Word Problems

harpazo

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Jan 31, 2013
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Most students fear word problems. I myself have been struggling with word problems for most of my life. There are gifted people that do not have a math degree but can solve a word problem by simply reasoning their way to the right answer. I am not one of those individuals.

Pretend you are taking a math test and come across a word problem that you've never seen before. What would you do to find the right answer? Take me through the steps, the process leading to the right answer. Of course, you need a problem to work with.

Here is one:

Two numbers add up to 72. One number is twice the other. Find the numbers.

Note: I know how to solve this problem. However, I would like to see different ways of solving this problem STEP BY STEP. Keep in mind that you have never seen this problem before. Explain.
 

MarkFL

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If I were taking a timed test, I would immediately write:

\(\displaystyle x+2x=72\implies x=24\implies 2x=48\)

Obviously, \(x\) is the smaller number, and \(2x\) is the larger.
 

harpazo

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If I were taking a timed test, I would immediately write:

\(\displaystyle x+2x=72\implies x=24\implies 2x=48\)

Obviously, \(x\) is the smaller number, and \(2x\) is the larger.
I know what you did here but can you explain the steps, the process PRETENDING to never have seen this problem before?

My Breakdown:

Two numbers add up to 72.

Let a = one number
Let b = the other number

So, a + b = 72.

One number is twice the other.

a = 2b

I see two equations.

a + b = 72...Equation A
a = 2b...Equation B

Find the numbers.

I would then proceed to use the substitution method.

a + b = 72

2b + b = 72

3b = 72

b = 72/3

b = 24

To find a, I can plug b = 24 in EITHER equation A or B.

What do you say?
 

JeffM

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This is exactly how I teach those I tutor to approach word problems in algebra, but it is a very basic technique that applies only to certain kinds of problems in algebra. It is not going to work for a problem in differential equations or constrained optimization.

Here is a more general approach. What am I trying to do and is there a mathematical discipline that will help me do it? If there is a mathematical technique that I think will work, I then ask myself what is already known and what must be discovered. Next I begin translating from English into the appropriate mathematical notation.

Now what you did follows that general approach.

What do I want to do? In your example, you want to find numbers that satisfy given conditions.

Is there a mathematical discipline that answers many such questions? In your example, the answer is algebra.

With respect to algebra problems, I have a standard technique: I assign a symbol to each unknown, and then I look for equations that equal the number of unknowns. And that is exactly what you did.

But if I was trying to solve a problem in optimizing a differential function subject to constraints, I'd use a different standard technique.

Ultimately, solving word problems cannot fully be reduced to rule. It involves "seeing" that certain mathematical techniques may apply and then having a systematic way to translate the problem into the form required by the technique. Once you get enough experience, you may see short cuts for certain kinds of problem.
 

harpazo

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This is exactly how I teach those I tutor to approach word problems in algebra, but it is a very basic technique that applies only to certain kinds of problems in algebra. It is not going to work for a problem in differential equations or constrained optimization.

Here is a more general approach. What am I trying to do and is there a mathematical discipline that will help me do it? If there is a mathematical technique that I think will work, I then ask myself what is already known and what must be discovered. Next I begin translating from English into the appropriate mathematical notation.

Now what you did follows that general approach.

What do I want to do? In your example, you want to find numbers that satisfy given conditions.

Is there a mathematical discipline that answers many such questions? In your example, the answer is algebra.

With respect to algebra problems, I have a standard technique: I assign a symbol to each unknown, and then I look for equations that equal the number of unknowns. And that is exactly what you did.

But if I was trying to solve a problem in optimizing a differential function subject to constraints, I'd use a different standard technique.

Ultimately, solving word problems cannot fully be reduced to rule. It involves "seeing" that certain mathematical techniques may apply and then having a systematic way to translate the problem into the form required by the technique. Once you get enough experience, you may see short cuts for certain kinds of problem.
What an informative reply. Thank you, JeffM.
 

Otis

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Apr 22, 2015
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… Pretend you are taking a math test … a word problem that you've never seen before … What would you do …
I'm not sure what you're thinking there. For example, you can't possibly expect to have seen every variation of a distance/rate/time situation. On the other hand, if you're talking about a problem involving math that you've never seen before, then the first thing I would do is wonder, "Why is this question on the test?" I would skip questions like that, and, if I had time after finishing what I could, I would then use all remaining time to show some effort on the rest.

If you're talking about a problem involving a topic or methodology taught in class, then I would use knowledge and experience gained from instruction and lots of practice to recognize it, followed by doing what I'd been taught.

If you're talking about not recognizing something on a test that you're supposed to know, then I would say, "Darn it", let the chips fall where they may, and revisit the material afterwards.

Most students fear word problems …
I don't believe that.

\(\;\)
 

petercole

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Aug 21, 2019
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Most students fear word problems. I myself have been struggling with word problems for most of my life. There are gifted people that do not have a math degree but can solve a word problem by simply reasoning their way to the right answer. I am not one of those individuals.

Pretend you are taking a math test and come across a word problem that you've never seen before. What would you do to find the right answer? Take me through the steps, the process leading to the right answer. Of course, you need a problem to work with.
Exactly I saw many students they run away from math problems even I saw that math is the only subject in which student can get full marks for any question as if every step and problem is correct there is no chance to deduct the marks.
 

