Chapter 4b - The Math Behind The Madness

The Test Prep Math series is the cognitive skills approach to math.  The purpose of this workbook is to train the child so thoroughly in the foundational skills underlying success in math that the parent - starting with me - will not have to worry about grades, homework, test scores, or accelerated math ever again.  Or any other subject.

The introduction to the book covers the basics of the skills that the problems teach.  I've never been comfortable that I did a thorough enough job of explaining what these skills are, why school teaches the opposite of these skills, and how the question design targets each skill.  The book is innovative in its approach and extraordinary in results.   I'll leave this page for users of TPM as a supplement of the introduction and refine this page occasionally.  Someday I'll be satisfied, maybe by revision 23, and then I'll replace the published introduction to TPM with this one.

The challenge of Test Prep math is that the skills are all wrapped into a complete package in each question.  Learning the skills one baby step at a time is impossible*.  The asterisk is required because there's a section below where I explain how to take baby steps while your child is getting up to speed on multiple skills at once. Once you get back to the baby steps, we're back skills building in one complete package per question. A student is not going to learn to devote 20 minutes to understanding the question if the questions doesn't take 20 minutes to understand; a student is not going to learn to check their answer unless there is a high likelihood of an incorrect answer; the student is not going to build working memory unless the question puts multiple concepts into the brain that need to be solved.  In short, a student is not going to rise to the occasion unless the occasion is above their abilities.

The achievement of Test Prep Math is to redefine 'abilities' as the core thinking skills behind math achievement instead of knowing math concepts.  The math in these workbooks is quite simple*.  Another asterisk, because the child isn't going to see problems as hard as the ones in this book until middle school or high school.  Regardless, being ahead in math is not necessary.  This is not possible in Kindergarten and 1st grade, where the core skill of number sense cannot be separated from basic operations.  This is not possible after 4th grade where a the math student faces a growing list of concepts presented in increasingly complex ways.  School leaves a big gap between ages 8 to 11 and Test Prep Math fills this gap in dramatic fashion.

It is well known that strong readers are made at home.   If the child isn't doing an hour of reading a day at home, or at least 30 minutes, it is very unlikely that this child is on a path to solid achievement.  All schools now make reading at home a part of the curriculum.  Most schools send the child home with a daily reading log starting in 1st or second grade.   Reading is fundamental to everything.   It shouldn't take 4 years of graduate math to realize that math achievement isn't going to happen magically if there isn't some activity at home beyond grade level homework, especially if this homework is a math facts worksheet or the homework only shows up once a week.  Problem solving is fundamental to everything, including reading, by the way, and problem solving is synonymous with math.

What makes 2nd to 4th math grade so special is that it is devoid of thinking.  As the child works their way up to decimals and long division, the objective of school is to ensure that the student knows the rules of decimals and long division.  The best way to meet these objectives is to clearly present the mechanical steps of these operations and then to assign a long list of routine problems to ensure mastery. I call this 'explain and practice'.  If my salary was tied to test scores by law, this is how I would teach math to 30 students.  This law is called No Child Left Behind.  The approach is referred to as Teach To The Test.

Under No Child Left Behind, the child is mastering how not to think, because this is what they are practicing every day.  This is also the best way to teach the child to hate math.  I could add 20 pages of examples from my work with parents and their children.  Here is a summary of the 4 most common results of teaching methods and math curriculum for this age group in the United States:

  • A child is a math star in 1st grade.  Test scores might still be high in 2nd grade, but they fall each year thereafter.
  • A forth or fifth grade child with a track record of solid A's in math is introduced to math that requires an investment in figuring out, and totally doesn't get it.  Both tears and grades fall.
  • A parent hires a tutor year after year to explain math to their child, in the hopes that the child will get ahead, but the child falls behind without the tutor.
  • A parent has aspirations for their bright, enthusiastic student, but test scores can't get past the cutoff.   In a growing number of school districts, this cutoff is now 99%.

The antidote is not more math.  The antidote is more thinking.  In the long run, which turns out to be short 3 months with TPM, depending on the beginning skill set of the child and the time commitment, it is much easier to teach your child how to learn on their own than hire tutors or fret about test scores.

The Short Road To Success
There are two hurdles to overcome with Test Prep Math.  The first hurdle is the expectation of the child.  The child is being programmed in school to expect things to come easily, because school math curriculum is designed to spoon feed math concepts with the end goal of high standardized test score based on math concepts.  Memorization and speed are valued, thinking is not.  I'm slightly amused when a child quickly reads a question from TPM and is shocked to find that he didn't understand any of it on the first try.  The child didn't expect to have to think, because the word "Math" is on the cover. Ironically, this child might be struggling in school math and have ITBS, PARC, or MAP test scores lower than expected.  More of the same is not going to help.

The second hurdle is the parent.  Parent's hate to see their children struggle.  It's painful if a child doesn't know something right away or get the answer correct from memory.  In other words, most parents in the United States value how much their children know over how much their children are learning or thinking.  Our culture values being smart much more than the process of getting there.

When I watch a child step through TPM, I silently measure the child's skill set across many dimensions. I'm more than happy to see a tiny bit of one or more skills improve by the end of the question.  I've been through this whole book with mine and other children and have seen both beginning and outcome.

