Dan Meyer dissects the flaws of math textbooks (video) | Hacker News Reader

Dan Meyer dissects the flaws of math textbooks (video)

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81 pointsanuleczka54 comments

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.

During training, soccer players often run around the field in a big group, but you'll never see them do that during a game. Running laps around the field isn't the goal, but it does help them achieve their goals.

His basic argument is that the point of high school math education is to teach kids how to reason and understand the world quantitatively. I agree. The teaching of algebra to high school students is secondary to the real goal. Without a good education, most people are sloppy thinkers -- and they vote more frequently than they solve equations.

Teaching so called practical word problems is not accomplishing this goal.

I'm going to chime in from experience and say that your college algebra students are the bottom of the barrel in regards to math students. With that respects there can be many reasons why they are there: they didn't care about math, they never really understood math, and/or they never really tried to get math.

If you are judging the ability of finishing word problems in the textbook, I'm assuming for homework assignments, and they really don't understand the actual math I think your problem is probably that most are using a solutions manual or spending what would seem too much time solving the homework problems.

As for solving word problems or practical problems I believe that this is exactly what math should be for especially at the college algebra level. To be serious most students entering college who want or are going to pursue a degree in mathematics are going to start off closer to the calculus level and maybe trigonometry depending on underlying circumstances. What people in college algebra are going for is a degree where math has to be practical and the more diverse applications they see the better. To give a simple example take logarithms: if you were to teach logarithms on the sole basis of algebraic symbols and expressions when it comes time that these students need to apply logarithms to real world problems (like say compound interest) there is going to be a good chance that they didn't really comprehend logs when they were taught them and won't be able to use them properly.

Which brings me to another point for word problems: word problems exist because they make students think and relate math to the real world it is fundamental to learning for many students to apply mathematical concepts in a multitude of ways in order to actual grasp a concept and one of the easiest ways to do this is to place that concept into word problems.

I strongly disagree with the notion that college algebra ought to be the place where one learns practical applications of mathematics. A valuable skill for any educated person is the ability to plod through minutia and make sense of it. Being able to be focused on the point at hand and not let distractions cloud judgment is a valuable skill.

If one really understands the mechanics of logs then I think they will not freak out when presented with a formula involving logs. I'm all in favor of applications but isn't this the job of other subjects? The applications should be taught in the subjects that apply it. The goal of a mathematics class ought to be to understand mathematics.

This lecture,

http://www.youtube.com/watch?v=WwslBPj8GgI

is long but well worth watching. He provides strong evidence that students learn to solve word problems but haven't the slightest idea of what is really going on. They are memorizing steps. This happens at Harvard and the community college.

I've given my students the standard word problems on travelling problems. They get the problems right. Then I ask them this simple problem and they universally get is wrong:

You drive 1 mile at 40 mph and then the next mile drive 60 mph. What is your average speed? (Almost everyone says 50 mph.) We aren't teaching thinking skills. We are teaching memorization of problem types. Almost none of them is actually practical, correct, or useful.

But why are we teaching travelling problems in a math class? I think it should be taught in physics.

To answer your last question first I think traveling problems are taught in math because speed and time are easy for students to relate to, the same way that finding the height of a flagpole by its shadow problems are easy for students to visualize and relate to. As for average speed that is one of those concepts that really isn't that relatable and should probably require an introduction beforehand.

As for the goal of a math class being that students should come out understanding math I completely agree but the beauty of math is that it is applicable everywhere and the goal of teaching a math class should be that students understand mathematics and where to apply it.

I couldn't imagine being taught vector calculus without some mention of applications and examples, my professor catered more towards mathematics majors with an abundance of Tensor notation and proofs but he never failed to understand that many in his class learned well by physical and other practical examples such as magnetism or fluids and so on.

Although I failed to watch your video I believe there is most likely a counterargument to word problems and what effect they have on students because if students really are solving word problems with the math they are learning the issue isn't that they don't understand what is going on but rather how to apply it to an actual math problem. Either that or the word problems do not incorporate well enough with the math that a student can grasp it quickly without knowing underlying ideas.

The experience of the professor at Harvard in the video contradicts your last paragraph. They know how to apply the math but they don't understand the problem. And the traveling problem I gave underscores this point. It's a very simple problem and almost no one gets it correct.

Then perhaps you and him are assigning the wrong word problems or not giving proper introductions to said word problems.

If all students are taught to do is "memorization of problem types" (to borrow GP's term), then they will only be able to handle actual problems that match one of the necessarily finite set of types of problems they have memorized. A different selection of problem types to memorize will not provide general coverage -- that requires learning to construct a solution to a new type of problem.

Nowhere in the thread has it been demonstrated that it would be impossible to design a word problem that does require the construction of a solution.

One of the problems I had with my dif. eq. class was that the grad student teaching the class just focused on "tank problems" and other "x problems" sets. So I see why story problems alone are not helpful. But, if they're judiciously used to encourage the understanding of abstract mathematical concepts, I think they are invaluable.

Perhaps you should watch the video and then you would realize that your characterization of this professor and me is wrong.

His argument is that current HS math doesn't provide enough distractions. Students don't learn to focus on the problem at hand, because the problem is already in sharp focus. All the variables are provided. The word problem is a slightly jumbled description of the formula to use. All you have to do is look up the formula, then plug in the variables.

Your students can't solve the travelling problem for the very reasons he's talking about. That problem is completely unlike anything they've ever solved in high school, because it doesn't give them the numbers to average. It's a two step problem where they need to use the given numbers in one formula, to get the resulting numbers for the second. HS students don't learn to solve problems: they learn to match paragraphs to formulas, then plug and chug.

Also, don't be such a purist. It's not useful. There is a great deal of overlap between math and physics. Physicists often invent new math to understand physics, and old math is often found to describe some newly discovered physical phenomenon.

You wrote:

"HS students don't learn to solve problems: they learn to match paragraphs to formulas, then plug and chug."

I agree completely and this is why I advocate for an end to the focus on so called practical word problems in high school in algebra classes. It should be in physics where one learns about travelling problems not in algebra. The focus on so called practical word problems is not accomplishing the goal and it's distracting from the learning of mathematics.

I taught physics in a community college as a full-time prof for 10 years, and had the occasion to teach math there a few times as well. After that stint, I started work with an ed software company developing math content for algebra students in the cc.

In my experience, the "word problems" can be used to motivate the students as to the interesting part of what they are studying. My focus wasn't on making them learn how to reason through complicated applications, it was on getting them to master the skills. So in that sense I agree with you. However, building a story around it (how do you determine which cell phone plan to buy?) helps keep their attention and shows them that they- yes THEY (the "bottom of the barrel" as was noted above- but I'll clarify this below)-- can apply math to their lives and live more intelligently. And getting students to that stage is my life's mission.

I think where math goes wrong is displayed beautifully in the rational expressions units (I didn't go to the original link, so forgive me if I'm entirely redundant here). When I was building the math content for the ed s/w company I was struck that nowhere in the chapter where students learn to simplify fractions involving a polynomial numerator and a polynomial denominator (tough stuff when you are first learning it) nowhere in the textbooks did they tell the student WHY. Why do all this hard work? What is the answer that we seek from all this?

[rant] About the idea of "stupid teachers teaching stupid students" in community college: Anyone who truly speaks from experience knows better. In a cosmopolitan setting you will have downright brilliant students in your classes- especially in your night classes. I've had many ivy-leaguers in my classes, top accountants for Exxon-Mobile and other high-power operators, sometimes from government agencies (I taught in Alexandria Virginia, about a mile from the Pentagon, where Mrs. Biden currently teaches). A common reason for them to be there? They've burned out in their careers and want to do something that "makes a difference." Going to med or vet school was common. They've got the degree, they've got the grades, they just need a couple of classes that they didn't take as undergrads, they're too smart to spend the money on a school that won't help them any more than a little ol' cc will. Keeping their job for an extra year and taking night classes is a smart move, so they come to the local cc. [/rant]

"I think where math goes wrong is displayed beautifully in the rational expressions units (I didn't go to the original link, so forgive me if I'm entirely redundant here). When I was building the math content for the ed s/w company I was struck that nowhere in the chapter where students learn to simplify fractions involving a polynomial numerator and a polynomial denominator (tough stuff when you are first learning it) nowhere in the textbooks did they tell the student WHY. Why do all this hard work? What is the answer that we seek from all this?"

