Alan Schoenfeld: Learning to Think Mathematically (or like a scientist, or like a writer, or…)
Alan Schoenfeld holds the Elizabeth and Edward Conner Chair in Education and is Professor of Cognition and Development in the Graduate School of Education, as well as Affiliated Professor of Mathematics. He has served as President of the American Educational Research Association and Vice President of the National Academy of Education; he is a Fellow of AAAS and a Laureate of Kappa Delta Pi. Schoenfeld’s research deals with thinking, teaching, and learning — specifically in mathematics, but with broader implications. His books Mathematical Problem Solving and How We Think: A Theory of Goal-Oriented Decision Making and Its Educational Applications explain what makes for successful problem solvers and how people make decisions in complex settings such as classrooms. Schoenfeld has headed projects related to problem solving, teaching, and equity and diversity.
Summary of the Presentation
Professor Schoenfeld’s background is in mathematical cognition, and he began by describing the ways in which his study of teachers helping students learning to think mathematically could be applied to teachers helping students learn to think according to the norms of any discipline. He reminded us that students entering into an academic discipline are not merely “learning facts and procedures” (though of course they are doing these things also) but “developing particular disciplinary habits of mind” that help them use their knowledge “productively.”
Schoenfeld defines “problem solving” as confronting a situation that does not have a ready answer — not merely doing exercises which can be completed using known procedures. By this definition, problem-solving activities would include writing an essay trying to convince someone of your perspective or applying a historical or biological concept to a particular situation. According to Schoenfeld’s research there are four categories of knowledge that determine the quality and success of our problem-solving attempts:
- our actual knowledge base
- our problem-solving strategies
- our control, monitoring and self-regulation (i.e. metacognition)
- our beliefs and the practices that give rise to them.
Our Knowledge Base
Although we typically realize that the content of our knowledge makes a difference to our ability to solve problems, we don’t always consider how we know it, or recognize our interpretive filters. When working with students, Schoenfeld argued that we need to be able to evaluate the interpretive filters they have developed. That is, being a good teacher means more than being able to explain the same thing in multiple ways — good teachers need to recognize how students know the material, and be able to intervene when their understandings falter. Schoenfeld gave the example of six three-digit number subtraction problems solved by a child, four correctly and two incorrectly. He noted that, if we carefully analyzed the child’s errors, we would realize that the child only made mistakes when there was a “zero” in the first number and subtraction required “borrowing” across columns. This student would be best served by an instructor who intervened by noticing this consistent error and explaining the concept again.
Problem-solving strategies, also called “heuristics,” can be identified in every discipline. In writing and composition, heuristics include outlining, using topic sentences, and following basic argument and rhetorical structures. In mathematics, they include approaches like drawing a diagram, looking at individual cases, solving an easier related problem, and establishing sub-goals. Students can be taught these strategies — and, in Schoenfeld’s experience, as a result they can learn to solve problems that the instructor cannot.
In order to solve problems effectively, we must control, monitor, and self-regulate our thinking. What we know matters, but how and when we use our knowledge matters even more. In writing, common errors for students and scholars alike are losing track of the argument and meandering; failing to explain to readers points that are already clear in our own minds; and forgetting to consider the audience for the piece. In mathematics, students often begin using techniques and strategies they know without evaluating how appropriate those strategies are for the problem at hand.
To illustrate the importance of metacognitive work in problem solving, Schoenfeld gave the example of two students asked to solve the following problem:
Two undergraduate students solving this problem hypothesized that points describing an equilateral triangle would result in the largest area, and they divided a circle into arcs, drew a triangle, and began calculating its area. When Schoenfeld asked them, twenty minutes later, how knowing the area of the triangle would help them justify that it was the largest possible triangle using points on the circumference of the circle, they couldn’t answer his question.
Schoenfeld noted that the students had spent a short amount of time — a few minutes — reading the problem, and then jumped right into exploring it with calculations. Their exploration took up a lot of time — and seemed like it could go on forever. They hadn’t devoted any time to cognitive tasks such as analyzing, planning, implementing, or verifying a solution.
By contrast, a mathematics faculty member working a complex problem spent much more time analyzing the problem, planning and implementing strategies, and only did exploratory work for a short, controlled period. The faculty member also tended to oscillate between analyzing and planning, rather than getting stuck in one cognitive process.
Schoenfeld proposed that we teach students how to have good metacognitive control of their problem-solving and work processes. Classroom methods he has used include showing videotapes of different kinds of problem solving, so that the class can analyze them; role-modeling problem solving for his students; serving as the “control” for the class; and asking students to analyze their own cognitive processes , e.g. “What exactly are you doing?”, “Why are you doing that?”, “How does it help you?” In a course that Schoenfeld teaches on problem solving, these methods have helped undergraduate and graduate students to process mathematical problems with the same type of cognitive control shown by faculty members and advanced scholars.
In areas such as writing and math, and in every discipline, we and our students develop beliefs that may not serve us well. For example, in writing, we might believe that “you just write down what’s in your head” and so writing should be easy, or that “writing is like telling a story; you just start at the beginning and follow the narrative.” Schoenfeld told the group that he spent around 5,000 hours writing each of his monographs, and that unrealistic beliefs about writing being “easy” can hamper not only students but faculty members as well.
In mathematics, unhelpful student beliefs might include, “math is just about rules you learn,” “all problems can be solved in five minutes or less,” or “school math has nothing to do with the real world.” Because of these kind of beliefs, students are less effective mathematicians. Schoenfeld gave the example of students asked to solve this word problem:
29% of students, when faced with this question, simply divide 1128 by 36 and choose the multiple choice answer “31, remainder 12.” Another 18% round down and choose the answer of 31, and 30% choose the answer “other.” Only 23% give the correct answer of 32 buses. Many of the students who gave incorrect answers, Schoenfeld argued, did so because they were simply applying mathematical rules they had learned without analyzing the nature of the problem or considering its real-world implications.
However, students do not simply develop these unhelpful beliefs out of perversity. They learn their beliefs by abstracting from the typical practices of their classrooms. Students who answered “31 remainder 12″ to the problem of soldiers and buses likely did so because they had typically been expected to do simple division problems without real-world referents. Schoenfeld suggested that enriching our classroom practices, in any disciplinary context, can help students develop more productive beliefs and behaviors.
Ultimately, Professor Schoenfeld argued, we should not only teach the content of our disciplines — we should teach heuristics for problem-solving within them.