What is 3 + 2? When mathematics educator Liping Ma asked U.S. teachers that question, she was not seeking the answer 5.
“I stumbled upon the question by accident,” she explains (2001). One day when she was thinking about how to make the teaching and learning of word problems more meaningful, she noticed that an important term was missing in English-language math vocabulary. In China, the concept is called li shi. Stating the li shi is the key step to solving a word problem. For example, for the word problem—John made three paper airplanes and Mike made two. How many did they make in all?—the li shi is 3 + 2.
In China, if the student is able to come up with a correct li shi, that student receives partial credit even if he or she does not compute the answer correctly.
When Ma asked, What do you call 3 + 2? educators offered responses ranging fromnumber sentence to horizontal problem, but no answer got at her meaning. A textbook published in 1896 came closest, identifying 3 + 2 as a mathematical expression. For example, a sample question read, How many bushels of rye at 88 cents per bushel must be given for 33 hogsheads of molasses, each containing 63 gallons, at 80 cents per gallon? Students were then asked to write the mathematical expression. Interestingly, composing a mathematical expression is a basic skill taught in many elementary schools globally but rarely in the United States these days.
Why is this so important? Liping Ma explains that thinking about the mathematical expression of an idea focuses students less on coming up with correct answers and more on understanding mathematics. She writes:In American elementary mathematics education, arithmetic is viewed as negligible. Many people seem to believe that arithmetic is only composed of a multitude of “math facts” and a handful of algorithms. . . . Who would expect the intellectual demand for learning such a subject actually is challenging and exciting?
How can we help students value mathematics for its intellectual challenge and exciting power to solve problems? Unfortunately, this question seems to have gone off the radar screen. According to a 2007 Public Agenda report called Important, But Not for Me, the majority of students and their parents polled believe that studying higher-level mathematics is not essential for life in the “real world.” Students also said that they are most motivated to study higher-level math, not by the arguments about competing in the international economy, but by the need to fulfill college requirements. The poll did not even ask if they thought it important to learn math for the intellectual power it conveys.
Today's mathematical debates are complex and divisive. Among the questions of the day are, Should we require all students to learn higher mathematics? At what age should algebra be introduced? What can countries with vastly different cultures learn from one another? How can math be made relevant to today's students? Will subjects like trigonometry really be important to job-related skills of the future? And finally, what is the real purpose of learning mathematics?
Although our authors in this issue weigh in differently on some of the other questions, they do agree on the importance of making math more meaningful to students.
Marilyn Burns (p. 16) writes,Only when the basics include understanding as well as skill proficiency will all students learn what they need for their continued success.
And Lynn Arthur Steen (p. 8) writes,Unless teachers of all subjects—both academic and vocational—use mathematics regularly and significantly in their courses, students will treat mathematics teachers' exhortations about its usefulness as self-serving rhetoric. . . . Students in high school need much more practice using the mathematical resources introduced in the elementary and middle grades. Much of this practice should take place across the curriculum. Mathematics is too important to leave to mathematics teachers alone.
The question Liping Ma raises—How can we help students better understand math?—seems to count most of all.
