Day 1, Saturday June 1: Earth

On Day 1, we get the chance to meet each other and meet the place we will be working and learning from: the Boisé de la Santé ('Health and Wellbeing Woods') on the Laval University campus.

Activities for today:

Sit Spots (see this short piece by David Strich for some background)

A sit spot should be chosen as a place that somehow calls to you -- a spot that you will be happy to revisit frequently, observe with all your senses, notice things and fellow beings that share this spot, and be a kind of 'home base' for your experiences in this place. 

We will take about 15 minutes at the beginning of each day to visit our sit spots and notice, reflect and connect with what is going on there. There will be prompts and suggestions for a variety of ways to connect with senses, feelings and ideas each day.

(Some of our suggested prompts may involve:

• Writing mathematical poetry/ecopoetry
• Soundscaping
• Sketching with an eye to line and angle 

…and more…)

Land shapes (Nenad)

In this activity, we explore mathematics and develop the language of walking on a surface such as a path, hill or a depression. 

Things to try: 

-Walking at the same height. What is the shape of the path that you are tracing?

-Walking the steepest path (make sure it is safe). What does the path look like? How long is it?

-Walking the least steep path. What does the path look like? How long is it?

-Try optimizing other elements (length of path, speed, the angle your body makes with the surface)

Touch and texture (Nenad)

What kinds of language, geometries, diagrams can you develop for different surfaces that you see? Finding, drawing, photographing and patterning in different tree barks

Drawing leaves with compass and straightedge (Susan)

This activity was inspired by Kimberly Elam's book, Geometry of Design (2001/ 2011), PrincetonArchitectural Press. Elam looks at poster and painting designs and designed furniture, cars, etc. in terms of compass and straightedge compositions.

What if we did something similar with, say, a leaf, or other living things in the forest or garden?

Here’s an article written by Robyn Massel (then a teacher candidate) and me about just such an experiment with her high school math class. 

Let’s try it: first choosing a leaf (or blossom, or …?) that has an intriguing shape, and then working out a way to draw it as closely as possible to its real shape using just straightedge and compass.

Paint chips and colours in the forest: Using paint chips in a variety of greens, browns, yellows and other colours from a local paint store.(Susan)

Can you match the colours of plants (leaves, flowers, bark) with the paint chips? 

What does this help you notice about colours in plants?

Homework reading:

from Robin Wall Kimmerer (2003) Gathering Moss: pp. 1 - 13 -- "The Standing Stones" & "Learning to See"

Homework viewing:

Fields Medal winning mathematician Manjul Bhargava short film on the origin of the Hemachandra numbers

Link to spreadsheet with our contact info (so we can invite everyone to co-author on this blog)

 Here is a link to the Google Sheet for our contact details: name, email (gmail if you have it, because
Google loves Google...), phone number if you're willing to share, affiliation and city or town where you live.

By the end of Day 1, we will invite everyone to be a co-author on our blog, so that we can all add text, photos, video, ideas, responses, links, readings, viewings, etc. 

This should be helpful when we write up our account of the working group for the proceedings after CMESG as well!

Some of our big questions for this working group!

Here are some of the big questions we want to address in our working group. We'll refer back to these

throughout our three days together:

1. How to integrate outdoor, embodied, sensory experiences with more typical mathematical work (writing, diagramming, symbolizing, calculating, etc.)?
   
   Some starting ideas: Connecting representations in ways that promote inquiry through both embodied and symbolic ways. Moving between representations in an
   
   
   
   
   
   oscillation or transdisciplinary way (weaving to poetry to mathematical diagram to group theory and back to weaving…)
   
   
2. How to help people re-story mathematics – tell themselves new and more inviting, more generative, more helpful stories about what mathematics is and how they might engage with it, through the natural living world, through the arts, through sciences?
   
   Some starting ideas: Mathematics outdoors in the living world as a way to move beyond fear/
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   rejection of school mathematics that many people experience. Can we get away from the stereotype of math as super-nerdishly precise, obsessive with precise measurement and calculation? (These are a small part of math but not necessarily what we love about math.) Math as patterns and their behaviours based on experiences of the world. 
   
