7 ms·

Dan Meyer dissects the flaws of math textbooks (video)

81 points by anuleczka 17y ago

snowbird122 17y ago
I love the fresh thought here on how to best present problems to students. I wonder how much work it takes to redefine each problem in the textbook.
- yequalsx 17y ago
  A great deal of work. Some problems won't be easily changed to suit his paradigm. For instance, how does one go about redefining an equation like, sqrt(2x+1) = sqrt(x) + 1? I'd like to know what this guy does for these types of problems. I've been teaching community college mathematics for 10 years and we simply don't have the time to do what he says. Maybe I'm bad at motivating students but my anecdotal experience is that most of the students are solely interested in getting a degree and not in learning. It's understandable that their focus is no getting a degree but focusing on learning makes it easier to get the degree. It's very hard to get this point across. While I see many problems with the current system of teaching mathematics there simply is no cure to apathy. At some point one has to be willing to sit down and learn to solve problems like, sqrt(2x+1) = sqrt(x) + 1.
  - bshep 17y ago
    This doesnt make any sense: sqrt(2x+1) = sqrt(x) + 1 EDIT: Ahh I thought he was saying that he had factorized/simplified the left side into the right and it wasnt making any sense.
    - SlyShy 17y ago
      Solve for x: sqrt(2x + 1) = sqrt(x) + 1 2x + 1 = x + 2sqrt(x) + 1 x = 2sqrt(x) x^2 = 4x x^2 - 4x = 0 x(x - 4) = 0 x = 0 or 4
    - chrjozefharibo 17y ago
      I solved it numerically. x=4 or x=0
    - XFrequentist 17y ago
      Yes it does: x=4,0
  - SlyShy 17y ago
    Math major here: I think actually he is arguing for a cure to the apathy. The reason people are apathetic about mathematics is it is presented like a dry and idiotic subject. A common complaint about word problems is "this is contrived, why does the water tank have exactly that parabolic curve, pfft math sucks, I quit" whereas the common complaint about questions that are pure symbol manipulation is "this doesn't matter to me, I quit." By introducing questions more open endedly (Could you kick a door down?) you avoid making the question sound contrived. Open ended questions also lead to generalization. (Side note, I've worked as a tutor and coach, so yes, I have classroom experience). The students get to saying, "well, we can't solve this question until you give us numbers." They get several sets of numbers, and they begin to realize that the method for solving the problem is generalizable. They develop the formulas from the examples, and learn they could have answered the open ended question all along. That's in stark contrast to the current model, where the formula is taught, and then specific examples of close ended questions are presented. The key to teaching mathematics is fostering exploration. The very best illustration of this idea is the Ross Mathematics Program in Columbus, Ohio. It is a two month long ground up rigorous exploration of number theory and abstract algebra. And here's the best part, there is no background necessary. If you know arithmetic, you have enough to begin, because all you start with the axioms of the natural numbers. And yet, the students exit the program having proved that groups of order p are cyclic, the Quadratic Reciprocity Law, etc. Everything is learned through problem sets, which the students do at their own pace. Every theorem used is proved, and every new proposition is discovered. Yes, it is a great deal of work, but nobody ever said teaching was easy work. Passion should be the #1 hiring criterion.
    - yequalsx 17y ago
      Using word problems is one likely to discover why sqrt(2x+1) = sqrt(x) + 1 is fundamentally different than cuberoot(2x+1) = cuberoot(x) + 1? Give this problem to calc one students and many will do both problems incorrectly. They'll incorrectly cube both sides in the second equation. They don't understand why we solve sqrt problems but not cube root problems but they know how to compute doubling time for bacteria growing in a petri dish. Your fourth paragraph is an argument in favor of what I've been saying. The applications we teach ought to be applications to math; not to physics or biology. But back to the apathy problem. People aren't going to be motivated by number theory or bacteria in a petri dish if they possess too great a disdain for knowledge.
  - kscaldef 17y ago
    Step 1 is figuring out why someone would want, or need, to solve that equation. He's by no means saying that students don't have to learn how to do algebraic manipulations. He's saying that without motivating the process and letting students understand the process that leads to the algorithm, they aren't really going to learn how to use math in a way that will be helpful to them. I suspect that he'd also argue that an hour spent slowly and carefully exploring 1 problem like this is better than having them do 30 examples with no context and no motivation.
    - yequalsx 17y ago
      Do they really learn such motivation from solving word problems? Especially when many word problems are not practical or even correct? How about this for motivation, we solve such equations because we can. Solving equations is useful and the more equations we can solve the better. It's not really motivating when we give a word problem whose model is a radical equation and then say solve the equation. Most equations can't be solved by algebraic methods and anyone who really needs to solve an equation and trust the answer is better off having a computer do the computation. In solving the 30 problems one might get to a point in understanding why sqrt(2x+1) = sqrt(x) + 1 is slightly harder than solving sqrt(x+1) = sqrt(x) + 1. In solving 30 problems one might get to a point to discover why sqrt equations can be solved but why we don't solve cube root equations. Or why sqrt(2x+3) = x is fundamentally easier than sqrt(2x+3) = x + 1. You won't get this from solving word problems.
