(Disclaimer: today’s post will be partially written from an Amerocentric perspective. It’s my hope that our non-American readership will also find it interesting, however.)
Did somebody say ghosts of the 20th century?
The 20th century saw a lot of changes, not just to mathematical development, but to our very conception of mathematics. Major intellectual upheavals around the turn of the 20th century completely altered how we think about mathematics, mathematical reasoning, and the connection between mathematics, truth and other scientific fields. These advances in understanding, combined with perceptions of the global political competition of the time, exerted a strong influence on educational praxis. Today’s blog post will sketch two major influences on American mathematical education: (1) an early 20th century philosophy called “logicism” and its heavy emphasis on symbolic reasoning; and (2) Cold War style global competition. These will be seen as presenting certain constraints to the mathematics learner, constraints that needn’t be obeyed.
Symbolic logic was overhauled in the midst of math’s foundational crisis around the turn of the 20th century. The field of computer science was born, and the deep logical connection between incomputability and the incompleteness of arithmetic were discovered. Proofs became seen as constructions of computer programs. Geometry, long seen as the shining example of the axiomatic method, suffered a few adjustments to its reputation. Not only was it shown that pure logic is enough to construct the bulk of Euclidian geometry (without relying on any inherently geometric notions such as points or lines), but the world was revealed, by Einstein’s work in physical relativity, to be non-Euclidian in nature, violating centuries of philosophical elaboration and rather deep human intuitions about the world.
From a mathematician’s perspective, the world had both changed fundamentally and had become a much larger place.
Emerging out of this era was a belief, called “logicism”, that all mathematics is reducible to logic. This isn’t to say that mathematicians reason solely with symbolic logic; the claim is that, given enough time and effort, one could prove (construct) all mathematically true propositions from a set of incontrovertible logical axioms. While this ultimately proved false (something in addition to logic such as set theory or category theory turns out to be needed), the notion that the proper presentation of mathematics is the symbolic presentation stuck. Alfred Tarski’s purely logical development of Euclidian geometry further cemented this notion — if we don’t even need picturable geometric notions to do geometry, then maybe symbols are all we need in math.
Logicism, as a philosophy, has fallen out of favor. Some of its attitudes, however, live on. Like all schools of philosophy, philosophy of math has its zombies.
Many of us are introduced to math chiefly through symbols. While graphing is its own separate skill that’s taught and tested for, this is mainly used in the study of functions and not, for instance, in exploring the concepts (and proofs) themselves. A lot of the looseness in Euclid’s presentation of geometry became attributed to his use of diagrammatic reasoning (although nowadays the virtues and formal properties of diagrammatic reasoning appear well-understood and explored in post-graduate research) — this has bred an aversion of sorts among textbook writers.
It’s thought that if symbols are all it takes to construct mathematics, then symbols are all it should take to learn and to do mathematics. What does this ignore? Human psychology, of course — it doesn’t address whether or not any of these concepts could have been stumbled upon without geometric analogy. Furthermore, the logical sufficiency of symbols doesn’t rule out the psychological or pedagogical usefulness of diagrammatic and geometric reasoning.
What we’re left with is a subject that worships the symbolic and eschews the graphical. This is unfortunate for those of us who think more readily in pictures. Furthermore, it removes a basic notion of “making sense” from the learner’s intellectual vocabulary.
The stress in learning theoretical mathematics is shifted to demonstrations of formal competence — whether or not a particular symbolic proof (essentially a piece of syntax) can be constructed on the spot by the learner. Rarely, however, is the learner asked to actually illustrate (using diagrams or geometric notions) what is going on in the proof. Thus, learners can “stumble” upon the right formal proof (more common in basic subject matter) without grasping on a deeper level the truth illustrated by the proof. For a good example of this, check out Bayes’ Theorem. Most of us learned this through a simple chain of algebra. Stepping through it employing Venn diagrams along the way not only forces this identity upon the mind of the learner, it also (from the beginning) stresses and re-stresses basic features of probability theory (that it deals with measurable space). The fact is not just represented this way, it is presented.
Drawing pretty pictures isn’t suitable as a sole method for learning mathematics, but that doesn’t rule it out as a useful tool for exploring mathematics. The main flashlight that we hand students is the symbol, however, and it needn’t be so. Thinking and reasoning in diagrams, while it can often fall short of a formal proof, can be incredibly helpful for enabling exploration and learning and there’s no reason to avoid it. Simply following many popular text books, however, we are compelled to do so.
