Week 3 - Sustainable mathematics in and with living world outdoors - Critical Reflection

    I noticed that my ‘stops’ this week were not isolated to stopping and thinking/questioning/wondering in the moment when I stopped during reading or viewing or doing but became kernels for my mind to play with across the activities and after the activities were complete while I reflected on bringing things together. Maybe now we have a few readings, viewings, and activities under our belts I can draw upon different resources to make connections and allow my mind to revisit the ideas over and over when new information stimulates ideas.

Nellie's arbour and my noticings of its construction

    One ‘stop’ for me was when viewing the short film, Dancing Euclidean Proofs. Sam, the education student in the film, talks about how dancing the proofs makes them take time, revealing the proof slowly rather than how the proofs are laid out in their entirety all at once on a page in a book (Gerofsky, 2018). This ‘taking time’ made connections for me to last week and to this week’s activity. In the paper by Fernandes and Healy (2013) Multimodality and Mathematical Meaning-Making: Blind Students’ Interactions with Symmetry, the authors note that they observed the students using specific hand movements when they explored cardboard shapes and figures on geoboards that allowed the students to “intentionally see” (Fernandes & Healy, 2013, p. 53). Perhaps the dancing of proofs allowed Sam and Azul (the other student in the dance) to ‘intentionally see’ the proofs, seeing the parts as well as the whole. When I was sitting drawing living and non-living things for the activity this week, I made a connection to this intentional seeing, too. Slowing down and taking time to really look at the subjects of my drawings made me explore their boundaries and the internal structures (line, angles, colour tints and shades, connections, etc.) more than just noticing the object’s or being’s existence in my world allows. I know that the arbour exists in Nellie’s garden next to mine, but I had never looked carefully at how it was constructed. I know the general form of a maple samara with the seed heavy at the bottom and the fragile ‘wing’ full of veins, but I had never noticed before how the veins leave the seed clumped together and then spread over the wing (which was more noticed with the digital microscope but the process of drawing made me curious enough to spend time digging out the microscope to take a closer look). It has come up already in my group’s blogs and responses the difference between rushing through the curriculum and slowing down to offer meaningful experiences to students to explore and embody mathematics; the video and activity showed me that time is important and slowing down and taking time to really look or experience the world mathematically helps mathematical understanding.

Maple samara through the digital microscope

    It was interesting to take the extra step of looking at my living subjects of my drawings with the digital microscope. After drawing both living and non-living subjects, I was ready to declare (in answer to some of Susan’s questions) that humans love straight lines and right angles and that my living things did not exhibit 90-degree angles at all. This was reinforced in my reading when Doolittle references Leopold Kronecker and says, “the Creator gave us shapes, straight lines are the work of man” (Doolittle, 2018, p. 104). However, looking at the mesclum leaf through the digital microscope, I did see what are in the very least approximate right angles and this surprised me and challenged my initial stance. Looking closely, revealed some access to “the grid” within the fractal geometry of the veins, but as Doolittle describes, that grid fails at a larger scale, as when looking at the leaf in its entirety, “zoomed out.”

Noticing right angles

    I connected the grid and the drawing activity together in my experiences (and frustrations, as much as I was self-talking that these were unnecessary) of getting the living and non-living subjects onto paper. I wonder if some of my frustrations with drawing from ‘live’ things, 3D objects/beings, is that shift from 3D to 2D (I do a little better if drawing from a photo but I didn’t take that ‘cheat’ in this week’s activity). In my pre-reqs for education, I did a 300-level geometry course where we wrote computer programs to represent 3D objects on a computer screen (my final project was a Rubik’s cube…that failed after about 3 or 4 turns of parts of the cube). I can see how drawing onto paper imposes the grid onto beings and non-beings that exist beyond two dimensions. I’d be very curious to explore recreating my objects as 3D models to see if my brain welcomes that more than drawing. For example, I could model Nellie’s arbour with popsicle sticks or create the maple samara out of clay. These activities, I believe would still give me the opportunity to look closely and notice lines and angles while still maintaining the 3-dimentional nature and maybe not letting the grid do some of the looking for me in its imposition on the objects or beings.

