Saturday, March 17, 2018

The Test That Won't Be Named

In this article, I'm going to jam 7 articles into one because I'm really pressed for time on the weekends.

Review of Home Schooling Literature
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.

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.

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).

The Secret to Learning
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.

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.

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. 

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.

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.

The Secret For Parents
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.

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.

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).

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.

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.

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.

To be on the safe side, we did lots of vocab ( 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?

I think my casual approach to reading is the reason language arts thrived in Math House.

Yes, I grilled the kids at the Word Board (How would a commander on the battle field use the word 'dispersion' in a sentence?), 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.

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.

The Magic of Slow
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.

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.

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.

The triangle above is isosceles and AB > AC.  Which of the following is false?

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'.

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.

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.

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.

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'.

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.

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.

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.

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.

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.

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.

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.

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.

The Danger of Test Prep Classes
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).

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.

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?

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.

Sunday, March 11, 2018

The Kindergarten Challenge

Here's a challenge I received from a reader. 

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.

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.

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.

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.

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.

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.

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.

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:
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.
2.  Give them advanced material and let them do all the work before you don't grade it.  (No typo, read that again.)
3.  Walk through the super hard material together, one question at a time after they do it.
4.  Do it with them, one question at a time, mostly just asking questions.
5.  Give them simple material on a super advanced topic so that they can learn one step at a time on their own.
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.
7.  Lay 5 skittles on the table, one of each color, and provide a skittle each time your child gets a correct answer.
8.  Give them a skittle just for making an attempt.
9.  Do the problems yourself while they watch.

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'. 

Which approach do you use?  I used them all.

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.

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.

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. 

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.

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. 

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. 

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.

Saturday, March 3, 2018

Putting Skills To The Test

I'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.

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.

Here are my hypotheses.
  1. The COGAT measures the skills that predict academic success.
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.

Any parent who forgoes COGAT test prep (or a similar cognitive abilities test) has no interest in a child with cognitive abilities.  

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.
  1. The skills are age independent.
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.)

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.

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.
  1. The list of skills is boring and unremarkable.
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. 

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.

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.

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.

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. 

Here's a transcript of the experiment.

Me:  Solve this equation:  3 + 5 = ? (He responded 8, then looked at me like I was a moron.)

Me:  Solve this equation:  3 + 5 = ___  Does it matter that I changed the question mark with a blank?  (He responded no.)

Me:  Now solve this equation:  3 + ___ = 8.   Is it totally confusing that the blank has moved?  (He answered no.)

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.)

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.)

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:
  • x = 2
  • 3 + x = 8 - 5x
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.

Rules:  a) apply one of the 4 transformations to both sides, b) only apply one transformation at a time.

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:

  • 3 + x = 8 - 5x
  • 7x - 15x = x(x + 5)
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.  

Next, they forget how to add and subtract single digit numbers.

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.

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.

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.

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.

That is the biggest leap in skills.

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.

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.

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*.  

*It depends on what they draw.  Horses aren't good enough.  Needs something with lines and circles in it.

I prefer crafts for math training to prepare for algebra.

Anyway, the subject passed the 25 minute algebra lesson and his parents didn't complain yet about any signs of psychological damage.

Wednesday, February 28, 2018

Skills In Perspective

In 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.

Let me explain this bluntly.

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.

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.

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.

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.

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.

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.

Here is the short version of the problem solving method:
  1. Be Baffled (thanks Berkeley math department)
  2. Spend a lot of time thinking about and exploring the problem
  3. Make mistakes and try again
  4. Check your work (I added this because it raises test scores)
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.
  1. 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.
  2. 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.
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.

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.

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.

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.

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. 

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.

Saturday, February 24, 2018

Skills in Action

A 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.

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.

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.

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.

Question 1:  F - T - L - T - ? - ?   Find the last 2 letters in this series.
  1. 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.
  2. I think for a minute about adult IQ tests.  Friday, Tuesday, Something That Begins with L, Thursday.  Fail.
  3. 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.
  4. 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.
  5. 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.
  6. The answer set is:
    1. L - T
    2. L - B ( I think this B has no vertical lines.)
    3. L - M
    4. T - P
  7. 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.
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.  

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:
  1. You get used to working with baffling things and don't get put off.
  2. 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.
  3. 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.
  4. 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.
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. 

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.

OK, let's see what we get out of more baffling problems.

What is the remainder when the 15-digit number 444444444444444 is divided by 9?
  1. 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.
  2. 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:
    1. 4/9 ~ r 4
    2. 44/9 ~ r 8
    3. 444/9 ~ r 3
    4. 4444/9 ~ r 7  this is good practice for division but a fail in solving the problem.
  3. 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:
    1. 4*111111111111111/9 = ?  Still hard.  Fortunately, I can look back on the first fail and continue.
    2. 1/9 ~ r 1
    3. 11/9 ~ r 2
    4. 111/9 ~ r 3.  Get it?  Light bulb. 
    5. Continuing, I get to r 0 at 1111111111 which puts 111111111111111 (15 digits) at r 5.
  4. 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.
    1. 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.
    2. 11/9 = (1 + 2/9), so 4*(1 + 2/9) ~ r 8, again, just like above.
    3. 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.
    4. 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.  
    5. 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.
We've got 3 big solutions approaches that we start using when the child is about 3 years old.  

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.

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.

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%.

Moving on, how about this problem.  What is the value of 1 - 2 + 3 - 4 + 5 - 6 + ... + 81 - 82?

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.  

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? 

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.  

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 ab can take on how many different odd number values?
  1. 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 aoutside of 42.   I've got a whole exponent crash course (including negative and fraction exponents), but this seems to be a good starter exercise.  
  2. The rows are a and the columns are b.  I didn't calculate the *'s but I could have.  
  3. 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.
  4. 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.
  5. It's possible to jump to step 4 as a competitive math coach, but not a regular bright kid doing competitive math coach.
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.

Finally, the last question is this.  If x and y are integers and 360x = y3, 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. 

  1. 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.
  2. But I don't like 360, so I wrote 2*2*3*3*2*5x = y3.   Then I rewrote it to be 2*2*2*3*3*5x = y3.  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.  
I got the entire solution correct by following these steps:

  1. I had no clue what to do.
  2. 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.
  3. I tried again.
  4. 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.)
  5. 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.
  6. The answer emerged with no effort.
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.

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.

Saturday, February 17, 2018

Totally Doable If Done Right

In 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.

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.

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.)

Here goes.

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.

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.

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.

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.

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.  

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.

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. 
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 counter argument from Greg Ashman 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:

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.

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.  

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.

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.

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.

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.

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.

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.

Saturday, February 10, 2018

Visual Math Et Cetera

For 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 Visual Math.  These are not expensive books.  The authors are from a ground breaking group of researchers that I've been following since the beginning of  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.

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.

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.

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.

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.

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.

There are 2 parts to a good fraction problem. 

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.

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.

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.

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. 

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. 

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. 

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.

Someday is almost here in math.  In plumbing, my 13 year old routed the pipes right before his birthday.