Week 3 - Activity Reflection

It's been a long time since I last sketched. I greatly enjoyed it but due to the temperatures of the week There was a mixture of indoor and outdoor observations and sketches. I spent time outside just watching what was going on a few times, some during lunch at school as the students played outside in the snow some from my car in my neighbourhood. I live on a street near a park with a big hill so there is often a lot of foot traffic of kids heading out to sled and dogs going for their walks. When watching I wondered how to sketch some of the living beings with so many of the ones of intrigue to me were in motion. 

The lines of the living things I observed had a roundness to them they felt more fluid to me more soft almost. The angles felt they followed more of a flow as well, without the sharp or abruptness of a right angle. 

In the human made things there was more of a sharp or abruptness to them, following sharp lines, even when there were curves or cylindrical/rounded features there was still a sharpness to them. They felt more "orderly" maybe in a way. Perhaps since the majority of things I focused on that were human-made were stationary and stagnant. 

I think typically there is more soft edges and multi varying angles that are not usually 90 degrees in living things, but human-made have a more sharp angles and lines that contrast the living things. 

I think these patterns exist out of functionality, all of the human made things I sketched were stationary things, they are build not to move (the suspension bridge does sometimes sway in the high winds), where as two of the living things I chose to sketch I found to be challenging due to their frequent movement, the other living thing was a tree and while it is ultimately fairly stationary it does grow, move in the wind, get weighed down in snow so it is also in motion of its own. 

Close observation might be used more for pattern identification and perhaps noticing lines and angles to talk about the different types they see and where they see what kind of lines or what kind of angles, but I feel if only using observation students will need more guidance as to what they are looking to focus on and may not identify these components consciously if just left to observations. Drawing or sketching as a follow up to close observation could lend to students naturally making more connections to the line and angle types present in their observations, and I think would be beneficial as a natural lead to inquiry for them instead of explicitly asking them head of time to look and think of the lines and angles, enabling the conversation to be more reflective. 

An engaging way I think whole-body movement could lend to this experience would be trying to mimic observations through body movement. I find myself often thinking of how to gamify components of our activities (this year I have classes that love to turn things into a game or competition it seems) and I think of them taking time to do observations and sketching and then trying to be their objects to embody their lines, angles, and purpose. If I were to do this for the stacks (Tufts Cove Generating Station) I would stand up tall and have my arms going straight up with my hands bent at the wrist in towards each other palms down and would straighten my wrists and then return to the original position as if from the column created by my arms it was puffing out smoke this would be accompanied by a puffing sound. Could also be a collaborative body of movement done as groups with multiple people, similar to how the students in this weeks video Dancing Euclidean Proofs used collaborative movement between the two of them to turn three propositions from Euclid's Elements into dance. 

I enjoyed this weeks activity and felt it really created a multidisciplinary experience. There was art with sketching, mathematical components of perspective, lines, angles, the connection to the world around us through doing this activity in place and outdoors, science with considering the living and non-living, and linguistic components in the writing of this blog post trying to describe and communicate my observations and experience. After reading Williams' Sustainability Education's Gift: Learning Patterns and Relationships (2016) I found myself with questions around implementation of interdisciplinary activities and how to connect across disciplines while keeping the value of the activity for math but not taking away from the other components. I appreciate that I did this activity after my reading as it helped bridge some connections for me through a direct example and a firsthand experience.  

A picture of my dog enjoying one of the weeks moments of outdoor observations.

Week 2 - Activity Reflection

This weeks activity started with an optimistic trip to the dollar store where I had high hopes and my purchases had high sugar. I started with the hexaflexagon but tried to use sour straps to make an edible one. I tried and tried and rewatched the videos but struggled with visualizing the folds and directions while doing it, ultimately having to bring my laptop to the kitchen where I was experimenting with these activities so I could watch the How-To video while making it. I initially had measured and cut just a tiny bit into the sour straps to help guide my folding but found the straps started to tear as I was folding it more tightly. Second(ish) attempt, I managed to get the sour strap together and had wet the sides to stick together to get the sugar off and make the candy sticky, I then put a jar of honey on top to weigh it down while it started to stick together. This worked temporarily and did allow me to get a hexi-flexi flip but quickly fell apart where I had tried to join the sides. I did my best to reassemble before eating it and similar to my experience it was a bit of a tough chew, sweet, sour, but overall enjoyable.