Screen grab from Global Oneness Project (2018)

    Thinking about the grid, mapping and drawing from 3D to 2D, and the Doolittle article, reminded me of the video we watched in Cynthia’s EDCP 551 course about counter mapping. Zuni Elder (New Mexico) Jim Enote introduces viewers to counter mapping his people have done to represent the land, in ways that “counter and challenge the notion of what maps are” (Global Oneness Project, 2018). Both Enote and Doolittle talk about Indigenous lands being lost through mapping. Not only can the grid fail in representing our 3D world, but it may also be used as a tool of oppression.

Questions to Ponder

Here are some questions that I am working on answering for myself still and I invite others to help me with:

Where can we find room in the curriculum to challenge dominant mathematics through embodied learning? (There is room, I know it, but I want to look for specific opportunities.) How can we discuss dominant culture and oppression in mathematics classrooms, even with younger students?

Is there a binary of non-embodied learning and embodied learning? Where is the boundary between the two? Does a boundary need to exist? How much of the body needs to be involved for learning to be considered embodied? Was using a digital microscope in anyway embodied?

Doolittle says, in reference to Brent Davis, “Because we do not let go of the grid, we are internalizing it. We go from “straight” lines and “right” angles to well-defined “equal” plots, to “rules” governing our own behavior, and finally, to a sense of control and mastery. This progression is due in some degree to culture and to some other degree to innate human nature, and the weight of each cause is subject to debate” (Doolittle, 2018, p. 104). How can we, as teachers, let go of the grid?

 

References

Doolittle, E. (2018). Off the grid. In S. Gerofsky (Ed.) Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7

Fernandes, S. & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind stduents' interactions with symmetry. RIPEM, 3(1), 36-55.

Gerofsky, S. (2018). Dancing Euclidean proofs [Video]. Vimeo. https://vimeo.com/330107264

Global Oneness Project. (2018). Counter mapping [Video]. YouTube. https://www.youtube.com/watch?v=U7DQmTjpFI0&t=1s

Week 3 - Reading Summary

Doolittle, E. (2018). Off the grid. In Gerofsky, S. (Ed.), Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7

In Off the Grid, Edward Doolittle explores, through many examples, how grid-based representations of space can fail and challenges readers to consider other perspectives outside of Euclidean geometry for representing space. Grid-based systems are common in dominant culture and have a sense of “familiarity and comfort” (Doolittle, 2018, p. 102) and can be extended from the spatial to the temporal, such as marking out time on clocks and calendars. He argues that the uniformity of the grid can place artificial perceptions of equality on spaces and links the grid to concepts of “control and ownership” (p. 104). Failures of the grid-system include when geology is uneven or has sudden changes (an example of Hamilton, Canada where a grid-based map can’t incorporate the Nigara Escarpment’s 100 meter drop) or is round (projections of the earth ultimately skew, distort, or eliminate distances and features). Failures also happen with two or more grid systems meet, such as how the grid system imposed on the Six Nations is incompatible with the grid systems of the surrounding areas causing people to get lost driving when entering the reserve from these other grids.

Doolittle explores non-Euclidean geometry systems (explored in other chapters of the book from which this chapter originates). One example is Riemannian geometry which allows for inter-spaces as well as “accepting all kinds of straight and curved grids as equally valid ways to measure space” (p. 111) in a “democratization of frames” (p. 111). He also asks the readers to consider Indigenous ways of knowing as alternates to the grid. For example, in respect to time, rather than relying on clocks or calendars, Indigenous ways of farming may look to the timing of the sun during the day and the patterns of temperatures to know when harvest time is. In connection to land, defining territory may rely on demarcations related to land features and movement of water (e.g., a whole drainage basin) and incorporate fractal geometry rather than a grid system; in fact, imposing the grid may reduce territory size artificially (see the example of the Grand River Mohawk territory, p. 114). Doolittle offers a discussion of chaos theory as an example of a complex dynamical system that can be used to understand things like climate change and even teachable moments. He speaks of how non-Euclidean geometries, including Indigenous perspectives, may have their greatest influence in how they may “free the human mind from rigid and dogmatic viewpoints” (p. 117) and allow the seeing of problems in new ways that allow for new solutions, that we may be “opening our eyes and widening our perspective to see the world as it is, not as we might just imagine it to be.” (p. 119)

 

Here are some quotes additional to the ones included in my summary that I made note of:

“Too often, the specific life, qualities, and character of a particular place become subordinated to the forcefully imposed “evenness” and uniformity of the grid geometry.” (p. 104)

“To paraphrase Leopold Kronecker, the Creator gave us shapes; straight lines are the work of man.” (p. 104)

“You can smooth out bumps on the ground and remove rocks and stumps, but you can’t fill a valley with a hill, and you can’t move a mountain.” (p. 104)

“By allowing all grids on a n equal basis in a kind of democratization of frames, Riemannian geometry allows us to look past the particular grid we may be using, to refocus on the actual underlying geometry of the situation.” (p. 111)

“We must look through the structures imposed by our minds to the reality that lies below the surface grid we have drawn on top of it.” (p. 113)

“Examples of more down-to-earth complex systems in which chaotic control may applay are transformations in the context of education where we have the notion of the teachable moment, in which a few small “bursts” at the appropriate time might accomplish more than months of haranguing…The non-trivial question is how to identify those critical moments.” (p. 116-117)

Week 3 - Delving into Drawing and Mathematics (with a little microscope magic for some extra fun!)

 OK - right off, when I saw our instructions this week and that we have to DRAW, my fixed mindset about drawing rose up and screamed "Noooooooo!" Knowing I would have to share my drawings on the blog was the biggest part of the reaction, rather than the drawing process itself. Drawing is not something I feel I am good at, not something I ever pursued practicing, and not something I choose to engage with other than once in a while. But I scolded myself gently and took it as an exercise in perspective taking - how might I help a student who is a reluctant drawer if we were to use drawing with math? I won't know unless I experience overcoming the drawing trepidation myself.

My community garden plot

What a great week for needing to go outdoors to draw! Weather forecast was promising, at least later in the days after fog was supposed to clear. So, I set my sights on Wednesday, planning to leave work as soon after 3pm as I could and walk down to my community garden to sketch. The plan was thwarted by a second year teacher needing to talk through a difficult report card (but she was so appreciative, I couldn't begrudge this) and it was nearly 4:30pm by the time I got home to collect my things. Knowing sunset was in about half an hour and my westward bound drive home gave me a view of the fog bank rolling in, I drove to my garden instead and set up on a step stool between my garden boxes.

As recommended, I spent time sitting first for several minutes, observing with my senses and quieting my mind. The fog was already making its presence felt in dampness on my skin and halos around the street lamps that had just turned on. The dampness tickled the inside of my nose. My garden is right along the Arbutus Greenway and it was busy with bicycle commuters and after-work walkers and joggers. I could hear the whirr of bike mechanisms, clear, clipped footprints of pedestrians, and the engine sounds of cars as they drove by on the street side of my little plot. Stretching my ears further, I could hear planes taking off from YVR across the river and horns on tugboats, likely warning each other through the incoming fog. There was chirping coming from the blackberry bushes across the greenway - I think chickadee but not entirely sure. Just before I opened my sketchbook, I could hear bursts of opera being sung (turned out to be someone singing along to what they were listening to as they walked!).

First sketch - Nellie's arbour

The first thing that really captured my eye was an arbour in my neighbour Nellie's garden, or at least the side of it facing me. It is made of strips of wood criss-crossed between sturdy posts. I hadn't read Susan's instructions past section b, so I didn't actually know what we were to do with our sketches but I instinctively made some notes about what I observed looking more closely at this structure than I ever had before. I noticed that the strips cross each other at the top and sides of the frame, creating a little 'v' at 90 degree angles and the strips had been to a point to fit into the corners. My sketch doesn't show well, but the gaps between the strips formed squares, intersecting at 90 degree angels. When I got home and read the c) section of the instructions, I found that looking at these lines and angles was exactly what I should have done!

Second sketch - maple samara

I tackled a maple samara next. My garden is covered in partially decayed samara, whirligigs, or helicopters (what do you favour calling them?) due to a tree in the corner of my little plot. By spring I usually pull out several saplings of the maple variety (and one or two of the walnut variety that I did not plant but my squirrel friends did!). I find smaller things easier to sketch more accurately but my sketch doesn't well depict how fine the exposed veins were where the 'skin' had already decayed nor how the veins feed out of the seed part clumped together and spread from there.

