This definition is clearly meaningless drivel. How is it possible for such a masterful work, which is clearly written by a top-quality mathematician, to open with such junk?

Russo proposed a compelling answer to this conundrum. It goes as follows.

Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant, as indeed shown by the fact that the utterly useless Definition 4 is never actually used anywhere in the Elements.

Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where this was needed.

The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever.

Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it.

To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text.

To these new, dumber readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as in Archimedes’s postulate.

Heron therefore explained that “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.” The phrase “uniformly …” obviously refers to the universality of the shortest-distance property applying to any two points on the line.

Now fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms.

In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense.

In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated.

The compiler therefore makes the decision to simply cut Heron’s description off after the bit about “uniformly in respect to [all] its points.” This solves all his problems in one fell swoop. The only drawback is that the “definition” becomes utter and complete nonsense, but since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore.

This is how the ridiculous Definition 4 — a mutilated vestige of what was once a very good definition — ended up in “Euclid’s” Elements.

I have included excerpts from Russo’s paper in my History of Mathematics Reader. There you can find the above argument in his own words.

]]>I wrote these notes to supplement a conventional book, because in my opinion typical intro to proofs courses are fundamentally questionable in their very nature. The premise of such a course is rather like that of My Fair Lady: uncivilised students must be taught to “talk the talk.” It’s the dress code and table etiquette of mathematics. In their fanatical devotion to the clinical, sterilised, Bourbaki way of doing mathematics, these courses have forgotten that precision elocution is pointless unless you have something to say. These courses deem actual mathematical content “too messy” and instead feed the student only fake theorems specifically concocted for the sake of being amenable to the desired mould of what a mathematical proof should look like.

Consider these notes, then, my pauper’s rebellion. Down with haut bourgeoisie snobbery for snobbery’s sake, down with fake perfume and powdered wigs. Let us have the courage to tackle real mathematics as it occurs in nature. Let us put meaning and purpose and exciting ideas first, and let us accept the airs and graces of the mathematical aristocracy only after they have proved their worth in this enterprise.

]]>I maintain that it is crucial to keep the mathematician’s point of view alive in historical scholarship. Technical acumen and an empathic sense of how a mathematician thinks are essential tools for understanding past mathematics. And this is best done by one who lives and breathes mathematics on a daily basis in the classroom, at the colloquium, and by the coffee machine — in short, from within a department of mathematics.

Fields-medalist David Mumford put it well when relating his “personal experience reading Archimedes for the first time”: “after getting past his specific words and the idiosyncrasies of the mathematical culture he worked in, I felt an amazing certainty that I could follow his thought process. I knew how my mathematical contemporaries reasoned and his whole way of doing math fit hand-in-glove with my own experience. I was reconstructing a rich picture of Archimedes based on my prior. Here he was working out a Riemann sum for an integral, here he was making the irritating estimates needed to establish convergence. I am aware that historians would say I am not reading him for what he says but am distorting his words using my modern understanding of math. I cannot disprove this but I disagree. I take math to be a fixed set of problems and results, independent of culture just as metallurgy is a fixed set of facts that can be used to analyze ancient swords. When, in the same situation, I read in his manuscript things that people would write today (adjusting for notation), I feel justified in believing I can hear him ‘speak’.”

This way of doing history has widespread resonance in the global mathematical community. All mathematicians know the feeling of struggling to understand a mathematical work until it “clicks” and one feels certain that one has experienced the same idea as the author, regardless of whether he be centuries or millennia removed from us. Those of us who approach mathematical texts in this way know not to pay too much attention to superficial aspects of the presentation: scribbles of various kinds are merely imperfect representations of the author’s thought, whereas the digested “aha” insights we reach when we understand it are its true content. Mumford is right that modern historians, by contrast, are trained to categorically reject such a “gut feeling” approach and stick slavishly to the exact written word as if it were a veritable alien communiqué for which no concordance with our own ways of thinking may be assumed.

The new historiography has greatly advanced the field by offering more specialised perspectives than mathematicians alone ever could, such as histories deeply informed by broader social context and meticulous work on sources and editions according to the highest standards of textual critical apparatus. But amidst the zeal to exploit these new frontiers the field has been left with a leadership vacuum in its traditional core dominion. The time is ripe for a resurgence of the mathematician’s perspective, whose cross-fertilisation with modern developments will bring great fruits.

Here in Utrecht we keep alive this mathematical tradition of historical scholarship. We have a legacy of generations of quality history of mathematics being done in a Mathematical Institute that commands the highest international respect. We are widely recognised as the natural heirs of this way of doing history. It is not for nothing that Jan Hogendijk was awarded the European Mathematical Society’s inaugural Otto Neugebauer Prize, epitomising our continuity with the Göttingen mathematical tradition, while Henk Bos was awarded the Kenneth O. May Prize, the highest honour of the International Commission for the History of Mathematics.

