**Copyright 2013 - 2017, me**

Introduction

My first experience teaching math was a summer camp I created for soon-to-be Kindergarten kids. I called it 'Math Stars'. Each Saturday morning, I threw random topics at them from arithmetic, Roman Numerals, Venn Diagrams, and big complicated word problems. My contribution was letting them work through the material, have fun, and eat snacks. Every wonder what kids are capable of? More than anyone would imagine. Each got a copy of Every Day Math Grade 1 to do at home, and parents expressed shock and awe that their children did the work. Without being asked.
I was surprised in November by a teacher who suggested we take what is called 'a gifted test'. What parent doesn't want their child to be gifted? At the time, the only websites I could find on this topic provided this advice: Get a good night sleep and do nothing to prepare. So I created my own material. It was insanely hard and mostly nonsense. I was right on target, although it would take me 3 years of reasearch to figure out why.

A cursory glance at cognitive skills tests is very enticing. These tests are designed to measure the thinking and problem solving skills that predict academic success. The test scores are highly correlated with academic success. I want my children to have these skills. Math seems like an ideal place to learn and exersize these skills, even the analytic verbal skills, verbal fluidity and conceptual vocabulary.

Easier said than done. My research went off in a variety of unpromising directions (education literature is useless, intelligence is a myth, most cognitive skills researchers have never met an actual child) until I came across George Poyla's brilliant work 'How to Solve It'. His audience was teachers of high school geometry students and his target was geometry proofs. His research included 3,000 years of mathameticians asking if there's any science behind how they solve problems. It took me about a year to apply his methods to 4 year olds. It looks a little different, but it works.

While I never met a math I didn't like, I don't expect the same for my kids. In fact, there are a few really good books written by parent/teachers who spent their careers in gifted and talented programs. They both state in strong terms, from experience, the fields that you encourage your children to pursue, those that the parent shows interest in, will be the ones your children definately don't want anything to do with because you are lame. If you want your budding astronaut to become an actuary instead, send your child to space camp.

Therefore, I'm going to teach problem solving, thinking, creativity, how to have a discussion while sorting out confusing concepts, but I'm not going to teach math. It just so happens that we're doing math, but the child will be in charge of learning it on his own. I'm just there to teach the other stuff. There are quite a few kids marching through math training programs being trained to do complicated math; this short cuts learning and is a train wreck waiting to happen.

I committed to a principle early on and have stuck with it ever since. Children learn to become strong readers because they read daily. It seemed logical that to become a strong mathametician, a child should do some daily math. I've modified this over the years to accomodate homework and it has finally morphed into the first law of Math House: 'No math no computer'. This works for kids who are approaching their teen years. Other variants include 'No math no fun activities of any kind, desert, TV, laughing, or anything', 'You have to do you math 1st thing every Saturday morning', 'OK, you can do vocabulary first', and 'Get your math done because I said so'.

Principle #2 is simply whatever you do on a daily basis, you will get good at. Eventually. Even if it's taken you 3 weeks to get through the first page of the totally inappropriate math workbook I gave you. In fact, if it takes three weeks to come to some rudimentary understanding of the material, I can be confident that a boat load of learning happened along the way and grit skills are growing by leaps and bounds as a bonus.

The Right Age To Begin

The rest of this article walks through math age by age. The best time to begin is whenever you stumble across my blog and think 'I could be doing more'. Good times to start include pre-K, K, 1st, 2nd, and 3rd grade. During this period, math curriculum generally stinks and its the perfect opportunity to focus on cognitive skills and let math take care of itself. During 4th grade, useful well crafted math of all types is avaiable. For this reason, I like to concentrate my work on 3rd grade and under and just point the direction after that.
There are some important milestones in brain development (actual age may vary). Math curriculum in the US has its own quirks. If you put these two things together, there are some great opportunities to nearly permanent impart cognitive skills in your child at the right time.

