Saturday, June 13, 2015

COGAT Questions #5

Last night we went to a play with friends of ours who have 3 super bright kids in a competing GAT program. One of the parents is heavily involved in all things academic and neighborhood, and the other parent teaches math at a great highschool, typically an honors and a non-honors course, plus does some teacher coaching or mentoring.   After the play, we went to a local pub to discuss the play and I sat there thinking how much I want to talk about my math education research but can't because a) they both know more than me and b) their kids are smarter than ours and c) my concepts would be really hard to introduce in a classroom with 30 kids.

Nonetheless, I found out that Jo Boaler is coming to speak at Evanston High School.  You may recall last year that I heard Carol Dweck speak at Evanston High School.   All they need after Jo is the Pope.  Have you watched sports recently and seen the fans holding up a big picture of their favorite player's head?  I'm going to have a big picture of Jo Boaler's head made for the talk and use it to block the view of the people sitting behind me.  Now that we're studying for the 4th and 6th grade tests, I need to thwart the competition.

The Number Puzzle
Question 5 on the COGAT is the number puzzle.  If you prepare your child properly for this question, you are also preparing your child for 1st grade math, to pass the AP exam after senior year, and everything in between.

Here is what the question looks like:  3 + ? = 4 + 4.  For Kindergarten, the question has pictures instead of numbers, and for older kids it looks more like multi-equation algebra:  ? = % + 3 and % = 4.

These questions can get super hard, like 8 + ? + ? = 19 - 5.  These questions are very similar to number matrices, but unlike number matrices,only one correct answer is possible. Somewhere on this site about 2 years ago, I referenced a research project wherein researchers gave a problem like 3 + 10 = ? - 4 and 95% of 4th grade kids answered "13".

The Problem With the Problem

My recommendations for this question are identical to my recommendations for teaching math to kids ages 3 to 10.   As a frame of reference, start with the problem solving skills according to Poyla:

  1. Read the Question
  2. Strategy to Solve the Problem
  3. Solve the Problem
  4. Check the solution
The Reading the Question skill is key to all of the COGAT problems.  It's 50% of the battle.  I've talked about it before.  95% of kids won't see part of this problem.   On the test, examples are given to help, but I'm not sure how much. 

Check The Solution involves a strong number sense.  Every Day math teaches this skill explicitly and was it's primary goal.   Kumon does this the hard way by over-and-over-and-over solving.  The answer set is not really part of the problem because there is only one correct answer.

For kids in grades 1 to 4, the Strategy skill is called "Regrouping" and it's fundamental to thinking and fundamental to the COGAT.   Most math curriculum teaches number sense and regrouping.  Unfortunately, 2 things happen to undo this.  First, at some point, kids have to memorize their math facts.  The theory is that unless kids can add and multiple, higher order problems are impossible.  Secondly, all of the thousands of math problems that a child has to do in the first few years of school are simple and go fast.   Bright kids memorize the mat facts early, and that removes the need to think again for the next 3 years.   

If kids memorize their math facts and can do lots of math problems quickly, they will be a disaster on the COGAT.

Even worse, kids are basically told that school is about how well you memorize and how fast you get your work done.   In reading, my kids can memorize a whole book in an hour and recite it back 6 months later.   But if I ask them "why?" or "what does it mean?" they have no clue, because they don't think.  I'd like to drive a bulldozer through the front door of the US Department of Education to let them know how I feel about this situation.  If we can't effectively teach our Way Over Privileged children, kids in poor neighborhoods might as well just skip school and go right to jail. 

The Solution
With really little kids, teach them arithmetic with blocks, and show them 4 + 2 (blocks).  Instead of asking to answer 6, ask them to regroup the blocks and give you a different problem.  Then ask for 6.   As a bonus you'll be studying for the part of the Number Matrix skills question as well.

For kids in 1st through 3rd, get flash cards as usual for the four arithmetic operations.   When you show a card with 7 + 9, ask for the regrouping.  The kid can respond "6 + 10" or "8 + 8" or "4 + 4 + 4 + 4".  It's up to them.   No answer is better than another.  Sometimes you can suggest ways they haven't considered, but don't push your answer on your kid or you'll undermine their learning.  In math, they should be telling you stuff and not the other way around or learning isn't happening.   If you are telling them something, it's because you don't want to spend 25 minutes solving a problem, and you are teaching them impatience.

