**Montessori Math Trails**

*By Jan Cohen, Founder, UrbanMathTrails, with special thanks to Michelle Velho, STEAM coordinator, Hudson Montessori School, Jersey City, NJ*

Math trails are a natural extension to the Montessori principles of exploration and appreciation of the natural and man-made environment. Dr. Maria Montessori believed that children should learn from and treasure nature and the great outdoors. “There must be provision for the child to have contact with nature; to understand and appreciate the order, the harmony and the beauty in nature”. The rich sensorial experiences of nature are also encountered in art, which is why Montessori considered art as essential to the environment of the child, a way of experiencing life and appreciating beauty.

“Education is a natural process carried out by the human individual, and is acquired not by listening to words, but by experiences in the environment.” Montessori’s view of the environment included both classroom learning and extensions beyond the classroom into nature and the arts, so that children may see themselves as part of the world. Fostering an appreciation for these experiences is a cornerstone of the Montessori teachings.

This article explores the connections between Montessori’s philosophy of math education and math trails as a strategic extension to learning beyond the classroom.

**Montessori Philosophy Math Education**

Dr. Maria Montessori believed that children are innately drawn to precision to organize, compare, order, classify and quantify patterns and relationships. Exposing children to these concepts early in life builds a strong foundation for their later understanding of more complex mathematical concepts. Without this exposure, Montessori believed, children are unprepared for advanced learning and critical thinking.

Montessori promoted this belief through her idea of a prepared environment -- everything children experience should facilitate and enhance independent learning and exploration.

She understood that children learn best by interacting freely with their environment, working on activities at their own pace and following their own developmental needs. Guided by the environment and the teacher, children teach themselves and learn more than when adults formally teach them.

This insight led Montessori to develop specialized multi-sensory and manipulative math materials to stimulate children’s senses, character and social interactions. Using concrete, scientifically developed didactic materials, Montessori believed children would experience the concepts of order, sequence, measurement, calculations, and exactness eagerly and joyfully. Montessori math materials are proactive, facilitating children’s progression from the concrete to the abstract. She designed math materials to be works of art themselves, captivating children’s attention with their beauty and creating the desire to reach out and explore them.

Montessori also established the relevance of mathematics in human society by emphasizing a connection between the study of math and other curriculum areas. With an integrated exploratory approach, children discover mathematical ideas in a variety of contexts, gaining a deep understanding of the connections, relevance and underlying meaning in the real world. It is the accumulation of these experiences that develop a depth of understanding and reasoning abilities that ultimately prepare children for an abstract understanding of mathematical concepts and symbolic representations in later years.

**Building on the Montessori Philosophy through Math Trails**

Montessori’s classroom materials prepare children to explore beyond the classroom and venture out into the community and the world, where they can discover mathematics in new contexts and enable the acquisition of knowledge through their interaction with the environment. And, since math is a system of thought, organization and communication of shape, quantity and arrangement, math is not only the building block for everything, it exists everywhere in the environment.

Math trails are educational experiences that extend Montessori’s philosophy and curriculum into the natural and man-made environment. They are typically organized as “walks” where students reflect on and solve math problems about what they see in the environment. They include prompts that relate to numbers, objects, patterns, layouts, quantity, structure, space, change and/or relationships. Children respond to these prompts by using mathematical reasoning, abstraction and logic to provide insight about their observations. Math trails vary by age, topic, theme, experience, interest, season, and venue. They encourage intellectual independence and social development through cooperative and collaborative attention to elements in the environment. Children work at their own pace, non-competitively, allowing their natural curiosity to lead the way and towards a lifelong love of learning math. Children are free to explore, discover and celebrate the beauty and universality of math, while formulating, analyzing, predicting and explaining, enabling children to absorb and apply mathematical concepts naturally.

Math trails may occur in any venue: neighborhood, park, garden, museum, public station or terminal, zoo, aquarium, market, and may even occur in the classroom itself. Most mathematical topics lend themselves to math trails; however, for the purpose of this article, a) symmetry, b) measurement, and c) sorting and classifying will demonstrate how math trails extend beyond the Montessori prepared environment to reinforce learning in these subject areas.

