Let’s pretend you’re back in math class. You might need to figure out answers to the following questions: If your doctor gave you three pills and told you to take one every half hour, how long would they last you? In a certain family, there are seven sisters and each sister has one brother. How many siblings are there? Or, how much do you have if you divide 30 by one half?
Problem-solving is not only one of the most important components of the study of mathematics; it permeates all aspects of life, including the professional world. Problem-solving teaches students to be critical out-of-the-box thinkers, hone organizational skills, and build a rational thought process required for making logical decisions. Students who are problem-solvers will someday pursue technical careers and become the researchers, inventors, designers, and engineers of the future.
There’s only one problem with problem-solving: When we look at the math section of a standardized test like the PARCC (Partnership for Assessment of Readiness for College and Careers), particularly at the middle school level—where proficiency rates of students in grades 6-8 who met or exceed expectations in math fell under 35 percent in 2015-16—we can see that it is the word-problem portion that often trips students up.
Word problems tend to be complicated in part because of their descriptive language. Students often don’t understand what exactly they’re being asked, especially when the problem includes abstract concepts. Other issues arise when students lack the fundamentals of math and cannot formulate a plan for solving or separate an equation’s steps.
The Problem-Solving Plan
As a mathematics and computer science teacher with two decades of experience, I know how it feels to invest yourself into a class for an entire school year and not get the results you hoped for. It can be discouraging to spend hours preparing and teaching bell-to-bell only to have little or no improvement. Likewise, students themselves get frustrated as their confidence erodes.
I’ve proctored standardized tests and witnessed students clicking the “skip” button or typing the letters “idk” (short for I don’t know) as soon as they were presented with a multistep word problem. I’ve seen students give up after only 15 minutes on a two-hour section. We’d covered all the skills they needed to solve problems, but during testing, I could do nothing but look on in silent despondency.
In the classroom, too, I recall spending weeks isolating a single mathematical topic and approaching it from every possible angle, only to present a question in word-problem form and have students respond with blank stares as the crickets chirped in the back row. The class just could not seem to piece together the necessary steps to think on their feet. I began to wonder: Maybe the problem was not with the word problems themselves, but with the difficulty in teaching proper problem-solving techniques.
Problem-solving is not only one of the most important components of the study of mathematics; it permeates all aspects of life."
If we trace mathematical roots back to the earlier grade levels, we see that key words often help younger students create a problem-solving plan. For example, we might use the word “more” as a clue for when to add: “Three kittens curled up on a blanket. Four more kittens crawl onto the blanket. How many kittens are on the blanket now?” Similarly, the word “fewer” indicates subtraction: “Marcy has six fewer cats than Nancy. Marcy has twelve cats. How many does Nancy have?” The issues reoccur when there is a lack of key words to follow: “Some kittens were sleeping on the blanket. Seven woke up to have a snack. Now there are three on the blanket. How many were on the blanket to start?”
These kinds of basics might be worth refreshing before diving into other tips and tricks. Each problem is unique and teachers cannot provide a single, overarching algorithm to solve them all. So how do we teach students to read a problem in English and translate it into math?
Practice Makes Perfect
My success has been to expose students to daily examples of process in more intentional ways. Provide students with guided experience by practicing a variety of problems on the web, from math contests, and from previously published standardized tests. Sort the problems into levels of difficulty or themes, and start with a one-step problem before moving to those with two or three steps. As students are gradually exposed to open-ended word problems requiring multiple processes, teachers should provide worked-out examples for students to study. Some may even contain intentional mistakes for students to find and resolve. To begin, the basic principles of mathematician George Pólya are useful:
1. Read the problem
2. Understand what is being asked
3. Make a plan to solve the most difficult step
4. Execute the plan
5. Check your solution for reasonableness
You should also never underestimate the computer as a powerful instructional tool. Students must learn to translate words into a series of steps toward a solution by applying informational cues, identifying variables, recognizing the unknowns in expressions, and explaining their reasoning. One way to do this is by thinking about how the word problems might connect to problems in their daily lives. By creating realistic content based on real-world problems, computers tend to hold the interest of students longer than lecture. Computers can generate questions tailored to the needs and capabilities of individual students and provide immediate feedback and correction before students make numerous errors of the same type.
The representation of mathematical concepts in illustrations can also foster deeper understanding. Use drawings, figures, or symbols to show the visual connection between the data and the unknown. Make a conceptual map and practice outlines of the necessary steps. Students should also have the opportunity to explain their own problem-solving strategies to others in the classroom as they develop the strategies that work best for them.
These steps will give students the tools to figure out that the pills the doctor gave would last one hour (you take one immediately, the second pill in 30 minutes, and the last pill 30 minutes after that); that the family has eight siblings (the seven sisters share the same brother); and 60 is what you get when you divide 30 in half.