Chapter 4d - Leveraging Test Prep Math

When your child completes Test Prep Math, you might be wondering what to do next.   You might just be happy to watch the impact over the next few years on test scores and grades, somewhat confident that your child on the verge of something bigger.  The official goal of Test Prep Math is a child who is comfortable working with complicated material that she has to spend a lot of time to figure out and will make many mistakes on the way to a correct answer.  These are the core skills that drive academic success.   These are stated in the introduction in the form of a survival guide for the parent.

The unofficial goal of Test Prep Math is much more powerful and this chapter is going to unleash your child's new found potential.  While the short term target of the course is the COGAT and the ITBS, the medium term target is to live up to the score in the next few years.  There's much more in TPM than I let on in the introduction to the book.

Since this chapter is going to be so long, I have to resort to links to help people get right to the immediate needs.  Here they are:

If a link doesn't work or it looks like I'm only 12% into a section, it's because I haven't finished typing.  In fact, right now none of the links aren't there because I have to get to my full time job.  Feel free to post a question or comment to chide me into hurrying along.  It's February 2016 right now, and everything on the list above is happening short of Trig, but I have a clear path in mind and have settled on this as my eventual goal.  I also see the convergence of verbal skills and math skill happening, but you probably figured that out when you started reading the questions in TPM.  If you think this is nuts because your child is now 10 or 11 and I'm talking about AP Calculus, wait until I finish my bonus section on the SAT.

When my younger child finished test prep math, we did nothing for a year.  I'm a big fan of taking a year off of math, especially if that year is 4th or 5th grade.  He went about 18 months without missing anything ever after he started working with TPM.  Ironically, never getting a problem correct in a workbook is the path to perfection. The problem with taking a year off of math is that my children get really crabby all day if I don't give them chores or math to do on Saturday morning, and I've decided that me doing the chores while they do math is in the best interest of both my children and the cleanliness level of the bathroom.  Plus, no one is allowed computer time in the house unless they do math.

School Math
We approach school math in a slow, plodding way.  I don't want school to undo a child who considers it normal to spend 25 minutes and 3 attempts to solve a single problem.   For example, yesterday he was asked to divide 492 by 3.  Long division is forbidden in the house.  We discussed the goals for this problem, which is to avoid any thinking or effort.  What's the easiest, cheatyest way possible? Well, 492 is hard.  300 divided by 3 is easy. It's 100.  That leaves 192, still too hard.  I was hinting at 180 next but my son settled for 150 / 3 = 50.  That leaves 42, but 30 / 3 = 10, and 12 / 3 = 4.  All of this added back up is 164.  It could have been done in 3 steps, but it was done analytically, which is a plus, and mentally, which uses working memory.

There are three important things to note about this approach.  First, the only way we can work through a problem like this is because the months of test prep math created a culture where this is possible, a culture of analysis and patience.  It also requires more working memory to decompose and recompose a problem.   Secondly, continuing to foster these practices and skills will set the stage for higher order math.  Practicing long division sets the stage for the 4th grade train wreck, not to mention a kid who falls on his face in middle school.   Third, I noticed that our work in algebra and trig (again made possible because we know how to go slow and be patient with bafflement) usually points to about 3 parts and an iterative round trip between question and answer.

We've hit some roadblocks since TPM.  Complicated pre-algebra expressions can be solved mentally, and my kids are pretty good.  There's a small step between TPM and pre-algebra (it's called order of operations).   I was hoping to lay the groundwork for algebra with pre-algebra but my kids don't need the groundwork.  When we got to algebra, my children looked at it like they could solve it and wouldn't listen to me.  This is my fault.  When test scores skyrocketed after TPM I was left out of the math equation and it's a transition now that I'm back.  Also, they expected 'x' to be a number with a single value, when in fact it's something different.

But we're back on track with algebra and trigonometry, and the only reason is that they expect math to be slow, complicated, and require a lot of thinking to get through all of the moving parts until the math is slowly revealed.  During grades 3 to 5, school math, even in the country's top gifted program, is a mundane task of learning and practicing new syntax and concepts.   We basically skipped this part of education while doing TPM followed by other things I think are challenging.

At the time of writing, I have an 8 year old and 12 year old.  When I say 'back on track with algebra and trig', it looks like this.  We tackle a problem, backfill the mundane or new concepts we skipped, spend a lot of time confused and baffled, make sense of it, get a few wrong answers, try again, and suddenly we know something about polynomials or the sin function.  There's nothing special about my kids (compared to the specialness of lots of other kids), and yet they are capable of plodding along with some really cools stuff.  What I'm hoping to convey with this article is that a child/parent team who invested in Test Prep Math experienced a revolutionary pedagogy and really big things are now possible.

Competitive Math
If you search for "competitive math tests" you'll find sites like this one.  Most of these exams are multi grade, and if you print one for 3rd grade, you'll probably end up with 5th grade questions at the end of the test.  We tried these before.  I have the best books, most expensive competitive math books.  For years, we had nothing to show for it.  Recently, one reader asked why should she get Test Prep Math when a book like Challenge Math is available?   If I had kids interested in math and science and other great things, if they were truly gifted mathematicians, Challenge Math would have been preferable, or Mathletes, or any number of books.  These are great books.  But at the time these books would have been appropriate, I instead had falling grades and diminishing interest, and the solution was Test Prep Math.

And now we can just skip to competitive math, the SAT, and on to high school math.

The instructions for a competitive math test include a time limit, typically 45 minutes.  You know better than that, so if you do these, ignore the time limit at home.   Give your child 1 or 2 questions each session, and let the child take a lot of time on each question. There is an enormous amount of learning locked up in each question and rushing through each question by-passes most of the learning.  At the beginning it took many weeks to get through a test.

During practice, these are best worked in a team setting.  I almost never know what to do when I first read these.  I can't imagine a child getting most of these problems correct on the first try.