After the ups and downs of the Hexaflexagon I got out the rocket candies and decided to just start playing with them and see where I ended up, Vi Harts video was vary fast paced but felt very playful so I wanted to let myself experience the natural flow of experiencing the materials. I found myself thinking of how to consider symmetry with the rockets. I arranged them in various ways thinking about how many lines of symmetry each arrangement gave, considering the initial ones I gravitated to and their structure, which initially followed very clear lines of consideration for lines of symmetry along x and y axes before I tried to branch into more patterned arrangements. Some arrangements were made with eyes open before letting myself close my eyes to  focus on and consider the feel. Followed by trying to create some arrangements of symmetry with my eyes closed considering what this experience might feel more like for students who don't have the sight. Throughout this I gave little consideration for colour of the rockets in the arrangement thinking more about the feel of symmetry and the overall patterns that could be created with them, while I do not have any students currently who are blind or with limited sight, I have had students who had colour blindness. and with so many hues of the rocket candies being similar I felt it was not something I wanted as a central focus to this activity though it could be incorporated with the symmetrical arrangements if desired. 

Focusing on the experience of the activities from the videos was different than I expected. In particular I was surprised with how many challenges I had with the hexaflexagon. I felt more confused when I was making it than I did watching the videos feeling I could visualize what was happening well until I was attempting it. I believe that the experience of learning from real 3D things, object, with shape, texture, etc., make the learning more memorable, and easier to recall for students later on when their learning involves a more experiential or tactile component. I started the geometry unit with my grade 8 students this past week and I start off our exploration of nets with a dissection activity where students look at and hold wooden 3D objects, create a hypothesis for what they think the net might look like, and then they dissect paper versions of the 3D objects (cylinders, pyramids with various bases, rectangular prisms, triangular prisms, cones, and cubes with a jolly rancher hidden inside) and they compare their resulting nets with their hypotheses. I had a discussion with some of my students that are now in grade 9 about the activity as I was preparing for it and they clearly recalled that day and experiment where as often when working with them if I remind them of a concept we covered they are not as consistent in recalling those lessons. The tactile component of mathematical experiences can present challenges such as in supporting all students through the experience and making the intended connections, as well as considerations for what resources/materials are available for teachers and students to have these experiences. I think there is great benefit to using hands on activities and experiments, it can support students who have sensory impairment to have another route of connection to the materials, ideas, and concepts related to the experiments/activities and it can support other students to strengthen and/or broaden their understanding and connection making. 

Week 2 Reading Reflection

Article Summary: 

Multimodality and mathematical meaning-making: Blind students' interactions with symmetry, this paper by Healy and Fernandes, examines the interactions of two blind students' explorations of symmetry. The two students this paper focuses on have different experiences with blindness. One student, they have called Edson, lost his sight in one eye at age four and underwent many surgical attempts to save the sight in his right eye but lost all but two percent of his vision in his right eye. Edson's remaining sight can distinguish between light and dark and the direction light is coming from (Fernandes & Healy, 2013, p. 44). The other student, called Lucas, was born with a congenital disease and had complete loss of sight by the age of two years old.

The article begins with a discussion of the relationships between what is perceived through the body's senses and what is conceived by the brain considering the positions of philosophy and neuroscience (Fernandes & Healy, 2013, p. 37). Describing a divide between an empiricist perspective (proposed by Francis Bacon), arguing more for science based on observation and experimentation, and the rationalist perspective (defended by Descartes), centering reason as the source for scientific certainty ( page 38). Rationalists assert that knowledge is not gained through the senses but through intellect and deductive reasoning, and that the thinking and awareness of such thinking are central to being. In contrast, empiricists assert the "idea that all knowledge is a consequence of experience" (Fernandes & Healy, 2013, p. 38). Locke and Diderot (empiricists) both argue this, Locke with the determination that knowledge is from ideas generated on the basis of  sensed experience and reflection, and Diderot with his work publishing about an interview with someone blind from birth (Fernandes & Healy, 2013, p.39) discussing the acquisition of knowledge being much more abstract, stressing along with Merleau-Ponty the essential need of body movement for touch similar to the need of light for vision.

Shifting to geometrical structures, there is an acknowledgement of the challenges this predominately visually associated field of mathematics brings to blind mathematics learners. Piaget and Garcia conducted a historical-epistemological  study on the psychogenesis of geometry structures, characterizing the three stages of development: intrafigural, interfigural, and transfigural.

Intrafigural: learners do not attend to the transformation of figures in a structured space, they are focused on internal properties of isolated figures, or on comparisons of the internal properties of two or more figures. Learners use references internal to the system under analysis. (Fernandes & Healy, 2013, p. 40)

Interfigural: learners consider that any change in the shape of a figure is a result of the dislocation of its parts, involve comparison between initial and final locations. (Fernandes & Healy, 2013, p. 41)

Transfigural: this stage concerns not only the transformation of one figure onto another but involves operations with all of the points of the plane and the verification of variation and invariants associated with different applications and conditions. Represents operations on a set of elements in which all the transformations can be composed and decomposed. (Fernandes & Healy, 2013, p. 41)

Both boys worked through the tasks with the first set using intrafigural strategies predominantly, this was likely due to the structure of the tasks presented in the first set. In the second set of activities, Edson, with guidance from the researcher to make a connection, used his visual memory of mirrors to support his understanding for the mathematical concept of reflection in his activity. Lucas did not have the same visual memory as Edson to utilize, so Lucas has to use his hands to seek and recreate these reflections, identifying and utilizing points of reference in the space, demonstrating more of a interfigural level at this point in time. Lucas had to revisit initial attempts of tasks a few times with support from the researcher, recognizing and adjusting his reflections accordingly. For the progression of Lucas' understanding throughout these tasks, the interactions he had with the researcher were essential to bring the materials and meaning together. 

