## Friday, December 29, 2017

Last month, a reader asked me how to teach fractions to a 2nd grader.  I'm assuming that the reader is asking me to provide the magic formula for a child of a specific skill set (which I haven't measured) paired with a parent who has a specific skill set (which I haven't measured).  This is a tough problem that requires the Force.  I retreated to the island of Skellig Michael off the coast of Ireland so I could meditate.

This article is going to describe how to teach a math topic (in this case fractions).  The approach for fractions shares a lot with other topics.  Fractions is especially important in math because it is slightly abstract and always multi-step, follows 3 straight years of spoon-feeding one step problems in school, and therefore befuddles students and parents alike for want of cognitive skills.

The Parent

Before we launch into this topic as a parent/coach/Jedi Master, we need to take a step back to appreciate the importance of this math topic.  It is fascinating, alluring, captivating.  Without this appreciation, it's hard to pass our love of math to our children.  If your child is ready to take on fractions at a young age, you will convince no one by saying "Fractions are totally boring and pointless to me but I'm going to make you do them anyway".

What are our goals for fractions?  We want the child to know fractions so that they can look at a super hard problem and see the answer right away.  We want the child to emerge from these fraction studies with a formidable skill set that can be applied to other quantitative areas.   We want to present this child with something like exponents or algebra someday and the child will say "Leave me alone.  I can do this all by myself".

There are some years from K to 8 where you want your child to do 45 minutes of math a day and be really, really good at it.  In our case, there were 3 of these years.  I wouldn't do it every year because the child will learn to hate math, and it's not necessary anyway.  If this happens to be one of these years for your child, in addition to what I describe in this article, get them a decent workbook that includes fractions and make them do every single problem in the book no matter how long it takes.  Otherwise, just do what I recommend here.

Fraction Foundation
The first thing we need is a high level definition of fractions.   When you divide 20 by 4, you end up with 5.  This means splitting 20 into 4 groups gives us 5 in each group.  If you have 20 skittles, but I'm only going to let you eat 1/4 of them, you're only going to get 5.  These are two different concepts, but the exact same mathematical operation, namely 20 ÷ 4 = 5.

What does it mean to divide 7 by 2?  What does it mean to divide 1 by 3?

There are two times when we have these discussions.  The first time is when I think we're going to be studying fractions in a few months or next year.  I call this Power Bucketing.  This discussion will create a brand new bucket in the child's brain called 'Fractions', and when the child sees fractions in school, while the other kids are trying to come to terms with fractions, my child will already have an empty filing cabinet in their brain for fractions and will have a permanent head start in this area.

When I teach fractions, we spend the first week just asking what fractions are.  I will give the child 10 to 20 minutes for them to think through these simple problems, like 7 ÷ 2 = 3 1/2.  After we've exhausted the mental capacities of the child, I'll ask for a picture or show them how to diagram fractions.

If you look through 2nd, 3rd, and 4th grade math curriculum on the topic of fractions, fractions are introduced slowly.  I'm not going to speed through this process.  Please view this Ted Talk on J.J. Abrams from 2007.  Look at that box with a question mark (in the video).  As long as the box is sealed, your child's imagination is in play.  As soon as you open it and describe its contents, you've ruined your child.  Let the child figure out what is in the box on their own.

Fraction Lifestyle
At this point, you can introduce fractions into your conversation.  Think about a really smart parent with multiple PhD's who just talks their child into Stanford.  We want to be like this parent, only not as nerdy.  The two most obvious uses of fractions are time and baking.  Get your child a brownie mix and make them do all of the work.  Put post it notes on the refrigerator reminding yourself to talk about time only in fractions, as in 'it's 1/3 past 5, what time is it?'  By the way, my older child has been in charge of making desert for years thanks to fractions.

Fraction Overdrive
This page from IXL describes the basic fraction related skills expected of 4th graders.  You can also look at grades 5 and 6 because fractions is going to appear every year from now on.  I didn't read any of it because it's too boring.

Instead, like all topics in math before calculus, with the exception of geometry, we simply have to state the obvious.  How to you add, subtract, multiple, and divide fractions?  Throw in 2 more operators (greater than and less than) and transformations (aka equals) and that's pretty much our goal.  This is exactly 8 things to learn (transformations are 2 things - equivalent fractions and transformation to and from mixed fractions).

This little exercise is going to be repeated with rational numbers, exponents, complex numbers, and other pre-pre-algebra topics.  When this child is doing algebra for the first time at age 9, and is stuck while trying to reduce a really complicated algebraic expression, I say 'Dude, you've only got 4 possible operations - addition, subtraction, multiplication and division, just try all 4 of them to see which one works.'

