Monday, February 21, 2011

Solving math word problems

A teacher asked me a question re: strategies for solving word problems. This is what came to mind:

1. Consider the popular 4-step problem-solving strategy (adapted from Michael Polayi, appears in the MathThematics text; there is a good description on the this page) as an overall approach. The basic steps are:
  1. Understand the problem
  2. Make a plan (there are lists of basic strategies like "guess and check", "use a formula", "draw a picture"; see the Scholastic link above for a longer list).
  3. Carry out the plan
  4. Look back (review, reflect, evaluate)
2. As part of step 1, understanding the problem, Ms. Schumacher at our school has students mark the problem text in different ways to isolate the key parts of the problem. She has her students underline the portion of the problem that says what to do, or what the question is asking for. They circle numbers or number words that are important to the problem. And they draw boxes around the actions they need to perform on the numbers (see next point). Reading math problems is a kind of literacy, and this deconstruction of the text can help comprehend the problem.

3. Again related to step 1: Since reading math problems is a kind of literacy, students need to understand the vocabulary of math. There are addition words, subtraction words, and so on for each basic math operation. For example, addition words are words like "sum", "plus", "add", "increased by". Subtraction words are words like "less", "difference", "decreased by". Being able to recognize key math words, and what operations they are calling for (as well as what order to put the operands in) is important. Purplemath has a table of "key words". Here is a link to a table students and I put together last year. (It is also important to be able to determine what is important information for solving the problem, and what is extraneous or irrelevant information).

4. Students have different skill levels when it comes to what strategies they might use, but it might be useful to reinforce that a strategy that works is a good strategy, the main trade-off being effort to arrive at a solution. Guess-and-check (or using "brute force") is okay if you have the time and patience; algebraic methods are very flexible and powerful but may be prone to error if the student isn't comfortable with them. It is also important I think that students recognize that a combination of strategies is frequently used (e.g., I find it useful to first draw a picture or diagram, or make a table, to understand the problem before thinking of an algebraic solution).

5. The last step is very important. Is the answer reasonable? Does it actually solve the problem? Was there an easier way to tackle it? Will it work for other problems?

6. Regarding teaching strategies:
  • Whatever can be done to help students de-code the problem can help. I am thinking of comprehension strategies used with other kinds of texts.
  • A think-aloud of how the teacher might approach the problem could model the problem-solving strategy for the students.
  • After students have highlighted the key parts of the problem (what the problem is asking for, the key numbers, the operations), modeling how to put the pieces together can help.
  • A KWL chart might structure the process, especially if students are familiar with the format. The Know part, "what do we know already?" = the numbers and operations that the students have highlighted. The Want to know part = what the problem is asking for. The Learned part is the problem solution.
  • Having students make up their own word problems for other students might help them understand the structure or genre of math word problems.
  • Writing in a math journal about what they did to solve a problem might help gel the process for them.
  • Re: thinking about general problem-solving strategies, posing practical problems for the students might build their confidence in their problem-solving abilities and connect with strategies they already use, but don't think of them as such ("activating prior knowledge"). For example, a million dollars is waiting for you at the Lincoln Park Zoo -- how are you going to get it?
  • Having a context for understanding a word problem is important. If the problem deals with things the students don't know anything about, it is hard to understand what the problem is asking. This starts to slide into the area of math and social justice -- presenting math in terms that connect in deep ways with the lives of the students. Students need to be engaged enough with the material to want to solve the problem.
  • There are lots of other words that have math meanings, separate from different ways of referring to the basic operations and relations. See the ISBE Math Frameworks Glossary for words and definitions that are important to know when approaching word problems.
  • Being able to talk to students about how they solved or didn't solve a problem can be very useful for understanding their thinking and the source of possible misunderstandings.
This just touches on some strategies. Any comments are welcome. Literacy Strategies for Improving Mathematics Instruction, by Joan M. Kenney et al. (an ASCD book) has insights into different aspects of math and literacy.

Since most real-world problems that require math present themselves as word problems, I think it is important that students are fluent in translating them into math terms, and using math tools for finding solutions. It is one of the important ways that math becomes meaningful.