Thursday, January 3, 2008

Approaching word problems

My students tend to give up in frustration as soon as they see a word problem, and so I increasingly avoid assigning such problems for homework and make sure we spend class time on them instead. There's a strategy for working with word problems that I read about somewhere - can't remember where, unfortunately - that involves paraphrasing the word problem within the constraint of an upper word limit, then paraphrasing the shorter version with an even tighter word limit, and so on. After a sufficient number of iterations, use of mathematical symbols becomes necessary to condense the information further, and so the word problem becomes translated into algebraic formalism.

I have not tried this method as stated, but it would be interesting to do that some time. The graphic organizer* I used a few weeks ago for systems of linear equations is inspired by this idea, however. There are little boxes** for each of the following:
  1. What exactly is the question? (What are you asked to find?)
  2. What are your variables?
  3. What information is given? List it or write a table.
  4. What equations can you write relating the quantities?
  5. Solve the equations.
  6. What is your answer?
Circulating among the groups, I found that the instruction to be very terse needed to be repeated quite a few times. That might be due to the fact that I have otherwise encouraged full sentences and elaboration... it must have been confusing that I was now insisting that the students be brief to the point of ignoring rules for decent writing. Many students initially started copying the entire word problem into the first box, groaning as they did so, instead of writing down the shortest sentence fragment possible that would convey just what they were going to find out.

It took a while for most students to realize that the variables they were to define were directly related to the questions stated in the previous box, that the variables basically were symbols for these quantities. Many tried to assign variable names to known quantities instead. I might try and rearrange the worksheet to visually reinforce the idea that the box containing the question and the box where variables are defined belong together.

In response to the prompt to list the given information, students were again inclined to be somewhat long-winded, and we'll need to work more on extracting the essential information and writing a table. Maybe insisting on a table is moving a little too fast, actually - once that is done we're practically in the next box already. As an intermediate step, maybe just listing the numbers in the problem together with a key word for what they quantify might be better.

The next part, writing down equations relating the known and unknown quantities, remains somewhat hard - but at least it's easier now that the students don't jump directly from skimming the problem to this step! I've given the students 2-3 out of 5 points on test items just for completing steps 1-3 above. That may sound like watering things down, but it really has resulted in more students even attempting the word problems - and once they have completed the first 3 steps they are much more likely to be able to complete the rest anyway.

The "what is your answer" box is for a sentence answering the question in the first box, and this answer has to make sense in the real-world context of the problem: units are included, and answers of the kind "4 remainder 2 buses" wouldn't work there, of course.

*inconveniently on my school computer just now.
**there's nothing like little boxes for prompting students to write something and not skip a step!

When I make up my own "real-world" problems they often involve pink dragons with purple wings and silvery scales. Some students roll their eyes then, but the dragon problems make me happy, and at any rate it would take a lot to make problems more boring than the ones in the textbook. Why are they all about ticket sales, long-distance phone calls, and cars? Yawn.


halshop said...

I use a very explicit rubric, which is a variation of what you're experimenting with, for teaching and grading "word problems." I look for:

*A full sentence defining any variables
*An equation or equations that will be used to solve the problem
*A solution, with appropriate work, of the equation or equations
*A full sentence answering the question

It takes students a while to get the hang of this rubric, but most find it useful once they do. In addition, it's made grading the problems a lot easier for me. It also has the advantage of giving partial credit to people who can at least read the problem and identify what the variable(s) should be.

H. said...

Fun that we're breaking down the word problems in so similar ways. My students seem to flounder the most on the first two of the points you listed. I'm trying to break that part down even further.

Talking of rubrics - some day we should exchange rubrics/rubric ideas for graphs!

Jackie said...

I do something similar with word problems for my pre-algebra students. Initially I asked guiding questions for each problem, breaking it into steps very similar to yours.

I think this has helped them to slow down and actually read the problems.