Category: education

What Game Are We Playing?

(for 3 Quarks Daily)

I know teachers who imagine the tune is what they have on repeat in hell, but I myself am strongly pro-Kahoot. If (like me) you were born too early to have your own school experience center around a large screen, and (unlike me) you have one of those boring non-teaching jobs, a brief explanation is in order. Kahoot is an app that lets you ask multiple-choice questions on your class screen, and have students answer them on their own devices to earn points. With its bright colors and  some other bells and whistles, it hits a sweet spot in the teenage brain that magically makes it care about getting mathematical terminology right. It’s the best thing.

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The Proper N

(for 3 Quarks Daily)

“You are aware”, I ask a pair of students celebrating their fourth successful die roll in a row, “that you are ruining this experiment?” They laugh obligingly. In four pairs, a small group of students is spending a few minutes rolling dice, awarding themselves 12 euros for every 5 or 6 and ‘losing’ 3 euros for every other outcome. I’m trying to set them up for the concept of expected value, first reminding them how to calculate their average winnings over several rounds, and then moving on to show how we calculate the expected average without recourse to experiment. It would be nice, of course, for their experimental average to be recognizably close to this number. Not least since this particular lesson is being observed by the Berlin board of education, and the outcome will determine whether or not I can get a teaching permit as a foreigner.

In case they are reading this, I would like to emphasize that I plan all my lessons with care and forethought; but for this particular one, you can bet I prepared especially well and left nothing to chance. Except for the part I left to chance, that is. To be precise: I had neglected to calculate in advance how likely it was for the experimental average over roughly 80 games to diverge from the expected value by a potentially confusing amount. I relied on my intuition, which informed me that 80 is a large number.

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Neil Postman and the Two Cultures

(for 3 Quarks Daily)

In 2022, I worked harder than before to keep my students’ tables free of smartphones. That this is a matter for negotiation at all, is because on the surface, the devices do so many things, and students often make a reasonable, possibly-good-faith case for using it for a specific purpose. I forgot my calculator; can I use my phone? No, thank you for asking, but you won’t be needing a calculator; just start with this exercise here, and don’t forget to simplify your fractions. Can I listen to music while I work? Yeah, uhm, no, I happen to be a big believer in collaborative work, I guess. Can I check my solutions online please? Ah, very good; but instead, use this printout that I bring to every one of your classes these days. I’m done, can I quickly look up my French homework? That’s a tough one, but no; it’s seven minutes to the bell anyway and I prepared a small Kahoot quiz on today’s topic. (So everyone please get your phones out.)

As a matter of classroom management, some of these questions are more of a judgment call than eating and drinking in class (not allowed, with some exceptions immediately after a PE lesson) but less complicated than bathroom visits (allowed in principle, but in need of limits that I may never be able to express algorithmically). In spite, however, of the superficial similarities between these phenomena – all subject to teacher- and class-specific settlements, informed and assisted by school-wide institutions such as regulations and phone bags – it feels as if more is at stake when it comes to smartphones. I sense more urgency, as if I’m laboring to stop a tide from coming in; as if what I am inclined to view as ‘complex’ and ‘multi-faceted’ and ‘also an interesting challenge, actually’ is actually one big thing only: an external force threatening to infiltrate my classroom and undo what I am trying to achieve there (which is called ‘education’ and which is therefore plainly also one big thing). I don’t feel this way about chewing gum.

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Giambattista Vico Enters The Classroom

(for 3 Quarks Daily)

It’s my favorite topic of the year, I tell the kids, before scanning my conscience for signs that I have just lied to them. No, this feels about right, and in any case, they didn’t need my reassurance: after a dry unit about ratios, at my mention of the word “probability” I can practically hear the sound of neurons firing. Whether they know of any contexts in which chance turns up? Boy, do they. My twelve-year old students inform me about dice, cards, spinners, lotteries, casinos, about your chances of survival in Squid Game, about unearthing rare minerals in Minecraft, and about whether the stuff you want to buy is actually in stock because sometimes they run out and there’s no telling in advance when. Also the weather.

