Week 3 Reflections - Math Outdoors

 Changing Perspectives

This week I read all three articles. As I read each article, I kept coming to the idea of perspectives. (This idea has been rattling around in my head since watching the Antonsen talk in week one.) 

The article I am discussing is c) 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

The stated purpose of this article is to “apply stress to the grid concept, in order to discover some of its failures, and then propose alternative geometries better suited to our needs in many domains, including education” (Doolittle, 2018, p. 102).

In talking about the grid Doolittle asks, “How we can reflect reality better in our thinking?” (Doolittle, 2018, p. 102). Our reflection of reality comes from the perspective with which we see reality. How does our reflection change when we change our perspective? Are there perspectives that reflect reality in a more authentic way than the “grid” perspective through which we often unconsciously live our lives? Do we see the grid and its structures as being only beneficial, easy, comfortable, familiar, and giving us a sense of control; or do we also consider the possibility of dangers associated with using the grid?

Doolittle proposes failures of the grid, particularly with regards to roads and maps, giving examples of towns/reserves in which he has lived. He brings in Euclidean geometry as an example of a failure of the grid, its value is limited to “small, uniform regions of space” (p.108), essentially ignoring the power of nature. He discusses alternatives to the rectangular grid which include: the hexagon (consider the bees), Penrose tiling, Amman-Beenker tiling, and Riemannian geometry. 

Riemannian geometry gives us another way to consider perspective, taking us outside of the Euclidean geometric idea of grid. Riemannian geometry, Doolittle describes as applying the Copernican Principle that we are not the centre of the universe, to grids. “Accepting all kinds of straight and curved grids as equally valid ways to measure space” (p.111). No grid is better than another, rather we are encouraged to look past the grid to “refocus on the actual underlying geometry of the situation” (Doolittle, p. 111).

Indigenous perspectives are discussed through string figures and the time grids of Indigenous ways of farming. Indigenous perspectives are also brought in as Doolittle describes his approach to the Königsberg Bridges problem. Doolittle asked the question, “what after all this time, haven’t we done that we should have done?” Thus the problem is considered from a different perspective, opening up previously unconsidered possibilities and his answer to this question leads him to a solution.

The following quote, about the grid in relation to agricultural land caused me to stop and consider whether we could apply the same idea to students. “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. (Doolittle, p. 104) Are we “forcing” students into grid like conformity that goes against their natural “grain”?

Using a chaotic control analogy, Doolittle brings the discussion back to education. He states, “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, and in which direction to provide the nudge” (p116-117). How does this compare with the control inherent in living within the grid, where we (teachers) are in control, with little space of students?

How does seeing the grid as beneficial or as dangerous change the way we interact with our world?

In describing the path to the source of the river in the Königsberg Bridges problem, Doolittle maintains “the path… would be long and arduous… but it is available to the diligent, the brave, the caring, and the open-minded” (p. 119). It is my hope that as educators, we can be diligent, brave, caring and open-minded.

As I read through the introduction this week, the ideas of roots and connection stood out to me. Maybe it is because I love taking my students into the Delta Watershed and examining the roots of the big trees in the forest. We consider the interconnectedness of the roots, what is happening beneath the soil, the life under the soil, how deep the roots might go in comparison to the height of the tree, the path that the roots take etc… (lots of good mathematics related to roots!). I am drawn to the idea of students understanding the sources and roots of mathematical ideas as well as how this can be explored and discovered by being outside. We explore the fractals that we see everywhere! (Doolittle discussed complexity theory as a superset of fractal geometry.) “What do you notice” and “what do you wonder” questions draw students attention to how mathematics shows itself in the natural world of the forest. How do we move beyond these questions with young students? The land as an entity is a great teacher. 

This week I attempted the activity a few times. My drawings… perhaps I will keep those to myself for now, but… as I was exploring the world through mathematical perspectives, I used my camera to capture images that highlighted for me natural lines and angles – forms that would not fit easily with the grid. The complexity of patterns and lines found in nature are unmatched with human-made things. We only come close when we mimic nature. 

I have not used a lot of whole body movement in teaching mathematics outside, but I am inspired by the Dancing Euclidean proofs to consider ways to allow students to explore their understanding using their bodies.

