Little Quantitative Skills

Many parents contacted me last year about this time.  'Oh no, the test is in 2 days an no one warned me.'  This is probably the most effective weapon school districts have deployed so far in the war against test prep.  What's the big deal?  Well, from the perspective of a GAT administrator, a kid may 'learn' the test, get a seat in a GAT program, and not want to be their because the family doesn't really have a strong preference for academic work to begin with.  Then the child gets crushed and pulls down the rest of the class.  For the last 2 years, school districts have been surprising unprepared parents and cheating the child out of the opportunity to do test prep.

On the other hand, what if this child was taught the skills that the test is looking for, blows away the test, and enters the GAT program ready to crush it?  What if test prep really means picking up a range of powerful cognitive skills that drive academic performance?  That's what this blog has always been about, and when I use the term 'cheating', this is what I really mean.

I feel like my 2nd biggest achievement is accidentally discovering a cheaty way to teach quantitative skills in a short period of time.   This is where I've discovered an indirect correlation between working memory and cognitive skills.

I'll start with age 4, but I think the magic really happens with kids in the 8-10 year range and then 12-15.  These seem like the best time to super charge the brain.  If you do any outside work with a 4 or 5 year old, scores of 95% or higher come easier because there is so little competition.  Taking a C student and going on to high levels of math competition is much more gratifying when the other kids catch up.

For a 4 year old, Shape Size Color Count methodologically steps through all permutations of shape, size, color and count in both an analogy setting and a quantitative setting until the child sees the aspects of a shape (lines, points, dimensions, color) and sees 3 or 4 shapes as a whole with a mental snapshot.  Older children will never pick up this ability in the same way.

100 of these questions gradually become more complicated while they introduce conceptual vocabulary, and by age 5, this child will be ready to sit down with 2nd grade math and do the whole thing with little help.  That is exactly what happened.

The Big Skills were inspired by the COGAT.  The Big Skills are the ones you really care about, the ones that you need to worry about if you've only got a week to do test prep.  In short, here's a picture that you've never seen before that you have to look at for a while to make sense of.  To answer the question, you've got to figure out the 2 or 3 questions leading up to it and perform 3 or 4 or more mental operations.  That's were working + memory = working memory comes into play.

Why ask your child one question when you can ask 3?  Whether it's math, the OLSAT or the COGAT, the starting point (after getting past being OK with total confusion) is thoroughly analyzing the diagram and eeking out everything that can be eeked out.  I call this section 'whole language math' because I want enough word logic to address the verbal section of the test even though the focus is on the other two thirds.  Vocabulary is king, after all.

Keep in mind that vocabulary is the driver of the test.  Notice in this diagram that there is a right triangle and an equilateral triangle.  One rectangle is oriented vertically and one is oriented horizontally.  Kids pick up words that come up in everyday usage, so use them.   Once a child has a word that can be paired with a shape, their brain has a new tool for evaluating the world.   You can't use '90 degrees' when describing a right triangle.  In fact, a 'square triangle that has one part like a square' is a better term.  When a child sees something that other kids don't because this child has a name for it already, then you are one step closer to a 99% on the GAT.

After this section, which I call the verbal section, the quantitative section looks like this.  This is at about the 1/3 mark of this section, 2 diagrams per lesson.

At the time, I was just thinking that I could give my child an advantage on the test if he already went through every scenario that could possibly be on the test and a boat load of scenarios that wouldn't be on the test because they were much more advanced.  This approach works with 4 year olds but not 7 year olds.  In deference to the test, which explicitly provides 'novel and challenging problems that require thinking', I slowly raise the bar on each question so that nothing is ever routine, as if saying 'minus 9 isn't in the answer set' could be routine for a 4 year old.   Then the magic started happening.

At first, I saw counting.  After a while it was a mental picture of four, and then counting to verify.  I like this phase the best because it's called 'Checking the Answer' and it's worth 15 points on any test. Checking the answer is a Big Skill. This exercise evolved into seeing 4 in one shot with confidence, and then on to a mental picture of 3 is the difference between a picture of 5 and a picture of 3.  This is the little skill.   By 2nd grade, the Test Prep Kid was still translating arithmetic problems into pictures on his brain board, doing the operation visually, and translating this back to a number.

I was a bit reluctant to use 'negative two' in a work book for a 4 year old.   When the child says 'minus two' it's just a phrase that means 2 less, and the warm up problems in lesson 101 take it slowly. Just some new language.  A few years later, the child will come across ' 3 minus 5 ' and I have to sit there silently while the little eyes widen and the brain puts it together.

When I see Shape Size Color Count on Amazon, I always check the review and laugh at it.  I can only guess that I was slammed by a competitor, but I can't imagine what book on the market is competing in this space.  Shape Size Color Count covers conceptual vocabulary through 2nd grade math, and it does it pretty well.

At the other end of this age range with 10 year olds, a completely different set of subskills emerges with test prep.  At this age, all of the questions are at a minimum 2 step and most 3 step.  With 4 and 5 year olds, the multi-step problems have to be oriented linearly.  This is the first solid subskill that emerges.  Advanced math is always multi-step, and its a shame math until middle school is all baby step.  I'm working on a video to explain how this works, but my video making skills stink.

I'll come back to ages 5-6 in a minute.

Here is an example of a quantitative problem we work with for test prep.  This is right about in the middle of the quantitative section of Test Prep Level 3 after a thorough section of word problems that slowly build the skills needed to address this problem:

F is an operator and a number, like "+ 10" or "x 3".  The bold F is "not F", as in "-10" or "divided by 3" respectively.   It takes about a day or more for the child to come to terms with F in the first few problems, and the first problem takes about 20 minutes just with F while working memory grows and mistakes shrink.  When "not F" is introduced, that adds 10 minutes to the work.  And later F is paired with G, finally we've got F, "not F", G and it's opposite, with the occasional FFF thrown in just for confusion.

My original goal was simply to blow away the quantitative section on any test.  I know what these tests are like and I wanted to be 4,923% sure that we would never miss a quantitative question.  I didn't expect my kids to be able to attack problems like '23(x - 15) = 8x - (5 + 2x) without having to use either a calculator or algebraic transformations.  But that is what happened.  They just look at it for a while and then announce the answer. It make teaching algebra a pain in the neck, but that's what I got.  "Why bother transforming the equation when I can just see the answer?" they complain.  Years from now, in middle school, these kids are going to be a problem for a teacher than wants students to write down their work.  

There are about 9 separate subskills involved in solving an algebraic problem mentally, but two key subskills that play the lead role are:

• Try a small number.  Try a big number.  Try something in the middle.
• Gauge whether the solution is getting closer or farther away, and adjust accordingly.

I can't explain how they do that second part.  It's almost magic and comes with practice.  My job as an academic coach is to focus on the Big Skills, which consists of attempting to do this problem without crying, checking the answer, and getting it wrong without getting upset.  The little skills just show up through no fault of my own.

When kids become adept at this type of solution strategy, and learn to do simple arithmetic like the Kumon approach, they look like little math geniuses.  Cognitive researchers (who have never seen an actual child) are amazed and attribute this skill to IQ.   IQ is a myth.  It's a learned Little Skill and it's really powerful.

Back to 5 and 6 year olds.  For this group, I think the best strategy is just Kindergarten Math (best math before algebra) followed by Every Day Math Grade 2.  At this age, EDM Grade 2 requires the Big Skills.  It goes slow and is one new concept after another.  After this, school math is fairly routine, even if a 6 year old is doing 3rd grade math, and I don't think it teaches cognitive skills.