Ever single one of these questions requires application of one or more problem solving techniques after evaluating the question in detail, backward and forward. Warn your child of this.  At this level, most of the questions probably can't be solved by drawing a picture.   When we go through these tests, I'm looking for a problem solving technique as the answer. The actual answer isn't relevant.  I want the strategy.  If my child can't tell me the strategy, I don't even want to discuss the answer, which is probably wrong anyway.

The two most important strategies that I'm looking for are problem decomposition and rewriting the problem as a simple problem, solving it, and then trying a harder version.  At least one question will require an organized chart to solve. These techniques turn a single problem into a 30 minute session, and turn a math problem into a learning experience that can be applied to all subjects.

For example, most tests have a picture of layered squares or triangles and the question asks how may triangle there are.   The child inevitably misses the question because they didn't see a shape hidden in the question.  This is also a candidate question for an IQ test.   The ultimate solution approach is to create an organized chart, but the best way to solve this is to first draw a single triangle that has a count of 1. Then draw a line down the middle of the triangle and you've got 3 triangles.  Don't stop just because the light bulb went off.   Double the current drawing, and if you do it right, you'll see 7 triangles instead of 6.  Or you'll just see 6.  But you're child is working their way up to a thorough understanding.

What's great about doing competitive math problems is that the child is most likely to get it wrong the first time.  This is not just the recurring main theme of math, but the recurring main theme of learning.  During the draft stage of TPM, if a child got one of my questions right on the first try, in a short period of time, I "rearranged" the question to improve the expected error rate.

In many ways, I think competitive math is a good substitute for Test Prep Math for older children, provided that the children have a firm foundation of core skills.   Competitive math sets a very high bar, and Test Prep Math is the ladder.  The challenge is that higher levels of math require problem solving strategies, and problem solving strategies are in the middle of the pyramid.  But like anything, if the child is willing to concentrate and put in the time, these strategies can be learned.

Euler's Formula
Here's an example of an exercise we did after TPM that I used as a filler while looking for other math to do.

Before I present this formula, which is shockingly complicated, I'd like to point out that it's enormously rewarding to work through this formula with Test Prep Math graduates.  The parent's love it because it makes me look like I'm a wonderful magical math teacher, when I'm just leveraging the hidden, Second Foundation of Test Prep Math.   The magic is that the parents don't know what the Second Foundation is all about.  They think it's a myth.  TPM doesn't appear to have any of the elements required to understand this formula, but they were lurking there all of the time.  I'd also like to point out that it's going to take about a year to get through this formula to the point where the children really understand it.  Finally, a child who has a rudimentary understanding of Euler's formula is going to look like they are profoundly gifted in math.  Our little secret is just that they have the core skills.

In this section, I'm going to walk a 3rd or 4th grade child through this formula using nothing but their skill set.  No knowledge is required, except for understanding of the very first Star Wars Movie or the movie Frozen, because it makes for a more interesting story.  I'm going with Star Wars.  When I wrote Test Prep Math, I was determined to get 2 boys where they need to be in math (which is the end of this discussion), but the majority of positive feedback I get is from the parents of girls, so I'm going with Star Wars in full confidence.

Here is the formula:

This formula introduces the 5 most powerful numbers in math:
  • e, which is roughly 2.71, and we'll get to it last.  E is like the last Jedi, somewhere on a planet, and we don't have the map yet.  We're only on Episode 1, The New Hope, and e won't be back until Episode 7 Trigonometry Awakens.
  • Pi, which is 3.14, or so.  Pi is special, like a princess.  But like Princess Leia or Princess Anna, pi has enormous powers and is going to end up being the General of math, calling all of the shots and directing the action.
  • 1 is the smallest of all numbers, but is the foundation of all other numbers.  3, for example, is really just 3 ones.   
  • Zero is the Death Star of Numbers.  You take a big number like 5,393,885,427, and multiple it by zero, what do you get?  You get zero.  I tell the kids that the number zero was originally drawn as a Death Star but it was too wide for Gutenberg type so after the thirteenth century, it was just an oval with no ray gun.  They ask "Really?" and I say "No, I just made that up."
Before we move on to taking apart this equation, we have to remove zero or it will ruin the whole thing because it's so destructive.  Plus, we need something a bit harder, so we're going with:

eiϖ = -1

Now we've got 5 incomprehensible symbols, the equal sign, and the number 1.  We're going to take these one at a time over the course of the next few months.   In case you count only 6 symbols in this equation, you'll see the 7th below.

First, we're going to tackle negative numbers.  The second section of Test Prep Math looked like logic, but negative numbers were everywhere.

A negative number is just the opposite of a positive, and they cancel each other out.

So 3 + -3 = 0.  That's the end of the story.

4 + -3 = ?  (Let the student figure this one out.)

You may see 4 - 3 = 1, but what's really happening is 4 + -3 = 1.  The minus sign was invented because it takes less to say "4 minus 3" than "4 plus negative 3", but if you don't understand that minus is really adding a negative number, you won't understand what's really happening.

I like to think of positive numbers like Jedi and negative numbers like Sith.  Again, zero plays a key role here because if you add any negative number and it's positive part you get zero, like 5 + -5 = 0. They cancel each other out.

One of the big problems with school math is that really great math concepts, like Pythagorean's Theorem, take many years to understand with any depth and maturity, but school gives the child about a week to practice it on a worksheet.  If you introduce these concepts way ahead of time, this gives the child a few extra years to come to terms with it, so they won't have to walk into a test memorizing a list of formula's that they don't fully understand.  We're not going to do that here, by the way, any time soon.

Next, we have to tackle multiplication and square roots to understand i and pi.  I teach i first because it takes so much longer to comprehend, like many months.

