Imagining Math Outdoors

You don't need a lot of resources, except your mind, to be a mathematical explorer, and as a result you can embark on an adventure from anywhere.

- Francis Su -

Imagining possibilities

How do you imagine a math lesson outdoors? What kind of math thinking would be going on? What would be possible for you and your students? Starting with math, there are many ways to approach taking math outdoors. In some situations it might be as simple as moving an indoor math activity outdoors and working at a picnic table. Or it might be more complex such as an inquiry unit using math to explore a problem set in the outdoors. For us, the process of envisioning what math could be like in a different space helped us rethink the possibilities for how our students engage with mathematics. Moving math outdoors had us imagining bigger and richer possibilities for math learning.

Looking for Balance

"We found that our math instruction was not well balanced and tended to have a heavy emphasis on the procedures of how to do math."

Math Procedures and Concepts

One of the reasons we began to explore math outdoors was the research by Richard Skemp on instrumental and relational understanding in math. Skemp, an influential pioneer in math education, drew on his experience in psychology, mathematics and education to develop a theory how math is learned. Here is a brief explanation of the two ways of understanding math. When students have an instrumental understanding of math they are able to follow a set of procedures to obtain the correct answer. They know how to do the math. Relational understanding connects to math concepts. This is when students are able to use their understanding of math concepts and apply it to solve problems. They know why the math works.

We found that our math instruction was not well balanced and tended to have a heavy emphasis on the procedures of how to do math. Taking math into the outdoors gave us the opportunity to imagine how to engage students in understanding how math works. A study by VanDijk Wesselius (2020) says that “outdoor teaching requiring a different mindset .... teachers feel hindered by an instrumental, indoor view on learning, and teaching” (p.2)

In Skemp’s article, he states that procedural understanding is having "no awareness of the overall relationship between stages and the final goal” (p14); it is ”rules without reason..." (p.2). There are many examples of mathematical procedural understanding in the classroom - reducing fractions to lowest terms, the subtraction algorithm, finding the area of a rectangle, to name a few. Teaching only procedures is problematic because students are not encouraged to see math as a way of understanding the world around them. Math is reduced to an in-school skill of knowing how to follow a series of steps to get to the right answer. That's not to say we don’t need procedural understanding, we do, but it needs to be taught alongside conceptual understanding.

Contrasting procedural understanding is relational understanding. Skemp states “... relational understanding ...(is) knowing both what to do and why.“ (p.2). This understanding is conceptual, allowing students to make connections and apply their mathematical understanding to new situations. When students have a relational understanding, they can reason and test out ideas, take risks, experiment and use different strategies to solve a problem. This aligns well with the competency goals of BC’s K-12 Mathematics curriculum. “Students will develop a willingness to take risks, experiment, and make logical guesses. They will learn through both their successes and their failures, developing perseverance, competence, and confidence in mathematics.”

Inspiration from Others

An activity that we did in one of our courses encouraged us to imagine outdoor math opportunities. We were asked to create some sketches of living and human made things, and then look at our sketches to observe lines and angles. This is one example of how a student can deepen their understanding of a math concept. Below is a reflection Kim wrote after the activity.

Somehow, just seeing angles in nature makes them more real, they're not some abstract lines on a piece of paper. When I taught angles, I had one student who couldn’t even grasp what I meant by an angle …I wonder if she would have understood better if we'd gone outside and observed angles in nature.

Simply taking math outdoors will not necessarily improve the balance of conceptual and procedural understanding. The outdoors, however, can provide both teachers and students with a context to engage in meaningful real-world problem-solving experiences that encourage relational understanding.

References

Government of British Columbia. (n.d.). Mathematics goals and Rationale. Building Student Success - B.C. Curriculum. Retrieved July 28, 2022, from https://curriculum.gov.bc.ca/curriculum/mathematics/goals-and-rationale
Skemp, R. R. (2016, June 27). Relational understanding and instrumental understanding. Retrieved July 28, 2022, from http://www.davidtall.com/skemp/pdfs/instrumental-relational.pdf
Su, F. E., Jackson, C. (2020). Mathematics for human flourishing. Yale University Press. https://doi.org/10.2307/j.ctvt1sgss
Van Dijk-Wesselius, J. E., Berg, van den, A. E, Maas, J., & Hovinga, D. (2020). Green schoolyards as outdoor learning environments: Barriers and solutions as experienced by primary school teachers. Frontiers in Psychology, 10, 2919. https://doi.org/10.3389/fpsyg.2019.02919

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