Problem-based Learning in Mathematics

[This is an entry I wrote for my school’s blog in 2015.]

Five years ago, after three years of teaching Geometry at PRS, I went searching for a way to improve. I felt strongly that the method I was using – lecturing over a section of the textbook in class, going over some sample problems that practiced those skills, and then assigning practice homework problems that exercised those skills – was not only not satisfying to me as a way for my students to experience doing mathematics, but it was targeting relatively unimportant areas of learning for my students. Given that my students would almost certainly forget half of the theorems they learned after a year, and more than 90% of those theorems after four years, I wanted to identify teaching methods that would target deeper, more meaningful and longer lasting, areas of growth.

After a great deal of thought, I decided that the single most important area for students in a math class was problem-solving skills. This was something that they would never forget, and it applied to all areas of their lives. In my research, the teaching technique that addressed problem solving the best was to use problem-based learning. In this teaching method, students are given a relatively small set of problems to do for homework, usually with no lecture preparation from the teacher. In class the students present the homework problems to the class and a thorough discussion of each problem occurs. The problems are often difficult and sometimes a student is asked to present a problem he does not know the answer to – at such times, the student presents whatever ideas and work he has, and then the class works with him to finish the problem. The teacher’s role is to guide the discussion as needed, often with leading questions and gentle corrections, and to occasionally show deeper patterns and connections in the material. Any well-designed set of problems can be used with this method, and the wonderfully crafted Phillips-Exeter Academy (PEA) problem sets were the ones I chose.

I was amazed by the improvements I saw using problem-based learning. Some of those improvements are in the area I was targeting – improved problem-solving skills. However, many of the improvements were in areas that I had not anticipated, but were nonetheless important. In what follows I will describe the advantages I have seen.

Problem solving skills

Given that I chose this teaching method to improve problem-solving skills, it is not surprising that it delivered in this area. Even still, I was surprised by the degree of improvement I saw. My students were getting daily practice using problem solving strategies such as: persevering, learning from simpler versions of a problem, trying lots of examples and looking for patterns, and finding the value in having partial solutions. All of this practice was making a huge difference. Math teachers in subsequent years were seeing a difference in how these students went about solving problems. 

Without intending to, I ran an experiment that vividly illustrated the difference to me. Every year I have a mini lesson demonstrating the value of perseverance. I give my class a problem that they are not quite able to solve and wait to see how long they take to give up. Before using problem-based learning, my students would give up and look to me for the answer after 30 to 90 seconds. After I switched teaching methods, I was stunned to see all my students still going strong after five minutes – they were working in groups discussing ideas and many were at the board comparing methods. It was amazing!

Being mathematicians

One of the most beautiful things about teaching this way is watching my students finally experience what it is like to be a mathematician. There is probably no other discipline taught in school that is traditionally taught in a way so foreign to the way it is practiced by professionals in the field. Paul Lockhart in “A Mathematician’s Lament” compares the traditional way of teaching mathematics to teaching painting using a fantasy paint-by-numbers approach – students would spend their pre-college years learning the names of colors and how to fill in bordered regions of canvas, and only when they got to college might the be ready to actually make a painting of their own.

Practicing mathematics is a beautiful art, and students should experience that beauty. I want my students to learn how to play with mathematical ideas, to experience the pleasure of discovering and creating patterns, and feel the joy of solving a really difficult problem. This is exactly what I get to see on a daily basis. Students play with and struggle with the problems outside of class alone and in groups, and then we get together seated around a Harkness table to discuss the problems as a group of mathematicians sharing ideas and discoveries. 

This excitement in solving interesting problems carries over to everyone who is involved. Two of the teachers involved in co-teaching using the PEA problem sets this year commented that working together through the problem sets was creating one of the most rewarding years of teaching in their careers.

True learning is messy and requires effort

When I first became a teacher, I thought that when I became really good at teaching, I would make learning mathematics effortless and straightforward for my students. I now realize that I could not have been more wrong. Numerous studies have shown that educational methods that cause students to productively struggle are by far the most effective. Learning things of true value requires lots of effort, and the process is messy involving lots of exploration, false starts, and unpredictable moments of insights.

The folks at The Art of Problem Solving, a group dedicated to math education, once said that if you are able to do all the problems that you are given, then you are doing the wrong problems. An early criticism I received after I switched methods was that some students were now struggling who used to be able to easily zip through their homework of doing the odd problems from 1 to 31. These students had not been developing the skills they would need to solve difficult problems, and instead they were gaining a false confidence by spending all their time solving routine problems. They had learned to see being unable to solve a problem as a personal failure rather than as an opportunity to overcome an interesting challenge.

Active learning

One day when I was giving a quiz, I presented all the homework problems so that my students would have lots of time to take the quiz. After I finished laying out all the homework solutions, I asked the class how they liked my presenting the problems compared to their presenting the problems. I was startled to hear them say that they preferred my presenting the problems because I could do it very efficiently and that they knew everything I told them was true and the best way to solve the problems. I realized in that moment that their reasons were exactly the reasons why I should not present the problems.

Many valuable things occur when students present problems to their peers. The presenter and all involved gain invaluable practice communicating logical ideas. They learn how to listen to the ideas of others, work with those ideas, and add their ideas to the conversational mix. They are involved in active learning – they are constantly evaluating and challenging the ideas they hear, listening to whether the ideas are the best ways to do things or whether they are even correct.

A final benefit of this structure is that we do not use a textbook. The students are responsible for creating their own “textbooks” by collecting the homework solutions together with their notes from class. Notetaking moves from being a seemingly useless exercise duplicating what is in the textbook, to an essential task needed to be able to review material and study for assessments. This is great practice for what they will need to do in college.