Rand's project found that elementary algebra is the first level at which a major part of math success is choice of strategy. Take, for example, this very simple quadratic equation: 6x2+12x+48 = 3(4x+48)
This has at least six workable strategies for a solution: a fast in-the-head method, an excruciatingly slow process of resolving to normal format and grinding the answer out of the quadratic formula, and at least four others. The best strategy for this problem might not be right, or even possible, for another.
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In algebra, and even more so at higher levels, the ability to decide which strategies are available and probably best for each problem pretty much defines proficiency. But the Rand report found that in 1988, most educational software taught nothing about choice of strategy, though the same report demonstrated that a strategy-teaching program was possible, even then.
In the decade that followed, educational psychologists, notably Jane Healy, showed that modern childhood environments failed to stimulate executive function.
It was another, bigger piece of the same problem Rand had found. Executive function is the part of the mind that plans, follows, assesses and re-plans a pathway through a complicated process. It's the difference between following a recipe and cooking from scratch, painting by the numbers and painting, or running a checklist and fixing a motor. It's essential for all applied math above the most basic level, as well as for critical thinking and everyday reasoning.
Developing executive function by giving students a rich experience of choosing and using problem-solving strategies might be the single most important thing that required math courses do for students.
How far along are we in dealing with this problem, a quarter century after Rand found it?
The bad news: Most algebra-learning software out there is still just textbook/lecture material -- although with animation and graphics -- and the self-evaluation is mostly just checking an answer for right or wrong. That's what's easiest to write and program, and many companies and programmers have never gone further.
The good news: Some of the best software now teaches parts of strategy and executive function. Parents and teachers shopping for this kind of software can choose from three categories. I call them hinters, executors and steppers:
-- Hinters: These offer a strategy hint with each problem. They at least make students aware that there are strategies, and that your choice of them matters. Among the "hinters" I examined, I thought MathTutor offered the best suggestions.
-- Executors: These go a step further by asking the student to input a problem from a textbook, handout or other program, and then choose a strategy from a list. The software then follows that strategy to write a perfect show-your-work homework answer. Online, Webmath has a pretty friendly interface, although it's put together from many different sources and therefore varies between detailed exposition of steps for some situations and simple, less-than-helpful demonstrations of others. A more consistent but sometimes less-thorough executor, available as a download or CD, is Bagatrix's Solved!. This program demonstrates how to attack algebra by a user-specified method for a user-specified goal.
-- Steppers: These not only enable strategy selection but also let students verify each step sequentially, encouraging them to try on their own rather than just copy perfect homework. Softmath's Algebrator is a delightfully well-designed example of this.
The good-bad news: There is still very little software that specifically teaches the most important aspect of developing executive function: how to choose strategies. Hinters are generally limited to one hint and to problems they have on hand, while executors and steppers make all the choices except the initial choice of strategy for the student, and offer little guidance for that first one. A good human math tutor sitting at the student's elbow could use any good hinter, executor or stepper to teach how to choose strategies, but right now, the student still needs that tutor for this crucial step.
Yet there's no reason this instructional software shouldn't already be common. A quarter of a century ago, Rand's demonstration program taught some strategy, although imperfectly. Today's commercial chess and backgammon teaching programs, running on cheapie tablets, do a better job of coaching on different strategies than math software does.
For years, math teachers have written problem sets where a student must choose and justify a strategy; that's what my college calculus teacher did. All that's missing is the inspiration to envision the program, the will to carry it out, and then letting teachers know it's available. Designers and entrepreneurs, get busy!
This column was originally published on UBM's Educational IT site.