Mathematics and Society - OS X Lion Server

DIDACTICSTEACHING VIA LAB APPROACHES2. Multiple embodiments. The Dienes blocks fornumeration are an example of an embodiment of a mathematicalidea. An embodiment of an idea is a (lessabstract) model for the idea. For middle school purposes,embodiments usually are concrete materials or~=1pictures. Embodiments of fractions could be circularor rectangular regions, number lines, or collections ofblocks. Zoltan Dienes, a mathematician-psychologist12.43who has given extensive attention to concrete materialsin his ~ork with school children, feels that as many different embodiments as possibleshould be spread throughout students' work with an idea so that they see theessence of the idea. (All the embodiments should not be presented at One time, ofcourse.) Working only with circular regions as models for fractions, say, mighthinder a young child's ability to "see" one-half of a collection of gumdrops. Hence,Dienes would endorse our usual practice of using many empodiments for fractions:number lines, oranges, cakes, set models, other regions in addition to circular ones,etc. Note that multiple embodiments should also help students transfer their learningto other situations (see Teaching for Transfer in this resource). Finally, dif- (ferent students may be able to "see" a concept best with different embodiments;hence, multiple embodiments may help with individual differences.3. Concept clarification. Dienes also calls for as many variations as possiblewithin each type of embodiment. Forexample, in using circular regions forfractions, a teacher would use regionsof different sizes, cuts with differentorientations, a variety of numeratorsand denominators, and fractions greaterthan one. Looked at another way, theDOE",> THlS SI-\ow-!i:?~principle suggests varying irrelevant characteristics as much as possible. Activitiesfrom Mathematics in Science and Society can be used for topics like proportions, percent,etc., with these ideas of multiple embodiments and maximum variability in mind.4. Concrete materials = realities. Perhaps one of the strongest arguments forconcrete materials is that they link mathematics to real things. It is very easy formathematics classes to deal only with abstract symbols. Students who have generated,gathered, organized and studied their own data in a variety of real contexts shouldgain some appreciation of the relevance of mathematics.48