I was already getting very chilled and losing light fast. I was worried my paper would dampen with the creeping fog. However, I committed to sketching two more things - some 'chicken wire' on my fencing and the street lamp.

City street lamp and signage

Chicken wire fencing

That was it - wimped out and headed home to warm up! (Context - my work place is always cold and we have to also have the windows open to circulate air so I had already sat all day in cold and just wanted to be warm!)

Patio mesclum leaf

On Friday I decided to inspect my patio for 2 more living things to draw. I had planted some mesclum mix in the fall and it grew and gave me a couple salads and I left a few leaves out for the house finches that visit regularly to enjoy. I sketched out one of the limp leaves - it is purple and yellow (faded from a deep purple and green) but I kept in pencil sketch. I also found some lichen growing on my plant ladder (a "ladder" with ledges for pots that my dad made me). I sketched that by slowly following the outline with my eye while equally as slowly moving my pencil, then filling in a few more details. (A quick delve into my plant ID book inside makes me think this is Ragbag (Platismatia glauca).

Patio lichen - possibly Ragbag

Digital microsope

During my first sketching session, I decided to bring the maple samara home to look at more closely - the fading light at my garden made it hard to see all the details. At home, I was like "I want to enlarge this!" and excitedly recalled that in one of my yet-to-be-unpacked-boxes from my summer work-site move was a digital microscope.  I brought that home with me on Thursday and used it on Thursday and Friday to look even more closely at the living things I'd sketched.  Here are some of the pictures I took with the microscope's software (clicking on the image should open a larger copy for you to look at):

Maple samara

Maple samara

Maple samara

Mesclum leaf

Mesclum leaf

Lichen

Lichen

The inquiry project through which I got the digital microscope was a mathematics-related one with teachers involved in Reggio-Inspired Mathematics. Part of what we wondered was how looking closely at things, especially natural materials, could inspire children's mathematical noticings and thinking.

Drawing definitely made me look closely and observe in detail for both the living and non-living things. What benefits or drawbacks are there to employing digital microscopes (or ordinary ones) for close up observation? What would asking children to draw from these zoomed-in views do for their noticings, wonderings, connections, thinking?

This is how I approached parts a) and b) of the activity and although I did a little of c) in my notes next to my first sketch, I plan to address some of the questions in connection to my reading and this week's viewing in another blog post. What I really wanted to do here was document my movement through the activity up to the reflection point.

Week 2 - Multisensory Mathematics - Critical Reflection

  “A recursive model is an interesting way to think about using multisensory math learning experiences as resources to think with…use the sensory activities to experiment with imagery and metaphor…These activities can become learning resources that helps learners synthesize multiple sense images to learn, remember and reconstruct these patterns with a deeper understanding.”                    Susan Gerofsky, course materials, Week 2

         The paragraph containing the above quotation was my first big stop in the readings and viewings and it related to many of my other stops, especially in reading the Fernades and Healy (2013) paper. The term “learning resources” was what initially made me stop, think, and process. I appreciate the idea that when we offer students, as what Susan wrote, “a really engaging and well-designed, shared multisensory activity”, students can draw upon these kinds of experiences, as resources, just as they might draw upon physical resources in our classrooms. What I took from the Fernades and Healy (2013) paper in a similar regard was that even if these experiences are drawn upon in imagination, they support learning. For example, one of the participants, Edson, had visual memories of mirrors that he was encouraged to use to support his development of the concept of symmetry. Additionally, the other participant, Lucas, had some assistance in the form of questioning from the researcher and then was able to employ that assistance and his own multimodal representations from previous tasks to tasks moving forward. How students are supported in perceiving and conceiving mathematical concepts gives them tools (multimodal representations) to use in future tasks and enriches their understanding.

I ended up using this idea of drawing upon previous experiences as learning resources to reflect on two experiences with the hexaflexagons this week: when I worked with some grade 1 and 2 students during a morning when I was redeployed and in my own exploration of hexaflexagons.

I landed in an inner city grade 1 and 2 class on Friday morning at the last minute. The day before I had printed several copies of Vi Hart’s hexaflexagon templates to carry in my bag in case I was redeployed and needed something to do with students. After a long play outside to start the day and the cries of “I’m hungry” coming back to the classroom, snack time it was and not enough time before recess to start an art project that was in the plans for the rest of the morning. The assistants and I suggested the students read or draw when snacks were done. One boy, let’s call him Bowie, said, “Instead, can I teach you origami?” and I agreed, as long as he also let me teach him hexaflexagons to help me with my homework. Bowie and I settled in at one of the tables and he taught me how to make a leaping frog.