**Montessori Inspired Symmetry Math Trails **

In Montessori education, young children start tactile and kinesthetic learning through carefully designed materials that isolate one specific quality such as size, weight, shape, texture, color, smell or sound to help children develop their senses. This hands-on exploration of concrete materials facilitates and optimizes independent learning, while preparing children to progress toward abstraction and paper and pencil math. Building on this sensorial approach to forms and shapes, children are introduced to figures: lines, polygons, angles, circles, and geometric solids, learning about the mathematical properties of their sensorial knowledge. They learn the “Golden Elements” of geometry: congruence, similarity, and equivalence, and they learn about the concept of symmetry. Symmetry beckons for exploration in the real world.

That real world may start with a math trail in the in the classroom, where children are invited to notice the symmetry of windows, floor and ceiling patterns, cabinetry, furniture, and even Montessori materials and collections. This may be followed by a math trail in nature, where children explore examples of symmetry in leaves, flowers, plants, fish and animals, providing more hands-on learning of patterns supported by mathematical principles. Even the youngest children can begin to understand the concepts and vocabulary of symmetry given time to explore these patterns in nature. Older children may investigate symmetry in nature more explicitly by identifying symmetrical vs non-symmetrical patterns, types of symmetry, lines of symmetry, centers of rotation and orders of symmetry. When drawing, recording or documenting their observations in a nature journal, their learning about symmetry is reinforced. The experience sharpens their powers of observation and enriches their relationship with the natural world.

Of the many connections between math and art, none is stronger than the concept of symmetry. Various art forms employ symmetry, sometimes precisely and other times approximately. On an art math trail, children are exposed to the geometric patterns of symmetry in paintings, pottery, textiles and wallpaper and sculpture, as well as their cultural roots. This can very exciting for children who otherwise are not naturally drawn to either mathematics or museums. Symmetry provides various opportunities for children to enjoy learning math, visualize mathematical concepts and connect learning geometry to a real-life experience. Using math to explain visual composition, facilitates learning in both disciplines, while also building intuitive cognition. Children begin to perceive symmetrical arrangements everywhere, reinforcing their understanding, building knowledge and illuminating the ideas that shape our reality. Perhaps most important, children learn that art and math are not contrasting units.

A math trail through any neighborhood reveals that across all cultures and throughout all time periods, architectural compositions are symmetrically arranged. Architecture differs fundamentally from art because we not only see it, we move around it and through it as well. This means that architecture provides children a special opportunity to experience symmetry as well as to see it. Moreover, the types of symmetry employed in architectural designs include the more common forms of symmetry observed in nature and art, as well as the cylindrical symmetry of towers and columns; the similarity symmetry of repeated elements that change in scale but retain their shape, such as in layered roofs and cathedral windows; and spiral or helical symmetry such as in spiral staircases. Exploring the symmetry of architecture in a math trail through a neighborhood is yet another opportunity to awaken children’s mathematical senses, while building connections to the community and meaning in the real world. These experiences reinforce the key concepts of the Montessori philosophy that relate to the outdoor environment.

### Montessori Inspired Measurement Math Trails

Measuring belongs to every culture. Students learn to measure at any early age, using both standard and non-standard units. Standard units like inches, feet, miles, centimeters, meters, kilometers are accepted units that are uniform around the world. Non-standard units are units that vary, depending on what they are and who uses them. For example, paper clips, or colored tiles help children connect measurement with everyday objects, which prepares them for using standard measurement tools and units at a later stage. Another example of using non-standard units is step length or arm or hand span length. They are non-standard in a different way, because they depend on who is being measured. In any event, non-standard units are useful to introduce very young children to the concept of measuring without the need to read any scales, i.e., what it means to measure rather than how to measure. Non-standard units are also useful for older children to approximate lengths and distances and develop a sense of scale.

Prior to using materials to teach measurement activities, early childhood Montessori students learn the concepts of size and opposite extremes, through manipulative activities. They may focus on comparisons like long, longer, longest or tall, taller, tallest or wide, wider, widest or heavy, heavier, heaviest; while older children may explore what can be measured, how to measure and comparison between measurements.

A math trail is an excellent way to extend the Montessori classroom learning of measurement and reveal the function of measurement in the real world. Whether comparing and ordering, counting units and fractional units on a ruler, deciding among the best units to use in different circumstances, or learning about precision and exactness, children improve their understanding of numbers by learning measurement. When children leave the classroom and explore measurement in the real world, it becomes relevant, if not compelling.

