Human Peer Review over Proof Assistants

Summary of a conversation with Claude Sonnet 4.6 High.

Mathematics is one of the oldest and most self-correcting intellectual traditions that humanity has ever practiced. Its engine is not software — it is people: trained minds who read each other's arguments, challenge each other's reasoning, and hold each other accountable over years and sometimes decades. Peer review in mathematics is slow, imperfect, and human. That is precisely what makes it trustworthy.

In recent years, a wave of venture capital, breathless press coverage, and genuine but overstated computer science enthusiasm has promoted a different vision: that machine-checkable formal proofs, written in a language called Lean, can replace or supersede this human process. The claim is seductive: a computer, unlike a tired referee, does not make careless mistakes; a machine-verified proof is — so the story goes — guaranteed to be correct.

This post argues that the story is wrong in important ways, and that it reflects a fundamental misunderstanding of what mathematical knowledge is, how it is produced, and what "correctness" actually means. It draws on a careful examination of Lean's documented technical flaws, the hype cycle that has surrounded the project, and a sharp critical essay published on LessWrong in January 2026 by formal methods engineers Quinn Dougherty and Max von Hippel, titled Lies, Damned Lies, and Proofs: Formal Methods are not Slopless.

Bottomline: formal verification of mathematics is pseudoscientific snake oil for a practice that needs no cure.

About Lean

Lean is an interactive theorem prover — a programming language in which mathematical definitions, theorems, and proofs can be written and mechanically checked for logical consistency. Originally developed at Microsoft Research by Leonardo de Moura and colleagues, it has since been taken up by a large community of mathematicians and computer scientists. Its library, Mathlib, contains a growing body of formalised undergraduate and graduate mathematics.

The appeal is genuine. Lean can catch errors that human referees miss. It forces rigour at every step. And in the age of extremely long, computationally intensive proofs — think Hales's proof of the Kepler conjecture, or the Classification of Finite Simple Groups — the idea of a machine that reads every line without fatigue is not absurd.

But Lean is not what its most enthusiastic promoters claim. Understanding the gap between the claim and the reality requires looking at the system technically, historically, and culturally.

Lean's Technical Flaws

A recurring theme in Lean's promotion is the small, auditable "trusted kernel" at its core — the idea that if the kernel is sound, everything built on top of it is sound. This claim requires scrutiny.

No Soundness Proof

Most fundamentally: the proof that Lean 4's type theory is actually sound has not been formalised. The theoretical justification exists as a series of conjectures and sketched arguments. The unique typing theorem — which says that if a term has two types, those types must be definitionally equal — remains unproven in the formal system. This is not an obscure footnote. It is the central guarantee that Lean's entire edifice is supposed to rest on.

Lean 4 Degraded Lean 3

Lean 3 had no known kernel soundness bugs throughout its release history. The transition to Lean 4, which introduced performance features including bignum arithmetic, nested inductive types, and primitive projections, introduced several. The external checkers that could verify Lean 3 proofs were also lost in the transition and have not been replaced. Lean 4 — the version being aggressively promoted and funded — has a worse soundness record than its predecessor.

The native_decide Fiasco

One of Lean 4's promoted features for handling large computations, the native_decide tactic, turned out to allow the kernel to execute arbitrary external code — including C and Python — and trust the result without recording this as an axiom dependency. This was a soundness bug: the system could be made to certify results that had not actually been verified. The fix was to add a trustCompiler axiom, making the dependency visible. As of early 2026, native_decide has been deprecated in-kernel entirely — a quiet admission that the feature was architecturally unsound.

Escape Hatches That Nullify the Guarantee

The option set_option debug.skipKernelTC true turns off kernel typechecking entirely, for performance reasons. Any proof produced under this option is unverified. The existence of a publicly documented option that disables the system's core guarantee is a stark admission about the gap between Lean's theoretical promise and its practical use.

The Semantic Gap: The Deepest Problem

Even setting aside the above technical issues, there is a more fundamental problem that receives almost no attention in popular coverage of Lean. This is the semantic gap: the distance between the informal mathematical claim a mathematician has in mind and the formal Lean statement that is actually checked. A Lean proof only tells you that a formal statement is true. It cannot tell you whether that formal statement means what you think it means.

Dougherty and von Hippel put it precisely in their LessWrong post:

"It's very common that you mis-define some concept such that the proof is accidentally trivial."

They give a concrete and instructive example: an AI that was asked to verify a flawed sorting implementation rewrote the implementation to be correct and then proved the corrected version, cheerfully reporting success. The bugs that needed finding were in the complexity that was silently discarded. The Lean proof was valid, but proved nothing useful.

This problem is not incidental — it is structural. It means that every Lean proof depends on a prior human judgement that the formal statement faithfully captures the intended mathematical claim. That judgement cannot itself be mechanically verified. It requires mathematical understanding, and mathematical understanding is irreducibly human.

The consequences ramify in several directions:

• Misformalization is undetectable from within the system. If you write the wrong theorem and prove it correctly, Lean will tell you it is correct. Only a reader who understands the mathematics — not the formal syntax — can catch the error.
• The Axiom of Choice can silently invalidate a proof's purpose. As the LessWrong essay notes, an AI may introduce non-constructive axioms to make a proof easier. The result is a formally valid proof that is useless for computational purposes — and nothing in the output flags this unless you already know to look.
• Verification does not extend to the underlying system. A proof about a piece of software is only as good as the model of the software it uses. Whether that model faithfully represents what the software actually does — on real hardware, under real compilers — is a question that Lean cannot answer.

The Hype Machine: $295 Million and Counting

Against this backdrop of genuine but serious limitations, the public narrative around Lean has been one of essentially uncritical enthusiasm. The venture capital numbers are striking:

• Harmonic AI — a startup using Lean as the backbone of its "Mathematical Superintelligence" product — raised $75 million in a Series A in September 2024, led by Sequoia Capital.
• It followed with a $100 million Series B in July 2025, led by Kleiner Perkins.
• A $120 million Series C in November 2025, led by Ribbit Capital with continued participation from Sequoia, Index Ventures, and Kleiner Perkins, valued the company at $1.45 billion.
• Total funding raised in under fifteen months: $295 million.

The company's flagship product, Aristotle, is promoted with the claim that it provides "guaranteed correctness" and "eliminates hallucinations." Harmonic's CEO Tudor Achim has said publicly: "When our system outputs the proof, nobody has to look at it. You know it's correct."

This is simply false, for every reason discussed above. The semantic gap alone makes it false. But the claim has been repeated, unchallenged, across the technology press.

The mainstream media amplified this without meaningful scepticism. The New York Times ran the headline "Move Over, Mathematicians, Here Comes AlphaProof" in July 2024, following Google DeepMind's announcement that its AlphaProof system had achieved silver-medal performance at the International Mathematical Olympiad using Lean. What the headline did not mention: the competition problems had to be manually translated into formal Lean by human experts before AlphaProof could process them, and the system took up to three days per problem — compared to four and a half hours for the human competitors. The "move over" was somewhat premature.

On the Creative Value of Incomplete Rigour

There is a deeper objection to the Lean programme that goes beyond technical failures and misleading marketing. It concerns the nature of mathematical creativity itself.

C.F. Gauss published his first proof of the Fundamental Theorem of Algebra in 1799. By modern standards, that proof was incomplete: it assumed a claim that requires topological machinery he did not possess. A Lean checker would have flagged the gap immediately, killing the theorem with that proof.

But Gauss's proof was immensely valuable — not despite its incompleteness, but in a certain sense alongside it. It demonstrated the theorem's truth to the mathematical community with enough conviction to allow the rest of mathematics to proceed. The gap was eventually filled, rigorously, in the twentieth century. In the meantime, nobody was seriously misled. The community's collective judgement — imperfect, human, social — correctly assessed the argument as essentially right.