   
3. How does observation in and with the living world inform mathematical ways of knowing?
   
   Some starting ideas: How can we observe and interact with the living world in ways that connect to mathematical observation? Mathematics as about careful observation in a way that leads to questions. (Aspects of observation in a mathematical way? → angle, line, shapes, distance, connectivity, chirality, concavity/ qualitative aspects of geometry, number, repeating or non-repeating patterns, transformations of figures or patterns; planes, surfaces, gradients, like exploring the ground in certain ways; curves, slopes, contours/ contour diagrams as in geography and 3D math; mappings of places and other things; space-filling curves, shortest or other optimal paths, optimization, relationships of operations on elements, how slight initial changes have large repercussions (chaos theory,...)
   
   Blueberry picking analogy, or koalas and kangaroos anecdote: you can see things once you are alerted or aware of them – a mathematical ‘lens’ or hearing. How are you introduced to things, so that you can build awareness and start to see, hear, smell, notice things in a new way? “Math is everywhere” iff you know how to notice it! You have to know a bit about it to be able to notice it!
   
   With these outdoor, embodied and less conventional math activities, you are learning how to see and notice and be aware of mathematical things and relationships. It takes some orientation (from a more experienced mentor, sometimes) and then patience and thoughtfulness. This is where students can become experts too. There can be a sense of mutual learning.
   
   
4. How to integrate mathematical learning with Indigenous ways of learning, knowing, caring for and living in good relationship, knowing that we are part of the greater-than-human world?
   
   Some starting ideas: What potential harm does mathematics bring to the living ecologies of our world through separation of “us” humans from everyone and everything else that we mutually depend upon for our lives? What is gained and what is lost from that kind of separation? Is it possible to appreciate and promote the wisdom that worldwide mathematical traditions bring, while also feeling and valuing respectful and reciprocal relationships with ‘our grandfathers, the rocks, and our grandmothers, the plants’, as well as all the other beings here? How can mathematics educators learn and connect with Indigenous colleagues and teachings to begin to find ways to move forward together?
   
   
   
   
5. How to generate mathematics/ mathematical understandings through mathematics outdoors?
   
   Some starting ideas: Justin Dimmel: The sun produces perfectly parallel rays – nature generates mathematical objects that we can work with in so many ways! The sun brings life and warmth and also brings geometry. What about trees – what do they offer and gift us? What kind of logic and math is embedded in it?We are not applying mathematics to something (like estimating number of trees in the forest for data, etc.); we are generating mathematics in different ways. We can listen to the pattern of the wind in the trees. This is transdisciplinary work. Math is also about naming and classifying: so if you have certain sounds, can you classify how you discern them? New Yorker article: training AI to listen to the language of sperm whales to learn what the whales are saying by making some sort of classification of sounds. (Example, Beaufort wind speed scale: flag is hanging down, or is flapping gently, or is making a lot of noise and flapping hard – each corresponds to a different wind speed) Is there something like a Beaufort scale that helps us recognize, name and work with other observations of the world in mathematical ways? Ex: qualitative observation of what happens to umbrellas or ocean wave → 40 km/h winds. Observing different hues of green in Lake Ontario in different seasons and winds – what can you start to recognize with experience? In Charleston, the Gulla people of the Carolinas make and sell sweetgrass baskets: how do you classify different turns and different angles in the basket making? There is specialized vocabulary and ways of noticing… how do we connect this with mathematical ways of thinking?
   
   
6. a) How to understand the intrinsic interweaving of movement (rather than only static images) with mathematical patterning and mathematical interpretations? 
   
   b) How to have experiences in the outdoors that let us read backwards from the sensory gestalt phenomena to know what we might be looking at, smelling, hearing, tasting, touching?
   
   Some starting ideas: From Gerofsky & Milner sprang paper: “In a sense, a weaving is a record of movement, of certain actions and interactions, just as certain graphs are records of movements or actions. In the braid theory sense, a completed sprang weaving is a record of crossings, of chiral permutations; it is the composition of each enacted crossing that defines a particular sprang weaving. However, it requires a certain familiarity, initiation, or skill with weaving to look at a finished product and have a strong idea of how it was made or how another could be made. As we weave sprang individually and collectively, we experience and enact in real time the composition of simple braids which together form the sprang weaving. We develop the capacity to look at a finished weaving and imagine the crossings which make it possible, reading backwards from the result.”
   
   Partly mathematical (abstract) ideas but also sensory, holistic embodied experiences. Developing intuition through experiences – Ex.: Dr. Tom Banchoff who worked with hypercubes and 4D objects so much that he developed intuitions about them.
   
   
7. What are the qualitative differences between learning mathematics in a classroom and in the forest, garden, riverbank or other outdoor living places?
   