      - kscaldef 17y ago
        Your second paragraph contradicts itself. You say solving equations is useful. So, why not show your students the use?
        yequalsx 17y ago
        Having the ability to solve equations is useful. Randomly write down an equation. That particular equation isn't likely to have practical applications. Is it worthwhile to explore whether or not it can be solved by algebraic methods? Why is it that first, second, third, and fourth degree polynomials can be solved by algebraic methods but not arbitrary polynomials of higher degree? Is this worth studying only if it has practical applications? It does have applications today but it didn't for the first thousand years this problem was tackled. The requirement that something be 'useful' before it is worthy to be studied is too great a burden. It's a detriment to intellectualism.
lotharbot 17y ago
tl;dw summary: you know you're doing math wrong if students display (1) a lack of initiative, (2) lack of perspective, (3) lack of retention, (4) aversion to word problems, and (5) the few who understand the math just want to jump to a formula. Math textbooks encourage teachers to teach math wrong. The way they present problems is with a complex visual with mathematical structure already imposed, step-by-step handholding through the problem, and asking a question at the end (a question that can often be solved just by figuring out which number to plug into which part of the formula, without necessarily understanding why.) Suggested method for teaching right: (1) use multimedia. (2) encourage student intuition. Students will argue with each other about what they see and buy in to the problem. (3) ask the shortest possible question. Don't begin with a page full of numbers, measurements, and individual steps. Let the detailed questions come out through discussion. (4) Let the students build the problem. Students will recognize the need for mathematical structure (labels, coordinates, measurements, etc.) as they decide what information they will need to answer the question. They'll go through the steps on their own. (5) Be less helpful. The textbook helps in all the wrong ways, taking you away from your obligation for developing patient problem solving and mathematical reasoning. Example: he completely rewrites a question from a math book about filling a water tank. He produces a video of a water tank being filled from a garden hose, which takes excruciatingly long to complete. Students get uncomfortable, complain about how long it's taking, and then put in their guesses as to how long it will take. Then they decide what information they'd need to calculate the end result, ask for the measurements they think are important, do the calculations, and watch the rest of the video to see if their calculation was right and how close their initial guesses were.
- Estragon 17y ago
  Central soundbite: "The way our mass-adopted textbooks teach math reasoning and patient problem solving is functionally equivalent to turning on [a TV sitcom] and calling it a day."
yequalsx 17y ago
I'm a teacher of mathematics at a community college. It appears to me that the crux of the problem is that people, including the guy in the video, confuse problem solving with mathematics. The utility of mathematics comes in the remarkable fact that a great deal of phenomena can be adequately modeled mathematically. The focus of an algebra class ought to be in learning the language of algebra. That is, in manipulating numbers in the abstract. The application problems ought to be saved for physics, biology, economics, etc. The result of an emphasis on so called real world problems in high school mathematics is a generation of students who are incapable of correctly manipulating algebraic expressions and equations. I recently gave my college algebra class an equation. It was a simple equation and all of them could solve it. I then asked them for an example of an equation that had no solution. Not a single person could provide an answer. They can solve the word problems in the textbook but don't have the slightest clue about what these mathematical concepts actually mean. Perhaps it isn't important that one need to manipulate algebraic objects. I won't argue with this. But let's not call solving word problems mathematics. If you want to learn mathematics then grinding through the minutia and having the patience to understand the symbols is necessary. You can't get around this. If the goal is to solve word problems then....go ahead and change things.
- anuleczka 17y ago
  In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?
  - flatline 17y ago
    The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answer for guesswork or estimation, like a calculator.
sketerpot 17y ago
Keep in mind that if you're posting on Hacker News then you're probably better at math than, say, 90% of the population. It's hard to come up with ideas for teaching math to people who are profoundly different from you.
silviasaint 17y ago
I don't like maths http://www.squidoo.com/womensera http://www.squidoo.com/womensera