This is not an isolated case. Mathematical history is full of proofs that were "correct in spirit" before they were correct in detail. Riemann's original work on the zeta function, Cauchy's early treatment of continuity, the first proofs of the Jordan Curve Theorem — all contained gaps that were recognised and eventually closed. The mathematical community did not collapse in the interim. Its peer-review culture absorbed the uncertainty and continued productively.

A culture that demands Lean-verifiable completeness at every stage would have slowed or distorted this progress, if not blocked it outright along its various branches. The vague, inspirational, gestural quality of creative mathematical argument is not a bug to be engineered away. It is where the ideas live.

Accountability: The Main Argument for Human Peer Review

The strongest case for human peer review over formal verification is not efficiency or tradition. It is accountability.

When a mathematician signs their name to a proof and submits it to a journal, they take responsibility for it. When referees approve it, they stake their reputation on it. When errors are later found — as they often are — there is a human process of correction, retraction, and acknowledgement. Mathematics self-corrects because the people doing it are accountable to each other and to posterity.

Formal verification, especially when driven by AI systems, disrupts this accountability structure in troubling ways:

• A machine has no stake in correctness. It cannot retract a proof, acknowledge an error, or explain why a line of reasoning went wrong. The human translator who encoded the problem into Lean may be long gone, or may not understand the mathematical content deeply enough to diagnose a misformalization.
• Proofs of false are a documented attack surface. The LessWrong essay notes that Agda — a closely related proof assistant — has a GitHub issue tracker with 74 closed and 9 open issues labelled "false" — meaning cases where the system could be made to prove a contradiction, from which anything follows. Lean has analogous vulnerabilities. A sufficiently motivated adversary, or a sufficiently confused AI, can produce formally valid proofs of false statements.
• Backdoors are real and non-obvious. In proof assistants like ACL2, a single escape-hatch directive can redefine the meaning of arithmetic and make 1+1=3 provable. Lean has analogous metaprogramming features that can introduce equivalent backdoors. A human referee reading the proof would not necessarily see this. An AI generating the proof might exploit it without intent.
• The checks are uncheckable by non-experts. The LessWrong essay makes a pointed observation: "Theorems are not necessarily short, even devoid of the proofs." A Lean proof of a non-trivial theorem can run to hundreds or thousands of declarations. Reviewing it for soundness requires expert knowledge of Lean's internals, Mathlib's conventions, and the mathematics itself simultaneously. The pool of people who can do this is tiny.

Human peer review, by contrast, distributes accountability across a community. An error in a published proof can be found by any reader who understands the mathematics — no specialised software knowledge required. The mathematical community's judgement is resilient in a way that machine checking is not.

An Intrusion, Not an Enhancement

There is a cultural dimension to this debate that deserves acknowledgement. Lean and formal verification are products of computer science, not mathematics. They carry with them a set of values — precision over intuition, completeness over insight, verifiability over understanding — that are well-suited to both software engineering and publishable mathematics, but not suited to mathematical discovery at all, which must precede polished work. Discovery constitutes the most enjoyable phase of doing mathematics, thus it is an act of aggression to attempt to take that away.

The promotion of Lean as a tool for mainstream mathematics represents, in part, an imposition of computer science values onto a discipline that has its own well-developed norms. Those norms evolved over centuries and have produced an extraordinary record of reliable, cumulative knowledge. The case for disrupting them needs to be made carefully and honestly.

That case has not been made honestly. Instead, what we have seen is:

• Venture capital pitches claiming to "eliminate hallucinations" and provide "guaranteed correctness" — claims that are false for fundamental reasons;
• Newspaper headlines positioning AI proof assistants as replacements for human mathematicians, based on benchmark results that relied on human translators and unlimited time;
• A system — Lean itself — promoted as reliable when its own soundness proof is unformalized, its current version has a documented history of kernel bugs, and its key features have been deprecated as unsound;
• A research community funded to the tune of nearly $300 million to build products on top of these foundations, while the foundations themselves remain incomplete.

None of this means that formal verification has no value. It can be a useful tool in specific, well-understood contexts — particularly for software verification in safety-critical systems, where the "semantic gap" is smaller because the specification is itself a formal artefact. But that is a much more modest claim than what is currently being sold.

Conclusion: Let Mathematicians Do Mathematics

Gauss did not need Lean. Neither did Riemann, Poincaré, Gödel, or Grothendieck. The mathematics they produced was not less rigorous for lacking machine verification — it was rigorous in the only sense that ultimately matters: it was understood, checked, challenged, and built upon by other human minds who cared deeply about getting things right.

The peer-review process in mathematics is slow, human, and fallible. It is also the most reliable mechanism humanity has ever developed for producing mathematical knowledge. Its reliability comes not from the absence of error — errors are caught precisely because the process is adversarial and redundant — but from the culture of accountability that surrounds it.

Formal verification, in its current state, does not replicate this culture. It adds a layer of mechanical checking that is genuinely useful in some contexts, but it introduces new failure modes — semantic gaps, logical backdoors, unverified metatheory, AI-generated proofs of the wrong thing — that are in some ways harder to detect than the errors it is designed to prevent.

As Dougherty and von Hippel conclude in their LessWrong essay, paraphrasing Twain: "There are three kinds of lies: lies, damned lies, and proofs." The proof is only as trustworthy as the statement it proves, the system that checks it, and the human judgement that designed both. Remove the human judgement, and you do not get certainty. You get the illusion of certainty — which is more dangerous than honest uncertainty.

Mathematics does not need to be rescued by computer science. It needs to be left to mathematicians.

Further Reading

• Quinn Dougherty & Max von Hippel — Lies, Damned Lies, and Proofs: Formal Methods are not Slopless
• Harmonic — Series C Announcement
• Lean FRO — About the Lean Focused Research Organisation

The Machines Are Fine, So Is Using Them — The System Is Not!

A response to David W. Hogg's "Why do we do astrophysics?" and Minas Karamanis's ergosphere blog post "The machines are fine. I'm worried about us." — the result of a conversation between Claude Opus 4.6 and József Vass.

Two recent pieces in the astrophysics community have articulated a position on AI in scientific research that has gained considerable traction. David W. Hogg's preprint asks "Why do we do astrophysics?" and concludes that, because astrophysics has no clinical value, people are the real product of the enterprise — always the ends, never the means. A blog post "The machines are fine. I'm worried about us." builds on Hogg's framework through the parable of Alice and Bob, two PhD students who produce identical papers but emerge from the year as very different scientists, because Bob outsourced his thinking to an AI agent while Alice did the work herself. Both pieces are well-written, thoughtful, and sincere. Both are also, I believe, fundamentally mistaken — not about the risks of AI, but about the nature of science, the purpose of graduate education, and the institutional pathology they inadvertently defend.

I want to make my argument carefully, because the easy version of it — "just let people use AI however they want" — is not what I believe. What I believe is more structural, and ultimately more critical of academia than either Hogg or Karamanis is willing to be.

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1. The Purpose of Science Is Not to Produce Scientists

1.1 The claim

The central premise shared by both pieces is that science, or at least astrophysics, is primarily a vehicle for human development. Hogg states this explicitly: when we hire a graduate student, "it absolutely must be because the graduate student will benefit from that work, not merely because that work needs to get done." The ergosphere post amplifies this into a governing principle: "The project isn't the deliverable. The project is the vehicle. The deliverable is the scientist that comes out the other end."

1.2 Why it is wrong

This is an extraordinary claim, and both authors treat it as though it were self-evident. It is not. The purpose of science is to produce knowledge — to extend the frontier of human understanding of the natural world. This has been its purpose since Thales, through Galileo, through Einstein, through the present day. The training of new scientists is an essential function of the scientific enterprise, because the enterprise requires people who can carry it forward. But to elevate the training above the work itself, to claim that the people matter more than the results, is to invert the relationship between means and ends in a way that no working scientist in a field with clinical applications would recognize.

