2.1 The problem-centered classroom
A look inside a middle school math class based entirely on problem solving.
As part of LEARN NC’s May 2001 Problem-Centered Math workshop, Professor Grayson Wheatley demonstrated a problem-centered approach to mathematics with an eighth-grade class from a local school. Although the content was geared to the eighth-grade curriculum, the methods could easily be adapted to any grade level K–12.
Teaching problem centered math
In this approach to problem-centered instruction, the teacher pairs the students and gives them a series of problems to solve. The teacher visits pairs and may ask questions or watch silently but should not correct the students or guide them too obviously toward a solution. After a given amount of time, the students present their solutions to the class, explaining their methods as well as their answers. If another pair of students disagrees, they can present their own solution, and the class decides as a group which answer is correct and which method of obtaining it is easiest, most appropriate, or most clever. Students thus have the opportunity to construct their own strategies for solving problems and their own understanding of the mathematical concepts involved, and they are more likely to retain those concepts longer than if they had been given them by rote. They also learn valuable oral presentation skills.
At certain points along the way, it is reasonable — and often necessary — for the teacher to point out that mathematicians call a certain property by such-and-such a name or that the exact relationship between a circle’s diameter and circumference is actually π and not 22/7. Although students can approximate π or the formula for the volume of a cone experimentally, they will be unable to prove their results and come up with precise answers.
It is also perfectly reasonable to nudge students, while they are working in pairs, to try alternative methods of solving a problem. Rather than saying, "That’s not right — try this," however, try asking, "Is that the only way of thinking about a problem?" or "What would happen if you did this?" This is particularly valuable if students are growing frustrated and have stopped making progress on a problem
Tips on group work
Grayson Wheatley argues that it is best to pair students of similar ability. If one student is clearly stronger mathematically, the other student will tend to defer to him or her, and the value of collaboration will be lost. Students of similar ability, on the other hand, will be able to challenge each other consistently. The teacher can provide additional challenges for pairs that finished assigned problems quickly.
One might think that certain pairs of students would then tend to dominate the presentations and discussion, but Professor Wheatley argues that this is not necessarily the case. Quite often, he says, students whom the teacher deems less able mathematically will come up with clever or original approaches to problems, and they will relish the opportunity to "trump" their allegedly smarter classmates. Just in case, however, "I try to avoid some real sharpies getting up and explaining their solution first." The other students may well assume that those students have the right answer, even when they do not
"We can’t have students getting into the eighth grade and not knowing procedures," Professor Wheatley admits. "In mathematics, we cannot operate without computational procedures. The question is, how do students come to have computational accuracy, fluency, and efficiency?" By allowing them to develop their own procedures and to share those procedures with one another, they learn to compute accurately and efficiently, but also flexibly — they have more than one way of approaching a problem. "The procedures evolve out of the attempt to make sense of their experiences.
What if they don’t get the right answer?
It is entirely possible that no one in the class will find a correct solution to a problem, or — seemingly worse — that the class will agree on a solution the teacher knows to be incorrect. If this happens, do not correct them. Instead, return the next day with a similar problem that will encourage the students to find different approaches and eventually reach a consensus on the correct solution
The teacher’s role
Class work of this nature does not require a lesson plan per se, but only good problems. Problem-centered learning thus requires less short-term preparation on the part of the teacher. The difficulty comes in selecting appropriate problems — particularly in the case described above, where students agree on an incorrect solution and the teacher must find a problem that will nudge them to think differently. Selecting problems of the appropriate difficulty and complexity requires a fair amount of mathematical agility, and thus a greater amount of experience and long-term preparation than traditional instruction.
Professor Wheatley argues that the problem-centered approach makes the teacher more of a professional.
Problem-centered learning requires a pretty high level of functioning on our part. That’s one reason I like it — I don’t want to be a clerk … who is just passing out the work and collecting it up and grading and averaging scores. That’s not very inspirational for me. That’s not what I want to be doing as my life work. I want … to think and make decisions" in response to students’ thoughts and ideas.
Since problem-centered math is based on problem solving and presentation, a problem-centered assessment must be, too. This means that students should explain — in writing — how they reached a solution. The process of verbalizing mathematical concepts is not only important to help the teacher understand what the child is thinking, but it also encourages the student to think more deeply about the mathematics. On standardized tests, of course, written explanations will rarely be required, but presenting their solutions will make students better able to solve new problems whether or not an explanation is required