I have taken up the role of torch-bearer of this movement in both words and deeds. I offered a big-picture vision for its enduring relevance and importance in a programmatic paper on the historiography of mathematics, and my more specialised works instantiate these ideals. A notable example is my paper reviving and defending the geometrical algebra interpretation of the history of Greek geometry: an issue where the battle lines have traditionally run along departmental divisions, and the older interpretations being advanced by mathematicians like van der Waerden and Weil have been singled out for criticism as emblematic of the dangers of the mathematically-oriented approach to history. With no mathematicians forthcoming anymore to challenge them, the humanistic historians who dominate the field today had been lulled into a consensus, to the detriment of the vitality of our field. The same dynamic is at play in many other cases as well, which is why the mathematician’s point of view has much to offer, not only in terms of subject-matter insights, but also for stimulating diverse and critical thought in the field.

My first publication, on the isoperimetric problem, symbolises how my point of origin is mathematics itself. Although it is a work of history in that it gives an exhaustive survey of historical solutions of the problem, it is clearly driven by a mathematician’s delight at beautiful proofs drawn from wide-ranging fields of mathematics, such as complex analysis and integral geometry.

My conception of the history of mathematics as being first and foremost about the development of mathematical ideas gives my work educational and expository appeal for a broad mathematical audience. My work on the history of the calculus, for instance, is fertile soil in this regard, and I have made the most of this in my free calculus textbook and several papers making classroom-relevant aspects of my research accessible to a wide readership.

This grounding in the mathematician’s point of view serves me well in my more historical work, where an intuitive sense of what makes sense mathematically often leads me to different interpretations than those who are guided more by contextual considerations external to the mathematical argument itself. I have taken on historians and philosophers along such lines for instance in my Copernicus paper and my paper on Leibniz’s early work on the foundations of the calculus, to name but two prominent examples.

Thinking like a mathematician also gives me a perspective on broader issues missed by historians and philosophers who keep technical mathematics at arm’s length. For example, in my dissertation I studied Leibniz and argued that from the corpus of his technical mathematical works there emerges a clear and unmistakeable picture of his conception of the purpose and method of geometry. This has wide-ranging implications for understanding the scientific and philosophical thought of that era generally, yet this perspective has been neglected since it is expressed “only” implicitly in the mathematical works. But as Albert Einstein said: “If you wish to learn from the theoretical physicist anything about the methods he uses, I would give you the following advice: Don’t listen to his words, examine his achievements.”

Mathematicians conversant with the history of science are also much needed to analyse technical issues. My paper on Copernicus is a case in point. In this article I refute an argument due to Swerdlow that has been considered crucial for over forty years and has been widely cited as decisive by historians who had not themselves worked through its technical mathematical basis.

On the other hand, mathematical understanding is not all it takes to do history of mathematics. Mathematicians who turn to history without background and training in this field often make grave errors of their own. I expose and refute many such errors of anachronism in my dissertation and elsewhere. One example is my paper on what is often called Leibniz’s proof of the fundamental theorem of calculus but which is actually nothing of the sort. The notion that this is Leibniz’s proof of this theorem is widely repeated in numerous sources. It is a notion that seems very plausible to anachronistic eyes looking only at a short piece of Leibniz in isolation, but in reality it is simply false, as becomes clear when the work is studied in its proper context. Another example is my paper rehabilitating Jakob Steiner’s geometry from anachronistic misjudgements. To clear up these kinds of things the field desperately needs proper professional expertise in both history and mathematics.

A key theme emerging from my dissertation, which I intend to build on in future work, is the influence of classical mathematics on general scientific and philosophical thought in the early modern period, which was much more comprehensive than recognised today. It was a widespread conviction at the time that if you seek truth, you must do what the geometers did; you must replicate their method and extend it to other branches of learning and philosophy. Descartes’s Discours de la méthode (1637) is explicitly written for this very purpose; indeed this famous manifesto on the method of doubting everything clearly proclaims that “I did not doubt” that “only mathematicians” had struck upon the right way of reasoning. Likewise Hobbes writes in his Leviathan (1651) that “geometry is the only science that it hath pleased God hitherto to bestow on mankind,” and proceeds to expressly fashion his philosophy in its image. Spinoza’s Ethica (1677) declares in its very title that its is “ordine geometrico demonstrata.” Newton opens his Principia (1687) with a preface outlining what “the glory of Geometry” consists in, in order to use its example to justify his innovative scientific methodology.

But what exactly did these authors mean when they spoke of “the geometrical method”? The complexities of this question are poorly understood by scholars and historians today. The 17th-century thinkers who invoked geometry were not referring to some superficial idea of geometry as conveyed by Plato or Aristotle. They were referring to the rich picture of the geometrical method that emerges from a thorough study of technical corpus of Greek geometry, as conveyed by advanced technical writers such as Pappus. Indeed they frequently refer to this technical tradition even in works that go well beyond geometry itself: Descartes cites Pappus in his Discours; Hobbes does the same in his Elements of Philosophy; Newton cites Pappus even in the very first sentence of his Principia. These authors were thoroughly versed in the technical Greek tradition, as their mathematical works show. By citing Pappus and other technical Greek material they are signalling very clearly that when they call upon geometry they mean not some simplistic idea of axiomatic-deductive method but a rich methodology only conveyed implicitly in the finest ancient works of advanced geometry.

