If Ms. Smith’s 8th grade algebra class works through 10 word problems in an hour, and Ms. Jones’ class works through 10 equation problems during the same time, which class is likely to learn more math concepts by the end of class?
Please show your work.
Word problems are often considered one of the most challenging tasks in a beginning algebra class, with students likely to stumble over the move from the clean, basic formula to applying it in a real context.
Now, however, evidence from an ongoing series of experiments with students from middle school through college suggests that word problems might be easier and more beneficial for students when presented at the beginning, not the end, of a mathematics lesson.
“Early on, symbols are barriers to learning,” said Mitchell J. Nathan, an educational psychology professor at the University of Wisconsin-Madison. “Even with no context, word problems provide powerful informal problem-solving strategies, and language itself provides an entry point to mathematical reasoning that is highly superior to the algebraic equation.”
Mr. Nathan is one of a group of researchers who want to rescue word problems from the back of the textbooks. In an ongoing series of experiments he described at a symposium at the International Mind, Brain, and Education Society conference here on Nov. 6, Mr. Nathan and his colleagues are finding word problems may be most helpful for students just starting to learn algebra, rather than being saved until after they have “mastered” the formulas.
An algebra problem can be presented in several different formats, and a series of experiments at the University of Wisconsin-Madison suggest word problems may help students learn new concepts.
SYMBOLIC
Solve for n: n x 6+66=81.9
NON-NARRATIVE WORD PROBLEM
Starting with a number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with?
STORY PROBLEM
“When Ted got home from his job as a waiter, he multiplied his hourly wage by the six hours in his shift, and added the $66 he had made in tips. He found he had earned $81.90. How much does Ted make per hour?”
Source: Mitchell J. Nathan
Mr. Nathan, his University of Wisconsin-Madison colleague Martha W. Alibali, an educational psychology professor; Kenneth R. Koedinger, a professor of human-computer interaction and psychology at Carnegie Mellon University in Pittsburgh; and others are developing an intervention called “Bridging Instruction” to help students and teachers use word problems more flexibly.
New Starting Point?
In a prior study by the same group, high school teachers predicted students would have more difficulty with math problems presented as stories or non-narrative word problems than with those presented as symbol equations. Moreover, the researchers found that teachers with a higher background in math—those who had majored in mathematics or physics, for instance—were more likely to think students would struggle more with math word problems than equations. Teachers with a lower math background and those who struggled in math themselves were more likely to believe students would struggle with stand-alone equations.
“Maybe one’s knowledge of math gets in the way of your ability to predict what your math students will do,” Mr. Nathan said. “These high-knowledge math and science teachers hold this theory about how students should learn math, and it doesn’t match up to the student behavior.”
In a series of experiments—five separate high school studies, as well as some trials with middle school and college students—researchers, including Mr. Mitchell, found students more accurately solved word and story problems than symbolic problems in both arithmetic and algebra.
In each group, students were randomly assigned to solve different versions of the same problem: a symbolic equation, a story problem using that equation, or a non-narrative word problem of the equation.
From the get-go, students were more likely to even try to answer a word problem than an equation. Working through narrative problems also made students feel more empowered to explore different methods of solving a problem, rather than following a single sample process.
For example, a student might “guess and test” a variable in a word problem, using clues in the text to suggest a potential variable and then writing out the equation to test it. Or, the student might “unwind” a story problem to figure out the proper order of operations in the equation; for example, a mother sharing money with three daughters would divide by three, because subtracting three wouldn’t make sense in the context of the story.
“When we asked students how they solved a problem when they had done informal reasoning, they said, ‘I cheated’—because they hadn’t done it the way the teacher had taught,” Mr. Nathan said.
Looking across the different variations of the problems, the researchers found 91 percent of students could solve a given math problem in a story- or word-problem form, while only 62 percent accurately answered the symbolic equation. Further, more than one-third of students could solve the problem only in verbal form, compared with only 7 percent of students who could solve symbolic equations but not word problems.
Bridging Instruction
In a more recent project, the researchers piloted their intervention, Bridging Instruction, with 90 7th and 8th graders in four math classrooms in a middle-class school district in the Midwest that used the process over nine weeks.
Teachers employing the intervention started algebra units with story or word problems, including having students act out a story problem and design their own experiments to solve them.
While both groups of students improved in solving linear, symbolic equations over the nine-week period, students in the pilot program also improved in solving nonlinear problems and had more growth in problem-solving overall.
Mr. Nathan admitted that there’s a big caveat to the findings: A student has to have enough English-language proficiency to understand the problem as written. Students who do not understand academic vocabulary—the ability to translate the phrase “starting with some number” into a variable like “x,” for instance—would have significant difficulties with word problems.
Still, Mr. Nathan suggested that non-narrative word problems could hit a “sweet spot” of helping students organize their mathematical notation using a minimum of culturally dependent words.
And he warned that the tendency to wait before using word problems could exacerbate gaps for students who struggle with algebra early on, because they may not be exposed to many word problems at all.
“It is a real wake-up call,” Mr. Nathan said. “Should we be getting rid of formal equations? Of course not. But we should be asking: When should students be given tasks to master different types of mathematical reasoning?”
