This week’s theme of integrating arts-based and outdoor mathematics with more traditional ways of teaching has occupied a lot of my thinking this week. I am in a position in my district, where we have found that how we’ve (the district as a whole) been teaching mathematics has not equated with increased mathematical success for students. We need to make changes! The question is what should these changes look like? What do we need to do differently to increase student success in mathematics system wide? These are big questions, that encompass pedagogy, teacher efficacy, curriculum etc. There is not an easy answer and so we are trying different solutions, to see if we will find success for students. There are many bumps along the road. The answer will take time, which is difficult because we want to see student success increase now. Individually, I am drawn to teaching outdoors, to trying new ideas and increasing my capacity as a teacher. This system wide view puts me in a different position than I would be if I was only considering my own classroom, my own setting, my own students and my own strengths and limitations. It is through this lens that I considered the readings this week.
Teaching mathematics as inquiry often feels like swimming against the tide. I appreciated Dr. Gerofsky’s thoughts regarding what is often considered as traditional ways of teaching mathematics through worksheets and teacher lectures. She said in the introduction, “They (worksheets and lectures) are not bad, they are just not enough.” (emphasis mine). How different would our mathematics classrooms look if we taught math as a “way to think, to come to connect with the world?” (Dr. Gerofsky, introduction). It does not have to be one or the other… but both and…
Although I read all three articles with great interest this week, the article I am primarily reflecting on is
b) Kelton & Ma: Reconfiguring math settings with whole-body, multi-party collaborations
This article brings together theories of social space, embodied cognition and mathematical activity and development. Kelton and Ma propose the valuable interdependence of setting, embodied activity and mathematical tools and practices as related to meaning making.
The study upon which the article is based used an instructional design feature called “multi-party, whole-body collaboration.” This means students were using their whole bodies in interaction with one or more people, in situated physical actions involving mathematical content.
Discussion of Space:
Kelton and Ma discuss the relation between activity and space suggesting that space is not just a backdrop of experience, it is an integral part of the experience. Reading about their thinking on spaces as “complex, historically constituted, dynamically experienced and socially produced settings,” was a stop for me as I connected these ideas with Indigenous perspectives on the value of places and spaces (Kelton & Ma, 2018, p. 178).
Discussion of Embodiment:
The study suggests that we need to move beyond a primary consideration of hands and expand to including other parts of the body, including the whole body. Dr. Gerofsky was cited in the article as an aspiration in this area to, “a more holistic consideration of human bodies” (p. 180). The Sarah Chase video is an example of this in action.
Discussion of Multi-modality:
Kelton and Ma assert, “we are committed to the notion that all mathematical activity is always multi-modal” (p.181). Another stop for me was when the authors share the ideas of Hutchins (2010) who argues that ‘courses of action’ can be ‘trains of thought’. How does this play out in our classrooms? Collaborative mathematical sense-making is pushing beyond words as the main expression of mathematical concepts, to value mathematical thoughts unfolding through action, increasing the visibility of thoughts to others.
Other pertinent ideas discussed by the authors include the importance of design for mathematical activity, which can either “amplify or dampen various bodily multiplicities, possibilities and relations” (p. 182), the individual versus social use of space, and the value of movements both subtle and big.
The aim of this study was to examine how new ways of using the whole body in familiar spaces “can create new opportunities for mathematics learning” (p.182). The study examines two case studies.
Case 1: Walking Scale Number Line (WSNL)
WSNL took place in a gym with lines made on the gym floor using tape. Students became points on the number line and then had to determine the centre and how to navigate their way through changing positions on the line. (Grades 2-8)
Case 2: Whole and Half
This study took place in the student’s classroom with partners. Partner one spread their hands apart and partner two had to place on of their hands at the ? way point between partner one’s hands. This was repeated in different positions around the classroom and with partner one using different configurations. (Grade 5)
The authors determined that reconfiguring classroom space had important consequences for sense-making. Students were given new opportunities for exchanging ideas, boundaries were explored, and imagination activated resulting in “mathematically significant innovations” (p.191).
The study heralds the importance and value of the design of mathematical activities to “leverage inherently embodied and social nature of mathematical thinking and learning” (p.192).
Changing the use of space “appeared to set up different kinds of relationships between mathematical activity and setting…where bodies went determined where the mathematics was” (p.193).
This article, as well as the other articles and introduction from this week, speak to the salience of lesson design. One question I have is, how significantly do the places and spaces in which we DO mathematics factor into our design of mathematical learning experiences. How does WHERE we do mathematics change or influence how we DO mathematics? I believe as Dietiker states, “It necessitates shifting our primary focus from educational outcomes to moving experiences that have potential to compel one toward an end.” (Dietiker, 2015, p. 3). I also ask along with Dietiker, “How might the aesthetic dimension of mathematical stories be understood and improved?” (p. 6) Which leads me to a question I believe we need to ask ourselves as educators when designing lessons, “What is the mathematical story?”
