formula, the first is x^(2+3) and the second is x^(2*3). But we're not interested in formula's, because formula's produce math dummies. </div><div><br /></div><div>The way to do these problems is to work the question and not solve the problem. x^2x^3 is simply (xx)(xxx) = x^5, and (x^2)^3 is (xx)^3 is (xx)(xx)(xx) = x^6. Eventually, the child will memorize the formulas in the same way they used to count on their fingers for 5 + 3 and eventually knew that 5 + 3 is just 8. Before 4th grade, the best I can do is lay the ground work for decomposition, restating the problem, multi-step solution operations, but they still jump into more advanced problems trying to get to a number in one (hard) thinking step. I've noticed that after school program kids are drilled in multi-step solution strategies, but I don't want a child trained in math solving. I want a thinker.</div><div><br /></div><div>This is the biggest difference in my goals and methods. I don't want a child who is trained in math, a a child good in math, a child who knows (advanced) math topics or a child who is 99% because of this training. I want a child who does really well in math he has never seen before or mastered because he is a thinker and a learner and can apply thinking and learning to math. I've always said if you need a 99% because it is required for GAT entry, do what ever it takes this year and forget your principals. In 7th grade, I can't say this; it is not possible to short cut your way into a 99% without a solid learner-thinker. Also, we've never actually deviated from principals or practiced rote math and we have always either been at 99% or been within striking range (in a bad year). I will say that it's never too late to start. There are advantages to starting early, but starting late does not preclude achieving the ceiling on a test.</div><div><br /></div><div>The most challenging topic using my approach on pre-10 and post-10 children is parenthesis. I will illustrate with this problem: (6^2 + 18 + 2 + 4^2)) - 2^2. This is not a complicated problem, but it is not possible to do a page of these problems with a child still learning exponents and parenthesis without writing down at least 3 or 4 steps in order to check steps for errors. In other words, it's faster and easier to let the pencil do the work than the brain. Before 4th grade, I'm happy to endure 4 or 5 wrong answers from mental calculations because the impact to working memory (not to mention arithmetic practice) is useful. But with the problem above, working memory gets a work out and the child still has to write down each step to survive the problem.</div><div><br /></div><div>In our 1 year ahead math program, it is common for kids to fall to 85% by about 6th grade. The program administrators - geniuses way ahead of their time - are focused on the final result and this interim dip is a researched based way to achieve the final product. The extra 14 points are achievable with a bit of extra work. If you review this article from the beginning, you'll see 3 or 4 math education concepts that all work together to produce 99% without a lot of extra effort. I don't think this approach would work very well in a classroom situation without modification, but it certainly can at home. Once any topic above is presented above, the next step depends on the child's response in the context of the child's individual skill set. A parent who gets to know their child and experiments a little with backtracking, repetition, exploring the question will stumble toward success.</div><div><br /></div><div>Now back to the 7th grade challenge that introduced the article. We have a very ambitious goal but not a lot of time to achieve it given homework and nonschool activities. The topics, approach, learning environment, and general mess of our preparation is in my opinion an exact mirror of the test.</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com5tag:blogger.com,1999:blog-5703568807615263851.post-91501415172945658112018-01-21T18:31:00.000-08:002018-02-03T10:18:35.649-08:00A Decent Reading ListI’ve been asked to recommend good books to read for 2nd to 3rd or 4th grade. I’ve search for a decent source so that I wouldn’t have to do this, but there is no decent source. There are many recommended reading lists, including grade schools, libraries, and sites that bill themselves as good sources, and none of them provide even a tiny fraction of the good books that are out there.<br /><br />In the book Pre-K Phonics Conceputal Vocabulary and Thinking I provided a comprehensive recipe for strong reading. It was inspired by the very inspiring introduction to Susan Wise’s seminal work The Well Trained Mind. She said that she would take a laundry basket to the library. I did this for 3 years. It’s enough to put your child squarely into ‘Chapter Books’. In Pre-K Phonics, I took this to the next level, and maybe the level after that.<br /><br />By 4th or 5th grade, your child will be reading books of their own choosing, books that take a week or two to read and have 10 more in the series or genre, and your recommendations will likely be ignored for the next 15 years.<br /><br />That leaves a very important 2 year gap where the child needs help finding good books. This is also the last time that Read To (super important) will be easy to do.<br /><br />In this article, I’m going to lay out the approach, and then over the next few months, I’m going to fill in the blanks. The math work that I provided for the early years is now coming to fruition in 4-7th grade weekend math (because of homework it’s no longer daily). We simply need a 99% on both sections of the MAP test in order to get into high school. That’s not asking much. Some day I’ll tell this story, and it will sound a lot like age 4, only with much more advanced topics. In the mean time, it’s time for reading.<br /><br />Here are the buckets.<br /><br /><b>Mandatory Books</b><br />I’m convinced that the Hobbit and Roald Dahl’s complete works (including autobiography) should be mandatory reading during this age. The list is much longer and needs work. If your child reads the Magic Tree House somewhere between late K and early 2nd, you are where you should be. The mandatory books will get you to the next level. (When I say ‘reading’, I mean Read To as needed, especially with the Hobbit.)<br /><br /><b>Top Notch Books</b><br />Gifted programs have a formidable reading list that includes classics like Kira Kira. These books are easy to spot because if you query the book in Amazon, you will see teachers guides in the search results. I suppose that’s not easy if you don’t start with the list. This is probably the most important list for my readers and the one I’ll work on first.<br /><br />To put this list together, I’ll simply steal it from a dozen programs I’ve watched over the years. Feel free to add to this list in the comments. At some point, I’ll just move this to the permanent pages.<br /><br />[Feb 3 - I've been trolling through material and it's so bad I'm going to have to go through all of the publisher's websites.]<br /><br />2nd Grade<br /><br /><ul><li>Dear Mr. Henshaw</li><li>The Miraculous Journey of Edward Tulane</li><li>Bunnicula</li><li>Sarah, Plain and Tall </li><li>Charlotte's Web</li><li>A Long Way from Chicago</li><li>Harry Potter (not in school; I recommend the whole series spread out over the next 5 years)</li><li>Boxcar Children</li><li>The Story of Pilgrims Progress (not sure about the age yet)</li></ul><br /><br />3rd Grade<br /><br /><ul><li>Fair Weather</li><li>Mr. Tucket</li><li>The One and Only Ivan</li><li>A Wrinkle in Time</li><li>Bud Not Buddy (pair with historical context and there is a play on this as well that's pretty good.)</li></ul><br /><br />4th Grade:<br /><br /><ul><li>Chasing Vermeer</li><li>Because of Winn Dixie</li><li>Love That Dog (poetry)</li><li>Kira Kira</li><li>The Mixed Up Files of Mrs. Frankweiler</li><li>Call of the Wild (boring)</li><li>Amulet (graphic novel series - not taught but all kids read it)</li><li>The Hitchhikers Guide Guide to the Galaxy (I don't know why this is here)</li><li>The Watsons go to Birmingham (pair with historical context)</li></ul><div>This <a href="https://www.mensaforkids.org/achieve/excellence-in-reading/excellence-in-reading-4-6-list/" target="_blank">link from Mensa</a> for 4-6 grade is not bad, but dated.</div><br /><br /><b>New Books That Are Classics In The Making</b><br />We have two libraries near us that are the 2nd and 3rd largest in Chicago. Because of this, we get to see all the books worth reading somewhere in the shelves. We tried them all.<br /><br />Most of these are for girls and have a girl theme. We really enjoyed these, but being boys, ignored the girl themes and simply enjoyed the creativity and good story. I’m sure there are good boy themed books, and I’ll list these, but mostboy themes seem formulaic. These books are 3-5th grade.<br /><br /><ul></ul><br /><ul><li>Keepers Trilogy (2nd or 3rd grade advanced) </li><li>Savvy (my favorite, definitely a girl book)</li><li>Tale of Desperoux </li><li>Percy Jackson and The Olympians (my 4th grader also recommends the Magnus Chase series)</li><li>I have to wait for a few kids to get back from Boy Scouts to complete this list, including one boy (not mine) who read the Lord of the Rings trilogy in 1st grade and got a 100% when I grilled him on it.</li></ul><br /><b>Books You Didn’t Think About</b><br />I am a big fan of picture books and winners of foreign book awards. The ones we choose tend to be small in words and big in mind blowing concepts. I had to do inter library loans to get many of these.<br /><br />Shel Silverstien is on this list. We bought his books and read them daily. Jack Prelutsky is on this list. David Weisner (Flotsam) and Brian Selznick (all his picture books).<br /><br />One day my child was writing a few poems for school. They were really, really good. It wasn’t a fluke.<br /><br /><b>Books To Enjoy Reading</b><br />If you search for lists for a 2nd grader (or a 4th grader because your child is an advanced reader) you’ll see a list that includes mainly junk. It might as well be comic books or romance novels. But we read all of these because it guarantees that your child will have one or more books in hand at all times. The child is not gaining anything out of these books (think Diary of a Wimpy Kid and Middle School) other than the habit of reading all of the time. So we read all of these. But not at night, when it was reading time, and a quality book should be in hand.<br /><br />James Patterson (top selling author) started writing books for reluctant middle school readers because his son was one. This list includes some really great works for advance 2nd and 3rd graders, especially boys, such as Treasure Hunters. You can’t put one of these books down. There isn’t much cognitive value to his books. That’s not the point. It’s about becoming addicted to reading.<br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com11tag:blogger.com,1999:blog-5703568807615263851.post-18310837512869252132018-01-14T14:36:00.002-08:002018-01-14T14:36:17.359-08:00Struggling in MathI have gotten a lot of questions in the last 2 months that I will summarize and then answer:<br /><br /><ul><li>My child is struggling with their At Home Schooling math, which consists (usually) of me making them do a math work book that is 2 years beyond their grade level.</li><li>My child started school at 99% and is now at 85%.</li></ul><br />I tend to stay focused on preparing for a strong high school math experience; neither of these two issues ever bothered me and your children are smarter and better than mine. We did have a dip in test scores and I went into RED ALERT mode until it was corrected. Both of these topics have been covered over the years, but it's pretty hard to dig through my blog to find answers. In addition, I already deleted the 300 articles that had a play-by-play of my struggles.<br /><br />Both of these are linked, because in order to get to 99%, your child either has to go to expensive after school math programs that will gradually make them hate math, or your child will work ahead at home.<br /><br /><b>Struggling In Math</b><br />The answer to all of your struggling questions is called 'Backtracking'. We do it all the time. I can't imagine doing any math above grade level without a lot of it. Here are some examples that I've written about while we were doing it:<br /><br /><ol><li>If we were doing EDM Grade 2 in Kindergarten 3 days a week, at least one day a week we did a first grade math workbook that was just adding and subtraction. Some times this is a nice break, sometimes it's catch up, some times it's practice.</li><li>Sometimes I take 2 or 3 weeks off to cover a concept that we never had or a concept that we just plane stink at.</li><li>Sometimes an entire section in the workbook is almost all wrong. Sometimes it's just a page or an important problem. The kid just doesn't get it. I circle the pages and we move on. A month or 2 later, we'll come back to the circled pages and do them again.</li><li>When the child is younger, there are some bad days because of hunger/sleep/sickness issues and we just do flash cards or arithmetic worksheets. Bad days happen rarely at older ages (always the day after a sleep-over), but when they do, we do nothing at all that day.</li><li>Sometimes we take time off from math and do projects like a puzzle or sewing something or a craft or a writing project or art, a comic book, whatever. In each case, the child just starts doing it and I will not interfere. I am convinced that these activities will produce a stronger mathematician than actual math.</li><li>We like to do things backwards. So if the book does it one way, we redo the whole thing backwards.</li><li>We like to do things step-by-step. Identifying the mini-steps helps you find backtracking material. Here's a really simple example. 23 x 15. This has 4 separate multiplication operations and 3 addition operations. Maybe your child should just practice multiplying 3 x 4 and 30 x 40, 9 x 2 and 90 x 20 etc for a while before coming back, or 20 x 15 and 3 x 15.</li></ol><br />There are two difference between you and me. First, as previously mentioned, your children are smarter than mine. Secondly, we back track a lot. Why continue to struggle with the same material? Do something else, practice something, come back to it later. It will all get done in the end because we are both picky and uptight parents about math.<br /><br /><b>Test Scores</b><br /><br />Lately I've been getting a lot of feedback from many parents that test scores are falling. I get this from almost all parents (like 85% of the ones I talk to) at some point during grade school, usually right around the midpoint. Here are the reasons:<br /><br /><ol><li>Your school program teaches and practices math at about the 85% level. Over time 99% children will end up working at the 85% level.</li><li>Your child is sick of doing math and needs a year off.</li><li>You are not doing daily math at home at a suitable level and 15% of the country is.</li></ol><div>None of this is a bad thing. I think our program starts pushing math at the appropriate time and produces graduates who are really strong in math. This will not make a parent happy in the following 2 circumstances: #1 Your child needs a 99% right now on an annual standardized math test this year. #2 You have some other objectives in mind that requires a 99%.</div><br /><br />Here is my 3 part recipe:<br /><br /><ol><li>Get math at a suitable level.</li><li>Do it. Backtrack a lot.</li><li>Focus on problem solving techniques and not math. Math will take care of itself.</li></ol><div>I can now see that I need another article because the leap between 3rd and 5th grade and it's called problem solving skills. My particular approach can be summarized as focusing on nothing but problem solving skills during 2nd and 3rd grade and it works. Not just any set of problem solving skills, but the core skills that are the foundation of all others. That, in a nutshell, is 95% of the motivation behind Test Prep Math. The other 5% is making math less boring than it normally is. </div><div><br /></div><div>But I'm hearing from parents of 3rd and 4th grade children that didn't go this route. I've got some thinking to do. It's solvable. Anyone can catch up to any level you want to get to.</div><div><br /></div><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com3tag:blogger.com,1999:blog-5703568807615263851.post-78808336400197275182018-01-12T09:18:00.000-08:002018-01-12T09:18:00.327-08:00The Language of MathThere is a strong and important connection between math and language.<br /><br />Think about a child learning language before the age of 2. You point to a blue ball and say 'blue'. The child sees round, blue, rubbery, your finger, you making some weird noise, you're looking at him or the ball or both, and you're probably smiling. What is blue? Then you point to a blue wall and say 'blue' and the kid is more confused than ever. <br /><br />In order to figure out blue, ball, yellow, green, box, toy, your child has a lot of confusion to sort through, is going to make 5,462,298 mistakes, and you're going to be smiling the whole time, and on top of that the child is going to have to identify patterns, sort through permutations and eliminate candidates until he comes down to blue is an attribute of color. The child may not see round or plastic or squishy yet, maybe he can sense it, but when there is a word tossed out there for 'round', his ability to think logically will be substantially improved.