# Student-Centered Mathematical Learning in Nature: A Case for Student-Designed Math Trails in Parks and Gardens

*by Jan Cohen, UrbanMathTrails*

Student-centered learning empowers students to investigate, discover and integrate new mathematical learning with existing knowledge and skills, and apply them in new contexts. By making new observations in unfamiliar situations, working cooperatively and developing mathematical communication skills, students enhance their understanding and application of mathematical concepts and skills to new problems. Mathematical problems in the real world are ill-structured and ill-defined, yet textbook learning focuses on solving scripted problems, which do not account for student diversity, are not relevant and are limited in perspective. Not only does this constrain mathematical understanding, it stifles critical and creative thinking.

Students develop higher level mathematical thinking and problem-solving skills when exposed to authentic and meaningful contexts, like nature, which inspires open-ended questions, dialogue, new connections and interaction. Higher level thinking can be fostered by building student inquiry skills and providing a collaborative learning environment that includes the identification of causal relationships between mathematics and nature, analyzing data as evidence and reflecting on new knowledge. When students learn how to formulate their own questions about the evidence of mathematics in nature and take responsibility for their own learning, they become proficient communicators, advanced problem solvers and independent thinkers capable of generating new and original ideas.

However, learning how to ask mathematical questions about nature does not occur spontaneously. Students must be taught how to abstract from observations, and probe underlying mathematical concepts and properties, causations and consequences. Educators must invite student inquiry by directing them to a specific topic or subject matter and exploring their understanding of it. It is essential that educators collaborate with students in the formulation of questions about mathematical patterns and principles in the natural world, starting with the fundamentals, then teaching them to dig deeper and ask follow up questions, to explore the impossible, question the alternatives, inquire about the details, and make and test hypotheses.

Math trails in parks and gardens provide robust student-centered learning opportunities for all levels of budding mathematicians. The breadth and depth of mathematical concepts to learn and reinforce are significant. Although the scope can be endless, it can be tailored to fit any topic or curriculum unit(s). There are myriad benefits, including the joy of learning outdoors, showcasing new connections between math in the classroom and in nature, nurturing mathematical communication and collaboration, and inspiring mathematical reasoning. Math trails also engage students cognitively, physically and socially. Although math trails can be designed and implemented by educators, the experience of student-designed math trails empowers students to be responsible for deciding what can and needs to be learned and implementing a continuous learning process that is transferable to other experiences and subjects.

The following is a starting point for educators to facilitate student-designed math trails in parks and gardens. It is organized according to natural and man-made objects and designs found in parks and gardens, and includes some mathematical concepts to formulate questions, as well as ideas for analysis and extensions. Happy Trails.

### Topics for Student-Designed Math Trails in Parks and Gardens

Leaves: Measurement, estimation, area, perimeter, symmetry, congruence, ratio, proportion, shapes, patterns, arrangement (degrees per turn), self-similarity, data collection, classification. Data analysis, scientific basis, Fibonacci (Lucas) numbers, rates of change, statistics, graphs, tables, Venn diagrams.

Flowers: Measurement, estimation, symmetry, congruence, patterns, size, shapes, angles, arrangement (degrees per turn), spiral geometry, data collection, classification. Data analysis, scientific basis, Fibonacci (Lucas) numbers, Golden Ratio, packing, statistics, graphs, tables, Venn diagrams.

Trees: Measurement, estimation, symmetry, shadows, angles, trigonometry, right triangles, circumference, radius, self-similarity, data collection, classification, cross sections. Data analysis, statistics, rates of change, graphs, tables, Venn diagrams.

Landscape Design: Scale, ratio, proportion, slope, shape, symmetry, density, surface area, perimeter, angles, percent, estimation, polygons, congruence, similarity, trigonometry. Golden Ratio, graphs, tables statistics, scale drawing, topography, coordinate systems.

Paths and Pavement: Measurement, estimation, ratio, proportion, patterns, symmetry, tessellation, transformations, ratio, dimensions, angles, surface area, perimeter, percent, expressions and equations. Graph theory, coordinate systems.

Fountains: Measurement, estimation, shapes, angles, arcs, circumference, perimeter, area, volume, parabola, congruence, direct variation, expressions and equations, surface area, speed. Models, functions and relations, graphs.

Sculpture: Shape, volume, angle, dimensions, ratio, proportion, symmetry, surface area, similarity, congruence, vertices, equations, topology, polyhedra, patterns, perspective, tessellation, transformations, rotation, classification, quadratic surfaces, fractals, knots, cross sections. Models, coordinate systems.