Week 3

 This week I read the article “Dancing teacher into being with a garden, or how to swing or parkour the strict grid of schooling” by Susan Gerofsky and Julia Ostertag (2018).  I found this article interesting but difficult to read, as it has a philosophical, but also critical, take on the current structures (the grid) that have been ingrained into our society and therefore educational practices.  The article attempts to provide educators with ways of teaching inside/outside of the grid structures, or alternatively, ways of seeing the grid from a new perspective so as to reimagine the possibilities within the grid.

Essentially, our society is built upon the very linear (or grid) Western point of view.  Whether it’s time, data tables or architecture, evidence of the grid system is all around us.  It can feel comfortable, logical, and clean.  However, it’s actually a representation of our need to control and organize the unpredictable nature of our world.  Whether it’s exerting dominance over new territories through gridded maps, or sorting hundreds of students into timetables, we use the grid to help us feel in control and powerful.  

Some examples of grids in my life!

        (Kitchen linoleum)                      (School calendar)

Exploration of gardens and other outdoor spaces is proposed to be one way of rejecting the grid.  Outdoor spaces encourage multisensory learning, as opposed to the indoor classroom that depends heavily on vision for learning.  Making use of multiple senses alongside vision works against the “inherently visual lines of the grid”.  It also allows for students to explore their creativity, imagination, and the roles of teacher and student become blurred as students share their knowledge and experiences.

But how do teachers escape the rigid, time-based structures of their day?  Even if a teacher moves their class outside, they’re still bound by bells, schedules, and even the days of the week.  This love/hate relationship poses a problem for the teacher who is attempting to reject the grid.

STOP - At this point in the article, the authors delve into an exploration of stepping out of clock time and into dreamtime.  They describe a learning experience within the UBC gardens in which students and teachers napped together and then shared their dreams which were creatively captured through sewing, drawing, etc.  Maybe it’s just my inability to let go of conventional structures and my concept of time, but I found this very difficult to relate to.  I just can’t imagine myself completely letting go and not having eyes on my students at all times.  It made me wonder if this experience only worked because the students were adults?  Or am I the problem and this would be totally realistic in a typical highschool setting?

In acknowledgement that teachers must work within the very strict and rigid time and space structures that construct education, the ideas of swing dancing and parkour are used as analogies to describe how teachers can work within and outside of these structures.  

STOP - These terms require an explanation in order to fully understand the analogies.

“Swing is extreme coordination.  It’s maintaining balance, equilibrium….Everything about the swing is about some guideline and some grid and the elegant way that you negotiate through that grid.” (Marsalis, 2011)  In essence, swing dancing requires moving “from, past, and back in sync with the beat.”  When compared to education, the beat is the rigid grid system we are bound within, and teachers can swing in and out of the grid, but always moving in time with it.

“Parkour remaps and even recreates urban space, creating a city parallel to the one defined by rigid structures and grids” (Geyh, 2009, p. 158)  Parkour is a way of moving efficiently and fluidly through urban spaces in a way that subverts the structures intended uses.  It reimagines the grid system in order to move faster and more elegantly.  This analogy invites educators to reimagine the grid they are confined within, and if needed, to work against the hierarchies, systems, and structures. 

The article ends with more questions than answers.  In particular, how will teachers adapt to a rapidly changing society, environment and climate? Will they be able to step outside of the grid, or remain stuck in its familiar comfort?  What implications will this have on our students as they face an uncertain future?

Final Thoughts - I found this article to be both inspiring and depressing.  I connected with the authors’ descriptions of the rigid timetables and structures in education, as that is a source of constant irritation within my own teaching practices.  I often feel confined by the rotating block schedule, length of each class, and even the months that school is in session.  My teaching partner and I spend an inordinate amount of time trying to plan our lessons in a way that allows for more freedom and flexibility within these structures, and it’s exhausting.  So I was inspired by the author’s analogies, and suggestions on making use of school gardens to work against the grid.  I was simultaneously depressed by how prevalent the grid system is within my reality, the exhaustion of working around or against these structures, and how oppressive they are to those without a western world view.  

