The Way We Teach Math, Sciences, and Languages Is Wrong

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A few years after I learned German, I got the chance to learn French. That experience gave me lots of ideas for why our teaching of many subjects, especially science and mathematics, is so unsuccessful—and for how we can improve our learning.

I studied French in school for five years. However, when I went to France after college, I could barely buy a train ticket. The impetus to try again came a few years later, in the summer of 1993. Our whole family was going to spend two months in Lyon while my father took a sabbatical. The rest of us enrolled in a four-week language course at the Alliance Française.

While still in America, to get more benefit from the language course, I started relearning French. On the recommendation of a friend who is a linguist and mathematician, I got the self-study French course made by Assimil entitled Le Nouveau Français sans Peine (New French With Ease). (Many other self-study courses should also work well. I have not tried them, so I do not have the knowledge to draw out lessons for learning other subjects, which is my main interest here. But to learn about language programs, I recommend the excellent “How to learn any language” site.)

I did one French lesson daily starting from Lesson 1. I read a short, idiomatic dialogue out loud using the pronunciation key, then listened to it on the tape, repeating it sentence by sentence. The lesson finished with 2 minutes of fill-in-the-word exercises using the vocabulary from the dialogue. Each lesson took about 30 minutes. After three months of this preparation, when I landed in France I could converse with random French people on the train. There was plenty of time to try, for it was a summer of discontent with strikes across the country. That’s how I learned la grève (strike). When I took the Alliance Française’s placement exam, they gave me the choice of the intermediate or advanced level. I think it was a fair assessment: Although not fluent, I could survive well.

Thus, I learned far more from 3 months with the Assimil self-study course than from 5 years of school French. The absurdity of that comparison only grows upon comparing the hours spent. The Assimil method took 0.5 hours per day for about 90 days: That’s 45 hours. The 5 years of school French happened every school day for roughly 50 minutes. A school year has roughly 180 days, for a total of 750 hours over the 5 years. Including homework, the total is probably 900 hours.

That’s a time-invested ratio of 20 to 1. Thus, despite spending 5 percent of the hours that I spent in school, with the self-study method I became far more competent in the language. As I think back on what made this self study so efficient, I’ve come up with several reasons that can be used in redesigning many kinds of learning:

  1. Active. In the Assimil course, you talk for most of the 30 minutes by repeating the conversation on the tape or CD. In school French, you mostly listen to the teacher talk or do grammar exercises (e.g. you conjugate verbs).

  2. Daily practice. The Assimil course happened every day, including weekends, holidays, and summers. (Stephen King writes every single day, including Christmas, July 4th, and his birthday.)

  3. No fear of mistakes. Speaking to myself, at home, I didn’t worry whether I was saying it right or about my grade. I just tried to match the syllable sounds and sentence intonations on the tape. As I found when learning German, and as Benny, the Irish polyglot, explains so well—for examples, see here, for learning German—the willingness to make mistakes is the stuff of learning. (Thanks to blog reader Jack for pointing out this excellent site.)

  4. Inductive (rather than deductive). In the Assimil course, you learn grammar first by using it in the dialogues. However, every seventh lesson has no dialogue. Instead you review, and slightly extend, the grammar that you have been using for the past week. By which time it has become easy! In school French, at least in my day, you mostly learn and practice grammar in isolation. Heaven for-fend that you might use a construction incorrectly. Better not to speak at all.

  5. Idiomatic. The dialogues used phrases that living humans might say, rather than, “My hat is green but my dog is brown.”

  6. Fun! The dialogues were amusing. Almost twenty years later, I still remember the punch line of the first dialogue. (And I have heard that earlier editions of the Assimil program were even more enjoyable.)

If we learned our first language like we usually learn second languages, it might look like this. A young child says, “I am hungry.” The parent replies, “Wait! Before saying am, you first must learn to conjugate to be in all persons and number, in the indicative, imperative, and subjunctive moods, and in the past, perfect, and future tenses.” After a few months, or maybe weeks, of this teaching, the child would conclude that it has no aptitude for languages and become mute. And human culture would perish in a generation.

