Saturday, February 3, 2018

Innovations in Math Education

I promise as soon as I complete this article I'm going to start populating the reading list in the prior article on reading.  But first...

My 8 year math program is about to come to fruition.  In April, math Experimentee #1 (a newly minted teenager who started K with a bunch of my newby missteps) is going to take the MAP test, and after a long interval of not caring, this score counts and it has to be a 99%.  In this article, I'm going to summarize where we are, demonstrate the leap in math skills that happens in 4th grade, demonstrate how my math program is dramatically different than regular programs, and present it in such a way that I lose most readers before I get to the end because that 99% is competing against about 10,000 other kids in Chicago who's parents are all googling 'How To Get 99% on the MAP So My Child Gets Into A Decent High School'.  Also, I'm going to discuss my approach in purely in 4th grade terms to help parents of younger children plan ahead, and explain why Test Prep Math is the way it is. 

Let's start at the beginning.   My first goal back in K was to conquer Every Day Math.   We didn't have to pick everything up at once, just a lot of hard work to show 'You Can Do This'.  My goal was simple.  For Experimentee #1, the goals focused on entering a GAT program in 1st grade, given that we were totally behind because we did nothing to prepare for it, not even phonics or learning to read, but at least the math would be familiar (it would be EDM Grade 2 - a complete repeat) and he would have some confidence.

After crying, forgetting, getting them all wrong, spending a week or two on a single 6 question worksheet page, having to find 1st grade books to practice concepts and skills we never saw before, I transformed the following survival skills into Academic Coaching Skills that we would use for the rest of our lives and pass down many generations (hopefully) of bright descendants until one actually wants to study math in graduate school.  Here they are:

  1. Set Your Expectations To Zero.  Don't expect your child to get anything correct, understand it, remember it, work on their own, or anything.  Even if you do the same problem every day for a week and it's 7 + 6 = ?  This is the parent skill.  The child-parent team skill is to enjoy 'Being Baffled' on totally hard work that has never been encountered before that will take a lot of time to sort out.
  2. Make Mistakes.  Mistakes are the key.  After a while I stopped looking at solutions because I expected mistakes.  
  3. Take A Long Time.  When we slowed down to 30 minutes per problem, we started making progress.  This is also known as 'Read The Question' where we spent more time thinking about 7 + 6 and what it could mean and how to work it before solving it.  
  4. Other tips I put in the blog over the years, but the top 3 were the key.
So here's what we got.  At one point, we sat down and looked at Student Journal #1, with every single problem answered.  Every single problem.  No child anywhere does every single problem in a math book, or every page or even every chapter.   This is a rare and invaluable life lesson.   Experimentee #1 has an extremely high tolerance for work, chores, painful work, hard chores, ridiculously hard chores.  Even better, Experimentee #1 is not put off in the slightest by being totally confused on material that is way beyond his abilities.

Somewhere during this process, the speed of learning and work accelerated to match the challenge, and by about 1/2 through Student Math Journal #2, we quit because the challenge was gone.

Experimentee #2 experienced hard core phonics (Pre K Phonics Conceptual Vocabulary and Thinking age 4.0) and hard core math (Shape Size Color Count age 3.9) because I wanted to address any gaps I found in GAT preparation and more importantly COGAT prep, and did it with a sledge hammer the size of an SUV.  Experimentee #2 has math skills that Experimentee #1 will never have, like a child who learns to play the violin from birth will always outplay a child who picks it up at age 6, but Experimentee #2 has a completely different work ethic.  Experimentee #2 will sit down with something quietly for hours and master it, but not without a lot of complaining about the fact that he can't pick it up immediately.  Experimentee #1 never complains.

There is a completely different path for K and 1st grade that will produce almost identical short term results.   Many parents enroll their children in an after school math program.   In a good program, the child learns problem solving skills and solution strategies as well as practices math daily.  This is not a bad approach, but it is not consistent with the goals I mentioned above and a few I am going to add shortly.

After 1st grade, we stopped learning math and went more hard core into Test Prep Math.   This series is not about becoming adept at advanced math topics, but becoming adept at navigating convoluted questions, staying in the 'math game' because the questions are somewhat on the goofy side and don't include boring, manufactured math book type questions, and building working memory.  This book is not designed for children already at the 99% level for math, it's designed to get them there shortly thereafter.  I've had a few parents who's kids finish 2 years of after school math (and are at 99% already) complain that the beginning of the book is too easy.  Kind of a 'duh' moment for me, but one I need to mention for those kids, Test Prep Math Level 3 in 2nd grade is preferred.  The purpose of this book is to lay the groundwork for 99% thereafter, not to put a 99% kid at 99.9%, except by accident (which is what we experienced, by the way). 