81 points by anuleczka 17y ago

I love the fresh thought here on how to best present problems to students. I wonder how much work it takes to redefine each problem in the textbook.

yequalsx 17y ago
A great deal of work. Some problems won't be easily changed to suit his paradigm. For instance, how does one go about redefining an equation like, sqrt(2x+1) = sqrt(x) + 1? I'd like to know what this guy does for these types of problems. I've been teaching community college mathematics for 10 years and we simply don't have the time to do what he says. Maybe I'm bad at motivating students but my anecdotal experience is that most of the students are solely interested in getting a degree and not in learning. It's understandable that their focus is no getting a degree but focusing on learning makes it easier to get the degree. It's very hard to get this point across. While I see many problems with the current system of teaching mathematics there simply is no cure to apathy. At some point one has to be willing to sit down and learn to solve problems like, sqrt(2x+1) = sqrt(x) + 1.
- bshep 17y ago
  This doesnt make any sense: sqrt(2x+1) = sqrt(x) + 1 EDIT: Ahh I thought he was saying that he had factorized/simplified the left side into the right and it wasnt making any sense.
  - SlyShy 17y ago
    Solve for x: sqrt(2x + 1) = sqrt(x) + 1 2x + 1 = x + 2sqrt(x) + 1 x = 2sqrt(x) x^2 = 4x x^2 - 4x = 0 x(x - 4) = 0 x = 0 or 4
  - chrjozefharibo 17y ago
    I solved it numerically. x=4 or x=0
  - XFrequentist 17y ago
    Yes it does: x=4,0
- SlyShy 17y ago
  Math major here: I think actually he is arguing for a cure to the apathy. The reason people are apathetic about mathematics is it is presented like a dry and idiotic subject. A common complaint about word problems is "this is contrived, why does the water tank have exactly that parabolic curve, pfft math sucks, I quit" whereas the common complaint about questions that are pure symbol manipulation is "this doesn't matter to me, I quit." By introducing questions more open endedly (Could you kick a door down?) you avoid making the question sound contrived. Open ended questions also lead to generalization. (Side note, I've worked as a tutor and coach, so yes, I have classroom experience). The students get to saying, "well, we can't solve this question until you give us numbers." They get several sets of numbers, and they begin to realize that the method for solving the problem is generalizable. They develop the formulas from the examples, and learn they could have answered the open ended question all along. That's in stark contrast to the current model, where the formula is taught, and then specific examples of close ended questions are presented. The key to teaching mathematics is fostering exploration. The very best illustration of this idea is the Ross Mathematics Program in Columbus, Ohio. It is a two month long ground up rigorous exploration of number theory and abstract algebra. And here's the best part, there is no background necessary. If you know arithmetic, you have enough to begin, because all you start with the axioms of the natural numbers. And yet, the students exit the program having proved that groups of order p are cyclic, the Quadratic Reciprocity Law, etc. Everything is learned through problem sets, which the students do at their own pace. Every theorem used is proved, and every new proposition is discovered. Yes, it is a great deal of work, but nobody ever said teaching was easy work. Passion should be the #1 hiring criterion.
  - yequalsx 17y ago
    Using word problems is one likely to discover why sqrt(2x+1) = sqrt(x) + 1 is fundamentally different than cuberoot(2x+1) = cuberoot(x) + 1? Give this problem to calc one students and many will do both problems incorrectly. They'll incorrectly cube both sides in the second equation. They don't understand why we solve sqrt problems but not cube root problems but they know how to compute doubling time for bacteria growing in a petri dish. Your fourth paragraph is an argument in favor of what I've been saying. The applications we teach ought to be applications to math; not to physics or biology. But back to the apathy problem. People aren't going to be motivated by number theory or bacteria in a petri dish if they possess too great a disdain for knowledge.
- kscaldef 17y ago
  Step 1 is figuring out why someone would want, or need, to solve that equation. He's by no means saying that students don't have to learn how to do algebraic manipulations. He's saying that without motivating the process and letting students understand the process that leads to the algorithm, they aren't really going to learn how to use math in a way that will be helpful to them. I suspect that he'd also argue that an hour spent slowly and carefully exploring 1 problem like this is better than having them do 30 examples with no context and no motivation.
  - yequalsx 17y ago
    Do they really learn such motivation from solving word problems? Especially when many word problems are not practical or even correct? How about this for motivation, we solve such equations because we can. Solving equations is useful and the more equations we can solve the better. It's not really motivating when we give a word problem whose model is a radical equation and then say solve the equation. Most equations can't be solved by algebraic methods and anyone who really needs to solve an equation and trust the answer is better off having a computer do the computation. In solving the 30 problems one might get to a point in understanding why sqrt(2x+1) = sqrt(x) + 1 is slightly harder than solving sqrt(x+1) = sqrt(x) + 1. In solving 30 problems one might get to a point to discover why sqrt equations can be solved but why we don't solve cube root equations. Or why sqrt(2x+3) = x is fundamentally easier than sqrt(2x+3) = x + 1. You won't get this from solving word problems.
    - kscaldef 17y ago
      Your second paragraph contradicts itself. You say solving equations is useful. So, why not show your students the use?
      - yequalsx 17y ago
        Having the ability to solve equations is useful. Randomly write down an equation. That particular equation isn't likely to have practical applications. Is it worthwhile to explore whether or not it can be solved by algebraic methods? Why is it that first, second, third, and fourth degree polynomials can be solved by algebraic methods but not arbitrary polynomials of higher degree? Is this worth studying only if it has practical applications? It does have applications today but it didn't for the first thousand years this problem was tackled. The requirement that something be 'useful' before it is worthy to be studied is too great a burden. It's a detriment to intellectualism.