Multiplication is a given, like 4 x 3 = 12.  If you multiply 3 x 3 = 9, you have the square of 3.  The reason it's called a square is because when you draw a picture of 3 x 2 or 3 x 5, you get a rectangle. You only get a square if the numbers are equal, like 4 x 4.  You have to point out that drawing the picture shows the area of a rectangle.  Multiplication and calculating the area are exactly the same thing.   The definition of multiplication is a picture of 4 x 3 with 12 cubes.  This is also the definition of area.  They are exactly the same thing, and the Common Core annual test is going to ask this explicitly.

So if you have the square of 3, as in 3 x  3 = 9, what is the square root of 9?  It's the number that when you multiply it times itself, it gives you 9.  So the square root of 9 is 3, and the square root of 4 is 2, since 2 x 2 = 4.  What is the square root of 16?  It's the number that solves the equation ___ x ___ = 16.

At this point, we have just uncovered a big stumbling block to kids of all ages in math.   They overthink definitions for new terms, terms like "square root".   They try to solve square root of 16 instead of solving ___ x ___ = 16, like there's some mental magic that is supposed to happen when the term square root is used.  Make a big deal of this, because you're going to need this theme again and again, especially when we get to algebra and the child sees x and forgets how to multiply, or prealgebra when the parenthesis wipe out all prior knowledge or arithmetic.

So the months pass, and we've got an understanding of square root and of negative numbers.  Almost.

What happens when a Jedi and a Sith end up in multiplication?   Weird things.
  • First of all, 3 x -4 = -12. 
  • -3 x -4 = 12
My Star Wars analogies fail.  It looks like any time 3 Jedi fight 4 Sith, we end up with 12 Sith.  I'll let school fill in the blanks. For now, I've got a kid who knows the two rules and in the next 6 weeks, I'll just check once in a while to make sure it's still there. School can close that gap later.

Once the child can solve square roots and remember the rules of multiplying with negative numbers, ask this question.  What is the square root of 4?  The answer is usually 2.  Or negative 2.  I'll try this question next:  What is the square root of -1?  What number satisfies the solution to __ x __ = -1?

At one time, when there were only 30 or so fields in math, the mathematicians had a bunch of theorems that they knew were right, but they couldn't prove.  Then one enterprising mathematician made up the number i.  i stands for imaginary, because the number doesn't actually exist.  With the power of i, all of the unproven theorems could be proven and math just exploded into 96 fields.  We decided that i is the dark saber, because it is so powerful but has something to do with negative numbers.

i is the square root of -1.  It is nothing else.  It can't be calculated.  What is the square root of -1?  It is i.  What is i x i ?  It is -1.  That's it.  Don't overthink it.   I'm a big proponent of "don't overthink it" because I have to use this phrase so much in getting younger children to really understand algebra. So don't overthink it.

What is the square root of -4?  ___ x ___ = -4.  It is 2i.  2i x 2i = 2 x 2 x i x i = -4.  It turns out that i appears on the MAP test in 7th grade for a child that happens to be a little ahead, where "a little" is defined as 99% plus.  This probably doesn't matter unless you live in Chicago and 99% is desperately needed to get into high school.

i is going to take a few months of practice, along with square and square root.

Moving on, what does it mean that the i and the pi are in superscript in Euler's Formula?  That is the 7th mystery symbol that I mentioned above.  Back to squares.

So 3 x 3 = 9.  This is written as 32 = 9.  The 2 means to multiple 3 x 3, or 2 threes.  What does 33 = ? Expect something goofy like 18.   33 = 27.  There is too much going on with 3's to derive patterns, so we'll work with 2 to the power of 2, 3, 4, and 5.  When you write these out and solve them, you'll get the light bulb.

By the way, what is 21 = ?  It equals 2.   I don't have a great 10 year old explanation for this, nor can I explain that 20 = 1.  But these facts are worth noting to a child comfortable with the unknown.  At this point, I usually ask for the child to solve the square root of every number before 100.   There are two ways to do this exercise, and we do both.  The first way is to find numbers with integer square roots, like 4 and 9, and then calculate the square root for numbers in between using fractions, which may require a few months working with fractions.  The second way is to use the prime factorization of each number and just leave the square root of 48 as 4 x the square root of 3.  I like exercises like this because it adds an element of Kumon repetition to the mix.

The" power" is the hidden 7th element in Euler's Formula, eiϖ = -1.  That means we have 2 left. Well, it's really 3 more elements when you have to figure out eor maybe 10 more, because of e.   At this point, the child is so advanced in terms of math that we might just take off a year and come back to this.  It's possible to get through most of this in an afternoon, but the child really needs about 6 months to understand square roots, powers, negative numbers, and i.  If the child is in 4th grade, we've got a whole year until 5th grade summer kicks off the next level, and there's always competitive math.

On to pi.  If your child isn't in a gifted program, the best way to introduce pi is a long string, a ruler (using cm and not inches), and ever pot and pan and jar in the house.  Save a jar, a bowl, and a pot for later.  Then ask your child to give you a chart of diameter versus circumference and tell you what the magic number is.

But I define pi formally as the area of the unit circle.  It's not so magic.  It just is the area of the unit circle.  The fact that it has super powers and shows up almost all of the time in tricky math situations against the villain of some unsolvable problem belies its humble origins. It was pi who took e, put it on a card, and stuck it in R2D2 for the rest of the movie.  Here is the diagram:

There's a lot you can do with this diagram, but the important point is that the radius is the distance from the center to the edge of the circle at all points in all directions.  Let you child figure out the area of the square (2 x 2 by definition) and guess the area of pi.  This is also a good time to discuss rational versus irrational numbers, which means introducing fractions, another tangent that could take 3 weeks, and also transcendental numbers, a select group that includes e and pi.

I could write another 50 pages on pi, but I suggest you get a stack of graph paper which you will need for later math, and look at how pi predicts the area of the circle when the square is 3 wide, 4 wide, 5 wide, and 6 wide.