   Some starting ideas: Affective aspects and emotional aspects: Nature experiences come with all sorts of feelings: discovery, awe, excitement, familiarity, peacefulness, calm, feeling that we ARE part of the living natural world, not separated from it.

Bienvenue au groupe de travail A du GCEDM 2024: Nous sommes d'un monde dynamique et vivant : enseigner et apprendre les mathématiques en plein air, en relation avec la terre, le ciel et l'eau

 Bonjour, collègues du GCEDM, et bienvenue ! Nous deux, Nenad et Susan, sonts les co-responsables de ce groupe de travail. Nous sommes enthousiastes d'explorer ces sujets dans le cadre des « mathématiques en plein air » avec vous. Nous prévoyons de tenir la plupart de nos séances à l'extérieur, au Boisé de la Santé juste en face de notre bâtiment principal du CMESG --si la météo le permet, bien sûr !   https://www.ulaval.ca/mon-equilibre-ul/bien-vivre-mon-campus/boise-de-la-sante. Au plaisir de passer un moment passionnant ensemble! 

Nenad Radakovic est professeur agrégé d'enseignement STEM à l'Université Queen's de Kingston, en Ontario. Susan Gerofsky est professeure agrégée d'enseignement des mathématiques et d'éducation environnementale à l'UBC, Vancouver, Colombie-Britannique. Nous sommes tous deux très intéressés par l’enseignement et l’apprentissage des mathématiques de manière incarnée, via les arts et en plein air, dans et avec le monde vivant.

Susan parle français et anglais et peut faciliter les discussions dans les deux langues officielles du CMESG/GCEDM. (Nenad et Susan parlent également un certain nombre d'autres langues, notamment le croate, le portugais et le mandarin.)

Voici la description de notre groupe de travail :

Groupe de travail A

Co-responsables: Susan Gerofsky et Nenad Radakovic

Nous sommes d'un monde dynamique et vivant : enseigner et apprendre les mathématiques en plein air, en relation avec la terre, le ciel et l'eau

Les mathématiques sont souvent considérées comme abstraites et désincarnées, distinctes du monde physique et de notre expérience du monde. Le dualisme corps-esprit platonicien/cartésien s’inscrit dans une tradition mathématique considérée comme « purement cognitive ». La façon dont nous faisons habituellement les mathématiques en classe reflète cela, avec des élèves silencieux, statiques et obéissants travaillant à leurs pupitres, sans aucun lien réel avec leur corps et leur sens ou avec le monde vivant qui les entoure.

Mais réellement, nous faisons tous partie du monde vivant. En nous inspirant des visions du monde et des modes de connaissance et d'apprentissage autochtones, nous commençons à comprendre que « les roches sont nos grands-pères et les plantes sont nos grands-mères » (Kimmerer, 2021; Cole & O’Riley, 2017). Nos idées mathématiques proviennent (et s'appliquent) du monde de nos expériences avec tous les êtres vivants, dont nous avons des liens de parenté.

Dans ce groupe de travail, nous explorerons diverses perspectives théoriques sur nos relations avec le monde et sur l'apprentissage mathématique découlant de ces relations. Nous reconnaissons avec gratitude que nous allons nous rencontrer et apprendre sur les terres ancestrales traditionnelles des peuples Hurons-Wendat, Wabanaki, Innus et Wolastoqiyik. Nous nous appuierons sur diverses sources, y compris les scientifiques et mathématiciens occidentaux traditionnels (par exemple, Kepler, 1611/2010; Galileo, 1632/2010; Weyl, 1952), les artistes et les poètes (par exemple, Sakaki, 1999; Major, 2018) ainsi que les théoriciens autochtones des sciences et des sociétés humaines (par exemple Bartlett, Marshall & Marshall, 2012). Grâce à des observations approfondies et réfléchies, nous nous entraînerons à remarquer des motifs mathématiques dans la terre, le ciel et les eaux. Nos activités comprendront des explorations en plein air en lien avec les plantes, les paysages, les ruisseaux, le temps, le soleil, la lune et les étoiles. Nous explorerons les structures et les représentations mathématiques qui découlent de notre observation et discuterons des implications pour l'enseignement et l'apprentissage des mathématiques en plein air.

Références

Bartlett, C., Marshall, M., & Marshall, A. (2012). Two-eyed seeing and other lessons learned within a co-learning journey of bringing together indigenous and mainstream knowledges and ways of knowing. Journal of Environmental Studies and Sciences, 2, 331-340.

