Week 1 - Mathematics and the Body

 Reading Summary

Goldin-Meadow et al (2009). Gesturing gives children new ideas about math.

This research report describes a quantitative study with the goal of examining the role of gesturing in learning mathematics. The subjects were 9- and 10-year-old children and a pre-post-test format was employed as well as coding of the children’s explanations of how they solved the problems. The children were assigned randomly to one of three conditions: no-gesture, correct-gesture, and partially-correct gesture. The children were taught a phrase as a strategy for solving equivalence problems and a gesture as needed for their condition group. All children had the same math lesson between the pre- and post-tests. The study found that the children that were taught gestures used them in the post test problems and the more correct the gesture, the more correct answers given on the post-test. Additionally, children who used gestures were more likely to include language about grouping (a reliable strategy for these kinds of equivalence problems) in their explanations. Based on the results, the authors posit that gestures not only help people learn “old ideas” but help people develop new ideas and that educators can help students prepare to develop new ideas by teaching them hand gestures.

 

Critical Reflection

Perspective Changing

“…understanding has to do with the ability to change your perspective. If you don’t have that, you don’t have understanding.” Roger Antonsen

This was a major stop in this week’s activities, the start of Roger Antonsen’s TED Talk. I believe I stopped because this idea was familiar and reaching back into my brain, I realized that perspective taking/change was the main message in Francis Su’s keynote for the 2021 Northwest Mathematics Conference. I enjoyed how Antonsen encouraged a change of perspective outside of just viewing things differently, but that we can attend to things with different senses as new perspectives, too. His examples of 4/3 with pitches (perfect fourth) and drumbeats were amazing illustrations of this perspective change. I laughed out loud when he exclaimed, “I love four-thirds!” because I could see with each additional way of attending to 4/3 he was getting more and more excited. (Did anyone else find Antonsen as intellectually attractive as I did?)

So where did change of perspective help my understanding this week?

My community garden plot

I engaged with change in perspective in my community garden. I jumped off of Susan’s idea of using something practical as a measurement tool in gardening. I recalled how my dad sometimes uses a rake as a measuring tool for spacing rows or for determining the width of rows when he is planting. A tool that is generally used for breaking up and leveling soil can be a useful measurement tool. I could use the width of the rake from back to front to mark the width of narrow rows; the length of the rake from side to side is a bit wide for rows in my raised boxes. I discovered that the distance between the tines of the rake is about an inch or between 2 and 3 cm. This is sometimes a good distance for seeds to be placed from each other and is a good depth for some seeds. The tines now have a use to me beyond breaking up soil – they are a handy spacing tool!

Length of rake ~ 16"

Width of rake ~ 4"

Space between tines ~1"

I believe I was also changing perspectives when I was trying to measure my step and pace. I looked at the measurements in 3 different ways:

o   Measure from back heel to front heel

o   Measure from back heel to front toe

o   Measure from back toe to front toe

I had to think through all 3 possibilities and consider the benefits and drawbacks of each. Looking at each way individually and in comparison, I believe made me have a deeper understanding of each measurement. In the end, I decided on the “heel to toe” measurement because, upon reflection, that is how I would use step or pace to measure – where one foot begins and the other ends. By the way, my step was approximately 82cm and my pace about 152cm.

 

Binaries and Bodies

In Susan’s introduction this week, she writes about binaries in relation to “a realm of abstract disembodied mathematical perfection, and…a realm of physical, bodily experience” and this was one of my ‘stops’. I stopped to make connections to readings from our EDCP 552 course with Cynthia last term. Specifically, I recalled one of the Data Feminism (D’Ignazio & Klein, 2020) principles, “challenge binaries and hierarchies.” What might challenging the binary of the abstract vs. physical look like, sound like, feel like in mathematics education? We also read Untangling the Web of Deficit Discourses in Mathematics Education (Adiredja & Louie, 2020) which speaks to how a limited view of mathematics means there is a “narrow space for students to demonstrate competence” (p. 43). I connected both the Data Feminism principle and the Adiredja and Louie article to Susan’s statement of “This division of the world into two realms is also clearly elitist and non-egalitarian. It does not promote democratic access to mathematical beauty.”

While I was at my community garden in waning daylight the other day, measuring it out with my shoed foot (which, very conveniently, is nearly exactly one imperial foot long), I thought about how measuring with my body was impacting my thinking and perception of the processes of measurement. The study described in the Goldin-Meadow et al (2009) article, suggests that merely making a gesture promotes new thinking about concepts and mathematical processes (though I had some problems with that claim because the study did not include a “completely incorrect-gesture” condition). So, by extension, moving the body or using the body during measurement activities should allow students to develop new understandings about measurement. While I worked, I thought of how young students might develop the idea that the measurement tool can be used over and over for a linear measurement and how they might begin to understand some concepts about fractions when the chosen unit/tool does not fit perfectly. I found myself having to estimate how much of my foot would fit in a remaining space when I got to the end of a line of measurement. I have a hard time picturing how the same concepts might be developed as richly by a purely abstract approach to mathematics that discourages the use of the body.

 

Sketch of my garden after measuring with my shoed foot

Questions to Ponder

When have you taught your students gestures to support their mathematical thinking and understanding? What led you to use gestures? How do this week’s materials make you think about gestures and mathematics now?

How can we help others to understand that using the body with mathematics teaching and learning is supportive of our students’ understanding and not a pedagogical fad? Is it important to speak about the inequalities promoted by holding up an ideal of mathematics as an abstract, perfect language removed from bodily experiences? Can changing perspective support how we explain this to other educators, parents, families, students?

References

Adiredja, A. & Louie, N. (2020). Untangling the web of deficit discourses in mathematics education. For the Learning of Mathematics, 40(1), 42-46.

TED. (2015). Roger Antonsen: Math is the hidden secret to understanding the world. [Video], YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o

D’Ignazio, C. & Klein, L. (2020). Data Feminism. MIT Press.

Goldin-Meadow, S., Wagner Cook, S., Mitchell, Z.A. (2009). Gesturing gives children new ideas about math. Psychological Science, 20(3), 267-272.