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Problem Solving

Being able to solve problems is one of the most important aspects of mathematical reasoning, and unfortunately, it is one of the areas least well taught in k to 12 schools. It is an important skill because it is useful for life, and a generic problem solving ability is useful across a wide variety of domains of knowledge. 


Be less helpful

Our curriculum developers seem to think that students need to be able to solve word problems, and so have chosen textbooks which give students great numbers of "problems" to solve, forgetting that the purpose of solving these problems is to develop general problem solving strategies. Some textbook companies have clued into this issue, and so have provided helpful "hints" as to how to solve these problems.  If the "problem" has a solution, and that solution is either given as a step-by-step process one can use, or as an answer in the back of the book, then the original thing is not really a problem, and our students know this.

This has been articulated well by many people, but I think Dan Meyer has done the best job explaining this issue in ways that a non-mathematician or mathematics educator can understand. In a riveting TED talk, Dan Meyer outlines some of the problems with our current mathematics education model. The most important idea in his TED talk is "be less helpful." The structure and scaffolding we have built for students around problem solving leaves too few places where students can struggle. While there is an important balance that should be struck between being completely frustrated and completely bored, too often I think we err on the side of boring our students.

TED talk by Dan Meyer



Start anywhere

We have also taught students that problem solving is a linear process, when in reality, many problems in the world outside of school are not solved through careful step-by-step processes. In the Agile Programming Development model, for example, programmers iterate between writing code (solving the problem) and responding to changes in the collaborative environment. This process is much different than the linear process we teach students to expect. So I've started developing a non-linear model which includes some of the basic phases of problem solving students might need.

The intention is to have students start to learn that problem solving is a process, rather than a sequence of recipes one can learn for each mathematical concept. Students can start by experimenting, writing, diagramming, collaborating, researching, or just by walking away from the problem. In a k-12 setting, you may want students to choose a different activity (instead of just doing nothing) from problem solving for the 'walk away' step, but this is an important step for solving many difficult problems. The Eureka moment, wherein a problem solver has a sudden insight into solving a challenging problem, is often preceeded by a completely different activity. The key thing here is that the problem solver chooses how they move between the different problem solving strategies depending on their needs.

Problem solving as a nonlinear process



Find problems

Problems for students to work in a mathematics class typically come a textbook, and more recently, a website. This teaches students (and teachers!) that mathematical problems are things imposed on the learner from an outside source. While it is true that there are a great many interesting mathematical problems for students to work on, it is extremely valuable for students to learn that they, and their experiences, are a source of rich mathematical problems. One of our objectives should be to have students see that the world is full of mathematics, and that what we teach them are tools for accessing this.

I have been advocating for students and teachers to pool their resources, and when they see photos of mathematics, to add their photos to a common sharing space. One such space is the Math in the Real World Flickr group. It does not really matter where exactly this shared space is. A wall in your classroom could be an excellent place to start. The idea is, build a set of mathematically rich tasks and ideas together with your students. Help them see that mathematics is a way of describing the world.

Many of these multimedia resources can be easily turned into mathematic problems. One can collect ideas for mathematical problems as you spot them, particularly if those problems are either in the community and culture of your students, or will help extend that community and culture to encompass the outside world.


Work together

I had a professor in university who had a habit of making mistakes when he would work out problems on the blackboard in front of us. He would go away after class, and come back the following class with his error found, and the problem tidied up. While this approach  gave us confidence in our instructor's ability to do mathematics, it did not help us learn the process he used to fix his mistakes. It would have been incredibly valuable for us, if at least once, he had talked through finding the error in his approach at least once.

The problem solving process I outlined above is not really complete. There are wrinkles to every problem, and an experienced problem solver finds ways to solve problems that are not typically listed in a textbook somewhere.  One of the primary ways that students will learn how to handle these wrinkles is by working on solving their own problems. Another way they will learn this skill is by watching an experienced problem solver work through a few challenging problems and explaining their thinking. We learn so much from what other people model - and the ability to patiently solve a problem is rarely modelled for students.

What instructors normally do is model problem solving for problems that have no wrinkles, and for which there is a well defined solution. While this approach obviously works for a great many cases, I think it sends the wrong message to students. Instead of students seeing their math teachers as fallable and as people who are willing to struggle to figure something out, they see us as sort of demi-Gods who are incapable of making mistakes. Here's a little experiment to do with your class on Monday if you don't believe me. Make a mistake in class, something obvious that the students will catch, and watch their glee as they get to point out your mistake to you. Their glee comes from having seemingly bested the infallible teacher. Students need to see math teachers as humans, so that they are more likely to see mathematics as a human activity.

An example of mathematical problem solving