<br /><br />By 1915 or 1911, I'm still debating, cognitive psychologists determined that the process of reading uses 100% of all cognitive skills. 100%. This will never happen again.<br /><br />If you want to know why I'm so over the top obsessed with reading and vocabulary during age 4, so much so that I created Pre-K Phonics Conceptual Vocabulary and Thinking to jam as much 2nd grade material into the brain of a child who can't pronounce C-A-T, you now know why.<br /><br /><b>Don't Lose The Magic</b><br />Learning to talk and learning to read, not to mention learning to walk, are much harder by a factor of a gazillion than anything a child will learn thereafter, including Pre-Algebra. But somewhere after learning to read, maybe around addition, the parent loses the Magic Learning Environment that allowed your child to overcome insurmountable learning objectives. You used to sit there smiling dumbly mistake after mistake totally happy every time your child rose an inch off the ground and then fell. Now you're yelling at your child for forgetting what 8 + 4 is or struggling with x<sup>-1</sup>. At least I am. We ALREADY discussed the exponent graph 3 times. Would you just pay attention once?<br /><br />The magic was that you were willing to try to teach your child what words mean, despite not having the slightest clue how this works, through mistakes and trying over and over and over again, usually smiling the whole time, and learning just exploded.<br /><br />This is the first connection between language and math and it's pretty lame compared to what follows.<br /><br /><b>Reconnect the Two Dots</b><br />If math uses a certain sub set of cognitive skills, but learning to read (definitely) or learning word definitions (probably) used 100% of cognitive skills, wouldn't it be great if you could bring the missing cognitive skills back to the math learning process?<br /><br />I think this is theoretically possible and in practice I just ask them to explain verbally to me how to what the question is asking, what do they know, is there anything they have learned before that can help, can you articulate your solution strategy? I also throw in anything I can think of related to a problem, like 'Polyhedron' or some other word to get that verbal section of the brain working.<br /><br />But mostly I like to talk through problems and concepts.<br /><br />Recently, we came across this question: What is 42% of 66? This is an advanced post TPM problem. I got it off a high school Algebra I final that has 190 questions and would be very hard for high school We're doing about 5 problems per session and learning a lot. This is an opportunity for a long discussion involving fractions, decimals, and %, as well as problem decomposition and lining up multiple steps, followed by cheating with algebra. In other words, in addition to math, it's going to be about 25 minutes of talking.<br /><br />Here's some fun verbal math discussions for a younger age. In these cases, I did very little talking and just left key questions out there for 3 or 4 weeks while the math sank in. Then we discussed, and I asked why? or prove it to me.<br /><br /><ul><li>The definition of 'square root' is this. 2 is the 'square root' of 4 because 2 x 2 = 4. What is the square root of 9? Does 10 have a square root? (Not yet, but it will later).</li><li>What is the square root of negative one? It's call 'i'. What is i * i? Why is this important (because the Fundamental Theorem of Algebra does not hold true without i in case you're wondering). What is the square root of - 4? </li><li>What is 2 - 5? I love this discussion. It goes like this: "Three". If 2 - 5 is 3, what is 5 - 2 and why are they both 3? This can't be right. If you have 2 and you give away 5, what do you have left? "You can't do it". Oh yes you can my friend, yes you can. </li></ul><div>If I can't find something to discuss in math work, I'll start looking for more math. y = mx + b and f(x) = mx + b are great topics for discussion and not writing. That's why we end up covering advanced math at a young age, simply to have something to <i>talk</i> about. How's this going? About as well as learning how to talk in the first place.</div><div><br /></div><br /><b>Is Any of This Going To Help?</b><br />I'm not 100% sure yet, but it might help with math learning. It's definitely helping with writing. Trying to compose an explanation for a complicated mathy topic just learned is really hard. It is a foundation leadership skill. It's similar to a reading comp skill, but only vaguely. It's easier than any classroom speaking task. I'm certainly not going to end up with a wall flower, what with me demanding a thorough explanation to a complicated explanation.<br /><br /><b>Product Recommendation</b><br />I highly recommend IQ Twist or IQ Puzzler Pro. We've had these sitting around for the last few years and my kid and his class are now obsessed with them. His 4th grade teacher is buying them for the classroom.<br /><br />It wasn't until I solved a problem myself that required turning and flipping multiple shapes when I realized that it's NNAT and somewhat COGAT training. We started talking through the solution to one tough problem and how one shape could only go in one certain place before I realized that this is all logic, visualization and math. If you run out and buy these for a 1st grader like I did, feel free to reach out for help because it took me a few years to figure out how to use these with a younger child.<br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com7tag:blogger.com,1999:blog-5703568807615263851.post-25662792469561710372018-01-08T10:58:00.000-08:002018-01-08T10:58:03.775-08:00Real MathMy son complained about his daily math. It was some problems from two Pre-Algebra topics.<br /><br />If we do pre-algebra every day it's going to get boring. I refuse to do either decimals or long division or math facts or anything between kindergarten math (totally engrossing) and pre-algebra (marginally useful) because it's all boring and useless. <br /><br />I need a fall back plan. He's been playing IQ Twist lately (highly recommend this game even though I don't get paid for any of my recommendations) and that got me thinking. There is this great math book called Mathematics 1001 that has 1000 math topics in addition to 2 pages on trig that allowed us to cheat our way through it. One of the topics in this book called 'Net's looked like the shapes in twist, and a little reading later uncovered this idea.<br /><br />Here are two <b>Nets</b> for a triangular pyramid. If you cut out either Net, you can fold it into the triangular pyramid. <br /><br /><div style="text-align: center;"><a href="http://2.bp.blogspot.com/-vADMGurqR30/WkKWHHS5UuI/AAAAAAAAAr8/uiRmA0v50BYJvMpbRNftaj-K5uCzFyFBACK4BGAYYCw/s1600/net%2Bof%2Bpyramid.jpg" imageanchor="1"><img border="0" height="128" src="https://2.bp.blogspot.com/-vADMGurqR30/WkKWHHS5UuI/AAAAAAAAAr8/uiRmA0v50BYJvMpbRNftaj-K5uCzFyFBACK4BGAYYCw/s320/net%2Bof%2Bpyramid.jpg" width="320" /></a></div><br />There are 11 nets for a cube. Draw them.<br /><br />I watched two sets of skills in action. First, there was geometric visualization, including rotating, flipping, and 3 dimensional manipulation of shapes which trumps the two dimensional manipulation on a cognitive skills test. If we were facing a test this year, I would have only shown the diagram on the left above and asked for 2 more nets for the triangular prism (even through there is only one because cognitive skills tests test your ability to come to terms with incorrect questions).<br /><br />Secondly, there were budding permutation skills at work, which is an extremely important math skill. Since no kid is going to get to 11, this gives me the opportunity to suggest permutations. "What's a permutation?" Well, take the letters a, b and c. I can write them as abc, acb, bac, bab, cab, and cac. There are 6 permutations of the letters a, b and c. Please give me the permutations of 1,2 and 3. This should be pretty simple. Then look at the basic T shaped Net for a cube, and start permuting the squares, one square at a time.<br /><br />We got to 7, which is pretty good for my 25 minute time limit. I need to stop at 25 minutes to save room for follow up questions, like telling me the rules for building a Net while staring at the 11.<br /><br /><b>Real Math</b><br />I expect this child to go far in math. He's not going to go anywhere without some intervention. Here is my intervention.<br /><br />I showed him a web diagram of the 11 nets for a cube. I stated that some guy (Albrecht Durer) asked how many ways you can create a folding diagram for a cube, and he came up with 11.<br /><br />Then I showed my kid the pre-algebra worksheet of about 20 equations. <br /><br />I asked this question. If there is a mathematics professor and researcher at some university asking questions and writing papers and going to conferences and helping his colleges in the Physics and Information departments apply abstract math to their work, which math is this professor doing right now? (And by way of association, which math are the physicists and computer sciences clamoring for?) Does it look like this (pointing to pre-algebra) or does it look like this (point to the 11 nets for a cube.) <br /><br /><b>The Answer</b><br />The answer is the net stuff. And why is it that your school curriculum looks like pre-algebra, the type of math that mathematicians don't do?<br /><b><br /></b>Here is my (mostly inaccurate but totally) true history of math curriculum in the United States. In 1930, a vice president at Ford Motor company created a list of skills needed by factory workers and accountants and dealers to create and sell cars. This skill set was widely applicable to industrial work of all types. A curriculum was created to teach it and used throughout the United States. Lots of cars were produced and everyone was happy. This curriculum is still used in 2017 in the midst of the Information Age.<br /><br />Of the 96 maths out there, school is going to consist of the 5 that would help you build cars by hand or build a bridge, which you are never going to do. The maths that you actually need to get through your life - starting now - are not taught at all.<br /><br />What I find most interesting is that the 5 maths taught in US curriculum are almost devoid of skills compared to the maths that could be taught. <br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-91083294334007340452018-01-02T09:17:00.000-08:002018-01-02T09:17:02.708-08:00The Train Wreck RevisitedThe 'Train Wreck' is one of the 5 things that should keep a parent awake at night worrying. Three of these are reserved for children over the age of 12, and that only leaves 'Lacking Motivation to Read' as the two things you need to worry about right now.<br /><br />The term 'Train Wreck' is used in situations where a child who previously got all A's in math is now getting a C. This most commonly occurs between 4th and 8th grade. It can also occur in Algebra or Geometry. This term also applies to a child's test scores falling from 99% to anything below 90% and is somewhat related to regression to the mean.<br /><br />None of this is very shocking. You want shock? Let me define this term formally.<br /><br />Train Wreck: At one point, your child held a formidable skill set and did well in math. A few years later, when your child faces a new math, the child doesn't do well because the child does not possess the skill set required to do well. You are left wondering what happened and either correctly blame yourself or incorrectly blame the teacher.<br /><br />The most common cause of the 4th grade Train Wreck is a child who is overly endowed with skills entering 1st grade and spends the next 3 years at school not thinking. By 4th grade (depending on the school district and curriculum), there is a jump in complexity, and the child has no tools in the tool shed. The train wreck in middle school or freshman year is usually caused by a catastrophic failure of curriculum, but can also be the result of a bright child languishing in an average curriculum.<br /><br />Regression to the mean is an empirical consequence of the level of instruction in school. Kids who score below the mean usually catch up test-score-wise while experiencing instruction at the mean, and kids who do much better than the mean usually slow down while experience instruction at the mean. I'm waiting for the field of cognitive psychology to have a 'duh' moment and figure this out, but tat the time of this writing, they are still baffled. Anyway, Regression to the Mean is a less dramatic version of the Train Wreck but is caused by the same factors.<br /><br />There are at least 2 leaps in cognitive requirements that take place in grade school math, and at least two in language arts. In high school, a really great curriculum will have at least one leap every year (most don't). Are you happy with your child scoring well this year, or are you really concerned about their score in 2 or 3 years? Thanks to No Child Left Behind, teachers are mandated by law to be concerned only about this year at the expense of next year. Thanks to having 30 kids with a variety of skill sets, the teacher can only do so much. You're going to have to pick up the slack. <br /><br />Happily, I've found that being only concerned with 2 years from now tends to take care of this year and next year for no additional effort. By when we work ahead 2 years, I stay focused on the skills, not the math.<br /><br />There's a really great book by a psychologist to deal with the Train Wreck. There's a lot of great 'Yoda' in this book, but it's downfall is that the author doesn't address the skills issue. He has a valid excuse because he has a PhD in Psychology, a field who thinks IQ magically happens. I've done a little work in this area, but not enough to write a book. My market is shooting for 99% (if you're reading this, this is now your official goal if it wasn't before) and we're going to need the whole bag of skills to get from X% to 99%.<br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com4tag:blogger.com,1999:blog-5703568807615263851.post-16240347444335764862017-12-29T18:00:00.000-08:002017-12-29T19:01:40.859-08:00Fractions in 2nd GradeLast month, a reader asked me how to teach fractions to a 2nd grader. I'm assuming that the reader is asking me to provide the magic formula for a child of a specific skill set (which I haven't measured) paired with a parent who has a specific skill set (which I haven't measured). This is a tough problem that requires the Force. I retreated to the island of Skellig Michael off the coast of Ireland so I could meditate.<br /><br />This article is going to describe how to teach a math topic (in this case fractions). The approach for fractions shares a lot with other topics. Fractions is especially important in math because it is slightly abstract and always multi-step, follows 3 straight years of spoon-feeding one step problems in school, and therefore befuddles students and parents alike for want of cognitive skills.<br /><br /><div style="text-align: center;"><b>The Parent</b></div><br />Before we launch into this topic as a parent/coach/Jedi Master, we need to take a step back to appreciate the importance of this math topic. It is fascinating, alluring, captivating. Without this appreciation, it's hard to pass our love of math to our children. If your child is ready to take on fractions at a young age, you will convince no one by saying "Fractions are totally boring and pointless to me but I'm going to make you do them anyway".<br /><br />What are our goals for fractions? We want the child to know fractions so that they can look at a super hard problem and see the answer right away. We want the child to emerge from these fraction studies with a formidable skill set that can be applied to other quantitative areas. We want to present this child with something like exponents or algebra someday and the child will say "Leave me alone. I can do this all by myself".<br /><br />There are some years from K to 8 where you want your child to do 45 minutes of math a day and be really, really good at it. In our case, there were 3 of these years. I wouldn't do it every year because the child will learn to hate math, and it's not necessary anyway. If this happens to be one of these years for your child, in addition to what I describe in this article, get them a decent workbook that includes fractions and make them do every single problem in the book no matter how long it takes. Otherwise, just do what I recommend here.<br /><br /><b>Fraction Foundation</b><br />The first thing we need is a high level definition of fractions. When you divide 20 by 4, you end up with 5. This means splitting 20 into 4 groups gives us 5 in each group. If you have 20 skittles, but I'm only going to let you eat 1/4 of them, you're only going to get 5. These are two different concepts, but the exact same mathematical operation, namely 20 ÷ 4 = 5.<br /><br />What does it mean to divide 7 by 2? What does it mean to divide 1 by 3?