The article concludes with questions of whether the similar symmetric path both participants used to initially explore the tasks were coincidental or a conventional approach. Emphasizing that, depending on what we know about sighted learning may not support the best learning design for the blind. 

Stop 1:

"If cognition is multimodal and if imagining involves reliving - and re-feeling - previous doings, then concepts cannot be seen as mental representations in which the abstract, logical universal properties of an object are stored in a somehow transcendental form stripped of the particularities of the settings in which it was encountered" (Fernandes & Healy, 2013, p. 40)

From early in this article, I was trying to recall the phrase where when you imagine something but don't get an actual visual image in your mind. From my googling, I believe the phrase I was looking for was aphantasia. I wondered when reading how it would tie in or relate to these understandings of knowledge and cognition in both people who are blind and those who are not, but experience this mental visualization. I wondered how the components of multimodal learning impacted the take-aways of it and the recollection, memory recall of participants, does the memory highlight the sensory experiences?

I liked how this quote acknowledged the relationship of experience and learning, as well as the challenge of mental representations in a vacuum void of the features present when introduced to or previously thinking of/ experiencing the thing trying to be visibly represented. 

https://aphantasia.com/guide

Stop 2:

"This intentionality shows the active quality of their touch and how images are made, not passively received." (Fernandes & Healy, 2013, p.43)

This quote was interesting to me, it was followed by the observation that both participants explored the figures in similar ways, "with an initial tendency to move both hands together, following symmetrical trajectories - something [they] had by no means anticipated when planning the tasks." (Fernandes & Healy, 2013, p. 43) It feels expected to me that there would be a methodical way in which people who are blind would explore a physical object or figure, and that it would likely be similar to each other. Just as often, there is a similar pattern to how sighted people would observe something with their eyes, looking up and down or side to side to gain their initial bearings. At least within the initial exploration, and with more time, it may lose some of the pattern of it as people narrow in on the details in different ways. 

Stop 3:

"... it would be a mistake to expect those who do not see with their eyes to necessarily follow the same learning trajectories as those who do." (Fernandes & Healy, 2013, p. 52)

I wonder if there is a trend to the trajectories that students who do not see with their eyes follow similar to each other? Or would it have a significant difference based on if they have visual memories or how much visual memory them have, like we saw the difference in the exploration of tasks with Edson and Lucas. I wonder how learning trajectories would look if considering the learning experiences of students who are deaf, would there be a trend amongst students with that same lived experience?

Question:

What methods do you use when engaging with symmetry as a sighted person? How accurately do you think you could explore or produce symmetry through touch, without use of your sight? What methods or processes do you think you would go through? 

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

 

Week 1 - Body Measurement Activity Reflection

I loved looking more directly and intentionally at body measuring through this activity. I have often resorted body measuring as a means of convenience when "eyeballing" something just wasn't enough. Prior to calibrating my body measurements for this activity there were some measurements I felt fairly confident about as I had at one point in time measured them, I knew from hand to hand and it was approximately a meter and a half, in this activity I measured my fathom to be 165.5cm, so it was close-ish. Most frequently I use body measurement to create consistency in spacing than for determining a specific distance.

Body measurement is so physically rooted in our everyday world as well as it being accessible mathematics for many people. The overarching idea of using the body as a means for measurement in place of manufactured measuring tools connects clearly the discussion of the ability to change perspective relating to understanding that Roger Antonsen's video demonstrated. 

I used body-based measurement to help space the gallery wall my partner and I have in our home that we were re-doing. I used the width of my hand to ensure there was a more uniform spacing between the art pieces and to maintain enough distance from the doorframe. The original gallery wall we had we would just add a new piece wherever it fit and it ended up feeling crowded and too close to the door frame. my partner had loosely grouped the art combinations they wanted together for the wall and we moved from the top left corner and made our way down and across maintaining the spacing to the best of my ability. 

This activity got me thinking about the next unit I will be exploring with my grade 8 students, which is surface area and volume. A common challenge I encounter in this unit is having enough measuring tools for them to use when working with physical manipulatives. I think I will be planning some activities around using body measurements as a tool for their calculations. Do you think having them calibrate their body measurements before or after having them use it as a means of measurement would be more beneficial? I can see that letting them calibrate it first might show them the direct relation quicker but I am curious if letting them create units out of their body measurements without its corresponding cm value would support them in shifting perspectives to using their body as a mathematical tool. 

[TED]. (2016, December 13). Math is the hidden secret to understanding the world Roger Antonsen [Video]. YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o

Week 1 - Reading Reflection

 Article Summary: Seeing the graph vs. being the graph

This exploratory study navigates ideas of gesture in graphing from Gerofsky's reflection of its natural presence in their teaching practice and in their students communications to its formation into initial exploration with people they knew by having them video themselves gesturing to describe graphs and its development into the more formalized research with grade 8 and grade 11 students of varying abilities using gesture to describe a varying set of 5 graphs. Predominately there were three types of responses that emerged from the students gestures that were broken down into 3 categories: 

"Category 1 students were precise and followed rules carefully, but often depended on memorization and algorithmic thinking rather than engaging fully with math concepts. ... Category 2 students' visceral, experiential approach to the graphs and multiple metaphors and verbal/kinesthetic/visual representations allowed them multiple potential entry points for sense-making and the creation of more robust mathematical conceptual objects. ... Category 3 students were in urgent need of help in learning to see graphs as whole objects and in bringing attention to those features of graphs considered mathematically salient." (Gerofsky, 2011, p. 253)

After reviewing and reflecting with the participants it was revealed which students were considered top students, average students, and struggling students and this strongly aligned with which of the categories students fell of the three described. This leads to further opportunities for exploration into the impact of intervention and the ability for observation of students gesturing in graphs as a a way to determine students patterns of noticing and engagement with secondary mathematics.

I had initially read this article in August during the MACAS conference after attending Gerofsky's keynote Opening up New Liminal Spaces for Learning, Research and Creativity with Mathematics and the Arts. I looked back to my original notes from my first read through of the article after reading it again this week to see what things caught me my first read through and what caught me this time, which ended up being very similar. 

Stop 1: An observation within the exploratory study about gestures in response to given graphs, "Although the gestures produced were not spontaneous gestures accompanying speech and were produced deliberately at the researcher's prompting, subjects were largely unconscious of features of the gestures they made." (Gerofsky 2011, pp.248-249). 

Throughout this article I found it fun to stop and engage in the physical gestures myself, and even when doing so with full intentionality I still was not consciously considering many factors that surprised myself. The origin being a significant point that I was not consciously considering or identifying in my choice of movements, upon reflection I believe I have identified the center of my chest to be where my gestures for this correlated the origin to be, though I don't think that is consistent when I have gestured in discussions of graphs in front of classes in the past. 

Stop 2: Figure 1: The five graphs used in the second pilot study (in schools).

Note: From Seeing the graph vs. being the graph: Gestures, engagement and awareness in school mathematics by Gerofsky 2011, Integrating Gestures p. 250. Copyright 2011 by John Benjamins B. V.

I took a minute to try to gesture the graphs used in the study myself. I found that I used a mixture of speed, sounds to communicate features of the graph, I consider the straight parts of the graph to be faster so my motions were faster when describing them. Sounds were often indicators of turning points as if the line was bumping into something to change its direction and turn. Reading through the categories I had features of both category 1 and category 2 in my gestures of the graphs.  

The gesturing of graphs as an alternative way of expressing them connected with the video of the week and its discussions on perspective and your ability to change perspective being tied with your understanding. The language used in Rodger Antonsen 2015 TED talk video Math is the hidden secret to understanding the world, discussing thinking of equal signs like metaphors felt like a light turned on, the connection felt so applicable, having just been finishing up a unit on fractions with my grade 8 classes so many conversations reviewing and discussing equivalent fractions I wonder how those could have gone with more integration into different perspectives and using language like metaphors in our conversations around equality. 

Teaching grade 8 and 9 where graphs are more formally entering the students math class worlds I have been wondering about ways to possibly gamify the gesturing to make it feel like something they may be more willing to engage with in a more full bodied way, almost like a graphing charades. Wondering what this would look like in practicality and if it would push for them to participate in the bigger motion exaggerated way or if they may start to find a uniformity and fall more to the category 1 grouping from the article. 

Question: In the gestures you make to communicate, how consciously are you considering the details that make up your gestures? When you read through this and saw the graphs did you also immediately start trying to gesture through them yourself as well? 

Citation: Gerofsky, S. (2011). Seeing the graph vs. being the graph: Gesture, engagement and awareness in school mathematics. In G. Stam, & M. Ishino (Eds.), Integrating gestures (pp. 245-256). John Benjamins Publishing Company. https://doi.org/10.1075/gs.4.22ger

Welcome All!

I am Colleen Nicholl. I teach Grade 8 and 9 Math in Halifax, Nova Scotia, originally known as Kijpuktuk which is a part of Mi'kma'ki the ancestral and unceded territory of the Mi'kmaw people.