Week 2 Reflections - Multisensory Mathematics

 As I was reading the introduction for this week, I stopped to mull over the question, “What would mathematics lessons be like if we took multisensory learning seriously?” The principles of UDL came to mind. Ideas about discovering mathematical relationships and using a variety of representations, supported this proposal that multisensory mathematics lessons embrace the principles of UDL (Universal Design for Learning). UDL has 3 guiding principles which include providing multiple means of: 1) Representation – the “WHAT” of learning 2) Expression – the “HOW” of learning and 3) Engagement – the “WHY” of learning.  You can look here for more information. Universal Design for Learning Guidelines

UDL proposes that we start with the belief that all students are capable and aspires to take away stigmas that have separated students with disabilities from the group. Gerofsky wrote in the introduction, “innovations supporting learning for students with sensory impairments will support learning for all” (Gerofsky, 2022). This is UDL in action. How would schools change if we stopped thinking of students as being disabled and instead considered how our environments – social and physical – were disabling. It changes the conversation completely. Can we stop thinking through the binary (false binary) of abled and disabled?

The article I am summarizing this week is c ) Angelika Stylianodou & Elena Nardi (2019), Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. This study’s aim was to “contribute to inclusion and challenge ableism in the mathematics classroom” (Stylianidou & Nardi, 2019, p. 343). The study challenged the effectiveness of using only the typical senses of sight and hearing in the mathematics classroom. Citing the Convention on the Rights of Persons with Disabilities, the paper defined two different types of inclusion: reasonable accommodation and universal design. The study involved a grade 5 classroom with a visually impaired student and drew upon Vygotskian sociocultural theory and the theory of embodied cognition, among others. The study engaged students in exploring shapes through touch first, prior to exploring the shapes through sight. The authors considered the wholistic experience of sight compared to the gradual experience of touch. Results confirmed that both VI and sighted students developed a deeper understanding and more accurate description of the properties of the shape through the sense of touch. A sighted student commented that using touch revealed, “hidden facts on the shapes.” (Stylianodou & Nardi, 2019, p. 348)

Mathematics is probably the subject that teachers struggle the most with implementing the UDL principles, likely due to the Platonic, Cartesian worldview that has been the basis of most mathematics instruction in schools. Multisensory, embodied activities create connections for students, allow students to understand relationships between ideas, and can enable a variety of representations. Using a variety of representation can lead students to see different perspectives and, as Antonsen proposed last week, result in deepening understanding. This may occur through the enhanced metaphors that are develop as students interact in these multimodal, multisensory activities.

These thoughts lead me to consider the difference between knowledge and understanding in mathematics. Has what many would consider to be traditional mathematics instruction, led to knowledge without understanding? 

In multisensory mathematics I see the coming together of theories of learning, use of imaginative education tools (see here for a mathematics example Learning Math through Stories) and practical applications that build rich and engaging opportunities for students to make sense of the mathematics they are learning.

Reflection on Activities:

Although I am acutely aware of “playing with” food, when there are food insecurities all around us, I can see the potential for using the sense of taste (and multiple senses) in mathematics. (And if the food used is actually eaten as it was with these activities, no food insecurity issues.)

One of my boys and I tried the Mathematically Correct Breakfast activity of cutting the bagel together. We were both successful on the first try (although my success was directly related to Emmett helping me!). This activity revealed to me the increase in surface area (and thus a greater cream cheese to bread ratio - Mmmm) in a way that would have been difficult for me to perceive without actually experiencing it. It helped me to "see" this and "taste" it and has me considering it in a way I wouldn't have without experiencing with my senses. 

Making the hexaflexagon was easier for me this time. I had made hexaflexagons for a previous class, worked my way through the productive struggle and found success came easier this time. Eating the hexamex flexagon burrito was just fun and engaging – ripe with possibilities for conversation and thinking mathematically. 

As I considered the Rockets video, I started to brainstorm different kinds of activities with candies. One of my boys is a huge Reese’s Peanut Butter Cup fan. For Christmas, we gave him a box full of all things Reese’s (and there are a lot of Reese's products!). Many activities could be done comparing the different types of cups – the chocolate to peanut butter ratio (it would be interesting to know the difference between the 1/2 pound cup, regular cup and the thin cup), weight to calories etc.

Although we would be unlikely to complete a Reese’s comparison in class, I have used candy to graph and practice addition with Kindergarten and grade one students and they loved it.

These real 3D multisensory objects offer multiple ways for students to experience the mathematics in the world. 

The activities this week have given me lots to think about!