<br /><br />There are two times when we have these discussions. The first time is when I think we're going to be studying fractions in a few months or next year. I call this Power Bucketing. This discussion will create a brand new bucket in the child's brain called 'Fractions', and when the child sees fractions in school, while the other kids are trying to come to terms with fractions, my child will already have an empty filing cabinet in their brain for fractions and will have a permanent head start in this area.<br /><br />When I teach fractions, we spend the first week just asking what fractions are. I will give the child 10 to 20 minutes for them to think through these simple problems, like 7 ÷ 2 = 3 1/2. After we've exhausted the mental capacities of the child, I'll ask for a picture or show them how to diagram fractions.<br /><br />If you look through 2nd, 3rd, and 4th grade math curriculum on the topic of fractions, fractions are introduced slowly. I'm not going to speed through this process. <a href="https://www.ted.com/talks/j_j_abrams_mystery_box" target="_blank">Please view this Ted Talk on J.J. Abrams from 2007</a>. Look at that box with a question mark (in the video). As long as the box is sealed, your child's imagination is in play. As soon as you open it and describe its contents, you've ruined your child. Let the child figure out what is in the box on their own.<br /><br /><b>Fraction Lifestyle</b><br />At this point, you can introduce fractions into your conversation. Think about a really smart parent with multiple PhD's who just talks their child into Stanford. We want to be like this parent, only not as nerdy. The two most obvious uses of fractions are time and baking. Get your child a brownie mix and make them do all of the work. Put post it notes on the refrigerator reminding yourself to talk about time only in fractions, as in 'it's 1/3 past 5, what time is it?' By the way, my older child has been in charge of making desert for years thanks to fractions.<br /><br /><b>Fraction Overdrive</b><br /><a href="https://www.ixl.com/math/grade-4" target="_blank">This page from IXL</a> describes the basic fraction related skills expected of 4th graders. You can also look at grades 5 and 6 because fractions is going to appear every year from now on. I didn't read any of it because it's too boring.<br /><br />Instead, like all topics in math before calculus, with the exception of geometry, we simply have to state the obvious. How to you add, subtract, multiple, and divide fractions? Throw in 2 more operators (greater than and less than) and transformations (aka equals) and that's pretty much our goal. This is exactly 8 things to learn (transformations are 2 things - equivalent fractions and transformation to and from mixed fractions).<br /><br />This little exercise is going to be repeated with rational numbers, exponents, complex numbers, and other pre-pre-algebra topics. When this child is doing algebra for the first time at age 9, and is stuck while trying to reduce a really complicated algebraic expression, I say 'Dude, you've only got 4 possible operations - addition, subtraction, multiplication and division, just try all 4 of them to see which one works.'<br /><br />When your child sees decimals and percentages some day, we'll have 2 additional transformations involving fractions.<br /><br />Where did I get all of this material? I spent a month thinking about it. Your child's teacher does not have a month to spend on fractions because there are typically 6 to 8 topics each day, plus statistics. With some math topics, I also wiki and read about Egyptian or Babylonian history. Your child's teacher won't have time to do this either. She has 8 subjects and 30 kids of cognitive profiles to teach. You have 1 child and fractions.<br /><br /><div style="text-align: center;"><b>The Student</b></div><br />Children are natural learners. Once the parent is prepared (95% of the battle), the rest is easy. Just give your child as long as it takes and don't help at all. Ask the questions and expect your child to work things out mentally, when your child doesn't succeed, ask them to draw the picture. Help as needed, but only after the child has exhausted their mental faculties. I generally observe mental exhaustion takes place at about 20 or 25 minutes (because I always choose really hard material), and I'm prepared to sit there, sometimes silently, for 20 or 25 minutes.<br /><br />If you hand a 4th grade book to your child, there will be gaps hidden in 2nd and 3rd grade material. The child will get stuck on a problem, and the way forward is material that they either never had or never mastered. Be prepared at all times to go back to 2nd or 3rd grade material as needed. Suppose they get stuck on a really hard problem, and you can see that it involves transforming from mixed fractions or comparisons. Take a few days off and do some problems involving mixed fractions or comparisons. IXL is good for this.<br /><br /><b>Step 1 - Comparisons and Transformations</b><br />I'm not sure why a book would be needed at all. The most important fractions are 1/2, 1/3, 1/4 ... 1/10. If the denominator is greater than 2, then you've got 2/3, 2/4, 2/5 ..., 3/4, 3/5, 3/6... and so on. Then you can multiple any fraction by 2/2, 3/3, 4/4 and you've got a set of un-reduced equivalent fractions.<br /><br />Pick any 2, and ask for >, < or =.<br /><br />If your child was adept at division and had a really strong number sense, I would not create flash cards to drill my child on fraction comparisons. If your child did not have a strong number sense because they never had really great curriculum at age 4 or 5 that built number sense, I would not only create flash cards, but I would create spread sheets with 100's of problems from the fraction list and drill the child until their number sense was invincible. In our case, we did this at age 4 with SSCC and never looked back. Except when we did this again. And one other time.<br /><br />You can search the internet for "comparing fractions worksheet" and see thousands of examples. If I never met your child and you only gave me 30 minutes to address "comparing fractions", I would print 3 of these: One with pictures, one with simple fractions, and one for harder fractions (involving primes versus composite numbers, like 12/13 versus 10/12) and I would find out quickly where they are.<br /><br />This exercise requires transformations - like comparing 5/6 and 10/12. This is a two step problem. 5/6 and 9/12 would be a more obvious two step problem.<br /><br />Throw an integer in, like 2 1/3, and you've got the other transformation to get to 7/3. We never go in the other direction, from 7/3 to 2 1/3. When I see this in a book, I comment that this is lame. In higher order math, we only work with 7/3, or 142/25, and never mixed fractions. Also, as I mentioned before 6 ÷ 3 = 6/3 (this is impossible to write in a vertical line, but basically I'm writing division problems as fractions and never using ÷ again).<br /><br /><b>Step 2 - The Other Arithmatic Operators</b><br />Once we 'get' fractions and practice transformations, we have to tackle addition and subtraction, then multiplication and division.<br /><br />Addition and subtraction involves transformation. We can't add apples and oranges. We have to transform one or both. This is why transforming and comparing fractions is a prerequisite.<br /><br />Pictures might help if you didn't spend any time doing step 1. We usually just skip to the hard parts, but you need to read Step 4 below to see why.<br /><br />Note that this is a 2 or 3 step problem. These types of problems reward a child who works slowly and a parent who doesn't expect correct answers. If the child is expected to do a lot of problems, expected to get them correct, and expect to do them quickly, the child will fail at multi-step problems. Because of this, I have settled on one or 'a couple' of problems as our daily routine until the child builds speed.<br /><br />If this child was 10 years old, I would expect the child to devise and explain a formula for adding fractions. Before this age I never even hint that there is such a thing as a formula. I want the child to go through the 5 or 6 substeps every time, using working memory, because amazing and surprising subskills will develop in that child's brain that will pay off in a big way later on.<br /><br />For an 8 or 9 year old, I would want to see a picture and an explanation of what is happening. I would also try out 1/2 + 1/3, 1/3 + 1/4 etc from the list I explained above. But I would do this every time he was stuck on something like 5/11 + 2/3, because this age desperately needs intuition number sense and now's the time to develop it. This is really going to slow down the topic, but if you do it right, you'll save many years later on not having to explain math topics.<br /><br />Multiplication and division require starting all over again with this article, both parent and child section, with each operation. What does it mean to divide 3 by 1/2 or 7 by 1/2? What does it mean to divide 1 by 1/3? How about dividing 4/5 by 2/3? The same basic questions are asked about multiplying fractions. What is 1/3 times 3/4? Before algebra in about 6th or 7th grade, I would want this child to think through the meaning of these problems every time instead of just turning 4/5 x 2/3 into (4 x 2)/(5 x 3), because if the child skips thinking through these each time, they will get to algebra ready to calculate but unable to understand. This approach precludes some problems and precludes lots of practice. This approach involves a few problems over a much longer period.<br /><br />Diagrams work really well in understanding multiplication and division. These will be articles on their own so I'm not going to cover it here. Have you ever read a history book that starts with the beginning of time, evolution, 40,000 years ago etc until it gets to the main topic, which might be 1972? That's how I handle these topics.<br /><br /><b>How Bad Can It Be?</b><br />The biggest challenge with teaching your child math is coming to terms with how stupid your child is. You're doing something that you just did the day before, and your child not only forgot what he learned the day before, he can't even add. He does a single problem in 30 minutes and it's totally wrong. There are 29 problems on the page that are not completed. It's a disaster.<br /><br />This is the make or break moment in your child's academic career. You have the choice between a future surgeon with join doctorate degrees in Sumerian literature and Bioengineering, or a kid who drops out of community college to form a rock band. The choice is yours.<br /><br />I'm usually pretty pleased and announce that will pick up problem #2 the next day. I can do this because in the futile mess I see cognitive skills developing. Within a few months, my child is making adequate progress and I'm looking for books on Sumerian literature on Amazon.<br /><br />Sometimes I am discouraged and ask how he could possibly screw up such a simple problem. After I say something like this, he will spend the next few weeks perfecting a base guitar riff.<br /><br /><b>How Good It Can Be</b><br />Once you've taken on a few topics like this once, each successive topic is easier and more fun. The key is that 6 to 8 months of hard work pays off, and you can see that doing a single problem for 2 weeks and getting nowhere is normal and leads to ripping through pages down the road and eating math for lunch. For a parent, it requires nothing short of faith to get through the first few weeks.<br /><br />For those of you who took my advice to do EDM Grade 2 in Kindergarten, you already know this. For those of you who do TPM, which is not all that mathy but is really thinky, you're ready to start. Unfortunately, in both cases, nothing ever gets easier and you still have to go through the whole painful learning curve with new maths. But doing Algebra II with a 9 year old and going through a painful learning curve is much more gratifying than doing decimals and going through no learning curve.<br /><br />Last week, my child was struggling on a problem from his Algebra I final exam. We stopped using math books altogether and just take tests, figuring things out on the spot. Sometimes, we'll take a break and do some worksheets on a new topic. Anyway, there were 4 maths involved in this topic, and he didn't know 3. He didn't even know the formula for the area of a circle. It took us over an hour to do a single problem, what with all the backtracking.<br /><br />Then I realized I accidentally grabbed the Algebra II final. When we went back to the Algebra I final, he had 6 questions of the form "What is 42% of 66?" and <i>didn't know how to do them</i>. Arrrgggghhhh!<br /><br />In each case, we took apart, figured out, and mastered new topics on the spot. This is the skill set that I want. This is the skill set behind the MAP test, for very important reasons. If you can get this skill set down early on, say fractions, then it's just a matter of plowing through pre-algebra, functions, algebra, geometry, trigonometry, calculus (AB and BC), linear algebra, real analysis and series, and then statistics. <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com8tag:blogger.com,1999:blog-5703568807615263851.post-77000328344178175832017-12-26T08:45:00.002-08:002017-12-26T08:45:33.767-08:00Post Holiday MathWe are in day 2 of a 2 week holiday break. Day 1 was a holiday and I have a hard time convincing anyone to do any math. My kids sat around all day having fun, eating, chatting and helping with chores.<br /><br />Math starts today.<br /><br />Daily math is a prerequisite of the kids doing anything fun. The kids say, "I don't want to do anything fun and I'm not doing any math!" Then they read, do crafts, engage in an imagination-building-problem solving activity like Legos in order to not do any math. It's quite amusing to me when I walk by their room, and they are sitting there reading for hours, and they look at me like 'Ha, ha, I don't have to do your stupid math, I'm just going to read. I win.'<br /><br />Reading is way more important than math. The jokes on them and I'm not telling.<br /><br />Daily math started with the simple thought, "If a child becomes a strong reader and thinker because he reads daily, how is he going to become at STEM?" The answer was daily math. Around third grade, I thought "There is way to much homework to do each night. We'll just do daily math on the weekends" and that's where we've been ever since except summer and breaks.<br /><br />Math contains more than math, of courses. It contains anything I think they need to succeed at the time. This usually contains math. On Saturdays in the summer, this can be math, vacuum the basement, practice your instrument and do a reading comp question, fix the toilet, replace light bulbs. <br /><br />This year, the April MAP test is on our radar and I'm becoming slightly more organized with daily math. We overdid vocabulary between SSCC and 2nd grade and haven't done much in this area other than define and discuss any unknown word found in reading or reading comp. I am reintroducing vocabulary as part of math. With a vengeance.<br /><br />I like the MAP. It has a lot in common with the COGAT. The cognitive skill set is slightly different, but in both cases there is an advantage that can be gained from working on these skills simply because school doesn't really teach cognitive skills. Doing lots of practice, ala Kumon doesn't help at all, and learning algorithms ala Singapore sets up a train wreck (like ending up in the 90th percentile or less - I never really defined what a train wreck is but that's it). The problem with any program at all is that the child can get ahead and doing well, and the parent thinks that success has been obtained. The clock is ticking. Any time a child is practicing or applying or using things taught, learning may or may not happen, but skills building is not part of the deal.<br /><br />I remember when my goal was simply to cheat my kids into a GAT program. What actually happened was that we just ended up spending a lot of quality time together and I learned how to be a parent. The long term formula for academic success is Cognitive Skills + Interest + Will. At this age, and in the succeeding years while we caught up, it was all Cognitive Skills at the expense of Interest and especially Will. You can burn a kid out with daily math every day every year, so I tried (and failed) to take some years off. To compensate, I completely changed the approach to my formula of Baffled + Spending Time on the Question not the Solution + Get it Wrong + Check the Work. This created an environment of Zero Expectations and No Progress, and in that environment magic happened.<br /><br />Somewhere along the way, 'Will' came back, most likely because of chores or instrument practice, and I'm doing my best to stay as far away from 'Interest' as I can so as not to ruin it. A child can only develop interest in a vacuum that does not include the parent. Unless the parent is super sneaky.