Tuesday, March 21, 2017

Figure Matrices and Folding and Lie Groups

Lately I've been dissecting folding and figure matrices and evaluating the core skills that go into these processes.

The goals of all questions on a cognitive skills test are to:

  1. Present the student a brand new problem
  2. With unspoken rules that can be derived by doing the problem itself
  3. Using rudimentary skills that all students should have
  4. To test the student's figuring out skills
  5. Because these figuring out skills determine how well a student will do in school.

GAT programs across the country have been raising the cutoff score.  In Chicago, it's approaching 99%.  It's been 99% in New York for a while.   I used to get emails about lower cutoff scores from around the country, but lately the emails from parents mention scores above 95%.  I'm surprised how many times 98% is mentioned.   In this context, the list above can be refined to include 'really hard problems' with 'complicated rules' using 'advanced skills' to determine who will get straight A's in school on their way to Stanford.

In theory, a student should be able to manipulate a square by flipping it or making it wider.   Older kids will also have to rotate a shape.   Flipping a square is tricky business, because as you know, once it's flipped, it doesn't look like it changed.

My core approach to coaching students works really well, so well in fact, that after studying for a few months for the COGAT, there is a huge leap in academic ability, as in a 4 year old goes from counting and coloring to tackling 2nd grade math on their own with little effort by age 5, or a 10 year old C student fills the report card with A's and reverses declining standardized tests cores.  Of course, context is everything, and the term 'little effort' in the context of Test Prep math takes a long introduction to explain.

But in some cases, there's one fatal flaw in the whole process.  Some kids don't have rudimentary visual spacial skills.

To review, the test prep process looks like this:

  • Work on a few hard problems over a long period of time.
  • These problems should be complicated and require a range of skills.
  • Mistakes should be common.
  • The student should work things out without the help of a parent

When I work a COGAT problem, for example, we spend a lot of time analyzing the picture and the potential answers before jumping into the solution.  If the child doesn't 'see' yet, I make them describe the diagram to me, and point out that a) I'm blind and b) I'm going to draw what they say.  Of course, when they try to describe a pentagon, for example, my drawing looks nothing like the picture they are describing.  For my own kids, I added c) I'm also 3 years old so don't use big words.  I added this last condition because they tend to miss the obvious in their complicated explanation, and the solution rests on the obvious.  When I get to this point, the blind 3 year old academic coach with poor drawing skills, it's because they just don't see the problem in enough detail to solve it.  When we do Test Prep Math, I'm a 4 year old who doesn't understand the problem so please explain the word problems to me, because I'm 4 years old, in terms I can understand.

The fatal flaw in this whole process is a student who doesn't have the rudimentary visual/spacial skills that a test like the COGAT assumes the child has.   Teaching the figuring out skills is a straightforward recipe which I've expounded on here at length, but teaching the "seeing" skills is something else.  My goal was that anything would be easy after the skills developed by the Test Prep Math word problems and quantitative sections, then I met a child who couldn't flip a shape. Apparently this book needs a visual spacial section before I can guarantee 100% on all tests of all type.  That and you child has to be familiar with 5th grade math to get a 99% on the 3rd grade MAP test.

The material behind Shape Size Color Count imparts the rudimentary seeing skills to 4 year olds in a very methodical way at a very high level.   After this, practice tests take on new meaning, or the student can graduate to Building Thinking Skills Level 1, at least for the nonverbal material.

For 1st grade, there's quite a bit of material on the market, and for 2nd grade and older Test Prep Math covers a whole new level of skills with the high bar in mind.

The issue I've been thinking about is the older kids who don't have any of the fundamentals of shape transformation.   Apparently there are 9 year olds who have never seen a Lego set or taken an art class.

One of the math books at the 8th grade level that we're almost finished with in preparation for 7th grade is devoted to geometric transformations.  The whole book would be great COGAT test prep if a 10 year old could understand any of it..   Arthur Tresse created a graduate level treatment of transformations that ended up being a brand new math before he tragically died in a dual over a girl. These are called Lie groups.  There is a Lie Group for all possible rotations of a circle, for example. The concept is simple, but the presentation is so complicated and full of terms and notation that I'm the only person who can see how this field is of direct importance to passing the COGAT if you are lacking visual spacial skills.

Some readers may doubt my summary of history or the fact that Lie Groups represent a field of mathematics.  Nonetheless, there is a big gap in systematic evaluation of transformations between Shape Size Color Count and 8th grade Algebra 1.  My goal between now and the 2017 test prep season is to fill this gap.  I'm just starting.  Give me about 3 months.

Of course it will be thorough like Shape Size Color Count, and much more complicated, but I've got one more goal in mind by tackling #1 from the list above by making brand new problems.   I've created COGAT Transformations 101 in the past, but it seems to be counterproductive by "training" the child to solve COGAT  problems instead of training the child to see and think.  The last thing you want going into a thinking exam is a child who has memorized formulas.  What you really want is a child ready to tackle new and impossible problems, preferably with a graduate level understanding of transformations.

The nice thing about training in transformations at the system level is that it's a really powerful way to approach math.   Entire topics can be disposed of in weeks instead of months.  I realize that most readers have a much shorter time frame in mind, like passing a GAT test, but I have to deal with the longer term consequences of test prep, like 192 grades on report cards and 52 standardized test scores before college applications are complete.  A few months of test prep has a much bigger payoff than you realize if you do it right.


  1. Hello,
    My son is in the 1st grade and has been identified as gifted in math and reading. He just took the COGAT for superior cognitive ability and did just average. He has one more time to take it this year in 2 months and then 4 times to take it in the 2nd grade and another 4 more times to take it in 3rd before he needs to pass and have it count towards the accelerated programs. What work books do you recommend for Superior cognitive ability test? Thanks

    1. There are 2 reasons why an academically astute child does not do well on a cognitive abilities test. The first reason is that the skills that make for good grades are the opposite skills that are tested on a cog ability test. The second reason is that some children have no visual spacial skills at all, which is what I'm currently working on. For 1st grade, the best you can do is first grade is BTS grade 2/3 and then BTS grade 4/6 (first section only). BTS has its gaps and has to be done with the parent GAT skills which you'll find in past articles if you google them. For 2nd grade on, the first problem is the focus of Test Prep Math, and I'm belatedly working on the second problem which should be done by June.