Saturday, November 12, 2016

The Skill Pyramid in Math

A while back, a reader asked me to elaborate on my comment that there is a shift in skills at the 4th grade level.   In my last article, I introduced some reading skills to set the groundwork for this discussion.

The skill set is fundamental to cognitive skills tests, to success in school, and many other things. Beyond the 2nd grade level, it's hard for me find material that exercises these skills as well as math, so I'm going to discuss these skills in the context of math.  As I mentioned in the last article, 100% of these skills are present during the process of learning to read at age 3 and 4.   The base skills are present throughout, but the next level happens at about 4th grade.

Level 0 and Level 1 are repeats of topics I've covered at length, so if you're familiar with my work, feel free to skip to Level 2.

Level 0
The skill at Level 0 is seeing things.   I haven't written about this skill in about 3 or 4 years.  It came back to me when I wrote about reading skills.

A child with this skill will see a trapezoid and notice that the base is wider than the top.  It's more like a rectangle usually than a square in that the base and the height are typically different.  A child without this skill will see a 4 sided shape that is "big" or "small".

The way to teach this skill is to take the 250 math vocabulary words from K through 2nd grade and make sure your child knows all of them at the earliest age possible.  Early readers get this skill naturally, and it should be obvious why - they are exposed to lots of vocabulary.  I don't know how a child is going to learn to see if he isn't presented with a trapezoid and it's name.

A parent can encourage "seeing" in everyday conversation as long as a parent is seeing things from the point of view of the child's limited skill set and determined to improve it.  Successful parents take the spaghetti approach and just throw lots of words on the wall in the hopes that some of them stick. I used the Word Board approach to make up for not doing this early enough.

If your child learns to see, skills further up on the pyramid are self taught - like comparisons, similarities, and patterns.  If I were going to write a book on how to pass the COGAT without test prep, this content would be in it.

Another skill I don't talk about because it's a subskill is the ability to visualize something, like a rectangle containing 2 squares or 4 + 7 = 11.   It's hard to separate this subskill from working memory.  There's a skill list in Level 3 that will make up for an deficiencies here.

Level 1

Level 1 begins as soon as your child sits down with a workbook that is beyond their skill set.

Here's my list of the core skills at the base of the pyramid:

  1. Totally comfortable being baffled by a problem.  Without this, the child can't do material that involves learning, which is by definition a problem that is baffling.
  2. Spending a lot of time investigating the problem.  In math, this is spending more time on the question than solving the problem.  Another way to think of this skill is not being in a hurry to finish.  Ironically, kids who exercise this skill finish faster with less mistakes.
  3. Trying again without frustration.
  4. Check your work.  This little skill is the difference between the 85th percentile and the 99th percentile.
  5. Working memory.  There's no way to get through a challenging problem without it.  It's not actually a skill, more of a brain muscle, but it's absence undermines everything else.
The sane approach to teaching a child, especially in math, is to give them the material one digestible step at a time so that they don't have to experience total bafflement and have a breakdown.  If I had to teach 28 kids math, I would do this so as not to lose half the class in each lesson.  Unfortunately, none of them would learn advanced skills it would be only a matter of time when they would be lost in math class.

These skills in math and any other subject are the foundation on which learning takes place, but not the actual skills used in learning.  My approach all along (aka Test Prep Math Levels 2 and 3) is to hammer away at these skills until they are at the 99% level and then let the child learn everything, including the learning skills, especially the learning skills - on their own.  It works great.  Want a child at the 99.9999% level in math?  Get them to 99% in the core skills.

Level 2

Level 2 happens when your child faces 4th grade arithmetic.   It's totally pointless and boring and if they don't come up with some other skills to cope, they're going to end up with a C in math.   Plus, there's a lot more notation and syntax to translate.  No skills are involved in 4th grade arithmetic at all.

If you see this abysmal grade on this report card, and you haven't been doing math at home for 3 years, then it's most likely caused by the fact that they are missing the skill set above (because school doesn't teach it) and you now have 2 problems on your hands.

At this level, a child is going to be introduced to relationships and patterns once again.   I call the skill set involved "cheating" and it's vital to math.   How to get through a worksheet with 30 problems involving adding 8's and 9's?  Order them, i.e., 8 + 1, 8 + 2, 8 + 3 and do no thinking at all.   

I make my kids cheat on every problem, because in reality there is a lot more work involved to cheat in math than not to cheat. Instead of counting and trying to memorize 8 + 7, the child can do problem decomposition (8 + 2 + 5) or devise some other cheating algorithm like the one above.  This is good preparation for the next level, where the skill set develops.

What makes 4th grade so interesting, therefore, is that a child really needs the Level 1 skills to get through more complicated problems, and they need to start practicing for Level 3 skills.  Level 2 is really just sandwiched in between Level 1 skills and Level 2 skills, and doesn't have much to contribute other than the skill of cheating and coping.

I told my child that 4th grade is just a write-off year in math and we'll skip it and just start preparing for 5th grade.  I learned the hard way by doing this that there can be a 3rd reason for your child getting a C in math.  What I know now, and should have said then, is that the skill we are practicing in 4th grade is the ability to complete assignments neatly and turn them in on time whether your care or not, and whether or not your parent just told you that 4th grade is a write-off year, which it is in math.

Level 3

Level 3 is pre-Algebra.   Pre-Algebra has 2 parts.  The first part is all of the math needed for algebra, like functions, graphing, some geometry principles, decimals and fractions.  The second part includes all of the skills needed to solve math problems and an introduction to abstract thinking in function notation.

The best approach to math problems is Kumon Pre-Algebra.  The worst approach to teaching Pre-Algebra is the same book, which in the introduction provides the solution algorithms for all of the problems so that the child doesn't have to learn problem solving skills.  Therefore, the best book on the market is Kumon Pre-Algebra with both the introduction and the solutions ripped out of the book.