Do you feel confined by the grid system?  In what ways?  How do you see yourself swinging (working in and out of the grid) or parkouring (working against the grid)?

References

Gerofsky, S., & Ostertag, J. (2018). Dancing teachers into being with a garden, or how to swing or parkour the strict grid of schooling”. Australian Journal of Environmental Education, 34(2). 172-188

Geyh, P. (2009). Cities, citizens, and technologies: Urban life and postmodernity. New York, NY: Routledge.

Marsalis, W. (2011) PBS Newshour interview with Jeffrey Brown, June 11, 2011. Retrieved May 2017 from http://www.pbs.org/newshour/bb/entertainment-jan-june11-marsalisjazz_06-01/>

Activity

I chose to stay inside to complete my sketches, as the weather was very wet this week!  Disclaimer here, in that I’m a terrible artist and these took me longer than I’d like to admit.  I’ve titled them in hopes of clarifying what they’re supposed to be.  

I started with a human made thing first, as I believed it would be easier with the straight lines and predictability of an immobile object.  I was quickly proven wrong when I struggled to mimic the rigid pattern of lines on my placemat.  It turned out to be a lot harder than I predicted, and I think it’s because of my desire to make it perfect.  I was less forgiving of myself when sketching the human made objects.  

When I sketched the living things, I started with another immobile object in the form of a cactus.  I gave myself grace when I couldn’t mimic the curved lines exactly, or recreate the shading along the ridges.  Finally, I did a quick sketch of my husband as he sat at the kitchen table on his phone.  I’m the most proud of this one, as I feel it turned out the best despite him moving around occasionally.  I remembered a lesson I had the art teacher do with my math class on proportions a few years ago.  In that class we practiced doing 30 second sketches of her while she posed.  She encouraged us to think of the overall shapes and their proportions to each other, rather than obsessing over the small details such as facial features.  She then showed us how to blend the shapes together to create a more lifelike figure.  I used these tips when creating this sketch, and I think it worked out much better in the end.

Week 2

    This week I chose to read the article “Tactile Construction of Mathematical Meaning: Benefits for Visually Impaired and Sighted Pupils” by Stylianidou and Nardi (2019).  The article summarizes the study conducted by the authors with elementary math classrooms in the UK.  The goal of the study was to determine the impacts of tactile learning on both sighted and visually impaired (VI) students.  Traditionally, visually impaired students are provided with adapted tasks and materials that are typically used for sighted students.  These adaptations are often limited, and can be exclusionary rather than inclusive.

STOP - An example of an adaptation with limitations is making use of counters with different textures that would represent the different coloured counters typically used for sighted students.  This can lead to a processing overload for VI students and complicates communication with their sighted peers during the mathematical task. In order to overcome these limitations, the authors suggest transforming the visual task to an auditory task.  This would allow students to experience the task with the same sense, and improve communication between students.  This made me pause in my reading and think about a couple of things.  One, as someone that is sighted, I am incredibly unaware of how VI students are interpreting the adaptations that are being provided.  I honestly would have assumed that having different textures to represent colours would be a great adaptation, and not even considered how hard it would be to communicate that information with someone who is using a different sense.  The other thing I thought about was how many of my students have some form of impairment (physical and/or cognitive) and whether it’s possible to have a truly universally inclusive math task?  As a teacher, how do you ensure that the math tasks/problems are inclusive of all when there are so many considerations to keep in mind?

The tactile learning task that was studied, involved the exploration of shapes through touch.  In particular, a shape that is a circle minus a circular segment (see the figure below!).  Both sighted and VI students were asked to describe the attributes of the shape using touch, with sighted students also using their sense of sight.  The authors were interested in determining if vision may generate a misinterpretation of the shape's attributes, since at first glance the shape appears to be a circle.  They were also interested in whether touch would generate a more accurate and detailed interpretation of the shape.  

