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When I tutor, I often see people who kind of know the stuff they cover in school at the moment and succeed at straight problems like:

Find the derivative of $f(x) = \frac 12 x^2$

But when it comes to text problems, they struggle to get started with the problem at all.

A hill that can be modeled with a parabola like $f(x) = \frac 12 x^2$. Somebody wants to walk up that hill but cannot cope with slopes more than 10%. To which point can that person walk?

My problem is that I am not really sure what their issue is. When I ask them what the derivative is, they can tell me. If I ask what the tangent is, they also know that. But they somehow cannot solve this problem.

When I solve such problems I start with "What do I know?", then "What does that mean in mathematical definitions?" and then try to think what the result has to look like. The steps in between are easily filled then.

How can I teach people this general, diffuse "art of problem solving"?

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Maybe their difficulties begin with "A hill that can be modeled with a parabola like $f(x)=\frac{1}{2}x^2$" –  Salech Alhasov May 11 '12 at 11:40
    
I think this question is extremely difficult to answer. Personally, I don't believe that there is a "one size fits all" approach to problem solving which will work for everyone. Your general method of: (1)what do I have? (2)what do I want? (3)how do I turn what I have into what I want? Is as good as one can do, in general (in my opinion). The problem in this particular question seems to be that the student is unsure of what the question is actually asking them to calculate. –  Sam Jones May 11 '12 at 11:43
    
That is just an example question that I made up to illustrate the problem. The „not sure what is asked“ seems to be the biggest issue, but do not have an idea how to help them understand questions better. –  queueoverflow May 11 '12 at 11:45
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Queueoverflow, I've upvoted this question because I find myself in the same situation. I've worked as a tutor for five years and the only conclusion I've come to is that I fail to interest the students in the subject and their brains won't start because of that. When they can't start doing an exercise, I'll often give them hints. They'll get it finally but when confronted with another similar problem, they'll have the same problems. –  user23211 May 11 '12 at 11:50
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A 10% slope means that when you go 100 meters (seen from above), you will gain 10 meters (10% of 100 meters) in height. That is a slope of 0.1. So you want to find the $x$ where $f'(x) = 0.1$. –  queueoverflow May 11 '12 at 16:07
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5 Answers 5

up vote 4 down vote accepted

For this particular case: We may understand that the definition of a 10% slope is mathematically obvious.

But, by couching it in these natural English language terms, it could give the impression to students that they require outside knowledge (whether or not they actually do need outside knowledge).

Students may be aware that "10% slope" has a particular meaning in English, having seen it on road signs, but probably have not directly associated that to a mathematical meaning. This could be one source of their confusion. When I see 10% used in natural English, I wouldn't immediately think about how that converts to a decimal, say.

More generally: Plenty of words have a slightly different meaning in natural English and mathematical language - including 'slope' itself. Students frequently (and correctly) understand that they have to handle their use mathematical language in a very careful way. By phrasing a question using natural language, this link with precision might be broken and confusing to students.

It may not be clear to students how to cross-interpret a question that is partly in mathematical language and partly in natural English. It doesn't help that questions written in natural language are sometimes badly phrased! Students might have developed a fear of these questions because their understanding of the reality of the situation doesn't match their own internal mathematical knowledge.

Indeed, there may be a disassociation between mathematical questions and actual reality. Question setters tend to insert mathematics into 'real' situations where they don't belong (or don't apply exactly correctly). In reality, as I understand, it would be unlikely and impractical to model a hill as a parabola. Apart from the apparent sign error, a hill is 3D and a slope can be overcome by tracking a zigzag ascent up diagonally. It is (perhaps) an unnatural and contrived question, phrased in natural language in a context which is unfamiliar to students. This could be a further source of confusion.

I think that interpreting (and writing) natural language questions is a difficult skill that has to be learned (and taught). I think all mathematicians come across situations where they have suddenly realised how a particular skill can be used in an unfamiliar or surprising situation - it's clear that this might happen more often to novice students.

To answer your question:

I think you could help students by teaching this specifically:

When I solve such problems I start with "What do I know?", then "What does that mean in mathematical definitions?"

Interpreting an English statement is not straightforward, and students would benefit from learning how to do this. Explaining how you do this, yourself, is a good idea!