Unfortunately modern scholars do not share these 17th-century thinkers’ excellent technical mastery of advanced classical geometry. Consequently, current scholarship has failed to appreciate the extent to which conceptions of the geometrical method permeates 17th-century thought. One indication of this is that the crucial Book 4 of Pappus’s Collection was translated into English for the first time only in 2010. Even more deplorably, Leibniz’s published mathematical works have never been translated into English at all, even though their crucial importance is universally acknowledged. Meanwhile, any philosophical treatise of even a fraction of the importance of these works has invariably been translated multiple times, betraying the skewed and anti-mathematical emphasis of modern scholarship.

Much of the richness and impact of the mathematical perspective has therefore been missed by modern scholars since it is not spelled out in philosophical prose, neither in Greek nor in early modern times. But mathematics speak loud and clear to anyone who cares to listen, and anyone who was serious about philosophy in those eras was obviously expected to know their geometry — much in the spirit of the famous inscription above the entrance to Plato’s academy. To understand 17th-century thinkers it is time for us to start taking their appeals to geometry seriously and recognise the full scope of the rich methodological picture they drew from advanced Greek geometry.

]]>This proves that there is gender bias in academic science hiring, or so we are supposed to conclude.

My concern is this: Many faculty members want studies to prove that there is gender bias. It fits their own political and ideological beliefs. They are happy when they see studies prove this. They like to refer to such studies. I know because I follow them on Twitter.

This raises the question: Did the faculty members in the study answer truthfully, or did they “second guess” the purpose of the study and submit whatever answer would produce their own preferred outcome? They may indeed have thought to themselves: “Although I’m not biased, I am convinced that a lot of my colleagues are, so I better answer as if I was too, so that attention is drawn to this important problem and progressive measures can be taken.”

Of course the faculty members knew they were being studied and of course they had no actual stake in their replies, unlike when they’re doing actual hiring. And if we look at the prompt the faculty evaluators received, the purpose of the study is quite transparent. So they had no incentive to be truthful, but some incentive to ensure the study produced the results they favoured.

Which hypothesis is right, mine or the authors’? We could try to test it by looking at gender differences among the evaluators. If the authors are right, and there is unfair discrimination against women due to bias and prejudice, one might expect this bias to be stronger among male evaluators, since women who are themselves established scientists might be expected to be open to promising female students. If my hypothesis is the operative one, on the other hand, one might expect the opposite; that is, that female evaluators would be even more biased than men, since they arguably have a greater stake in “gaming the study” to make sure it shows gender bias. The latter is in fact what happened, though the difference is not great. Meanwhile, if one looks at real data instead of contrived experiments, “actual hiring shows female Ph.D.s are disproportionately … more likely to be hired” (source, page 5365).

]]>That is what I and others suspect. An article in the latest issue of Isis claims to disprove us:

> This essay seeks to explain the most glaring error in Ptolemy’s geography: the greatly exaggerated longitudinal extent of the known world as shown on his map. The main focus is on a recent hypothesis that attributes all responsibility for this error to Ptolemy’s adoption of the wrong value for the circumference of the Earth. This explanation has challenging implications for our understanding of ancient geography: it presupposes that before Ptolemy there had been a tradition of high-accuracy geodesy and cartography based on Eratosthenes’ measurement of the Earth. The essay argues that this hypothesis does not stand up to scrutiny. The story proves to be much more complex than can be accounted for by a single-factor explanation. A more careful analysis of the evidence allows us to assess the individual contribution to Ptolemy’s error made by each character in this story: Eratosthenes, Ptolemy, ancient surveyors, and others. As a result, a more balanced and well-founded assessment is offered: Ptolemy’s reputation is rehabilitated in part, and the delusion of high-accuracy ancient cartography is dispelled. (687)

As an aside, here’s a pro tip for academic novices: If you have little of substance to offer, make sure to lay it on thick with self-congratulatory posturing about how your work is supposedly based on “more careful analysis” and “more balanced and well-founded assessment,” showing everything to be “much more complex” than others think. After all, who would dare disagree with someone who is so careful and balanced and ever so sensitive to complexities?

Contrary to his smug proclamations, the author’s case is flimsy. For one thing he immediately admits that, indeed, Ptolemy’s error can be solved in a single stroke by recalculating his map with Eratosthenes’s excellent value for the circumference of the earth, yielding an “uncannily accurate” map (692). But he alleges that this is a mere “coincidence” (691). What is his evidence for this?

His main argument concerns the value of the length unit “stade.” A “‘short’ stade … is implied by the high accuracy of Eratosthenes’ value for the circumference” (694), for which there is no explicit evidence in the record, unlike a “long” or “common” value for a stade that is mentioned in some sources. Of course this is not strange since our hypothesis is based on precisely the claim that many excellent sources are lost. But there is in fact implicit evidence for the short stade, as the author himself in effect admits:

> The main argument for the “short” stade is based on comparison between ancient and modern distances: those measured on a modern map are divided by their ancient counterparts in stades, giving the length of an average stade. This comparison has been undertaken repeatedly, … and invariably the average stade comes out to be much closer to the “short” value of 157.5 m than to the “common” one of 185 m. On this basis, many researchers suggest that for practical purposes ancient surveyors used a special short stade, one never directly attested in extant sources. … This stade is often termed the “itinerary stade.” This result might have been regarded as a brilliant confirmation of the “short stade hypothesis” were it not for one “but”: strangely enough, in comparing ancient and modern distances, a crucial factor has been lost sight of——namely, “measurement error.” The proponents of the “itinerary stade” proceed from a tacit assumption that ancient distances were measured almost as accurately as modern ones. However, this cannot be true, for two main reasons. First, with rare exceptions, there is no indication that distances given by ancient sources were actual measurements on the ground, rather than rough estimates deduced, for example, from the duration and the average speed of travel. Second, and most important, even when ancient distances do derive from actual and quite accurate measurements, they were certainly measured not as a crow flies but including all the twists and turns of the route. (701-702)

The author’s counterargument is very weak. It merely asserts what he is trying to prove, namely that distance measurements in the time of Eratosthenes would have been poor and naive. The obvious reply, which the author does not consider, is that it is very possible that Eratosthenes and others used much more sophisticated mathematical methods such as triangulation. Instead the author expects us to believe that the generation that gave us the geometry of Archimedes was too stupid to account for “twists and turns of the route” when estimating distances for geographical purposes. There is also some reason to think that Eratosthenes defined a new stade based on his earth measurement——another possibility ignored by the author.

> We can apply a simple test: the same comparative approach may be used to determine the length of the Roman mile. Since it has been firmly established as equal to 1,480 m, such comparison will yield the average accuracy of Roman distance measurements. … For this purpose I have examined more than 160 distances given in Pliny’s Natural History. … If these distances are believed to be accurate, then, by the same logic that has led us to the “itinerary stade,” we have either to conclude that Pliny’s mile was equal to circa 1,190 m, which is impossible, or to assume that Roman measurements of distances were much less accurate than Greek ones, which is hard to believe. Otherwise, we have to conclude that Pliny’s distances were overestimated by 25 percent on average——that is, by approximately the same amount that Eratosthenes’ and Strabo’s distances must have been if they were expressed in the “common” stades of 185 m. (702-703)

Apparently it is “hard to believe” that the greatest generation of geometers who ever lived were better at measuring distances than a second-rate encyclopedist from a civilisation that never contributed an iota to mathematics during its entire lifespan.

The author also maintains that “the hypothesis … contains numerous logical fallacies” (693), namely:

> The match between the recalculated Ptolemy coordinates and the modern ones, however close, does not in itself mean that the longitudinal distortion in Ptolemy’s map was due entirely to a single cause—namely, the wrong value for the Earth’s circumference. Nor does it warrant discarding two other possible causes: the exaggeration of distances and the lengthening of the stade. Another crucial point to stress is that Ptolemy’s recalculated map turns out to be accurate only in terms of spherical coordinates. This does not mean that the actual distance measurements underlying it and Eratosthenes’ value for the Earth’s circumference were equally accurate. (693)

Calling these things “logical fallacies” is just ridiculous. Obviously no one ever claimed that these things were logical implications, only that they were the likeliest explanations. The author does however commit a logical fallacy himself when he draws the non sequitur that our hypothesis is a “delusion” from the fact that Ptolemy’s map errors can be explained by other means.

Altogether, the author’s preposterously overblown claims exaggerate his case by a mile and then some. I am surprised that such wording was allowed to stand in a respectable journal. The author has not taken our hypothesis seriously, let alone given a “careful” and “well-founded” demonstration that it “proves” to be a “delusion.”

]]>Our coach wanted to make us master passers so that we could play tiki-taka. So we spent hours and hours every week on one specific drill: playing football without goals. Without the “distraction” of trying to score goals, we could focus purely on passing and possession play. That was the idea.

In reality, what happened is that we got sick and tired of this boring drill that took all the fun and excitement out of the game and turned it into a pointless drudge. Instead of making us master passers it made us disinterested slackers who didn’t see any reason to put in our best effort.

Unfortunately much curriculum planning in mathematics is based on the same hare-brained logic. Again and again we see the same pattern: in order to do B you need A, but teaching A and B together in one course would be too much, so A is detached and drilled at length in a prerequisite course.

The problem is that B is the only reason anyone is interested in A in the first place, so now you are teaching an entire course on a topic A which serves no purpose whatsoever in and of itself. By severing A from B you are guaranteeing that your course has zero intrinsic motivation. You are not helping students by giving them “a good foundation in A” before moving on to B. Rather you are obliterating the meaning and purpose of mathematics and forcing your students to approach it as an empty chore.

A notable example is the plague of “intro to proofs” courses where the pedantry and mechanics of proof writing is detached from any context where these skills serve an actual purpose. In the same way we detach integration rules from their purpose, which is solving differential equations, and we detach rings and ideals from their purpose, which is number theory, and so on.

The fallacy is one of short-sighted, non-organic thinking. “I’m a teacher of B. I find that my students are lacking in A. Let’s solve my problem by having them do tons of A in a prerequisite course.” This is B-centered thinking that zooms in on one particular issue and wreaks havoc with the cohesion and integrity of the curriculum as a whole in order to “fix” it.