What do you think are the answers to your questions?

One thing that was quite easy to do is to set the rational expression equal to something and solve for one of the variables. We did a lot of problems where there was only one variable: (x^2+9x+20)/(x^2-25)

In motivating them you don't necessarily have to give every reason to learn something, just reason enough to buy into what you are trying to teach them.

But the motivation for solving complicated rational equations does not come from word problems. The word problems for this topic involve very simple rational functions. My point has been that the motivation for studying and doing much of mathematics ought not come, solely, from practical word problems.

I agree with you that you (we) shouldn't feel the need to have a one-to-one mapping between problems we solve and word problems. Even in elementary physics you could argue that the word problems in many cases aren't practical, but we teach it to teach the flavor of the approach; the attitude of how we attack problems (or, less poetically, what exactly we mean when we say "cause and effect").

That said, the motivation is still greater when the students see some hope that what they are learning is meaningful.

What that motivation is depends on the level of the class. For algebra 2, it might be "we learn to manipulate rational functions because it gives us a tool to understand (or solve) a certain class of problems. Someday you may own a business where you have to worry about the average cost of something that depends on things that change alot, or variables as we call them. And you know what an average is: it's the amount of something divided by how many there are. Well if your amounts are represented by an algebraic expression, and your total is represented by an algebraic expression, then the quantity you will be interested in will look something like this (writes a rational function on the board). Now, what the fuck do we do with this?"

In response to yequalsx:

I see 2 applications of rational expressions off the top of my head: 1) From an abstract viewpoint, the message is "We can do the same stuff with polynomials that we do with integers. The only issue is that factoring is harder." This message gets very garbled 2) Solving ratio problems A/B = C/D come up very often. For example: I've won 43/87 freecell games. How many in a row do I have to win to get to 70%?

I agree with both of these examples. And here is my point. The sort of practical applications of rational functions involve very simple rational functions. So if our motivation comes solely from word problems then the natural question is why are we learning to manipulate

( x^2 + 2xy + y^2)/(x - y) + 1/(x^2 - y^2)

Has there ever occurred a practical application in which this expression has come up? If our motivation is expanding our understanding of manipulating algebraic objects then this expression is very much a practical application. It's an extension of an existing idea - rational numbers - to a new class of objects. Be able to do this sort of abstraction is useful in mathematics, and in fact is the essence of mathematical thought. It's also useful in things like programming.

The motivation for learning to manipulate rational expressions has to come well before you get to this.

You can give yours students an incredibly useful skill that has inordinate practical application, exciting their minds with possibilities, intrigue, and a sense of empowerment. Or you can engage them in rote mechanics, fail to impart the usefulness of mathematics, and send them all to sleep.

Great generalization and oversimplification. How about this oversimplification?

Or you can teach a generation of students that if isn't 'practical' then it shouldn't be done and certainly isn't worth any effort or struggle.

Hopefully then, we can eliminate poetry, music, art, literature, and p.e. We have practical problems to solve!

I consider engaging and consuming forms of art akin to play. At first sight to spend lots of time playing might be seen as a waste of time but the fact that a child prevented from playing would fail to develop properly says something about its purpose. Basically I don't agree that those subjects are purposeless.

That's great that you consider this. Many do not consider poetry, music, art classes worthwhile because they consider those classes impractical.

I don't agree that learning how to abstractly apply rules and manipulate objects is useless. It's particularly important in programming, for instance.

I think the other poster was suggesting that students will perceive abstract symbolic manipulation as useless unless they're taught why it isn't useless, probably via application problems. Considering that the majority of students will never have a genuine interest in math for its own sake, I think that textbooks and curricula designed for public education should include high doses of application problems, with enough theoretical problems to interest the true mathematicians and verify the students' understanding of the abstract concepts.