As an aside tell your child the area of 2 circles, say with radius 3 and with radius 9, and ask the child to calculate the area of a circle with radius 5 or 6 without using pi.  This is a 3 or 4 week project and will involve fractions.  You could use decimal and a calculator if the child has decimals in school, but I generally don't because fractions have much more meaning.  Here and there, usually on weekends, we just spent 6 months learning about math the way mathematicians did.  It used to frustrate me in school that we got a week to learn and prove some formula or result that took mathematicians 200 years to uncover.  No wonder kids hate math.

I probably should write a book on this topic because none exists for younger age groups.  It should be called "Learn Math The Way Mathematicians Did" but it would have a small readership because I need children who are comfortable working with a steady stream of the unknown for long periods of time despite mistakes.  Maybe I should call it Test Prep Math Level 4.

I think we do more art at home leading up to real math than math.  Art is great preparation for math and a very good way to practice the soft skills learned in TPM.  But our home has an iron clad rule:  "No one gets to touch a computer unless they do math," so we end up plodding along in math.

Summer After 5th Grade
Up to this point, we've been biding our time.  Now the fun begins.

Out of curiosity, I bought the College Board's SAT Prep book, with 10 practice tests.  The College Board publishes the SAT, so I guessed correctly that their test prep book would be a lot easier and less serious than offerings from other publishers.

I assigned my child any easy problem he could do in the first practice math book.  He found about 5 and got some of them correct.  Of course, incorrect answers are nothing new to a TPM graduate and didn't slow us down one bit.  Then we did the same thing on the second test.  Next, we used the TPM approach for the next level of problems.  I asked for him how to figure out how to solve a problem with unfamiliar material by analyzing the problem.  This required a lot more thinking and rereading the question.  We uncovered a treasure trove of tricks and techniques hidden in the problems.  For example, we found that some questions require the student simply to step back and think or solve a small part of the problem to get the answer instead of the brute force approach.  Since TPM hammers away at reading the question instead of solving it, this came quite naturally after a few weeks of practice.

Any time we came across pre-algebra concepts like parenthesis or fractions, a bit of backtracking with a pre-algebra book was needed, but mostly the college board's version of the test is pretty light on math concepts and heavy on analytical thinking and basic problem solving skills like decomposition or staring with an easier problem.

At some point, a test was completed, and then another.  I reassigned each test, fully marked with answers and notes, and asked my child to explicitly identify the trick or technique required to get the solution in the easiest way possible with the smallest amount of thinking or calculating.   While the book only cost $10, this exercise meant that we actually had 2 $5 books to work through.

I don't want to spoil this chapter, but I'm going to anyway.  Our target is the reading test not the math test.  Reading is so much more important than math and is much harder.   I think that the average reading score is 10 percent points less than the average math score on the SAT, the MAP, and the ITBS.   Think about that statement and how wrong it must be for a second, and then let me add "for children who excel in math".  It's also more important because TPM put us so far ahead in math that we had nothing to occupy our time other than competitive math and Euler's formula.  I had been worrying about this for many years, and it was a growing concern during my semi-annual review of standardized test scores in the 2 years leading up to creating TPM. If you flip open one of the TPM books to the middle, it should be obvious where this is heading.  But first, we need to get 8th grade math out of the way.

8th Grade Math in One Chomp
We skipped pre-algebra at home.  It seems like the continuation of the misguided focus of US math on calculation.  It's a pointless and boring exercise.  I figured my child would get enough useless calculation at school, so we just moved to the last chapter.  The explicit reason we did 8th grade math was that the program was going to cover it in 7th grade and this was an exercise in Power Bucketing. Read the article from Feb 2016.

The best way to learn a math calculation concept is to use it in a problem.  This gives meaning to the more boring parts of math.  That' why we skipped pre-algebra.

The school uses CPM Math, which is perfectly designed for accelerated math.   This course has versions for 5th through 8th grade.   The course was motivated by very recent research with the goal of moving math from the industrial age to the information age, from boring calculation and memorization to thinking.  This research also included ways to get lagging, under-educated kids from impoverished school districts up to speed.  As far as I can tell, CPM math requires few prerequisites and a bright child can just jump in and learn.

We got 6 of the 8 books.  My stubborn child would choose any book that I didn't want him to do, so we ended up doing a different book each day during the summer.  By the midway point of 6th grade, he did 30 to 50% of each book and we skipped a lot of redundant material.  I think that was probably enough.

Since I don't have an under-educated child in an impoverished school district, I could add a bit of bravado to the mix.   For example, we saw y = mx + b for the first time in a series of problems designed to impart intuition and understanding of math used in the real world, but they neglected to actually point out y = mx + b for most of the book.  So I assigned an exercise similar to ones I outlined a few sections ago.  I handed my child a piece of graph paper and asked him to show me the 12 interesting values of m, set b = 0, and graph them.   After a few hours of misfires, he understood on his own that "interesting" as 0, 1/2 1, 2, -1/2, -1, -2, and infinity.  Maybe it's not 12.  My contribution was mostly to say "that's not interesting" when he added 3 or 4, since 2 was already interesting.  Then I simply asked for an explanation of what b does.  We did this with each section.

Back To The SAT
Once the College Board Math book was finished, I bought one from Sylvan.  It was a much more serious book and required not only a higher level of thinking but some understanding of high school math concepts.   By this point, we are easing into algebra in order to tackle some of the harder problems.  Algebra isn't the slightest problem, because the second section of TPM lays the ground work, and but we have to do more pre-algebra backtracking because parenthesis are a problem. Unbeknownst to me, while we were skipping pre-algebra at home, his program was skipping pre-algebra at school.  Oops.

I really enjoy the exercise where we go back over solved problems to name the solution strategy. This is a great learning experience.  The children seem to enjoy it because it seems easier to them and more interesting than solving the problem in the first place.  I enjoy it because it is actually much harder for them than solving the problem in the first place.  Plus there's more learning involved.

