Saturday, December 16, 2017

Problem 123

Testing season is in full swing in Chicago right now with the majority of test takers in K grade, followed by 1st grade. 

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 be feel better.

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.

I owe a reader a discussion of fractions, and I'll use this article to warm up.

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?"

Here is my response:
  • 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.  
  • The 1st few pages in the book took us about 3 weeks.  Any page could take a week.  Acceleration happens later in the process.
  • Our error rate was about 50% on a good day.
  • 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.  
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.

Page 123, Lesson 5-6, #3:
Connect the points in order from 1 to 3.

Find and name 3 triangles
__________________________
Try to name a fourth triangle
________________________
Color a four sided figure.


Step 1:  Be Baffled
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.

Step 2:  Backtrack
The first challenge is that section 5-3 discusses the naming of line segments, like AB, 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.

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.

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.

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'. 

If this were a problem like 72 - 49 = ?, backtracking might be a 1st grade workbook for a day or two.

Step 3:  Dig into the question. 
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.

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.

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.

Step 4:  Give Up
You will give up on something.  You are not working with a 2nd grade, 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 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.

After doing this for 8 or 9 months, children should be completing the work with reasonable accuracy in a reasonable amount of time.  My goals were 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.

Think carefully about what I did.  I got a child to sit and work alone for 20 to 25 minutes on material he wasn't taught and didn't know.  Later he isn't going to get any help.  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).

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. 

On the other hand, what if I trained and drilled 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 early 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.


Friday, December 8, 2017

Fractions One

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.


So we’re not going to learn fractions this way.


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.  


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.


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?


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.  


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.


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.


Tuesday, December 5, 2017

Advanced Math and Little Kids

I 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.

Let's take the first question first.  How do I teach my child fractions?

Here is my step-by-step*:

  1. 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.
  2. 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.
  3. 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.
#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.

*I will present a more detailed step-by-step but we've got a lot of ground to cover first.

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.  

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.

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.

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.

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.

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 to get the child to 99%.  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.  

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.

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.

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.

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.

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.

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.  



Friday, December 1, 2017

Pick the Right At-Home Math Curriculum

I 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.

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":
  1. You've only got so much time to help and you're not a teacher
  2. You need a high MAP score and teacher recommendation for the GAT program
  3. You may or may not have to pass the COGAT this year
  4. This child is only 7.  And not necessarily good in math. 
  5. You can't afford a tutor or an after school math program.  Plus you hate driving.
  6. If your child does get into a GAT program, you want them to be the best.
  7. If you run into problems, you're going to send 19 emails a day to getyourchildintogat@gmail.com so this advice better be good.
  8. Math curriculum from US publishers stinks
I updated the article How To Create A Math Genius 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. 

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.

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.

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. 

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 strong academically in the future, 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.

#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.

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%.

Thursday, November 30, 2017

Teaching Algebra To A Fourth Grader

I'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.

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.

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.

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.

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.

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.

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. 

Here we go with the harder question.

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? 

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.

Step 1:  Backtracking
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.

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) = x1/2 so they quit guessing and ask for help.

The missing element is the equation x = 3(13 - 2).

What makes x = 3(13 - 2) a better problem than 1/x(23 - x) = x1/2 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.

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.

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.

Step 2: Derive the Equations
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."

For some kids, backtracking might include writing equations from word problems.

So we got 3B = A and A - 2685 = 1/2B. 

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. 

The three important principles for this step that we haven't come to terms with fully are:
a) The best way to determine the correct equation is to write down the crap you know is wrong and fix it
b) don't write the two equations buried in a bunch of pictures
c) if the older brother wants to interrupt math with the new Avengers trailer, you're going to lose 20 minutes

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.

Step 3: Wait for the Leap
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.

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.

My son pointed out that he can't solve the equation, and then complained and glared at me.

Why?  "Because it's got a B and an A.  It could be anything."

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.

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.

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.)

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.

What did I do?  I did three things.
1.  I didn't look up the solution and explain it to him.
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.
3.  I waited, and waited, and was prepared to wait for the next 3 weeks if that's what it takes.

I was rewarded in 3 big ways.
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.
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.
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.

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.

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.



















Wednesday, November 29, 2017

Teaching 4th Grade Math to a 2nd Grade Child

I 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. 

I'm struggling with 4th grade math materials. What's the best way to teach my second grader how to solve these questions? 

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? 

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.

I'm going to provide a step-by-step guide, and this is going to be a long article.  Brace yourself.

There are many great reasons to teach a 2nd grade child 4th grade math.  Here they are ordered from most important to least important.
  1. 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.
  2. 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.
  3. You want your teacher to notice that your child is bored in math and recommends your child for an advanced or accelerated program.
  4. You are blatantly cheating your way to a high score on the MAP test.
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.

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.

Rule #1:  Don't, under any circumstances, teach math.  
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.

Rule #2:  It's going to go painfully slow at the beginning.
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.

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.

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.

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.

Rule #3:  Let's teach something besides math.
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 understanding of math and language are linked.  I'm not sure math itself is linked to language, probably, but understanding math definitely is.

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.

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. 

Rule #4:  You need solution strategies to survive.
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.

Now about that solution.

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.

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.

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.  

Here is the parent tool set:
  1. 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.
  2. 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.
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?

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.

There were three times as many jelly beans in Jar A as in Jar B. After 25 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?

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.

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).

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.

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.

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.

On to the next question.  Solve these in order:
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.
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.
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.
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.

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.

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.

On that note:

Rule #5:  Get a Fallback Book for Bad Days
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.

Plus, I can't help every day, and it's nice to have a worksheet that I don't have to grade.

Plus, we may need it to backtrack on missing math topics and a 3rd grade book would do it.

Rule #6:  You'll Never Succeed
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.

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.

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.

Monday, November 27, 2017

Add 20% To Your Child's Score

Here is a thorough 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.

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.

These authors ask an open question that I have already answered.  It's a really big question and has a big answer.
...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. 

"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.

Since I don't have any at risk kids in my home, I asked a different question "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?"

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.

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. 

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.

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.

But the work is not done. There are two problems I'm dealing with in my own research. 

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.

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. 

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.