- During the process of learning to read (aka phonics), 100% of cognitive skills are in play. This will never happen again.
- In Pre-K, the child's ability to pick up conceptual vocabulary is at its peak. You can cover math concepts through 2nd grade.
- The best math book of any type is Sylvan's Kindergarten Math. This is the last time cognitive skills will be tightly coupled with math curriculum. You won't see decent math again until middle school.
- Because of the slow start for US math curriculum, combined with a child's naivete towards it's parents, a Kindergarten can actually lick most of Every Day Math Grade 2 during Kindergarten. This is more of an exersize is executive skills and grit than math, but it's a heck of a start if you can pull it off.
- First graders are no less capable than Kindergarteners, but they have more work, less sleep, and 3rd grade curriculum is the low point for math in terms of it's boringness and uselessness. If you got off to a strong start earlier, you can take all of first grade off. If not, you should look toward an alternate curriculum. Keep in mind that some publishers put out grade level books that are already 2 years ahead of US curriculum. Going up even a single year could be a disaster. You will know that you have the wrong level if you are trying to teach your 2nd grader algebra because your 4th grade workbook is at the level of 6th grade or 7th grade. I'll discuss this more in the 1st and 2nd grade sections below.
- Second and third grade should be a tour de force of thinking, thinking, thinking. This is a great time to tackle logic, complicated analysis, and the other forces underlying future math. Math facts, decimals and long division are the opposite of math, so avoid them at all costs. This will pay off down the road.

Let me share a word of caution. If your goal is simply to pick the right book or after school program so that your child knows what to do on the MAP test or ITBS that is 6 weeks away, this article is not for you. My advice is consistent with a strong performance on the COGAT, and I expect no less than 99% on both sections of the MAP within a year or two of targetting thinking skills, provided your child bothers to read the reading comprehension passages instead of just answer the questions.

My goal is a child who teaches himself advanced math at a high level. If you want a math genius, spend every minute working on learning skills and stop worrying that your child keeps getting the wrong answer. Math knowledge will take care of itself.

In the rest of this article, I'm going to use the terms "school math" and "test math". School math is boring spoon-fed concepts like arithmetic, decimals, and anything you see on a curriculum page like this page from IXL for 2nd grade. "Test Math" involves problem solving and critical thinking. The reason I like the MAP test so much is that it is chock full of "Test Math". Before second grade, being ahead in math provides a huge advantage, but when your 5th grade child comes home from the MAP test and asks you what "sine" and "cosine" mean, you'll know that the test crossed the boundary over into thinking and problem solving and your child thought and problem solved into trigonometry. It starts in 2nd grade when kids near 99% have to deal with exponents. Teaching exponents and trig is not going to work. Teaching problem solving will.

Age 3 - Beyond Letters and Numbers

At age 3, my goals for both math and reading are very simple. I introduce letter sounds and numbers to see if anything sticks. It took all year to get from C-A-T to "cat", and to introduce counting to 20. We spent all year reading stacks and stacks of picture books. A solid math foundation includes learning to read. By the way, learning to read involves 100% of cognitive skills, including the math related skills. I am personally offended by children who are adept math learners at school simply because they read a lot before school. It seems like cheating. You want a strong mathematician by middle school? Create a strong reader.
My only goals during this time was to gauge when the brain had developed those sections that understood the basics of reading and math. In our case, it was always an average time. If I read Welcome To Your Child's Brain or The Read Aloud Handbook I would have done lots more picture book reading, lots more talking with big words, starting at age 2 and I might have seen a brain ready to go earlier.

The most important math skill for 3 and 4 year olds is the ability to see their world, to see it in detail, and compare and contrast what they see. These skills are at the base of the pyramid. There are a number of craft books that involve basic folding, cutting and pasting that contribute to this ability and are fun for 3 year olds. The best part of these books is that it's a pleasant way to help the child develop Executive Skills while they sit alone working on projects.

Age 4 - The Critical Age

**Test Math - Age 4 years 0 months**

This is the most critical time of learning and the most underserved and ignored phase of development. If you are scrolling through this list on your way to a later grade, I promise we're going to catch up big time for everything you missed. But if you have a child in the vicinity of 4 years 0 months, you have an opportunity to get out in front from the start. As a reminder, when I say 'Test Math', I'm thinking of the skills that are foundational academic success. Adding and subtracting aren't on the list.

I already mentioned above that the process of learning to read exercises 100% of conitive skills and this fulcrum of learning is never going to happen again. The question becomes how to maximize this process and eek out every last little iota of benefit.

The key to math this age is vocabulary. Vocabulary is therafter 75% of everything. Vocabulary is 75% of math, writing, school, cognitive skills tests, thinking. Vocabulary happens to be 100% of vocabulary, obviously, but it is also 99% of math before first grade. The other 1% is learning to count. To illustrate this point, think of the words 'big' and 'small' to differentiate two objects. That's not just a tool for the brain to use, it's a tool that sprus thinking at a variety of levels. Now think of the words 'narrow', 'wider', 'tall', 'half'. The brain that just picked up these new words has a lot of work to do both internally and exploring her world. That, in a nutshell, is the relationship between cognitive processes and vocabulary. Vocabulary doesn't accomodate thinking at a higher level, it

*creates*thinking at a higher level.

