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.