Initially, there might be unbearable culture shock to many parents.  Some children react with tears of frustration when faced with a problem that they don't get immediately.  To help meet a lower pain threshold in these cases, I have a section below that describes how to take baby steps if a parent want to.  I recommend this approach at the onset for first kids, because I've found that skipping the tears saves a lot of time down the road.  There are 100 questions in this book.  It takes about 70 questions to meet its objectives, usually, so why not use the first 5 or 10 questions to ease the transition?

The short road to success is simply to know this up front and explain it to the child.  All children except for one little girl in my neighborhood react by trying everything they can to get out of it. Sometimes tears convey 'You're making me actually think and my brain hurts because it's never had to think before.  Maybe if I cry you'll back off.'  I'm OK with these tears.  Sometimes tears convey 'I used to think I was so smart because I could do easy work in seconds, but now I realize that I'm not as smart as I thought I was.'  These tears are hard for me to deal with.  From what I gather, children love TPM, some children struggle, and my kids were the worst of all, experimenting with a range of emotions to avoid having to do any thinking and trying every argument, evasive maneuver or ploy to avoid mental effort.  When coach other children, the parents have tears of joy because their child likes math so much.   I'm a great coach because I know how to deal with the worst case scenario.

Before we get to baby steps to avoid the struggle, you will need to understand the skills that will be struggled with.

The Core Learning Skills

I had the decency to provide a satisfactory explanation for these skills in the introduction of the book without the space to elaborate.  The purchase price of this book is determined by the number of pages and therefore I didn't have the luxury of going into detail.

Skill #1 - Baffled
The core learning skills are simply the list of skills a child needs to sit down with impossible work beyond their current academic knowledge that they've never seen before and probably won't get to for 2 years and make some progress on.  These skills are prerequisites for a cognitive skills test.  The skills are required for 99% on standardized tests because standardized test makers found that using the same approach of cognitive skills tests shrinks the test and makes it a better predictor of academic performance.

I spent 5 years researching the skills.  The very first experiments were blind.  I did not know the skills existed and just handed 2nd grade math to a Kindergartner who had a smattering of 1st grade math and watched what happened.  The skills are readily observable in their absence with most children.

The first skill is being comfortable working with material that is totally baffling.  It is usually baffling simply because it is new.  The approach to the 9 sections of the COGAT is to present the student 9 sections of novel problems, aka problems that have never been seen before.  After the COGAT makers found out that being familiar with the question types is good for 4 or 5 points, they released a brief paper that describes the questions with simple practice questions.  They didn't do this to level the playing field.  They did it because really bright students, those in the 90's, risk doing much worse on the COGAT, as in 50th percentile, even though they should be at the 99th percentile, without first seeing how the questions work because bright students are perfectly capable of making up their own logical rules that are the opposite of the rules of the test.

If a child has he first skill, which I call "comfortable being baffled", the child will continue with the work.  Children who don't have this skill will burst into tears and any hope of progress is thwarted. The only way to get this skill that I know is to tell the children up front that they are going to get work this is baffling. That's the whole point of learning.  If they knew the stuff, why waste time doing it?  This usually doesn't work.  So I set aside regular school math, which is typically the opposite of baffling - it's blatant spoon feeding - and proceed to assign baffling work on a regular basis day after day week after week.

While we are developing this skill, I usually have a time frame in mind, and after about 20 minutes of suppressed tears, I'll jump in and we'll figure it out together.  The best way to approach baffling work from grade school through graduate school is to put the child on a team of kids equally baffled. When they see that everyone is baffled, the tears are gone.  In a home situation, the parent has to join the team.  I have an advantage here because as the author of the question, I'll read it and announce that I have no idea what the heck I was thinking when I wrote the question and it will take me about 5 minutes to figure out what it's asking.

Skill #2 - Spend Time on the Question
The next step is the skill of spending 20 minutes trying to figure out what the question is asking. While the complete list of skills defines academic progress and high levels, this skill is the key to it all. Children who think it's normal to do some thinking on new and complicated material make progress, and children who give up after spending 30 seconds assuring themselves that they don't understand the question can not move forward.  

I started experimenting with this skill after I discovered How To Solve It by George Poyla.  He wrote this book to mentor high school teachers on teaching students how to do geometry proofs.  I translated this book into strategy for teaching 5 year olds how do COGAT questions and 2nd grade math.   Half of this book can be summarized by "read the question until you understand it".

While working with older children, I realized that this skill can be translated to how much brain power a child is willing to expend in doing their work. Most kids never experience real thinking. Math books take the child from one easy step to the next with enough examples to prevent even the laziest thinker from getting lost. By tradition, teaching methods emphasize comprehensive examples followed by routine calculations to master the skill that was spoon fed in its entirety.

When I sit down with the slightly above average child for the first time, the child is totally offended that I expect them to do a problem that requires thinking.  Sometimes, the child recognizes right away that the question is incomprehensible without a bit of work and determines that it is because the child is a dummy.  We need to get the child beyond the expectation that they are going to be told everything.

The original design of the book was simply 100 problems to break this habit and the next one.  I want my children conditioned to expect thinking and mental effort.  3 months of convoluted word problems involving multiple equations at the pace of one problem a day will do the trick.  I'll say more about this below because there is a lot more to it.

