Saturday, December 19, 2015

"Hands On Equations" is Bullshit

I'm going to try to make this rant coherent, but frankly I'm going to confess that I'm foaming at the mouth right now.

If you have a child in a gifted program or in a school at between the 3rd grade and 5th grade (US) level, you've probably encountered the product from Borenson and Associates called "Hands On Equations." The goal of this product is to teach Algebra. I repeat the word "product" here, because this is not something that involves novel design, new processes or the like. In fact, I'm pretty sure the notion of an equation as a "balance scale" was used as far back as the Greeks. The only other part of this system is some red dice found in any board game, and some blue pawns, which also be borrowed for Sorry!, Clue or Life.The product website makes some pretty bold claims, and schools have made big money buys, but in reality, this is a bit of a scam.

Here's the thing. Algebra requires abstraction. Normally a child can understand a mathematical operation like 2+2. They can also understand that adding an equals sign requires them to balance out the equation, i.e., 2+2=4. These are rule based actions, and children from about the age of 4 rely on episodic memory - memory of facts, figures and events which are explicit in nature. Episodic memory is useful in recalling rules and sequences and carrying that out. Anything rule based can be understood by a child. I can teach a kindergartner Einstein's General Relativity, because it is rule based.

Algebra, however, involves breaking the sequence, and having the child develop the rule using "abstraction". So, 2+a=4 follows the familiar pattern of a rule, but it breaks a rule in that "a" is not a number. Until the development of a region of the brain known as rostrolateral prefrontal cortex (RLPFC), the type of abstraction which allows a child to figure out that a=2 has not developed. This generally occurs between the ages of 10 and 12 for most children. (Girls develop it in the early part of that range, boys towards the end. Girl Power.) The bottom line - biology will determine when children are ready for algebra.

What Hands On Equations does is that it creates a set of rule-based processes that mimic abstraction by allowing the student restructure the algebraic equation into a arithmetic equation that needs solving. So in this case, 2+a=4 becomes a=4-2 or 4-2=a. A child whose RLPFC has not developed can easily see that "four take away two equals 2".

Now, I like to play devil's advocate. Does any of this matter? If a child can solve algebraic problems, even if it is not true abstraction, it at least allows them the function, right? Unfortunately, no. What it does is create the expectation that everything has a rule-based analog. Some concepts in later forms of mathematics (trigonometry) cannot be represented with rules. Concepts like asymptotes in analytical geometry, limits in calculus or confidence intervals in statistics require a child (or more likely an adolescent) to think not in concrete terms that can be quantified absolutely, but in abstract terms that need to be related through metaphors. A math student's first exposure to this is in learning that "a" can be any number, but that only one number makes 2+a=4 true.

Frankly, schools that push Hands On Equation's on their students are doing them a terrible disservice. It is not algebra, and it does not replace it.

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