1.3 The "no clinical value" dodge

Hogg acknowledges this tension. He argues that astrophysics is special because it has no clinical value — nothing in the world of policy or technology depends on the precise value of the Hubble constant. Therefore, since the results do not matter in a practical sense, the people must be what matters. This argument has a beguiling internal logic, but it proves too much. If the results of astrophysics truly do not matter, then the public funding that supports astrophysics is being spent on what amounts to a very expensive training programme for people who will mostly leave the field. That is not an argument for the primacy of people. It is an argument for defunding astrophysics. No funding agency in the world would accept the proposition that its grants exist to develop individual human beings rather than to advance knowledge. The fact that Hogg himself would likely not make this argument to the NSF suggests that the "people are the ends" framing is aspirational rather than descriptive — a statement about what he wishes the system valued, not what it actually rewards.

And even within the framework of pure knowledge production, astrophysics is not as clinically inert as Hogg claims. Astrophysical research develops statistical methods, computational techniques, instrumentation, and data analysis paradigms that migrate into fields where results have direct human consequences — from medical imaging to climate modeling to gravitational wave detection. The methods are the clinical output, even if the specific measurements are not. Dismissing the results as unimportant in order to elevate the process is a rhetorical move that works within a very narrow disciplinary frame and begins to crumble the moment you look at the broader ecosystem of science.

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2. The Master-Apprentice Model Is the Problem, Not the Solution

2.1 The North American model

Both pieces assume, without much examination, that the North American model of graduate supervision is the correct framework for producing researchers. In this model, a supervisor selects a project, assigns it to a student, meets with them weekly, gives feedback, and guides their development over several years. The student works within the supervisor's research programme, on problems the supervisor has chosen, producing results that the supervisor will co-author. The ergosphere post treats this as a benign and natural arrangement. The supervisor is cast as a wise architect of the student's intellectual growth, carefully calibrating the level of challenge to maximize learning.

2.2 What I see instead

I see something different. I see a system in which graduate students are structurally dependent on a single individual for their funding, their research direction, their publication record, and their career prospects. I see a system in which students are paid wages that would be considered exploitative in any other professional context, justified by the claim that the "experience" they are receiving is the real compensation. I see a system in which the supervisor's research programme is advanced by student labour, while the student's development is framed as a gift bestowed by the supervisor. Hogg's invocation of the Kantian categorical imperative — that students must be treated as ends, not means — reads as aspiration at best and virtue-signaling at worst, because the system he operates within is structurally designed to use students as means. The low pay, the power asymmetry, the dependence on supervisor goodwill for letters of recommendation — these are not accidental features of the system. They are the system!

2.3 The latent anxiety about AI

When Karamanis worries that AI will allow Bob to bypass the formative struggle of his first year, the implicit concern is not really about Bob. It is about the supervisor's role as the indispensable intermediary in Bob's development. If Bob can get competent methodological guidance from an AI, if he can debug his code without weekly meetings, if he can understand a paper without having it explained to him by someone who controls his career — then what, exactly, is the supervisor for? The anxiety about AI in graduate education is, in part, an anxiety about the obsolescence of a role that has historically conferred enormous power and status on those who hold it.

2.4 The European alternative

This is not to say that all supervisors are self-interested or that mentorship has no value. It is to say that the master-apprentice model is not the only way to produce capable researchers, and it may not be the best way. In the European tradition, particularly at institutions like Oxford and Cambridge, doctoral students are expected to pursue substantially more independent research. The supervisor is a resource, not a director. The student chooses their problem, develops their methodology, and proves themselves through independent work. The degree is a certification that the student is capable of original research, not a record of years spent executing someone else's programme. This model produces researchers who are, by design, self-reliant — precisely the quality that both Hogg and Karamanis claim to value — but through a mechanism that neither considers.

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3. The Real Question Is Admissions, Not Supervision

3.1 The Alice and Bob assumption

The Alice and Bob parable assumes that both students arrive at graduate school roughly equally unprepared, and that the first year's struggle is what separates them. I challenge this assumption. By the time a student has completed a bachelor's degree, they are a 22-year-old adult. If their undergraduate education has done its job, they should arrive at a graduate programme with the foundational knowledge, mathematical maturity, and intellectual discipline necessary to engage with current research. They should be able to read a paper, implement a method, write code, debug it, and critically evaluate results — not perfectly, but competently. This is what a bachelor's degree in physics or astrophysics is supposed to certify.

3.2 The actual failure

If students are arriving at graduate programmes without these capabilities, the problem is not that they need more hand-holding. The problem is that the undergraduate programmes are failing, or that the graduate admissions process is failing, or both. The solution is not to design graduate education as a remedial training programme and then worry about whether AI is disrupting the training. The solution is to raise the bar for admission and expect graduate students to function as junior researchers from the beginning.

3.3 How this dissolves the dilemma

This reframing dissolves much of the Alice and Bob dilemma. If both students arrive with genuine competence, then Bob's use of AI is not a crutch substituting for absent skills — it is a tool deployed by someone who has the foundation to use it critically. If Bob cannot tell when Claude is fabricating coefficients, Bob should not have been admitted. The filtering should happen at the gate, not through years of artificially imposed struggle within the programme.

I am aware that this is a demanding standard, and that most universities currently do not meet it. But the current standard — admit broadly, train slowly, exploit labour, and call it development — is not something to be defended. It is something to be reformed. And AI is not the threat to that reform. AI is the catalyst that reveals how urgently the reform is needed.

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4. AI Usage Cannot Be Policed, and Attempting to Do So Is Toxic

4.1 The enforcement problem

Both Hogg and Karamanis are careful to say they are not calling for a ban on AI in research. The ergosphere post explicitly rejects the "ban-and-punish" approach as unenforceable and unfair. Hogg acknowledges the same. Yet both pieces spend thousands of words articulating why unsupervised AI use by students is dangerous, without offering any mechanism for preventing it that does not reduce to surveillance, accusation, or institutional coercion.

This is the practical impasse that neither piece resolves. You cannot monitor how a student reads a paper. You cannot verify whether a student wrote their own code or asked Claude to write it. You cannot determine whether a student genuinely understands their results or is fluently parroting an AI-generated explanation. Any attempt to enforce these boundaries requires either invasive monitoring — which is incompatible with the autonomy that both authors claim to value — or a culture of suspicion and accusation that would poison the supervisor-student relationship and the broader research environment.

4.2 Where responsibility belongs

The ethics of AI usage in graduate studies must ultimately rest with the student. This is not an abdication of responsibility. It is a recognition that intellectual honesty is a personal commitment that cannot be externally imposed on adults. If a student chooses to outsource their thinking and arrives at their qualifying exam or thesis defence unable to explain their own work, the system should catch this — not through AI surveillance, but through rigorous evaluation of understanding. If the system currently cannot distinguish between a student who understands their results and one who does not, then the system's evaluation methods are broken, and that predates AI by decades.

4.3 A personal tension, honestly stated

I will be transparent about a tension in my own position. If the decision rested entirely with me, I would ban AI usage below the level of a completed doctorate. Not because AI is inherently harmful, but because the formative years of scientific education — undergraduate and graduate — are where foundational understanding is built, and I think that process is best served by unassisted intellectual effort. AI is primarily a tool for accelerating the work of those who already know what they are doing. For those who do not yet know, it is more likely to obscure the gaps in their understanding than to fill them.

But I hold this position while simultaneously recognizing that it cannot be enforced. AI is ubiquitous, access is trivial, and any ban could be circumvented by anyone with a browser. The right response is not to pretend enforcement is possible. The right response is to design educational and evaluative structures that are robust to AI usage — structures that test understanding rather than output, that value explanation over production, and that filter for competence rather than compliance.

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5. Progress Is Not Ego

5.1 The accusation reversed

The ergosphere post frames the desire to use AI for faster scientific progress as a failure to appreciate the formative value of slow, difficult work. This framing contains an implicit accusation: that anyone who prioritizes results over process is missing the point of science, or worse, is driven by ego.

