I have been asked to facilitate a workshop on teaching and learning problem solving. The audience will be university professors looking to design or update courses that include the goal of problem solving. It seemed appropriate to share my plan here - that there might be some interest in this beyond the workshop. As always, I am also interested in your feedback.
Schema Activation: Journal Jot
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
How does this compare to problem solving in your discipline?
Focus: Teaching and Learning Cycle
While some aspects of problem solving may seem intuitive to us, intentionality in our teaching efforts increase the likelihood of students successfully learning this process. The model that I use as I plan my courses is the Teaching Learning Cycle. You can read more about the Cycle and how I have applied it here. For this workshop, we will simply focus on the need to identify what we are assessing and how; how we will analyze the results of our assessment; a plan for providing the necessary support needed for students to achieve our goal; and the instruction that will provide that support.
Activity: Demonstration and Discussion
In my courses for preservice teachers, I use the Process Standards from the National Council of Teachers of Mathematics to inform some of my targets. For problem solving, I use these:
Some of these may be difficult to assess using traditional tests. Therefore, I have developed an approach that I call Metacognitive Memoirs (read more about them here). Essentially, in an effort to have students make their thinking visible, I ask them to do more than simply give an answer or show their work. They are expected to share their thinking, which allows me to better analyze what they can do and what they are trying to do.
Dr. Alan Schoenfeld researched college students' self-regulation during non-routine problem solving to problem solving. (A summary of this research is provided here (pdf), beginning on page 60.) The figure below shows his analysis of two students trying to solve this problem: find the largest triangle that can be inscribed in a circle.
when students are doing real problem solving, working on unfamiliar problems out of context, such behavior more reflects the norm than not. In ... videotapes of college and high school students working unfamiliar problems, roughly sixty percent of the solution attempts are of the "read, make a decision quickly, and pursue that direction come hell or high water" variety.
When I evaluate my students' metacognitive memoirs, I am looking to see that they are able to do break from this trap. I use a holistic rubric that uses a trip from Grand Rapids to Detroit as an analogy. The students add another layer of metacognition to the assessment by highlighting where in their memoir they are using the NCTM's Problem Solving Standard. If the students are stuck somewhere short of their goal, then I have more information that can inform what comes next.
Dr. Schoenfeld compared the approaches used by novices to expert problem solvers. He found that an expert "spent the vast majority of his time thinking rather than doing (1987, p. 194)." Therefore, Dr. Schoenfeld began to plan ways to make the thinking done while problem solving visible. That is, the thinking of both the novices and the experts.
One of the ways I plan to do this is by sharing this page from How to Solve It by Dr. George Pólya - a seminal piece of work in the area of mathematical problem solving.
With Dr. Schoenfeld's research and Dr. Pólya's four-phases of problem solving in mind, I plan lesson resources that will support students in sharing their own thinking and recognizing the thinking of others. Below are a few examples from a College Algebra course.
These are a bit more procedural than I would like, but the circumstances surrounding the class meant that I needed to meet the students where they were at in order to facilitate their progress toward out goal - problem solvers.
Dr. Schoenfeld says that one way that thinking processes related to problem solving can be brought
out in the open is to model them, presenting "problem resolutions" rather than problem solutions. At times I work a problem as though I were working it from scratch, going blow-by-blow through the solution process (1987, p. 200).Thanks to the Mathematics Association of America, we have an example of the other expert, Dr. Pólya, modeling his problem solving approach to some college students.
I did my own "problem resolution" (what is sometimes called a think-aloud) around this real-life problem I encountered. Unfortunately, the video has not yet been digitized.
I decided to participate in an indoor triathlon over the weekend. In this event you swim for 15 minutes, bike for 15 minutes, and run for 15 minutes. For each activity, you accumulate yardage. At the end of the event, the person with the most combined yards wins. Although I am quite competitive, I am also a realist and therefore my goal was not to win but to break 10,000 yards. So how many yards ought I try to get in each activity?
This is the handout I provided to the students so that they could monitor my thinking and expand upon it.
Once problem solving methods have been modeled, students need to have plenty of opportunities to practice monitoring and regulating their thinking. During this time it is important for teachers to monitor students' progress and mentor them through the tough patches. The problems need to challenge students' thinking. Exercises that challenge their ability to remember and follow directions are not enough. It also helps if the problem solving is happening in the classroom so that the observation of students efforts is fresh and any necessary feedback more immediate.
What was the main ideas of this workshop?
Hopefully participants will recognize something related to the idea of making thinking visible and doing it intentionally.
So what was important about these ideas?
Dr. Schoenfeld found that by applying the idea of making metacognition (thinking about our thinking) an integral part of a problem solving course, students were able to better monitor and regulate their efforts when encountering non-routine problems.
Now what will you do with these ideas in your courses?