To prepare for a quick head start in math, I created Shape Size Color Count. The goal of this book is to provide math vocabulary through 2nd grade to a 4 year old, and to endow the child with the ability to visualize numbers and visualize number operations. The target child is somewhere between 3.9 and 4.4 or 4.5 years old. Parents who have multiple PhD's and an extensive home learning environment from birth have reported that it's not hard when they started at 4.5. Parents with kids in the high 90's on COGAT/MAP have complained that it's really hard to get little brother at age 4.1 past the first few pages. It breaks my heart when a child picks up the book too late in the development cycle, but I'm really excited when I hear 'struggle' because big things are on their way.

Both sections are inspired by the COGAT. Section 2 should be an obvious attempt to prepare for number and figure analogies. Section 1 is actually much more important because it is much more vocabulary rich and applicable to other sections. It shouldn't surprise anyone that I had another COGAT in our future. I am enamored with the concept that there are fundamental skills that underpin academic success. This book directly targets those skills that can't be gained by reading Bereinstein Bear. It does so quite efficently, although I can't guarantee that a 4 year old will complete all of the excersizes in 90 days. When I work with 4 year olds, I expect maybe 3 or 4 good days each week when they are not hungry, tired, sick or otherwise not able to sit up and think at a high level. I also expect to see the brain tire out at about the 15 minute mark. Some weeks are better than others.

The first part of this book is for parent and child to talk through the problems, questions and answers. The more talking the better. In my permanent pages I list suggested "solutions" for the first 40 problems to give you an idea of how to unlock the dialogue. It's not about the answers, it's about noticing and talking about what you see. Seeing skills will develop during this discussion; seeing skills are the key, not the answers. Cognitive skills tests before 2nd grade differentiate children based on what they see or don't see. Keep in mind for the long term as you develop your discussion skills and learn to ask questions is that our pre-algebra, algebra, geometry, trig, and calculus work is heavily discussion oriented. I ask a lot of questions, but when I state or demonstrate something, I usually cut 2 months off our timeline for new material. When I present new material and instead of teach it I ask the child to explain it to me, we cut 10 months off the timeline. Before 7th grade, I picked up a rigorous Chemistry book and said, 'Why don't you knock off all of the concepts in the first 3rd of the book to get a head start.' It's not exactly 'Whole Language Math', although sometimes I refer to it by that term, but it's more 'A Whole Lot Of Language Math'. It starts here.

Keeping in mind the importance of the vocabulary section, I was taken by surprise by the huge impact of the quantitative exercises. Appearently this material must line up with brain development because the result is a visual numerical sense that is stunning and unlikely to be developed at a later time. I noticed early on my child was assigning personalities to the shapes in each diagram and seeing who showed up or who was missing or who changed in the next box. I didn't think was important with the animals, but it happened with the shapes as well. By the time were working full matrices on practice tests, he was seeing '4' and '6' and the difference of '2' as concepts in one shot instead of counting. As predicted by cognitive skills research, this temporary gap was not at all temporary and has continued to widen. When I talk about catching up below for parents who start late to the game, I've got a strategy for picking up other skills to reach 99%. Kids who get an early start with number sense are like really early readers. You'll never get the same skillset and you can't catch someone who is accelerating.

Take a look at this figure matrix.

The problem we immediately encountered is that a 4 year old had no idea what to do. Maybe at 4.5, but not at 4.0. Try explaining to a 4 year old that the transformation in the top row is 'plus 1', please apply the transformation to the bottom row and 3 + 1 = 4. The answer is four. The result is officially called 'deer in the headlights', as in a deer on the highway stairing in the dark mezmorized by headlights as it gets run over.

So I switched to something we could work with.

Now we've got a workable problem. In the first box some cats are at a party. What changed? A new cat showed up. It turns out that kids have no problem picking up +1, +2, 0, -1, and -2. Yes, I read the answer choice as "minus 1" and kids just go with the flow. "Minus 1" to them means one cat left, or there's one less box. Is this going to help when they are 8 and I ask "What is 2 minus 5" for the first time at about age 7? No. But they'll know negative numbers in the next few days.