But by this point the need to transition to algebra is slowing us down.  We're back to just a few math problems each day.  We should also go back and take another chomp out of 8th grade math.  But I've got 2 SAT books and we are only doing half of each.  I'm gearing up for algebra, but I haven't decided yet.

So I got out the College Board SAT Test Prep book and assigned a reading section.  It turns out that a child who was forced to read my convoluted, vague and misleading math questions in TPM for 2 years is perfectly qualified to get at least a 50% on high school level reading comprehension questions after a bit of practice.   But something was definitely missing.   It turns out that the SAT is measuring specific reading skills, and not only didn't we have these, but we didn't know what they were. We hit a wall at 50% on easy reading comprehension problems (at this level).

The Sylvan book is much better for reading comprehension because it is more carefully designed than the College Board book.  The downside of Sylvan is that it is much harder.  The upside is that you can see the skills.  We applied the same 2 step exercise that we did with math to Sylvan's reading comprehension.  It is more tedious and painful for the children because it requires rereading, but only partially so.  TPM makes re-reading par for the course, not to mention making mistakes and trying again, not to mention comfort being baffled.  But to keep everyone in the game, we just focused on wrong answers.

What I never did ever with the SAT test prep books is to read the sections that explain any of the concepts.  I probably should some day, but this content is spoon feeding and not learning, and there is no grit in just applying what someone else tells you to do.

Last weekend, we got to the advanced essays in Sylvan, and by this point, there were only 2 wrong answers.  And a few months later, we're down to 1 wrong answer.

That's the punchline to TPM.

For those readers who have never seen Test Prep Math and can't understand why a math course for 2nd to 4th grade leads to high SAT test scores in reading comprehension (for an 11 year old), I should explain what the skills are that are explicitly covered in TPM, and why reading and math naturally reach a convergence point at the top of the pyramid that starts with the skills at the bottom of the pyramid. But I'm out of time and will have to finish this after we get past the 7th and 8th grade tests for high school.

I suppose the punchline should have been that we moving through the math section in the second SAT book 6 years before the big event in high school.   But you can't beat math AND reading scores.

I Need a 99% on the ITBS Right Now
There are 4 parts to a 99% on the ITBS by 4th grade.

There is nothing on any standardized test including the SAT that is as complicated or mentally challenging as TPM.  No word problem in school math through 8th grade will come close.  I have the full grade school curriculum (lots of variants too) and compare word problems for my own amusement.  The working memory demands of TPM are above cognitive skills tests if you make your child do the problems mentally, which you should even if the work goes slower and results in more mistakes. Especially if the work goes slower and results in more mistakes.  So this is a good start.

The child should become accustomed to checking their work and expecting mistakes.  This single skill is worth 15% on standardized tests, even for the brightest of students.  Back in the good old days when the solutions of TPM were full of errors, because I spent more time reformulating the problems to make them harder than checking solutions, children learned to check answers the hard way. Apologies to all parents who use solutions.  You should stop doing it, by the way.  When your child announces the correct answer, tell them you are skeptical and make them prove it.  This is the 15%.

But there are two more skills needed for the ITBS.  First, the child has to be a year or two ahead in math.  I don't know why this is, but this is the case empirically.  Secondly, the child has to be adept and school math concept calculation, like decimals or long division.   TPM intentionally avoids this topic because it will inevitably destroy a child's interest in math.

There is good news. First, TPM develops a hard worker if nothing else.  Secondly, you can get away with making your child do boring school math every night for at least 5 months without ruing your child permanently. Plus, it's easier than having to do TPM, if less exciting.  If we have a big testing year, leading up to the test, we get out a school math book at least one year ahead, and do the problems.  It's a necessary evil if test scores are required for program entry.

I Need a 99% on Reading Comprehension
When I devised the word problems for TPM, my main goal was a problem that was hard to understand in order to teach the skill of thinking about a problem before answering it.  In order to maintain the challenge, I rearranged the way problems were presented, worked in vague or poorly chosen words.  I started to see permutations in logic among the 'is' and 'is not', so I rearranged things to cover a wide range of sentence structure and flow.  The best questions don't make sense without a discussion or argument.

I didn't realize the unexpected achievement until we started dissecting SAT reading comprehension problems.  Since we're a math house, I feel like this is my biggest achievement.   My children can read SAT reading comp problems at young ages because they are accustomed to reading convoluted word problems.

What more would you need to handle 4th grade reading comprehension?

The answer is practice.  In looking at reading comprehension questions, I've noticed some unexpected tricks that aren't found in a math book.  You may pause to think 'duh' if you want, which is what I'm thinking.  Some questions in reading comp present 4 bad answers for the child to pick the least worst one.  Some require leaps of intuition.  I've seen a key sentence disguised as the instructions (put in instruction italics) so that the child doesn't read it.  There are other tricks, and these are easy to spot if you get a reading comp book, practice it, and figure out why your child got a question wrong.  While you do so (hopefully with a more advanced book than grade level), get the Word Board going again on the refrigerator, because vocabulary is king in reading comprehension.  If you haven't done Vocabulary Workshop until you are two years ahead, do it.  You might even resort to Wordly Wise.

What Does A Word Problem Have To Do With A Figure Matrix
The main problem with prepping for a cognitive skills test like the COGAT is that those figure matrices can be distracting.  Many parents that the approach of training their child on figure matrices and lose site of the purpose and methodology of these tests.  Facility with figure matrices is not a predictor of academic success.  You might notice that there are no subjects in school that use figure matrices, and figures only play a minor role in geometry.  The point of these tests is to identify strong thinkers, and strong thinking follows the core skills.

The word problems were explicitly designed to be confusing to read.  The nature of the math in the problem is conducive to mistakes, whether from hidden information, subtle relationships, or just the fact that there are 3 equations in play.   This forces the child to slow down, read the question multiple times, and live in fear of mistakes.   My theory is that a child with this type of training can do anything, whether it's the COGAT or advanced literature or math.