To circle back to my opening comments on changes needed to increase system wide student success in mathematics, I agree with Riley et al, who discuss the lack of teacher efficacy in implementing connections in mathematics (connections between mathematical concepts, connections between mathematics and other subjects, connections between mathematics and multi-modal experiences). Riley et al. discuss the key benefits of EASY Minds approach to teaching mathematics which incorporates physical activity into mathematics. They say, “key benefits perceived by both students and teachers were increased enjoyment and enthusiasm for mathematics and enhanced opportunities for students’ social, emotional, physical, and cognitive development” (Riley et al., 2017, p.1667).
The benefits of integrating embodied arts based and outdoor mathematics education with more traditional ways of teaching have been documented. How can we increase teacher efficacy in creating and implementing mathematics programs that encompass these ideals?
I also really appreciate Susan's (Dr. Gerofsky's) observation that traditional paper and pencil methods of teaching math are not bad, just not enough. I'm in a very similar position as you. I see so many teachers struggling to understand the big ideas of mathematics, clinging to programs like drowning people to bits of flotsam (yes, that is a reference to my blog URL) if they have them. Many who don't have no sense of mathematical narrative that they are trying to impart. Instead, they fling random bits of math at students and hope for the best. There is no doubt we need a base program to guide and support teachers. Then we, as system leaders and influencers, need to show how we can make it so much more.
ReplyDeleteFunny you should choose the Kelton and Ma article to summarize in detail. It was by far my least favourite of the bunch. I wondered if all the fancy words and convoluted writing were trying to artificially elevate the significance of two very simple lesson reflections. I know: a bit harsh. However, I think it might be true.
Love your visual of flinging random bit of math... So true!
ReplyDeleteIn my group, the Kelton and Ma article was the article I was responsible for this week, thus it was the centre of my reflection. I agree that the writing felt convoluted and a bit over the top for the topic of the study. It did give me things to consider... especially the use of space, and moving embodiment beyond hands. My favourite of the three articles was by far the Dietiker article. "I wanted students to experience mathematics in a way that showed that it could be exhilarating, surprising, captivating, and full of wonder." (p.3)
Hi Joy,
ReplyDeleteI had a few issues to deal with and finally I was able to read the article by Kelton and Ma. Then I read your reflections. You have done a fabulous job highlighting the key ideas of the article and bringing in the discussion other research findings, like Riley et al.
I found Kelton and Ma’s design of the research and interpretation of the whole-body and multi-party activities for embodied math learning, in and out of the classroom, interesting. I myself once came up with an activity in which students in grade 7 measured the length of the basketball court in a gym using measuring tape. Then they calibrated their steps. After that, they determined by pacing the measure of different lengths (such as radii of circles and arcs marked on the court, and the horizontal distance from free throw to the hoop). Students found the activity enjoyable, engaging, and useful. Kelton and Ma in their review of the literature stress the contrast between their work and previous studies. They emphasized their treatment of the role social space and the environment played in embodied math learning. The question that comes to my mind is how the activity conducted by my students could have been used similar to that in Kelton and Ma’s pedagogical design.
In Kelton & Ma, the first activity involved seventh and eighth graders walking a giant number line in a gym and students experienced embodied interactions as they applied number concepts, such as middle of the line and opposites. The second activity in classroom fifth graders performed pair tasks to show whole and half measures. The embodied activities were appropriately designed to the math concepts studied at the grade level. I believe these activities are important as we consider a policy of inclusion, particularly for those who become disengaged in math if pencil and paper calculations are involved. On the other hand, when I was teaching high school math (grade 11 precalculus), there were many students who preferred the traditional pencil and paper exercises and an approach that would help them achieve a high score in standardized assessment of their math skills and knowledge. I have always wondered about the challenge of striking a balance between having such activities (which can be time consuming in terms of planning and implementation) and the traditional problem-solving method, which may work very well for some.
Thanks for your comments Zaman. Striking the balance in developing mathematics pedagogies that integrate embodied, multisensory, outdoors and arts-based modalities is what this course is all about. It is important to remember that while some students prefer the paper and pencil activities that result in high scores in standardized assessments, there are other students who are unable to access deep conceptual understanding of the mathematics this way, not to mention, experience the wonder and joy of mathematics. It’s also important to remember that both have a place and importance. It is hard to play with mathematical concepts, if we lack understanding of the skills and yet we develop understanding as we play. With each activity we participate in, I am being stretched beyond my comfort level, and yet I am finding the value in the activities. As we stretch ourselves each week, I am making more and more connections between the activities and the mathematical value they add.
ReplyDeleteThank you very much, Joy, Zaman and Jen, for this great discussion and the critical thinking and teaching experience that inform it. It's not easy changing things, and it's especially tricky to re-balance, rather than just throwing out the old and running with the new!
ReplyDeleteI've wondered so often why we don't have math specialists in elementary schools in BC! We have specialist teachers within elementary schools teaching other 'technical' subjects including French, music and PE: they are elementary teachers with extra background and comfort levels in these areas, and they swap classes for their specialty subjects. Many, many countries in the world do this with elementary math too; why don't we consider it, rather than having teachers flinging flotsam (as you so aptly put it, Jen)? I simply cannot understand it. Have math teachers with a strong grasp and love of mathematics from at least Grade 4 on (intermediate years), and preferably throughout elementary. What do you think? It doesn't necessarily require an undergraduate math degree, but at least a demonstrated love and understanding of math, and perhaps a post-baccalaureate diploma?