Skill #3 - Check the Work
The next skill is simply to check the work before the child has determined that the work is completed. The importance of this skill comes from a variety of sources.  The Grit version of this skill is to be comfortable getting an answer wrong.  I stressed the grit skill in introduction of TPM.   Resilience in the face of failure is a powerful life skill.  In TPM, this resilience comes quickly, and checking the work without a feeling of frustration provides a series of academic benefits beyond a child who does what it takes to succeed:
  • Children at all levels get 15% higher on tests when they check their work.  This is especially true of the ITBS.
  • All learning comes through mistakes.  If the material is easy enough to get on the first try, it's unlikely that the child learned anything.  I'm not talking about "learning" to count, as in learning to add 3 + 4.  This is not really learning.  I think I'll need to justify this statement at some point or else parents of young children will have no idea what I'm talking about.
  • Being comfortable with mistakes raises the bar on what type of material the child is working with.   Easy material has a low error rate.  More challenging material has a higher error rate.
  • Both children and their parents have been conditioned to judge performance based on accuracy, but learning doesn't take place in terms of right or wrong.  Learning takes place in the process. An enormous amount of learning takes place in between the start of the question and the answer.  Wrong answers extend the learning process.
  • Mastering math facts are an unfortunate stepping stone to high level math.  Kumon figured this out many years ago.  However, practicing math facts undoes the learning process.  To put calculation practice back into the learning process is doable if the question requires redoing the calculations 2 or 3 times or more.
All of these goals are met when a child has accepted the fact that the problem ins't finished until the answer is checked.  If the child expects to get the answer correct on the first try and bursts into tears when he gets the wrong answer, this child has some growth ahead of him.

There is an even more powerful life lesson lurking in TPM that has to do with checking the answer. I've tried to tell my children that it is much easier and quicker just to do the work slowly and carefully the first time, instead of rushing through it, getting it wrong, and having to do it over and over again. In process engineering, we call this the one-touch principle.  Children will listen to me, but in their skepticism and arrogance they usually speed through things and learn the lesson the hard way.  I watched this lesson sink in over a period of about a month until they finally got it.  They've been reaping the benefits in school and on tests ever since.   I'm not sure if "carefully and slowly" means checking each step as you go or just applying enough brain power on each step to guarantee the right answer.  I'm not going to look a gift horse in the mouth.  The results are phenomenal.

For those of you who didn't spend time on a horse range, the condition of the teeth and gums of a horse are a good indication of how it was cared for and it's overall health.  If the horse is free, it doesn't really matter what you got, because it's free and you don't want to offend the giver.  You can secretly check later.  In the case of a child who has decided it takes less time and effort to do things right the first time, it doesn't mater whether the teeth are pearly white because of lots of brushing or because of a diet rich in calcium.

All learning takes place with a foundation of these three skills.  Learning at the 99% level or learning beyond grade level is not possible in the absence of the next two skills.  It was the next two skills that paved the way for a question design that taught the first 3 skills and mastery of math facts.

Skill #4 - Working Memory
This skill is incorporated into almost every question of the COGAT.  Most kids have an adequate working memory and strong readers have pretty good working memories.  My kids had fairly normal working memories.  Two things happened that made this skill a central part of Test Prep Math.

I didn't realize how vital this skill was until I met a kid who didn't have any.  By the time he got to question mark for a 1.5 part math word problem he would have forgotten the first half of the content. It made progress impossible.  Over the course of about 4 months, as we did work on a daily basis, his working memory increased to the point where he not only retained the question, but I could throw additional questions at him before he began working on the problem and he didn't need to read the question again.  In working with this child and other children, I could see over time in each case that if we worked with material that taxed working memory, working memory would increase over time.

The second thing that impacted my esteem for working memory was middle school.  At this point, all subjects tax working memory.   Pre-algebra and algebra throw many concepts at once at the child and the only way for a child to sort these out in a relatively painless fashion is to hold the concepts in memory while they are addressed in order.  If we look at any subject, for example reading or science or social studies, children who can retain the bits and pieces of a learning experience in memory while they sort it out have a distinct advantage over other children.  Working at higher levels requirers a strong working memory, as does 99% on a cognitive skills test, as does college, which is, by the way, reserved for the top 90% of students for the best universities.

One part math questions do not exercise working memory.  Sue has 3 apples, and Joe has 4 apples. How many do they have together?  This is level 1 working memory.  There is a single problem to solve.  This problem doesn't have 3 parts, it has 1 part.  It's 3 + 4 = 7.

The COGAT generally sticks with between level 1.5 and level 2 of working memory.

A track star training for the 1500 meter race is not going to spend the week running 1500 meters each day.   He's going to run at least 3000 meters per day, probably more.  Unfortunately, getting beyond working memory level 3 in math requires writing down a little table or a set of equations, and therefore level 3 becomes the optimum level to work with in math.  Test Prep Math Level 2 starts with some level 2 working memory problems but goes to level 3 quickly and stays there.  There is no concept of gradual increase in working memory.  To get there, you need to start with level 3, and it will take 5 tries and 45 minutes at the start.  As working memory grows this will take less time and less attempts.