This approach went so well that I started nominating myself for various awards.  The heck with the golden apple, this book should win me the diamond apple.  Then I met a few TPM graduates who got close but not close enough.  I reviewed their scores and started working with them (thanks to Skype) directly on figure matrices.  I found 2 problems.

First, the kids learned to cheat their way through my problems in both sections by writing down the equations as they went.  This is a big problem and can undermine the course, but is easily fixable in a few months of working memory building - via the problems.

But I found a second problem.   Some kids have no idea what a flip or rotate is.  It didn't occur to me that a parent wouldn't buy a practice test, even though I've recommended it as mandatory about 194 times in my blog, and the cheaters don't go through the questions like they are supposed to, one shape, one line or curve at a time question and answers before tackling the problem.

I can't solve the first problem since I can't be in everyone's house at the same time, but I'm working on a solution to the second problem.  I have always been a fan of the underdog, since one of my kids was the underdog, and the more underdoggy the better.

My Child Is Bored In Math
One of the consequences of teaching math at home is that the child is bored in math at school.  This will be inevitable.  If you go ahead a year in math, or you teach your child the fundamental skills of learning, she should skip 2 grades in order to be challenged.

A parent with a single child can learn at a super fast accelerated breakneck pace in when the parent tailors the entire curriculum to the learning needs of a single child on a daily basis in an environment.where there are no tests, where there is no time limit, where mistakes are expected, and where tomorrows assignment can be harder, easier, or something entirely different as needed.

Our rule is that math class is a good place to meet friends, take a break, or learn to do homework that you don't want to do.   It's my child's opportunity to learn to take ownership, even if the learning experience includes a C in 4th grade math because my child likes to learn things the hard way.  If you're that great in math, maybe you should be working on your budding social skills (to put it in a positive frame), or to learn to understand where other children are.

How To Take 2 Years Off of Math
I'm not sure why I posed this question because its a lie.   Once we got past 2nd grade math in K, I decided that I would just let my child coast along in math until after 4th grade, when the combination of their developing brain and the math curriculum make math worth doing again.

During 1st grade, we did very little math.  But I ran into two problems.  First, if my children don't wake up on Saturday morning and do some challenging academic assignment, they are rotten the whole day.  If this challenging academic assignment is to read for 2 hours, they are sluggish the whole day and get chubby.   Chores never really caught on until later in the day, and they both decided that practicing an instrument is more of a night thing after seeing the Buick commercials where the protagonist plays cards with Jazz musicians at 3 am.

What I mean by take time off of math is to step away from grade level or grade level plus workbooks and just look for math problems for the sake of math problems.   This is how we ended up doing competitive math and going way ahead.   Taking 2 years off of math means doing a single problem and having no time limit, not 10 minutes, not an hour, not 2 months.  We'll just do it slowly and think and talk.

One of my parenting rules from test prep came back to haunt me.  Start with something that is so hard for your child that it takes them 3 days to do a single problem which they get wrong.   Revisit the material periodically.  Eventually they get it. Eventually, after they get it, the pace magically picks up, and before you know it, they have some sort of mastery.

Algebra, graphing, proofs, and of course competitive math all fell one after another while we were biding our time until we would do math again.  Standardized test scores also fell during this period, because before 6th grade, standardized tests are calculation heavy and we did very little arithmetic practice.   I suppose what I really mean by taking 2 years off of math is taking 2 years off of arithmetic.

It takes about 3 months of arithmetic practice to get back to an acceptable grade/test score, and by this time, we have something much more valuable.

I originally designed Test Prep Math to be a year off of calculating, but to meet my objective of inducing mistakes to teach "try again", it ended up being calculation heavy.  Section 2 was supposed to be a precursor to thinking, but it is also calculation heavy

We're Doing Competitive Math
Competitive math is the next best thing to test prep.  The difference is that one competitive math problem might end up being 5 problems before you understand all of the unspoken steps and unspoken rules.  On cognitive skills tests, these sub problems are usually obvious.

Because of this, in doing competitive math and teaching algebra I identified a missing skill.  It was always implicitly there, but I think it needs to be explicitly spoken in these contexts.   Every problem in both fields is really 3 or more mini-problems.  Unlike Test Prep Math, where it's obvious that there are multiple problems in every problem, it's not so obvious in a cryptic or terse problem like 'Prove the sum of two even numbers is even'.

I noticed that kids look at a problem that is short, expecting to know the answer, and unable to solve it.  The reason is that it's 3 problems, and they haven't read the question long enough to figure out what the missing problems are.  In my hubris, I assumed that after TPM, it would be obvious, but a year later we're back to the same problem.  Because of this, I refined my skills list for competitive math and algebra:
  1. Be comfortable being baffled, because you will be.
  2. Take a long time to read the question, like days or weeks (varies by age),  You may have to look at it differently than you expect to.
  3. Every problem hides multiple sub-problems
  4. Mistakes are good.  Different approaches that don't pan out going to be inevitable until you get more practice.
Occasionally either of my kids will look at a super hard problem that took me 30 minutes to work out ahead of time and solve it in 2 minutes.  I can barely keep up with their defense of their answer, but it is correct.  This is the working memory factor from TPM.   It's really powerful.  But most of the time, each problem presents a brand new way at looking at math and we might take multiple days to solve it.

Competitive math and algebra seemed undoable to me because we never really made progress in a 45 minute session.  One day I just decided to see if we made any progress in 8 sessions of a 4 week period, and it worked.  I don't know why I ended up with a fixed mindset.  Fixed mindsets are bad.

