“It’s fine to work on any problem, so long as it generates interesting mathematics along the way – even if you don’t solve it at the end of the day. (Andre Wiles)
This quote reminds me of a couple of key ideas that I have encountered through my work as a numeracy facilitator. I think we have to get away from thinking that “speed” is an indicator of success when it comes to problem solving. Yes, we want students to become efficient problem solvers, but we want them to have the time and experiences needed to construct deep understanding. We want to expose them to a variety of solutions in order to learn different ways of thinking and to think outside the box. Thus, coming to the understanding of the efficient ways through educator moves that make connections for them and build procedural fluency. Students who believe that speed is an indicator of success may have a tendency to give up easily or defer to those students who come to an answer quickly. As an educator or parent do you inadvertently honour speed in the problem solving process?
This notion of generating interesting mathematics is another important idea because the focus in our math classes and at home doing math homework needs to be on the math. As you read this you are likely saying, “of course!” Through my many years of classroom experiences I became aware of and noted a pattern of student behaviours that took their learning away from the math. Some of these include:
- drawing a picture when they did not need to in order to solve the problem
- drawing pictures with added details unnecessary to solve the math
- highlighting key words which becomes “most” of the words
- writing out important information before starting to solve the problem because “I’m supposed to.”
- making the work look “pretty” with different coloured markers
It is not to say that we do not want students thinking about the key information, for example, but when the focus is on the highlighting and not on the understanding of how this information is helping them, then it just becomes a make work project that moves students away from the math. As educators and parents when we have students follow a formula or model for problem solving it is imperative that we observe who this is working for and who it is just providing extra unnecessary work that takes the place of thinking. One size does not fit all!
I have noticed that many students who have a tendency to struggle in math or do not know how to get started tend to put their focus into behaviours that look like work but move them away from the math. They usually do not want others to know that they are not getting it and they are very good at looking busy. Those same children will persevere by trying to remember a rule (standard algorithm or formula), rather than thinking about alternate ways. These are often our “students of mystery”. It is important to observe them carefully and provide the specific, timely feedback that they need in order to be successful. Students who are flexible thinkers feel confident in finding alternate ways when their memories fail them. For that reason we have to be very careful about what we as educators and parents honour in math and ask our children to do. Yes, we want a solution to be written in a way that others can understand, but we have to remember that while we are grappling with something our own work is often messy. We would never expect an author to write the story or poem out perfectly the first time. Does that mean that we never work on representation or organization of thinking? Absolutely not, but once again timing based on the students in front of us is important and that is the art of teaching!