Lakatos, the Jack Kerouac of math

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“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.

Wizard math… day 2

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WE’D LOOKED at the sequence generated when you take a hexagon and steadily add more hexagons. It gives a linear rule. Some students got this easily, others found it challenging.

For 20 minutes the task was to work through a handful of similar problems in the textbook.

The issue was how to give an extension challenge for those who could easily do this sort of problem. Jo Boaler in her What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject advises open-ended problems are one solution. She also argues that so-called low-ability students benefit from doing hard problems, or, at least listening and eventually participating in finding solutions in mixed-ability group discussion. Talking math is learning maths. The higher ability students benefit by explaining their thinking to other students.

On the board I drew a square with diagonals drawn in red, a pentagon with diagonals drawn in red, a hexagon with diagonals, a heptagon and octagon, also with diagonals drawn in red. I also put up an incomplete polygon labelled n. I numbered the polygons 1, 2, 3…

I also wrote: Challenge question, spot patterns, how many red lines?

Breanna was out of her seat immediately counting the lines. Within seconds she announced the number of lines coming from each vertex was the same as the number above the polygon. Then she sat down. So? I asked.

“I can’t do any more,” she replied. I explained she needed to write down her discovery so she had something new to look at and work on to get the next step. No, she said.

Shane was busy counting lines. Rebecca stared at the diagonals in the heptagon and octagon and said it was too complicated. “What about this one,” I said pointing at the pentagon “start here… what you find out here will work for those.”

Breanna had started to build a table. “Breanna,” I said, “tell the others about how you counted the diagonals.”

Shane spotted the number of sides of each polygon was 3 more than the number above the polygon. Breanna explained the number of diagonals was the number of vertices multiplied by the number of lines coming out of one vertex. “And the polygon labelled n?” I coaxed. “The number of sides is n + 3,” said Jonathan from the other side of the room.

Robert who had earlier struggled with the basic exercises, sat watching the board and listening intently to the discussion on Breanna’s table .

As others finished the textbook exercises I drew on the board a 4×4x4 cube made up of 64 small cubes. Next to it I wrote, “if the cube is painted, what proportion of the small cubes have paint on them?”

Stopping the class, I focused everyone on the new problem. On our wall we have a list of Polya’s problem-solving strategies. I pointed to the list with my hand next to “Make it simpler”.

Shane was walking from desk to desk debating with other students the number of cubes. Rebecca asked, “you paint the back too?”

Jonathan and Nick were back and forth at each other, Jonathan slicing out invisible cubes in the air with flattened hands. Nick drew out the net of cube and cut it out: “Look,” he announced, beaming, “I’ve made a cube.”

Jonathan just wanted to explain how he had worked it out, how he got the total number of cubes, how he excluded the cubes inside the large cube, how he decided to not to double count cubes with paint on more than one side… all the time his hands slicing out cubes in the air.

What are the dimensions of the cube? I asked Jonathan. 4 he responded. 4 what? 4 times 4 times 4. How would you write that? 4 to the power three… 4 cubed… Oh! he exclaimed as a giant lightbulb flashed in his brain.

As the students left, Jonathan and Shane were still telling each other about how to solve the problem.

“Do you want to see my work?” said Robert showing me the textbook problem he had completed. “I saw you watching and listening Robert,” I said. Robert smiled.

The Calculus class tries the polygon problem to relax after an intense hour wrestling with implicit differentiation. OK, I say, tell me the number of lines in the 99^th polygon.

Becci runs to the board closely followed by Jared. Megan is shouting how to count the lines, but Becci and Jared are engrossed in mathematical disagreement about how to move forward. Josh, Jordan, Jesse and Nicole sketch out the pentagon and hexagon.

Soon the room of nearly 20 students is loudly split between those insisting the rule includes (n −3) and those who say it is (n + 3). As agreement settles on (n + 3), good-natured boasting and mocking ensue.

But, everyone was talking math. And talking math is learning math.

Grading gets an F

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THE PRINCIPAL of my school has bravely initiated a discussion about whether or not D and F grades should be used by teachers. The argument goes along the lines if D and F grades are ditched then teachers will need to work with their pupils to find the success within every student.

Ds and Fs don’t motivate or promote learning. Finding success does.

Ds and Fs have enshrined the out-dated pedagogy that grading is about sorting and ranking, that students need to be judged in comparison with one another.

The result of such alpha-numeric grading is that the traditional idea of the “good” student — the students who most closely resemble the aspirations of their teachers — is the scale against which students are judged; learning is demonstrated by turning homework in on time, doing well on quizzes and tests and putting your hand up to answer questions while not talking out of turn.

The result of such behavior is turned into a mathematically nonsensical percentage inside a computer, which then spews out the grade. And for many teachers that’s it.

If a student does not fit into this rigid mould, or cannot demonstrate learning by these criteria, then the result is F for Failure.