When your child sees decimals and percentages some day, we'll have 2 additional transformations involving fractions.

Where did I get all of this material?  I spent a month thinking about it.  Your child's teacher does not have a month to spend on fractions because there are typically 6 to 8 topics each day, plus statistics.  With some math topics, I also wiki and read about Egyptian or Babylonian history.  Your child's teacher won't have time to do this either.  She has 8 subjects and 30 kids of cognitive profiles to teach.  You have 1 child and fractions.

The Student

Children are natural learners.  Once the parent is prepared (95% of the battle), the rest is easy.  Just give your child as long as it takes and don't help at all.  Ask the questions and expect your child to work things out mentally, when your child doesn't succeed, ask them to draw the picture.  Help as needed, but only after the child has exhausted their mental faculties.  I generally observe mental exhaustion takes place at about 20 or 25 minutes (because I always choose really hard material), and I'm prepared to sit there, sometimes silently, for 20 or 25 minutes.

If you hand a 4th grade book to your child, there will be gaps hidden in 2nd and 3rd grade material.  The child will get stuck on a problem, and the way forward is material that they either never had or never mastered.   Be prepared at all times to go back to 2nd or 3rd grade material as needed.  Suppose they get stuck on a really hard problem, and you can see that it involves transforming from mixed fractions or comparisons.  Take a few days off and do some problems involving mixed fractions or comparisons.  IXL is good for this.

Step 1 - Comparisons and Transformations
I'm not sure why a book would be needed at all.  The most important fractions are 1/2, 1/3, 1/4 ... 1/10.  If the denominator is greater than 2, then you've got 2/3, 2/4, 2/5 ...,  3/4, 3/5, 3/6... and so on.  Then you can multiple any fraction by 2/2, 3/3, 4/4 and you've got a set of un-reduced equivalent fractions.

Pick any 2, and ask for >, <  or  =.

If your child was adept at division and had a really strong number sense, I would not create flash cards to drill my child on fraction comparisons.  If your child did not have a strong number sense because they never had really great curriculum at age 4 or 5 that built number sense, I would not only create flash cards, but I would create spread sheets with 100's of problems from the fraction list and drill the child until their number sense was invincible.  In our case, we did this at age 4 with SSCC and never looked back.  Except when we did this again.  And one other time.

You can search the internet for "comparing fractions worksheet" and see thousands of examples.  If I never met your child and you only gave me 30 minutes to address "comparing fractions", I would print 3 of these: One with pictures, one with simple fractions, and one for harder fractions (involving primes versus composite numbers, like 12/13 versus 10/12) and I would find out quickly where they are.

This exercise requires transformations - like comparing 5/6 and 10/12.   This is a two step problem.  5/6 and 9/12 would be a more obvious two step problem.

Throw an integer in, like 2 1/3, and you've got the other transformation to get to 7/3.  We never go in the other direction, from 7/3 to 2 1/3.  When I see this in a book, I comment that this is lame.  In higher order math, we only work with 7/3, or 142/25, and never mixed fractions.  Also, as I mentioned before 6 ÷ 3 = 6/3 (this is impossible to write in a vertical line, but basically I'm writing division problems as fractions and never using ÷ again).

Step 2 - The Other Arithmatic Operators
Once we 'get' fractions and practice transformations, we have to tackle addition and subtraction, then multiplication and division.

Addition and subtraction involves transformation.  We can't add apples and oranges.  We have to transform one or both.  This is why transforming and comparing fractions is a prerequisite.

Pictures might help if you didn't spend any time doing step 1.  We usually just skip to the hard parts, but you need to read Step 4 below to see why.

Note that this is a 2 or 3 step problem.   These types of problems reward a child who works slowly and a parent who doesn't expect correct answers.  If the child is expected to do a lot of problems, expected to get them correct, and expect to do them quickly, the child will fail at multi-step problems.  Because of this, I have settled on one or 'a couple' of problems as our daily routine until the child builds speed.

If this child was 10 years old, I would expect the child to devise and explain a formula for adding fractions.  Before this age I never even hint that there is such a thing as a formula.  I want the child to go through the 5 or 6 substeps every time, using working memory, because amazing and surprising subskills will develop in that child's brain that will pay off in a big way later on.

For an 8 or 9 year old, I would want to see a picture and an explanation of what is happening.  I would also try out 1/2 + 1/3, 1/3 + 1/4 etc from the list I explained above.  But I would do this every time he was stuck on something like 5/11 + 2/3, because this age desperately needs intuition number sense and now's the time to develop it.  This is really going to slow down the topic, but if you do it right, you'll save many years later on not having to explain math topics.