By the way, Shape Size Color Count has a partner called Pre-K Phonics Conceptual Vocabulary and Thinking. It does for general vocabulary what SSCC does for math vocabulary. It has most phonemes through 2nd grade and vocabulary is lined up to maximize relationships by the end of the book especially synonyms which hold a much higher cognitive payload than opposites. It was our only phonics book and put us into second grade chapter books by Kindergarten. I want to warn readers what to expect. We had a stack of library books of all types on hand and referred to this phonics book 4 or 5 times a week. It's more like a graduate text book of phonics than what's on the market today. It has no pictures or color. I patterned it on phonics books from the early 1980's before they were watered down. It was hard at first, and when it became fun, it was because the words themselves came alive. Our district has cutoff scores above 98.8% and this book is under the heading 'take no chances'.

**School Math - Age 4 years 6 months**

Age 4 is the happy time because school math and test math are very similar. Math curriculum for Kindergarten is full of vocabulary and thinking skills. My favorite math book for any age is Sylvan's Grade K Super Math Success.K Math is a great companion for Shape Size Color Count. K Math can be done on bad days, or when the parent is busy, because it's easy. SSCC is parent intensive, and Sylvan is a nice break while the child learns to hold a pencil. Maybe this is why I liked the book so much.

The governing premise at this age is 'everyday'. If reading every day builds an expert reader, how are you going to build an expert mathematician? The answer is a little math every day. If your child is tired and hungry and not capable of forming a coherent thought, find some easy math and build the routine. I think of K math as establishing the routine. In addition to the primary workbooks, I kept 2 other ones that were easier, or had more appealing content like mazes and puzzles. Doing super challenging advanced math homework everyday can be frustrating for a little kid, and when he was most frustrated, or simply refused to do his math, I would lay out three books and give him a choice of which one to do. This always cheered him up, and he would always choose the easiest one. Easy math is better than no math. Unfortunately, being sick or tired happens every 3rd day on average and illness or lack of sleep can set a kid back 2 years mentally. Parents aren't immediately in tune with this. The backup workbooks addressed this situation and give both parent and child an easy out.

**Test Math**

While other years might have only 1 Test Math section, this year has 2. I am convinced that the best 'next step' to learning arithmetic or quantitative math at this age is the COGAT practice tests and critical thinking material. I would consider this visual math, and while we did these, I witnessed a range of problem solving skills developing. The problem with practice tests is that it's too big of a step for children at age 4. Many parents have reported to me that their child gets all of the questions wrong. That's why started with SSCC, mentioned above. It's much more rewarding and productive to do a practice test with all of the starter skills first, and SSCC is all starter skills. If you missed the cutoff for SSCC, start in with a more standard critical thinking book first before tackling a practice test.

I recommend Building Thinking Skills Primary. We found the non-verbal half to be highly doable, but the verbal half is more true to the grade level printed on the cover. If you are facing the OLSAT, COGAT, or NNAT, the test prep process for age 4 is going to be a big investement in academic skills. There are quite a few books on the market for this age range, most of them stamped 'K-1st Grade'.

**School Math - End of Pre-K**After the test, we went back to school math in order to have something to do and keep our skills sharp. More specifically, my son will keep his math skills sharp, and I'll keep my "not getting aggravated because he can't add" skills sharp. I picked Spectrum Math 1st Grade. It's on par with Every Day Math, but instead of building a strong number sense, Spectrum just skips along much more quickly and spends much more time on arithmetic. Spectrum is more of the 'math skill' list and not the 'core thinking skill list' which is my target. It makes up for being not as well designed by being twice as long. There are far more arithmetic problems in the Spectrum workbook than Every Day Math. It would make a great companion to Every Day Math for a First Grader who is struggling to learn arithmetic.

As I mention repeatedly on my blog, there is a big disclaimer with this work. A older 4 year old doing first grade math does a fraction of the work of a first grader, at a much slower pace, and has a higher error rate. In the Spectrum book, I would spread a page of 18 arithmetic problems over 3 days or more. By the end of the book, there are pages with 36 problems, which we do 12 per day. The 125 page book is going to take about a year and we won't do the whole book. Only as needed.

Age 5 - The Beginning

**Test Math**Kindergarten and first grade are the targets of almost every single test prep book on the market. All you have to do is refer to my curriculum page to get started.