Going from TPM to Algebra
In the second edition of TPM, I added Section 2.  The official reason for including it was to make 100 questions harder than anything on the quantitative section of any test and make short work of pre-algebra.   While this section is a great refresher on number sense, I designed the questions with pre-algebra in mind, specifically the working memory and complicated figuring out of equations in the absence of algebra tools like transformations.

The proper way to do these questions is to do them mentally with no writing.  This takes a lot longer and results in more mistakes, but during this process the brain is devising strategies that are really powerful when applied later to advanced math.

When I was writing these questions, I was also struggling with a 4th grader who just couldn't get the concept of 'x' or 'y'.  I wondered if I could just build abstraction into Section 2.  If I had a child who just survived at least the first half of Section 1, then I had a child who was prepared to spend a lot of time reading the question, and that is the secret to abstract thinking.   In deference to the COGAT, Section 2 has 'F', which is " + 3" or " - 4", and not 'x', which would be 3 or 4.   For some reason that I never bothered to research, little kids tend to get F even if they don't get x.

After Section 2, I've got a child who's ready for algebra.  The only thing standing in the way is Pre-Algebra, which involves a lot of boring syntax and rules, and tends to ruin math, and 8th grade math, which handles math concepts in a more challenging and useful way.   Pre-Algebra tools are best learned in the context of Algebra.  It gives meaning to the tool set and is a lot less boring.

But first, there's 8th grade math to deal with.   We took a single piece, y = mx + b, and spent a 3 weekends doing this exercise:   Graph the 8 interesting values of m, setting b = 0.  After this, take one of the m's (not 0 or infinity) and graph the 3 interesting values of b.  Explain what m and b do to a graph of a line.

With that out of the way, we started our algebra course with the last page from our 8th grade math book.  Solve x in these 2 equations (one at a time, these are not simultaneous equations):
5x - 23 > 3x + 10
5(x + 14) < 9 - 6 x

Without writing a book on this, it's totally doable and introduces a lot of concepts, including x being a whole range of numbers and not just a single value.  This still leaves about 5 major concepts from 8th grade math, but we're Bucketing here for later.  The main value of algebra, in my opinion, is to apply the 4 operators to the equation to get x on the left and a number on the right.  For example, the first thing to do is to subtract 3x from each side.  Trial and error show which operations work and which just make the equation more complicated.

After this, I try to encourage my kids to apply transformation to their math homework.  A nasty side effect of TPM's Section 2 is that my kids find it easier just to look at an equation and solve it in their brains, which is an enormous amount of mental work, instead of just doing some quick transformations that require no work at all.   I'm not sure which is the bad habit here, but when they can avoid algebra, they do it to spite me.

The next step is even harder, which is to solve x2 + 9x + 20 = 0 and the common variants of this equation.  The goal is to get it to (x + 4)(x + 5) = 0, and see that x has to be -4 or -5.  This will take a few weeks.  Once this is out of the way, replace 0 with y.

You might note the fundamental theorem of algebra here.  This theorem was standing in the way of a lot of math.  Take x2 = 1, for example.  This has 2 solutions.  But what about (x - 1)2 = 0?  This is where i comes into play, which is something I introduced as soon as possible (age 7) with square roots to set the stage for this discussion.

The next step involves graphing polynomials of order 2, which is replacing 0 with y on the last equation.   We need to find the zeros, which we can do from above, but we also need the low point on the curve.  I'm thinking about a quick lesson in calculus.  Maybe after that we'll go back to 8th grade math to do transformations and symmetry or another random topic to let the algebra loaf rise a bit before putting it in the oven.

Some Math Notes
After a few years of experimenting, I've finally perfected pre-algebra through calculus in small amount of time.  The first graduate of TPM hasn't exited 4th grade at the time of this writing.  I'm switching back and forth between him and his brother on the same material.  The brother is in 7th grade and needs a 99% on a certain test for high school enrollment that I'm not going to mention for fear of the competition.  This section describes each topic that needs to be covered and how to do it.

I don't know how this can be accomplished outside of the unique approach of TPM.  There are children who are already at 99+% who really don't need TPM, but their math training is not leading in the direction of what I want to do next.  I've tried what follows with bright kids who went through other programs and I can't get it to work.

I'm writing this while I have time.  When this is complete, I'll simply remove this line.  I've got so much to put in here.  We've spent the last 3 years working on these topics.

I think of algebra as a worthy reason to cover pre-algebra.   Pre-algebra on it's own seems like doing long division or decimals.
  • See the bottom of this article for a narrative of me teaching algebra to a 9 year old friend in 25 minutes or less.
  • The starting point is equations with one unknown named x.  There are plenty of worksheets on the web for this.  The goal of these worksheets is to fix the equation.  By this, I mean that the equation needs to end up with x on one side and a number on the other, like x = #.  
  • There are usually 4 transformations - multiply both sides of the equation by the same value, divide, subtract and add.  
    • The most common mistake is to forget multiplying/dividing the whole side by the expression or number.
    • The other common mistake is not to write the transformed equation down in its entirety, which is necessary because you're going to screw it up and need to find your mistake.
  • I also like multiplying one element of the equation by 1, and 1 usually takes on the form of 3/3 or x/x.  One son pointed out that this is multiplying both sides of a fraction by the same expression.  I like that.
  • To determine which transformations you need to do, you have to ask what you don't like about the equation.  What you want is x on the left and a single number on the right, like x = 3.  Anything out of place is a candidate for a transformation.
    • So if 7x - 5 = 23, the problem could be the 7 or the 5.
    • Little kids get stuck on which to pick like there's a correct answer and they are going to get it wrong.  There are only 4 options (x,-,+,/) so try all 4 and see what happens.  If it looks better afterward, you picked the right one.  If not, try another.  This is a big shift in thinking from rote school math.
  • Parenthesis are usually a killer and we have to back-track with a pre-algebra worksheet or two or three, or I just have to explain how they work.  
  • I like to work between (x + 2)(x + 3) = x2 + 5x + 6 and back.  This is important to either find the two solutions of a two order polynomial or find the zeros on an equation if y is involved.
  • This is a good point to introduce the Fundamental Theorem of Algebra, the first of many times you will have to point this out.
  • Work through (x + 2)(x + 2), (x + 2)(x - 2), (x - 2)(x + 2), (x - 2)(x - 2) and see what happens.
  • The biggest problem with algebra is that it takes a long time before the kid reluctantly agrees that writing out the equation after each transformation is much quicker and easier than trying to do multiple steps at once.  Let the pencil do the work.
  • SAT practice books have good algebra problems.  I warn the kids (again and again) that as soon as you see x in a problem, you're going to have to rearrange the equation.  Each type of question on the SAT has an approach.
  • I like to start with a line and ask the kid to prove that it is straight.  We then prove everything we can after that.  I help a lot, but not with the thinking.  I should write down these proofs.  It's fun to get an adequate proof from a 4th or 5th grader that a straight line is straight.  
  • Geometry is useful to teach proofs and the thinking that goes with it.  Proofs are hard.  They are one good problem a day while they last.
  • Like proofs, a geometry problem on a test requires a special approach - solve everything you can before you read the question.    There are harder ways to do a geometry problem, but none easier.

