Back to Natural Math® workshops

January 20th , 2004
Cary Homeschoolers

This workshop was conducted at The Scrap Exchange in Durham, North Carolina. Children created their own projects with their parents' help. Then Maria D. looked for mathematics in the projects, talking to parents about ideas for more games and activities.

Proportions in nature and art

What creature is this? How do we know?

The topic of proportionality is one of the cornerstones of any mathematical curriculum. It bridges arithmetic with higher math, since it requires algebraic, abstract reasoning. For example, when you set up a numeric proportionality problem such as 3:4=6:?, you are necessarily solving an equation, finding an unknown, and exploring relationships that span not only given numbers, but any quantity. In our example, the same relationship can encompass 6:8 or 30:40.

When and how do children begin developing proportional reasoning? Even a toddler will probably laugh at "out of proportion" pictures from Alice in Wonderland or Dr. Seuss or many other authors and artists who address the subject:

Even a small child will notice that Alice's neck is too long, or that she became huge compared to the house. In the sculpture above, the workshop participants easily recognized a giraffe, and the author confirmed and added that the giraffe's name was Lovey :-) Children told me they knew it was a giraffe because it had a long neck. This recognition of the relationship between the length of the neck and the size of the body is a beginning of proportional reasoning. At this stage, reasoning is qualitative, that is, children use "non-quantity" words such as "long" or "too big" or "smaller." Later, children can build on the same ideas when they make quantitative estimations. How many arm's lengths should be from the floor to the ceiling in the house? Definitely not one (poor Alice), but how many, really?

When children draw, their creations reflect "psychological proportions" rather than real-life ones. For example, if children draw their play group, they are likely to draw their best friend larger than other people. By looking at proportions in children's drawing, you can figure out a lot about the child's relationships with friends and family. Children also tend to draw details that fascinate them larger. For example, many children draw hands and fingers too large for the size, because they pay more attention to them.

When children draw, or work with construction sets, or make sculptures, parents can talk about proportions in children's creations. Use both qualitative language (longer, shorter, wider, thinner) and quantitative language ("five width to the length"). Sometimes children are touchy about comments to their artwork. I found that it may work better if I create a similar object alongside the child, commenting on my work instead of the child's work.

A "lava lamp" looks realistic, thanks to its realistic proportions!

"Out of proportion" sculptures and drawings are considered funny, whimsical, artistic. Children usually like them, and are happy to do math with them. Proportion is one of the bases for image recognition by the human brain.

• Relationships between parts of well-known images <~> proportions

Symbols and abstractions

A robot princess. Lace and the crown symbolize the princess part. Cylinder shapes and buttons symbolize the robot part.

Symbols and signs are everywhere around us. Clothes are widely used to symbolize social status, occupation, religion or simply today's mood. Children become aware of such everyday symbolism quite early. They will recognize the cowboy, the soldier, the sailor and the clown by their clothes:

Parents can use the word "symbol" in such situations. Also, the word can be used to describe children's art. Above, I said that the crown was a symbol of a princess. There is an interesting game to play with children. Pick up some symbolic clothes, such as hats - a baseball hat, a witch hat from Halloween, a winter fur hat. Or maybe shoes: snickers, ballet slippers, house slippers, high hill dress shoes. Then make up your own symbols for characters such as "cat" or "robot" or "princess" or "dragon" and draw them on pieces of paper. Create your "costume" by attaching paper to yourself with tape, and also picking symbolic clothes. This way, you can be a "ballerina witch cat" if you put on ballet slippers, witch hat, and a paper with a symbol for "cat." Children can make up little skits about their newly created characters. They can change the characters by adding more symbols. For example, you can draw a symbol for "tiny" or "red" or "young" on paper and attach it to yourself in addition to whatever symbols you already had.

Also, work with symbols is a good way to introduce the idea of abstraction. Some of the symbols children see around them will resemble the objects or actions that are symbolized, for example, the flagman road sign or the audio button:

Such symbols are called "iconic."

Other symbols do not resemble the symbolized action or object, such as the stop button on the VCR, or the plus sign for addition:

+

Such symbols are called "abstract." Much of children's art does not resemble anything, since they are experimenting with media and their own movement. When parents introduce the idea of abstract symbols and abstract art, it helps children to be more accepting of their experimentation and attempts at drawing. If the child tried to draw a cat and the picture came out incomprehensible, there is always an opportunity to say that it was an abstraction, or an abstract symbol of a cat. In my experience, such approach is helpful to children who have emotional problems around learning to draw or to sculpt.

An abstract sculpture. Also note repeating patterns in red and white. Some of the modern abstract art uses patterns as well.

Another abstract creation. This girl used five or six different ways to attach things in her sculpture, including tape and knots. Such method of developing objects or ideas through a collection of different techniques is quite important for children (see also "psychologically important games" below). For example, a child will often try to use every crayon in a new box, creating a picture where it makes sense to do so, such as a meadow with different flowers.

• types of clothes or other objects <~> symbols
• symbols not resembling the symbolized <~>abstraction

Symmetry:

A model for number construction and multiplication

Psychologically important games:

Webs, collections, cubism

Some of children's games and activities are played by most children during certain stages of development, supporting the child's growth and learning. Peek-a-boo or playing house are well-known examples of such psychologically important games for toddlers. During the workshops one could see elements of such developmental activities in children's projects.