My goal for Kindergarten is to do every problem in the two Every Day Math Grade 2 Student Journals. This will be the last time we do any school math until the summer after 4th grade. It's the last time that school math has any relevance to thinking or learning before Algebra. The two times we did this two things happened. First, we got off to a very slow start. Second, about 8 months later we stopped at about the half way point because things got 'too easy' and it seemed pointless to continue. Many parents who tried this reported the same thing. Like me, they considered giving up early and are glad they stuck with it.

The biggest lesson I learned in all of this was that doing a single hard problem for 20 minutes is 100 times the learning of doing 30 easy problems in the same time. This characterized math in our house (except on bad days or when we were backtracking) from then on. As we moved through more advanced math in subsequent years, I found that this approach directly leads to amazing results. Teaching math in this way results in a formidable skill set is useful in all subjects.

First, your child will battle through every problem, getting many wrong. There will be bad days and there will be whining. At the end of the first workbook, you can hand your child a completely finished book that they did on their own that is 2 years ahead, and you will be handing them a permanent life lesson in grit. If you were an experienced academic coach, you would get through the first page with no crying. But you are not. I can tell you from experience that the whining will go away on it's own sometime in the first 4 to 6 weeks as both parent and child acclimate to this level of work and reset their expectations, especially the parent.

The first few problems are going to be painful and go slow. Expect a high error rate, maybe 100% wrong. Just plod on. The pace goes up on its own and the accuracy rate improves on its own. You just have to watch. Being anxious or frustrated while overseeing this work will teach your child to be anxious and frustrated, so stop it. Also, when you help, don't help. Your goal isn't for your child to understand 3 + 4 = 7, but for your child to figure out how to help himself figure it out. Ask a lot of questions. Be baffled and stumped. Enjoy not knowing and discovery. Expect absolutely nothing from your child.

The second way it paid off was in school, and in our case, it was first grade. In first grade, which was a killer year for reading and other subjects for my first child, in math the class did Every Day Math Grade 2. You can imagine my child's reaction when he was handed the Student Journal, which he already completed in its entirety, and was assigned a subset of problems.

There's a third way this pays off. It teaches the parent how to be a top academic coach for the rest of grade school. The key things to know are: set your expectations at zero, ignore wrong answers, and go really slowly. The pace will speed up on it's own, and once you see it for yourself, the pressure is off. Mistakes are awesome.

Every Day Math focuses on number sense. Number sense is not the only an important skill for this age, but it's the most teachable of the important skills at this age that you can put into a classroom math program. It is perfectly reasonable for a parent in the United States to expect their child to be able to work ahead 2 years in math because the rest of the world does it. Number sense is the foundation.

There are quite a few alternatives to this approach. There's Mathasium and Level One and Singapore math. Under certain circumstances, these are not bad ideas, but they are very different in approach and outcome and it would take customer work to untangle it. In short, I'm putting the kid in a very difficult (but supportive) situation and he has to grow his problem solving skills to get out. A teacher in a program that you actually pay money for can't do this for obvious reasons.

Age 5 - First Grade

**Test Math**Again, kindergarten and first grade are the targets of almost every single test prep book on the market. You can refer to my curriculum page to get started.

**School Math**

This is a tough year for school math. First of, math in 3rd and 4th grade (US curriculum) stinks. I feel like teaching it makes a child dumber. To elaborate, "smart" to me is a set of cognitive skills. School math teaches the opposite of these skills. Secondly, at this point a quick way to get a good score on the ITBS or MAP is to be a year or two ahead in math. The good news in all of this is your opportunity. Keep reading.

There are some alternatives. I recommend grade level Singapore or current+1 Kumon word problems. Rip out any solution guide or how-to manual. Let your child take 2 weeks to solve a problem if that's what it takes. I'm already a fan of Kumon word problems. I'm going to start experimenting with Singapore. The only problem have with either curriculum is the rush by teachers and parents to explain how to do each problem so that the child can just do it. That's called learning and not training. If the child can't figure it out on their own, the work begins not ends. When I work with Singapore kids, they solve problems like they know exactly what to do, like they've done this hundreds of times. If your first, second, or third grader is not struggling through math problems, you've got a real problem. The skills that they need to develop over the next 3 years are not being exercised. There is going to be a jump in the level of math. I call this the 4th grade train wreck. Sometimes it happens in middle school and sometimes in high school. Any time you are hiring a tutor for your child, you've got a kid who is missing skills, and the tutor is there to ensure they don't have to develop them.

