Suzana Kaori Ura, Alzira C. M. Stein-Barana and Deisy P. Munhoz share the resources they have been developing to facilitate the understanding of multiplication.
The multiplicative principle is the tool allowing the counting of groups that can be described by a sequence of events. An event is a subset of sample space, i.e. a collection of possible outcomes, which may be equal to or smaller than the sample space as a whole. If event A can occur in m different ways, event B in n different ways and event B is independent of event A, then the event (A followed by B) can occur in m�n ways.
It is important that students understand this basic principle early on and know how to apply it, because it is the basis for more complex subjects such as Combinatorics, Probability and Statistics gathering.
Our objective is to facilitate children's understanding of the multiplicative principle through a cooperative, entertaining and enjoyable activity: making fashion! Here we propose the use of paper dolls for teaching the multiplicative principle. The paper dolls we refer to are paper cutouts, with separately cut clothes and accessories, an inexpensive toy used by many generations. In the early 20 th century its popularity increased as they were presented in newspapers and magazines. The use of clothing combinations is certainly not new in order to exemplify the application of the multiplicative principle, but the use of paper dolls as a didactic resource helps the understanding of mathematical concepts through the manipulation of concrete objects. With this toy children's imagination, creativity and logic can be stimulated by the various possibilities it offers.
Materials and preparation
We'll use paper dolls, clothes and accessories. The dolls, clothes and accessories are designs that can be done in the classroom or even found on the internet and printed out. Students are encouraged to cut the designs themselves, as the act of cutting stimulates motor coordination. The tabs are important elements in the setting of clothing and should be folded over the body of the doll. The clothing ranges from dresses, blouses, skirts, shorts and trousers, while the accessories are shoes, hats and purses. Figure 1 illustrates a doll and clothes.
Making fashion
After a period of free exploration of figures, students should choose a set of blouses and shorts and find out how many combinations they can find with them to dress up the doll, an activity that can be done in cooperatives groups. Throughout this investigation, students should be listing the outcomes and counting the possibilities. This material allows students to construct knowledge about the multiplicative principle in an active way.
If, for instance, the student separates three blouses and four shorts, the event A would be the three blouses (m = 3) and event B the four shorts (n = 4). Therefore, the number of possible combinations of shirt and shorts is as follows:
m�n = 3�4=12
This result can also be found using the tree diagram shown in the illustration on the page opposite, or by counting.
If also two shoes are included then the possible looks are, making s=2:
m�n�s= 3�4�2=24.
It is known that images are important tools in communicating scientific ideas and also in their conceptualization. Paper dolls work similarly to book illustrations, stimulating childrens' minds without giving out the answers right away, thus encouraging them to complete the creation with their own imagination. The construction of maths skills can be complemented by playing like this with paper dolls, in this case used to explain the multiplicative principle. It is not just the girls that will be making fashion - boys will also be dressing up their figures, creating uniforms' combinations for sports teams.
By dressing up dolls, children are involved in a meaningful context instead of simply memorising definitions, so students can develop the concepts of multiplication and combinatorics based on their own concrete experiences.
[Author Affiliation]
Suzana Kaori Ura, Alzira C.M.Stein-Barana and Deisy P. Munhoz work with development of materials for the teaching of Physical Sciences and Mathematics at the Universidade Estadual Paulista, Brazil.
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