Wednesday, February 28, 2018

Skills In Perspective

In the last article, I went a little overboard on the technical detail with some middle school competitive math.   I tried my best to lay out problem solving so that you can see it is consistent with little children and consistent with the high school, college, graduate school and post doc experience.

Let me explain this bluntly.

You want your child to have problem solving skills.  This is much better than having to help with math or hire a tutor to spoon feed your child steps from question to answer.

But if you try to teach your child problem solving skills in the hopes that these spur cognitive growth, you will fail.  It's as bad as having your child memorize formulas and rote practice applying them.

Here's a brief history of skills.  In 1945, a researcher at Stanford named George Poyla took 3,000 years of research into how mathematicians solve problems from philosophers, ancient Greeks, and mathematicians themselves, and wrote a book called How To Solve it to help high school teachers mentor their students on solving geometry proofs.  The emphasis of How To Solve it is 'mentoring', not doing any work for the student or teaching problem solving algorithms or heuristics.

By the 1970's problem solving was turned into a pre-packaged, spoon feeding program to help students apply problem solving methods to pre-algebra and more advanced maths without the need to understand anything that they are doing, let alone math.

The #1 problem in problem solving is that the defective learning approach that emphasizes a speedy, correct answer that has been memorized and practiced has evolved into a defective learning approach that emphasizes a speedy, correct answer using a problem solving technique that has been memorized and practiced.

When I finished translating How To Solve it into a method suitable for parents of 4 year olds, I was stunned to find a solid approach that also works for graduate school.   I added a step that researchers at Berkeley identified as the #1 success factor for surviving their first year calculus courses.  The first experimentee of the program is now 9 year's old, and needs about 10 minutes to get a score of 50% on SAT reading comprehension tests.  Obviously, we have a way to go, but the method is so general that if pretty much works everywhere, including assembling Ikea furniture and fixing plumbing issues.  I would recommend it simply for the benefit of not having to call a plumber.

Here is the short version of the problem solving method:
  1. Be Baffled (thanks Berkeley math department)
  2. Spend a lot of time thinking about and exploring the problem
  3. Make mistakes and try again
  4. Check your work (I added this because it raises test scores)
In between #2 and #3 sit the process of problem solving.  In the last article, I demonstrated the most powerful problem solving techniques from the standpoint of a baffled parent trying to help their child learn some new material that is way beyond the child's skill level.  Think figure matrices, multiplication, fractions, exponents, algebra, trig or whatever.  I'm going to continue the numbering from the above list and explain why shortly.
  1. Start with a much, much easier version of the problem, like 1 x 2 = 2 and just keep adding to it and iterating until you are back to the original problem.  This can take weeks if you're trying to teach multiplication to a 5 year old.  In some cases, the child is missing something fundamental from material we skipped, so we just backtrack to an easier math book to practice the prior material and then come back to the problem.  Backtracking happens a lot in Math House.  Ironically, I can teach basic Trig in about 30 minutes, but it takes months to teach basic alegra.
  2. Translate the hard problem into 2 easier problems and solve the easier problems instead.  This approach usually involves decomposition or regrouping in the early years, and gets trickier in high school math.
There are other good approaches for more advanced topics outlined in Poyla, like solving the problem backwards, applying some theorem or proof that you just learned in the prior problem (which works for both Geometry and the COGAT), filling in the missing word or shape.  If you give the child enough space to explore the problem and make mistakes, the child will learn these methods on their own, or even better, make up their own methods however inefficient.

When I combine the two lists, which is why they are numbered contiguously, I end up with 90% of my teaching method for math until we get to Algebra and Geometry.

There is a great deal of contemporary discussion on the topic of why students are struggling in Physics.  The consensus of physics teachers is that students are more interested in getting to the solution (using the internet to find the method) and less interested in learning physics.   You can find many, many books written to demonstrate the step-by-step approach to solving every class, subclass, and subsubclass of algebra problem if you wish to be an algebra expert without knowing what you are doing.  If a parent would just take a step back from Teach To The Test, you'd find that it takes a fraction of the time to get a 99.9% based on thinking and learning than a 90% based on practice and memorization.  To emphasize this point, we tend to do 2 to 5 problems a day and make much more progress more quickly than children who do 30 or 40 easy problems a day.

Learning happens from the start of the first problem until the student realizes that there is a formula or method that can be used to solve problems of this type.  When the child struggles with 2/3+ 5/7, lots of learning is happening.  But once the child realizes that each fraction has to be transformed to share common denominators, we're done with learning.  Learning also stops when the solution is checked as well, right or wrong.

The biggest complaint I receive from parents who start down the path that I recommend is that it doesn't work.  By 'doesn't work', it means that their child is frustrated, lost, and getting nowhere.  To me, this is a description of the initial stages of the process a not a defect or shortcoming in the approach.  Some stubborn kids need about 6 weeks to undo the programming from school, programming that you must know what you are doing, do it quickly, and obtain the correct answer without effort or challenge. It takes a while for the child to realize that expectations have changed. 

Sometimes it takes 2 or 3 weeks on a half dozen problems to teach the child that we are going to go slow, think a lot, be confused, hit dead ends, have to backtrack, and get things wrong a lot. To accelerate this process (meaning show the student that the rules have changed), I'm usually confused, get the wrong answer, and don't check the solutions. Once the child gets past this hurdle, the pace begins to go very quickly, and if you stick with this approach, the child will in a few years teach themselves entire subjects very quickly, or if you insist on teaching your 9 year old algebra, not very quickly but adequately.

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