The A-course is rarely a success, because it’s hard to learn something well without knowing what it’s for, and it’s hard to stay motivated and excited when there is no purpose to what one is doing. But, unfortunately, the same blinders that led to this course in the first place also means that its proponents are blind to its failures. The disastrous outcomes only leads them to double-down on their short-sighted scheme. “Now they have a whole course on A and they still don’t get it! Obviously this proves how essential it is to drill and drill and drill A before going on to B.” Thus, as with medieval bloodletting, the failure of the treatment is taken as evidence that more of it is needed.

The naiveté is the same one that led to the myth that vitamin C cures and prevents the common cold. Compare: “Vitamin C is essential to the immune system. Therefore, taking tons of vitamin C pills will keep us from ever catching a cold.” “In higher mathematics it is essential to understand concepts and techniques relating to writing proofs. Therefore, making all our students cram these skills in a dedicated course detached from any content will have them flying through later courses without impediment.”

Just as a well-rounded diet gives us all the vitamins we need in a natural way, so also a well-rounded mathematics curriculum automatically incorporates any necessary material in an organic and natural manner. Unfortunately mathematical curriculum planners do not believe in teaching mathematics they way it grows naturally and organically. Instead they would rather play the role of a hubristic doctor in a sci-fi dystopia who thinks he can “improve” on nature by replacing all organic foods with artificial capsules.

]]>At the level of lecture discussion it makes students eager to attend class and attentively follow your reasoning since this will give them “free answers.” For instance, you are introducing multivariable functions and want to convey the idea that a great way of analysing them are by their cross-sections with horizontal and vertical planes. So you pose this problem and work out at least part of it on the board. Students are primed to appreciate your point since it answers a direct need of theirs. And by stopping short of giving away the final answer you force students to pay attention to the underlying method since they will need it to complete the problem.

You also want to create substantive students discussions in pairs or small groups. To this end it is nice to have conceptual questions that allow for multiple reasonable standpoints. An example is this “paradox” on how one integral can have several “different” answers. You can ask students to work in pairs, one checking one method, the other the other, and then try to convince each other that they are right. Heated discussion ensues, ultimately leading to some reflection on the meaning of the answer––a lesson that cannot be taught often enough.

Full-class discussion or group work is especially stimulating for problems that involve more open-ended conceptual thinking, interpretation, and reflection, rather than single-track computation. This and this are examples that work very well.

Exam-oriented thinking is a plague that prevents students from learning and teachers from teaching. Many a traditional course shoots itself in the foot already before it starts by being structured around the idea of a final exam consisting of a fixed number of highly standardised, computational problems. This corrupts the teacher, who in this mindset asks questions that are “good practice for the exam” instead of asking what lines of inquiry are best for actually learning mathematics in a meaningful way. It also corrupts the students, who quickly conclude that rote computations is all they “really need to know” and hence zone out at any attempts by the teacher to explain underlying reasoning.

The worksheet model frees us from this tyranny. Teachers are no longer crippled by the straightjacket of having to ask only “exam-type” questions, and students find that a large part of their grade comes from a variety of questions involving genuine thought rather than a restrictive set of archetype calculations. We are free to pursue interesting “one-off” problems that make you think, instead of having to discard them as “unexaminable” just because they are not replicable ad infinitum with different formulas and numbers. Since a large part of the graded work takes part in a formative, discussion-oriented setting, we are not constrained to ask self-contained, unambiguous questions of a fixed level of difficulty, as a traditional high-stakes exam requires. Instead of designing our course with the exam in mind, we can design it with mathematical thinking and learning in mind.

Here are a number of examples of problems in this vein. These questions make you think about the material from various vantage points: the “why” behind certain formulas; visual, intuitive, qualitative interpretation of what you are doing; and at the end even some “cultural interest” connections.

These types of problems can be incorporated in a class in various ways depending on the format of the class. In a larger lecture setting they can be used as the basis for the lecture, in which case the students have an extra incentive to follow along since they need the answers. They can also be used to break up the lecture for a few minutes of reflection and discussion among students. In some settings the boundary between class discussion material and exercise assignments need not be sharp: one can assign a number of these kinds of problems and let student requests determine which get discussed in class and which are left as homework. A small class could even be entirely student-driven thanks to the structure that a well-thought-out sequence of questions affords.

The worksheet format is also suited for longer “story” problems such as these, which allow us to work out substantial problems from first principles, such as setting up a differential equation before solving it. These problems too can be incorporated in various ways, from making homework more interesting to making extended discussions of applications viable in a lecture (since it is now truly part of the [graded] course content rather than “enrichment” material as in a traditional course). In classes of moderate size one can also assign such problems to individual students to present to the class. Since everyone needs to enter the answers in the online system, it’s everyone’s points on the line and the class will listen attentively and try to catch any errors. Assigning such problems based on individual student interests is also a way of drawing on existing expertise and connecting the course with other parts of their study program.

The worksheet format also allows us to break down the traditional division between “theory” and “practice.” Again, this very unfortunate and harmful aspect of conventional teaching is in large part a product of examination needs: having students run through computation problems that can be multiplied at will is very convenient for examination purposes, whereas asking for explanations and conceptual reasoning is very messy. But with the WeBWorK worksheet model we can make the latter realistically implementable.