Instead, I prefer either Go Math from Houghton Mifflin or Eureka Math. I've used both of these and they are both recommended by school districts in New York*. When I looked at the 3rd or 4th grade math, I thought 'this is easy'. Eureka is spoon feeding. But we're not talking about 4th grade math, we're talking about a 6 year old at home with problems and no text book and no examples. Perfect. Expect a slow start and be prepared to look for supplemental material, of course, but if you dropped your 6 year old off at my house and wanted her to be a math expert 8 months later, this is the material I would use. *Anything recommended by New York, Illinois, California or Florida is way too easy at grade level.

To help everyone along, here are the core problem solving skills I derived from Poyla. It's a really slimmed down version with some research from Berkley and Stanford thrown in. Sometime between now and 5th grade, most of the missing steps are going to be thrown in to tackle higher order math, and the last one comes in time for middle school.

- It's OK to be totally confused and baffled by a problem.(Berkley)
- Most of the battle is learning to figure out what the question is asking, not the actual answer.(Poyla)
- Mistakes aren't bad. Mistakes are part of the process. Learning happens in mistakes; learning doesn't happen with correct answers.(Stanford)
- Check the Answer. This comes naturally over time from a child who expects to get the problem wrong because they make so many mistakes.(Poyla)
- Be prepared to backtrack and have material standing by, wiki, google, Khan, IXL, and any how to you can find. When your child someday sees double digit division in a problem and they don't know multiplication yet, this will be an indication that you will spend a few weeks filling in gaps.(Me)

Forget how quickly the child adopts these skills. Scores are going to diverge at this point based on how the parent responsds to problem solving. The parent has a big problem when the child can't do a problem and the parent is not a qualified math guru. Is the parent going to sit down for 45 minutes and figure it out, talking through it slowly, looking for supplemental material, or does this parent want a quick fix so that the kid 'knows it' or 'can do it'. Answer this question and I'll tell you how the child will turn out in the future.

Second Grade and Third Grade

**Test Math**There's almost no cognitive skills material left on the market by this age. The Building Thinking Skills books are still good for verbal, but we did the Grade 2 & 3 books in Kindergarten and got through 75% of the non-verbal section of the Grade 4-6 book. I compared some of the material in SSCC to the half way point in the Grade 2 & 3 book. This is a slightly unfair comparison if you are just starting out and are eyeing 99% as a goal. If they could put the 4-6 grade nonverbal with the 2-3 verbal, it would be perfect.

I created Test Prep Math Level 2 and Test Prep Math Level 3 to address this gap. The oldest edition started with 100 word problems. I wanted language and logic rich problems that aren't lame, which is a nice way of saying convoluted but goofy. Everything should build Working Memory, which is the magic bullet for cognitive skills tests, not to mention learning. That's a nice way of saying the child has to untangle multiple equations from the problem and solve them, preferably without using a pencil because this will induce mistakes and do-over, built in arithmetic practice. The next edition contains 100 quantitative problems that are, at a minimum, 150% as hard as the cognitive skills tests. Finally, in the third edition, I added a fairly comprehensive set of figure matrices and related question types, set up so that there's work to do untangling the figures to transform on top of determining all of the alternate transformations that qualify on top of choosing the answer.

All of these problems are impossible until the child gets the core skills I listed in the First Grade section, then they are easy, then I just ramp up the complexity to keep each question at about 15 to 20 minutes if possible. Once the child learns to explain a problem, even the matrix problems, and simply state what is happening, the answer is usually apparent. This is a huge advantage on congnitive skills tests and the primary method we use in tackling practice tests as a ramp up to the big event.

My favorite part is the word problem sections. I think it has the most lasting value because when we started doing SAT prep I realized that the SAT reading comp questions were as convoluted and nonsensical as the word problems in TMP and we got off to a fast start. In 5th grade. I call the SAT Test Prep Math Level 4 in Math House. Anyway, I'm a big fan of the research that demonstrates the differences between why girls and boys learn math differently, how girls like to see relationship and think more open endedly. So I connected everything with a plot and had questions that had more than one answer or an open answer. I was thinking 'boys should have these skills' but what I got were little girls who loved the book. At least there are two boys who picked up the target skills. One has announced he wants to go into a writing field, and the other is a really great poet. Both are good at math.

From experience, I've worked with kids at the 99% level who come from a Singapore background. These kids should just skip to Level 3 in 2nd grade. Don't let them use pencils. You're on your own after that. There are kids who would benefit from TPM during 4th grade, but none of their parents read my blog.

