Study Suggests Math Teachers Scrap Balls and Slices

<p>I have no comments, but figured some of you would.</p>

One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other?</p>

<p>Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn.</p>

<p>That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train.


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<p>Yay, my first posting of an article.</p>

<p>I will use apples and oranges. The whole discussion is apples and oranges as far as I’m concerned. I always thought the point of teaching students to do word problems is not to make it easier for them to understand the underlying abstract equations, but to teach them how to translate physical or real world situations into equations. First they learn how to work the equation symbolically, then how to translate a concrete example into an equation. It's an additional skill, not a learning tool. Although I suppose for some students it might help cement the idea of the equations, but not for most. At least that’s how I remember learning to do word problems.</p>

<p>Some problems are particularly counterintuitive to me. I can't remember examples specifically, but for example the types of problems where speed or rate of operation change midway through a process and the variables are in the denominator of fractions. Ditto boats going into a headwind. Maybe some USAMO type kid can see the application of all this stuff and learn better that way, but for a more "workmanlike" person such as myself, I would just go about translating the words into an equation. Trying to learn by seeing the big "application" picture would just confuse me.</p>

<p>I can see the argument that maybe this is something better taught in a physics class or something like that (balls rolling down inclines or cannon-firing ballistics), but there is no way getting around learning to apply the equations at some point unless you want to be some sort of pure mathematician.</p>

<p>^^Hmm - I always thought the exact opposite - to translate equations to use in real world situations.</p>

I don't understand. I've never seen a problem where they give you an equation and ask you to come up with a word problem that fits the equation. Care to share an example?</p>

<p>Because I think we're talking a semantic difference here.</p>

<p>Most generally what I have seen is you start with a physical system or circumstance and try to describe it with an equation, or system of equations. In many cases you can't just pull out a standard equation without making a bunch of assumptions, and in a lot of situations you end up with complicated partial DEs or non-linearities and end up having a computer solve it numerically.</p>

<p>^I used to have to create word problems to fit an equation ALL THE TIME in middle school and high school, but we had a very unusual math program when I was growing up that the school has since abandoned. I had to do it the other way around, too.</p>

<p>Word problems definitely do help me understand the abstract math concepts. In fact, that is one of the only ways I can grasp them. The idea though is that once I've come to understand it, I don't need a word problem anymore to understand. It's just a sort of stepping stone I guess.</p>

Okay, I’m curious. I have never heard of this. I can see this for very simple equations (Three birds are sitting on a fence and 2 fly away).</p>

<p>But say for a quadratic equation – what word problems would you develop from them and how would it help you learn to solve them?</p>

<p>When I learned about quadratic equations we learned the basic form, then how to solve them by factoring, completing the square, or the quadratic formula. I cannot imagine a word problem that would have helped me learn to solve them. After that we learned they were applicable to physical systems (eg kinematics).</p>

<p>Edit: I will modify my opinion slightly. For some equations, like quadratics, logs, or exponentials, I can see how, after introducing students to the form a teacher might say - "This is applicable in these physical cases- blah blah." It is a good time to introduce analogs too. But I still don't think that's much help in learning the mathematical maipulations for a vast majority of students and equations, and to get a nexus between the word problem and the math equation, I think you generally have to start with the physical system and attempt to describe it mathematically in most instances. </p>

<p>So a lot of the disagreement may be semantic.</p>

<p>regardless of what's the best way for college students to learn math, I don't think young children can learn abstract concepts and equations without FIRST learning and mastering more general and concrete concepts, that's just not how their brains work... although I suspect Baby Einstein will be excited to hear about this study ;)</p>

<p>I dug up the article (The Advantage of Abstract Examples in Learning Math, Science 25 April 2008: Vol. 320. no. 5875, pp. 454 - 455). Sweet monkey fritters but that is some dense jargon in there! The example used for testing has nothing to do with trains. A word problem version would be "If N people order a strawberry smoothie, and each smoothie calls for 2/3 cup of orange juice, write an expression for the number of full cups of juice you would need, and the fractional remainder you would add to make N smoothies." Then the example shows a graphic of things like two two-thirds cups resulting in a one-third cup.</p>