One useful way of getting students engaged with proof-oriented thinking is asking them to evaluate purported proofs, like this. In a traditional course, the teacher and textbook may model many examples of good proofs, but students are seldom confronted with erroneous reasoning. Therefore they often come to associate proofs more with superficial aspects such as phraseology than with actual content. Reasoning-evaluation problems like this forces them to look deeper and cultivates a healthy critical mindset for reading proofs in general.

Here’s another theory example: I introduce the fundamental theorem of calculus, give an intuitive proof, and then ask some follow-up questions that should be easy if you followed the proof, but often prove not so easy since students have so little experience with this type of mathematical reasoning––which is exactly why we need these kinds of questions. In a classroom I might go through the given proof on half a board and then ask the students to complete the proof of the follow-up case in parallel on the other half of the board, mimicking the steps of the first proof with minor adjustments as needed. I might ask for a volunteer student to come to the board to carry this out with the help of suggestions from the class and maybe some leading questions from me if needed. If I refuse to do it any other way (i.e., explain it myself) the students will be pushed to go along with it: after all, if they don’t they will have to do this as a homework problem, which will be much harder and more work.

The second follow-up problem asks for much greater conceptual insight. I marked it with a dagger , signalling that it is a challenge problem. I like to include some problems like this for ambitious students to puzzle about, while others do not need to worry about them since the grading scale will reflect that these kinds of problems are for those aspiring to the very highest grades.

Much other theoretical material will be of this “ type”: it is not at all required for average students in introductory courses, but on the other hand you want to encourage students with substantial mathematical aspirations to start reflecting on more theoretical aspects as early as possible. One way of doing this is to include some of the more theoretical material as -marked readings with various interspersed comprehension questions. Here are a number of examples of how this can be implemented in WeBWorK. With such an “interactive textbook” type of presentation, ambitious students are rewarded for reading the theory and given a “training wheels” guide to reflective reading of mathematical texts. These are some examples.

All of the above problems I have written myself. WeBWorK comes with a large library of standard practice problems (which I also use), but to reach all the goals I highlighted above we must go beyond this restricted notion of what an online homework system is for.

]]>I always knew I wanted to study mathematics and I followed a full-time mathematical course load from my first day to my last back home at Stockholm University. But I also craved more. I had a broad intellectual appetite. So in addition to my mathematics courses I took art history, philosophy, and so on. My first semester I took an evening course on the history of classical music. The lecturer would weave a grand narrative and ever so often sit down at the piano and play some select bars to illustrate a point. This remained with me as an image of how I longed to experience mathematics: the technical masterpieces and their broader context each heightening the appreciation for the other.

My mathematical courses, meanwhile, were plain-vanilla technical courses. I could do the exercises and pass the exams easily enough, but I was frustrated that bigger why-questions were not being addressed. I could play the formal game of definitions and proofs, but I wanted to know why anyone would want to define, say, principal and maximal ideals in the first place, and what the purpose of these theorems about them was supposed to be. So I went to the library to find out. And that is what brought me to the history of mathematics, for it was only there that I could find the answers to my why-questions. It was only by studying the history of the subject that I realised, for instance, that all the abstract gobbledygook about rings and ideals that I was being fed in my courses was really just number theory with all the interesting applications left out. Thus my interest in the history of mathematics was always a means to an end; it was always subordinated to teaching and mathematical understanding. And, fundamentally, it still is. In this way the problem of teaching has remained the root of all my scholarly work ever since.

It was no accident, of course, that I found the answers I was looking for in the history books. Mathematicians do not make up arbitrary definitions and start proving theorems about them aimlessly. They work on interesting and natural problems, and introduce new concepts only when they serve a credible purpose in this pursuit. That is how history works, and I think a good argument can be made that it is how teaching should work also. This is why, to my mind, mathematics and its history—and thus in my case teaching and research—form an organic whole.

It was in this mindset that I finished up my masters degree, so naturally I was more excited about teaching than going on to do a Ph.D. right away. I got a two-year position teaching at Marlboro College in Vermont, which was a lovely experience. I had long perceived my affinity with the liberal arts ideal, and I at once felt at home in this environment. With these people I debated Kant in the dining hall, traced conics in the snow, cast horoscopes according to ancient principles, and, most of all, taught mathematics in a thinking rather than robotic fashion. The small class sizes and absence of centrally fixed curricula gave me ample opportunity to translate my long-held ideals into teaching practice. From this moment, if not sooner, I knew that this was a passion I wanted to make a career of.

But first I had to get my Ph.D., of course, and I was still more excited about understanding mathematics historically and contextually than about joining the frontiers of research. So I went to the LSE in London to do a second masters degree in philosophy and history of science. This afforded me the opportunity to study in their own right the kinds of big-picture questions that had kept cropping up in my mind as a reflective mathematics student. My view of mathematics was greatly enriched by these perspectives, but doing such a Ph.D. would have taken me too far from home. I was now a full-fledged humanist who could speak to philosophers and historians like a native, but in my heart I was still a mathematician.

So I went back to the root, the problem of teaching that had started it all. I enrolled in a Ph.D. program in educational mathematics at the University of Northern Colorado. My two years spent there made me a native in the world of mathematics education research as well. I studied what research has shown about learning and cognition, I was inducted into social science research methodology, and I saw reform teaching of all kinds in action at the hands of its most enthusiastic advocates. This was all very valuable, and I was always prepared to devote my life to mathematics education. But ultimately I felt that the current paradigm of mathematics education research was too intellectually restrictive for my tastes. So I left.

