Manipulatives and Metaphors
By Dr. Eugene Maier
"Manipulates," the saying goes, "are a bridge to the abstract." Presumably, I suppose, from the concrete. But. given the image the word 'bridge' conjures in my mind, the metaphor seems wanting to me.
"Bridge" does suggest a connection, and I would agree, that the concrete and the abstract ought to be connected. But, "bridge" also suggests to me a sense of separation and departure. A bridge may indeed take me to some new place, but once I'm there the bridge has served its purpose; it becomes a thing of the past and may as well be forgotten. Thus, in the bridge metaphor, manipulatives provide a means of escaping the concrete to get to the abstract.
For me, the role of manipulatives is not to move one from the concrete to the abstract, but to provide a concrete support for the abstract. A metaphor that is more appealing to me is that of the studs in a wall that provide support for the structure. As the structure is being erected, the studs are in full view, however as the structure gets built, they become covered over and are no longer exposed to our view. However, they still remain there, continuing to support the structure, invisible to the naked eye, yet still visible in the mind's eye.
Take, for example, the notion of arithmetic average. One way that I've found effective for introducing this notion to students is to have students erect some stacks of blocks and then, without changing the number of stacks, level them off. Thus, if they create 5 stacks, containing 3, 8, 10, 4, and 5 blocks respectively, when leveled off, each of the 5 stacks will contain 6 blocks. I tell them that, mathematically speaking, one says the average of 3, 8, 10, 4, and 5 is 6. ( The numbers in this example were chosen so that things work out nicely—in most instances, one finds that one has to cut some blocks into parts to make the stacks all the same height, which leads to a nice discussion about fractions.)
As students become familiar with this model for finding averages, they determine that one way to find the height of the leveled off stacks is to determine how many blocks there are all together and divide that total by the number of stacks, that is, they discover that the average of a set of numbers is the sum of that set of numbers divided by how many numbers there are, which is the usual textbook definition. At this point, one could leave the blocks behind and cross the bridge to the abstract and henceforth deal with averages in the abstract world of pure number. But to do so, loses all the power of the image of leveling off stacks of blocks which gives much more insight into how averages behave then the abstract definition does.
For one thing, the "leveling off" image can be quite useful when computing. It's much easier to average 93, 89, 95 and 94 by leveling off these numbers than by adding them up and dividing by 4. Thinking of these numbers as stacks of blocks, moving 1 block off the last stack and 2 off of the second last stack, and putting these three blocks on the second stacks gives us stacks of 93, 92, 93 and 93. So we have 4 stacks of 92 with 3 extra blocks to be divided among the 4 stacks. Hence the average is 92 3/4 , and very little arithmetic was required to determine this. For another example, consider this typical school problem about averages: Helen has grades of 83, 75, and 90 on three math exams; what score must she get on tomorrow's exam is she wants an average of 85 on the 4 exams? What Helen wants are 4 "stacks" that level off to 85. The first stack, 83, needs 2 more, the next stack needs 10 more, while the third stack has 5 to spare. Hence the first 3 stacks need a total of 7 more to level off at 85. Hence the last stack needs 7 more than 85, or 92, for all 4 stacks to level off at 85.
The image can also be adapted as the curriculum progresses. (If the examples that follow come from territory that is foreign to you, skip this and the next paragraph. You can get the point without following the details in the examples.) All those mixture problems encountered in algebra courses are nothing but averaging problems in disguise. For example, to find the amount of a 40% sugar solution that must be added to an 85% sugar solution to create 1800 ml of a 60% solution, think leveling: stacks of 40 and stacks of 85 are to be leveled off to get stacks of 60. Since 85 is 25 more than 60, 4 stacks of 85 provides an excess of 100 over 4 stacks of 60. These 100, spread over 5 stacks of 40 will bring these stacks to 60. So 5 stacks of 40 and 4 stacks of 80 will provide 9 stacks of 60. (The figure may be helpful.) Thus to get 1800 stacks of 60, take 1000 stacks of 40 and 800 stacks of 80. In terms of the original language of the problem, one needs 1000 ml of the 40% solution.