The spectacular achievement in working with level 3 is to train math facts without having to give a child a worksheet of math facts and undo core skills #2 and #3.  Level 3 means nothing more than 3 equations at once that have to be held in working memory and solved in order to get the answer.  This is impossible on the first try, especially in the first part of the book.  The child will come to expect the wrong answer on the first try, the question will have to be read again, working memory will be burned in a bit more, and the child will have to do 3 calculations again.  Probably again and again. There's a math facts worksheet built into every equation, and even better, it requires solving the same equations.  I've out Kumoned Kumon.

Children are reluctant to write things down during math.  They should, but for now I take advantage of this reluctance and encourage children to try to work out the math mentally.  It makes it harder, requires more attempts, and give working memory a full workout.

Skill #5 - Verbosity
I don't yet have a satisfying term for this skill.  I think verbosity covers it for now.   It means untangling a blizzard of words, but this blizzard of words holds all things thinking and all of the ways possible to think.

Test Prep Math is fairly methodical in coverage of all permutations of math problems.  I carefully charted these out in order and built this into the questions.   As I wrote and rewrote the questions, I referred to my list.  Somewhere I've got over 60 versions of word documents with names TPM0.docx, TPM1.docx, TPM2.docx etc.  If you peel back the verbiage and write out the equations, you would see this for the questions in each book:
  • a + b = c; d + e = f; c + f = g;
  • a - b = c; d + e = f; c + f = g;
  • a + b = c; d + e = f; c - f = g;
  • This continues on an on while I move around the pluses and minuses in each permutation.
  • Occasionally I incorporate multiplication and division, but only to throw off the child in the same way that the COGAT occasionally incorporates doubling and halving to throw off the test taker.
  • Then I remove one of the letters, like b or e, and replace it with a sentence that suggests what it is but doesn't state it outright.
Verbosity comes into play when the variants of the equations is turned into a word problem.  If I used the term "all together" in each problem when I meant adding in the 3rd equation, the problems would be boring to write and boring to solve.   So I created a list of all the ways I could say "all together" and used the more complicated ones.

At once point with TPM Level 3, I wondered if I needed to do this with multiplication.   The answer is no for a two reasons.  With multiplication, the problems take way longer, but it is not time spent on the core cognitive skills, and instead represents time spent on calculating. Division is even worse. The core skills are so powerful that a child should learn multiplication on their own.  There is no reason why a child can't add and subtract in their brains, which is required for the full working memory work out, but I can't say this with 3 multiplication problems for 8 to 10 year old children.

Word variants conveying the math equations is extended to achieve other goals beyond a question that takes 20 minutes to understand.  I start each book with a few simple problems, but the progression of problems carefully expands the mental effort required to read the question.

Skill #6 - Logic
Somewhere in the process of making questions complicated enough I crossed the line into another skill.  This skill is more prevalent in TPM Level 3, but neither book over does it any in any case because of the target age group.  I did my best to spoon feed introductory logic skills one baby step at a time.  At some point before the half way point of TPM Level 3, there were so many baby steps that question is nearly jogging.  When my younger child did TPM Level 3, he was too young, so went back 20 questions and did them over while I waited for his brain to catch up.

The COGAT questions are explore the child's logic abilities, especially inference, deduction, and ambiguity.  School takes a leap in skills during 4th grade or later, depending on the school, and certainly by middle school.  The leap is in logic skills.  I don't know how children can get past this leap with no preparation.   I call this the 4th grade train wreck.  Instead of a child poised to take on bigger and better subjects, the child has hit a wall and is floundering.  The primary culprit is lack of the 1st 3 foundational skills, but another prime suspect is also the lack of logic skills.

The first subskill is inference.   A few sentences point to a result needed to solve a problem, but these sentences don't explicitly state in spoon feeding terms what this result is.  A careful reading and rereading and rereading again process will point the way.  It's obvious once the child sees it, but it takes work to get there and some patience.

The second subskill is deduction.   If a implies b and b implies a, then a implies c.  I don't think TPM explicitly teaches deduction in the same way that inference is routinely used, but the 3 step nature of each question hammers away at the underlying thought process so I hope it sets the stage.  I think that's about the best that can be done with the age group.

The third subskill is ambiguity.  Ambiguity is gently introduced in TPM Level 2 and more forcefully in TPM Level 3.  I do this in the Super Bonus questions routinely so as not to demoralize my own children, let alone the children of people brave enough to put their trust in this material.  In this way, the child can get an answer to the required question without worrying about whether or not the bonus or super bonus questions are doable, which in some cases, they are not. The way that cognitive skills tests use ambiguity is to introduce a problem that has 2 possible solutions.  The child will probably not see the expected answer in the answer set, so the child tries a different approach, gets the second solution, and sees that in the answer set.  The way I do this is a Super Bonus question that is to present an almost almost unsolvable without 'inferring' a continuum of possibilities from the question and realizing that there is no solution at all.  In other words, the child has to use their imagination.

The Super Bonus questions achieve other objectives that may or may not list fully.  The Bonus Questions and Super Bonus Questions are first a bonus for the child, but when I created the titles I was thinking this is a bonus for the parent.  One objective was to avoid lame, boring math that is predictable and boring in its boring boringness.   My older son put himself in charge of cutting lame questions from the book.  I had to start adding bonus and super bonus questions so that my hard work didn't end up on the chopping block.   My next objective was to demonstrate how much imagination goes into math, and how debatable it is.  This may surprise you, but there are papers that rethink counting numbers.  The older child was on the verge of higher order challenges in school.  During 4th grade, we used TPM Level 3 to catch up on his base cognitive skills, but he was brining home assignments at the 5th and 6th grade level that he was totally unprepared for.  The bonus questions contain little starter bits of higher order challenges, especially addressing reading skills. For example, he was expected to figure out what was happening in an assigned book, and he had no clue, and no one, including his current teacher, ever told him that he was expected to do this.  So bonus questions call out a ridiculously goofy scenario that the child totally missed when they were deriving the 3 equations.

