I think the Standards for Mathematical Practice (SMPs) have the potential to be the game changers that we need in mathematics education. They by themselves create opportunity/demand for changes in instructional practice. At worst they call for instructional practices that have not accompanied standards previously. The NCTM Principals and Standards were the first step, but they never received the formality of being attached to content standards like the SMPs have.
I always start with the first standard, Make sense of problems and persevere in solving them. I believe this standard is the first standard because it addresses what mathematics can be. In order for students to be able to persever, there is a need for students to look beyond just finding the answer to learning to find multiple pathways to finding answers, so that when the path is blocked, they have an option other than quitting. I like asking the question, “what does persevering in problem-solving look like?” to teachers that I work with. I often provide one potential answer, “To think harder (accompanied with a grimace on my face).” I’m not sure many teachers understand what it takes for most students to learn how to persevere in solving problems.
There are some really nice resources available for identifying practices that help teachers address perseverance. I think the most important aspect of developing perseverance is for teachers to understand that it is NOT about the answer but about:
- how student view problems,
- how students learn to see relationships in problems,
- how students learn to solve problems through multiple methods, and
- how students think about struggling with problems
I have long thought that too many times in mathematics education we make it about the answer when the answer is irrelevant. When teaching students to solve equations, we value the answer (because that’s how they get points) when we all know it is about the process. When learning fractions, we give positive feedback when the answer is right, not necessarily when understanding the concept was right. I can give any number of examples, and often when we do honor the process (i.e. FOIL) we build it up as the only/best way of doing it when in fact it is only one way.
I preface my discussion of perseverance in problem-solving with this information, because I don’t think developing perseverance is difficult. It won’t take any new innovative approaches, but it will necessitate, intentional use of practices that we struggle with currently. My greatest concern is that people will think this is beyond their students ability, when in fact, I think it is more about what we ask our students, and how we get them to share their understandings. I also, have grave concerns that it will be something that teachers think they can do once a unit when it must be embedded in the core of all mathematics instruction to be effective. It is how we need to teach students to think mathematically.
This will be a two part series and in this first part, I want to make the case for figuring out what perseverance looks like for students, and partially what it looks like for teachers. In the second post, I want to focus solely on examples of what it looks like in the classroom, and strategies for achieving these goals.
Perseverance in Problem-Solving
I will use the definition of perseverance in problem-solving from a student’s perspective to include;
- being able to make sense of the information in a problem through a variety of approaches,
- text interpretation
- written/verbal description of relationships between quantities/actions in a problem
- symbolic representation
- attempting to solve a problem and explain why the approach being used makes sense in terms of the problem
- being able to use an alternative approach to solve a problem is a viable solution cannot be found using a first approach
- being able to justify the validity of a solution to a problem.
That is a very aggressive definition because it isn’t simply about finding the answer as much as it is about the journey to finding the answer.
With that definition in mind it becomes imperative that we as teachers don’t focus on the answer. The numerical answer is not important! Let me repeat that statement, the numerical answer is NOT important. If we want to students to really value perseverance, then we have to focus our efforts on the pathway to the solution and never give the idea that the answer is as or more important than the process to finding the solution.
In my mind then, one of the first things that must go as a result of this shift is any worksheet that focuses on repetitive approaches to solving problems (i.e. 25 problem worksheet around solving 1 step equations with +/-). These kinds of worksheets value the answer and not the process.
Another shift must be from the teacher providing the solution path. If it’s about the solution path and not the answer, then this must come from them. They must initiate it, in fact they must initiate multiple paths. They must learn to reason about the approaches validity and learn to validly question another student’s thinking. If the teacher is the origin of that in the classroom then students will never learn to make sense of problems themselves. The shift for teachers must be about providing questions that students ask along the way and holding student accountable for asking those questions, and learning to be able to answer them when it is not evident.
I have always prided myself in developing a student’s capacity to problem solve, but with the resources that are now available for developing a student’s problem solving skills, teaching problem solving has the opportunity to progress light years very quickly.