Reading Reflections on Nathan, M. "Expert from Foundations of Embodied Learning

 "Education is basically about engineering learning experiences." (Nathan, p.3)

This quote caused my first stop, as I considered the simple yet complex nature of the statement. Teacher development is about the kinds of learning experiences we present for students. I believe all teachers want to create and present the best learning opportunities for their students, yet what that looks like varies greatly. Herein lies the dilemma of education - about which much is written, argued about and copious resources developed to support the learning of students. How do we engineer the best learning experiences for students? Does this change depend on the time period during which we are teaching or the culture in which we teach? How does the composition of the classroom and the individuals therein (students and teacher) affect the best learning experiences? So many questions... can we even agree on how to define learning? The author of this article includes a definition of learning as "changes in our behaviour." The author goes on to explain that, "a pattern is emerging...that embodied learning is a natural human activity, and it is possible to design for it and harness it in ways that can inform educational practices and policies in order to usher in a new era of educating the embodied mind." (p.5)

Another stop for me (there were many in this article) came with this statement, "Meaning and sense making through personally grounded ways of knowing are not the primary objectives of these scholastic experiences. It justly matches what students most often say when asked what they are doing in school: 'I don't know'." (p.6) It is so true, that many students don't know how to explain what they are learning, and yet isn't it true that to truly understand, one should be able to explain understanding to others?

The last stop I will discuss has to do with assessment. As educators, we often create "restrictive environments" (p.7) to assess students. Does this really allow teachers to develop understanding of what students know? I have recently been experimenting with more open-ended ways of assessing students' knowledge and have come to see that traditional assessments only give us a tiny glimpse of student knowledge, knowledge which may go well beyond what students are able to show in traditional assessment.

As I read this article, I found myself often in agreement with the author. The article discussed the use of cognitive tools, particularly grounding and linking metaphors, based in embodied experiences. Ideas of equity and enthnomathematics were discussed. I have asked lots of questions above, feel free to answer (or add your thoughts about) any of them but I will also end with a final question. How does embodied learning support students in sense-making and meaning-making in mathematics?

Activity Reflection - Body Measurement, Outdoors and In

 I completed the activity this week (on calibrating body measurements) with one of my own boys and it reminded me of a body measurement activity that I did with my class last year. (Since I have moved out of the classroom and with Covid restrictions, I don't have the same kind of access to students as I did in the past.)

With my grade one students, we began a measurement activity by discussing height and different words we use when we are measuring. I showed students a photo of an old doorframe from my house (we saved it when we renovated our house) where we had kept track of my boys height each year. 

Doorframe Measurement

With the help of my EA, we measured the height of each student with string, cut the string and labeled it with the students name. I gave the students a 'Field Journal' which we set up prior to heading outside with categories like "shorter than me, longer than me and the same as me." We headed to the forest by our classroom and student were directed to use their string to measure whatever they chose in the forest and make notes in their journal. In the photos you can see students happily measuring and recording.

When we returned to the classroom, we debriefed our measurements and shared our finding. We discussed the benefits and drawbacks of measuring in this way, and we used the language of measuring in our conversations. We furthered the activity by lining up in our classroom - shortest to tallest and then we took the strings from each person - in height order - and made a graph on the floor with the strings.

This was the first lesson of several on measuring using non-standard units (in this case our height). Students were involved in the lesson and the embodiment of the lesson furthered their understanding, their connection to the learning and their enjoyment.

Later on in the year, we used our knowledge of measuring in the practical application of measuring the growth of potato plants we were growing,. This time we used rulers and then metre sticks to measure. The young students were able to use standard measures of instruments because they had built up their knowledge of measurement, starting with using their own bodies. 

These activities demonstrated, as stated in this week's introduction "mathematics coming from and going back to our bodily experiences of the world." (When we were done growing the potatoes, we counted how many we had grown from the seed potatoes and then we ate the potatoes - which helps our bodies grow! A full circle experience, so to speak.) The mathematics were meaningful to students because it was connected to their lives. We needed the symbols of mathematics - for recording and keeping track of growth and comparisons but we connected these symbols with our real world experiences and who we are - a far cry from the Platonic prejudice, from the ideas of mathematics as being solely mental and other-worldly. As the title of Antonsen's  Ted Talk, "Math is the hidden secret to understanding the world," I believe these measuring experiences helped my young students to see not only the world but the mathematics with in it from different perspectives, with different mediums, and hopefully in a way that developed their understanding, empathy and care of the world.

                                                 

 

One of the most important math facts about me....

boys

I have

three