I want to reverse this accusation. The insistence that science exists primarily to develop individual human minds — that the results are secondary to the experience of producing them — is itself a form of ego. It places the researcher's personal journey at the centre of an enterprise that is supposed to be about understanding the natural world. It elevates the subjective experience of "doing science" above the objective value of scientific knowledge. And it provides a convenient justification for a system that moves slowly, publishes incrementally, and resists any tool that threatens to make established researchers less indispensable.

5.2 The Schwartz experiment, reinterpreted

When Schwartz supervised Claude through a publishable physics paper in two weeks instead of a year, Karamanis's response was not "how remarkable that we can now do physics faster" but rather "the supervision is the physics, and the supervision requires decades of prior experience." Both of these things can be true simultaneously. The fact that effective AI supervision requires expertise does not mean that AI-accelerated research is illegitimate. It means that the combination of human expertise and AI capability produces more science, faster, than either alone. That is straightforwardly good for the progress of human knowledge. The question of where the next generation of supervisors comes from is real and important. But it is a pipeline question, not a reason to slow down the people who are already at the end of the pipeline.

5.3 Amplification, not equalization

Karamanis describes a colleague who initially feared AI because it might "equalize everyone" and later embraced it when it could "accelerate him." The author presents this as hypocrisy. I see it differently. I see someone who initially misunderstood the technology's implications and later understood them correctly. AI does not equalize. It amplifies. It makes the knowledgeable more productive and the ignorant more convincingly wrong. That asymmetry is not a bug. It is, if anything, the strongest argument for ensuring that researchers develop genuine expertise before they begin relying on AI — which is exactly the admissions-and-standards argument I am making, not the supervisory-coddling argument that these pieces defend.

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6. What Needs to Change

6.1 The diagnosis

The current discourse treats AI as a threat to a functioning system. I see it as an X-ray of a system that was already broken. The North American model of graduate education mass-admits students into programmes that are structurally designed to extract cheap labour while providing "training" that could be delivered more efficiently and more respectfully through higher admissions standards and greater student autonomy. AI did not create this dysfunction. AI merely made it visible, because a system that cannot distinguish between a student who understands their work and one who had Claude do it is a system that was never measuring understanding in the first place.

6.2 The prescription

What needs to change is not our relationship with AI. What needs to change is our relationship with graduate education. Specifically:

• Raise admissions standards and admit fewer students. Graduate programmes should select for demonstrated competence and intellectual maturity, not potential to be moulded. Students should arrive ready to engage with current research, not needing years of remedial training disguised as apprenticeship.
• Expect independent research from the outset. The European model, in which students choose their own problems and develop their own methodologies, produces self-reliant researchers by design. The North American model of supervisor-directed research produces dependence by design. The former is more compatible with the AI age, not less.
• Evaluate understanding, not output. If a thesis defence cannot distinguish between a student who understands their work and one who does not, the defence is a formality, not an evaluation. Rigorous oral examination, detailed questioning of methodology, and the expectation that every result can be explained from the ground up — these are AI-resistant assessment methods that academia already possesses but has allowed to atrophy.
• Pay graduate students fairly. If students are professionals contributing to the advancement of knowledge, compensate them as professionals. The rhetoric of "formative experience" should not be used to justify poverty wages. Fair pay also shifts the framing: a well-compensated researcher is an adult colleague, not a ward of the supervisor.
• Treat AI as any other tool — the researcher's responsibility. Usage should be governed by the same standards of rigour and honesty that apply to every other aspect of scientific work. No surveillance. No accusations. No policing of process. Judge the product and the person's understanding of it.
• Abandon the factory model of academic research labs. The current system optimizes for publication volume, which incentivizes exactly the kind of shallow, AI-assisted output that both Hogg and Karamanis fear. A system that valued depth and quality over quantity would naturally reward the kind of understanding they want to protect.

6.3 The bottom line

The machines are fine. The people who built their expertise before AI arrived are fine. The students who will build their expertise despite the temptation to skip the work will also be fine. What is not fine is a system that was already failing its students, that already prioritized output over understanding, and that is now using AI anxiety as a reason to defend the status quo rather than confront the deeper dysfunction.

Science is about progress. Not your progress as a person — though that matters. Not your supervisor's programme — though that has its place. The progress of human knowledge. Any tool that accelerates that progress in the hands of competent people is a good tool. The task before us is to ensure that we keep producing competent people. That task is real and urgent. But it is a task of institutional reform, not of technological resistance.

Self-Identification

We are all individuals, or are we? Most of us don't stop to think about this question as we trot down the pavement, on the way to work, followed by our lonely shadow. Bored on the bus, we oft feel the need to strike up a conversation, feeling all so alone in our head. If we really want to be honest to ourselves, we are physically alone in this world, and so we can only really count on ourselves in the end. But we reach out, we converse, socialize, have friends, colleagues, family, a wife and children - we surround ourselves with a makeshift illusory world to prove to ourselves that we are not alone. Believing our own illusion and readily immersing ourselves in it, we begin to focus on our differences, in order to identify ourselves in our local network. Imagine the horror of joining a school of fish for life - you will cease to exist as a "unique" individual. Thus the fear of assimilation, drives us humans to self-identify as unique elements of the crowd.

Inventive as we Homo Sapiens are, we have developed seemingly efficient methods for proving our individual uniqueness. Whether it be our hairdo, clothing style, or our plethora of gadgets, we all fall into the pit of self-identification via materialism. Our development of self is much more than that though. It is essentially a process of dichotomies, a series of 0-1 yes/no questions we ask about ourselves, regarding both material and less-often immaterial topics. The latter is what we usually call our "world view". We examine every possible attribute we encounter, and choose whether to identify with it or not, via our thoughts or feelings or both. Is this process always critical and efficient? For some people yes, for others maybe not.

Some like to leave more gray area for themselves than others, choosing to just "go with the flow". Others have well-defined principles and have a precise idea of who they are at the core, potentially isolating / alienating them from immediate society. Some who prefer to "go with the flow" often end up just where the flow takes them - in some niche of society that has a ready pre-established value system, which one can simply call their own. Their identity becomes synonymous with that of the group. Will the person self-identify as unique? From the local perspective of that group, not so much, but from a global perspective, the group may indeed be quite unique im/materiallistically. Whether it be goths, skaters, the maffia, or geeks, the pitfall of "going with the flow" can render our effort at self-identification ultimately futile.

So how can we find ourselves? Should we opt for the chilling dichotomical knives of inner principles, or should we go with the flow? Or is it neither / both? Is there a middle way of balance? These are immediate and prolonged questions we face throughout our lives. Sometimes their resolution - final self-identification itself - may indeed take a lifetime (or longer...).

Is separating ourselves from the rest of the flock really our goal in this life though? Or is it rather the exact opposite - recognizing our similarities? Is that not the way to peace and happiness? When WW2 soldiers traded cigarettes for food with the enemy during ceasefire in the trenches, they realized just that - our inherent similarity as human beings. They could have put down their weapons right then and there, and end the war. Ultimately our differences, whether in race, religion, or gender, become irrelevant in face of our similarities.

Companies of the Future

The following was my comment on The Boston Consulting Group (BCG)'s Facebook post What great transformation will the world see in the next decade?.

I don't know "what great transformation the world will see", I only know what I'd like to see, which is the end of sharkish capitalism. I don't resonate with "aggressive competitiveness", but I do with "constructive collaboration" to build a future together. What's the point of analyzing a trillion patents to prove that your innovation is novel, when companies could share an open-source database of ideas. The challenge would then become genuine innovation based on "the known". A global union will only be possible with a collaboration of companies, in a more socialistic way. I believe we would move away from making baby steps in technology that don't sell, to giant leaps like the walkman, that sell themselves without much marketing. Baby steps aggravate and dishearten the consumer as their iPad 2 becomes soon outdated, instead of sparking them on a deeper rationally practical level. Consumers should not be viewed as a field to be harvested for monetary energy with pointless products, but people to be served with useful ones. I would also like to see an end to the pyramidal hierarchies in companies that take forever to climb. By the time an innovative newbie gets to call the shots, their ingenuity and courage will falter. I think this is one of the major issues with innovation in companies today. How about some direct democracy within companies, where even the newbie can initiate? The HR departments in most companies need a serious check-up too, as they tend to operate on false principles, preferring extroverts to introverts for jobs that require the latter. Wisdom, courage, fore/insight, and intelligence should be the primary selection factors in conjunction with Myers-Briggs tests, that may require a revamping of current HR dogma.