Otis

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Exactly I saw many students they run away from math problems …
Hi Peter. Did you run away from harpazo's pretend problem? ;)

For me, it's a question that involves three equal pieces which add to make 72. That is, the smaller of the two numbers asked for must be one-third of 72.

I used the multiplication table (6×12=72) and this property

\(\displaystyle a \cdot b = \frac{a}{2} \cdot \frac{2b}{1}\)

to mentally calculate 72/3.

If anyone interested does not understand my approach, please ask! :)

\(\;\)
 

JeffM

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Most students fear word problems.
'I don't believe that.
I would believe that if the statement were re-worded to say "A great many high-school students feelings about word problems range from dislike through hatred and fear."

But the statement begs the question of why is that so. And it is that question that is important. If you have difficulty doing the highly stylized and unnaturally simplified problems called "word problems," you will never be able to apply mathematics to the actual problems to which mathematics can give answers or help to give answers.

My personal feeling is that the inability to master word problems comes from a defect in teaching. We do not stress, at least not enough, that mathematics is a language the purpose of which is to find an answer from what is already known. Now I do not believe that there is a universal technique to translating a problem described in a natural language into the language of mathematics. Indeed, many problems cannot be translated at all, such as what should I get my wife for her 75th birthday. So perhaps the first step outside a classroom is to ask yourself, "Might I be able to translate this problem into a mathematical problem that I know how to solve?' So, I might suspect that a problem could be solved by measure theory. I have not a clue about measure theory. When I hear that some theory is based an atomless measure space, I know that I must (1) give up, (2) take a course, or (3) find an expert.

Now this of course is not an issue for students in a class or reading a text. Their word problems are specifically designed to let them exercise techniques that they have already been exposed to. So, given that I suspect math can help me, my next question is whether I can translate what I do know into mathematical language. If the answer is yes, then I do that and write it all down.

students should be strongly encouraged to do this. I might go so far to say that it ought to be a requirement on homework for high school students. At least it would stop them from saying "I don't even know where to start."

My next question is whether I can translate what I am looking for into mathematical or quasi-mathematical language. If the answer is yes, then I write that down. In an algebra class, that would usually be something like

\(\displaystyle x = \text {a numeral}\) or \(\displaystyle x = \text { an algebraic expression.}\)

Now we no longer have a word problem. We are now concerned only about what mathematical techniques are in our tool box let us go from what we know to what we want.

I have worked with students who can do mechanics til the cows come home, but (to retain my bovine metaphor) are buffaloed by translation. That indicates to me a serious deficiency in how the subject is taught.
 

Otis

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Apr 22, 2015
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… My personal feeling is that the inability to master word problems comes from a defect in teaching …
I think most of us would agree that instruction is a major reason, Jeff. (Too much public education in the USA is clearly problematic.) Yet, my feeling is that students lack also motivation to read and practice sufficiently outside of class. Either they're not interested in the subject (or school in general) or they don't get decent mentoring and encouragement at home, or both.

Poor instruction notwithstanding, for those students who are competent at solving exercises once they've been provided expressions and equations, what else is preventing them from translating word problems into math statements if not a lack of exposure/practice?

Can we agree that, in addition to serious issues with instruction, the situation is borne also by societal failings and basic human nature?

🤔
 

JeffM

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My academic training was in history. I am consequently highly allergic to monocausal explanations of social phenomena: 99% of the time such explanations are simply wrong. Moreover, it is frequently difficult to assign qualitative, let alone quantitative, weights to causes. So arguments about what is the most important cause can seldom be resolved and distract attention from what can be done to improve things..

There are obviously cultural and genetic causes that account for differences in the educational performance of different individuals, and perhaps they far outweigh deficiencies in the educational process itself. But we can do nothing whatsoever to change the genes of individuals, and culture is susceptible to only very slow change. Moreover, to the extent that parents suffering from poor education themselves do not know how to facilitate their children's education effectively and do not know what they should demand from the public schools, we are essentially blaming the victims of past poor education for current poor education.

The things that we could do are (1) start improving the curriculum immediately, (2) recognize that permitting ineffective teachers to stay in classrooms is to sanction politically and to reward economically the systematic abuse of children, and (3) alter the ridiculous pay structure that rewards people for administering rather than teaching, discourages young adults with high economic potential from considering public teaching as a career, and rewards seniority rather than objective metrics of performance.

Perhaps most parents are now and always will be ignorant or oblivious, and almost certainly children will always be easily distracted and bored. Those obstacles may be forever unrectifiable, but what is not unrectifiable are the obvious deficiencies in US education. I prefer to focus on what can be done before we raise another generation of ignorant and oblivious parents.

A typical algebra text will have 20 or 30 problems that are pure mechanical drills and 6 or 8 word problems. The drills are admittedly easy for teachers to grade and do not require much paper and ink to present. I would encourage texts to have far more (and more diverse) word problems than is currently the case. Furthermore, I would be asking that what is required for word problems is setting the problem up for mathematical solution rather doing the mechanical work of solving it. The mechanical work is boring and error-prone and so frustrating. We already have more than enough drills in mechanics,
 
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