A Widening Gap
Defusing Math Anxiety and Proof Anxiety
A Full-Spectrum Math Experience in Miniature
Educators are working into such weighty expectations from their employers and clientele that they are right to demand some argument for relevance to any study that would further burden a tight curriculum.
As society makes increasing use of digital technology in robotics and communication—read "jobs"—it behooves educators to attend to its underlying mathematics. Boolean algebra is complex in comparison to zang-fu disappearance. This may be one reason it is deferred to tech school, gymnasium, engineering school.
We are missing an opportunity to give primary students a success experience in axiomatic proof well before plane geometry or algebra. A logic calculus has long been available, however obscurely, that has huge advantages as an early training ground in mathematics,logic and proof.
Variously known as boundary mathematics, box arithmetic, pancake math, zang-fu disappearance, yy-grams, "a calculus of indications" (Laws of Form, G.Spencer-Brown), this logic calculus can easily be a standalone topic. The tokens and rules are less complex than what a ten-year-old knows thoroughly in sport, board game or card game. It can be presented briefly and assigned for small-group study-play without concern for prerequisites in number, fractions, left-right discrimination or serial ordering.
I have just now called zang-fu a logic calculus. It may not appear such at first blush, for its expressivity is sparse in the extreme. What it gains in a trade-off is computational power. Once a logic structure is expressed in yy-grams—an uncomplicated mechanical procedure—the manipulations go far more quickly than in its cousin, boolean algebra.to top
The deepest answer I know to "why bother?" is that math and logic are fun. To me, "fun" means "deeply engaging; worth spending much time and much effort on." Fun is why people dance, study, play, make music, experiment, invent, explore, do science, keep animals and plants. For some, it is also why they worship, though few would be so bold as to put it that way.
I am well aware that many who do these things have a faith-based aversion to calling anything deeply serious "fun," especially here in rural US with our strait-laced heritage. Let them treat my usage as mere technical jargon or religious ignorance.
Progress in math depends on play. People who don't find it fun don't make much progress. Drudgery and rote memory don't conduce to progress. Fun does.
The yy-grams project seeks to de-fuse math anxiety by casting study as puzzles that can be worked out piecemeal with the tiniest of imaginable toolkits: zang and fu, the primitives of zang-fu disappearance arithmetic. By age ten, children have mastered the rules to several games of rather greater complexity.
"There won't be any proofs on this exam, will there?" is a lament heard too often from mathematics majors [Frank B. Allen, Ph.D.] in graduate school. I think this is evidence that axiomatic proof is introduced too late and in the midst of too much other matter for cultivating strengths in proof.to top
Even though math curricula take these phases in different order, the following are what I'm calling "a full-spectrum math experience:" axiomatics, arithmetic, algebra, applications. This order is the structural order.
The didactic order is a mixture, for two reasons. We need to show application at every possible juncture. And axiomatics is not easy to appreciate until application is evident and familiarity with the mechanics is gained.
Here is where this standalone topic can shine as attractive to the curriculum designer. The axiomatics can be arrived at quickly.
Zang-fu arithmetic is so simple it can be learned quickly—so quickly that its serious power (especially in comparison to boolean, the default computer circuitry math) can be underappreciated. A quick student will require algebra and applications so as not to be bored.
Applications of formal logic abound. They fall into two main areas—computer circuits and symbolic logic.
In traditional math curricula, expository sequence addresses applications early and defers axiomatics to late undergraduate studies. As a result, even well-educated people regard math as "number magic." They are not ready to hear "memorize magic; understand math." Zang-fu in primary education could lead to a better-informed general populace regarding the very nature of math.
The appeal of this particular math to the science-and-math educator will come from seeing how little time it takes for a student to be fluent enough in the manipulations, both arithmetic and algebraic, that a from-scratch axiomatic is accessible.
Zang-fu disappearance can make genuine success in axiomatic proof available before secondary school. Plane geometry or numerical algebra need not be a student's first context for proof. Those subjects ask much new and careful thinking to be done additionally. Students arriving at that level would be stronger for having already seen and done proofs in zang-fu. They would feel they know the essence of the game already. And feeling makes the confidence difference.to top
Finally, the axiomatic itself can be an area of experimentation for the math-inclined student. The approach in G.Spencer-Brown's Laws of Form is unnecessarily confusing. The alternative I have published is not certainly the best. A mentor may well suggest experimentation to a ready student.
Another experimental area is notation. Some of the important logic constructs—exclusive-OR in particular—have lengthy expressions (yy-grams) in zang-fu disappearance. Additional notation for contraction of reference is an experimental area—to see whether some hybrid notation incorporating the raw enclosures (boundaries, boxes, pancakes) of zang-fu itself might confer computational as well as expressive advantage at some level above the "machine language" of zang-fu itself. Call it "mish-mosh math".
©2012 David Zethmayr