<br /><br />I'm thinking about 'Interest+Will+Skills' a lot because for the older child, my goal is that he does really well in AP Language Arts and/or History, with assumed A's in math of some kind. All of the math education is pointing in that direction for this child. I found that at one of the selective enrollment high schools in Chicago, a child can take Calculus as a freshman, followed by Linear Algebra/Multivariate Calculus, a course that's no longer on their website which I will demand be reinstated, and AP Statistics, and assumed A's in AP Language. This is 4 years of college credit math. We're going for it.<br /><br />The only way I can possibly think of achieving these goals is to do something creative, unusual, and different. Something that is more looking at things from a fresh perspective than hard work. Hard work is not going to do it.<br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com3tag:blogger.com,1999:blog-5703568807615263851.post-9369155278222733262017-12-23T11:55:00.000-08:002017-12-25T04:32:58.648-08:00The Makings of a ThinkerHere's a rough non-copyright violating approximation of a figure matrix question from my favorite COGAT practice test, grade 2. <br /><br />This is the last question in the book and the hardest.<br /><br /><div style="text-align: center;"><a href="http://3.bp.blogspot.com/-2t4B0QJsm9c/Wj6l3aWZ4CI/AAAAAAAAAmI/y3ih_HrqDTEw__f2_Z4twVk8PQCfU_A6wCK4BGAYYCw/s1600/testq.jpg" imageanchor="1"><img border="0" height="222" src="https://3.bp.blogspot.com/-2t4B0QJsm9c/Wj6l3aWZ4CI/AAAAAAAAAmI/y3ih_HrqDTEw__f2_Z4twVk8PQCfU_A6wCK4BGAYYCw/s400/testq.jpg" width="400" /></a></div><div style="text-align: left;">In this article, I'm going to show you how much mileage you can get from a single question.<br /><br />When I coach, usually at the behest of a parent who provides a compelling reason or academic puzzle that I want to add to my research, I'll start with whatever material they have available and do a single question. There are many other things I do with a practice test besides a single problem, but my favorite Academic Coaching Session Agenda is the Single Problem because the student picks up the most skills.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">This may be the only time I'm working with the child, and my primary goal is to train the parent who is lurking nearby, and I want an impact, so I do it exactly like I would with my own children. Like this:</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Step 1:</b> I instruct the child to do the problem. Take as long as you like, and before you answer the question, I want you to tell me that you're ready to answer the question but not what the answer actually is. I will probably announce that this is a really hard answer and I'm totally confused so I hope that the student can do it because I sure can't.</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Step 2:</b> The child either announces the answer or announces that they are ready to answer. a) If they announce the answer and it's correct, I'll tell them I think it is the 2nd one and be prepared to prove your answer* b) if they announce the answer and it is not correct, my favorite case, I announce that they are wrong - try again and c) if they just announce that they have completed the question and are ready to answer, I announce that they probably got it wrong so go back and double, triple, and quadruple check the answer, followed by a) or b) when they announce the answer.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">*At some point during this training, the child will learn to check their answer. I am going to encourage this behavior in multiple ways including saying 'Check your answer'.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">This approach is the birth of skills. If the child answered incorrectly, then we're going to get double the skills from this exercise. It's not clear to most parents what these skills are. These skills are the skills of kids who will go into an accelerated history or reading course, teach themselves, and do well. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">When I announce to the child that they are wrong, they are probably wrong, or their answer does not agree with mine, the child can sense that I'm happy about this situation, and I genuinely am happy because we can learn something. I love mistakes, even the ones I make. Mistakes drive learning and it's one of the 5 core skills.</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Step 3:</b> Explain the question to me. First of all, I want to know what the transformation is. The first shape undergoes 3 transformations. Zooming through problems is the way to miss subtleties like the height of the shape diminishing by about 10% before it is rotated 1/4 turn counter clock wise. Some kids say rotated 'to the left' which is OK with me provided 'to the right' always means clockwise.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">In this phase, we're learning how to see, the names of things (like rotate 1/4 turn counter clockwise or decrease in height slightly). I will correct the child's grammar or terminology, expecting that they eventually use the adult level words that I do in adult level sentences with multi-clauses. It's the opposite of Baby talk and the reason why my books have that awful looking graduate text book themed covers.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">When the child thinks they are done, I'll point out that explaining the question includes explaining what is happening in each and every answer. I would like to know what transformation took place to make each of the answer choices, or what transformation failed to take place. That's 4 additional problems as far as I'm concerned. </div><div style="text-align: left;"><br /></div><div style="text-align: left;"><i>I've never found a problem in a COGAT book that can't be solved with a thorough out loud explanation.</i> Sometimes when I'm working with my own material, I get the problem wrong, repeatedly, and I look at my answer and wonder what the heck I was thinking. Then I go through it the way using the steps I expect a student to use, and oh year, it makes sense again. When you say the transformations out loud (problem and answer choices) hard problems are turned into easy problems. I can't over stress the importance of this technique. This is why Shape Size Color Count is so verbal<br /><br />I call this skill 'Reading The Question' because most kids can't do it without a lot of training, and most parents lack the patience to wait. I know as a parent I used to lack the patience, and sometimes I still do. To accommodate my coaching inadequacies, I'll just turn over the material and go clean for 20 minutes before the teamwork begins, shouting out things like 'Read the question again' while I do my work. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">There is a prerequisite skill I call 'Seeing' that children have to develop. In this case, 'seeing' is visual and includes proportions and the ability to mentally rotate images. It takes some practice. In an academic household, those places of non-stop learning that produce GAT standouts, this practice started at age about 2. For the rest of us, COGAT practice is as good a time as any.</div><div style="text-align: left;"><br />I should point out that this is not a hard problem because it's missing the magic of the COGAT. The quadrilateral lacks symmetry. A problem like this would be practice for K. This is why practice tests are practice for the format of the test and not the thinking of the test. Also, there are 3 transformations, which you'd think would be good for working memory, but the shading transformation removes answer choices right away, making the problem easier, not harder.<br /><br /></div><div style="text-align: left;"><b>Step 4:</b> If the child can't get to the correct solution on their own, I'll mark the page and come back later. This question is still holding learning. If I have to announce that the shape is shrinking in height before turning, I just destroyed the learning opportunity. If there are 10 more questions with this transformation, I'm stuck having to announce it. It's a judgement call and depends on how much time is remaining before the big event. If you have a lot of time, you can back track by drawing 10 or 12 shapes, and ask your child to shrink one dimension and turn it 1/4 turn in one direction. Backtracking in this way is a version of finding an easier problem to solve before tackling the harder problem to solve. No branch of mathematics can withstand this approach, and every single super hard complicated advanced problem can be solved in this way if needed.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">For one child, we spent a solid 4 months doing cognitive skills training (including BTS and much much harder material of my own making). When we finally came back to math, we followed this approach from that point forward through SAT and calculus. I learned that these core skills are universally applicable. This is probably why the COGAT is such a great predictor of academic success. Take any topic, like fractions or exponents or roots of a 2nd degree polynomial, or multiplication or anything, and at one point we slowly went through a few problems using this approach and learned months worth of material with a small amount of effort.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">At some point during the actual test, the child will come to the questions that differentiate the 97th percentile from the 99th percentile. These are the questions that differentiate those kids who probably would do well at Stanford with a little effort from those kids who will be sitting in a GAT program next year because of the ridiculously high cutoffs in almost all states. The kid who gets these question correct will either be the child who is already 99% because his parents both have PhD's from those who have learned the skill set and go super slow on these problems:</div><div style="text-align: left;"></div><ul><li>They are not the slightest bit discouraged by not knowing the answer right away or being confused.</li><li>They take a long time to thoroughly investigate the problem</li><li>They have a few techniques to fall back on when it gets really, really hard. </li><li>They are not discouraged when they don't see their expected answer in the pick list. They try again as a matter of course.</li><li>They check their answer, and all the answers, at least twice if not more.</li></ul><div>I think the best way to teach these skills is to approach the training in the way I described above. You should see how the approach is consistent with these skills. It should also be clear that the other approach, I call this the school approach - explanations and lots of routine practice in the hopes of memorizing or mastering a set of question techniques - is not consistent with the skills needed at the top.</div><div><br /></div><div>For parents a week or two from the COGAT who reach out to me the first time for help, and have done zero of anything before that, this approach is the way to go. Of course, if you plan ahead, you'll be able to go much, much farther, but the approach is roughly the same.</div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com1tag:blogger.com,1999:blog-5703568807615263851.post-32897442340329005342017-12-16T09:01:00.002-08:002017-12-18T07:03:40.024-08:00Problem 123Testing season is in full swing in Chicago right now with the majority of test takers in K grade, followed by 1st grade.<br /><br />While sitting in the testing center, you may notice a members of a tiny but super intelligent articulate species talking to their adoptive parents about the composition of the earth's core. Then on the drive home, your child may sit in the back seat telling you in explicit detail about each problem he missed. These are both good reasons to buy a math book that your child won't see for 2 years and make him do it. It made me feel better.<br /><br />In this article, I'm going to demonstrate how to help your child work through material two years in advance. Problem 123 is short for the last problem in EDM Grade 2 book on page 123, and the context is going to be a 5/6 year old in Kindergarten who made it to page 123 despite not completing K math and having skipped 1st grade math. You can apply this context to other grades and other material (like a 2nd grader doing fractions), but if your child has been going to an after school math program for the last 2 years this is not going to produce experience for the child nor the same set of cognitive skills and you'll have to find a different challenge to achieve the same results.<br /><br />I owe a reader a discussion of fractions, and I'll use this article to warm up.<br /><br />Let's begin with my favorite email from parents and my common response. Here is a brief summary of the email: "This isn't working and I don't know what the heck I'm doing. I don't know how to teach math. What should I do?"<br /><br />Here is my response:<br /><ul><li>You are not teaching math. Focus on teaching the core learning skills and the child will teach herself math in the case you are blessed beyond belief with daughters, or himself math if you're like me and stuck with a bunch of boys. </li><li>The 1st few pages in the book took us about 3 weeks. Any page could take a week. Acceleration happens later in the process.</li><li>Our error rate was about 50% on a good day.</li><li>After about 30 minutes on this exact problem, I just gave up and made a note to come back to this topic at some point in the future (which was next week). I'm going to do it fully below because it shows you how to teach math to yourself which will make you a better math coach in the future. </li></ul>At this age, we're going to focus on the most important skill of Being Baffled, which is comprised of numerous subskills. Then I'll talk about the 'Reading the Question' subset which you will focus on through 4th grade. The other core skills like Getting the Problem Wrong (aka Making Mistakes) and checking your work are not discussed. <br /><br /><div style="margin: 5px; vertical-align: top;"><div style="align: left; float: left; width: 50%;"><b>Page 123, Lesson 5-6, #3:</b><br />Connect the points in order from 1 to 3.<br /><br />Find and name 3 triangles<br />__________________________<br />Try to name a fourth triangle<br />________________________<br />Color a four sided figure.<br /><br /></div><div style="float: right; width: 50%;"><a href="https://2.bp.blogspot.com/-fhG43yU5wjc/WjVCv3QMNYI/AAAAAAAAAk4/3xCF4i4noDQZ-VEyKnXqKCcDjhsPX2FVACLcBGAs/s1600/Drawing1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="641" data-original-width="1328" height="154" src="https://2.bp.blogspot.com/-fhG43yU5wjc/WjVCv3QMNYI/AAAAAAAAAk4/3xCF4i4noDQZ-VEyKnXqKCcDjhsPX2FVACLcBGAs/s320/Drawing1.jpg" width="320" /></a></div></div><div style="clear: both;"></div><br /><b>Step 1: Be Baffled</b><br />Say 'This is a hard problem' then leave your child alone for a minimum of 15 minutes to do the problem. I started this approach on page 1. Somewhere between page 1 and page 123, 15 minutes of doodling, yelling, and complaining became 10 minutes of thinking and trying and 5 minutes of doodling, yelling and complaining.<br /><br /><b>Step 2: Backtrack</b><br />The first challenge is that section 5-3 discusses the naming of line segments, like <span style="border-top: 2px solid black;">AB</span>, problems 1 and 2 in this lesson connect shapes with lettered dots, but it's left to the child to make the leap to naming triangles. A Kindergarten kid is not only not going to make the leap, but by this point they never mastered (or even got) the whole line segment naming business.<br /><br />Over the years, I've come to appreciate that 'Being Baffled' is a mandatory problem solving step, because it sets up the rest of the process, especially in BC Calculus. Being baffled relaxes everyone (especially the parent) and opens the brain to thinking. The opposite of 'Being Baffled' is frustration, impatience, and a subpar performance.<br /><br />Fortunately, the example at the top of this page (not shown) has the same triangle without the numbered points, so we need to backtrack a bit. Ask the child to name the line segments in the example triangle. We should get AB, AC and BC. Then ask the child to come up with a way to name the triangle.<br /><br />I'm rarely severe on vocabulary. At some point, I might just say that a triangle is named just like a line segment. A line segment is AB, but a triangle is ABC. What is the difference between BCA and ABC? Does this triangle have any other names? If the child is 8 years old and a boy, I would be disappointed if the child didn't say 'Bob'.<br /><br />If this were a problem like 72 - 49 = ?, backtracking might be a 1st grade workbook for a day or two.<br /><br /><b>Step 3: Dig into the question.</b><br />What is a triangle? Ask you kid to define it. It's a shape with 3 sides. How do you make a triangle? You put three sides together. Show your child 3 lines that don't touch and announce you created a triangle. Each side has to touch 2 other sides at its end point. I'm meandering through the question starting with the Stone Age and working my way back to 2017.<br /><br />There is a whole set of skills that formulates the skill of 'Seeing'. Some kids can do it, other kids have a lot of work to do. In this particular problem, there are 4 triangles. Two are obvious, one is not obvious, and one is hidden. This problem will show up on most competitive math tests in one form or another. Seeing is a big part of math and reading and science and innovation and internet startups. It's also one of the main skills of the COGAT.