One very, very long project! Some objects are connected with a rope. The whole thing was longer than the room.

A project by a workshop participant made out of various objects connected by strings reminded me of "webs", a math-related activity I research. If children get some long thread and free reign of space, they tend to make webs. Webs can be of several kinds. One kind looks like a spider web and is usually made between chairs, between table legs, or on any other suitable place. Another kind goes around objects, usually large objects such as the whole house, bushes and trees, or large pieces of furniture. Children often run or walk fast, unwinding the thread as they go, and moving around and around the chosen object. In a third kind of webs, children visit all the rooms of the house unwinding thread behind them, or mark their path around all interesting spots in a single room with the thread.

Playing with webs, children create visible traces of their movement in space. Thus they make the history of the spatial explorations visible. Distance "from here to there" may be too much of an abstraction; feeling it as thread running through your hand makes that distance more concrete. The idea of a circumference or a perimeter is even more difficult; a child running around and around an object while tracing the perimeter with a thread is exploring this concept. Parents can support web creation by supplying children with inexpensive thread, such as acrylic that is often available on sale or at thrift stores. If children have covered a room with webs and do not want to cut them down, taking photographs may help children feel better. I collect photographs of webs; if you would like to share any, please send them to me!

A sea worm made out of several "slices" of tubing, carefully connected. A tube worm. This is an example of an extension of the same theme, a beginning of a theme collection. Another sea creature - a contribution to the theme collection.

Creating collections is an activity important both for the general development and for the growth of mathematical understanding. If you give a toddler a toy, she will play for some time. If you give her several toys of the same kind - four blocks, three stuffed bears, five plastic cups - she will play with them for much longer, exploring their relationships, arranging them in different systems and creating role-based games. Businesses sometimes exploit this psychological necessity of collecting by selling objects children are supposed to collect, or by advertising sets of toys that go together well. Here I discuss how parents can become aware of the children's need for creating collections, and can support the idea in meaningful and healthy ways.

There are several types of children's collections. In theme collections, shown in the photographs above, children put together objects that are related to a common idea or theme. This powerful associative thinking mechanism is used, for example, in the unit study method of learning, where subjects are collected around a common theme to form a unit. To support learning around theme collections, parents can organize "theme centers" around the house. For example, if children are interested in sea creatures or in butterflies and flowers, as in the examples above, their artwork on the theme can be displayed on a special low table in the corner. Seeing their previous creations, children remember the techniques they have developed, and build on, instead of starting from scratch every time. Parents can also arrange books on the topic, and any other learning materials, in such a theme center. As children create more work on the topic, it can be added to the collection, the process that is usually deeply satisfying and motivating for children.

In the "many of the same kind" type of collections, children pick a type of objects and try to obtain or make many such things. Sometimes advertisers urge children to collect all objects of a particular kind. There seems to be a psychological need for such type of collections as well. For example, when my daughter was a baby, she collected teddy bears on yard sales. She probably had thirty of them by the time she lost interest. To satisfy this psychological need, parents can help their children collect "less commercial" objects, such as different kinds of local rocks, photographs of different birds, pressed leafs, homemade masks, or drawings of particular things. When I learned to do this, instead of buying toys I, for example, helped my daughter create a collection of her drawings of fairies, and to supplement it with poetry about fairies found on the Internet for a beautiful homemade fairy book.

When children create collections of similar objects, they are learning valuable mathematical ideas. Obviously, they are learning to categorize, since this type of collection is built on the idea of similarity. Moreover, they are learning to organize their sets into subsets, such as sorting by colors or sizes. This is a beginning of work with various data structures such as tables or trees, which can also be explored used children's collections. For example, using a collection of pictures of different dog breeds, parents can help their children create a tree showing the history of breeding. Such structures will come very handy in algebra, number theory and computer science! In fact, a multiplication table is just such a collection of facts, organized in a table:

Parents can help younger children play with their collections, or decide how to expand collections, using similar data structures. Here I organized a collection of photographs of bears using a table. Having a digital camera instead of money while visiting shops can be much more satisfying for kids!

	Polar bears	Grizzly bears	Black bears
Small
Medium			A missing kind of a bear. Using this table, a child can easily see what to search for!
Large

 

Another big idea that came up in workshop participants' creations is style. The idea of a style relates to symbols, as discussed above. For example, dressing up means choosing clothes in a particular style that symbolizes a special occasion. One particular style of art that is relevant to mathematics is cubism, an abstract movement developed in the early 1900th. In cubism, forms are simplified and broken apart into planes, as in this collage "Three Musicians" by Pablo Picasso:

Monart, a popular methodology for helping children learn drawing, is also based on analyzing shapes that form a picture. In this method, children learn to perceive all objects in terms of basic elements of shape. In mathematics, breaking objects down into basic elements is used in calculus for integration, and in numerical methods for approximations.

Workshop participants had to work with available shapes to create their projects. The style of cubism, and the study of approximation, thus occurred naturally. In most projects one could see simple shapes used to approximate and to build up more complex shapes.

This puppy sculpture can be described mathematically, as approximation with cylinders.

Various shapes made out of cylinders.

A robot made out of cylinders: another approximation example.

• webs <~> paths in space, topology
• collections <~> data structures
• cubism <~> approximations