Multiplication and division require starting all over again with this article, both parent and child section, with each operation.   What does it mean to divide 3 by 1/2 or 7 by 1/2?  What does it mean to divide 1 by 1/3?  How about dividing 4/5 by 2/3?  The same basic questions are asked about multiplying fractions.  What is 1/3 times 3/4?  Before algebra in about 6th or 7th grade, I would want this child to think through the meaning of these problems every time instead of just turning 4/5 x 2/3 into (4 x 2)/(5 x 3), because if the child skips thinking through these each time, they will get to algebra ready to calculate but unable to understand.  This approach precludes some problems and precludes lots of practice.  This approach involves a few problems over a much longer period.

Diagrams work really well in understanding multiplication and division.  These will be articles on their own so I'm not going to cover it here.  Have you ever read a history book that starts with the beginning of time, evolution, 40,000 years ago etc until it gets to the main topic, which might be 1972?  That's how I handle these topics.

The biggest challenge with teaching your child math is coming to terms with how stupid your child is.  You're doing something that you just did the day before, and your child not only forgot what he learned the day before, he can't even add.  He does a single problem in 30 minutes and it's totally wrong.  There are 29 problems on the page that are not completed.  It's a disaster.

This is the make or break moment in your child's academic career.  You have the choice between a future surgeon with join doctorate degrees in Sumerian literature and Bioengineering, or a kid who drops out of community college to form a rock band.  The choice is yours.

I'm usually pretty pleased and announce that will pick up problem #2 the next day.  I can do this because in the futile mess I see cognitive skills developing.  Within a few months, my child is making adequate progress and I'm looking for books on Sumerian literature on Amazon.

Sometimes  I am discouraged and ask how he could possibly screw up such a simple problem.  After I say something like this, he will spend the next few weeks perfecting a base guitar riff.

How Good It Can Be
Once you've taken on a few topics like this once, each successive topic is easier and more fun.  The key is that 6 to 8 months of hard work pays off, and you can see that doing a single problem for 2 weeks and getting nowhere is normal and leads to ripping through pages down the road and eating math for lunch.  For a parent, it requires nothing short of faith to get through the first few weeks.

For those of you who took my advice to do EDM Grade 2 in Kindergarten, you already know this.  For those of you who do TPM, which is not all that mathy but is really thinky, you're ready to start.  Unfortunately, in both cases, nothing ever gets easier and you still have to go through the whole painful learning curve with new maths.  But doing Algebra II with a 9 year old and going through a painful learning curve is much more gratifying than doing decimals and going through no learning curve.

Last week, my child was struggling on a problem from his Algebra I final exam.  We stopped using math books altogether and just take tests, figuring things out on the spot.  Sometimes, we'll take a break and do some worksheets on a new topic.  Anyway, there were 4 maths involved in this topic, and he didn't know 3.  He didn't even know the formula for the area of a circle.  It took us over an hour to do a single problem, what with all the backtracking.

Then I realized I accidentally grabbed the Algebra II final.  When we went back to the Algebra I final, he had 6 questions of the form "What is 42% of 66?" and didn't know how to do them.  Arrrgggghhhh!

In each case, we took apart, figured out, and mastered new topics on the spot.  This is the skill set that I want.  This is the skill set behind the MAP test, for very important reasons.  If you can get this skill set down early on, say fractions, then it's just a matter of plowing through pre-algebra, functions, algebra, geometry, trigonometry, calculus (AB and BC), linear algebra, real analysis and series, and then statistics.

## Tuesday, December 26, 2017

### Post Holiday Math

We are in day 2 of a 2 week holiday break.  Day 1 was a holiday and I have a hard time convincing anyone to do any math.  My kids sat around all day having fun, eating, chatting and helping with chores.

Math starts today.

Daily math is a prerequisite of the kids doing anything fun.  The kids say, "I don't want to do anything fun and I'm not doing any math!"  Then they read, do crafts, engage in an imagination-building-problem solving activity like Legos in order to not do any math.  It's quite amusing to me when I walk by their room, and they are sitting there reading for hours, and they look at me like 'Ha, ha, I don't have to do your stupid math, I'm just going to read.  I win.'

Reading is way more important than math.  The jokes on them and I'm not telling.

Daily math started with the simple thought, "If a child becomes a strong reader and thinker because he reads daily, how is he going to become at STEM?"  The answer was daily math.  Around third grade, I thought "There is way to much homework to do each night.  We'll just do daily math on the weekends" and that's where we've been ever since except summer and breaks.