For kids in 3rd and 4th grade, be wary of computation.  Almost all the problems in good math books and math work books were designed for regrouping.   Ask your kids to solve all of the problems in their brains, and if they can't, you have some work to do in the regrouping department.   A problem like 22 + 39 should never be solved with "carry the 1" but instead regrouped into 21 + 40.   Some 3 digit problems are designed for regrouping, and some just for old calculation methods.  Skip the ones that are simply calculation.

I plan to forbid memorization of math facts and assign problems that should take more than 5 minutes each.  That's why we work ahead 2 or 3 years.  Not because I think my kids are at that level, but because it guarantees that they have to think really hard and go really slow.  I'll take a C in 3rd grade math if it means my child is developing real skills

Regrouping is the stepping stone to algebra.   Just thought I'd throw that in if it's not obvious.

Back to Number Puzzles
This question has some of the hallmarks of a good test question.  It is multi-step.  It is confusing and unfamiliar.   It requires a strong working memory with multiple storage buckets.  It is also easier than other question types; unlike the number matrix (question 4) and the figure matrix (question 7) there is only one possible answer, and there is nothing vague or confusing about the answer except in one case. 

This question measures whether or not your child is learning math properly, and thus whether or not your child will succeed in higher math or accelerated math.

I recommending spending 10 minutes a day 4 days a week for a kid preparing for the K or 1st grade COGAT.  Use flash cards, blocks, and challenging problems as practice.   Practice regrouping with a special emphasis on doubles, triples etc for Question 4.  If you do really well on this question type, it will make up for your child's deficits in the vocabulary section because you didn't know he was supposed for read chapter books by age 5.  

How long it takes will vary.  If you start at age 3, the first problem with take 115 days.  If you start a month before the test, at this age it should go quickly and you'll have less to do, but hopefully your child has an adequate math background.  If you start at age 3, on the other hand, your child will be day dreaming during math practice and when you ask him what the heck he is doing he'll respond "I came up with a better solution to Fermat's Last Problem and I was just seeing if I could extend it to the Hodge-Conjecture".

If your child is at the 95% level, you don't really need a practice test, unless he needs to get to 99%, in which case you need a practice test, because a practice test is good for 4 points on the test at this level.

I don't think a Test Prep Tutor can do this.   Not in 30 minutes a week.  Workbooks aren't necessary.  So again, here's a question type that you can address for free.


  1. A question about cogat...
    Are more points are taken off if the question was answered incorrectly than unanswered?
    Do question carry different weighs?

    1. I presume you are asking this question to provide guidance to your child on how to take the test. It will backfire in horrible ways. Tell your child to pay attention like their life depends on it and answer every question correctly. As we went through practice tests, the kids learned that they will get the problem wrong if they don't read every answer. They learned how to answer a question correctly without really knowing the answer from vocabulary workshop. They learned that every question wants to trick them. I should write a post on this.

  2. Thanks so much for replying.....will look forward to your post..:)
    Your blog is an invaluable resource for us.
    Always look forward to reading your posts...please keep them coming!

  3. I'd love to see my kids getting into GAT and they're working hard but I also think about a possibility they don't make it because it's too competitive and their test skills are not the best of the best. Any advice for kids who unfortunately dont make it to GAP and what they can focus at home to make sure they can possibly be as successful as GAT students in middle school or high school?

    1. For the last 4 years, I've been working diligently on keeping my kids gat once they got in the program. That's the darker side to passing the test based on effort. Lately, I've been looking at prepping for the 4th and 6th grade tests, and the 8th grade test is the biggie. Prepping for me is getting them the skills they need to actually be at that level, but you figured it out already. I spent an hour in my favorite bookstore reviewing workbooks, and once you get past the building thinking skills series, there is very little out there. I'll be posting soon on this topic, but this blog already has the building blocks of my program..