If we taught math or science like we normally teach languages…oh, wait, we do! (And I believe, although with less direct knowledge, that we teach most subjects this way.) Look at the factors on the preceding list:

  1. Active? Most of the learning is spent passively copying down what the teacher puts on the board or, in the high-tech version, using ghastly PowerPoint slides. This method of knowledge reproduction made sense 800 years ago, when a book cost $20,000 (in today’s dollars). The invention of the printing press has changed book prices but not how schools and universities organize learning.

  2. Daily practice? The practice, at least in many college courses, usually happens once a week on the homework problems.

  3. No fear of mistakes? The biggest fear students have is making mistakes, for they lead to punishment: bad grades.

  4. Inductive? Rather than teaching inductively, we teach students a system of axioms or rules (the grammar)—for example, Maxwell’s equations of electromagnetism or the rules of arithmetic. Then we ask students to construct grammatical sentences, which the “good” students learn to do. But they cannot speak: That is, they cannot use the language of mathematics or science to understand or express their ideas about the world.

  5. Idiomatic? Hardly any of the usual examples (such as Atwood’s machine) help students explain any feature of the real world. I know from my own experience. After four years of getting A’s in physics courses, if you had asked me, “Why is the sky blue?” I would have been flummoxed, for neither skies nor colors were among my course topics.

  6. Fun? When I used to go to parties (before becoming a parent), I used to tell people that I was a physicist or a mathematician. However, I stopped because the invariable response, “Oh, I was never good at physics [or math]!” illustrated how rarely anyone enjoyed learning physics or math. When I start going to parties again, I have my rejoinder: “It’s not your fault. It’s because of the teaching… and I’d like to do something about it.”

Instead of teaching physics or mathematics as we teach second languages, then blaming the victims for not doing well, and expecting them to internalize the blame (an example of the Stockholm syndrome), why not use physics and mathematics to ask and answer questions about the world? Rather than starting the course with “motion in a straight line at constant speed” (it’s hard to imagine a topic more dull or, alas, more typical), we can use physics concepts such as force and energy to estimate the gas mileage of a 747 by dropping coffee filters—thereby giving the concepts depth and meaning.

Following these thoughts, I stopped teaching physics the usual way. Now on the first day of class we make a related estimate, of the world-record cycling speed. Later we find out why toast always lands butter side down (and it’s not because the butter side is heavier!).

These changes are but a start. Let’s take inspiration from skillfully constructed language courses to design a whole new physics and mathematics curriculum—and extend the principles to all areas of learning.

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  1. Mr.Richardson says:

    “Later we find out why toast always lands butter side down (and it’s not because the butter side is heavier!).”

    I [i]really[i] want to know why!

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    • frankenduf says:

      duh- murphy’s law

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    • Sam_L says:

      The height of our tables and the size of our toast.

      If we had taller tables, the toast could complete more of a rotation, thus landing on the non-buttered side. We could also change the size of the bread so that the moment of inertia is different – thus changing the rate at which the bread spins – and appropriate for non-buttery landings. Obviously, a combination of these approaches could also be used.

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      • Steve O says:

        To clarify, for those who aren’t crazy about probability: it’s not necessarily that it always lands after a half rotation, but that a half rotation has maybe a 40% chance, a full rotation has 20%, 1 1/2 is 10%, etc. In the end, the cumulative probability of ending butter side down will be significantly higher.

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      • john says:

        I think I like that one even better than confirmation bias, though I think they actually work in tandem… But, maybe this means it would be better to eat toast upside down!

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    • Bernoulli says:

      It’s probably because the difference in texture (between buttered and non-buttered sides) create air flows that lead to pressure differentials that tilt the slice of bread. Air planes fly on related principles.

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      • Sam_L says:

        Bernoulli’s principle has little influence at the speeds toast falls. In an absolute sense, yes, this will influence the movement; in a practical sense, it has no impact.