Math contains more than math, of courses.  It contains anything I think they need to succeed at the time.  This usually contains math.  On Saturdays in the summer, this can be math, vacuum the basement, practice your instrument and do a reading comp question, fix the toilet, replace light bulbs.

This year, the April MAP test is on our radar and I'm becoming slightly more organized with daily math.  We overdid vocabulary between SSCC and 2nd grade and haven't done much in this area other than define and discuss any unknown word found in reading or reading comp.  I am reintroducing vocabulary as part of math.  With a vengeance.

I like the MAP.  It has a lot in common with the COGAT.   The cognitive skill set is slightly different, but in both cases there is an advantage that can be gained from working on these skills simply because school doesn't really teach cognitive skills.  Doing lots of practice, ala Kumon doesn't help at all, and learning algorithms ala Singapore sets up a train wreck (like ending up in the 90th percentile or less - I never really defined what a train wreck is but that's it).  The problem with any program at all is that the child can get ahead and doing well, and the parent thinks that success has been obtained.  The clock is ticking.  Any time a child is practicing or applying or using things taught, learning may or may not happen, but skills building is not part of the deal.

I remember when my goal was simply to cheat my kids into a GAT program.  What actually happened was that we just ended up spending a lot of quality time together and I learned how to be a parent.  The long term formula for academic success is Cognitive Skills + Interest + Will.  At this age, and in the succeeding years while we caught up, it was all Cognitive Skills at the expense of Interest and especially Will.  You can burn a kid out with daily math every day every year, so I tried (and failed) to take some years off.  To compensate, I completely changed the approach to my formula of Baffled + Spending Time on the Question not the Solution + Get it Wrong + Check the Work.  This created an environment of Zero Expectations and No Progress, and in that environment magic happened.

Somewhere along the way, 'Will' came back, most likely because of chores or instrument practice, and I'm doing my best to stay as far away from 'Interest' as I can so as not to ruin it.  A child can only develop interest in a vacuum that does not include the parent.  Unless the parent is super sneaky.

I'm thinking about 'Interest+Will+Skills' a lot because for the older child, my goal is that he does really well in AP Language Arts and/or History, with assumed A's in math of some kind.  All of the math education is pointing in that direction for this child.  I found that at one of the selective enrollment high schools in Chicago, a child can take Calculus as a freshman, followed by Linear Algebra/Multivariate Calculus, a course that's no longer on their website which I will demand be reinstated, and AP Statistics, and assumed A's in AP Language.  This is 4 years of college credit math.  We're going for it.

The only way I can possibly think of achieving these goals is to do something creative, unusual, and different.  Something that is more looking at things from a fresh perspective than hard work.  Hard work is not going to do it.

## Saturday, December 23, 2017

### The Makings of a Thinker

Here's a rough non-copyright violating approximation of a figure matrix question from my favorite COGAT practice test, grade 2.

This is the last question in the book and the hardest.

In this article, I'm going to show you how much mileage you can get from a single question.

When I coach, usually at the behest of a parent who provides a compelling reason or academic puzzle that I want to add to my research, I'll start with whatever material they have available and do a single question.  There are many other things I do with a practice test besides a single problem, but my favorite Academic Coaching Session Agenda is the Single Problem because the student picks up the most skills.

This may be the only time I'm working with the child, and my primary goal is to train the parent who is lurking nearby, and I want an impact, so I do it exactly like I would with my own children.  Like this:

Step 1:  I instruct the child to do the problem.  Take as long as you like, and before you answer the question, I want you to tell me that you're ready to answer the question but not what the answer actually is.  I will probably announce that this is a really hard answer and I'm totally confused so I hope that the student can do it because I sure can't.

*At some point during this training, the child will learn to check their answer.  I am going to encourage this behavior in multiple ways including saying 'Check your answer'.

This approach is the birth of skills.  If the child answered incorrectly, then we're going to get double the skills from this exercise.  It's not clear to most parents what these skills are.  These skills are the skills of kids who will go into an accelerated history or reading course, teach themselves, and do well.

When I announce to the child that they are wrong, they are probably wrong, or their answer does not agree with mine, the child can sense that I'm happy about this situation, and I genuinely am happy because we can learn something.  I love mistakes, even the ones I make.  Mistakes drive learning and it's one of the 5 core skills.

Step 3:  Explain the question to me.  First of all, I want to know what the transformation is.  The first shape undergoes 3 transformations.  Zooming through problems is the way to miss subtleties like the height of the shape diminishing by about 10% before it is rotated 1/4 turn counter clock wise.  Some kids say rotated 'to the left' which is OK with me provided 'to the right' always means clockwise.

