*“It is also acknowledged that transmission of the virus will likely be attenuated in outdoor settings and outdoor play and learning have many benefits for children and youth. School boards and educators should therefore incorporate outdoor learning activities into the curriculum.**”*– Sick Kids and partners, Covid-19: Guidance for School Re-opening, July 29, 2020, p. 9

As school starts up again, it can be challenging to plan amidst all the uncertainty. Teaching in the age of Covid forces us at times to pare down our practice to what is most essential and re-focus our priorities in the light of our fundamental values and purposes as educators, reminding us of what matters most and what can be most easily dispensed with. One thing is clear – despite everyone’s best intentions, education (and life) are unlikely to proceed as usual for quite some time.

A rich and structured math learning environment can exist with few of the classroom structures and resources we have come to rely upon. Taking children outdoors to alleviate risks of contagion provides new opportunities to rethink our pedagogy as we learn with, on, and from the land. The world beyond the classroom offers a wealth of interesting, relevant materials and spaces with opportunities for counting, sorting, patterning, reasoning, calculating, measuring, surveying, constructing, visualizing, problem-solving, modelling (to name just a handful). In short, most of the important components of a math program – at any level – can be rigorously covered in outdoor settings. Yet, despite frequent reminders that we are not only safest but also thrive in the open air (as in this recent Globe and Mail piece), there are limited suggestions for teachers beyond vague allusions to the health and well-being benefits of time outdoors. These are well-substantiated and crucially important benefits, but educators also need pragmatic curriculum plans for designing thoughtful, systematic math instruction in outdoor settings.

#### WHY TAKE MATH OUTDOORS?

Beyond mitigating risk in a pandemic, there are compelling reasons for taking math outdoors:

- Mathematical cognition is rooted in the body and its relation to the space in which we live and move. Movement through space offers children an essential mode of understanding that can’t be easily replicated at a desk. Physically grounded understanding through kinesthetic and other senses – especially vision – provides a basis for the symbolic models and abstractions of complex mathematics.
- Coming to know disciplinary concepts (learning math) goes hand in hand with doing math (engaging in mathematical activity), and using math, applying concepts in the real world.
- The documented impact of outdoor experience on engagement, well-being, and executive function has particular relevance for math learning.

Investigating math outdoors attunes students to the mathematical structure of the physical universe. Math takes on resonance and relevance as children gain a new and enduring lens upon their close and distant surroundings. - Working with natural forms in their individuality and imperfection highlights the relationship between idealized mathematical forms and the approximations and messiness of the real world. As children classify objects and build concepts, they develop a sense of which differences are relevant in a given context and which are merely expressions of individual variation. They start to better understand the statistical nature of our grasp of the world.
- The natural environment – the land – is an inherently integrative setting. In uncovering the mathematical structure of natural forms and materials, integral links between math and science are cemented and elaborated upon.
- The culturally and racially discriminatory side of STEAM subjects and professions is of pressing concern for education. Moving beyond the classroom has huge potential to level the playing field and re-jig classroom hierarchies. It offers opportunities for children’s skills and knowledge to emerge in new and surprising ways. This is especially critical as we commit in Canada to incorporating Indigenous perspectives – many of which connect deeply to the land – in all our teaching.

#### APPROACHES TO MATH IN OUTDOOR SETTINGS

There are a number of ways to approach math in outdoor settings, ranging from traditional instruction moved outside with photocopies, clipboards, and pencils to investigating problems and activities more intimately tied to the structure and content of a school’s surroundings. One currently popular approach is to create “math trails” in which students explore the math concepts embedded in a location, identifying mathematical structures and performing operations, such as estimating heights, noting forms of symmetry, calculating frequencies, finding shapes and patterns, and so on. While these activities are generally engaging and help sensitize children to the enticing variety of math in their lives, they don’t always extend beyond the occasion or lead into more rigorous or lasting concept development.

Other investigations may be more focused on building deep networks of conceptual connections. Out of limitless possibilities, we have listed a few examples along with some resources. We often find that once teachers get started with outdoor learning, the unprecedented engagement, curiosity, and focus of their students engender a surge of further possibilities.

#### IN CONCLUSION . . .

Bringing school outdoors offers new ways of seeing children and their learning. Released from the confines of a classroom, children show themselves in a different light. Time outside can be a uniquely joyful experience, allowing children space and freedom to pursue their interests in their own time and in their own ways, take risks as they test the limits of their bodies, be solitary or social, ask and pursue the questions they most care about. Curricular concerns need never overwhelm the potential of the outdoors to support the kind of rich connected play with materials and ideas which characterizes mathematical thinking at its peak. Watching students as they invest so deeply in their own learning can open a teacher’s eyes to unimagined strengths and ways of thinking.