From there I went to my current position at Utrecht University, where I completed my Ph.D. in the history of mathematics. This has been the perfect Ph.D. for me. It has brought together all the themes that drove me during my searching years. It has allowed me to be an intellectual and a philosopher, a humanist and a reader of books, while remaining with both feet on solid mathematical ground. My path may not always have seemed the most direct but in retrospect its logic is clear: in writing my thesis on the history of the calculus I have gone full circle back to my early university days and made it my daily bread to explore precisely that elusive richness that I always felt was hiding behind the austere façade of the modern mathematical curriculum.

]]>If, like me, you find the face-value implications of this evidence rather depressing, you may want to look for some way of explaining them away. That’s what I tried to do, and here is what I found.

First we may ask ourselves: Do the test scores really measure quality of learning? Maybe direct instruction is a good way of “teaching to the test,” leading to good scores on artificial standardised tests that do not really measure what we really aim for in education. It seems to me that this is not a convincing answer in this case. Judging by the published samples of actual test items used, the questions seem sound and certainly like the kind of thing one would want students to know. They don’t seem at all focussed on “cram”-type knowledge, like standardised vocabulary and rote calculations. If anything, they seem like exactly the kinds of questions advocates of enquiry-based learning would prefer. So we got beaten at our own game, as it were.

Next we can ask ourselves: How does PISA define and measure teacher-directed and enquiry-based learning anyway? Again, this seems to have been done in a very reasonable way that does not leave any room for invalidating the findings on methodological grounds.

To measure how teacher-directed a class was, “PISA asked students how frequently … the following events happen in their science lessons: The teacher explains scientific ideas; A whole class discussion takes place with the teacher; The teacher discusses our questions; and The teacher demonstrates an idea.” (63)

Enquiry-based was measured instead by the following statements: “Students are given opportunities to explain their ideas”; “Students spend time in the laboratory doing practical experiments”; “Students are required to argue about science questions”; “Students are asked to draw conclusions from an experiment they have conducted”; “The teacher explains how a science idea can be applied to a number of different phenomena”; “Students are allowed to design their own experiments”; “There is a class debate about investigations”; “The teacher clearly explains the relevance of science concepts to our lives”; and “Students are asked to do an investigation to test ideas.” (69)

This seems quite in order. In fact, once the analysis is split into these sub-statements, the case for direct instruction is even much stronger than the above table suggests. For it is the most direct-instruction-y part of each group that works best, and the most equiry-y part that is least successful, as we see in the tables on pages 65 and 73.

We may also ask: what is “adaptive instruction”? This sounds like something reformers would approve of, and it is highly correlated with success. However, once we look into the details it is not so uplifting: in a nutshell, it seems “adaptive instruction” may in practice be more like the traditionalist tricks of “teaching to the test” and “dumbing it down,” for the statements PISA used for this measure were: “The teacher adapts the lesson to my class’s needs and knowledge”; “The teacher provides individual help when a student has difficulties understanding a topic or task”; and “The teacher changes the structure of the lesson on a topic that most students find difficult to understand”. (66)

Finally, we must ask ourselves whether the results highlighted in the table are false correlations somehow, or at least not causations. After all, we see in the table that “after-school study time” is strongly associated with negative scores. Surely this is a matter of correlation rather than causation: students who spend a lot of time studying outside of class are weaker on average, but this is not the reason they are weaker, one would hope. Could something like this hold for enquiry-based learning too? This would have to mean that weaker students are exposed to enquiry-based learning to a greater extent. Is there any evidence for this? I’m not sure. The study includes tables on which countries do most direct instruction (64) and which do most enquiry-based learning (72). It turns out that leaders in enquiry-based learning are not the self-proclaimed avant-garde in the rich West but rather the Dominican Republic, Peru, Jordan, Lebanon, Algeria, etc. The correlations control for other variables so this does not appear to be the whole explanation, unless these controls operate only on a within-country level, which seems a possibility (e.g., socio-economic profile seems to be defined relative to the country, not relative to the world). So it seems possible, but I can’t tell how likely, that the disastrous results for enquiry-based learning are due to between-country differences rather than within-country differences. If so, that would undermine the face-value implications of the data, since within-country differences would be much more relevant for pedagogical decision making (as it would better measure what happens when the two teaching methods are applied to comparable students).

More confusingly, these tables actually seem to show that the dichotomy of direct instruction versus enquiry-based learning is a false one according to this data. Because many countries do either lots of both or very little of each, which makes no sense if we are picturing it as and either-or situation. Korea, for instance, are dead last by a wide margin in the use of teacher-directed instruction, yet they are somehow also second to last on enquiry-based learning. What on earth are these Koreans doing in their classes then, if it’s neither one nor the other? Many other countries exhibit similarly paradoxical results. This suggests that this data is poorly suited for making judgements about one teaching style versus the other. For this purpose, it would have been better to have asked the students questions that forced them to pick a point on this continuum instead of asking them about each separately, in a non-exclusive way.