In this phase, we're learning how to see, the names of things (like rotate 1/4 turn counter clockwise or decrease in height slightly).  I will correct the child's grammar or terminology, expecting that they eventually use the adult level words that I do in adult level sentences with multi-clauses.  It's the opposite of Baby talk and the reason why my books have that awful looking graduate text book themed covers.

When the child thinks they are done, I'll point out that explaining the question includes explaining what is happening in each and every answer.  I would like to know what transformation took place to make each of the answer choices, or what transformation failed to take place.  That's 4 additional problems as far as I'm concerned.

I've never found a problem in a COGAT book that can't be solved with a thorough out loud explanation.  Sometimes when I'm working with my own material, I get the problem wrong, repeatedly, and I look at my answer and wonder what the heck I was thinking.  Then I go through it the way using the steps I expect a student to use, and oh year, it makes sense again.  When you say the transformations out loud (problem and answer choices) hard problems are turned into easy problems.  I can't over stress the importance of this technique.  This is why Shape Size Color Count is so verbal

I call this skill 'Reading The Question' because most kids can't do it without a lot of training, and most parents lack the patience to wait.  I know as a parent I used to lack the patience, and sometimes I still do.  To accommodate my coaching inadequacies, I'll just turn over the material and go clean for 20 minutes before the teamwork begins, shouting out things like 'Read the question again' while I do my work.

There is a prerequisite skill I call 'Seeing' that children have to develop.  In this case, 'seeing' is visual and includes proportions and the ability to mentally rotate images.  It takes some practice.  In an academic household, those places of non-stop learning that produce GAT standouts, this practice started at age about 2.  For the rest of us, COGAT practice is as good a time as any.

I should point out that this is not a hard problem because it's missing the magic of the COGAT.  The quadrilateral lacks symmetry.  A problem like this would be practice for K.  This is why practice tests are practice for the format of the test and not the thinking of the test.  Also, there are 3 transformations, which you'd think would be good for working memory, but the shading transformation removes answer choices right away, making the problem easier, not harder.

Step 4:  If the child can't get to the correct solution on their own, I'll mark the page and come back later.  This question is still holding learning.  If I have to announce that the shape is shrinking in height before turning, I just destroyed the learning opportunity.  If there are 10 more questions with this transformation, I'm stuck having to announce it.  It's a judgement call and depends on how much time is remaining before the big event.  If you have a lot of time, you can back track by drawing 10 or 12 shapes, and ask your child to shrink one dimension and turn it 1/4 turn in one direction.  Backtracking in this way is a version of finding an easier problem to solve before tackling the harder problem to solve.  No branch of mathematics can withstand this approach, and every single super hard complicated advanced problem can be solved in this way if needed.

For one child, we spent a solid 4 months doing cognitive skills training (including BTS and much much harder material of my own making).   When we finally came back to math, we followed this approach from that point forward through SAT and calculus.  I learned that these core skills are universally applicable.  This is probably why the COGAT is such a great predictor of academic success.  Take any topic, like fractions or exponents or roots of a 2nd degree polynomial, or multiplication or anything, and at one point we slowly went through a few problems using this approach and learned months worth of material with a small amount of effort.

At some point during the actual test, the child will come to the questions that differentiate the 97th percentile from the 99th percentile.  These are the questions that differentiate those kids who probably would do well at Stanford with a little effort from those kids who will be sitting in a GAT program next year because of the ridiculously high cutoffs in almost all states.  The kid who gets these question correct will either be the child who is already 99% because his parents both have PhD's from those who have learned the skill set and go super slow on these problems:
• They are not the slightest bit discouraged by not knowing the answer right away or being confused.
• They take a long time to thoroughly investigate the problem
• They have a few techniques to fall back on when it gets really, really hard.
• They are not discouraged when they don't see their expected answer in the pick list.  They try again as a matter of course.
• They check their answer, and all the answers, at least twice if not more.
I think the best way to teach these skills is to approach the training in the way I described above.  You should see how the approach is consistent with these skills.  It should also be clear that the other approach, I call this the school approach - explanations and lots of routine practice in the hopes of memorizing or mastering a set of question techniques - is not consistent with the skills needed at the top.

For parents a week or two from the COGAT who reach out to me the first time for help, and have done zero of anything before that, this approach is the way to go.  Of course, if you plan ahead, you'll be able to go much, much farther, but the approach is roughly the same.