I was also thinking that I didn't want to a child to ever trust their answer again, not even after 20 minutes.  How better to make checking the answer a normal part of the math process than to make answers impossible or nonsensical.  I only did this in the bonus and super bonus questions.  At first, I thought I overdid it.  These turned out to be the most fun in our At Home math.  Then the mother of one of my testers reported that her daughter loved the idea of a math question that didn't have an answer or could be any answer.  I was sold and the problems weren't cut from the book.  These are greatly scaled back in TPM Level 2, but all heck breaks loose on some questions in TPM Level 3.  I spell this out as best as I could in the solutions.  For the parent who reads the solutions and wonders what I'm doing, the answer is trying to create a genius out of your child.  This is one of the ridiculously insane experiments that paid off in a big way.  The ridiculously insane experiments that failed, like competitive math and brain teasers, are not part of TPM.

Back to verbosity.  The verbose nature of every question is a blatant mechanism to change math from a 15 second math fact exercise to a 20 minute thinking exercise.  A question can't require 20 minutes to understand if it doesn't require 20 minutes to understand.  With one problem a day, is it possible that this type of verbal exercise is going to have a positive impact on reading and reading comprehension test scores?  The jury is still out.  Scientific research doesn't jump to conclusions even if the conclusion is logically obvious.   What I've determined thus far is that TPM improves verbal fluidity and the certain basic skills of tackling reading but reading comprehension tests add a list of additional reading skills that still need to be practiced and aren't covered in TPM.  I started working through these in December of 2016 and will have a thorough list in a few years.  These additional reading comprehension skills are all beyond 4th grade, even for gifted children.

I should explain 2 of my failed experiments.  Before I wrote the questions for TPM, I tried competitive math and brain teasers.  Competitive math was a failure for us, even with children 1 or 2 years ahead in math and already at the high end of the test score scale.   It's simply too much math. It requires a child who loves math just for the sake of math. Even worse, competitive math is a nonstarter without the fundamental skills that TPM is designed to impart.  Brain teasers were somewhat fun in their silliness, but cognitive skill development was not possible because the answer usually could not be derived from the content of the question.   TPM includes some of the silliness but the answer (or lack of answer in the case of certain Super Bonus questions) can be determined from the content of each question.

Making It Fun and Easy

Putting It All Together
TPM buries 3 equations into a verbose word problem.  As the book progresses, and the child picks up some skills, the verbosity begins to include some basic elements of logic, especially inference and ambiguity.   The result is a question that takes a long time to read to understand.  The question has 3 equations, and this makes it very unlikely that the first attempt will result in a correct answer. 

As the child gains figuring out skills and builds their working memory, variations in the words and the subtle introduction in inference prevent the case where a child will spend less time and less attempts to get a correct answer until later in the book.  For example, the difficulty in the terms use and the appearance of inference make it likely that children will solve the answer and get it wrong because they skipped a word or missed a subtle hint in the question.

This cohesive bundle of skills will be a challenge for some parents.  There are a variety of ways to make work less daunting.

Pulling It All Apart
With my own children, I prefer a heavy handed approach of sitting on the iPad until the child has finished a page in TPM.  If you take this approach, this is what you might encounter:
  • The child will complain that they don't understand the problem and ask you to explain it. Instead you tell them to read it again, followed by read it again, followed by explain it to me, followed by explain it to me one word at a time.
  • The child will yell at you for being a mean, unhelpful parent.
This approach requires the child to build all of the skills at once, undoing years of spoon-feeding, and is quite painful.  Alternatively, this approach requires the parent to behave in the opposite way of their behavior from the birth of the child until now.  For some parent-child teams, this is too much all at once.

Approach #1 - Just Go Slow
It turns out that there are baby steps you can take to avoid the tears.  One of my readers once called me "Tiger Dad".  I think I'm the opposite of that because I'm so undemanding.  We have a learning explosion in this house without me caring about perfection, correct answers, or finishing the work. There's always tomorrow, and if I just saw 30 minutes of learning taking place, we're done for today.

I went from zero progress and lots of tears to lots of progress and zero tears just by slowing things down.  Since it would be about 20 problems until the child could work at all independently, I broke things into stages:

  • Stage 1 - We read the problem together and talk through each little bit of it.  I'm doing some of the work, but my primary job as parent is to watch the clock so that I drag things out to at least the 20 mark before we're ready to start calculating.  I ask questions about each sentence and each word, suggest silly and incorrect answers, and just enjoy sitting their with my child.
  • Stage 2 - The child gets 5 minutes to read the question first, and then we proceed as in stage 1.
  • Stage 3 - The child has to do most of the work.  They read the question, and then come to me because they don't understand it.  I make them read it again, then read it to me outloud.  In this stage, I'm doing less helping and more questioning on key words and overall flow of the problem.
  • Stage 4 - I give the child a full 20 minutes to work the question and then they have to come to me to prove they understand it.
  • Stage 5 - I just wait until the child is finished and tell them they are wrong and try again.  In this stage, I start explaining how much easier it is to do the problem right the first time.  I might even suggest they write down questions as they go.
In other words, as a parent who just bought TPM, I'm committed to a long learning process with a very big payoff, instead of a short, painless non-learning process with little payoff.  I don't want a kid who can get an A in 3rd grade math because they have memorized their math facts, I want a child who goes to Stanford for a graduate joint degree in biochemistryengineeringlaw, or at least a child who gets 99% on standardized tests for the next few years and doesn't need help with their homework.
By the way, going slow is the key to gifted and talented.  A worksheet with 30 routine math problems decimates the child's skill set, whereas one big thinking problem endows the child with learning super powers.  Going slow is required in the latter case, but not the former.

Approach #2 - Baby Steps

The end goal of each question is usually to uncover 3 equations and solve them.  If your child can uncover one equation, and acknowledge that 13 + 8 = 21, then he made some progress.  If you have to help by reading the question, appearing baffled and in need of some help, and work with your child to get that 13 + 8 out of the text, and this process just took 25 minutes, you just made progress.   Start the rest of the question tomorrow.  How long will it take your child to get to the point where they can do a whole question on their own, with multiple attempts at the answer, and not show any frustration? I can't answer that.  Some times its immediate, and some times it's not.  It doesn't matter, because it's not a race.  I can say this because I've witnessed it happen many, many times, and in the end it just magically happens much faster than you would expect when you're only on question 5 and you are dealing with tears.

You can start by doing one part of one question in 10 or 15 minutes, and increase from there.  That's the message of The Homework Trap by Kenneth Goldberg, a book I both agree with and don't agree with. Keep in mind the goal of TPM is a child who thinks on their own, so behave accordingly. Always give your child progressively more time to think on their own before you contribute.  In the meantime, contribute with questions and learn to present the facade that you are the dumbest parent in the world before you provide useful help.

Approach #3 - Stop Caring
I've tried this and it works.  It sound crazy but it really works.

In many articles, I was stress the importance of changing the home culture from right or wrong to learning.  Right/wrong and learning are opposed to each other.   Normally, we check answers and do the work over as part of the session if we have time, or we do it over the next time.  I stopped checking answers.  It's been a few years since I last did this, so I did it again in the last month just to see the magic again.

I noticed that my children started to pick up on things they did wrong from the previous time.  Or I would ask, "Can you check if you got the answer correct?"  Sometimes they complain that they can't because I ripped out the solutions, or with material that they don't think has solutions because I very carefully ripped out the solutions, they'll just do the work over.

The difference when I'm not checking answers is that there is no clear measure for them to determine that they've succeeded in earning computer time.  The iron clad rule in this house is no math, no computer.   If I assign a page, they know that 5 correct answers earn computer time, and they've learned to work very carefully and efficiently because mistakes will turn a 30 minute math session into an hour or more.  You'd think that it would only take 10 minutes to redo 2 problems, but if they make a mistake when I check, I start asking questions about the work and time goes out the window.

When they bring their work to me and say "I'm finished can I play Overwatch or some other inappropriate video game?"  If the right/wrong factor is not present, I'll just look at the work skeptically and ask them to prove that they really earned computer time.  This is the math version of the Word Board at in it's most powerful form.  They have to justify their learning achievement for the day.  If they can prove to me that they learned something or understand how the problem works, the discussion progresses smoothly.  Otherwise, I'll start checking answers with raised eyebrows and they're stuck there for another 30 minutes.

Can you imaging your child explaining what is challenging or unusual in a math problem and how they derived the solution?  This is a major theme of the Common Core, and parents of older children warned me that this is the big challenge on the PARC exam.  Look at Skill #5 above and look at some of the questions in TPM.  It is unlikely that my children will major in math in college.  There are way too many exciting fields to study, some that use math, some that don't.   In all cases, even with math, the child has to write, expound, justify, explain, and verbalize.  After we got through TPM, writing, expounding, justification, explaining and verbalization hit new levels.  It should be obvious why.  The discussions about whether or not they had a good learning experience really helped.

At this point, you're probably thinking that I'm the Yoda of academic parenting.   The truth is that I was a total disaster, the opposite of everything I recommend, when I started taking an interest in At Home schooling.   For those of you unfamiliar with my term, it means doing some academic work on the side at home a few times a week.  It's like after school Home Schooling, and it's most common form is doing next year's math at home.  It takes a solid year for a parent to change their own attitudes and behaviors.  The original and current introduction of both TPM books has this as a running theme.

Approach #4 -  Problems Solving Skills
This is a whole topic on it's own, in addition to being a great fallback plan for a struggling parent-child teach.

Problem Solving Skills
The introduction lists math problem solving skills.  These are very powerful ways a child can teach herself math.  These skills are like a tow truck when you're car ran off the high way into a ditch full of mud.  You will probably need to use these very early in the book.

Problem Solving Technique #1 - Draw A Picture
There are two ways to look at this skill.

First, drawing the problem gives the child something to do when they are stuck on the question.   It's better than sitting there crying.  If the child can't understand the question, make her draw it.  It's like a magic elixir potion to drink that makes her smarter, only the trick is that it's just colored water.   While she's trying to draw a picture of the problem, she's actually just taking the time to understand the problem.  She should have done this to begin with, but she has expectations that it should go quickly but is taking forever.  Kids know that drawing takes a long time, especially younger children, and now the pressure is off.