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One idea is cognitive load. The student knows enough to establish any particular piece of the problem, but has trouble juggling everything when there are too many components. I think word problems have a higher cognitive load than drill problems because the reader has to read through the whole thing and keep several ideas in their head at once before beginning to synthesize a response. This is related to math anxiety where in a sense a student's own negative perceptions about math and their ability to do it start to drain cognitive energy. I agree with Krishnakanth that this problem has been exacerbated by the educational system in some places, which teaches math as a list of recipes to follow and doesn't emphasize the analytical, critical thinking aspects.

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This is a very interesting question. On the one hand, we're told that we need to make maths more interesting by relating it to real life as in your sample question, but on the other, doing so creates the heavy cognitive load that Jim Conant refers to. To try to answer the question rather than just discuss this topic, these guys suggest that showing worked examples can lessen the load and help students to learn how to solve similar problems. And since problem solving is mentioned, there's Polya or John Mason. (I forget the source now and can't track it down, but I recall reading a quote by a teacher of problem solving that his approach was to give his students problems and not tell them the answer!)

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I remember, from my own student years, feeling rather bored and irritated by some math problems that were formulated as "real world applications" ... as this "hill than can be modeled by a parabola". It's not that I was allergic to applied math (i'm an engineer), rather that in that context I felt that it was an unhelpful, artificial and somewhat patronizing "motivation". Just noise. –  leonbloy May 11 '12 at 17:32
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When I used to tutor Algebra at a community college one of the biggest challenges that students encountered was the abstract thinking needed to solve word problems. Just as you describe the students would have no trouble with the mechanics of the math but would stumble when they were asked to move between the abstractions present in the mathematics and the concrete realities of the problems. So what do do about it?

My approach was to explicitly try to to explain the concept of abstraction. That the math is a representation of a reality that we wish to explain. I would do this initially outside the concept of math. A typical approach might be:

  1. Ask the student to draw a tree. (this usually resulted in a typical lollypop type tree representation)

  2. Ask them to explain why their drawing represents a tree. (During this discussion I am trying to get them to identify the "treeness" that makes a tree a tree - e.g., leaves and a trunk)

  3. Once we have identified that a tree can be represented by leaves and a trunk, ask them to simplify and generalize (i.e., abstract) their drawing further (at the end of this we usually wind up with a circle on top of a rectangle that represents a abstract tree)

  4. Depending on how well the student is getting it, we may go through a few more examples (e.g., a car, a house, etc...) until it makes sense that reality can be abstracted by identifying essential components.

  5. After this I introduce math as a similar method of simplifying and representing the complexity of reality. Depending on the student's experience you could build off the tree example to show that the abstract tree could be represented by the diameter of the circle of "leaves" and the height of the trunk.

  6. Finally I try to show how by abstracting the reality with math we can use the tools of math to ask and answer questions.

The key in my experience is to get them to practice abstract thinking as a skill independent of the mechanics of solving the mathematical problems.

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The problem lies within the educational system, we're feeding the students bits and pieces and expect them to connect the dots... I remember in school I was derivatives as just that, derivatives... later on I just found out that a derivative signifies the change in a function based on the input changes... that's just an example... So, I'd suggest our methodology of educating students should change from providing formulas and theories to making them understand the significance of the formulae and the practical use of them... My opinion, making them realize how it can be used in real life, would be the best way to teach...

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I think feeding the students bits and pieces and expecting them to connect the dots is a wonderful way of teaching! –  user1729 May 11 '12 at 14:39
    
When I talk to people who are fascinated about some topic, I just need to give them pieces and they are all over it. Like I told another Vim enthusiast about the rectangular selection with <C-v>. I only mentioned it and he read in the manual how to use it. But I think this approach does not work, if the person is not interested and curious. –  queueoverflow May 11 '12 at 16:10
    
Although I understand your point, it's still a great leap to go from "signifies the change in a function based on the input changes" up to "realize how it can be used in real life". How would you make that connection? –  Ronald May 11 '12 at 16:53
    
So both the problem and the solution lie within the education system... It is good to know that students are not required to actively participate. –  Joshua Shane Liberman May 11 '12 at 18:11
    
The problem is that the self-motivated students do not require tutoring. So only the ones that are not interested end up with a tutor :-/ –  queueoverflow May 11 '12 at 18:44
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