## Saturday, December 16, 2017

### Problem 123

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

While sitting in the testing center, you may notice a members of a tiny but super intelligent articulate species talking to their adoptive parents about the composition of the earth's core.  Then on the drive home, your child may sit in the back seat telling you in explicit detail about each problem he missed.  These are both good reasons to buy a math book that your child won't see for 2 years and make him do it.  It made me feel better.

Let's begin with my favorite email from parents and my common response.  Here is a brief summary of the email:  "This isn't working and I don't know what the heck I'm doing.   I don't know how to teach math.  What should I do?"

Here is my response:
• You are not teaching math.  Focus on teaching the core learning skills and the child will teach herself math in the case you are blessed beyond belief with daughters, or himself math if you're like me and stuck with a bunch of boys.
• The 1st few pages in the book took us about 3 weeks.  Any page could take a week.  Acceleration happens later in the process.
• Our error rate was about 50% on a good day.
• After about 30 minutes on this exact problem, I just gave up and made a note to come back to this topic at some point in the future (which was next week).  I'm going to do it fully below because it shows you how to teach math to yourself which will make you a better math coach in the future.
At this age, we're going to focus on the most important skill of Being Baffled, which is comprised of numerous subskills.  Then I'll talk about the 'Reading the Question' subset which you will focus on through 4th grade.  The other core skills like Getting the Problem Wrong (aka Making Mistakes) and checking your work are not discussed.

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

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

Step 1:  Be Baffled
Say 'This is a hard problem' then leave your child alone for a minimum of 15 minutes to do the problem.  I started this approach on page 1.  Somewhere between page 1 and page 123, 15 minutes of doodling, yelling, and complaining became 10 minutes of thinking and trying and 5 minutes of doodling, yelling and complaining.

Step 2:  Backtrack
The first challenge is that section 5-3 discusses the naming of line segments, like AB, problems 1 and 2 in this lesson connect shapes with lettered dots, but it's left to the child to make the leap to naming triangles. A Kindergarten kid is not only not going to make the leap, but by this point they never mastered (or even got) the whole line segment naming business.

Over the years, I've come to appreciate that 'Being Baffled' is a mandatory problem solving step, because it sets up the rest of the process, especially in BC Calculus.  Being baffled relaxes everyone (especially the parent) and opens the brain to thinking.  The opposite of 'Being Baffled' is frustration, impatience, and a subpar performance.

Fortunately, the example at the top of this page (not shown) has the same triangle without the numbered points, so we need to backtrack a bit.  Ask the child to name the line segments in the example triangle.  We should get AB, AC and BC.  Then ask the child to come up with a way to name the triangle.

I'm rarely severe on vocabulary.   At some point, I might just say that a triangle is named just like a line segment.  A line segment is AB, but a triangle is ABC.  What is the difference between BCA and ABC?  Does this triangle have any other names?  If the child is 8 years old and a boy, I would be disappointed if the child didn't say 'Bob'.

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

Step 3:  Dig into the question.
What is a triangle?   Ask you kid to define it.  It's a shape with 3 sides.  How do you make a triangle?  You put three sides together.  Show your child 3 lines that don't touch and announce you created a triangle.  Each side has to touch 2 other sides at its end point.  I'm meandering through the question starting with the Stone Age and working my way back to 2017.

There is a whole set of skills that formulates the skill of 'Seeing'. Some kids can do it, other kids have a lot of work to do.  In this particular problem, there are 4 triangles.  Two are obvious, one is not obvious, and one is hidden.  This problem will show up on most competitive math tests in one form or another.  Seeing is a big part of math and reading and science and innovation and internet startups.  It's also one of the main skills of the COGAT.

Ask the child to find all of the line segments in this picture.  I see A1, 13, 3B for example.  Then how many ways can you take 3 line segments that each touch 2 others at the end?  We gave up after 3 named triangles.

Step 4:  Give Up
You will give up on something.  You are not working with a 2nd grade child, but a 5 or 6 year old.  At some point, it's time to move on, and you have not achieved mastery over some math topic.  Fortunately, EDM has some repetition so you'll see some topics again, just not this one.  Fortunately, your child is going to get this material again in school, and they'll look like the smartest person on the planet when they see it again and figure it out quickly.

After doing this for 8 or 9 months, children should be completing the work with reasonable accuracy in a reasonable amount of time, but I need to stress this child will never complete the work like an 8 or 9 year old would.  My my goal of 'reasonableness' was met, and we stopped at about the 1/2 way point of book 2.  That was good for 99% on the MAP for a while.

