Earlier this week Pam Harris wrote a thought-provoking article called “Strategies Versus Models: why this is important”. If you haven’t already read it, read it first, then come back to hear some additional thoughts…..
Many teachers around the world have started blogs about teaching, often to fulfill one or both of the following goals:
- To share ideas/lessons with others that will inspire continued sharing of ideas/lessons; or
- To share their reflections about how students learn and therefore what kinds of experiences we should be providing our students.
The first of these goals serves us well immediately (planning for tomorrow’s lesson or an idea to save for later) while the second goal helps us grow as reflective and knowledgeable educators (ideas that transcend lessons). Pam’s post (which I really hope you’ve read by now) is obviously aiming for goal number two here.
Models vs Strategies
In her article, Pam has accurately described a common issue in math education – conflating models (visual representations) with strategies (methods used to figure out an answer). Below I’ve included a caption of Cathy Fosnot’s landscape of multiplication/division. The rectangles represent landmark strategies that students use (starting from the bottom you will find early strategies, to the top where you will find more sophisticated strategies). Whereas the triangles represent models or representations that are used (notice models correspond to strategies nearest to them).
In her post, Pam discusses 3 problems that arise when we do not fully understand the different roles of models and strategies:
- Students (and teachers) think that all strategies are equal.
- Students are left thinking that there are an unlimited, vast number of “strategies” to solve a problem.
- Students get correct answers and are told to “do it a different way”.
I’d like to discuss how this all fits together…
Liping Ma discussed in her book Knowing and Teaching Elementary Mathematics four pieces that relate to a teacher having a Profound Understanding of Fundamental Mathematics (PUFM). One of these features she called “Multiple Perspectives“, basically stating that PUFM teachers stress the idea that multiple solutions are possible, yet also stress the advantages and disadvantages of using certain methods in certain situations (hopefully you see the relationship between perspectives and strategies). She claimed that a PUFM teacher’s aim is to use multiple perspectives to help their students gain a flexible understanding of the content.
Many teachers have started down the path of understanding the importance of multiple perspectives. For example, they provide problems that are open enough so students can answer them in different ways. However, it is difficult for many teachers to both accept all strategies as valid, while also helping students see that some strategies are more mathematically sophisticated.
As teachers, we need to continue to learn about how to use our students’ thinking so they can learn WITH and FROM each other. However, this requires that we continue to better understand developmental trajectories (like Fosnot’s landscape shared above) which will help us avoid the issues Pam had discussed in her original post.
If we want to get better at helping our students know which strategies are more appropriate, then we need to learn more about developmental trajectories.
If we want teachers to know when it is appropriate to say, “can you do it a different way?” and when it is counter-productive, then we need to learn more about developmental trajectories.
If we want to know how to lead an effective lesson close, then we need to learn more about developmental trajectories.
If we want to know which visual representations we should be using in our lessons, then we need to learn more about developmental trajectories.
If we want to think deeper about which contexts are mathematically important, then we need to learn more about developmental trajectories.
If we want to continue to improve as mathematics teachers, then we need to learn more about developmental trajectories!
While I agree that it is essential that we get better at distinguishing between strategies and models, I think the best way to do this is to be immersed into the works of those who can help us learn more about how mathematics develops over time. May I suggest taking a look at one of the following documents to help us discuss development:
- Fosnot’s landscape for addition/subtraction
- Fosnot’s landscape for multiplication/division
- Fosnot’s landscape for fractions/decimals/percents
- Lawson’s continuum for addition/subtraction
- Lawson’s continuum for multiplication/division
- Clements’ trajectories
Might I also suggest reading more on similar topics: