Lakatos, the Jack Kerouac of math
“WHEN will I need any of this in real life?” is such a common question in a math classroom that it is a cliché.
At one level it is a tried and tested way to stop a lesson. At another it is a genuine question. After all, if a student isn’t going to use the likes of the Quadratic Formula, why learn it? And spend so much time, over so many years, learning it all? And suffer so much stress1? Why, indeed, does an average student need much more than the numeracy required to navigate life?
The question does not come so often from the student who enjoys math, which also usually means the student who is more than competent at it.
The answer used by myself, and some other teachers I know, is that math is about creative thinking skills, and, going further, the ability to think in abstract terms, to examine problems from different perspectives, the bending of minds. The other oft-used term is problem solving. Problem solving, creative and logical thinking skills are essential life skills.
Some look convinced.
Those who don’t need to ask the question (but may out of genuine, disinterested curiosity) carry on, almost entirely driven by their own motivation.
The question facing math educators is how to make math relevant — and fearless — for the rest?
Students may not need, or ever use, the Quadratic Formula, but finding the Quadratic Formula with the aid of algebra tiles, completing the square and computer manipulatives as well as some help from other students does need skills that are useful for life… the sort of math thinking skills needed in an increasingly complex technological world… an increasingly complex political, economic and social world.
Proofs and Refutations, The Logic of Mathematical Discovery by Imre Lakatos raises not just a question of how math is taught. It is also raises questions that go to the core of what education is about. Like the world, it is complex, technological, social and political in nature. There are many stakeholders, besides the students.
The pressures on teachers (and students) are contradictory.
The modern math teacher is meant to create rigorous, interesting, relevant, fun, diverse, engaging, multi-cultural, technology-based, investigative lessons that allow each student to search out their own path of discovery, at their own pace, based on their independent learning plan in co-operative group situations tailored to each student’s own learning style.
At least three times a day.
Plus relevant daily assignments with meaningful and timely, individualised feedback. Fully covering the curriculum. And preparing the student for repeated, on-the-record, high-stakes, multiple-choice computer testing… the results of which are the meat for the political-funding grinder.
So. No pressure then.
Teachers and students find themselves caught between the traditional deductivist approach and the vision of an heuristic future.
The demands of the curriculum, rigorous testing regimes, daily assignments, textbook styles and time all push the teacher back in the direction of deductivism. And as most parents are the product of deductivism, they too are fans— even while assuring the teacher they themselves hated math at school.
Competition from new interactive media, educational research and sheer desperation and frustration push the teacher to at least dream of an heuristic world where students are engaged and thinking for themselves.
Lakatos could easily have chosen a different title for this delightful read: Proofs and Refutations, the Heuristic of Mathematical Discovery.
Here Epsilon, Pi, Omega battle it out with Alpha and Beta: Adventure and the search for knowledge versus battalions of formalists, logical positivists, conceited infallibilists, dogmatists, Euclidean rigourists, monster barrers.
In the final few pages2 Lakatos abandons his pseudonymous allies and speaks directly with Shakespearean passion, anger and open contempt for the deductivist approach.
Lakatos summarises this “Euclidean ritual” as painstaking lists of axioms, lemmas, unseemly definitions followed by carefully worded theorems, often loaded with heavy-going conditions. The theorem is followed by the proof.
“The axioms and definitions frequently look artificial and mystifyingly complicated.”
The student of mathematics is obliged “to attend this conjuring act without asking questions either about the background or about how this sleight-of-hand is performed.” Should a student wonder or discover by chance that mathematics could not have developed in such a fashion, “the conjuror will ostracize him (sic) for this display of mathematical immaturity.”
Lakatos complains, “mathematics is presented as an ever-increasing set of eternal, immutable truths.
“Counterexamples, refutations, criticism cannot possibly enter.” Conjectures are suppressed.
This “authoritarian” deductivist style “hides the struggle, hides the adventure.
“The whole story vanishes, the successive tentative formulations… are doomed to oblivion while the end result is exalted into sacred infallibility”.
And then the coup de grace buried in the footnote (p142, n2): “It has not yet been sufficiently realised that present mathematical and scientific education is a hotbed of authoritarianism and is the worst enemy of independent and critical thought.”