Think carefully about what I did.  I got a child to sit and work alone for 10 to 15 minutes on material he wasn't taught and didn't know before I would jump in and start helping.  As the months go by, he gets less and less help, just more questions.  I taught him (because math is a team sport and I was the missing team member as needed) to be baffled, to spend a lot of time on the question and to backtrack as needed, to make mistakes and be totally OK with that, to try over and over again and to check his work because he got most things wrong on the first try (not demonstrated above).

With that skill set, and continued refinements over the next few years, it is reasonable of me to expect that he gets 99% on both sections of the MAP from this point forward, can handle accelerated work in all subjects with little or no help, can teach himself instruments and other things of interest to him, and go to Stanford for graduate school.

On the other hand, what if I trained and drilled him on math topics during this period?  What would I expect from a child who spent 4 years zipping through math because he was expertly taught and trained on math concepts?  This is what school does really poorly and what after school math programs do really well.  But it's not the skill set I want. You wouldn't notice a difference between either approach if you just looked at math and you just looked at a 2nd or 3rd grade performance on a math test of some kind.  The difference will show up elsewhere and it will show up later.

## Tuesday, December 5, 2017

### Advanced Math and Little Kids

I have about a dozen questions from readers that have been swirling in my brain, all on the topic of casual work-ahead At Home Schooling in math.  I've been trolling parent forums and reading amazon reviews while a new round of 1st through 3rd curriculum shows up from my latest buying spree.

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

Here is my step-by-step*:

1. You do a complete inventory of all of your child's skills and your skills as a parent that are required for your child to teach herself fractions.
2. You fix the ones you can fix immediately and work on the rest at the appropriate pace and the appropriate material.  You can work on fractions if you want while you do this.
3. Your child teaches herself fractions.  You help by reinforcing the 5 core skills which you can see while your child struggles with the material on her own, with no help learning the actual math.
#1 is the problem, of course.  It's also the problem with parent forums and helpful parent advice.  It is also a problem with teachers, even good ones, but not the really great ones who have taught for 20 years.  #2 is easy once you get it, and looks impossible before you actually see it work, then it's total magic.  #3 is our goal.

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

Back to parent forums and book reviews.  Parents are blind to the cognitive skill set of their child and where this fits relative to other children, not to mention their own skills as an At Home academic coach.  They find something that works and then state with no further thought that it should work for other parents.  Maybe, maybe not.  If the parent mentions either a) my child reads 6 hours a day or b) my child got 99% on both the COGAT and the MAP or c) my child got 99% on the MAP but didn't do so well on the COGAT then I have a pretty good idea where this child is on the skill spectrum.  a, b, and c are three totally different places, but I've spent enough time investigating so many children in these three cases that I can just prescribe the medicine.  The rest of the world needs more analysis.

Wouldn't it be great if you could follow really 100's of successful parents around for 10 years and take notes and build a program based on what they did to put their kids at the top of the heap?  That's exactly what I did, and not just in math.

Recently I've been getting questions related to a certain famous math curriculum.  I haven't seen this material in 5 years since I reviewed it and then gave the books to a tiny little test case and followed up every week.  It wasn't right for my children, but I found a little girl who I thought would benefit for her specific case and she did.

The books are arriving and I'm really disappointed.  It's not about the core skills at all.  It's about explicitly showing the child how to do mathematical operations.  It skips learning.  Even worse, the questions tend to be the one-shot deal, as in one sentence that is pretty clear that the 2 numbers have to be added.  The inevitable result is a child who is told how to do math, never develops the skill set for #3, does pretty well on tests, and then has to be taught fractions.

In the last few months, I've gotten to personally know the Amazon drivers in Chicago because they show up at my house so much delivering material.  The last time I did this I was so disgusted that I wrote Test Prep Math.  Not much has changed. I've also pulled down at least a dozen curriculums (sic) from the web and gotten to know their creators from doing a little research.  I've come to the conclusion that the Test Prep Math series is the best material math material anywhere.

This is hard to say.  Authors have warned me that once you publish, you face a life of insecurity from that point on.  They were right.  I've freaked out when one mother told me that her child who's at the 99% found the books easy.  OK, I can deal with that.  The book is designed to get the child to 99%.  Just skip ahead until it gets hard. There is a review on Test Prep Math 2 where the reviewer slams me because the book is confusing and the answers are wrong.  As explicitly stated in the introduction, it is supposed to be confusing, and even I get the answer wrong when I speed through it and forget that it was designed for multiple readings on purpose, for you to see you skipped something or blindly assumed the wrong thing.  Those are core skills #1 (dealing with confusion) and core skills #2 (spend more time with the question - a lot more time - like 3 weeks if that's what it takes for the skills to emerge for the first time).  The book was returned and I feel personally responsible that the reviewer's child is going to eventually fall short in school.