<br /><br />Ask the child to find all of the line segments in this picture. I see A1, 13, 3B for example. Then how many ways can you take 3 line segments that each touch 2 others at the end? We gave up after 3 named triangles.<br /><br /><b>Step 4: Give Up</b><br />You will give up on something. You are not working with a 2nd grade child, but a 5 or 6 year old. At some point, it's time to move on, and you have not achieved mastery over some math topic. Fortunately, EDM has some repetition so you'll see some topics again, just not this one. Fortunately, your child is going to get this material again in school, and they'll look like the smartest person on the planet when they see it again and figure it out quickly.<br /><br />After doing this for 8 or 9 months, children should be completing the work with reasonable accuracy in a reasonable amount of time, but I need to stress this child will never complete the work like an 8 or 9 year old would. My my goal of 'reasonableness' was met, and we stopped at about the 1/2 way point of book 2. That was good for 99% on the MAP for a while.<br /><br />Think carefully about what I did. I got a child to sit and work alone for 10 to 15 minutes on material he wasn't taught and didn't know before I would jump in and start helping. As the months go by, he gets less and less help, just more questions. I taught him (because math is a team sport and I was the missing team member as needed) to be baffled, to spend a lot of time on the question and to backtrack as needed, to make mistakes and be totally OK with that, to try over and over again and to check his work because he got most things wrong on the first try (not demonstrated above).<br /><br />With that skill set, and continued refinements over the next few years, it is reasonable of me to expect that he gets 99% on both sections of the MAP from this point forward, can handle accelerated work in all subjects with little or no help, can teach himself instruments and other things of interest to him, and go to Stanford for graduate school.<br /><br />On the other hand, what if I <u>trained</u> and <u>drilled</u> him on math topics during this period? What would I expect from a child who spent 4 years zipping through math because he was expertly taught and trained on math concepts? This is what school does really poorly and what after school math programs do really well. But it's not the skill set I want. You wouldn't notice a difference between either approach if you just looked at math and you just looked at a 2nd or 3rd grade performance on a math test of some kind. The difference will show up elsewhere and it will show up later.<br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-31454467610756704862017-12-08T16:18:00.001-08:002017-12-08T16:18:37.563-08:00Fractions One<b style="font-weight: normal;"><div dir="ltr" id="docs-internal-guid-e44e0e40-389f-81bc-4e85-d71f912b9bf0" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">There are a lot of good math curriculums that teach the mechanics of fractions. I’ve seen step by step diagrams to add fractions with different denominators and add mixed fractions. With a thorough explanation and lots of practice, a young child can do fractions without any increase in academic skills or knowledge of math whatsoever.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">So we’re not going to learn fractions this way.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">The MAP test distinguishes kids who are ahead in math from the rest in the early grades. In later grades, it distinguishes kids who can figure out new math on their own. That’s what we want. </span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">The starting point for fractions is for the child to tell me what they know about fractions. Some kids have not learned to articulate math, so we can work on this gap, It is most likely going to take some time for their brain to digest fractions on its own WITH NO HELP so I’m willing to wait. Plus, i need to find out where they are. Plus, they need to figure out what they already know because they are going to have to use it.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Start with ½, ⅓,¼ etc. What are these? Order them biggest to smallest? Can you draw it? If we put 2 in the numerator position, what do we get?</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">If you wanted me to teach fractions to your 7 year old who has never seen fractions before, we wouldn’t do more than 1 or 2 problems a day. Each problem is on par with a really good science experiment that spurs the imagination. Doing a bunch of problems is pointless to the learning process. Once the imagination is engaged, we’re learning, and during the thinking process WITH NO HELP learning skills are being generated that I’ll need in 3 years when I plunk down an SAT book.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">What is the difference between ⅖ and 2 divided by 5. I want to know. Let’s do it. Suppose we divide 2 by 6 and then by 7. What’s going on? I want to know. Tell me, or we can figure it out together.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">By the way, mathematicians never use the “divided by” sign. We always use ⅘ and say ‘4 divided by 5’ when we mean divide by or four fifths when we mean fractions, because these are the same and the divided by sign is lame.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Over the next few days or once a week, we’ll continue forward or repeat this conversation while it sinks in. If this kid is learning fractions now, then we’ll be decomposing 2nd degree polynomials soon and I won’t be in the mood to help. That’s why I won’t assign a fractions worksheet. Instead, I’ll ask them to decompose every number 1 through 100 and circle the prime numbers. When they need this, they won’t know it so I’ll have to tell them, but they are just kids.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">From experience, the most important thing the kid needs to know is the answer to this question: If i add 3 pieces of cloth to 2 T-shirts, how many T-shirts do I have now? (10 minutes later) It’s the same with fractions. Either you make a T-shirt out of the 3 pieces of cloth and add it to the 2 T-shirts to get 3 T-shirts, or you rip each T-Shirt in half and add it to the 3 pieces of cloth to get 7 pieces of cloth. But you can’t add T-shirts and pieces of cloth without doing something.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Then I would take a single question of each type and we’ll do it together and look at it. By ‘together’ I mean I’m not going to help at all. Maybe I’ll give hints. Once they get it, we can do a harder version of that question type later. Or we try a different one. Ore we draw pictures, try an easier version, split it into 2 problems, or sometimes just iterate through all integers with that version of the question, starting with 1/1 and ½, ⅔, etc until patterns emerge. Or turn it into a word problem that is relevant to their world. Or all of the above.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Can you imagine what a little child who wants to be a piano expert does to become better? They practice the same piece over and over and over again. They drill and drill and scales and scales over and over.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Math is not like the piano at all. Math is learning to think, to analyze, to find patterns, to impute and make logical deductions, inferences, leaps. To put 2 unrelated things together. Drilling teaches none of this. Doing a single hard problem for 15 or 30 minutes while the parent is silent or asks questions is the prerequisite of thinking.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">If I were starting from scratch with your child, I’m guessing this might take 1 to 2 months, maybe more to get to the really hard fraction problems. It would require very little effort on either of our parts. Just a lot of staring, questions, and thinking.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Where did I get the ability to teach fractions? We were doing fractions for the first time and I had 25 minutes of silence to stare at the problem while the work was in progress. I asked ‘what are fractions anyway’ and started to look at them anew. </span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">At some point, you might want to assign a workbook page or the whole thing to get the ball rolling. When and how is your preference. I would never assign a fractions worksheet ever because a 7 or 8 year old doesn’t need fractions, and they will get smart enough by doing fractions to determine that math is useless, boring, and lame. This is my personal opinion. What I do instead is assign material that has lots of problem types, including fractions, and I assign that. It’s more sneaky. I just download tests of all kids and we do the problems that are appropriate. On these tests, either the problem is within reach, we skip it, or I’ll do it because they won’t see it again for a year.</span></div><br /><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">The big issue to keep in mind is what your child did in the last few years. By the time we got to fractions, we had already been through this type of experience a few times and had done material that was less math topic and more hard core thinking. If you have less practice with this, then fractions will be your boot camp.</span></div><br /></b><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-3083065750599007432017-12-05T04:15:00.002-08:002017-12-05T04:15:50.839-08:00Advanced Math and Little KidsI have about a dozen questions from readers that have been swirling in my brain, all on the topic of casual work-ahead At Home Schooling in math. I've been trolling parent forums and reading amazon reviews while a new round of 1st through 3rd curriculum shows up from my latest buying spree.<br /><br />Let's take the first question first. <i>How do I teach my child fractions</i>?<br /><br />Here is my step-by-step*:<br /><br /><ol><li>You do a complete inventory of all of your child's skills and your skills as a parent that are required for your child to teach herself fractions.</li><li>You fix the ones you can fix immediately and work on the rest at the appropriate pace and the appropriate material. You can work on fractions if you want while you do this.</li><li>Your child teaches herself fractions. You help by reinforcing the 5 core skills which you can see while your child struggles with the material on her own, with no help learning the actual math.</li></ol><div>#1 is the problem, of course. It's also the problem with parent forums and helpful parent advice. It is also a problem with teachers, even good ones, but not the really great ones who have taught for 20 years. #2 is easy once you get it, and looks impossible before you actually see it work, then it's total magic. #3 is our goal.</div><div><br /></div><div>*I will present a more detailed step-by-step but we've got a lot of ground to cover first.</div><div><br /></div><div>Back to parent forums and book reviews. Parents are blind to the cognitive skill set of their child and where this fits relative to other children, not to mention their own skills as an At Home academic coach. They find something that works and then state with no further thought that it should work for other parents. Maybe, maybe not. If the parent mentions either a) my child reads 6 hours a day or b) my child got 99% on both the COGAT and the MAP or c) my child got 99% on the MAP but didn't do so well on the COGAT then I have a pretty good idea where this child is on the skill spectrum. a, b, and c are three totally different places, but I've spent enough time investigating so many children in these three cases that I can just prescribe the medicine. The rest of the world needs more analysis. </div><div><br /></div><div>Wouldn't it be great if you could follow really 100's of successful parents around for 10 years and take notes and build a program based on what they did to put their kids at the top of the heap? That's exactly what I did, and not just in math.</div><div><br /></div><div>Recently I've been getting questions related to a certain famous math curriculum. I haven't seen this material in 5 years since I reviewed it and then gave the books to a tiny little test case and followed up every week. It wasn't right for my children, but I found a little girl who I thought would benefit for her specific case and she did.</div><div><br /></div><div>The books are arriving and I'm really disappointed. It's not about the core skills at all. It's about explicitly showing the child how to do mathematical operations. It skips learning. Even worse, the questions tend to be the one-shot deal, as in one sentence that is pretty clear that the 2 numbers have to be added. The inevitable result is a child who is told how to do math, never develops the skill set for #3, does pretty well on tests, and then has to be taught fractions.</div><div><br /></div><div>In the last few months, I've gotten to personally know the Amazon drivers in Chicago because they show up at my house so much delivering material. The last time I did this I was so disgusted that I wrote Test Prep Math. Not much has changed. I've also pulled down at least a dozen curriculums (sic) from the web and gotten to know their creators from doing a little research. I've come to the conclusion that the Test Prep Math series is the best material math material anywhere.</div><div><br /></div><div>This is hard to say. Authors have warned me that once you publish, you face a life of insecurity from that point on. They were right. I've freaked out when one mother told me that her child who's at the 99% found the books easy. OK, I can deal with that. The book is designed<i> to get the child to 99%</i>. Just skip ahead until it gets hard. There is a review on Test Prep Math 2 where the reviewer slams me because the book is confusing and the answers are wrong. As explicitly stated in the introduction, it is supposed to be confusing, and even I get the answer wrong when I speed through it and forget that it was designed for multiple readings on purpose, for you to see you skipped something or blindly assumed the wrong thing. Those are core skills #1 (dealing with confusion) and core skills #2 (spend more time with the question - a lot more time - like 3 weeks if that's what it takes for the skills to emerge for the first time). The book was returned and I feel personally responsible that the reviewer's child is going to eventually fall short in school. </div><div><br /></div><div>I've gotten a lot of emails and a few comments from readers who state 1) my child finished TPM Level 2 and is finishing TPM Math Level 3 and 2) what do I do next? When I get this type of email, the questioner probably has no idea that they have a friend for life. I'm planning to put TPM Level 4 on a free website, mostly because it's going to take me a lot of time to piecemeal the material out there and my new friend for life won't have time to wait, and I'm still weeks away from TMP Level 1 and it's taking up time.</div><div><br /></div><div>By the way, in my ongoing effort to make kids so ridiculously smart that they blow away the COGAT, which was my original goal before I decided a math chair at MIT was also a good idea, I've finally perfected my ability to deliver figure problems to 6 year olds that are 3 times harder than anything they'll ever see again. It's much easier with older children to take away the net. Never underestimate the importance of the COGAT. It measures skills that kids need to teach themselves fractions. It doesn't care if they can actually do fractions or any other type of math. The COGAT wants kids who already know how to learn and can go from Kindergarten to fractions in one year, which is what happens when you enter certain gifted and talented programs.</div><div><br /></div><div>Test Prep Math 4 launches the math career. It's all about math. The skills continue to refine and develop, and the fifth core skill (problem solving skills) becomes wider and deeper on it's march to passing the AP exam in BC Calculus. When you child chooses a joint major in English and Music instead of a STEM career, those problem solving skills explode yet again and you discover why so many CEO's and law firm partners have English or music backgrounds, but you wanted a doctor so we blew it.</div><div><br /></div><div>Here are the Test Prep Math Level 4 milestones. By 6th grade, your child will have finished all of the practice math tests in at least one SAT book. You will have administered at least a rigorous Algebra 1 final where they will encounter some pre-algebra and many algebra topics for the first time. They will have been introduced to important concepts in high school geometry, Algebra 2, trigonometry and calculus and you're holding off on the ones that require maturity to grasp. If you've ever seen TPM, you won't be surprised to find out that TPM 4 includes the reading comp portion of the SAT as well, but you have to go a bit slower because of all that unfamiliar vocabulary. If you were fortunate enough to do Pre-K Phonics Conceptual Vocabulary and Thinking, and followed the directions with regard to the Word Board, the SAT vocabulary goes pretty quickly. Some day, when my youngest completes his 7th and 8th grade high school enrollment nightmare, I'm going to spell out in detail why we're doing this. Until then, just go with the flow.