In contrast Lakatos models a more open and optimistic approach, a world where imperfection is a virtue. For Lakatos it’s not the answer that counts: it’s how you get an answer, which only leads to the next question, that matters. Learning as a journey. And it’s the road that’s interesting, not so much the destination… which is only a starting point of another road.
Lakatos is the Jack Kerouac of mathematics.
“Literary criticism can exist because we can appreciate a poem without considering it to be perfect; a mathematical or scientific criticism cannot exist while we only appreciate a mathematical or scientific result if it yields perfect truth.”
Lakatos has not yielded perfect truth. But this work helps put us on a road of discovery in the classroom. For Lakatos education was about fostering independent and critical thought, and for him that would mean adopting the road of discovery and not the Euclidean ritual – “this good and evil spirit of nineteenth century mathematics”.
But Lakatos was a political man3. And much of Proofs and Refutations has the passion of a revolutionary political manifesto.
Lakatos would have recognized the tensions and politics that mire modern-day teaching.
On the one hand: open-ended investigations, discovery, problem solving and self-learning, assessment for learning. The heuristic method.
On the other: the insistent political pressure of test scores, pushing teach-to-the-test strategies. The rigidity of the curriculum map. Students trained to view education as a production line for collecting points. Assessment of learning. (Assignments means points. Tests means points. And as the most amusing, satirical show on BBC radio4 for many years always said, “points means prizes…” Grades, GPAs, scholarships, college, career.)
But there is a synthesis out of this Euclidean thesis and heuristic anti-thesis. Behind the closed doors of classrooms teachers are experimenting with counterexamples and stretching concepts to open up new conjectures5 which may yet give greater depth and breadth to real learning.
1 I did a Google search some time ago for “math phobia”. It returned 527,000 links. I put the same words into amazon.com and a list of 234 self-help guides was returned with names such as Overcoming Math Anxiety, Conquering Math Phobia: A Painless Primer, Danger Long Division, Overcome Your Math Phobia and Make Better Financial Decisions. And many, many more.
2 Lakatos puts clues in his opening pages. Heuristics is mentioned in both the Acknowledgements and the Author’s Introduction. And in the opening pages Lakatos teasingly makes clear, in a lengthy footnote (p9), that for pre-Euclidean Greek mathematicians porisms, results that appeared by chance, springing from the proof of a theorem, were considered a windfall, or bonus. “The heuristic precedence of the result over the argument, of the theorem over the proof, has deep roots in mathematics,” says Lakatos. As the mathematicians in Lakatos’ sites often pointed back to some notion of a golden age of Greek mathematics and rigid method, this Lakatos footnote is an opening salvo.
3 Lakatos was a member of the Hungarian Communist Party and an active member of the anti-fascist armed resistance during the Second World War. After the war he was a part of the Communist administration and was involved in the reform of Hungarian higher education. He spent six weeks in solitary confinement and three years in prison. The reasons why are unclear— He was rehabilitated in 1953. After Lakatos fled Hungary and the Soviet invasion of 1956 he was supported by the Rockefeller Foundation and the London School of Economics. At LSE he remained a close friend and colleague of Georgy Lukacs, widely accredited as the father of western Marrxism. He also befriended Paul Feyerabend who formulated an anarchic theory of knowledge.
4 I’m sorry I haven’t a clue, BBC Radio-4, presented by legendary jazz trumpeter Humphrey Littleton. Littleton arbitrarily awarded points based on no stated criteria for games that appeared to have neither logical conclusions nor rules. No one knew what the prizes were or why points were awarded— the only real prize for all concerned being an addictive dose of hilarity that somehow commented on much of the nonsense of current events.
5 In my own still-mostly-deductivist classroom we have dumped the textbook as unintelligible, barred points as monsters, and have incorporated self-assessment and words such as “On your way”, “Getting it”, “Almost there”, and “Got it” instead of meaningless points, percentages and letter grades. Verbal in-class contributions count on a par with written assignments. Lower-end students say they get hope. Higher-end students are challenged to demonstrate thinking with their explanations. All students are challenged to demonstrate some learning, that they have improved their understanding. That, at least, is the aim. Some days it works. Some days not.