Sunday, May 31, 2015

COGAT Question #4

The Quantitative Matrix
Question type #4 on the COGAT is the quantitative matrix.  We started practicing this question type early because I thought it would be easy.   It was a lot harder than anticipated.   At the time I my primary goal was to beat the test.   Much later I found out that I was developing advanced cognitive skills in my child.

Here is the beginning version of the quantitative matrix for grade 1.  Everything I'm going to talk about applies to the other grades that uses numbers instead of pictures.  I will describe in more detail the numerical versions of these questions at the bottom.

This is a well formed problem.  To begin with, it is 2 step.  First, solve the top row.   Second, do the same transformation on the bottom row.  The arithmetic is very simple, so it's just a matter of careful thinking.   Even though this question is simple, the test makers can pile on complications and surprises.   I can even stump adults with this question and not use a number greater than 2.

My 3 year old couldn't do the easiest of the problems.  We waited almost until the age of 4, and he still couldn't do it.   This was a major setback to my test prep schedule.

My solution was to cover the bottom row with my hand and concentrate just on the first row.  I ended up with problems that looked like this:

This was still too hard.  In desperation I came up with this formula:

My story was this:  "In this square, these cats are having a party.  Then what happens next?  Either more cats come or go.  What happened?"  Now he could see that a new gray cat showed up.   Much progress.  We did about a hundred of these and then moved on to this version before going back to the boring shapes on the COGAT.:

When I made these, I simply used all permutations for numbers 1 through about 6 which is all I could fit on the page.  There was special emphasis on anything that was also division and multiplication, since 2 - 4 is plus 2 and doubling as well.  I could usually stump him on one of these.   These questions can get harder, but you don't need me to tell you this.

How To Do This
If I had to do this all over again, I would just use blocks and a bag of skittles.   The main reason I ended up with all of these pictures was because I was researching intelligence theory and early childhood development and thinking that there was some mystery code I needed to crack.  The  only mystery is why any intelligent person would think intelligence is hereditary.

The Results
The first result was a really strong number sense.   I could see as he was figuring out the top row that he was assigning identity to each shape in the top row to see who just came or just left.   I found this odd.  Later, when he did his first few triple digit addition problems in his head at 6 1/2, I knew that he had developed a strong number sense, because this is one of the predictions.  The other thing I noticed was that he would solve problems with his own algorithms instead of memorized calculations.  Finally, he can figure out most math concepts for grades 1 through 6 with a little effort.  If you read this blog from 2 years ago, you'll see that we did a lot in math, but most of his quantitative skills came from our practice on this question.

The second achievement was a permanent boost in working memory capacity.   I think he permanently burned in 4 buckets into his brain for numbers, and 2 buckets for operations.   That's six additional working memory buckets, and probably 2 additional processors.  This makes for a huge advantage in early academic work.

Other Grades
Page 18 of the COGAT review presents a few of the more advanced versions of this question.

Question:  1-2, 3-4, 5-?  Answer Choices: 6  5  4  or 3
This one is primarily interesting because it can distinguish addition from multiplication.   1-> 2 could be plus one or doubling, and 3- > 4  is a third more and plus one, so the answer is definitely plus 1.  These problems are in a the form [x->y] = ax + b = y where either a or b is zero.

Question:  [2->5], [4->9], [3->?]  Answer Choices:  4  5  6  7  or 8
This one is kind of cool because it uses 2 equations and 2 unknowns.  [x->y] = ax + b = y, and just plug a bunch of a's and b's into a spreadsheet, and you've got your own test prep book for this question.  I have no idea how a 1st or 2nd grader could do problems like this, but if you start with easy ones where a or b is zero and work your way up, allowing for a 50% error rate, it's painful but doable, and lots of cognitive skills will bloom in the process.

I've also seen problems in COGAT test prep books of the form [12 - 3  =  4 + ?  ].   This form was used in a study of 4th graders that found only 5% of them even recognized that there was a 4 in the equation.  I've used worksheets of this type to teach number facts, since it involves a 2 step process and does more for working memory and number sense than simply memorizing 12 - 3 = 7.   Simply memorizing 12 - 3 = 7 is a math train wreck, by the way, and you should never allow it.  If you are forced to use math fact flash cards, ask your child to make 12 - 3 into an easier problem (6 + 6 -3 or 10 - 1) and ask for the answer secondly. Also, don't provide any cheats or formulas or mnemonics.   If your child doesn't figure these out for himself, he'll lose out on developing cognitive skills and will end up at a disadvantage to his Princeton bound peers.

Teaching Math with COGAT
Because of the 2 step nature of the COGAT questions, this is my preferred method of teaching arithmetic.   For a solid year somewhere in grades 1 to 4, we would drill COGAT questions until a deep numbers sense exists and we have expanded working memory.  I would normally teach math with a lot of thinking, problem solving, and conceptual vocabulary, but for a brief training period we mainly do calculation.  This training period coincides with the period in school when kids have to memorize their math facts.

There are 2 other quantitative formats on the COGAT.  Ideally, with the skills picked up doing these problems, your child will excel at the others, but as with most things, it's not that simple.

The strategy I just outline requires some time, patience, and diligence.   The cutoff for most GAT programs is about 95%.   My guess is that the kids scoring in the 95% are have this score solely because they have parents in the top 95% of time, patience, and diligence.

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