In higher level math, the picture provides insight into the nature of the problem and provides different ways of viewing the relationships that will uncover a different path to the solution.  In lower level math, aka grades 2 to 4, drawing a picture just helps organize the problem before solving it.

The second way to view drawing a picture is that it is blatant cheating.  Take a 4 year old trying to learn 3 + 7 = ?.  If the child draws 3 circles and 7 circles and counts them, she avoided having to learn adding and is back to counting.   Is this good?  Well, yes, the child just took over the learning process and is now calling the shots on how she progresses.  Does she move to counting on fingers next? Will she start drawing 6 as two groups of 3, and then start drawing 7 as 2 groups of 3 plus 1?  It's up to her. Real learning will take place with 100% certainty.

With TPM, the challenge is determining what the first equation is.  Then determining the second equation is usually harder after about problem 20.  Finally, hardest of all is working out the third equation, which uses the first two results.  Drawing the problem is not cheating.  Drawing postpones progress in working memory, but as I've said before, I think there are at least 30 extra questions in TPM so the child has room to get up to speed on their own terms.  What I mean by this is that a child who starts out question #1 with the direct approach - solving it on their own even though it takes 2 hours - will get to about question 60 or 70 and will have met all of the goals of the book.  Since this is extremely unlikely, the book has 100 word problems.

The great thing about drawing the picture is that the child is keenly aware of the cost.  All of that pencil movement is taxing and the 20 or 30 minute session is penalized by 10 minutes.  As a process engineer, I know the overall time will be shorter because rework will be eliminated.  I've actually told my children that over and over but they had to learn it the hard way.  In any event, the child will eventually tire of drawing the picture and solve the problems mentally at some point, re-engaging working memory.

Problem Solving Technique #2 - Problem Decomposition
Math problems can be hard.  They are hard because a good math problem is really multiple math problems all rolled up into one big problem.  By 5th or 6th grade, all math problems will be like this. Later in middle school, all reading, social studies, and science work will be like this as well.

This problem solving technique is not really going to help your child because every single problem needs it and the book is nothing if not a course in problem decomposition.  The 'Baby Steps' approach directly addresses problem decomposition.

Problem Solving Technique #3 - Do An Easier Problem
This is the most powerful of all problem solving techniques and the most powerful of all learning superpowers.   If the problem is too hard, rewrite the problem as an easier, solve the easier problem, and then once you get it, do the harder problem.

I use this technique to teach any new topic in math.  For example, a child who sees double digit addition for the first time as '36 + 43' and has no clue what to do can start with '10 + 10', '20 + 30', '22 + 30' and just work their way up to '36 + 43', learning the whole time.  Why wouldn't you just start the child with something simple and spoon-feed them up to '36 + 43'?  Because they won't learn how to learn, they won't learn how to take leaps, earn grit skills, or become an academic powerhouse on autopilot to success.  They'll just be stuck at average and you'll need to hire a tutor.

There are a few cases in TPM where this skill comes in handy.  If the arithmetic is too cumbersome and is getting in the way of a right answer, change all of the numbers from 13 and 7 to 2 and 1.  This is a great exercise for understanding the problem in the same way drawing a picture is, but it takes some practice for the child to understand what they are doing.  It doesn't come magically.  If the inference is too challenging, remove it, solve the problem without it, and then put it back.

I don't think the other problem solving techniques apply to this age group in the context of TPM.  I mention them in the book but they're not really appropriate for this age group.  The 4th technique is to just guess and see if you got the answer correct.  This is not a good lesson for the age group and not a good way to induce learning.  It's great for doctoral work, but not fundamental cognitive skill training.  The 5th technique is to work the problem from different angles, such as backward.  Training in both start-to-finish logical flow and problem decomposition are required to successfully work a problem backwards, so technique #5 is premature.  On the other hand, both of these techniques are fair game for Section 2, but the child will naturally see where they might be useful so an exposition for the parent is not necessary.

Section 2
Section 2 was not in the first edition of the book.  It didn't even occur to me that this type of material was possible for a child.   A variety factors fell into place all at once that resulted in a perfect storm of awesomeness, and level 2 was born.  Here they are, in order:

  • We did TPM during the summer, and by about October I was out of math to do.  I have a big gap to fill between useless counter productive 2nd to 4th grade math and the more thinking oriented middle school math.  The school switched to a revolutionary math program that I had been tracking for 5 years, and I was disappointed to find that it fell short on mechanics.  Plus, the verbal and thinking content was at a level beneath TPM.
  • My younger child did math phonics at age 4, and this wasn't even invented when the older child was that age.  If I didn't do something drastic, my 8 year old was going to catch up and surpass his older brother.  I needed a math phonics course for the older child or I would spend the rest of my life feeling like I'm a bad parent. 
  • Because of the word problems in TPM, I've got children willing to struggle through something bigger and better.
  • Many readers were reporting that their children were not getting past the cutoff score on the math ITBS and COGAT to get qualify for the gifted and talented program.  I wanted to recommend that TPM is good for at least a 99%, but I wanted an insurance policy especially for desperate parents short on time.
The question design starts with the same basic format of the word problems without the words.  For TMP Level 2, most of these are of the 2 step nature.  The goal of Section 2 is identical, a problem that takes 20 minutes, most likely results in the wrong answer on the first try, and give working memory a work out.