Complex Numbers
I like complex numbers because it's just silly fun, good for imagination.
  • What is the square root of negative one?  It's i.  What is i2?  It's negative one.
  • What are the two solutions (according to the fundamental theorem of algebra) for x2=1?
  • What are the 4 solutions (according to the fundamental theorem of algebra) for x4=1?  For a very long time, mathematicians knew that 4 solutions existed but could only think of 2.  Complex numbers solved a lot of math mysteries.
  • To round things out, take (4 + 3i) and (2 - 5i).  Add, subtract, multiply and divide.  Division is tricky.

Exponents are introduced somewhere in pre-algebra and used with polynomials.   Later, exponents are explored on their own terms:
  • 2x is a function to be graphed, and during this process x1/2 and x-1/2 can be dealt with properly.
  • Next, there is x2x3 and (x2)3.   I've found through trial and error never to ask kids to simplify an equation with the 2 formulas behind these equations or even tell them that there are formulas.  Instead, I ask them to prove every problem.  They would have to then write x2x = xx * xxx and (x2)3 = (xx)(xx)(xx).  The reason I do this is that they need a useful problem solving technique and a way to slowly acclimate themselves to complications with exponents.
  • Finally, I'll throw in negatives everywhere and let the kid work it out for himself.
  • Also, once it's proved that x2x-3 = x-1, you can see why x0= 1, because x3x-3 = 1, and x3x-3 = x0.  Which is kind of cool.
I like to follow exponents with logs simply because it's good for thinking.  
  • I demonstrate logs by showing the problems that astronomers face while working with really large numbers.  It's just too much typing.  Logs make things shorter.
  • Then we list the powers of 10 as a warm up.
  • There are great introductory worksheets on the web to show Log250 or Log50.  I haven't gotten to ln yet.
  • I'll ask for a graph of log base 10, graphing log10, log100, log1000 etc.
  • Finally, I write down explicit formulas for x2x3 and (x2)3 and ask the child to derive on their own commensurate formulas for logs.  No kidding. 
I think this is perhaps my best achievement yet. 
  • First, we graph the sine/cosine triangle on the unit circle.  The hypotenuse is always 1.  We can derive the case of 45 degrees.  I need to research 30 and 60 degrees.  90 degrees is obvious.
  • I define sine as the horizontal value and cosine as the vertical value.  (I have to fact check this - we don't memorizing things in Math house.)  I mention that these are so powerful super math concepts that they show up solving all sorts of things in more advanced math.  They're like Pi, e and i.
  • The we jump right into the law of sine and the law of cosine, which is a cool generalization of the pythagorean theorem.  Add to this that the sum of the three angles on a triangle equals 180 degrees.
  • The fun begins next.  I'll upload images to describe this soon.   Take a triangle, with 3 angles and 3 sides.  Put numbers in for any 3 of these 6 and ask a) if it can be solved and b) how to solve it.
  • It's worth mentioning that any time there is a single variable in an equation with a bunch of numbers, then there is a solution.
  • After this, look for a stack of triangles or a tower of triangles worksheet and practice.  These usually require the geometry axioms I mentioned above.
Reading Comp
I've been working on this for the last 3 years with SAT practice test books.  I've always thought of TPM as a math book that will help us get 99% on the reading comp.  I'm about ready to provide the how to guide, but a certain 9 year old who just did section 7 on practice test 2 is demanding the computer so I'm not going to get very far.
  • I've noticed that kids learn more and pay attention more when we do it together.  I alternate between 3 methods - do it alone, and then we'll review and correct it, do it alone, but before you answer, tell me everything you know about the author, and do it together.  It's a lot like learning to read.
  • I've noticed a lot of mistakes can be avoided if you take a break after reading the passage and then go back and ask 'who is this author and what's their message'?  Is it a discussion, argument, etc.?  Is the author a scientist in 1960?  Is he mad about something?  What does he like and dislike?  The questions are going to expect you to know this, and it took me 3 years to figure it out so it's unlikely a child will without being told.
  • Advanced vocab can be a problem.  Sometimes, we take a break and introduce new words.  Just like the Word Board from days long ago, a decent vocab word needs a story and facial expressions, maybe me acting out regret or guarded, skeptical.  In about 40% of the cases, I can just add y to the end of the word.  Decisive becomes decisiony.   Those 4 words were in a single question from today's exercise.
  • Many of the questions are totally obvious if you step back and think about them but really hard when going from line 30-33 to the question and back.   If the question asks did the scientist 'support', 'refute', 'collect evidence' or whatever,  you can simply take it at face value.  If there's no refute, support, collect whatever in the passage, cross out the answer.
  • More to come.

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