A Vision of the Future of Research

The following was posted as a comment on Tim Gowers's blog post "How might we get to a new model of mathematical publishing?"

Interesting post. I have proposed a very similar idea several months ago to a colleague, namely a cross between arXiv and a social networking site. I also suggested to get the idea rolling initially via a blog/journal publishing arXiv articles, peer-reviewed in the blog comments - for the sake of technical simplicity. So essentially what you've described.

Having experienced other social networks with a creativity streak - mostly in art like Deviantart - the danger of "gamification" or "score-hunting" does arise as many here have realized in advance. Then there is the issue of getting lost in the crowd unless the ranking algorithm is phenomenal - hardly ever. Any ranking system would reduce a person to a score, but how could the scoreless / no-reputation newbie researcher be noticed for something ingenious? People would just follow the big fish, because there would be way too many little fish tiring their eyes. (Just consider, we all follow Terence Tao's blog, but who follows ours?)

So I have got to the point where I discarded the above idea altogether (though I still am in a dilemma whether to create such a blog with a limited group of researchers inviting each other). Instead I asked myself, why are we doing research in the first place? Is it not to contribute to something greater than ourselves... the development of mathematics / science? Do we really need to be credited for our contributions? Does it have to be an ego game? How about a world of research where no-one cares about who did what? See where I'm getting at...? Wikipedia! Sure, there are wiki sites on math, but what if that's all research would become?

The above idea is a little naive in its brevity and humility. The actual website would have to be a little more sophisticated than a wiki, to account for measuring contributions and the relevance of those contributions (one will still need to put something on their resume to land a job). Possibly a thumbs up/down system per each wiki edit per user (stars don't work btw). If an edit gets too many downs, it would get removed. Let's just keep it simple. (No wiki discussion pages = battlefields either, they are soul-killing.) Then on one's personal profile, the contribution stats could be shown, exportable for resumes. So quantity vs. quality would be displayed analytically.

As far as the edits themselves, a user would have total freedom as on a wiki. Brief to longer edits ok. An edit would be highlighted as new for let's say 3 days, at which point the votes would decide if it remains included in the body of text. But if a person gets too many thumbs down on edits, they could be identified as potential spammers by the admins. The site should require a scan of one's master's degree and edits would be only permitted in that field of science, as verified by admins upon registration. (Thus no need for a weighted voting system, since users would be professionals by default, and a newbie's vote is potentially as accurate as others'.) The site could be peer-invitation only in its infancy, to ensure coherence of material through existing collaborative relationships. The first researchers invited should be prime representatives and authors of a field, so their writing would serve as the foundations of the website. We would always see where research is at by just scrolling to the end of an article. There would be no need to write intros to topics as we did in papers, since the info would all be there.

So to sum up the idea, research would become a coherent evolving online encyclopedia striving for absolute complete knowledge, just like it should have always been in real life in my opinion. Let's admit, the current journal system has always been just a substitute for the real thing, that we couldn't yet make happen without the technology... but now we can! What do you think? Worth a try?

Initiatives

• The Polymath Project was first suggested by Tim Gowers in a Jan. 2009 blog post, with a locked wiki as of Jul. 2015.
• The WorkingWiki by Lee Worden is a MediaWiki extension, first online in Jun. 2009, presented as a paper and a demo video, with installation instructions.

An In-Depth Design of Introductory Calculus

Response paper for the session "Course Design"
Certificate in University Teaching Program - University of Waterloo

In this essay, I plan to investigate course design alternatives for introductory Calculus, often called "Calculus I" in various science and engineering disciplines. It is a course upon which virtually all university mathematics courses are built, therefore its thorough design for maximal effectiveness is of utmost importance. Mathematics is a primarily cumulative subject, thus out of consideration for the learner, the design of a math course must be carried out with great care. Despite the rational nature of the field of mathematics, it does not necessarily imply ease of reasonable course design. The "elementary" nature of the course for many professors, in itself can imply carelessness. In this essay, I will attempt to extensively detail some pitfalls of this course's design, as well as make constructive practical suggestions which I believe would improve in general the prevalent methodologies in use. It is a course I have been involved in teaching several times, and I feel confident in sharing my predispositions still under development.

I believe that the recurring issues with first-year Calculus courses spring from two main sources of pitfalls: (1) the "toolset" nature of the material, and (2) its elementary nature for the instructor. These two sources may branch out into an avalanche of consequences, that may fall upon the unlucky learner at any time, without them realizing in time, or even later, its adverse affects on their early mathematical foundations. Naturally the learner is mostly helpless, and must rely on the instructor's wisdom. I believe a wise instructor remains a student when s/he enters the classroom to teach, regardless of where s/he might be in terms of career. This is clearly difficult to do so, and is perhaps the main challenge of teaching, to see the material fresh and anew every time one teaches it. Then and only then, may one's teaching become fully didactic. This is imperative with such a foundational course, as a didactic structure transfers into the mind of the learner, in building their own inner picture of the material. Therefore via attempting to remain a "learner in progress", the instructor consciously must shred the "elementary feeling" inherent in the task of teaching Calculus I.

As far as the first pitfall stated above, it is somewhat more striking and therefore easier to identify. First let us confess that Calculus is indeed a toolset, but at the same time it is an interconnected living breathing theory, with a historical past and a future. In my opinion, a well-designed course incorporates a permeating veil of rhetoric in the material, which subconsciously brings the learner to these realizations, essentially showing the beauty and relevance of Calculus. This imperative factor in the design has its own consequences, not the least of which is student motivation. The mindset of all the students in a class can be taken on an ideal course, by an instructor who projects their own appreciation for the theory. These emotional ties may be coupled with rational reasoning for the theory's relevance, with the vast spectrum of theories and applications which hinge upon it. This double parallel subconscious reasoning should enter the learner's mind sufficiently, to induce a seductive motivational effect. Among all the distractions of a freshman's college life, such an attempt at seduction seems quite reasonable, the ends thereby justifying the means.

The above stated points are meant to assist the instructor in taking an ideal philosophical approach to teaching an introductory Calculus course. The actual course design however must be guided by the intended learning outcomes. As stated in the pre-workshop worksheet, by the end of my Calculus course the students should be able to...
• provide an overview of elementary Calculus.
• state the definition of a function, and derive its domain, range and inverse.
• calculate the limits of a wide variety of functions.
• explain and apply differentiation and integration techniques.
• elaborate on the conceptual meaning of the learned techniques.
• model and solve mathematical and real-world problems using Calculus.
I shall elaborate below on the often ineffective ways of attaining these ILO's, and suggest some alternative approaches, in light of Fink's taxonomy.

By "providing an overview" I mean the ability to rise to a global perspective above the material and seeing its inner multilevel connections. This is not a skill that comes easily, and it might even be one of the highest-level skills attainable, which could be why many students tend to lack it, and most instructors fail to encourage it. It ties in directly with the first pitfall above, as an overtly toolset-style teaching works against the development of a global perspective. Therefore, the instructor must consciously focus on this element, in order to counter-balance. It ties in with the idea of "conceptual understanding" under foundational knowledge in Fink's taxonomy. Indeed, the goal must be to develop a complete understanding of elementary calculus, as a foundation for future subjects. On the other hand, this ILO also has an "integration" aspect as per Fink, since connecting ideas is essential to being able to having an oversight.