The equation format is best practices for cognitive skills test design.  Since children are conditioned to think in left to right terms, as in 3 + 4 = ?, cognitive skills start by throwing in an extra term to identify performers, as in 3 + 4 = ? + 7.  Then the tests extend the thinking to an additional question part, which basically amounts to 3 + 4 = x + 7, x - 3 = ?  Like the track star in the 1500, this isn't going to get us to the finish line for the COGAT, so I extend this to a third step, and thus hit the limit of working memory as a bonus.  Level 2 takes its time getting to 3 steps.

Between the first problem in TPM2 and the last problem in TPM3, the secret door to abstract thinking is magically opened.   You won't appreciate this right away, but y = mx + b is in your future, as is ax2 + bx + c.  It turns out that between 132 x 73 and algebra, abstract thinking will show up more than you think.  It shows up as soon as Suzie has more than 3 apples, which is just a way of saying that she has 'x' apples.

In this context, where is logic?  Logic shows up when F is replaced with the cursive F, which is not F, as in the opposite of F.  It's not algebra, which is way beyond 8 to 10 year olds.  It's logic.

These questions are quite ambitions and TPM clearly states that progress with the core skills in Section 1 is the prerequisite of Section 2.  Readers who defied my warning were rewarded with tears. The child will actually learn to navigate the tricky word problems despite my best efforts to keep the questions at a challenging level, and then find that Section 2 requires starting from scratch.  I can guarantee that a child won't be able to do the first question, will struggle as the questions get harder, and will grow way beyond their current skill level by the end of the book.  Or sooner.  Some braniacs will actually do well on the first problem.  These kids are like some of the braniacs in the program my kids are in, and I didn't write the book for them.  I wrote it for everyone else.

More Advanced Skills

I have the privilege of watching graduates of TPM in action in later grades and with more advanced classes.  There's definitely a limit to the skills that an 8 to 10 year old can master.  It's the base of the pyramid, and a parent or teacher can't just throw a fully developed pyramid block on top of air.  Each of the skills grows both independently at first, but once the skills are present with sufficient strength, much greater things are possible.

When working memory is established, and the child becomes comfortable working with unknown and confusing material over a long period of time, the door is opened on the next level.  At an older age, these skills congeal into something more powerful that a younger child can handle.  I tried this higher level work before TPM and it doesn't result in a satisfying learning experience.  After TPM, it's quite rewarding to witness.  It takes way to long to get the basic skill set down when the child has to master additional content or addition skills.   I think Kumon knows this.  My basic disagreement with the Kumon approach is that the child is wasting time on the wrong skill set.

This is why competitive math is so disappointing when presented at the wrong time.  Competitive math has really challenging problems, but it also carries new and advanced math material.  That's too much at once when you only have half a skill pyramid.

The difference in learning after age 10 with a solid foundation is that the child can handle the basic process of bafflement, spending a lot of time figuring things out, holding a list of unknown concepts while they are learned and sorted out, and struggling with repeated attempts to crack the mystery, the basic skills behind TMP, over a period of weeks and months.

It's really hard to make progress with an 8 year old if the lesson can't take place start to finish in one sitting.   Even when 8 year old's are struggling with the first problems in TMP, we 'finish' it in the same day and I don't bother to tell them they got the answer wrong if they just spent 35 minutes getting there.   I'll just wait a few days and a few more problems and then we'll go back and do it again.  By age 10, it's less likely we won't get through the problem in one sitting, but I won't push the next level until working memory is finished and the core skills are there.

What I'm going to present next demonstrates all of the core skills at play, plus long term memory, over a longer period.  Note that longer term memory is a feature of a good phonics course (aka Test Prep Phonics) and Vocabulary Workshop, which I can't recommend enough.  Long term comes from reading.  I don't really see the opportunity to build let alone use long term memory with math before high school.

First, back to competitive math.  If you hand a 4th or 5th grader a competitive math test, which you can download for free online or spend $75 to purchase in book form, the child will see a list of really hard problems requiring the problem solving techniques, and math for 2 or 3 years in the child's future peppering the problems.   A problem may not be a one day activity.   I assigned my younger son a single 19 question competitive math test and it took him 3 weeks to finish.   The instructions clearly state that the time limit is 45 minutes, but it's a 5th grade test and he's only in 3rd grade.  One of the middle school children I coach had never seen some of the math concepts in this test.

I'm not really all that impressed.  Math is the least challenging part of a gifted program (well, after TPM it is anyway) and only a small part of all of the learning to come.  Maybe the most important part, but the smallest.  Reading and science are much more daunting.  Maybe science is the most challenging of all, because it's math, history, and reading combined.

We started the 7th Grade Project the summer after 5th grade.   I am going to have a whole page dedicated to this experiment on my list of permanent page some day, but I had to unpublish the page. Typing as I go is not good for clarity and I don't want to steer anyone in the wrong direction.   During this effort, I'm witnessing the next level of basic skills.  [I've run out of time so I'll have to finish this later, but it's way more amazing than you can imagine, especially if you're just trying to get past 4th grade right now.]

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