Mathematics Where Students Learn by Doing

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Upper School students learn to solve new math problems by applying what they already know

From the Winter 2011-12 Caller

By Jim Wysocki

In a progressive school, the methods by which courses are taught will often differ greatly from what we teachers experienced as students. One such method is problem-based learning in mathematics, a popular example being the Harkness Method, which originated at Phillips Exeter Academy in New Hampshire. Catlin Gabel’s goal of producing young adults who are independent learners and resilient students can be seen in many aspects of this problem-based learning method. Asking questions, both by student and teacher, is a fundamental component of this method. In that vein, there are several questions to consider when introducing it. What is problem-based learning? How is it uniquely used at Catlin Gabel? How is it similar or dissimilar to the way other schools are approaching it? How can it help students become more successful mathematics students?

 
“What do you mean, we have to do the problems before you teach us the material?” asks a student at the beginning of a course taught in a problem-based format. This is then followed by, “Wait, we have to present the solutions? Aren’t you going to teach us?!” the next day. Students initially struggle with the method because they have come to expect certain practices in a math classroom. Although this is an overgeneralization, many students have come to expect, rightly or wrongly, that a math classroom is about taking notes, writing down procedures, and then practicing those procedures. Even when they have not been successful with such an approach, they cling to it because it is familiar.
 
However, in problem-based learning, students learn content and skills through their application—rather than apart from it. Whereas students already do this often in English, history, or modern languages it is less common in mathematics, where the assumption is often that you must learn skills before applying them. Imagine English classes that teach students about language decoding, grammar and syntax, and the writing process maybe years before they begin to actually read and write. The approach to problem-based learning being used at Catlin Gabel right now is to present students with an ongoing series of problems that alternately introduce, provide practice for, and ultimately apply mathematical concepts to new and different problems.
 
No matter what method is used, two primary components of the problem-based method are the importance of asking questions and the development of the skill of transfer. While getting students to ask questions in the beginning is difficult, they come to recognize their value. One student recently wrote, “It is always better to ask a question than not know its answer.” While questions are an essential part of the method, the ability to apply knowledge to new and different problems, on a regular basis, is fundamental. This is the nature of problem solving, and although challenging in the beginning, the students adapt. One student commented that problem solving “comes very naturally now, and I think that in many cases it seems like after working through it for a bit I understand it well enough to have learned it from a teacher.”
 
Problem-based learning is used right now in Upper School in courses that include Year Two of the integrated program, Accelerated Precalculus, and Calculus 2. Each of these classes approaches the method in similar, yet different ways. The Calculus 2 curriculum is a set of over 400 problems, organized in a logical progression of skills and concepts. Although they are not arranged into units, certain themes come and go throughout the course. In the Year Two and Accelerated Precalculus courses, the problem sets are much more explicitly unit-based. Because of the nature of Catlin Gabel’s own curriculum we create the problems ourselves, using our experience in teaching many of the topics as well as considerable resources gathered over the years. In addition, other techniques help students adjust to the method, including returning to traditional lecture format periodically to “wrap” things up and allow for specific review of topics before assessments, and the use of material they developed as part of previous courses.
 
It is becoming more commonly accepted and realized that students need to have an opportunity to work through ideas with feedback from others in order to master concepts. This does not merely need to be feedback from the teacher, although their role is critical to the success of the method, but from the students as well. In fact, as the year has progressed our students are beginning to recognize the value of their peers’ feedback, and their ability to provide it. As one student said, “I like how in class we share our work on the board, because I like to see how other people decide to do different problems. It gives me insight on other possible ways to do something, and I learn a lot.”
 
Learning mathematics in this way builds students’ confidence and resiliency. One student said, “I have learned to jump into any problem and try anything I can to make a dent in it,” and another, in commenting on classroom presentations felt that “when I have to explain something, I have become more confident with this throughout the year.” Resiliency can be summed up in one of two ways. First, it is the willingness to persist in the face of frustration and adversity. Secondly, it can be thought of as the ability to learn from failure. When students learn math as a “recipe” of algorithms to be applied given the right circumstances, they become accustomed to the idea that they can only solve math problems that look a certain way. In addition, if they do not produce a correct answer, often with minimal work, they give up and wait for someone to show them how to do it. As we know, any math that most of us encounter outside the school setting often bears little resemblance to anything we did in school other than perhaps basic arithmetic, as in counting money or determining a tip. It just is not possible to teach students all the little ways that math intrudes on our daily lives and give them an algorithm for it.
 
Problem-based learning recognizes this, and thrives on it. Not all the problems are “real-world” ones, but students are given a carefully designed set of problems they have the tools to solve, without necessarily having learned an algorithm for them. One student’s comment was reflective of her efforts when she said, “I think over the course of these months I have become a more creative thinker.” And, in recognizing that the teacher’s goal is to develop independent learners, one student realized what was behind the teacher’s willingness to give students room to think and work by acknowledging that “it means that we almost control our education.”
 
Jim Wysocki, chair of Catlin Gabel’s Upper School math department, has been at the school since 2010. He previously taught in California at Chadwick School and the Irvine Unified School District, and was a Math-Science Fellow with the Coalition of Essential Schools.