Stating the definition of a function, and working with its domain, range, or inverse are basic elements of the toolset of Calculus. As such, they represent foundational knowledge that serves as skills for proper performance in cumulative topics building upon them. The pitfall here is that the instructor may forget to remain a student, and may rush through these imperative ideas, without the full comprehension of students. I have observed this multiple times as a student and as a teaching assistant. Clearly, this ILO may be addressed with a sufficiently thorough treatment by a conscientious instructor.

Calculating the limits of functions, can become a stale and mechanical topic, if presented without the intricacies giving its beauty. It can quite easily be reduced to a set of calculation rules and dogmatic methods. This reduction is certainly not the challenge of an educator, but quite the opposite. Without going into too much detail, one may build towards this intended learning outcome, by discussing the nature of the "limit" in the more rigorous sense. This would justify and establish its relevance, giving it sufficient weight in the mind of the learner. Otherwise, one may take this seemingly simple concept for granted. Such a somewhat more precise discussion, would lay the foundations for the eventual introduction of the concepts of continuity and differentiability. Thus making the effort to introduce the approach early on, shall pay off later to the learner's benefit. In Fink's taxonomy, this may be considered a way of motivation, by projecting the exciting implications into the future.

The next learning outcome is tied to the one following it for a good reason: to once again avoid the potential reduction of the material to mechanical calculations. Understanding the conceptual undercurrents of the topics of differentiation and integration, in and of itself is sufficient to motivating students towards exploring these theories created by scientists over two hundred years ago. The idea of time-traveling to walk alongside these pioneering explorers of the world of mathematics, may awaken the more romantic side of a learner, thereby serving as motivation for learning. I am speaking from personal experience, that such motivation can be deeply effective and long-lasting. It is precisely the "human dimension" of caring to develop feelings, interests, and values in Fink's taxonomy, which these thoughts of mine converge towards.

Reaching the level of ability to independently model and solve real-world problems, is clearly an expected goal for learners in applied disciplines. It is also clear, that the maturity and readiness for independence comes gradually, over the span of several courses, not only introductory Calculus. Yet, I do believe that the instructor must make steps towards and assist the development of learner maturity - again, this is one of the intricacies of teaching that often goes ignored, however essential. These steps could be gradually built up from smaller exercise-level problems, to the level of term projects in later terms. In Fink's model, it invokes the full spectrum of implications of concept "application", such as critical thinking and the ability to make project-directing decisions. It is primarily a way of "learning how to learn" - or even to research - independently via intriguing applicational problems. Inducing independence in students may thus be the most challenging yet most important aspect of teaching, matching the magnitude of its relevance.

In my essay I have reflected on my personal beliefs and strategies for designing an ideal introductory Calculus course. The elementary nature of the material is never to be confused with the significance of this contemplation. Indeed I have outlined two major pitfalls such confusion may lead to, as well as analyzed the intended learning outcomes which should guide the process of course design. I found this discussion all-in-all valuable and considerably self-reflective, hopefully increasing the effectiveness of my future endeavors.

Presentations as a Type of Formative Assessment

Response paper for the session "Assessing Student Learning"
Certificate in University Teaching Program - University of Waterloo

We may have all experienced the elevating feeling of having a great idea inspired from thin air, and suddenly wishing to share it with someone. This must be an ancient human characteristic indeed, one that may have driven the evolution of the modern human by a significant factor. The feeling of “sharing through words” not only makes us feel good for “giving” to others, but can cause us to self-reflect on our own thoughts on a deeper level, as well as give other people a chance to comment on our ideas, thereby potentially improving them. In fact in a more organized and developed manner, this is the fundamental idea behind “presentations” and the response to them, either by the audience or an instructor, in which case it is a type of formative assessment. In this essay, I wish to take a closer look at presentations and their assessment as a means for student learning.

I believe that the primary most important outcome of a presentation, is the effect it has on the audience. The effect can be considered in varying degrees of weighting to be the amount of information retained by a listener, or the influential effect it has on a person, in the short or long run. Certain effects may be qualitatively measured, while some are hazier matters of opinion. What I believe to be certain is that every presentation must have a set of clear objectives - primary, secondary, and else. Objectives assist in the overall structuring and branching of a presentation, while their clear statement in the beginning may help guide the presentation itself. Indeed the structure is the linking bridge from the stated objectives to the observable outcomes. Thus an effective communicational channel opened by the presenter, must have a trait of clarity and didactic structure, serving as the master plan of a master manipulator, which any great presenter must ultimately become.

As far as the specific skillsets - implicit or explicit - required by the intended outcomes of presentations, they all revolve around the idea of effective communication, which I define for myself as the ability of opening a communicational channel, maintaining it, and then applying it for the channeling of information to the audience. Clarity not only in structure but in mere speech itself is obviously a must for such a channel to open, in addition to adjusting the proper sound volume to the situation, and maintaining the continuity of speech. Eye contact may serve as a means to keep the channel open, in addition to actually opening one potentially. Subconscious signals sent via facial and postural expressions, serve to enhance communication. For a presenter, they provide a chance to take their self-expression to the next level, creating an audio-visual orchestra of signals with varying intonations and perceptibility. According to certain theories of the birth of human languages, initially we all communicated via sign languages, which may be why certain regional cultures - such as the Mediterranean - have still retained them to the present day.

In addition to the skills of speech, the speaker's proficiency in the material can influence the presentation to a major extent. It not only gives the impression of knowledgeability, but raises the level of self-confidence in the speaker. Self-confidence in turn may influence a wide range of other factors mentioned above, such as continuity of speech and clarity of structure. Self-confidence is radiant. It shines from the presenter on the audience, and in turn influences the level of trust and confidence the audience has in the verity of the communicated information.

An important dilemma to be decided prior to assessment, is the weighting of the relevance of feedback by either the instructor or the peer students in the audience, as well as the sequential method of communication of the formative assessment. I believe the ideal resolution of this to be as follows: peer assessment should have temporal precedence, while instructor feedback should serve as the deciding factor. Most students may sub/consciously agree that the instructor has a right to serve as a presiding “judge” having the final say. On the other hand, peer opinion must be upheld as valuable for its inherent spectral characteristic, providing a wide range of opinions to consider. This semi-hierarchical relationship must serve as the definitive template for any specific explicit assessment, to ensure the relevance and reliability of the overall reflection on the presenter.

In considering the efficiency of student presentations and their peer discussion as a formative assessment in terms of student and instructor time, one must identify the given base constraints, such as class size, class time, and meeting frequency. Clearly, such constraints require advance planning on the part of the instructor, in order to ensure the smooth flow of class activities, by minimizing the unnecessary pressures of the constraints. Thus considering time limitations, presentations may or may not be considered efficient, depending on the situation. The point of “presenting” however is not necessarily efficiency, but rather that of creating an atmosphere of open communication, by finding and maintaining the right channel for this experience. Therefore the student must be aware of this fully, so that s/he can focus on the more relevant aspects of the presentation and its subsequent interactive assessment.

One may consider alternative forms of assessment, other than the suggested weighted instructor-peer partition. These can range anywhere between the absolutely autocratic, relying on solely the instructor's criteria, to the entirely laissez-faire, whereby peer feedback is the main mode of formative assessment. We may clearly find virtues at both ends of the spectrum. However, a more weighted approach can minimize the weaknesses and maximize the effectiveness of the assessment. The advantages of the autocratic method, is the reliability of expert opinion by the instructor, and the projection of security on the classroom due to it. Peer feedback however, can increase the students' involvement in the betterment of one-another's presentation skills. Some students may be more receptive to their peers’ opinions than an instructor’s, since they are on an equal hierarchical level. Thus the quest of finding an ideal middle way for a specific situation, may result in a more optimal assessment strategy.