After analyzing the results, the authors concluded that tactile exploration of the shapes lead to better inclusion of VI students within typical math classrooms, while simultaneously benefitting sighted pupils by changing their perception of the shapes.  Both VI and sighted students were able to distinguish the defining features of Shape X through touch, but the sighted students had a difficult time noticing the flat edge with just their vision.  The contributions of the VI student were noted to be “practical” and “refreshing”, and were used by the teacher to further the class discussion.  They propose that designing tasks that allow all students to experience mathematics through the same sensory tool will create more inclusive classrooms and challenge ableism within mathematics.

STOP - The conclusions of the authors in this article support the ideas found in the introduction to this week’s topic.  Students with a sensory impairment can provide a different perception of the same pattern/task/problem that students without the sensory impairment may otherwise have missed.  This reminded me of the Antonsen video from last week, where a change in perception was linked to developing a deeper understanding.  It made me wonder, if multi-sensory experiences are helpful in changing perceptions, how do we ensure that students with a sensory impairment also get to experience multi-sensory learning?  How do we support them in developing multiple perceptions if a multi-sensory experience isn’t an option? 

While I appreciate that multi-sensory learning opportunities lead to increased understanding amongst all students, I still have the following questions:

1. Is the concept of multi-sensory learning in math being taught within teacher training programs?  It definitely wasn’t when I went through about 15 years ago, and I’ve not heard any of my younger colleagues talking about it.  

2. Where can teachers find resources or information on incorporating multi-sensory tasks within their classrooms? 

3. How do we incorporate these multi-sensory tasks in a sensitive way so as to celebrate the different perspectives of the impaired students without making them feel targeted, or singled out unfairly?

Activity Discussion

For the activity this week, my youngest child explored the data within snack sized smarties boxes.  When it’s not Halloween, the candy selection isn’t great in my town!  We started by opening just one box, and counting up the colours and then put them into a bar graph.  You can see her work in the photo below. 

Eventually we determined that the average number of smarties in a snack size box is 10, and that the colour distribution is completely random.  However, for some reason she ended up with mostly brown, purple and pink.  She also noticed that within each colour group, some of the smarties are faded while others are more intensely coloured.  Sadly we ran out of time to explore that further.  

I also made (and played with) a couple of hexaflexagons out of paper.  It was really hard at first, as it turns out I can’t fold an equilateral triangle to save my life.  I did however, create a lot of right triangles which then fold up into squares.  They don’t flex in the same way as a hexaflexagon, but I still thought they were pretty neat.  I did eventually master the triangle folding technique, and my best hexaflexagon can be seen here.  Given my folding struggles, I decided not to attempt the taco hexaflexagon.

Week 1

 Article Summary and Stops!

This week I read the article titled “Seeing the graph vs. being the graph” by Susan Gerofsky (2011).  In this article, the author describes the first two years of a study conducted around the use of gestures within graphing, and their relation to one’s overall understanding of the concept.  Historically, gestures have been studied alongside oral language as a form of communication when observing the process of concept formation.  The math concepts that are typically represented in those studies include patterns and relationships , size, space, quantity, and time. This study differs in that it focuses on the gestures used to describe graphs as a whole, and that the gestures were elicited through the use of pictures. 

*STOP - This part of the article caused me to stop and wonder what other areas of math could be included in a study about gestures and communication.  I started trying to formulate a list of math topics not listed above by going through the different concepts that I teach in grade 8 & 9. It struck me how many of the topics that are typically included in these types of studies are measurement based, or easily depicted with visuals.  Are gestures helpful in learning, communicating and understanding math concepts that are not inherently visual, spatial or measurable?  What would those topics be?

Participants (students) were asked to describe the graph in each picture using gestures and non-mathematical words.  Their responses were coded and categorized for the types of gestures/movements that were used and the descriptive language or metaphors that were used.  These categorizations were then compared with how the students performed in math class over the course of the year.  It was found that the students who used whole body movements and rich metaphors to describe what they saw in the graphs, were consistently identified as “top” students within their class.  In other words they were “being the graph”.