Again, my approach is that if you have the skills down, the learning can take place on it's own, but these are high level skills and are hard to learn without direction.

The skills to be learned during this stage are the problem solving skills.  I listed these on my math page on the right.   The best treatment of these skills is a book called "How To Solve It" by George Poyla, a guy who worked at Stanford in the 1940's.   I was so taken with his approach, that I dumbed it down to the 5 year old level and the result is the material above.

His list of skills roughly looks like this:
  1. Understand the problem.
  2. Devise a strategy to solve the problem.
  3. Execute the strategy.
  4. Check your work.
You'll see a lot of similarity with the list at Level 1 and this list except for #2.  The math subskills for Level 3 are the details of "Devise the strategy to solve a problem".  This is the third level of the pyramid, if the first level is "seeing" and the second level is the list that starts with "Baffled".  At each level, a child has an opportunity to go from great to awful if they can't work to the next level.   The parent is going to add to the frustration if they don't see it coming.

At this level there is a shift from applying things you know to solve a problem, to figuring out how to solve the problem in the first place.  There's no mystery in "8 x 3".  Both parent and child know how to do it.   Every pre-algebra problem should have a mystery, and the parent seeing it and telling the child is undermining the child's ability to learn the problem.

Here are the strategy skills:
  1. Decompose the hard problem into 2 easier problems to solve.
  2. Use the result of a previously solved problem and plug it into the problem at hand to make the problem easier and remove some of the work.
  3. Visualize the problem or think of it from a different angle.  This is usually presented as solve the problem backwards, but I find that a more useful technique is to solve it from the inside out. I define "more useful" as "shows up on the SAT more often". 
  4. A slightly different approach is to draw a picture of the problem to gain insight or just solve the picture instead of the equation.   If your child has mastered abstract thinking and understands the question, this usually isn't necessary, but it's a life saver otherwise.
  5. Solve the entire solution by just solving a part of the solution because you noticed that you're given a big problem, but the question is just asking about a single element of the problem.  (Again, stole that from the SAT.)
  6. The most powerful solution technique of all is to restate the really hard problem as an easier one and solve that one first, then work up to the harder problem.  For example, how far apart are the fence posts if the fence is 240 feet long and their are 13 fence posts?   A 3rd grader this morning tried to solve 240 / 13.   When you start with a fence that only has 2 posts, then 3, then 4, you'll eventually see that the answer is 240 /12.
  7. Sometimes you can restate the ridiculously hard problem as an equivalent but far easier problem.   This effect is really applying one or more of the techniques above.
Somewhere between pre-Algebra and Algebra, abstract thinking is added to the skill list.  It's really frustrating when your child is baffled by y = mx + b.   Like working memory, this is some sort of brain muscle, and the best way to teach it is exercise over a long  period of time in the year leading up to when your child needs the skill.  If you've ever seen a problem from Section 2 of Test Prep Math, you might wonder if I was having a problem at home with abstract thinking in 5th grade when I was writing the 2nd edition of those books. Officially, these problems are designed to be 150% as hard as the COGAT quantitate section, and they are at least that.   But yes, I also addressed some of my other frustrations with working memory and abstract thinking to edition 2.   

My 3rd grader is not ready for pre-Algebra, but we're out of math books for this kid, so I'm going to teach these skills using competitive math.   I'm reluctant to jump into competitive math because it's really math and I don't want to ruin any budding love of this subject by overdoing it.   On the other hand, I've got an 8 year old who wants to play video games while eating halloween candy, and the iron clad rule in this house is that there is no enjoyment whatsoever if you don't do your math first.

I downloaded a few sample tests and "invited" him to complete these as a prerequisite for fun.  It didn't go well.  He has no problem solving skills.  We worked through the fence problem together, and I showed him technique #6.  I told him that we are only going to work problems that require this technique (which is most of them, as far as I can tell) and then once he get's it, we'll move on to applying the other techniques.  It's a great way to really learn math, like own it internally and earn every piece of it.

With the older child, I got a copy of an SAT test prep book, which is mainly pre-algebra problems, and we did the other techniques after 5th grade.  This worked really well.

Level 4

Level 4 should happen leading up to high school math.  It's not officially taught as far as I know.  I didn't even know it existed until I read the books of Jo Boaler.   According to Boaler, girls approach math differently than boys.  While boys are intent on solving the problem, as fast as possible, almost competitively, girls want to know how the material relates to the whole.  According to Boaler, if a teacher doesn't realize this, their course will turn girls off to math.

Oh my gosh, I thought, that's how professional mathematicians think.   That's how all children should think.  I didn't think like that until I got out of college.  I don't want to make this mistake with my kids.  Boaler focuses on middle school.  I focus on pre-middle school with the hope that by middle school I'll be out of the picture, job well done, and can retire.  But this is a middle school skill.

Since my other child is taking middle school math right now (see my page on the 7th Grade Project while I describe what I'm up to), this is the 3 part exercise I assigned.  Part 1, assigned 2 weeks ago, was to provide an example for each interesting and relevant application of each of the 4 primary function types.  The only reason why I did this was a warm up to Part 2.  We did Part 2 today, starting with linear functions:   Tell me everything you know about these function, everything important, useful, relevant, interesting.   We'll work on this over the next few weeks, and then move on to the other functions, and then I'll follow it up with Part 3, which I haven't devised yet, but it will focus on the Level 4 skill of see the relationship to the whole.

Parenting Skills

If you reread the articles I wrote on parenting skills, they are all motivated by teaching a parent to create the proper learning environment at home that fosters these skills.   I've got a bit more to say about parenting in the next few months, including some surprise skills.

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