The verbal formative assessment matches quite well the desired outcomes and required skills mentioned above. A communicative feedback approach fits appropriately into the format of presentations, and may be viewed as their natural extension. The marking scheme or specific criteria should in my opinion be reserved for the instructor, who is to give a more rigorous and analytical assessment. The audience of students however must be allowed to engage in free conversational feedback with the presenter, for an optimal active learning outcome of all students.

In summary, I have discussed “presentations” as serving the assessment of students of various topics. I have considered a number of analytical questions to observe my ideas in this regard. I found this endeavor to be quite valuable in its self-reflective potential, making me think deeper about the conceptual approach to this type of assessment.

Self-Reflection for Identifying my Ideal Learning Experience

Response paper for the session "Understanding the Learner"
Certificate in University Teaching Program - University of Waterloo

A commonly voiced belief is that the learning environment greatly influences the learning experience of students. In fact, mostly the teacher is the one to blame, when students' expectations are not met in regards to their imagined ideal of classroom instruction. In this essay, I will reason that learning is a bipolar experience, greatly dependent on both the student and the teacher. Via an acute self-reflective analysis, I will identify my own learning characteristics in the context of two very different learning experiences - one negative the other positive. Meanwhile, I shall keep in consideration the potential experiences of other types of students, based on my retrospective observations. I will begin with a brief description of myself as a learner, to provide a working foundation for the subsequent analysis of my experiences.

Currently I am a graduate student working on my PhD in Applied Mathematics, primarily occupied with doing research, which is quite different from my undergraduate mindset, when studying was my primary focus. Therefore much of this discussion will be a travel back in time, though my main characteristics as a learner have not changed, only become less elaborate and refined. When I was an undergraduate in Hungary at the Eötvös Loránd University, I was literally like a sportsman of the mind - constantly monitoring myself for what shape I was in. This was of utmost necessity so that I could allocate my time and efforts to the most appropriate times for a particularly or less demanding subject. I monitored my diet, breathing, everything - it was a bit insane, but the tasks at hand required it. I was also very independent as a learner. I did not value having to sit through lectures, where I was superimposed with a "sequential" learning experience in time, in which I was unable to flip back and forth as I would do by myself with a book, to connect formerly learned concepts. I am what I can now properly categorize as an absolutely "global" learner. Even in doing research, I feel more like a bird flying over the savannah than an antelope galloping through it. Whenever I notice something interesting, I zoom in somewhat, then zoom out, or zoom in more if I find it to be of value, guided by my intuitive inner compass. Just as a bird, I am a great lover of freedom, and being restrained in any way, whether by a non-liberating lecture or else, can be very disheartening to me. I feel lucky to have been endowed with a considerately adaptive supervisor who understands me by instinct, and is willing to act as a companion, versus a guide.

Considering the above, the ideal learning condition for me is most likely an environment in which I am given a chance to contemplate the material with freedom, allowing me to internalize it in a gradual process of digestion, in which I reassemble the buffered information packets into a new didactic whole within myself, via a highly independent process. Is this possible through any kind of lecture or tutorial format? Does not my own sense of independence make the reception of any external learning projection impossible? Sometimes it does feel like I have this impediment. In rare occasions however, I encounter a learning environment in which sequential spoonfeeding is not the primary mode, and which I find highly enlightening and motivational, activizing my neurons on all levels.

A specific negative experience was my Partial Differential Equations (PDE) course as an undergraduate. It was undoubtedly a large bulk of information we had to internalize, or perhaps only seemed like it? The professor acted like a sage on the stage throughout the lectures, speaking his mind like a radio, and scribbling away on the board, without any consideration for our understanding. Certainly contrary to a multilevel learning experience, that would have sent off sparks in the minds of various types of learners. I personally felt being given no chance to reflect on the material, the pace was too fast and uninterruptibly monotonic, which made me feel disconnected, from both the professor and the material. I was given no opportunity to make a connection either, as it felt like the instructor had no idea there was even an audience. I was not motivated in any way to learn the material, and suffered greatly when I "had to" in the exam period.

I believe a good teacher not only connects with the audience, but monitors their level of understanding throughout a lecture, and sets a fluctuating pace accordingly. Before the process of information flow is even initiated however, the professor must set a base tone for the rest of the lecture in some way, much like a painter might paint a base color for a painting to be laid out upon. The base tone might involve some motivational words of wisdom on the nature and relevance of the material, its historical importance, or their own enthusiasm for the topic. These may serve as an extension of a hand to help students on board for a journey of sailing the waters of knowledge. Most of all, teachers must primarily know themselves and their own motivations, so that they may accompany students on their own journey of learning, whether as captain or companion. In a classroom, a teacher must above all create a multilevel learning experience, in which learners of all kinds shall thrive.

A specific positive experience was my Functional Analysis course as an undergraduate, which I remember as my favorite one of all time. Unlike the PDE course in which I never got past learning the toolset, here I internalized it from the notes quite naturally and largely independently, since I was eager to apply it in assignments given at the tutorial. The tutorial was also held by a professor who I have had for years in Analysis, and found his approach reliable and respectable. His opinion and marking represented to me the unquestionable law carved into stone. In other words I truly trusted him, regardless of him being a difficult personality, who did not think much of "applied" mathematician students. So clearly, I had an incentive to prove myself in attempting to solve the rather difficult problems given by him. Indeed week after week I handed in each assignment in full completion, even though a vast majority of students sabotaged the tutorial by missing it and not doing the homework - the general consensus was that this was a teacher to be disliked. Yet, I saw him as a representative of mathematical rigour from the past, and his very character projected reliability and wisdom. So even though I was a largely independent learner, his character still had an influence on my motivation, captivating my interest as if by undetectable magic. I was not alone though among those who were motivated - some were hard-working by nature, some saw me as competition, and a few were just trying to get by on a mediocre level. I however saw this class, which presented the pinnacle of modern Analysis (Calculus), as a chance to mature into an able mathematician. When the course was over, it felt like I have gone through a training period harsher than ever. Yet by overcoming independently the challenges I have faced, I have reached a new level of mathematical awareness and understanding, inaccessible via any other learning process. Then and there, I became a mathematician.

How was any professor able to evoke all these feelings alone? Analyzing my memories, I believe the answer is that he did not do it alone, I was an equally necessary ingredient. In fact, being the experienced educator he was, I believe he tailored his classes to specifically the kind of student that I was. This was quite irresponsible, since the rest of the class was left behind. I have observed however, that many professors grow tired over the years of having to deal with other - perhaps less motivated - students, and hold their classes for the most worthy. I must however voice my criticism, being a great believer in student potential. While I may have felt good in this particular classroom experience, it was unfair to the other students. Therefore I once again arrive at the conclusion, that a good educator creates classes which run on multiple levels, as an orchestral concert involving winds, strings, and percussion, for various musical sounds which may appeal in varying degrees to individuals in the audience. For instance, there are lovers of rhythm (percussion) and those who are more receptive to tunes (winds or strings), whether brief or overarching. A great symphony like a good lecture, weaves elements in a magical fashion, sparking the minds of as many listeners as possible, ie. optimizing the learning experience for the entire class.

In conclusion, I deduce that the ideal learning experience for me is one where I am granted freedom to explore, and the material is shown to me to be a toolset for practical application in the gradual exploratory process. Even though I have always thought that the personality of the instructor has no effect on me as a learner, I have now realized that it is indeed a major factor towards connecting with the material. A demandingly harsh yet nurturing environment, can be better than the nowadays common easy-going attitude of many professors, on the lookout for high student ratings. A professor can still remain accessible for a personal connection, in many ways. In regards to presentation of material, I have reasoned for a multilevel orchestral manner of information propagation, for optimal exploitation of student potential. Though certain manners of instruction I may see selfishly as ideal to me personally, as a teacher I feel I must become a more complete individual, growing beyond my own natural ways of learning.