*STOP - In the video, Roger Antonsen states that the use of metaphors, imagination and storytelling indicates true understanding of a concept.  This seems to support Gerofsky’s (2011) finding that the “top” students were able to make use of multiple rich metaphors to describe what they saw in the graph.  This made me stop and wonder if there was a cross-curricular opportunity here between mathematics and humanities courses?  For example, both teachers could work on storytelling, descriptive and figurative language, and creativity within their classes as a way of helping students make connections between the 2 courses. 

Those who used small, controlled gestures that often involved eye level tracing of the line shape were consistently rated as “average” students.  They “see the graph” but haven’t yet achieved a deep understanding of the graph as a whole.  Finally, the third group struggled to describe the graph as a whole, and often used inaccurate language or gestures to represent it.  These students (except 1) were consistently identified as “struggling learners” and were unaware of the features of graphs and other concepts such as symmetry.  The author concludes that “being the graph in a fully embodied way fosters engagement and attentiveness far more than merely seeing the graph” (p.254)

Final Thoughts/Questions - I wonder why the “struggling learners” are not able to see the graph?  Is this a learning disability?  Do they have poor attendance and therefore major gaps?  Do they also struggle in other classes such as English that require the use of similar skills?  How do we support these learners in a different way?

Response to Introduction, Video, and Activity

First of all, let me just say that I feel very vindicated with this week’s activity!  I am well known within my house for using anything but a measuring tape when buying furniture, hanging pictures, gardening etc.  My husband is often exasperated with my insistence that I don’t need to use a measuring tape, and he’s convinced that when I do use a measuring tape, I usually end up with the wrong measurements.  He very much subscribes to the idea that math equals precision and accuracy, and that there is one right way to do everything.  That using my body to measure is childish and somehow lesser than his way of measuring.  His way of thinking is reflective of the current methodologies often found in modern math classes, in which only young children should need to rely on such “simple” strategies. 

My daughter Hannah (age 7) volunteered to help me with this week’s activity.  She’s had some prior experience with using her body to measure objects at school, but had not taken the extra step of using a ruler to measure herself first.  Below is a picture of the measurements that she recorded!

It was interesting to see her interpretation of the measurement descriptions, as well as watch her use a ruler for the first time.  She was unaware that there were 2 measurement systems on a ruler, and that there would be 2 different numbers as a consequence.  It also took awhile to convince her that “0” was not the same as the edge of the ruler, and that it was OK to have numbers that were not whole numbers. 

Initially, we were planning on measuring my car and the garage (it has oddly sized doors as opposed to the standard garage door), as a means of finally answering the debate on whether I could be parking in the garage or not.  However, the -40⁰C wind chill forced us to stay inside and bake cookies instead.  So she decided that she would measure the size of a cookie using finger widths, which you can see in the picture below.  She determined that the perfect cookie size was 4 finger widths or approximately 2 inches.  Anything smaller was too crisp, and anything bigger was too soft.  Perhaps not practical to most people, but in the world of a 7 year old who’s learning how to bake independently, it was a pretty big deal to get the right sized cookies.

Overall she had a lot of fun and really enjoyed having creative control of how and what she measured.  As a child with ADHD, she also found that being able to move while learning was helpful in sustaining her focus for a longer period of time.  Ultimately, she was very proud of herself for solving the perfect cookie problem!

References:

Gerofsky (2011).  Seeing the graph and being the graph, pp. 245-256.

Roger Antonsen (2015 TED talk 17:04) Math is the hidden secret to understanding the world. <https://youtu.be/ZQElzjCsl9o

Hello!

 Hi!  My name is Courtney Lepetich, and I work and live on the traditional territory of the Lhtako Dene Nation.  I teach math and science at Quesnel Junior School (grades 8 & 9).  I have 3 children aged 14, 11 and 7.