I love introducing them to the math of probability precisely because it is a topic that they have already thought about so much, but that I know many of them have incoherent intuitions about. Not that I don’t, of course, but I am a little bit ahead of them and I know from experience what I am working with. Somebody will opine that the chance that it’s going to rain next month is “fifty-fifty” because it either will or it won’t. Somebody will say that there is a 1/7 chance that someone’s favorite day of the week is Thursday. Somebody will reason that since the highest sum you can roll with two D6-dice is a 12, the chance you roll a 10 is 1 in 12, which…

Oh wait, is it? Anyway, you get the point: the process feels properly Socratic, taking slightly muddled concepts that students already feel strongly about, and providing the right nudges to make them reconsider some of those concepts and make others click together. It is a significant source of satisfaction every year to see just how fast ‘the group’ moves from guessing that a sequence of three coin flips has six possible outcomes (unless it can land on its side, which they also unerringly point out), to calculating the number of four-digit pin codes where all the digits have to be different.

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Why I’ll change how I talk about proof

New 3QuarksDaily column:

I have some very simple New Year’s resolutions, and some that require an entire column to spell out. One example of the latter is that I want to make a subtle but meaningful change in how I talk to my (middle and high school) math students about proofs.

First, I need to be open about the fact that talking about proof in mathematics comes with an especially strong impostor syndrome: as a secondary school teacher, who am I to talk about the nature and limits of deductive reasoning in a discipline of which I have barely scratched the surface? I know high school proofs of high school mathematical concepts; I am vaguely aware that people much cleverer than I have tried to reduce all mathematical claims to analytical (logical) truths, or at least to a limited set of axioms; and I believe that after having mentioned this, I have to name-drop Gödel as the person who has supposedly put a stop to these projects, even though I have never studied his incompleteness theorem and doubt I would be able accurately to judge its relevance and meaning. That’s about it.

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River-Crossing Riddles Through the Ages

New 3QD-column:

Have you ever been in this situation where you had to get a group of 3 men and their sisters across a river, but the boat only held two and you had to take precautions to ensure the women got across without being assaulted?

This problem is one of 53 puzzles in the oldest extant puzzle book in the Western (Latin) tradition: the Propositiones ad acuendos iuventes or problems to sharpen the young. Its authorship is uncertain but it is often and plausibly attributed to Alcuin, who possibly sent them to the Frankish ruler Charlemagne in 800 AD. I hope you will allow me a brief introduction of these puzzles, before I go on to do what I hope will by then be redundant, namely spelling out why I think you should be thrilled by their existence.

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Why Teach Math? Two Voices From The 1920s

New 3QD-column: Tatiana Ehrenfest-Afanassjewa and Eduard Jan Dijksterhuis on intuition and abstract thought in math class.

“Am I ever going to use this later?” As a math teacher, I seem to be getting this question about once a month (which is actually less frequently than I would have predicted). It is asked with varying degrees of openness to the idea that a satisfying reply is even conceivable, but almost invariably by students who are probably justified in believing that their tertiary education or future career is going to involve preciously few linear equations indeed.

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Math as an Art Paul Lockhart’s Mathematician’s Lament

Paul Lockhart starts his “Mathematician’s Lament” (later expanded into a book under the same title, but I’ll be discussing the shorter article here) by comparing math class to a misshapen music or art class. Suppose that in music class, enjoying actual music is supposed to be too advanced for children, so they are made to start with memorizing the circle of fifths and pointing the stems of quarter-notes the right way; or suppose that in art class, painting is postponed until after preparatory “Paint-by-Numbers” classes. This, Lockhart suggests, is how math class works; it stifles creativity and natural curiosity and therefore goes against the spirit of mathematics.

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Blended Learning: Kwadratische verbanden in 3HAVO Afsluitende blogpost voor Onderwijskunde(-IV) aan de lerarenopleiding Windesheim

(Update achteraf: Neem de eerste letter van elke zin.)

Onderwijskunde-4 wordt naast het gezamenlijke dossier afgesloten met een individuele conclusie en blogpost. Hieronder volgt mijn reflectie op onze ‘blended’ leeromgeving in blogpostvorm.

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Lof van het gedisciplineerde denken

Ger Groot schreef deze week in Trouw een opiniestuk naar aanleiding van een eerder artikel in Trouw (6 april, p. 7 van de Verdieping) van Hanne Obbink over de profielkeuze van middelbare scholieren.

Kort gezegd sluit die profielkeuze steeds meer aan bij het stereotype dat C&M een profiel is voor losers dat ook daadwerkelijk studiemogelijkheden afsluit, en is N&T het profiel dat je doet als je slim bent en dat alle deuren opent – inclusief talen en cultuurstudies, die beter lijken aan te sluiten bij C&M. Dat leidt ertoe dat leerlingen steeds vaker de N-profielen kiezen als het maar even kan.

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