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  2. Chris says:

    Great review and breakdown of the Assimil course. I think we’re all due for a new way of approaching learning. Could it be that it was *because* you already had five years of training previously that you were able to learn French quickly with Assimil? Granted, there’s likely a lot of room for improvement in what’s taught in those five years but it would have given you a broad base from which to start re-learning (which would actually be closer to remembering).

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    • Laura says:

      Historically, even the methods described for Assimil are old and out of synch with what we know about cognitive theory these days. Rote memorization is never the answer.

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  3. Uncle Big Steve says:

    My kids’ schools started teaching math using Math Expressions. I haven’t had the pleasure of helping with any homework yet but from what they tell me, it is a more real world, applied way of teaching math. Anybody that grew up with the dreaded word problem won’t like it from the description.

    I look forward to seeing what it’s all about. As an engineer, the challenge for me will be trying to help them in the context of that program, not trying to figure out the correct answer. It should be fun.

    I’m bookmarking the Learning Languages site you linked to. I had 4 years of German in HS and I couldn’t speak it now (20 years later) if my life depended on it. I got good grades mostly because I could learn the rules and the conjugations and such but speaking it was a nightmare. Midlife crisis goal of learning another language may have just gotten a jump start. Thanks!

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  4. Philippe says:

    You do not address that the first time you learned French, you probably thought it a chore and simply a requirement (like, I would venture, most people being forced to learn a foreign language in school because it is a requirement, view it). Whereas the second time, you were learning for the great adventure to come.

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    • john says:

      Perhaps my very similar experience will address that.

      I took two quarters each of German and French in College, with no prior preparation unless you count learning to count to 10 on the German side and three years of highschool Latin on the French side. These classes happened within an 18 month period, and featured four different teachers.

      Q1 German: Elderly Gymnasia System professor, much inclined to the “learn all the grammar and tables of declensions and conjugations” method. At the end of the quarter I had a D, and knew how to count to 10 and the proper name of the umlaut.

      Q2 German: Somewhat crazy native speaking son of a retired baker from Bavaria made some reference to the book but spent most of the time having us write things on the board, and working out elaborate comic skits and dialogs in which we had to improvise most of the content while he corrected our German and embarrassed us within the context of the skit rather than about our German.

      Q1 French: Very decent middle aged guy who tried very hard through the je suis, tu est, something or other thing and occaisional readings from the book to practice pronunciation.

      Q2 French: Young woman who said, “Well, second quarter? Guess we don’t need English anymore then…” and never said a single word besides “oui” and “non” which I could understand for the rest of the quarter.

      When I arrived in Europe for the first time 10 years later I could read signs, order beer, buy stuff in shops (if I could figure out the name) and ask directions in poor broken German. In French I could look confused, shrug, and say (learned by rote my first morning there) “Je ne parle pas Francais”. Later I learned to ad “Je suis desole.”

      I agree completely with Mr. Mahajan (something I think I’ve not done in the past) since my Calculus classes were taught exactly in the style of French Q2. I learned more watching one 30 minute podcast about limits than I did in 9 months of Calculus.

      The educational system is broken in many, many ways, but this is definitely one of them.

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  5. Mike B says:

    Didn’t the Mythbusters bust the whole toast falling butter side down thing? Anyway, while I an sure new teaching methods can help improve understanding somewhat, the current methods help identify those people who are innately good at a subject so that they can then specialize. It’s optimal to have people utilize their natural hardware acceleration instead of simulating natural skill through software tricks.

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  6. Lauri Jon says:

    Why does toast always land butter side down? This I’d really love to know!
    (And I was one of those people who most always did exemplary in math… took a Level 5 Logic class in my freshman year. Thanks, LauriJon

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  7. Dan Marcus says:

    Not discounting the very valid points made in this post, I think Sanjoy misses one very important variable – the fact that he was a repeat learner when self-learning. We may not consider ourselves competent or confident at languages when we leave school but we are often left with a base level of knowledge that makes returning to the language later in life considerably easier!

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  8. Chess Piece Face says:

    My experience with Montessori math curriculum is that it’s very inductive. Students learn the Pythagorean Theorem the same way Pythagoras did, analyzing triangles, etc.

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