Since children naturally absorb their environment experiences when they act on their environment, a measurement math trail in nature or the neighborhood provides a tactile and kinesthetic learning experience for children, as well as an opportunity to develop estimation strategies and refine estimation skills. Young children may estimate how many pencils long is the bench, sidewalk square, fence, staircase, door, etc. They may analyze comparative lengths/distances by counting steps. They may estimate the number of individual units of something are needed to cover a surface.

In the neighborhood, older children may estimate and compare actual measurement of doors, windows, mailboxes, pavement tiles, benches, fire hydrants, signs, fences, stairs or distances between locations. They may measure the size of a brick and use it to calculate the height of a wall. They may measure the size of a sidewalk square and use it to calculate the area of the entire sidewalk on the street. They may measure the height of one stair and use it to calculate the height of the stairway. They may name objects they see that correspond to specific lengths, area or capacity, and in the process, learn words and units of measurement that reinforce math communication.

In nature, there are many measurement opportunities: circumference of trees, bushes, pinecones, tree fruit or rocks; diameters of flowers, tree stumps or shrubs; length of leaves, twigs, pods, or exposed roots; width of paths; volume of soil or boulders; length of shadows; area and perimeter of signposts; and, the height of fences, flowers or shrubs. Young children may compare and order such specimens of nature and use non-standard units to measure them. Older students may estimate their sizes and then compare them to the actual measurements taken using a tape measure. This experience is again consistent with the Montessori philosophy, which advocates for the outdoor classroom and emphasizes the importance and value of children connecting with nature.

### Montessori Inspired Sorting and Classifying Math Trails

Children have a natural desire to make sense of their world and for this reason, sorting activities are appealing. Montessori sensorial sorting activities teach children to organize their world. Children learn that some things are alike, some are different and may be organized into groups, which prepares them for numerical concepts and set theory when they are older. By comparing and contrasting, children learn how to apply logical thinking to objects, mathematical concepts and life in general.

Using Montessori sorting trays, children learn how to discriminate visually and stereognostically (sense of touch), enhancing their abilities to perceive and understand like and unlike sizes, shapes and forms, including distinguishing how similar objects differ from each other. These sensorial experiences focus on differentiation as a necessary means for children to understand their environment through their own personal experience.

Math trails naturally extend sorting and classifying of the environment into new contexts. Nature abounds with sorting and classifying opportunities. Trees, for example, may be sorted and classified according to their primary categories: deciduous versus coniferous. Leaves are easily sorted and classified in a few ways. Leaves vary according to the arrangement of leaves around the stem, either alternate, opposite, or whorled. Another sorting and classifying opportunity may focus on simple single leaf blades versus compound leaflets. As children get older, they are ready for more careful discrimination. A math trail may introduce the concept of sorting and classifying leaf edges or margins: crenated, lobed, entire, or serrated; or leaf blades (laminas), i.e., the shape of the broad part of the leaf.

Flowers may be sorted and classified in a few basic ways. Are the number of petals odd or even? Do the flowers bloom as single flowers or in a cluster? An interesting sorting and classification scheme for flowers relates to their position on the branch. Are they terminal or axillary flowers? Flowers may also be sorted and classified according to their anatomical arrangements. Are they flowers with partially or fully joined petals or are the petals separate and distinct? Are the flowers characterized by their radial symmetry or by their bilateral symmetry? A math trail through nature is the ultimate resource for a hands-on sorting and classifying learning experience.

Sorting and classifying specimens in nature also lends itself to data collection and graphing opportunities. When children keep a record of things they have observed, gathered and sorted, the experience turns into a way to organize their discoveries and communicate their knowledge in a natural and uncontrived way. Children are led from the concrete experience of observing and collecting to the abstract representation of their findings, learning which graphs are best under which circumstances, how to interpret their findings and draw conclusions.

### Conclusion

“To assist a child we must provide him with an environment which will enable him to develop freely,” said Dr. Montessori. Math trails are an excellent vehicle to fulfill this goal. They are not only a natural extension of the Montessori classroom, math trails may be the new Montessori “schoolyard”. They are a way to harness the environment as a new and inexhaustible educational tool that showcases the math of the real world, while exciting the mathematical imagination and electrifying the mathematical senses of every child.