Another boy, we’ll say Tim, joined us wanting to do origami but was being ignored by Bowie and my hexaflexagon template caught his eye. So while Bowie proceeded to make more origami (and not help me with my homework as agreed), I helped Tim with a hexaflexagon. This was quite difficult, even just to have him fold on the dotted lines and still so after I gently creased them for him. We got it constructed, he coloured the two sides, and I taught him how to flex and open it. I wish Bowie had fulfilled his part of our agreement because I was so interested if the folding would have been as difficult for him. My feeling is that Bowie would have folded it really well. His folding for origami was accurate and the creases were very sharp. In reflection, this is because Bowie has experiences of paper folding to draw upon if he were to make a hexaflexagon. My interpretation of Tim is that he avoids a lot of things that might be hard for him and relies on other people to entertain him. What experiences has he had that he draws upon to tell him to avoid things? Maybe his hexaflexagon experience will now be something he uses in the future to inform new learning.

My hexaflexagons from Vi Hart's templates

            I have to admit, my first reaction to knowing I would have to try out hexaflexagons this week was a bit negative. Jen and Jen did hexaflexagons for the Virtual Math Fair last year and I was in their room. I tried making a hexaflexagon both during our practice session and during one of their math fair sessions and was unsuccessful (now I know I was really over complicating the initial folding to make the triangles). I gave up and have not tried again since. However, I reset myself after my first reaction to it this week, used Vi Hart’s template, and got it! So, I asked myself, not only can great multisensory experiences affect future learning, but negative ones can also come into play. I alluded to this above in relation to Tim and after considering him in this way, I realized that I had initially been blocked by seeing all my failed hexaflexagons in my imagination. I sucked it up and got my hexaflexagon made and then made the sturdier one. Now I can make them without the template out of any size scrap strip of paper. I saw a video online for a hexa-hexa-flexagon I’d like to try, too (If I try it, I’ll edit my post and add it in!). How do we make sure that our students store their experiences with multisensory mathematics activities positively, so they are learning resources, not road blocks?

 

(I didn’t explore the bagel cutting myself, as cool as it looks, because, although I had some bagels in my freezer, they were pre-sliced ones!  No Flex Mex either – no tortillas and no need to consume any this week so I didn’t make the purchase. My imagination went to a Flexi-Crepe though – with Nutella or other such yumminess!)

 

Questions to Ponder

As in my last paragraph, how do we make sure that our students store their experiences with multisensory mathematics activities positively, so they are learning resources, not road blocks?

The snacks were fun to work with, but I resist using food in classrooms for a multitude of reasons. What other engaging materials that might stimulate the senses could be used for data analysis or geometric explorations? How might you use them?

In regard to the paper I read, what expectation is there on researchers to seek advice from experts in the impairment they are addressing to be sure their methods do not skew their results? For example, Fernades and Healy (2013) were surprised that both students approached examining figures in a symmetrical action – both hands moving on each side of the figure in a mirrored way. However, this didn’t surprise me as it is what I have observed people with visual impairment to do. How do they tease out what is a pre-learned strategy from what is developed through the tasks of the study?

References

Fernandes, S. & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind stduents' interactions with symmetry. RIPEM, 3(1), 36-55.

Week 2 - Reading Summary

 Fernandez, S. & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind students' interactions with Symmetry

This article describes a quantitative study that explores the experiences of two blind students through multiple experiences with symmetry and geometric translation.

The authors review and critique different theories and conceptions of the body and cognition, including rationalists (e.g., Descartes) and empiricists (e.g., Locke and Diderot) and explain their adoption of the understandings of Merleau-Ponty’s phenonmenology and its alignment to current work in embodied learning. The study is based in the perception that experiences are stored as “multimodal representations” that are “reactivated to simulate states of perception, action and introspection associated with an object, even when it is not present” (p. 40). The authors recognize that neuroscience can be used to support findings of phenomenological studies.