<p>I've completely confused you now, haven't I? :) That's because this is the old-fashioned complex version of the game. The purely symbolic version shows two diamond shapes producing a circle. </p>

<p>For what it's worth, I could immediately pick up the "rules of the game" from the measuring cup graphics. That might be because I have a few decades of cooking and baking behind me. Or maybe my brain works oddly...erm, differently.</p>

<p>This reminds me of the eternal whole language/phonetics reading debate. Each approach works better for some kids, and many kids end up using elements from both methodologies.</p>

This reminds me of the eternal whole language/phonetics reading debate. Each approach works better for some kids, and many kids end up using elements from both methodologies.


<p>This is (part of) the eternal math version of the same debate. And it's been waged off and on for well over a hundred years. And rather violently (at least in the name-calling sense) for the last forty or fifty years or so. And particularly violently (in the name-calling sense and in the "spread unfounded rumors about your oponent" sense) in the last twenty years or so.</p>

<p>Some kids are content with studying numbers and equations without any context and come to both love the subject and are able to apply it when necessary without any real struggle. [I was one of these kids myself.] "Stupid" applications can turn these kids off---my own response to the trains "why should I care?"---if "parallel" tracks was even mentioned in the problem. And if the problem didn't mention "parallel" tracks, I was the smartypants who would ask "Do you mean * when will the trains collide?*" But the applied problems in calculus were the only ones (other than Riemann Sums) that really caught my fancy 'cause they were the only ones that required me to think through the situation.</p>

<p>But other kids really dislike studying numbers and equations without any context. They often are the ones who come to dislike math intensely because they see math as a bunch of made-up rules that make no sense and have no connection to real life. And this group often has real trouble applying the math they know in other classes ...</p>

<p>Well, there may be some merit in learning math in a totally abstract numbers-and-symbols way, if that's all you're ever going to do with math.</p>

<p>But I remember two things:</p>

<p>The first is my son, as a second grader, complaining that everything they did in school was "stupid." He said, "Mom, we spend all this time learning to do subtraction. When am I ever going to use subtraction?"</p>

<p>Obviously for him, real-world examples of how to use subtraction would have been a motivating factor.</p>

<p>The second example involves my daughter, now a college economics major who works as an undergraduate teaching assistant for a finance course. She has told me that many of the students whom she tutors during office hours struggle with the course material because they don't have a good grasp of basic algebra, including word problems. "Finance is basically all word problems," she says. </p>

<p>For my daughter's students, prior experience in doing algebra word problems would be an aid to success in a very practical, career-oriented course.</p>


<p>Your d is facing the major complaint that math department's have been hearing from their client departments off and on for a hundred years: Why can't students * apply ** the mathematics they learned in Course X in the context of our courses?*</p>

<p>It's a serious and significant question. It's driven a whole lot of educational research in the last 50 years. It's not been solved by anybody yet. In my humble opinion it won't ever be completely solved.</p>

<p>But---it is clear that for many students, learning the rules abstractly is not enough: They compartmentalize that information into a box in their brain called "Useless math facts" and never think to access it when doing a word problem in a finance class or a chemistry class or a biology class ...</p>

<p>And it's also clear that many students---regardless of the pedagogical approach used in their K-12 courses---never seem to master enough of the routine symbol manipulation to be comfortable with it. And its no surprise that these students have trouble applying the mathematical knowledge that they don't really have.</p>

<p>My own humble opinion based on my 30 years of teaching college level math? Somehow math teachers and math professors need to work on teaching our students how to be flexible in thinking about mathematics. They need to have a whole bunch of tools, but they also need to see the connections between those tools so that they don't try to mindlessly memorize numerous special cases as a way of simply getting through the current course. But I don't have a magic solution on how to actually do this for all students at all levels in all courses :(</p>