**School Math**

After the first TPM, maybe the second, we start easing back into school math. Anything after at current+1 and current+2 is fair game. It's all on the web. Here, you can go back to IXL or Khan Academy for a list of topics, and you can find as many problems as you want. By the end of third grade, we started to work on competitive math under the first Law of Math House, 'No math no computer', and start to expand our list of problem solving skills toward the ones peculiar to math.

For those of you who have never done current+2 (see Kindergarten section) or even current+1, now is a good time to get started. Keep in mind that a 5th grader who has already gone through this program will be using material that most 7th graders can't do. When you are choosing material for current+2 for the first time, you have to pick the appropriate material, and by approriate, I mean either Eureka or Go Math (for 2nd grade) or CMP Math from Michigan State for 5th grade. When we do current+2 after 3rd grade, it's CMP Math. There are more rigorous grade level programs that I mentioned in previous sections that are inappropriate for current+2 because the grade level work is already current+2.

**Power Bucketing**

At some point during second grade, I introduce a new category of math. It's called Power Bucketing. This concept introduces a new topic like exponents that the child will see next year or the year after. We start with a simple definition of what I know, or I'll ask a simple question and just let it hang out there for a few weeks. It started with "What is 2 minus 5". It's not 3. 5 - 2 = 3, so 2 - 5 can't equal three. Eventually, we'll do a few problems, and then maybe I'll look for a worksheet. If it took brilliant mathematicians thousands of years to define zero, our children should get a few months to think about negative numbers, square roots, exponents, the definition of 'i' which is super fun, prime roots, linear functions and on and on.

It's called Power Bucketing because pre-introducing the topic creates a brain bucket to store the material that will follow. The goal is that the child shows up in class, the topic is negative numbers, and he's got a bunch of negative number slots in his brain ready to file the instruction as it comes in. In practice, the child slowly comes to terms with the concept on his own and can just skip to the problems. Or move to the next one. Power Bucketing will play a role staring in 4th grade when the next 8 years of math is going to be eaten one chomp at a time. Speaking of chomp, one form of Power Bucketing is where we sit down and comprehensively derive entire subjects, like linear functions or geometric principles. It starts with a big question, followed by the 4 skills above in order.

General Guidelines for Teaching Math At Home

The classical education has 3 phases. In 1st through 4th grade, the child is a knowledge sponge. The more facts, concepts, words, etc that you can cram in the brain, the better. There is no limit to the child's ability to acquire information. The next stage (5-8) is the thinking stage, where children learn how to think. By highschool, children are beginning to have opinions and one of their opinions is how lame their parents are. This is a really rough summary. I recommend introduction to The Well Trained Mind for inspiration and a better definition.

The secret to gifted education is that all of this can happen from the very beginning in high volume and at a fast pace. Gifted education simply recognizes that a child in the right environment, aka a home with insane parents, is going to pick up everything at once. There are some limits of course, but it pretty much is all there in the beginning. You can be lame by as early as 2nd grade.

The secret of getyourchildintogat.com is that I watched both my children display zero hints of promise, or even intelligence, for the first 3 to 6 weeks of everything we set out to do, stuck with it (I don't know why) and was rewarded with children who succeeded beyond my expectations. Of course, you hand a 2nd grade math work book to a 5 year old and ask him to do the work and explain it to you as he goes, you're going to have to re-adjust your intitial expectations. To zero. But then after a while you begin to expect your child to enter MIT's graduate math program any day now.

My official goal is that I hand my 4th or 5th grader an SAT practice book and he starts working through it and learning new math based on the problems. This is just one of the many insane things we do. I came up with a 1 day exercise to self-teach half of the material in a 125 page eighth grade text book. Anyway, in order to get to this point you have to prepare for this goal. The math concepts are just a distraction until after 3rd or 4th grade. The more 'math' you teach your child, the less time you have to teach 'skills he needs to crush the SAT in 5th grade' or 'do 8th grade math on her own during the summer after 6th grade'. Also, the more math 'concepts' you teach, the less 'learning and wonder' of math is going to happen.

Here is a really rough summary that took me 500 articles to cover in more detail, but it will get you started.