So there are some grounds for casting doubt on the data, but by and large I think we have to admit that this PISA report is very damning evidence for the fashion of the day in educational reform ideology.

]]>The data is as follows (Table 1, page 18):

The final exams were double-blind graded by a third party (48), so they can be considered objective.

In a nutshell: Male students like male teachers much better, even though they do not do any better under them. Female students like male teachers a bit better, even though they do a bit better under female teachers.

Thus: Evaluations are biased in favour of male instructors. “The first main result is that gender biases exist.” (5)

Maybe so, but let’s look at the more fine-grained evaluation components in the table. To which specific aspects of teaching do students attribute their higher ratings of male teachers? Note that the answer is quite unequivocal: men are significantly better on “animation & leadership,” “current issues,” and “intellectual development,” and pretty much equal to women on the rest. Both genders of students largely agree on this. This clear pattern suggests that there are underlying differences in teaching approaches between male and female teachers, not just crude, across-the-board gender bias.

In fact, note that “current issues” and “intellectual development” are things that may be very desirable in an education generally but not directly relevant to a specific course examination. So it could be that the male teachers were better at imparting those more general skills, but no better at course-specific aspects like “organization,” “instructional materials,” etc. This hypothesis would give a rational explanation for virtually the entire table without the need to assume that the students were biased. According to this hypothesis: male and female teachers were equally good at the course-specific aspects; therefore they produced equal outcomes at the final exams; but male teachers were better at the bigger-picture aspects “current issues” and “intellectual development,” which, although not directly relevant to the exam, were nevertheless valued by students.

Even differences by student gender can explained by this hypothesis: male teachers and male students alike have a particular interest in those bigger-picture aspects, while female teachers and students alike are more focussed on the particular course at hand. If so, that would explain why male students gave the male teachers a greater “boost” in evaluations than the female students did.

It seems to me that this hypothesis is a simple and unified way of explaining many nuances of the data that are completely unexplained on the crude hypothesis of across-the-board gender bias.

I do not particularly believe this hypothesis. My point is not that it is true and that there is no gender bias. I think there is a fairly decent case to be made for some gender bias here. My point is only that if we are to work objectively with data then we must have an open mind toward alternative hypotheses such as this, instead of jumping to the gender-bias conclusion with blinders on. Gender bias may be a problem, but so is confirmation bias in favour of predetermined conclusions.

The only main point of the data that my hypothesis does not explain is the evaluation item called “quality of animation & ability to lead.” This seems to me a very stupid item, and I am inclined to dismiss it altogether. First of all, animation and leadership are surely very different things, so they should not be combined in one item. Furthermore, in the appendix where the author supposedly gives the actual questionnaire given to the students, the corresponding questions simply reads: “How do you evaluate your teacher’s class leadership skills?” (61) Is this what students were asked? Then why has the author inserted “quality of animation” in the data table at this item heading?

What is “class leadership” anyway? Does it mean leading discussions? If so it would seem to be related to many of the other items, like “communication skills,” “organization of classes,” “usefulness of feedback,” etc. But men score high on “leadership” but not on these other things, so it seems leadership must mean something else. Could it mean something like being strict and firm with deadlines for example? But then shouldn’t it be related to “clarity of assessment,” which it is not in the data? In sum, the question of “class leadership” seems to me much too vague and poorly formulated to be taken seriously. With such a vague question, it would not be surprising if students, lacking any way to answer it meaningfully, fell back on some stereotype about men being “leaders.” But this would say nothing at all about the course.

The author in fact has her own way of explaining the more fine-grained scores on the student evaluations, which goes as follows:

“The second main result I find is that students rate teachers in different dimensions of teaching according to gender stereotypes of female and male characteristics. … Students give more favorable ratings to women for teaching skills that require a lot of work outside of the classroom, such as the preparation and the organization of the course content, the quality of instructional materials, and the clarity of the assessment criteria. … Male teachers, however, tend to obtain more favorable ratings by both male and female students in less time-consuming dimensions of teaching, such as quality of animation and class leadership skills.” (5)

In my opinion, this explanation doesn’t fit the data at all (and certainly not as well as my hypothesis above). The author is trying to force gender stereotypes into the picture, but her attempt is quite absurd. Why would “clarity of assessment criteria” “require a lot of work outside of the classroom”? On the contrary, a lazy teacher will typically make up simple assessment criteria that are very clear indeed, since this makes life easy for the teacher. And I suppose no one ever works on their “leadership” and “animation” skils, they are just innate? What a strange assumption. And in any case it is very dishonest and deceptive to focus on them and claim they are representative, when, as we clearly see from the data, male teachers also scored much higher on “ability to relate to current issues,” which is clearly something that could cost the teacher a lot of preparation time: it is much easier to just “teach from the book” the same way year after year.

In conclusion, I believe the research literature is biased in favour of the gender bias interpretation. Although, as we have seen, other interpretations are perfectly plausible, the author pushes only the gender bias interpretation. And she pushes it too far when she writes that “male students give much higher scores to male teachers … in all dimensions of teaching” (5), and when she claims that gender stereotypes explain item-by-item variation in the evaluations. Yet these unwarranted conclusions are the takeaways from the study picked up and cited by others.

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