One important reason for studying mathematics is to nurture an aptitude for and a confidence in problem solving. To be good at problem solving, one must be able to sort through the unknown, make sense of it, persevere in it, and come away with a solution. This ability is not only critical to mathematics, but learning to solve problems successfully crosses over all aspects of life, in school and out.
George Polya’s book, How to Solve It was published in 1945. Polya, frequently referred to as “The Father of Problem Solving,” is best known for the following four basic principles of problem solving.
|George Polya’s 4 – Step Problem Solving Principles|
|1. First Principle: Understand the problem.|
|2. Second Principle: Devise a Plan.|
|3. Third Principle: Carry out the plan & Solve the problem.|
|4. Fourth Principle: Look back & Check for reasonableness & accuracy.|
These principles still stand today as reliable strategies for problem solving. Unfortunately, over time, the original significance inherent in each of Polya’s steps became less like real problem solving and more like a trick for quick calculating.
As a result unfortunately, an approach to solving word problems known as “find the magic words” was born. Instead of actually reading the problem, students just followed a scripted formula – underline numbers, find and circle magic words and Presto, put them together and problem solved. But is it?
Tracy Johnston Zager (@TracyZager), a highly respected math coach and author, shared the following two problems with her twitter followers.
First grade word problem
Lois has 16 counters.
She gets 10 more counters.
How many counters does Lois have in all?
Now here is another first grade word problem.
There are 20 pens.
10 pens are in each box.
How many boxes in all?
Several of the students she observed chose D. This answer proves more students are adopting the “magic words” method to problem solving. In fact, this is just a small sampling of what is now seen as routine in solving word problems; students don’t see a need to even read the words in context any longer. They simply circle the magic words; in this case, “in all,” underline the numbers; in this case 20 and 10, and then just “put them together”; in this case 20 + 10 = 30.
But 30 boxes of pencils is not a reasonable solution to this problem. The truth is, when students only use finding magic words as their method for understanding a word problem, the meaning behind the math-at-hand is missing.
Here’s another dilemma; as children get older and the problems get bigger, the magic-word approach rarely works.
When students mindlessly pull out and put together isolated words and numbers, the essence of the problem gets lost and the meaning within that context gets watered down to a few scripted steps, regardless of what the problem really means.
The acronym CUBES as a similar pick-pull-and-calculate method to problem solving is getting popular. Check out the premise behind the problem solving; circle, underline, calculate.
The problem with CUBES and similar strategies is that as students get older, the problems require more thinking, with layers that cannot be solved with a simple pull and put together method.
Without a real understanding of how to comprehend a word problem, students are not only led astray, but grow up thinking that problems are not worth investigating. In fact, students using these approaches can get the impression that problems are neat and tidy; when actually problems can get quite messy requiring logic, persistence, and creativity as trying alternative strategies can help clear the muddy waters a bit.
What’s a teacher to do?
One answer is for teachers to first spend valuable instructional time investigating and modeling what it means to actually understand a word problem.
Below you will find a sample anchor lesson for modelling how to think like a real mathematician when understanding word problems.
Anchor Lesson for what it looks like and sounds like to understand a word problem like a Real Mathematician
- Students will notice how a mathematician understands a word problem.
- Students will relate the skills a mathematician uses in understanding to that of a reader, scientist, historian, etc.
- Students will identify how using the strategies makes comprehension easier.
Setting the Stage:
- Put up the following text (or something just as difficult) on Smartboard.
(Students should not have a copy of this.)
We now know that we can find equilibria solutions to a differential equation by finding values of the population for which dP/dt=0. But we also observed that the two equilibria solutions for the logistic population model differ: for equilibrium, nearby trajectories (solutions to the differential equation) are “attracted” to the equilibrium; for the other equilibrium, nearby trajectories are “repelled.” We call the first case a stable equilibrium (also called a sink) and we call the second case an unstable equilibrium (also called a source).
- Challenge your students to switch roles with you. They are now the teacher and you are the student. They will be “grading” you in how they think you did as a reader.
- Make sure you have read this ahead of time several times. Really impress the students with your oral reading. Exaggerate your reading so it sounds like you really understand the piece.
- Have the students talk to a partner and come up with a grade on how you did as a reader.
- Ask them what grade they gave you and why they gave you that grade. They will more than likely give you an “A.” (I always get one “C+” from the student who is going for the laugh.)
- Here’s the powerful part – reveal that you have no idea what you read. Share what Cris Tovanni calls “fake reading.” That is reading without thinking. Reveal that if they asked you anything about what you read, you would not be able to answer it because you really have no idea what you read. Add that real reading means you are actively involved. It takes both text and thinking.
- Let the students know the same thing happens in mathematics. Sometimes we just keep moving along in our problem solving without thinking. When that happens, we really can’t get any real meaning out of it. Sometimes we even just circle magic words and underline numbers thinking this alone will help us make meaning of the problem. But, without any real connections to what we underline, we rush into strategies without thinking and that can sometimes lead us astray.
- Ask students how many of them have ever faked their way through a math problem – they didn’t think, they just kept moving along. Raise your own hand. We’ve all done it.
- Put the following problem (or any open-ended math prompt) on your Smartboard. Students should not have a copy of the problem. That can be distracting. You want them to pay attention to your Think Aloud.
