“Problems with a Point”

The Internet is full of math problems, but many of them are pointless, says EDC’s Paul Goldenberg. They exist solely to practice what a student already knows, without leading to or developing larger concepts or questions. In such cases, he says, “the individual problems don’t matter, and neither do the answers.”

Problems with a Point, a new EDC Web site developed by a team of eight current and three former EDCers, along with an active teacher advisory board, is just what its name implies: problems that lead somewhere useful. They build mathematical understanding while also providing students an opportunity to practice mathematical skills. As an example, Goldenberg offers a sequence of problems for elementary students: if you ask young children what happens when you add two odd numbers they will add many pairs of odd numbers together before concluding that the sum is always an even number.

“They’re getting the practice,” he says, “but it’s in the context of research. The answers matter: each one is a piece of information about the pattern. And when they get the pattern, you can go on and ask, ‘why does this pattern have to be always true?’ Now they’re solving a proof problem, even if the ‘proof’ involves blocks or hand gestures.”

Problem Sequences Develop Understanding

In a similar spirit, the Problems with a Point are organized in sequences that approach a mathematical idea in a variety of ways, so that a student’s understanding develops as he or she works through the problems. Designed for students, teachers, parents, home-schoolers, and others, the site contains an extensive, and growing, body of problems. A powerful search tool lets users select problems by mathematical topic, problem-solving method, available computational tools, and lesson lengths.

A brief synopsis precedes each problem, describing the kind of problem it is, the kind of math concept it introduces or develops, and what prior math knowledge it requires. Sections containing hints, answers, and solutions (explanations of how answers are arrived at) follow each problem. Most problems are presented in both HTML and Adobe Acrobat format, so that they can be viewed quickly on the Web or printed out for use in class.

The latter option was designed for teachers. A special teachers’ resource on the site suggests other ways teachers can use it, such as integrating problems into an existing curriculum, choosing problems for extra-curricular work, or using problems to assess student understanding of math concepts and operations. With the search function, teachers can find problems suited to an individual student’s needs, problems that work with both traditional and reform curricula, and models of problem sequences teachers can use as examples for creating their own sequences.

“Each sequence, ideally, tells a bit of a math story,” an unfolding of relationships, Goldenberg says. What’s important about math, he explains, is not only the facts and results that have been developed and collected over time, but “the ways of thinking that lead people to reach results. Those ways of thinking—using logic, making connections, seeing causality, narrowing variables—are universal. They’re useful to everyone, not just mathematicians.”