Geometry at an Early Age

Two articles look specifically at the role of geometry in school, from primary through to university level, and discuss the difficulties with trying to create a universally valid curriculum.

In The Place of Experimental Tasks in Geometry Teaching: Learning From the Textbook Designs of the Early 20th Century, Taro Fujita and Keith Jones write:

“A characteristic feature of geometry is its dual nature, in that it is both a theoretical domain and perhaps the most concrete, reality-linked part of mathematics.”

It is this duality that creates difficulties for teachers who are faced with trying to bridge the gap between theory and practice within the classroom, two parallel studies that many children struggle to fathom at the same time (Fujita, Jones, 2004). It is interesting that the two threads are difficult to weave together in a mathematical setting when theory and practice so often converge in nature and in art and design.

In The Role of Intuition in Geometry Education: Learning From the Teaching Practice in the Early 20th Century, Fujita Taro, Keith Jones and Shinya Yamamoto question the role that geometry has in a teaching curriculum and suggest that a slight reshuffle of the specification of geometry might be in order, specifically focusing on intuitive geometry–where children show an ability to “see” geometrical shapes and manipulate them in the mind to solve problems (Fujita, Jones, Yamamoto, 2004). Interestingly, their article highlights an address given by J. Perry, Professor of Engineering at the Royal College of Science, in 1901, who questioned the value of teaching Euclidian geometry to children (Fujita, Jones, Yamamoto. 2004). Perry was an advocate for experimentation and a more intuitive approach to geometry. This is fascinating to me as a creative; the idea that mathematics can be approached in a more creative way, by being encouraged to simply observe our surroundings. This method would likely appeal to children who do not think analytically or struggle in a scientific or mathematical setting. It also opens the floor to a wider selection of children being brought into geometry study and develops the ability to think outside the box or push expectations.

This is another aspect of my own graphic design that I want to develop further: rather than simply designing how I think I should design (trying to anticipate the expectations of others), I’d like to grow my own ability to “see” design in new ways, try new approaches and not concern myself with only doing what is popular in the moment. Seeing a subject as far removed from art and design as geometry in a new and creative way encourages me to push my own boundaries and methods.

Back to geometry. This intuitive ability to see shapes and use them to solve problems ties in with the paper Modularity and Development: The Case of Spatial Reorientation by Linda Hermer and Elizabeth Spelke. Hermer and Spelke explore children who, during development, share some similarities with animals in how they spatially place themselves in their surroundings using geometric cues. Their study revealed that the developing brain is more in tune with room geometry than with lesser-defined patterns or other defining factors (like odour, colour, etc). At an early age, we reorient ourselves in accord with the shape of our surroundings (Hermer, Spelke. 1996), meaning that geometry is playing a large part in our natural development, before it is formally introduced or taught. That we are subconsciously aware of geometry relative to ourselves is an exciting concept and something that could be explored in advertising–particularly advertising aimed at children.

Attempts at tying geometry with a more creative, intuitive practice have been made in a number of key learning equipment. Not all of the following toys were intended for children but have been adopted over the years as useful developmental tools.

The Spirograph, whose origins date back to the mid 19th Century, was originally created to help prevent bank note forgery, but it eventually became a popular 2D geometry drawing game. Spirographs involve the creation of hypotrochoid (a circle rolling inside a fixed circle) and epitrochoid (a circle rolling around the outer edge of a fixed circle) roulettes. A similar method known as Guilloché is still used on paper currencies, passports and security seals today, which means we have all come into contact with spirograph patterns.

The Lego Group was founded in Denmark in 1932 by Ole Kirk Kristiansen and, since then, Lego has become one of the most popular toys of all time. Providing children and adults alike with Automatic Binding Bricks of different shapes and colours, it teaches us to think not just creatively but also geometrically and in 3D. Lego is an effective puzzle-solving tool which exposes us to science, technology and engineering. For example, constructing a complex object like a bridge takes thought, development and trial and error. These small plastic geometric shapes teach children to patiently think through problems to reach a desired outcome. Through playing with different coloured Lego bricks, children can also learn about symmetry, colour combinations and patterns, and really let their imaginations run wild.

The Helix Oxford Maths Set was launched in 1935 by Helix and offered its famous geometry set. It wasn’t until 1959 that they began to mould their own plastic tools, which are the tools I used when growing up. The set includes a ruler, set squares, protractor, compass and pencil, pencil sharpener and eraser.

Finally, the paper Fortune Teller was first introduced in the book Fun with Paper Folding (Murray and Rigney; 1928) and is a more informal geometric toy that many will recognise from their school days.

fortuneteller01

The fortune teller was originally called the “salt cellar” and was intended to stand on a table while the four pockets contained condiments. Fortune Tellers were a popular game at school and I remember creating them with my friends (usually asking most pressing questions such as “Which member of Take That will I marry?”).

References

Fujita, T; Jones, K (2004): The Place of Experimental Tasks in Geometry Teaching: Learning From the Textbook Designs of the Early 20th Century. International Congress on Mathematical Education (ICME-10). Copenhagen, Denmark.

Fujita, T; Jones, K, Yamamoto, S (2004): The Role of Intuition in Geometry Education: Learning From the Teaching Practice in the Early 20th Century. 10th International Congress on Mathematical Education (ICME-10). Copenhagen, Denmark; 4–11 July 2004.

Hermer, L; Spelke, E (1996): Modularity and Development: The Case of Spatial Reorientation. Elsevier – Cognition 61. Department of Psychology, Cornell University, Uris Hall, Ithaca NY.

Murray, William D., Rigney, Francis J (1928): Fun with Paper Folding. Fleming H. Revell Company.

Spirograph | Denys Fisher | Science Museum

Guilloché Patterns | Ed Pegg Jr., February 9, 2004

Lego

Helix Oxford Maths Set

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