Instead of more math, we went directly from Test Prep Math into reading comp questions.   This should be obvious from the problems in Section 1.   Section 2 takes us directly into competitive math questions (because I need something to fill the gaps before ramping up real math in 4th grade).  But the MAP score is only half math; the rest is reading comp.

From 1st through 4th grade, we only stayed a year ahead in math while I put together the basic skill set that we need.  This basic skill set is very similar to the skill set that kids would use to survive an advanced engineering or abstract math course in college but it's missing formal solution strategies.  College is the other goal, and I'm thinking ahead as usual.

At inappropriately young ages, while we were biding our time putzing around with current + 1, I started introducing advanced topics, just for fun, just to exercise thinking and start to explore the wonder of math. It was enormously enjoyable to surprise a kid with these types of questions:
  1. What is 5 minus 3?
  2. What is the square root of 4?  9?
  3. What is 1 divided by 2?
  4. If 1/3 of my donuts are chocolate, what percent of these are not chocolate?
If your child sees any of these questions for the first time in school, I guarantee the wonder, fun, learning and enjoyment of math will be totally crushed out of the experience because your child will be presented with definitions, comprehensive examples, and a long list of routine problems that have nothing to do with anything.   It then just becomes a pattern matching and lookup referencing exercise.  The child will 'learn' math, but not know how to learn.

Sometimes we would resort to backtracking, which is finding a workbook or online resource to practice the material during the learning process.  If we got '1/4' kind of but not really, a worksheet might fill in the gaps.  If a concept (fractions in this case) requires an understanding of division that is not there, we would certainly backtrack to a division worksheet and then come back to fractions.

Over time, however, I discovered the power of bucketing, which I subsequently labeled 'Power Bucketing'.  This is very similar to what I witnessed with Experimentee #1 going into 1st grade and being handed the same EDM Grade 2 workbooks that were completed the previous year.   Math is much easier to understand the 2nd or 3rd time than the first time, and quick mastery is the likely result.

With '3 - 5', I would just leave the question out there and not answer it.  Or maybe I would answer it, but then a month later I would ask it again and watch the same process starting over again from the start, but going a bit faster and progressing a bit farther.  When this come up again out of nowhere the third month, we might end up with mastery with almost no work and exactly zero practice.  Even better if the child sees negative numbers on his own in a book, he dives right in and the result is not only self-mastery, but he owns it.

SQRT(4) and also 5x - 13 = 2 will demonstrate the leap in skills that takes place around 4th grade.  Kids coming off work like Section 2 of TPM can calculate both of these without understanding how they do it.  Good little mathematicians iterate through possible solution values until they arrive at the answer, and great little mathematicians add weighing with high-low bands that narrow to the solution strategy to arrive at the answer more efficiently.

After 4th grade, when the brain is capable of higher order thinking, these two exercises gain new meaning.   The definition of SQRT(4) is the number when squared that equals 4.  In other words x^(1/2) is solved backwards.  Square roots present the opposite syntax of squares, and the solution is to back into the answer.  This is critical for topics that are going to come later. 5x - 13 = 3 is a simple introduction of y = mx + b, which is an important framework for characterizing more complicated problems, and the elements of y = mx + b have additional meaning besides finding a number.

There are also new skills that come with these math concepts.   A 3rd grader will jump in and solve either problem to get a number.  It's all one step.  A 4th or 5th grader will decompose the problem, spend more time analyzing the question, and learn more during the problem.  I've introduced younger children to the next level of math skills, like problem decomposition and making a hard problem easier; this exercise can take 20 minutes and is really good for thinking.   It requires a lot of working memory which is why in 2nd and 3rd grade working memory is most of the focus.  But older children do this intentionally, quickly, and know why they are doing it.

Let's look at some pre-algebra concepts that have been a real struggle for me to teach.   

First, x^2x^3 versus (x^2)^3.  Per formula, the first is x^(2+3) and the second is x^(2*3).  But we're not interested in formula's, because formula's produce math dummies.  