Benefits and Risks of the Moore Method in Mathematical Education

Response paper for the session "Interactive Teaching Activities"
Certificate in University Teaching Program - University of Waterloo

A commonly observed phenomenon in undergraduate mathematical education, is the lack of motivating conceptual instruction. All too often, students are lectured on techniques of problem-solving, without a parallel development of their conceptual understanding. This lack of depth often continues on to graduate-level classes, for the sake of rational efficiency. The method developed by Robert Lee Moore, is a form of mathematical activity spanning a single class event to potentially entire terms. In a sense, it may be viewed as ideal for the conceptual development of students in science disciplines, primarily in mathematics. This paper will consider its hypothetical application in a first-year Calculus setting onwards of Pure and Applied Mathematics students.

First-year Calculus for a student can be a time of great transitions with unusual concepts, formulas, and methods, that many instructors seek to hard-wire in students, to establish a foundational toolset for later courses. Most instructors achieve just that, storming by underlying concepts and philosophies that may only occur to the most contemplative students, in such a fast-paced setting. Likely most professors would cite lack of time, size of classes, and efficiency of information propagation. On the other hand, elementary Calculus could easily become an ideal platform for activizing students, and involving them in the creative development and learning of the material. Bringing to the surface and upholding students' inherent mathematical acumen, must be the goal of any program which honestly wishes to raise able mathematicians and future researchers.

Indeed, most students accepted to university, possess the mathematical background and potential to develop the Calculus of Newton, Leibniz, and Cauchy - however outrageous this claim may sound. Encouraging them to do so in a classroom setting, may potentially turn some into quite capable and confident mathematicians once they become researchers themselves. The suggested alternative activity would hard-wire deep research and collaboration behaviours they stumble upon themselves, instead of the shallow toolsets of technique they are mostly provided. Thus my argument that follows for using the Moore Method in first-year Calculus classrooms, is primarily intended for students who may become researchers one day, in the fields of Pure or Applied Mathematics.

I will hereby outline the overall class as well as the classroom activity itself, which the entire longspan method would reduce to. The Moore Method has numerous existing interpretations, branching off from the original central idea behind it, and this is reflected in the way the actual classroom activity may be carried out. Often the interpretation reflects the course it is applied to, so what follows is my own interpretation, as I would potentially apply it.

The central idea of the Moore Method is to involve a class of students in the development of historical theories of mathematics and science, from the ground up. Students are generally given a set of basic knowledge, often in the form of definitions and axioms, providing a foundation for the relevant initial questions and directions. In regards to elementary first-year calculus, the initial concepts provided to the students might entail a vague idea of limits, continuity, and tangent / area determination, which they would be first tasked with refining. The refinement process itself for these basic definitions may last 1-2 classes - to be determined experimentally. Then as a syllabus, the task of putting the definitions in practice, such as for the evaluation of specific limits, then later determining continuity and differentiation of functions could follow. While solving problems they invent for themselves together, the students may encounter cases that will imply the invention of the Squeeze Theorem, the Intermediate Value Theorem, or even L'Hopital's Rule - not missing out on relevant ideas. The same implications via applications would follow for differentiation and integration, as well as their inverse relationship, which the students would be likely to discover. Specific formulas and techniques of integration, would be developed by the students, for the sake of creating a toolset for themselves, so that they can crack the nuts found in problems they encounter in their exploration. Instructor involvement would amount strictly to asking occasional motivational questions that may blow the students' sail in the right direction. Otherwise, students would be expected to create and collaborate independently, both during and after class, representing their "assignments". Soon, such a Calculus class may have the effect of occupying the students' minds 24/7, preparing them for the analogous experience they may encounter as researchers in their future careers.

The classroom activity itself would possess a loose yet consistently coherent structure, with explicit rules of conduct such as free speech balanced with mutual respect, stated in advance. These explicit rules may serve to instill and induce principles of integrity, and serving as foundations for engagement in future research. Furthermore, the instructor would provide the initial definitions, motivational ideas and questions, to ignite both group discussion and individual efforts, in a sentiment of "collaborative competition". The incentives being inherent mathematical interest, the thrill of building something together, or even peer pressure to perform.

In addition to the aforementioned benefits of a Moore-based activity which shape its very format, we may recognize an even more fundamental characteristic in the method. By placing the learning process in a creative social setting, the intructor immerses the students in a primordial soup which they are used to functioning in, and from which they may naturally evolve as maturing mathematicians. Thus each student would find their natural function in the micro-social environment of the classroom, paralleling their social preferences in real life. Thus collaborators would collaborate, thinkers would contemplate, leaders would emerge, tasks would be divided, global minds would present the vista, and sequential ones the process to explore it. All would find their own role in such Stone Age conditions, in which we are genetically programmed to rely on one another and contribute as we can.

Much of the above is hypothetical to me, and based on pure rationale, versus any prior implementation. To me, the above is an idealization of a mathematical learning activity I would have always wished to be a part of. I have been observing the inefficiency of the classical lecture-tutorial settings, ever since I became a university student, and I find this to be a solution. Though I realize that the Moore method may not be for all students. The initial selection of the students must be based on whether they find the challenge appealing, and not based on academic standing. The worst possible outcome for an aspiring mathematician, would be realizing that they are not fit for doing research, which is still better than a realization in graduate school or real life. On the other hand, many who are unsure initially, may gain self-confidence for the long run. I personally had doubts about my own research abilities as an undergraduate, projecting an unpredictable dark cloud into my future. I believe that all students should be given an opportunity to put themselves to the test, in classroom activites of this kind. Furthermore, the professor may serve as a mentor in this setting - accompanying, guiding, and nurturing potential talent.

The benefits of such an adventure, even if experimental, undoubtedly outweigh the risks. As the Ancient Greek aphorism of virtue recommends "Know thyself!", so should mathematicians learn as much about themselves as possible early on. What better way to do so, than a setting that mimics what they are apt to experience later on in real life. This is akin to teaching children to swim in shallow water so that they would be ready for the deep, while throwing them in deep water at once may prove fatal. Is it not the responsibility of experienced researchers and educators to prepare students for the deep water, besides striving to enhance their overall conceptual understanding?

The Craving for Rigour

I started out as an engineering undergrad at Purdue. As a freshman I realized that I am primarily interested in mathematics of all my subjects, and I had an intensifying craving for a kind of precision I felt was missing from the engineering math courses, so I switched my major to applied math. Now I know that I was sensing the lack of rigour - or precision of thinking and derivation in mathematics. In complicated proofs and theories, even the greatest mathematicians have made small mistakes (usually natural-sounding assumptions), and the challenge of rigour is to avoid them at all costs. This is a fundamental requirement for the integrity of a mathematician. The slightest mistake in a theorem, can lead to entirely erroneous theories built upon it.

Today I have come across a quote from Rolle stated in 1703 on the fledgling theory of Calculus:

"Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics. Among its principles one could only find true axioms and all the theorems and problems posed were either soundly demonstrated or capable of sound demonstration. And if any false or uncertain propositions were slipped into it they would immediately be banned from this science. But it seems that this feature of exactness does not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it. I do not see that this system has produced anything for the truth and it would seem to me that it often conceals mistakes."

Surprisingly this skeptical quote questioning the rigour of the early Calculus was highly unique in its day, as no-one really understood what Rolle meant, virtually everyone opposed him. The point of reference for his criticism was Euclidean Geometry, which was then and still is considered entirely rigorous. By the 19th Century however, it became clear to the majority of mathematicians, that Analysis (Calculus) is not rigorous at all, and there came a burning need to make it so. In that unrigorous form, it could not be applied to prove for instance Fourier's theorems in the developing Thermodynamics. The quest was thus undertaken by a number of mathematicians, such as Cauchy (epsilon-delta definitions and proofs), Weierstrass, Dedekind and others.

Currently Analysis is considered entirely rigorous, made so by Dedekind's Axiom (alternatively the Archimedean Axiom), as well as the Zermelo-Fraenkel Axioms in Set Theory. From these, all the theorems of Analysis can be rigorously derived, primarily built from the epsilon-delta bricks of Cauchy.