Additionally, the authors draw upon Piaget and Garcia’s stages of development of geometry in learners: intrafigural, interfigural, and transfigural. In the intrafigural stage, learners focus on the properties within a given figure or structure, the internal features. When working in an interfigural stage, learners draw upon references outside of the figure or structure, such as an axis of reflection, and compare separate parts of the figure to this reference point. The final stage, transfigural, sees figures or structures that result from transformations as “objects in their own right” (p. 41). These stages develop linearly and recursively. In this study, the authors reference intrafigural and interfigural stages in relation to the participants.

The participants were two blind students, Edson who had visual memories having lost his sight at the age of 15, and Lucas who lost his sight at the age of two and claimed no visual memories. Neither student had been taught about symmetry or geometric reflections. The method used is based on Vygotsky’s double stimulation where subjects were given tasks beyond their current level of knowledge but also provided assistance from the researcher to develop strategies for the tasks. The tasks in the study were designed to develop a sense of symmetry through folding cardboard geometric figures and reflecting figures across an axis of symmetry using a geoboard. Data was in the form of transcribed interviews with the participants as they worked, including the assistance the researcher provided.

Results of the study were discussed in relation to Piaget and Garcia’s stages of geometric development and to embodied cognition. The authors found that in the task of folding cardboard, the participants worked in the intrafigural stage, attending to how points on the figure related to each other, not to a point of reference outside of the figure. In this task, particpants also drew upon experiences; Edson made reference to folding clothes and the authors were clear that both participants used multimodal images developed within the task as they moved to a new iteration. They also note that while the participants developed a symbolic representation, it did not replace the physical memories of experiences. In the geoboard activities, the authors claim that the participants changed their multimodal resources and were able to work in the interfigural stage. Edson had visual memories of mirrors that aided his work with the assistance of the researcher in pointing out this set of resources he had. Lucas, with no visual memories to draw upon, drew upon guidance from the researcher to complete initial tasks which he then was able to use, in conjunction with his experiences on previous tasks, to complete further tasks. The researchers used their findings to question the linearity of Piaget and Garcia’s stages, suggesting that what stage learners use depends on the resources available. The article concludes with stating that this research helps confirm the link between the body and mathematical abstraction.

Week 2 - Data-able Snacks

 I thought I'd start this week by blogging one of my activities separately from my reflection piece.

Munchies Mathematics

The inspiration was found in Vi Hart's Math Improv: Smarties video. Oh, and that this was homework for the week!

I started with a bag of Original Munchies Snack Mix.  I LOVE Munchies too much so I don't bring it into the house often but having to do math with a snack was a good reason to go buy some!

I wondered about the proportion of the parts to each other - the pretzels, Doritos, Sunchips and Cheetos. Are the treats equal in representation in the mix? Is there one treat that is included more than the others?

I decided to go with weight to check this out, considering the ingredients are different sizes and shapes. The bag says there are 250g in the bag. Bag and snacks weighed in at 260g and when I weighed the empty bag, it was 10g so 250g written on the bag is accurate. (And now I have to believe the manufacturers reaction to complaints that the bag is mostly air when they say it is true to the weight indicated on the bag.)

Next step, sort the parts into bowls:

I actually had to swap the bowls for Sunchips and Cheetos part way through, having grossly over-estimated the amount of Sunchips.

Weighing in commenced and results recorded - and rechecked when they didn't add up to 250g.

Seriously, why bother with the Sunchips at this amount?!

Working out the ratios was next:

Finally, I thought, "What would my perfect proportions of ingredients be?" I can do without Sunchips and I wouldn't mind more Cheetos relative to Doritos and pretzels.  I decided on a Doritos: Cheetos: Pretzels ratio of 1:3:1.  I checked the bag for serving size and it was a convenient 50g, meaning I needed 10g of Doritos and Pretzels and 30g of Cheetos to make my perfect serving of Munchie mix and I weighed out the ingredients accordingly.

Finally, I thought, "How many perfect servings could I make with my one bag?" I have to look at the Cheetos because they are the ingredient requiring the most grams.  The total in the bag was about 78g and at 30g per perfect serving, I can only get 2 perfect servings out of a bag that contains 5 servings of 50g.

But Munchies are pretty good the way they are, so after enjoying one perfect serving, I threw all the remaining ingredients back together!

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.