Yet the research should be pointing us to question this approach: Not only does this traditional way of measuring learning not reveal the learning going on among many students, it is actually an obstacle to learning for all students… the achievers as well as those who appear not to be getting it.

As part of the discussion in our school I was challenged in a meeting to summarize the case against alpha-numeric grading. I mumbled a few sentences as best I could for as long as it seemed polite to do so.

Then, later, I kicked myself for forgetting the key reason grading does not work. So, I decided to summarize in short sentence bites the best case I could muster for a two-minute contribution:

❏ Grades tell students nothing about what they need to do to improve.

❏ Grades tell students nothing about what they have achieved.

❏ Grades focus students on grades and collecting points, not on what they are learning.

❏ Grades introspectively focus students on ability, or their feelings of lack of ability, not on how they can work to improve.

❏ Grades destroy intrinsic motivation.

❏ Grades don’t measure learning: grades measure obedience, compliance and how well a student can jump through a teacher’s grading-policy hoop.

❏ Grades discourage intellectual risk taking.

❏ Grades divert the attention of teachers and parents as well as students.

❏ Grades encourage rote learning, memorization not reflection.

❏ Grades pit student against student, ranking and sorting.

❏ A grades require F grades. Grades force teachers to give Fs to justify the As. Grades work against finding the success in every student.

❏ Grades increase stress. Stress is bad for learning.

❏ Grades don’t describe learning.

❏ Grades throw students off the back of the boat.

❏ Grades discourage student collaboration.

❏ Grades reward skills not valued in later life, such as memorization.

❏ Grades demoralize and demotivate.

❏ Grades label and stigmatize.

❏ Grades are part of an out-dated carrot and stick, rewards and punishment behaviorist approach to education.

❏ Grades lower the self-esteem of low achieving students and discourage risk taking among higher achieving students.

Readers will find plenty of links elsewhere in this blog on the research behind these statements. But a good start would be From degrading to de-grading by Alfie Kohn.

What’s on the PiFactory blog…

Welcome to The PiFactory blog. Here you will find notes from the classroom of a radical math teacher.

Below is a listing of all the blogs in date order. All blogs contain extensive links to relevant research and resources.

Relax and watch the kids shoot squids and crash cars reviews working with Geometry Expressions with Algebra 2 students.

We should be teaching mathematical thinking argues using technology in the classroom raises the question of the purpose of math education.

Avis foregoes the restroom and expresses some geometry looks how Geometry Expressions can be used on lower-level math courses.

Just weighing a pig doesn’t fatten — Obama hint on testing examines little-reported comments by President obama on his education policy.

UK moves closer to ending damaging tests in favor of teacher judgement reports on moves in the UK to reduce testing and put greater emphasis on assessment in the classroom by teachers.

Students learn what they think about reports on Dan T Willingham’s conclusions on how the brain works and learns.

Behaviorism 101 summarises how a supposedly dead and discredited theory is still at the heart of the US education system, doing daily damage.

Hope and change in my classroom assesses the likely educational policies of the new US president, and the balance between his education secretary Arne Duncan and key policy advisor Linda Darling-Hammond.

Assessment… a new sort of grade book looks at how classroom assessment based on descriptive feedback can be recorded.

Competitive grading still sabotages good teaching finds a decade-old description of poor teaching that gives a depressingly accurate description of current public school practice.

Win, win not fail, fail outlines the arguments against alpha-numeric grading and how grading undermines real learning.

Working inside the black box reviews the latest research from Prof Paul Black and the Asessment Reform Group now driving major changes in the UK education system.

What really was the point? reflects on what’s important for kids at school.

The final answer summarises an attempt at an alternative to the final exam.

Rigorously challenged rigorously challenges rigor.

Testing… a teachable moment argues if a student asks for help in an exam, then the teacher has a great teachable moment.

Open learning targets reports on PiFactory plans to go open source.

No prizes in points argues points and other behaviorist rewards in education diverts student attention away from what it is they are trying to learn.

Barack, me… and institutional racism in schools discusses the difficulties of raising the issue of racism in schools.

The business of parent conferences reflects on the shortcomings of student-parent-teacher conferences that focus on alpha-numeric grades.

My mistake, I didn’t read the question on getting stuck and unstuck.

JJ’s knock-out question describes one student’s demonstration of mathematical thinking.

Adolescence, a time for second, third… as many chances as it takes reviews research on the teenage brain and the implications for teaching.

Assessment… it’s all in the tone of the voice reflects on assessment aimed at helping learning.

Annah and Camilla get unstuck… and 5 million points describes a case of student self-assessment.

Student self-assessment &mdash the research says… reviews research on the gains to be made by using student self-assessment.

I’m stuck! — do I get 5 million points? reports on how two students got stuck and started to learn.

Assessment — when the numbers don’t add up outlines the case against numerical or percentage-based grading.

The research gives testing an F reports on the research showing the extreme damage caused by frequent testing.

Why I no longer set homework summarizes the case against homework.