I've gotten a lot of emails and a few comments from readers who state 1) my child finished TPM Level 2 and is finishing TPM Math Level 3 and 2) what do I do next?  When I get this type of email, the questioner probably has no idea that they have a friend for life.  I'm planning to put TPM Level 4 on a free website, mostly because it's going to take me a lot of time to piecemeal the material out there and my new friend for life won't have time to wait, and I'm still weeks away from TMP Level 1 and it's taking up time.

By the way, in my ongoing effort to make kids so ridiculously smart that they blow away the COGAT, which was my original goal before I decided a math chair at MIT was also a good idea, I've finally perfected my ability to deliver figure problems to 6 year olds that are 3 times harder than anything they'll ever see again.  It's much easier with older children to take away the net.  Never underestimate the importance of the COGAT.  It measures skills that kids need to teach themselves fractions.  It doesn't care if they can actually do fractions or any other type of math.  The COGAT wants kids who already know how to learn and can go from Kindergarten to fractions in one year, which is what happens when you enter certain gifted and talented programs.

Test Prep Math 4 launches the math career.  It's all about math.  The skills continue to refine and develop, and the fifth core skill (problem solving skills) becomes wider and deeper on it's march to passing the AP exam in BC Calculus.  When you child chooses a joint major in English and Music instead of a STEM career, those problem solving skills explode yet again and you discover why so many CEO's and law firm partners have English or music backgrounds, but you wanted a doctor so we blew it.

Here are the Test Prep Math Level 4 milestones.  By 6th grade, your child will have finished all of the practice math tests in at least one SAT book.  You will have administered at least a rigorous Algebra 1 final where they will encounter some pre-algebra and many algebra topics for the first time.  They will have been introduced to important concepts in high school geometry, Algebra 2, trigonometry and calculus and you're holding off on the ones that require maturity to grasp.  If you've ever seen TPM, you won't be surprised to find out that TPM 4 includes the reading comp portion of the SAT as well, but you have to go a bit slower because of all that unfamiliar vocabulary. If you were fortunate enough to do Pre-K Phonics Conceptual Vocabulary and Thinking, and followed the directions with regard to the Word Board, the SAT vocabulary goes pretty quickly.  Some day, when my youngest completes his 7th and 8th grade high school enrollment nightmare, I'm going to spell out in detail why we're doing this.  Until then, just go with the flow.

We're not even going to look at the SAT until the summer after 4th grade and really get into it a year later.  Before then, we've got a lot of ground to cover, and it includes fractions.

I'm going to need 2 articles to do it, and they'll probably be long.  The first article is going to lay down the ground rules that apply to math starting in Kindergarten and that you will use thereafter if you want your child to learn.

## Friday, December 1, 2017

### Pick the Right At-Home Math Curriculum

I spent the last few days thinking about the comment I received from Anonymous asking about current+2 curriculum for a 2nd grade child.  The last two articles on this topic were experimental and not helpful, and I'll delete them some day.

Taking a step back, here is a better version of the question: "What is the best way for a 2nd grader to work through 4th grade math so that she (or he) obtains all of the grit related benefits from doing so, learns more math, subject to the following constraints":
1. You've only got so much time to help and you're not a teacher
2. You need a high MAP score and teacher recommendation for the GAT program
3. You may or may not have to pass the COGAT this year
4. This child is only 7.  And not necessarily good in math.
5. You can't afford a tutor or an after school math program.  Plus you hate driving.
6. If your child does get into a GAT program, you want them to be the best.
7. If you run into problems, you're going to send 19 emails a day to getyourchildintogat@gmail.com so this advice better be good.
8. Math curriculum from US publishers stinks
I updated the article How To Create A Math Genius to be more clear about this situation. You might want to refer to the content starting at first grade.  In this article, I'm going explain why my curriculum choices are counter intuitive and logically valid.

My top 2 choices for curriculum are Go Math from Houghton Mifflin and Eureka Math.  A few years ago, a teacher suggested I review Eureka Math for 4th grade and I had a pdf of the whole book but I can't find it.  It's totally spoon feeding math, not only in the book but in the problems.  Go Math has a more intuitive approach, which means more concepts and less actual math.  For a kid who's already been through the advanced math exercise, he can do the Go Math homework for current+1 on the bus while playing Minecraft and discussing Star Wars memes.  And get them all correct.

If I was more worried about the MAP, I'd go with Eureka.  If I was more worried about the COGAT I'd go with Go Math.  I would probably pick Go Math anyway.