A total of 8,000 runners started a long distance race. The results of the race are listed below.
- 3/15 of the runners finished the race in less than 4 hours.
- 0.6 of the runners finished the race in 4 or more hours.
- The rest of the runners did not finish the race.
A. Calculate the number of runners who finished the race in less than 4 hours. Show and explain all your work, even if you used a calculator.
B. Calculate the number of runners who did not finish the race. Show all your work. Explain why you did each step.
- Give out the following T-Chart for students to complete during your Think Aloud. Make a copy on chart paper to use as an anchor reference in future problem solving.
|How to think & UNDERSTAND a word problem like a REAL mathematician|
|What did you notice your teacher doing to understand a word problem like a real mathematician?||How did what you noticed help your teacher better understand the problem?|
3. Tell students that you are going to think aloud through this problem. Challenge them to notice and write anything you did to help you make meaning of the text and think like a real mathematician.
- Read the entire problem through one time (Do not read parts A or B yet). Then say, “Wow, that’s a lot of information. I think I better reread it slowly and take it apart.”
- Start over and read the first sentence. Stop and make a personal connection that helps you visualize the scenario, but get off track a little. (Use this step if you are noticing students losing focus during problem solving). Say something like, “I can make a connection to this. I went to a marathon once. I can picture that day. That was a really hot day. Oh wait, I’m getting off track. That connection is not helping me make meaning of the text. I’m going to have to ‘turn down the volume’ to that and get back to the problem. I better reread it from the beginning to get myself back on track.”
- Continue reading. Share that you first want to remove the numbers in your mind so you can connect to the story. Now replace the numbers accordingly:
- with “the total number of runners” for 8,000
- “one part” for 3/15
- “another part” for 0.6
- “a third part” for the words “the rest” in the third bullet.
- Identify there are three parts that make up the total number the runners in the race. (This will give your students a sense of the part/part/whole relationship).
- Reread the problem again, this time with the numbers included. Make a connection to 3/15 of 8,000. Say something like, “Hmmm, 3/15 of 8,000. Let me make sense of this before I move on. I know that 3/15 expressed as a friendlier fraction is 1/5. Write 1/5 in front of 3/15. Draw a visual. It can be the race (an open number line) showing what 1/5 would look like. I know in my brain that 1/5 of 8,000 is 1,600. So 1,600 runners finished the race in less than 4 hours. That’s not a lot of the runners. Point out the remaining part of the open number line (leaving four more fifths).
- Read aloud the next sentence, “0.6 of the runners finished the race in four or more hours.” Say a connection to your math class and say something like, “I remember when we were working with fractions and decimals and I found it easiest to work with either all fractions or all decimals.” Now depending on your students, you choose what you would like to do. If you want to focus on the part/part/whole relationship using fractions, say something like, “I think I’d like to continue my drawing using fractions, so I started with 1/5 and now I have 0.6 of the runners. If I think of 0.6 as a fraction, I am thinking of 6/10. Since I already am working with fifths, I’m going to rename 6/10 as 3/5.” Now draw what three more fifths would look like on your open number line. And say “If 1/5 is 1,600, then 3 times that or 3/5 would be 4,800 runners. That means most of the runners finished the race in 4 or more hours. (Purposely do not underline numbers and circle magic words. Just connect to context and meaning. Estimate and make numbers friendly and reasonable.)
- Notice that “If 1/5 finished in less than four hours, and 3/5 finished in four or more hours, that leaves 1/5 that did not finish. You know this because 1/5 + 3/5 + 1/5 = 1. (Draw as a fraction bar or show it on your open number line). That means the same amount of runners that finished in less than four hours also didn’t finish the race.
- Saying something like, “Wow, this is easy. I really understand the story in this math. I’m ready to begin reading my questions and solving my problem.”
Reflecting on Student Noticings:
- Stop and have partners share what they noticed you doing to think like a real mathematician and not just do fake math.
- Discuss with the whole class and write down what the students noticed as a T-chart on chart paper to use as a reference. Take ideas and reframe them to include good thinking strategies. Do they notice you visualizing, making connections, and looking for the mathematical patterns in the numbers; ‘turning down the volume” to connections that distract; making sense of the problem; rereading to clarify; rereading to get back on track; making a model; asking questions; using friendly numbers to help with mental math; staying in the context; and estimating?
- Challenge the students to discover how this all helped you make meaning of the text. Remember to share that is how mathematicians think. In fact, it is how scientists think, and historians, etc. Good thinking is good thinking. It crosses over content area. Students should now record their reflections on the right side of the T-Chart.
- Point out that you didn’t even begin answering the questions yet; you were making sense of the problem first.
- Do not solve the problem. It can be a problem they solve later on.
Students Give it a Go:
- Hand out copies of similar problems and have partners use active reading strategies to think through the problems out loud to each other. While one student thinks aloud, the other student notices what critical thinking habits they used to make meaning of the text.
- If you have time, they can solve the problem. Be careful not to rush into this. The focus here is how to think critically through problem solving.
One of the best math consultants in the country, Donna Boucher (@mathcoachcorner), and I collaborated on a list of Fix-up Tools for Mathematics. We have found this to be very useful in helping students recognize when their attention wavers and then have the tools they need to get back on track in their problem solving.
Feel free to download and share!