#### IDEAS FOR MATH INVESTIGATION

Classification and Patterns

Many parents face children’s insatiable drive to collect – especially loose things in nature – as they daily empty the gritty contents of their children’s pockets. While a mix of aesthetic, narrative, and acquisitive motivations likely feed into this instinct, it is clear that spontaneously organizing, classifying, and building collections is one way that children work to make sense of their worlds. The potential of children’s collections for triggering intriguing questions about classification can be a natural starting point for teachers.

The more we learn about identifying and naming objects in nature, the more clearly we are able to see them, as a “screen of green” becomes increasingly differentiated. Western traditions of natural taxonomy embody complex classification schemes that require logic, fine-grained perceptual discrimination, and geometric reasoning of many kinds. For example, consider combinations of different leaf type categories. Are they lobed or unlobed, opposite or alternate, smooth, single- or double-toothed? Do they contain forked or simple veins, are their shapes simple or compound? Do the leaflets cluster in even- or odd-numbered groups? These few examples only hint at the vast variety of permutations and combinations, including those of bark types and twig forms, that assist with tree identification. This process can be pursued at various levels of sophistication using diagrams and logic charts (“tree diagrams”) in simple field guides. For some children, the intersecting Venn diagrams of taxonomic distinctions and associated vocabulary are endlessly fascinating.

Introducing very different but equally systematic modes of categorizing and naming – such as considerations of function and relationship essential to many Indigenous classifications – further deepens understanding.

The mathematically describable regularities in nature are quite astounding once you start to notice them. Many natural phenomena, such as the expanding spirals of a snail shell or radial symmetry of flower petals, reflect mathematical patterns such as Fibonacci number sequences or fractals. There are countless opportunities outside to explore the mirrorings and transformations of complex symmetries. Noticing and analyzing natural patterns mathematically awakens students to the structure of the universe.

Measurement and Geometry

Using movement to understand geometric concepts such as angles or the construction and transformation of shapes contrasts with the more static approach of most textbooks. For example, students can draw circles by incising the ground with a stick attached to the end of a string tethered to a center stake as they walk around the centre. They can then construct angles as fractional parts of circles based on motion, as they traverse a quarter of the circumference, drawing lines from their start- and end-points to the centre, continue half way around, and so on, repeating the process with further subdivisions of angles. They might also explore the constant relationship between circumference and radius as they construct and walk a series of circles with 1 metre, 2 metre, and 3 metre radii.

The British organization Learning Through Landscapes provides free, thoughtful lesson ideas for outdoor math at different grade levels, including a document on “Ten Ways to Measure Trees” which describes ingenious strategies for determining the height of very tall objects in the environment. For example, you might calculate the ratio of a friend’s height to the length of their shadow (or, more simply, measure the shadow cast by a metre stick), then measure the tree’s shadow and reason proportionally to calculate its actual height. Or use angles and sightlines, walking away from the tree and looking backwards through your legs until you are just able to see its crown. Your distance from the tree at that point will equal its height. Challenging students to figure out how the various methods work and how they are related will lead into an in-depth exploration of important mathematical ideas.

Further exploring perspective, sightlines and angles outdoors can range from considering bird’s eye views and proportionality in mapping to thinking about hiders and seekers in nature and who can see what from where. Exploring the shadows cast by simple three-dimensional objects such as cubes is another rich area for investigation that builds understanding of the mapping projections we use to represent a spherical earth in two dimensions.

Data Management

Enumerating the data beneath our noses on large and small scales offers authentic reasons for all kinds of counting, working with large numbers and fractions/percents, graphing, inferring causality, assessing probabilities. Students might count the number of blades of grass in a tiny square area and estimate how much grass fills the entire field. Or a class might set out to survey the biodiversity of an area through sampling, in which pairs of students tally the contents – animal, vegetable, mineral – of single, randomly located, 25 cm x 25 cm squares. The individual data can then be amalgamated to create a single graphic representation that shows the relative frequencies of biota in the area.

Citizen science invites members of the public to contribute to scientific studies by collecting local data, such as counting specimens of raptors, ferns or honeybees. As well as building patience, number sense and keen observation skills, this work allows participants to deepen their sense of place while seeing the local in a larger context. There are ongoing opportunities for schoolchildren to involve themselves in such projects.

Another readily accessible option is eBird Canada, an interactive website that invites birders of all ages and experience to list identified species and see which birds others have spotted nearby. Contributors see their own data as it dynamically shifts in relation to that of others.

#### DOWNLOAD ROBERTSON RESOURCES

Finding Geometric Shapes Outside

Students try to identify shapes while outside and make shapes with sticks.

Measuring the Depth of Snow

Students measure the depth of snow and discuss snow accumulation.

#### HELPFUL RESOURCES

Author

Julie Comay*Faculty in the Master of Arts in Child Study and Education program and former elementary teacher at the Dr. Eric Jackman Institute of Child Study *