</div><div><br /></div><div>We're not even going to look at the SAT until the summer after 4th grade and really get into it a year later. Before then, we've got a lot of ground to cover, and it includes fractions.</div><div><br /></div><div>I'm going to need 2 articles to do it, and they'll probably be long. The first article is going to lay down the ground rules that apply to math starting in Kindergarten and that you will use thereafter if you want your child to learn. </div><div><br /></div><div><br /></div><div><br /></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com2tag:blogger.com,1999:blog-5703568807615263851.post-42873330557808675252017-12-01T08:23:00.000-08:002017-12-01T08:23:48.789-08:00Pick the Right At-Home Math CurriculumI spent the last few days thinking about the comment I received from Anonymous asking about current+2 curriculum for a 2nd grade child. The last two articles on this topic were experimental and not helpful, and I'll delete them some day.<br /><br />Taking a step back, here is a better version of the question: "What is the best way for a 2nd grader to work through 4th grade math so that she (or he) obtains all of the grit related benefits from doing so, learns more math, subject to the following constraints":<br /><ol><li>You've only got so much time to help and you're not a teacher</li><li>You need a high MAP score and teacher recommendation for the GAT program</li><li>You may or may not have to pass the COGAT this year</li><li>This child is only 7. And not necessarily good in math. </li><li>You can't afford a tutor or an after school math program. Plus you hate driving.</li><li>If your child does get into a GAT program, you want them to be the best.</li><li>If you run into problems, you're going to send 19 emails a day to getyourchildintogat@gmail.com so this advice better be good.</li><li>Math curriculum from US publishers stinks</li></ol><div>I updated the article <a href="http://www.getyourchildintogat.com/p/chapter-6-math.html" target="_blank">How To Create A Math Genius</a> to be more clear about this situation. You might want to refer to the content starting at first grade. In this article, I'm going explain why my curriculum choices are counter intuitive and logically valid. </div><div><br /></div><div>My top 2 choices for curriculum are Go Math from Houghton Mifflin and Eureka Math. A few years ago, a teacher suggested I review Eureka Math for 4th grade and I had a pdf of the whole book but I can't find it. It's totally spoon feeding math, not only in the book but in the problems. Go Math has a more intuitive approach, which means more concepts and less actual math. For a kid who's already been through the advanced math exercise, he can do the Go Math homework for current+1 on the bus while playing Minecraft and discussing Star Wars memes. And get them all correct.<br /><br />If I was more worried about the MAP, I'd go with Eureka. If I was more worried about the COGAT I'd go with Go Math. I would probably pick Go Math anyway.<br /><br />The target of Eureka and Go Math, and the rest of US curriculum, are the 50% of below average kids in the US with parents who know nothing about math and don't care. This is perfect for a 2nd grader attempting 4th grade work, because the 2nd grade is starting way, way below average and her parent has zero experience teaching 4th grade math to a 2nd grader. Really great 4th math curriculum is designed for bright, talented, engaged 4th graders with a parent who knows something about 4th graders, or at least has had 9 months of experience with a 3rd grader. <br /><br />If your 2nd grade child works through 4th grade math, and you follow the rules, #1, #2, #4 and #5 are taken care of. #8 makes this possible. For #3, you need more material beyond advanced math. The COGAT is looking for kids with generalized problem solving skills who will be <i>strong academically in the future</i>, not kids who are ahead now. But if you want your advanced math to impact the COGAT score, start with 100 (average) and add 1 point for every leading question you ask, add 5 points every time your child makes a mistake and you just shrug your shoulders because you don't care, and subtract 1,000 points every time you tell your child how to do something. This will be an indication of their final score on the COGAT.<br /><br />#6 will happen on it's own. Most GAT programs only go 1 year ahead on math so your child would see the exact same math for a second time.<br /><br />I'll take care of #7 right now. "My son/daughter has been working on one of these books for 3 weeks and gets them all wrong and has only done 2 pages." This is exactly what I expect. This is the path to gifted. The secret is just to keep going even though it doesn't make sense. This is so counter intuitive that only about 10% of parents are willing to try it, and only 1% of parents are willing to follow the guidelines of an encouraging learning environment at home under these conditions. That's why only 1% of children make it into the top 1%.</div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com2tag:blogger.com,1999:blog-5703568807615263851.post-55102910129919907722017-11-30T04:36:00.000-08:002017-11-30T04:36:08.153-08:00Teaching Algebra To A Fourth GraderI'm really excited about yesterday's algebra post and the worked the followed last night proving I was totally wrong. I'd delete the post entirely if it wasn't chock full of good advice that sets the stage for today's topic, which is teaching Algebra to a 4th grader. Which is going super well, thank you, and after I complain that this is insane and ruining our children, I'll show you how.<br /><br />There is a branch of math curriculum that is not very mathy. It's fuzzy and intuitive and wholelistic and verbal and it doesn't really care if the kid currently or will ever know anything about math. I love this type of math. It's used in our school, chosen by a group of teachers who have about 9,282 years of teaching experience between the 4 of them. One is named Yoda, just so you know.<br /><br />What I've found is math that works for average or below average children in the US will work for your slightly above average child much earlier. I think the best math curriculum of all time, even surpassing Sylvan's Kindergarten math book is the middle school math Jo Boaler created. It involves no math at all, and then she just walks kids into 'WHAM' real math. Her problems show up in CMP math, which is what our school uses.<br /><br />On the other end of the spectrum is Singapore math. I've decided that Singapore math is now Public Enemy #1, replacing Kumon as the worst thing you can do to your child. One of my early famous articles describes Anti-Kumon, a program that I felt so strongly about that it ended up being the Test Prep Math series. I also hate Mathasium and Level One. I'm going to hunt down Singapore math grades 5 and 6 ASAP and start using them. I've already recommended Kumon pre-algebra grade 6, a good book once you rip out the part at the beginning that spoon feeds how to do each problem and the part at the end called Solutions.<br /><br />What Singapore found out is that you can train your child on advanced math and they'll look pretty capable as a result. The top high schools in the country are full of overstressed, overanxious kids spending long hours doing homework and beginning the teacher to just please please tell them exactly what to do to get an A because they've never been given the opportunity to learn and don't know how. The Singapore material itself isn't such a bad idea (which is why they're about to get an order for books from me), it's what parents do with the material that is detrimental to their child's future.<br /><br />Recently I conducted a search for problem solving books. I've developed my methodology based on Poyla's 1945 book and was wondering if anyone else came up anything else that was helpful. (The short answer was The Art of Problem solving that came out 10 years later and is helpful for wealthy people who live in suburban Connecticut.) For the 8 weeks prior, all of my Google searches were 'Algebra II problems', for the older brother, and Google's search engine is nothing if not intuitive, so it gave me an extensive list of books and websites devoted to detailing the step-by-step solution to every algebra problem ever devised. I was horrified. This is where math training leads. A drug addiction to solution guides.<br /><br />Anyway, I'm going to apply hard core Poyla to 2 problems. The first problem is below, and the second problem is that you have to get your child to learn it on their own, no cheating. In yesterday's post, there were two problems, and children naturally gravitate to the the second one. I should have recognized this one as a necessary step for the first one but I blew it. <br /><br />Here we go with the harder question.<br /><br /><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 12px; text-align: justify;">There were three times as many jelly beans in Jar A as in Jar B. After 2685 jelly beans in Jar A were sold, Jar B had twice as many jelly beans as Jar A. How many more jelly beans were there in Jar A than in Jar B at first? </span><br /><br />We followed exactly the problem steps from yesterday's article, including a diagram that was totally unhelpful, but there was no way to solve this equation without extensive backtracking and #7 the missing element. I hate the missing element technique. It is absolutely fundamental to geometry proofs, but it takes a lot of work for a child to derive the missing element in 1 hour that took brilliant mathematician's 250 years to derive. The secret is #8 use everything you ever learned that points to the missing element, and if you haven't actually learned it yet, you need backtracking.<br /><br /><b>Step 1: Backtracking</b><br />For backtracking, I used IXL and Khan Academy algebra problems with parenthesis. I did this a few months ago. These are real powder puff exercises, like 4(x - 10) = 23. I look for parenthesis because kids who grow up with wholelistic language-based thinking math take many months to remember how parenthesis work no matter how many ways you spoon-feed it to them. I finally created Kumon style parenthesis worksheets and told them just to memorize it.<br /><br />I'll tolerate estimate-iterate for a while (is x 3? How about 30? What about 12.2?) because it's good arithmetic practice and builds the type of number sense needed for statistics, but eventually I'll resort to something like 1/x(23 - x) = x<sup>1/2 </sup>so they quit guessing and ask for help.<br /><br />The missing element is the equation x = 3(13 - 2).<br /><br />What makes x = 3(13 - 2) a better problem than 1/x(23 - x) = x<sup>1/2</sup> or 4(x - 10) = 23? The answer might take multiple 30 minutes daily discussions. The answer is that in the easy equation, x is on one side and all of the numbers are on the right. In fact, the easiest equation of all is x = 33. x on the left, a number on the right. The goal of algebra is to get x on one side and the numbers on the other side, in the cheatiest least effort way possible. The goal of prealgebra is to handles all types of numerical operations including exponents. Algebra adds 'x' and mixes things up.<br /><br />From that point forward, we had an 8 week battle to see whether or not the stubborn kid could solve the problem without resorting to algebra, no matter how long it took, or whether he had to learn the fundamental principle of algebra: you can add/subtract/multiple/divide/power up/power down each side of the equation by the same factor (whether it's 5 or (x - 2)) and the equation will take a step in the direction of 'easier' if you didn't screw up the parenthesis again.<br /><br />We spent so much time analyzing what was wrong with equations (the x is not on the opposite side as numbers) that it qualified as a principle on which to build.<br /><br /><b>Step 2: Derive the Equations</b><br />In the problem above, this wasn't an issue because our math program is founded on convoluted complex word problems with double reverse logic. We lost a few minutes because one of the unfortunate side effects of this approach is a kid smart enough to point out how stupid the problem is. "Who buy's 2,685 jelly beans? Like they're going to sit there and count them. This is a dumb problem."<br /><br />For some kids, backtracking might include writing equations from word problems.<br /><br />So we got 3B = A and A - 2685 = 1/2B. <br /><br />The second equation was rewritten as B = 2A -5310. The reason is that at one point in our backtracking, I told him if he see's x in an equation (aka a variable), then there is a 100% probability he'll have to work the equation with transformations to derive the answer, so stop wasting time trying to solve it in your head. <br /><br />The three important principles for this step that we haven't come to terms with fully are:<br />a) The best way to determine the correct equation is to write down the crap you know is wrong and fix it<br />b) don't write the two equations buried in a bunch of pictures<br />c) if the older brother wants to interrupt math with the new Avengers trailer, you're going to lose 20 minutes<br /><br />No matter how many times I encourage mistakes and do overs, each new step up the math ladder is greeted with this expectation of getting things right the first time. Mistakes are the fastest way to the goal. Perfection is a hard stop on the road to learning. We would save a lot of time if he would just write 2,685 - B = 2A, realize it's wrong, and fix it.<br /><br /><b>Step 3: Wait for the Leap</b><br />At some point during this problem I started cursing Anonymous for putting me in this position. This would be a great problem for a long weekend. To solve it, my kid has to figure out how to solve simultaneous equations, on his own, and all we've got are our problem solving techniques.<br /><br />I am 100% sure that 100% of Singapore kids are told what simultaneous equations are and shown how to solve them, then they can practice this technique, get high test scores and great grades, without ever have experienced true learning. It's like taking a Grade A steak and grinding it into dog food. For my buddies from Southern India, I don't have a good analogy. I once made Indian food and proudly brought it to work. My coworkers told me it was 'bachelor food'. All those great spices mixed into a tasteless mess. That's what happens to Singapore math when it's trained and not learned.<br /><br />My son pointed out that he can't solve the equation, and then complained and glared at me.<br /><br />Why? "Because it's got a B and an A. It could be anything."<br /><br />I asked him to specifically point to what is wrong with the equation. After about 5 minutes, he pointed to "2A" in the equation B = 2A -5310. So I asked him to fix it.<br /><br />We had already established algebra is about fixing equations. He knew the way to do this was transformations. In the first 7 or 8 minutes, he just stared trying to determine how to transform the equation. No luck. Then he got really intense because somewhere in the pictures of his bear and a girl named 'Amy', he could sense 3B = A plays a role.<br /><br />In Poyla, one of the foundations of understanding the question is 'use ALL available elements of the problem'. This becomes really important in geometry. We haven't spent much time on it. I asked him if anything else could help. Since 3B = A was buried in doodles, I asked him to show me all of the pieces of this problem. I'm not sure this was necessary, but it was getting late and he had science homework and my spouse was yelling at me. (Solution strategy #9, when your spouse is yelling about how late it is, start asking questions that direct your child.)<br /><br />We had 7 or 8 minutes of silence and I could see he was becoming really excited in an intense concentrating way. He said "2A is 6B" and wrote 5B = 5310. When you're excited about learning, you can do 3 transformation in one step and I'm not going to complain. This is how brainiacs get to the point where they solve things mentally to the consternation of their teachers.<br /><br />What did I do? I did three things.<br />1. I didn't look up the solution and explain it to him.<br />2. I didn't help other than ask questions and suggest one of the 8 problem solving techniques. In this case, I suggested all 8 and we used all 8. I will continue to do so until I'm banned from helping by my son, which is scheduled for middle school, at which point I will solidify my role as the dumbest, lamest parent on the planet and my child will reach self sufficiency.<br />3. I waited, and waited, and was prepared to wait for the next 3 weeks if that's what it takes.<br /><br />I was rewarded in 3 big ways.<br />1. I concluded the whole session by mentioning that 2 equations with 2 variables is called 'simultaneous' equations. I pointed how that he taught himself how to solve simultaneous equations and this is a big deal. He already knew at this point that he taught himself and it was a big deal to him.