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?

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.

You contradict yourself. As you said, one does not need to be skilled in combinatorial search, er I mean, algebraic manipulations to be able to solve problems; one only needs a solid conceptual grounding and understand the symbols and the rules for their manipulation. I wouldn't be surprised if most students who pass college calculus courses with good grades are entirely unable to recognize the situations where calculus is applicable.

Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations? I don't know. There are compelling arguments for both points of view.

When you say problem solving do you really mean so called practical applications? If so then I disagree. If you mean solving problems to understand the essence of mathematical ideas then I agree.

For instance, linear equations are the easiest equations to solve. All of them can be manipulated to yield the solution (or be reduced to showing all numbers are solutions or have no solutions). Linear equations are really 1st degree polynomials. What about second degree polynomials? After a clever bit of insight and some work we get that all 2nd degree polynomials can be solved. The quadratic equation is a solution to a problem in mathematics. It's useful as it leads to a greater understanding of mathematics.

I believe that this is what mathematics courses ought to focus on. Not on problems that involve throwing a ball in the air and predicting how long it takes to hit the ground.

"Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations?"

We've had calculators for decades now, and my kids are still learning how to do arithmetic.

But is it worthwhile or important?

I think it is.

If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean.

When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have some clue as to what the calculations are supposed to be doing, there should be some alarms going off if the numbers are not at least reasonable.

Or what if you just wrote a program that includes some arithmetic calculations, and need to verify that you are getting the right answers for your test data?

I remember another time at that same bank, where a developer did not know that multiplication had precedence over addition, which led to errors in a financial calculation.

I'm sure you could easily think up many more examples like this.

There was a time when it was important to know how to use interpolating polynomials. Rationalizing the denominator was only useful before calculator became ubiquitous. Being able to make fire was without use of matches or flint used to be essential.

There are plenty of ways to teach algorithms without using arithmetic. Maybe a knowledge of doing basic arithmetic by hand is essential to understanding higher math. I don't know. It's worth exploring though.

Let me share a frustrating experience I recently had.

I hired some dude off Craigslist to install my TV wall mount because I don't have the proper tools. I tried to explain where I wanted the TV: I want it to be centered in the wall, but then moved a few inches to the right.

He simply couldn't understand what I was saying. It blew my mind, I had to just measure it out myself and mark the spot. If he had a solid of understanding of basic high school algebra, he would have been able to formulate the problem in his head.

Done.

While I'd agree that students often approach word problems by basically memorizing mappings between a problem class and a small set of equations, I'm not sure that merely teaching them the mechanics of mathematics in the abstract is sufficient for imparting a thorough, practical understanding of the subject.

My experience with math-related courses was that I had the greatest difficulty generalizing the word-problem aspects into being able to reason about novel problems. A lot of the lectures and texts kind of paid lip service to the idea of problem solving in a general manner, but I think there's a lot of room for improvement in terms of capturing and teaching the underlying requisite types of mental activity, in a way that's explicitly procedural.

I agree with you. The problem I've encountered is that students who lack intellectual curiosity aren't motivated by the practical applications found in algebra textbooks. They aren't motivated by any sort of problem. So there is a reduction to the lowest denominator. That is, we give word problems that are reducible to mappings to a small set of equations.

I wrote a bit about this a few years ago, before my stroke:

http://stanrogers.blogspot.com/2005/03/girls-math-stuff.html

Warning: the site in general is a personal blog, so don't expect to see much else of interest to anyone.

addressing this statement in a vaccuum: "The focus of an algebra class ought to be in learning the language of algebra"

i think learning the relevance of its applications goes hand-in-hand and is necessary for a deeper understanding.

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.

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."

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.

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.

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.

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.

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.

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.

Your second paragraph contradicts itself. You say solving equations is useful. So, why not show your students the use?

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.

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.

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

I solved it numerically. x=4 or x=0

Yes it does: x=4,0

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.

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