lotharbot 17y ago

tl;dw summary: you know you're doing math wrong if students display (1) a lack of initiative, (2) lack of perspective, (3) lack of retention, (4) aversion to word problems, and (5) the few who understand the math just want to jump to a formula. Math textbooks encourage teachers to teach math wrong. The way they present problems is with a complex visual with mathematical structure already imposed, step-by-step handholding through the problem, and asking a question at the end (a question that can often be solved just by figuring out which number to plug into which part of the formula, without necessarily understanding why.) Suggested method for teaching right: (1) use multimedia. (2) encourage student intuition. Students will argue with each other about what they see and buy in to the problem. (3) ask the shortest possible question. Don't begin with a page full of numbers, measurements, and individual steps. Let the detailed questions come out through discussion. (4) Let the students build the problem. Students will recognize the need for mathematical structure (labels, coordinates, measurements, etc.) as they decide what information they will need to answer the question. They'll go through the steps on their own. (5) Be less helpful. The textbook helps in all the wrong ways, taking you away from your obligation for developing patient problem solving and mathematical reasoning. Example: he completely rewrites a question from a math book about filling a water tank. He produces a video of a water tank being filled from a garden hose, which takes excruciatingly long to complete. Students get uncomfortable, complain about how long it's taking, and then put in their guesses as to how long it will take. Then they decide what information they'd need to calculate the end result, ask for the measurements they think are important, do the calculations, and watch the rest of the video to see if their calculation was right and how close their initial guesses were.

Estragon 17y ago
Central soundbite: "The way our mass-adopted textbooks teach math reasoning and patient problem solving is functionally equivalent to turning on [a TV sitcom] and calling it a day."

yequalsx 17y ago

I'm a teacher of mathematics at a community college. It appears to me that the crux of the problem is that people, including the guy in the video, confuse problem solving with mathematics. The utility of mathematics comes in the remarkable fact that a great deal of phenomena can be adequately modeled mathematically. The focus of an algebra class ought to be in learning the language of algebra. That is, in manipulating numbers in the abstract. The application problems ought to be saved for physics, biology, economics, etc. The result of an emphasis on so called real world problems in high school mathematics is a generation of students who are incapable of correctly manipulating algebraic expressions and equations. I recently gave my college algebra class an equation. It was a simple equation and all of them could solve it. I then asked them for an example of an equation that had no solution. Not a single person could provide an answer. They can solve the word problems in the textbook but don't have the slightest clue about what these mathematical concepts actually mean. Perhaps it isn't important that one need to manipulate algebraic objects. I won't argue with this. But let's not call solving word problems mathematics. If you want to learn mathematics then grinding through the minutia and having the patience to understand the symbols is necessary. You can't get around this. If the goal is to solve word problems then....go ahead and change things.

anuleczka 17y ago
In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?
- flatline 17y ago
  The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answer for guesswork or estimation, like a calculator.

sketerpot 17y ago

Keep in mind that if you're posting on Hacker News then you're probably better at math than, say, 90% of the population. It's hard to come up with ideas for teaching math to people who are profoundly different from you.

silviasaint 17y ago

I don't like maths http://www.squidoo.com/womensera http://www.squidoo.com/womensera