The way to do these problems is to work the question and not solve the problem.  x^2x^3 is simply (xx)(xxx) = x^5, and (x^2)^3 is (xx)^3 is (xx)(xx)(xx) = x^6.   Eventually, the child will memorize the formulas in the same way they used to count on their fingers for 5 + 3 and eventually knew that 5 + 3 is just 8.  Before 4th grade, the best I can do is lay the ground work for decomposition, restating the problem, multi-step solution operations, but they still jump into more advanced problems trying to get to a number in one (hard) thinking step.   I've noticed that after school program kids are drilled in multi-step solution strategies, but I don't want a child trained in math solving.  I want a thinker.

This is the biggest difference in my goals and methods.  I don't want a child who is trained in math, a a child good in math, a child who knows (advanced) math topics or a child who is 99% because of this training.  I want a child who does really well in math he has never seen before or mastered because he is a thinker and a learner and can apply thinking and learning to math.  I've always said if you need a 99% because it is required for GAT entry, do what ever it takes this year and forget your principals.  In 7th grade, I can't say this; it is not possible to short cut your way into a 99% without a solid learner-thinker.  Also, we've never actually deviated from principals or practiced rote math and we have always either been at 99% or been within striking range (in a bad year).  I will say that it's never too late to start.  There are advantages to starting early, but starting late does not preclude achieving the ceiling on a test.

The most challenging topic using my approach on pre-10 and post-10 children is parenthesis.   I will illustrate with this problem:  (6^2 + 18 + 2 + 4^2)) - 2^2.  This is not a complicated problem, but it is not possible to do a page of these problems with a child still learning exponents and parenthesis without writing down at least 3 or 4 steps in order to check steps for errors.  In other words, it's faster and easier to let the pencil do the work than the brain.  Before 4th grade, I'm happy to endure 4 or 5 wrong answers from mental calculations because the impact to working memory (not to mention arithmetic practice) is useful.  But with the problem above, working memory gets a work out and the child still has to write down each step to survive the problem.

In our 1 year ahead math program, it is common for kids to fall to 85% by about 6th grade.  The program administrators - geniuses way ahead of their time - are focused on the final result and this interim dip is a researched based way to achieve the final product.   The extra 14 points are achievable with a bit of extra work.  If you review this article from the beginning, you'll see 3 or 4 math education concepts that all work together to produce 99% without a lot of extra effort.  I don't think this approach would work very well in a classroom situation without modification, but it certainly can at home.  Once any topic above is presented above, the next step depends on the child's response in the context of the child's individual skill set.  A parent who gets to know their child and experiments a little with backtracking, repetition, exploring the question will stumble toward success.

Now back to the 7th grade challenge that introduced the article.  We have a very ambitious goal but not a lot of time to achieve it given homework and nonschool activities.  The topics, approach, learning environment, and general mess of our preparation is in my opinion an exact mirror of the test.


  1. Thanks to your amazing blog, my kindergartener qualified for our district's gifted programme with a 99% composite on COGAT 7. However, she managed only an 88% on non- verbal( 99% on verbal and quantitative). I am stumped as I thought she was ok at that section. She doesn't enjoy puzzles and Critical Thinking company type stuff , but could do well if she wanted to.We dont have any testing ahead of us, but I would like to help her develop her spatial reasoning abilities. Plus, after almost a year of on and off COGAT practice I feel like we need to keep working on test-prep.. I am not sure if my daughter feels the same way, though! To me the prep part is more rewarding than the actual result, which is rather unfortunate.Could you suggest some resources that could help with my daughter's non-verval reasoning? Thanks again for your wonderful blog that is such a blessing to many parents who want more for their child.

    1. I don't see a reply from me, but I've got a lot to say. You're ready for the big leagues. Go to Michaels and load up on crafts and origami and paint/marker things and sewing. Those are a step up from the COGAT. They're fun, cool, spatial, skill based. Be prepared to start with easy things and help or finish as needed, especially sewing. A 5 or 6 year old is totally proud of her work even if you do most of it. As children age they do more and more on their own. Getting a 88% on the non-verbal says a lot about your child's skill set, but projects and art take this leaps further. I'd also suggest everything else on the planet to at least try (every kid is different), but start with art.

    2. Thank you for your inputs! Will load up on crafty stuff..thanks!

  2. Hi Norwood:

    Thanks again for your great resources - always a big help with my now GATE qualified child. One quick question - you had earlier provided a link to good comprehension resources (not Vocabulary Workshop, but one more resource. Unfortunately, I am unable to find it.
    If not too much trouble, could you give me some good comprehension resources for a GATE qualified 4th grader (going to 5th grade soon)?

    Thanks much!

    1. Continental Press, Level E. Maybe even E & F, but at least E.