After all, ask yourself: if you can’t picture it, do you really understand it?
Nationalism? In my mathematics education? It’s more likely than you think.
The Cold War had plenty of lasting changes on the American psyche, and it would be odd indeed if these changes happened to miss educational praxis. The entire society was seen as being in frenzied competition with the Soviets; as a result, science and math education took on a heavy “applications heavy” character. The goal was to produce engineers more swiftly. Presumably, these engineers were supposed to build giant love rays that would bring the entire misunderstanding to a satisfactory end for all parties involved.
Better tech up faster, people, it’s time to fight Communism!
A basic insight of economics is that demand for consumption drives production — when people want products, the system can adjust to produce them. The United States demanded engineers, and the educational system shifted to try and provide these as quickly as possible. The result is heavy emphasis on teaching algorithms — rather than promote a deeper or more theoretical understanding of the math involved, students are drilled in what are essentially calculator functions. Those of you who remember being drilled on problem after problem, “plugging and chugging” values into formulas, remember this style of education well. I have come to refer to this as “post-apocalyptic mathematics”. Its chief usefulness would shine in the event every computer in existence simultaneously winks out of existence, forcing us to perform stochastic integrals by hand again.
Luckily for Star Fleet, Spock was gratuitously educated in calculator functions. It really shined that time he was stranded in the 1960’s and had to build a computer from vacuum tubes.
Not everybody finds the syntactic backflips that we learn in Calculus 2 to be that useful in aiding our understanding. In fact, it appears more like cruel hazing than anything else.
Stressing the algorithmic point of view (math as the ability to carry out an algorithm) led to a style of teaching that favored large and repetitive problem sets. Testing in such an environment essentially checks whether or not you can carry out the operations identically to a machine, not whether you understand the conceptual content that is animated by the symbols.
The absurdity of this style of learning math hit me squarely between the eyes in my post-secondary education. In a statistics class I once approached my professor, asking for help. “What sorts of practice problems should I be drilling?” I asked. She looked at me like I was from the moon. In her French education, she had (apparently) not learned math in this way. “Go somewhere, lie down, and daydream about it for a while” ended up being her advice.
I was stunned; at first, I thought she was kidding and I laughed. She was being serious, however, and it was some of the best advice I’ve received. I had been so thoroughly socialized in the post-apocalyptic method of teaching math that I was simply seeking to re-enact its abusive aspects. If I was to truly understand the material, however, I would have to step beyond repetition in the form of practice problems, and into the realm of speculation and imagination. Learning to sit patiently with the topic, to contemplate the objects, and to explore their relations in meditative thought proved to be just what I needed. It improved my ability to extend what I learned to formally novel applications, something we can only hope happens with repetition of “calculate this” style problems.
After all, now that I’m in the real world, I have a computer to do the math for me. In fact, if I were to attempt any of the calculations we run at Distil by hand, I should be immediately fired rather than lauded. Why is school the opposite, then? The focus should be on understanding the content of the mathematics, which is a far cry from being able to successfully carry out the operations by hand (in a limited space of time).
I ain’t afraid of no ghosts.
There’s no reason to accept the strictures of the past, or to be bound by old conceptions and constraints. There’s no reason to be scared into symbolic logic by a foundational crisis, or flogged into applications by Stalin. Free yourself! Reason in pictures! Scribble diagrams! Allow yourself the aid of geometry analogies when learning and using math. Step outside of the grind of relentless chains of practice problems. Remember that at its core, mathematics is a human activity, and that means that it must serve our needs rather than the other way around. This implies a necessary measure of rest and reflection. Grab hold of the problems in whichever way best suits your style of thinking, learn to take your time and meditate with the concepts. Mathematics, after all, is the only branch of mysticism that actually works. Just be sure to exorcise your world of ghosts.
About the Author
Mike Freenor is all about the numbers. As Senior Data Scientist at Distil Networks, he is responsible for designing and implementing a suite of statistical and behavioral analyses, built in Hadoop mapreduce. Mike is currently wrapping up his doctoral dissertation, critiquing modern macroeconomic methodology at Carnegie Mellon University.More Content by Mike Freenor