For many learners, mathematical word problems are the most dreaded type of homework. In an informal poll of college math students, I recently heard the following opinions about word problems:
- “I hate them! They’re so confusing!”
- “Why can’t they just say what they want me to do?”
- “I usually just find the numbers and then do whatever math we were doing in the chapter.”
One of the main methods that students use for solving word problems is direct translation – skimming the problem for key words and numbers, translating those directly into their best guess at a mathematical statement, and then solving. The guess may be based on what math operations were most recently covered in class, and what associations students have for the key words.
Does this sound familiar? Is it an effective strategy? Let’s see what this strategy might look like applied outside of the math classroom:
In real-life situations, we need to take time to size-up a problem and consider our plan of attack. Research shows that we have been trained and conditioned to look at math problems differently from other types of tasks. We approach the problems differently, and we even read a word problem differently than we would read a story.
Some of this has to do with the terminology of mathematics, and also with the artificial nature of many word problems. As soon as we see these familiar structures, we set our mind into math-problem-mode, and forget all of the other strategies and ways of thinking that we could draw on.
Applying SRL to problem solving
An approach that shows greater success than the direct translation method described above is a meaning-based approach. This strategy is based on SRL (self-regulated learning) principles, and can be used for both classroom and real-life problem solving.
If you’re not yet familiar with SRL, read through some of the other entries on this blog. The main phases of self-regulated learning include:
- Task recognition – figuring out what you’ve really been asked to do.
- Goal-setting and planning – deciding what you want to achieve and how you might get there.
- Task enactment – putting your plan into action.
- Assessment and adaptation – checking your status to see if you’ve met your goal, and adapting as necessary.
Understand the problem and determine your goal
This is most important stage of the process, and the most neglected. Instead of trying to dive right in to calculations, take some time to orient yourself.
- Read the problem multiple times. Skim first to get a general idea of what’s going on, then read again to reinforce. Start building a mental model of the scenario.
- Extract and organize the important information into your own words/diagrams/charts. Read through slowly. When you come to a piece of information, stop and process it. Write it down in a way that you understand. Make sure to identify the question that’s being asked – this is your goal.
- Visualize the information. Imagine the actions in your mind and consider relationships, then put them down on paper. Draw pictures or diagrams, make tables or draw arrows between items to make a visual model that matches your mental model. Assess your progress by rereading the question to see if your image matches what is there.
In the following video, I demonstrate my own task understanding process with an example word problem. Don’t worry too much about the specific math that I’m doing; try to focus on the SRL process as I work through it:
Make a plan and take action
Only now is it time to open your mathematical toolbox.
- Analyse what you have, what you need, and what external information might apply. Define variables for the missing elements. Try to work out a series of steps to get from your starting point to your goal.
- Calculate.Solve the equations, put in your values and find an answer.
Assess and monitor
You’ll want to assess your progress the whole way through. The final check will confirm that you’ve completed the problem.
- Check your answer. Are you answering the question, or do you still have work to do? Does your answer make sense in the context of the question?
In the example problem, I described what I was doing and thinking at every stage. This wasn’t just for the sake of the video. Self-talk – either out loud or internally – is a valuable tool for working through any process. One way of structuring self-talk is the “say-ask-check” method.
- Say: What I am doing
- Ask: Does this make sense? Why am I doing it this way? What should I do next?
- Check: Was my strategy succesful?
Testing as you work
It’s easy to get lost in the abstract nature of equations and variables, and forget what they represent. While you’re building equations to solve your problem, stop and check to see if they make sense. Plug some real numbers in to see if the answers you get match what your mental model would give. I did this in the video example when I tried using 5 ft for the length of the ladder, and thought about what number I would expect for the height.
Another strategy I used was to break the problem down into smaller, manageable pieces. Trying to process a statement like: “Three less than twice a number is fifteen” can be overwhelming and confusing. The trick is to look at the pieces of information one at a time. If you want to, you can even physically cover up the other parts with your thumb or a piece of paper. Let’s break this example down:
“Three less than […]”
One thing is three less than another thing. Can you think of a number that is three less than another number? Five, for example, is three less than eight. If we subtract three from eight, we get five, so “three less then” means we’ll be taking some mysterious quantity and subtracting three.
- [something] – 3
“Three less than twice a number […]”
Ah ha, now we know a bit more about the mysterious “something” of the last step. We know that it is “twice a number”. Twice means two times, so two times “a number” is the thing we’re looking for. We don’t know what “a number” is, but we still want to talk about it, so this is the perfect time to introduce a variable. Let’s let x = “a number”, then the “something” we were missing last step can be represented as: 2x.
- 2x – 3
“Three less than twice a number is fifteen.”
Oh, so that thing we were building before is the same as fifteen. If two things are the same as each other, that means they are equal.
- 2x – 3 = 15
By looking at the sentence one piece at a time, we were able to focus on individual details, and slowly build up our own representation of what the question described. We could check our representation against real numbers at each stage to see if it made sense.
Try it yourself
Now that you’ve seen the steps for approaching a problem, you can try applying them. If you’re taking a math class, choose a word problem from your current chapter, or use the sample problem below. Print this template (pdf), or add headings to your own paper to ensure that you are completing all of the steps. I’ve filled in a template page with the ladder problem (pdf) from the video to give you an idea of how to use it.
Sample problem: Empress taxi charges $3.30 per trip, plus $1.92 for each kilometer travelled. If Susan has $16 to spend, how far can she travel?
Once you’ve tried the problem, look back over your work.
- Did you spend time visualizing and summarizing the information in the problem before starting to build equations?
- Did you identify the goal or question being asked?
- Did you clearly define your variables before using them?
- Did you check that your equations made sense by putting in numbers?
- After calculating, did you check that you had answered the question?
- Did you check to see if your answer was a reasonable value?
If you answered yes to these questions, then you’re on your way to math problem-solving with SRL!
In case you’re interested, I’ve uploaded my worksheet for the taxi problem (pdf). Don’t worry if yours doesn’t look the same as mine – in fact, I’d be surprised if it did! Everyone processes information in their own way. Though there is only one correct numerical answer to a problem, there are all sorts of perfectly valid ways of getting there. Even if you didn’t get to the right answer, it’s the process that’s important right now. If you understand what you’re trying to do, can explain what happened in each step, and get to your goal, then you’re doing it right! It’s ok if someone else has a different approach.
Mayer, R. E. (1998). Cognitive, metacognitive, and motivational aspects of problem solving. Instructional Science, 49-63.
Montague, M., Krawec, J., & Sweeney, C. (2008). Promoting Self-Talk to Improve Middle School Students’ Mathematical Problem Solving. Perspectives on Language and Literacy, 13–17.
Pape, S. J., & Smith, C. (2002). Self-regulating mathematics skills. Theory into Practice, 41(2), 93–101.
Wiest, L. (2003). Comprehension of mathematical text. Philosophy of mathematics education journal, 17.