<br />2. 3 months ago, it was horribly painful for him to transform x - 3 = 6 by adding 3 to both sides. Now he was doing 4 steps in once (multiplying 3B = A times 2, substituting 6B for 2 A, subtracting 6B from each side of the equation and multiplying each side by -1). I have repeatedly told parents to look for this effect, starting with phonics and first math when you get 3 weeks of zero and want to quit. It's nice to see anyway.<br />3. As a parent, I took a big leap myself in problem solving skills under the problem of how NOT to teach my child how to solve simultaneous equations even though Anonymous put me into this awful spot.<br /><br />We are not going to have to practice simultaneous questions to perfect it. It's been earned, not trained. I don't like perfection, it removes the problem solving aspect that will be gained the next time the topic comes up, which will probably be this weekend with 8th grade simultaneous linear functions because I'm totally psyched.<br /><br />In the last 6 weeks, I've come to the realization that the approach behind Test Prep Math is not at all compatible with Singapore math before grade 4. Test Prep Math tries to avoid math at all costs while building up the skills underlying math, logic, and reading convoluted problems to earn the first 3 foundation problem solving skills that I covered in yesterday's article. These are 2 wholly different world views. I'm betting the farm that by middle school and then again in high school I will inevitably be proven correct. I'm standing on a mountain of research, logic, and common sense from qualified teachers that I stole (problems solving technique #10). By why wait until then? 4th grade is a great time to crush a few years of Singapore math.<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com0tag:blogger.com,1999:blog-5703568807615263851.post-21534618189805710602017-11-29T13:56:00.002-08:002017-11-29T13:56:20.465-08:00Teaching 4th Grade Math to a 2nd Grade ChildI received this great question from Anonymous that deserves at least one post, if not a book. Your child and skill level may vary, but from my stand point, it's the same question. <br /><br /><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">I'm struggling with 4th grade math materials. What's the best way to teach my second grader how to solve these questions? </span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;" /><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">There were three times as many jelly beans in Jar A as in Jar B. After 2685 jelly beans in Jar A were sold, Jar B had twice as many jelly beans as Jar A. How many more jelly beans were there in Jar A than in Jar B at first? </span><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;" /><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;" /><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">Aileen and Barry had an equal number of postcards. After Barry had given Aileen 20 postcards, Aileen had five times as many postcards as Barry. Find their total number of postcards.</span><br /><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;"><br /></span><div>I'm going to provide a step-by-step guide, and this is going to be a long article. Brace yourself.</div><div><br /></div>There are many great reasons to teach a 2nd grade child 4th grade math. Here they are ordered from most important to least important.<div><ol><li>You didn't think to teach your child 2nd grade math when they were in Kindergarten. Frankly, it won't matter by middle school when you begin, but the earlier you start, the more time you have to block out all of the memories of frustration until you just remember what a great idea it was. An earlier start imparts more technical skills and a later start imparts more grit, but grades are high in science and language arts, which is what you really want.</li><li>You want to imbue your child with unmatched grit and generalized problem solving skills so that the rest of their academic career will be easy no matter what the challenge.</li><li>You want your teacher to notice that your child is bored in math and recommends your child for an advanced or accelerated program.</li><li>You are blatantly cheating your way to a high score on the MAP test.</li></ol><div>This problem is from Singapore math. Be very careful with Singapore math because like Kumon, it shows the kids how to do the math and undermines a host of more important skills like how to think. There are problem solving guides that come with these types of math courses and they short circuit learning. You can destroy your child's thinking ability in one shot and it's hard to undo the damage.</div></div><div><br /></div><div>One more thing to keep in mind with Singapore math. 4th grade math compares with 5th grade or 6th grade math the way we normally refer to math curriculum in the US. I've seen 2nd graders do 2nd and 3rd grade level Singapore and come out ahead. You might want to think about switching to 3rd grade Singapore math or 4th grade lame standard US math.</div><div><br /></div><div><b>Rule #1: Don't, under any circumstances, teach math. </b></div><div>You don't want your child to learn math. If you focus on the more important skills, they will learn really advanced math on their own. But if you try to teach them math concepts to solve these two problems, they are not going to learn math or anything else. It's not about math. The child is in charge of math, and you are in charge in an environment and experience where learning will explode.</div><div><br /></div><div><b>Rule #2: It's going to go painfully slow at the beginning.</b></div><div>It's really hard to watch a child tackle a problem that requires basic problem solving skills while they pick up basic problem solving skills. It's painful. If you want your child to learn how to learn, you can help by being confused, by being patient, by asking questions, but you can't just tell them how to do it.</div><div><br /></div><div>It does not surprise me when a child takes 2 or 3 days to get past the first problem. It does not surprise me that they forget something we did or said 10 minutes ago. But I'm always totally shocked that in a few months they're zooming through 4th grade material like a slightly below average 4th grader, and I'm pleasantly surprised that test scores are now 100% across the board.</div><div><br /></div><div>I'm always happy to receive an email from a parent that starts out with "I was doubtful at first because we got no where in the first 3 weeks..." because I know exactly where it's going. If your child does ballet every day, they will probably become adept at ballet. In the same way, success is inevitable on 4th grade math. Give it time.</div><div><br /></div><div>I like to say "of course your child can't do 4th grade math, because she is only a 2nd grader". But she will. These problems, however, are challenging for a 6th grader. At Math House, we've worked through much more inappropriate problems, so I say go for it.</div><div><br /></div><div><b>Rule #3: Let's teach something besides math.</b></div><div>Language is probably the most important. In the 2 problems above, there are at least a dozen words that your child could read and not understand, at least not in the context of the problem. I'm going to provide some solution strategies that will help you in the first few weeks, but you need to get to a discussion of the problem as the primary way to work through it, not just because you want a high reading comp score as a bonus, but because <u>understanding</u> of math and <u>language</u> are linked. I'm not sure math itself is linked to language, probably, but understanding math definitely is.</div><div><br /></div><div>In the first few passes of each problem, invite your child to explain it to you, word-by-word and sentence-by-sentence. For many parent-child teams, this will be total culture shock. It takes changing gears and practice. If your child can't articulate the question on the 7th try, word by word, you may ask for a picture or try again the next day.</div><div><br /></div><div>Being confused, having to read a question 5 times, and getting it wrong are 3 important skills that have to be practiced and developed. If your child doesn't become an expert at these 3 skills, and you as the primary academic coach aren't totally on board, more advanced work is going to be a real struggle. </div><div><br /></div><div><b>Rule #4: You need solution strategies to survive.</b></div><div>You, the parent need the solution strategies. My kids know all of them and are ready to tackle graduate study of Lie Groups, but if they use them, they use them behind my back. I've never met a problem anywhere that can't be solved by these, so when they are stuck, I just shout out random solution strategies and we're back in business.</div><div><br /></div><div>Now about that solution.</div><div><br /></div><div>The challenge with the 2 problems above for a 2nd grade child is "2685" and "five times". I don't care if my 2nd grader picks up an understanding of 4 digit numbers and multiplication/division. That's his problem. I want him to understand the essence of the logic and problem definition.</div><div><br /></div><div>If the child understands the problem, in second grade, we're way ahead of the game. Moving forward with strategy and solution will follow in time. I prefer the child to get there when they get there, on their own.</div><div><br /></div><div>By the way, you can just google these problems, tell your child the solution framework, and set your child up for failure down the road. It's your choice. </div><div><br /></div><div>Here is the parent tool set:</div><div><ol><li>Draw a picture. This doesn't work really well with 2,685. Plus, this strategy is appropriate to geometry and should only be used as a fallback when your child is really frustrated. Drawing is relaxing. In this case, I would ask them to draw a diagram to show me the before and after (with colored bars instead of cards) just so I could see that they understand the problem. Given the difficulty level of these problems, a drawing is inevitable, or acting them out with a stack of pennies.</li><li>I tried algebra. Total failure in 2nd grade. The 4th grader is now starting to get it because I told him it's total cheating. Yahoo answers recommend algebra for 4th graders, but if you are successful, by the time your child gets to 4th grade they will just look at the question, stare at it silently, and announce the right answer. They will be using elements from the rest of the list.</li></ol><div><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">There were three times as many jelly beans in Jar A as in Jar B. After 2685 jelly beans in Jar A were sold, Jar B had twice as many jelly beans as Jar A. How many more jelly beans were there in Jar A than in Jar B at first?</span></div></div><div><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;"><br /></span></div>This problem needs #3: Make a simple problem. In competitive math and math after calculus (like infinite series), a simple problem is followed by incrementally harder problems until we've developed a generalized algorithm. In this case, we just want to understand the problem.<div><br /></div><div style="text-align: start;"><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">There were three times as many jelly beans in Jar A as in Jar B. After </span><span style="background-color: white; text-align: justify;"><span style="color: red;">25</span></span><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;"> jelly beans in Jar A were sold, Jar B had twice as many jelly beans as Jar A. How many more jelly beans were there in Jar A than in Jar B at first?</span></div><div><br /></div><div>Now we've got a problem that a 2nd grader can work through, although it's going to take a few days at 30 minutes of concentration time per day. I would recommend getting a bag of jelly beans after the first day. Hopefully, you have lots of pennies, but now we've got a problem that deserves a picture. Regardless, going back to 2,685 is going to add nothing to the problem for 2nd grade.</div><div><br /></div><div>How did I pick 25? I estimated and iterated (solution technique #4 which kids get really good at for problems like this after a few months of work).</div><div><br /></div><div>Lay out the 25 sold jelly beans, and ask your child what we don't know. (Many readings of the question later and some discussion) and we don't know how many beans are in Jar B and how many beans in Jar A were not sold. You can do this on a 3 part diagram and place the sold beans in part 2 of Jar A.</div><div><br /></div><div>Then invite your child to start putting down beans in the 2 missing places (#4 estimate) until we've got the beans left in Jar A to equal those in Jar B. Finally, have your child read the question out loud and explain the answer to you. Here's a tip. Start with 1 bean in A for the part left after the sale (solution strategy #5 - start with 1) and ask how many need to go in B to establish twice. Ask whether or not 1 in A and 2 in B satisfy the initial condition. Your child is going to go "What does initial condition mean" so you have to read the problem again and write down the 2 conditions the beans have to satisfy. As your child adds beans so that the part in A that is not sold is 1/2 of the part in B, see whether or not you got the solution.</div><div><br /></div><div>In this way, a 2nd grader will build number sense, learn multiplication/division from the ground up, and have to concentrate really hard to get through it. All great skills. If you throw in discussion skills, your child is going to make a lot of progress. It is unlikely that your child will get any where near competent on 4th grade Singapore math. This has never been part of teaching current+2, but eventually it will happen. The first year is mainly about grit.</div><div><br /></div><div>On to the next question. Solve these in order:</div><div><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">Aileen and Barry had an equal number of postcards. After Barry had given Aileen 1 postcard, Aileen had two times as many postcards as Barry. Find their total number of postcards.</span></div><div><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">Aileen and Barry had an equal number of postcards. After Barry had given Aileen 2 postcards, Aileen had three times as many postcards as Barry. Find their total number of postcards.</span></div><div><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">Aileen and Barry had an equal number of postcards. After Barry had given Aileen 4 postcards, Aileen had four times as many postcards as Barry. Find their total number of postcards.</span></div><div><span style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12px; text-align: justify;">Aileen and Barry had an equal number of postcards. After Barry had given Aileen 20 postcards, Aileen had five times as many postcards as Barry. Find their total number of postcards.</span></div><div><br /></div><div>In addition to problem decomposition (inherent in these problems), estimating+iterating, and diagramming, I recommend solving these problems in reverse. #5 Start with the end state and see if you can work your way backwards to the initial condition. It's good practice on an important solution strategy.</div><div><br /></div><div>Those 4 versions of the problem are not just a variant of start easy and work your way up, but have an element of what I call 'Backtracking'. When we do 'work ahead' these days, we'll come across something like arithmetic in the complex plane and have to take off time from the problem to practice adding etc complex numbers. It can happen on any problem. In your case, it could be arithmetic with multiple digits or decimals. Be prepared.</div><div><br /></div><div>On that note:</div><div><br /></div><div><b>Rule #5: Get a Fallback Book for Bad Days</b></div><div>I've used boring current+1 workbooks which just have pages of fill in the blank when we're having a bad day because at least I want daily math to be an established pattern during the current+2 year. In your case, I highly recommend Singapore Math Grade 3, or grade 3 if some publisher stole this question, because you may find that the grade 3 book is already 2 years advanced over 2nd grade and end up switching to it. Then get a boring 3rd grade fill in the blank book for bad days.</div><div><br /></div><div>Plus, I can't help every day, and it's nice to have a worksheet that I don't have to grade.</div><div><div><br /></div></div><div>Plus, we may need it to backtrack on missing math topics and a 3rd grade book would do it.</div><div><br /></div><div><b>Rule #6: You'll Never Succeed</b></div><div>You'll never succeed in a 2nd grader doing 4th grade math like a top notch 4th grader. You don't want to, so don't set out with this goal in mind. You want your 2nd grader to be an amazing kid in all subjects, prepared to take on the best of the best. But a great 4th grade mathematician will crush him. If you want a child to work quickly and accurately 2 years ahead at the end of 6 months (which may happen a few years in the future on its own), you'd have to spoon feed, memorize, and train, and you'd end up with a dummy who hates math.