Cole, P, & O'Riley, P. (2017). Performing survivance: (Re) storying STEM education from an Indigenous perspective. Critical Education, 8(15)

Galileo.(1632/2001). Dialogues Concerning the Two Chief World Systems. Modern Library.

Kepler, J. (1611/ 2010). The six-cornered snowflake. Paul Dry Books.

Kimmerer, R. W. (2021). Gathering moss: A natural and cultural history of mosses. Penguin UK.

Major, A. (2018). Welcome to the Anthropocene. University of Alberta Press.

Sakaki, N. (1999). Break the mirror. Blackberry.

Weyl, H. (1952). Symmetry. Princeton University Press.

Welcome to CMESG 2024 Working Group A: We are part of the living world: Teaching and learning mathematics outdoors, in thoughtful relation with earth, sky, and waters

Hello CMESG colleagues, and welcome! The two of us, Nenad and Susan, are co-leading this working group, and we are looking forward very much to exploring these topics in 'math outdoors' with you. We plan to hold most of our sessions outdoors, in the Boisé de la Santé just across from our main CMESG building --weather permitting!  https://www.ulaval.ca/mon-equilibre-ul/bien-vivre-mon-campus/boise-de-la-sante. Looking forward to an exciting time together!

Nenad Radakovic is an associate professor of STEM education at Queen's University, Kingston, ON. Susan Gerofsky is an associate professor of mathematics education and environmental education at UBC, Vancouver, BC. We are both very interested in teaching and learning mathematics in embodied ways, via the arts, and outdoors, in and with the living world.

Susan speaks French and English, and can facilitate discussion in both official languages of CMESG/ GCEDM. (Both Nenad and Susan speak a number of other languages as well, including Croatian, Portuguese and Mandarin.)

Here is our working group description

Working Group A

Leaders: Susan Gerofsky & Nenad Radakovic

We are part of the living world: Teaching and learning mathematics outdoors, in thoughtful relation with earth, sky, and waters 

Mathematics is often thought of as abstract and disembodied, standing apart from the physical world and personal experience. Platonic/Cartesian mind-body dualism plays into a tradition of mathematics as ‘purely mental’. The ways we customarily do mathematics in classrooms reflect this, with silent, static and obedient learners doing quiet desk work without any real connection to their bodies and senses or the living world around them.

But we are all part of the living world. Taking inspiration from Indigenous worldviews and ways of knowing and learning, we begin to understand that ‘the rocks are our grandfathers and the plants are our grandmothers’ (Kimmerer, 2021; Cole & O’Riley, 2017). Our mathematical ideas come from (and apply to) the world of our experiences with all our living kin.

In this working group, we will explore diverse theoretical perspectives on our relationships with the world, and the mathematical learning arising from these relationships. We acknowledge with gratitude that we will be meeting and learning on the traditional ancestral lands of the Huron-Wendat, the Wabanaki, the Innu, and the Wolastoqiyik peoples. We will draw from a variety of sources, including Western mainstream scientists and mathematicians (for example, Kepler, 1611/2010; Galileo, 1632/2010; Weyl, 1952), from artists and poets (for example, Sakaki, 1999; Major, 2018), and Indigenous theorists of science and human societies (for example, Bartlett, Marshall & Marshall, 2012). Through close, thoughtful observations we will practice noticing mathematical patterns in earth, sky and water. Our activities will include explorations outdoors connecting with plants, landscapes, streams, weather, sun, moon and stars. We will explore mathematical structures and representations that arise from our noticing, and discuss implications for teaching and learning mathematics outdoors.

References

Bartlett, C., Marshall, M., & Marshall, A. (2012). Two-eyed seeing and other lessons learned within a co-learning journey of bringing together indigenous and mainstream knowledges and ways of knowing. Journal of Environmental Studies and Sciences, 2, 331-340.

Cole, P, & O'Riley, P. (2017). Performing survivance:(Re) storying STEM education from an Indigenous perspective. Critical Education, 8(15)

Galileo.(1632/2001). Dialogues Concerning the Two Chief World Systems. Modern Library.

Kepler, J. (1611/ 2010). The six-cornered snowflake. Paul Dry Books.

Kimmerer, R. W. (2021). Gathering moss: A natural and cultural history of mosses. Penguin UK.

Major, A. (2018). Welcome to the Anthropocene. University of Alberta Press.

Sakaki, N. (1999). Break the mirror. Blackberry.

Weyl, H. (1952). Symmetry. Princeton University Press.