- Working memory is king. In the process of working through multi-equation problems mentally, a very powerful set of unteachable subskills will emerge in your child's brain. Take away the pencil. You'll get 5 wrong answers and mistakes, and this will just strengthen working memory.
- Do not, under any condition, let your child practice or memorize math facts. Having to come up with ways to add 7 + 8 because you are now 12 and never bothered to learn math facts is the path to a math chair at Berkley. This is well documented.
- Learning math facts or learning solution alogithms to common math problems (think carry the one or solve simultaneous equations) is going to shut down the type of learning we desperately need for the next few years. Any memorization or 'practice', as in repeating the same stupid math technique over and over is going to elminate the type of thinking we need right now to get ready for the future.
- Verbal discussions will provide a set of skills that are 10 times as important as math, but you won't realize this until your child is in high school. For now, work at an advanced level is best done in a team setting, and you are the defacto missing team member.
- You are not just using the short list of problem solving methods above, you are reprogramming your child to be these problem solving methods. Plus there are about a dozen more you'll eventually need to solve the actual problem which I didn't mention here.
- Use reading and language and vocabulary as an opportunity to tell your child everything you know, answer questions, talk, share, show. Use math to make them think on their own. Don't help. Don't explain. Don't tell.
- Learn to be a GAT parent. Have zero expectations, learn patience, stop caring about the solutions. Encourage discussion. Let your child own the process even if they go off in the wrong direction and make lots of mistakes along the way.

If you drop your 10 year old off at my house, and he's a little ahead in math because you've been focusing on learning and thinking and problem solving for he last few years, we'll get to our first calculus lectures by age 12. 45 minutes on derivatives on Saturday and 45 minutes on integrals on Sunday. It's magic to watch. It's as exciting as teaching complex numbers to an 8 year old. It's just silliness and wonder. Of course, by the end of the week he will have forgot almost all of it because what use is calculus to a 12 year old? On the other hand, if you spend the last few years training your child in math while chasing a MAP score, he wouldn't have any of the skills needed to stay above grade level in math for an extended period let alone do well on the MAP. Ironically, the highest MAP scores - the really really super high ones - are achieved by kids who stink in math. Knowing math is not a prerequisite for doing well in advanced math. Thinking is.

hi - your link to the math workbooks is not working. can you direct me to where i can find it? also, did you rewrite this page? i vaguely recall this being in table format? (yes, you have avid followers!)

ReplyDeleteI'm in the middle of rewriting it. I've got about 4 more pages to write. My contempt for math fact was recently confirmed by research. As for the links, if you mean the little booklets for 4 year olds? The links are blocked by some internet providers, especially at work and at school, and I haven't found a solution for that. Or did you mean other links?

DeleteYes, I was referring to the little booklets you created similar for Math that you say are similar to the Bob Books. I will try if I can access it through other devices. Any timing on the reading guide you are coming out with for 3 year olds?

ReplyDeleteWell, I found them in google documents. It brought back a lot of great memories. I will get the links back up in a week. I'm doing my final review on phonics, and I'm on about lesson 60 of the reading book. It's way more work than I expected. It was easier for me to do than to transfer. I'm intent on taking an average child and an average parent, and turning them both into GAT level powerhouses.

DeleteAre the links up yet?

DeleteHi, thank you very much for sharing your experience and research for free! I have a question about the everyday math grade 2 student journal books: which edition are your talking about? I have found different editions on the web and got confused.

ReplyDeleteHere's the timeline for the editions: http://everydaymath.uchicago.edu/about/em-history/timeline/

DeleteThe one I used was the third edition.

Thanks for your fast response.

ReplyDeleteI agree with every single word above. I constantly circle every important piece of data in the problem to my kids. And yes, kids that are not taught to check their work will be less successful than those who do. Right now, I am teaching my older one how to recheck. I give her a sheet and give her twice the amount of time she would need to finish up the sheet and put her on a timer. I tell her I will not check her sheet until the timer chimes. I call this reverse timing. Instead of timing how fast she can solve a problem, I am asking her to slow down and time herself. Also, one thing which my kids love doing is "be the teacher". I ask them to pick one topic and prepare for about 10 minutes and teach a simple concept to the other two in the room. That's their bonus exercise of course!

ReplyDeleteI like the reverse timer idea. We typically work with problems that can take anywhere from 15 to 90 minutes, so I'm not sure that would work. The "be the teacher" idea is a whole level or more up from gifted. I usually ask my kids to invent a problem and make it harder than the one they just did for me to take, and less lame. I think this similar but different. I'm going to try your approach.

Delete