</div><div><br /></div><div>Instead, after you get to about the 75% mark of the book (or the 3rd grade book in this super hard series once you come to your senses), when your child is only misses half of the problems and takes forever, look for amazing things in all subjects. Take a year off of math and do other things if you can. Then be prepared to spend the rest of grade school feeding your child advanced math so they aren't bored.</div><div><br /></div><div>The original experiment for current+2 never got beyond adequate, although he works nicely on his own. Sometimes he does really well with current+3 or current +5, and sometimes it's 100% wrong. Recently, I created a new website for our Boy Scout troop. He sat at the computer next to me because he wanted his own website. [Insert eye rolling here, because that's what I was doing.] I sat there stunned when he typed html from scratch. Who types html from scratch? He certainly didn't learn this in school. Then started adding detailed styling and animation like he has a programming gene. The level of learning skills when he's motivated is at about current+7. That's what I'm talking about. I didn't give him a fish when he was hungry. I didn't give him a fishing pole or a net. Apparently by focusing on problem solving skills and not helping or caring about the answer to a math problem, I gave him a whole fleet of fishing trawlers. That's what I'm talking about.</div><div><br /></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com1tag:blogger.com,1999:blog-5703568807615263851.post-14040730952255829212017-11-27T08:16:00.000-08:002017-11-27T08:16:37.895-08:00Add 20% To Your Child's Score<div><a href="https://www.minneapolisfed.org/~/media/files/publications/studies/earlychild/lessonslearned.pdf?la=en" target="_blank">Here is a thorough</a> paper summarizing early childhood studies. It's slightly dry if you're not in to this sort of thing, but it's very inspiring how much success has followed investment in at risk children living in poverty.</div><div><br /></div><div>The general conclusion is that taking kids who live in a home devoid of eduction and putting them in a top notch academic program is going to have a big impact. Early studies found that when you send a kid back into the original environment, the scores and grades plummet back to where they came from. It's nice to see later studies address this issue.</div><div><br /></div><div>These authors ask an open question that I have already answered. It's a really big question and has a big answer.</div><div style="margin-left: 20px; text-align: left;"><i>...many early child interventions are conducted with at-risk children living in poverty. There are many reasons to suspect that the same results may not occur if the same intervention were conducted with affluent children. </i></div><div><br /></div><div>"Affluent" in this case means a home with education and stability. I would agree a child from an "affluent" home may not see much benefit from a program designed for an inner city child with a single parent who didn't finish high school, even though some of the at risk kids in these programs saw IQ leaps from IQ = 92 to IQ = 130. That is one friggin' big leap.<br /><br />Since I don't have any at risk kids in my home, I asked a different question <i>"What type of radical dramatic change would I need to do around here to go from 110 to 125, or 125 to 135, or to 160 just for the day of the big test?"</i></div><div><br /></div>Step 1: I'm going to give myself 18 months. It turned out that it took 14 months just for me to get my act together as a parent, followed by 2 months for my child to get past radical core skill therapy (in one case barely in time for the test) and then 2 months to ramp up to a new level. You can get your act together on day 1, and I'd be happy to provide a list of mistakes not to make, but if you've been reading my blog I think you're past that.<br /><br />Step 2: I want my kids to experience the same shock that these at risk kids experienced walking out of poverty into an advanced academic program run by a bunch of PhD's and taught by their graduate students. <br /><br />Step 3: We're not going back. I am on constant watch against video games, surfing, online chatting, and fun of any kind as my kids try their best to have a normal life. It turns out that we only need about 20 or 30 minutes a day of heads down concentration on something inappropriately hard, but I've made those 20 to 30 minutes a prerequisite of fun.<br /><br />Step 2 is formalized into Test Prep Math. I want a single shocking 25 minute problem a day at first. (Yes, I ramp up slowly because some kids cry and more adept kids can just zoom ahead feeling confident before the 'wham'.) I want mistakes and confusion. This is the birth place of problem solving skills. If you present a child with a doable problem, there is no need for problem solving skills. How about just easing your child along with some step-by-step and scaffolding? You're not going to get a leap of 20+ points like these studies have found taking baby steps.<br /><br />But the work is not done. There are two problems I'm dealing with in my own research. <br /><br />On one end, I just got 1 started on Amazon. The reviewer complained "the book has so many errors". Those "errors" are alternate solutions. I stole this directly from the COGAT and love it. You do a problem, get it wrong, don't understand the solution, and then dig in for 20 minutes to figure out that you assumed adding but the only available solution uses multiplication. If you don't like confusion, don't by the book, because this is the most important skill and the base of the whole GAT skill pyramid. I'm always worried about printing issues, so I'm getting a new copy just to check the solutions for the 10th time. My other copies keep getting 'borrowed' by neighbors.<br /><br />On the other end of the spectrum are kids who have really great math training and skip right past the confusion and problem solving steps because they already know how to do the problems. The COGAT is a big stumbling block because it demands problem solving ingenuity. A child never learns to solve problems if they are formally taught math, and if your child goes to a great math program like Mathasium, Level One, or Singapore, they have completely different academic world view than the COGAT. It's not a bad thing, and could be a good thing, but it's the opposite of what I want for my children. I'm laying awake at night wondering how to fix these kids. It's one thing to lead a horse to water and they refuse to drink. It's another thing if the horse is drinking gallons of water and is still thirsty. <br /><br />I'm thinking of just adding more bonus question to Level 2. I am the master of giving a child a question that he can't answer without 20 minutes of logic and solution strategies, but it would just make people like the 1 star guy more baffled. I could spoon feed everything in the solution, but this will just help others avoid the learning process. I hate solutions. Too many parents think that the whole purpose of test prep and math is to have your child know something. It's not. Think more radically, like 20 points radically, whether this is from 79 to 99 or from 99.1 to 99.7. Step 0 is big goals.<br /><br /><br /><br /><br /><br /><div><br /></div><div><br /></div>Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com2tag:blogger.com,1999:blog-5703568807615263851.post-30097647317744366862017-11-25T11:00:00.000-08:002017-11-25T11:00:03.260-08:00Math Over 20 Minutes a DayI'm back in the daily math business thanks to the break. It beats listening to 2 boys make up video games and verbally describe the action because they are not allowed to play real video games unless they do some really serious math, read, practice their instruments (together without fighting) and clean the kitchen and vacuum the place.<br /><br />Each child began their official math career with 6 months of math at a level of current plus 2. (This was Every Day Math, hard but not too hard.) It's a right of passage and a way to impart senior executive functioning skills, problem solving skills, core learning skills, and grit. The exercise below is exactly the exact identical same situation, just with different math. Since then, I haven't worried about an organized program.<br /><br />When the video game talk became pushing and shoving, I upped the ante. The fourth grader would get a 6th grade math test from Virginia and the 7th grader got a rigorous high school Algebra II final.<br /><br />For math at an inappropriate level, we follow these rules. First, the kids do it. Since they are missing vocabulary and concepts, because I haven't done current+2 in a long time, we do it again together the second time. (We've been skimming current+4 and current+6).<br /><br />I'm going to describe how we do some of these problems so that you'll see that it really isn't about math at all. It's about a high degree of analysis and problem solving because a) the MAP is a big deal this year for us and b) multi-digit multiplication is 100% useless and distracting from important skills.<br /><br />Here are the rules for doing math at home:<br /><ol><li>Let the child do the work on their own first with no help.</li><li>When you go through it, pause to take the material on it's own terms because the child hasn't seen it yet and he started with a final and not the actual text book.</li><li>Figure out a way to cheat.</li></ol><div>#3 is the key to everything including academic and life success. It's why I expect my kids to open the book the first time the night before the Chemical Bio Organic Genetic Engineering Chemistry final and ace it the next morning.</div><div><br /></div><div>Here's how it works in action. As you can see from the picture, we got off to a slow start on the very first question.</div><div><br /></div><div style="text-align: center;"><a href="http://4.bp.blogspot.com/-cgpe_qNv49U/WhcJSlKI0fI/AAAAAAAAAjo/xHQNeyi0NDAoNyHO90imN6Ipp25d4ja0QCK4BGAYYCw/s1600/WIN_20171123_11_42_44_Pro.jpg" imageanchor="1"><img border="0" height="260" src="https://4.bp.blogspot.com/-cgpe_qNv49U/WhcJSlKI0fI/AAAAAAAAAjo/xHQNeyi0NDAoNyHO90imN6Ipp25d4ja0QCK4BGAYYCw/s320/WIN_20171123_11_42_44_Pro.jpg" width="320" /></a></div><div><br /></div><div>The way to cheat on this question is to note a) the answer is in the vicinity of 25 or 26, and since no addend ends with 9 in the 10,000th place, the answer has to be C. Much more gratifying than multi-digit addition which is used no where in life or in any other class or in college.</div><div><br /></div><div>Question 3: 6x + 3 = 3(2x + 3)</div><div>Here are my comments: <i>Every time you see x in an equation, be prepared to rearrange and transform the equation.</i> This is a good place to learn how parentheses work without the spoon feeding and repetition of Pre-Algebra. "x" requires a few long discussions under the heading of Power Bucketing (aka setting up future math) but we already had those discussions. <i>The equation becomes 3 = 9. Now pick the answer. </i>On the SAT, we'll switch to looking for a subset relationship to find the answer and other cheatiness, but for now they need to learn transformations.</div><div><br /></div><div>Question 5 was even more fun.</div><div><br /></div><div style="text-align: center;"><a href="http://3.bp.blogspot.com/-gttMLnREJzw/WhcL8w0evlI/AAAAAAAAAj0/csL2CtRdQrcAiEOfxgr-wdiLPgj3R9-PgCK4BGAYYCw/s1600/WIN_20171123_11_55_47_Pro.jpg" imageanchor="1"><img border="0" height="232" src="https://3.bp.blogspot.com/-gttMLnREJzw/WhcL8w0evlI/AAAAAAAAAj0/csL2CtRdQrcAiEOfxgr-wdiLPgj3R9-PgCK4BGAYYCw/s320/WIN_20171123_11_55_47_Pro.jpg" width="320" /></a></div><div><i><br /></i></div><div>It took a while, but we settled on putting 6.23 x 9.3 within the bounds of 6 x 9 and 7 x 10. You can see this work in the middle of the page, and you can see on the right the framework my son used to actually calculate 6.23 x 9.3 using successive digits. The cheaty-est way requires the most work, higher order problem solving skills, more creativity and more time. I love it when a student is excited that cheating turned a 5 minute problem into a 20 second problem, never stopping to think that it took us 20 minutes to get there. </div><div><br /></div><div>By the way, one reason to let the child do the work first is because children will most likely resort to calculations and they all need practice in arithmetic. The main reason is that whatever they answer is expected to be wrong, and Math House loves mistakes. </div><div><br /></div><div>Question 6 in the picture is awesome. It involved a wiki definition of "statistical" followed by an evaluation of each statement on those terms. 'Statistical' is a summary or characterization of the data, and 3 of those answers ask for a single number. The last one is seeking an average. Also, statistics usually follows the rest of the material in a math course and in terms of timing usually ends up being taught after the you-know-what test. I'm not sure how the little one ended up with the correct answer the first time.</div><div><br /></div><div>The Algebra II final was loaded with solution strategies, most of which were not discussed yet and ended up being more boring to re-work than the 6th grade test. The breadth of the Algebra II test was good so it was worth it. On both tests there was this little gem:</div><div><br /></div><div><div style="display: inline; float: left; width: 45%;"><div style="text-align: center;">Alegra II</div><div style="text-align: center;">Evaluate 3n/(n + 3) + 5/(n - 4) </div></div><div><div style="display: inline; float: left; width: 45%;"><div style="text-align: center;">Pre-Algebra</div><div style="text-align: center;">Order 1/4, 1/5, and 11/40 </div></div></div></div><br /><br /><br /><i>In both cases, you can't compare apples and oranges, or 1/5 and 11/40. And you can't directly add things with different denominators. What happens if you multiple something by 1? Does 100/100 qualify as 1? What about (n + 4)/(n + 4)?</i> These are my comments as we meandered to comprehension.<br /><br />Many parents complain that they don't feel qualified to understand math in a child's terms, let alone teach it. Here is my response. What they are really saying is that they don't want to take the time to understand and solve the problems of a) the math on it's own simple terms (the 'simple' takes time to get there) and b) the problem of working slowly with the child until the child has learned some skill. And yet they expect their child to magically acquire patience in analysis and problem solving skills?<br /><br />You've got a third problem, which is that at school the child is learning boring spoon fed repetitive work that values memorization, speed, and 100% accuracy. So if they don't learn the important skills at home from you, they won't learn it.<br /><br />Learning starts with unknown, slow, and 0% correct. I'm happy if you feel like you're at this point too.<br /><br />First, acknowledge that your baffled. You can be baffled on how to teach this while the child is baffled on what to do. This sets the right tone and makes everyone comfortable.<br /><br />Next, look at what you know about the problem. a) Your child stinks at parentheses, isn't 100% conversant with variables that span R, doesn't see the need for common factors and b) you don't know how to teach them.<br /><br />Third, come up with a strategy. It will have the following components: a) it will take a long time, b) it will be step-by-step, c) you have to back track on something simpler, like having the older child do the fraction problem first and having the younger child compare 1/2 and 1/3 first, d) you will have to reevaluate and try again.<br /><br />The preceding paragraph is going to be the child's take away. Someday they'll understand "n", common denominators, how to use parentheses, algebraic transformations and the rest of it. At least they'll be comfortable when they get to it in school or on a test. But those problem solving skills starting with 'Baffled' and ending with 'Try Again' are so powerful both of you are on a path to 99%.<br /><br />I'm going to conclude with a warning about accelerated math in school or after school. It is now common to teach Algebra II in 7th grade and Geometry in 8th grade, or to skip to Geometry in 8th grade, despite the compelling evidence that the result is many kids quit math early in high school and reporting hating it. At this young age, speeding along, the kids memorize and learn to use the concepts. There is little if any time devoted to the key skills that are taught in high school, especially in Geometry, learning adult level problem solving skills doing Geometry proofs. I would be surprised if any 8th grader could prove to me that a straight line is straight, let alone prove each building block of Euclidean Geometry all the way up to trig, but that's what a rigorous high school class teaches, because it is essential to Calculus. It's essential to thinking.<br /><br />We've been doing high school Geometry proofs for the last 6 months with both kids. Algebra II is a belated detour. You might imagine that the approach is dramatically different than an accelerated school math program. In 14 months I'll tell you why this is so important.<br /><br /><br /><br /><br /><br /><br />Norwoodhttp://www.blogger.com/profile/09462923179883891369noreply@blogger.com3