Thursday, 5 September 2013

Algebra as the Generalization and Formalization of Patterns and Contraints

Although pure computational arithmetic of the sort that dominates elementary school mathematics, the kinds of counting and sorting involved in combinatorics, and pure spatial visualization need not inherently emphasize generalization and formalization, it is difficult to point to mathematical systems and situational contexts where mathematical activity does not involve these two processes. The manipulations performed on formalisms (which I identify in this chapter as the second kernel aspect of algebra and which sometimes yield general patterns and structures—the essence of the third, structural, aspect of algebra) typically occur as the direct or indirect result of prior formalization. Generalization and formalization are intrinsic to mathematical activity and thinking—they are what make it mathematical. Generalization involves deliberately extending the range of reasoning or communication beyond the case or cases considered, explicitly identifying and exposing commonality across cases, or lifting the reasoning or communication to a level where the focus is no longer on the cases or situations themselves, but rather on the patterns, procedures, structures, and the relations across and among them (which, in turn, become new, higher level objects of reasoning or communication). But expressing generalizations means rendering them into some language, whether in a formal language, or, for young children, in intonation and gesture. In the case of young children, identifying the expressed generality or the child’s intent that a statement about a particular case be taken as general may require the skilled and attentive ear of a teacher who knows how to listen carefully to children.


We distinguish two sources of generalization and formalization: reasoning and communication in mathematics proper, usually beginning in arithmetic; and reasoning and communicating in situations based outside mathematics but subject to mathematization, usually beginning in quantitative reasoning. The distinction between these two sources (mathematics proper and situations outside mathematics) is especially problematic in the early years, when mathematical activity takes very concrete forms and is often tightly linked to situations that give rise to the mathematical activity. Whether the starting point is in mathematics (and therefore arising from previously mathematized experience) or from a yet-to-be-mathematized situation,
the source is ultimately based in phenomena or situations outside mathematics proper because, after all, mathematics thinking ultimately arises from experience and only becomes mathematical upon appropriate activity and processing. This view is the basis of many reform curricula.


Although students in traditional mathematics classrooms might model the same situations as students in classrooms that promote understanding, and reason similarly within mathematics to formalize those situations into algebraic differences and inequalities, students in traditional classrooms are more likely to generalize from objects and relations already conceived as mathematical (e.g., a student might generalize patterns in sequences of numbers in a hundreds table or multiplication table¾mathematics proper). In classrooms that promote understanding, students are more likely to begin by generalizing from their conceptions of situations
experienced as meaningful and to derive their formalisations from conceptual activities based in those situations. For example, a student comparing differences in prices between cashews (expensive) and peanuts (cheap) for two different brands, might generalize that a small increase in price of, say, peanuts of brand A will not change the outcome of a comparison.This difference of context can make mathematics involved more vital/important to the student both in the short and long term.


A Classroom Example of Early Generalization and Formalization
The following example from a third-grade class was observed . The teacher began by asking how many pencils there were in three cases, each containing 12 pencils. After the class arrived at a repeated-addition (12 + 12 + 12) solution, the teacher showed how the result could be seen as a (3 x 12) multiplication. She expected to move on to a series of problems of this type, but one student noted that each 12 could be decomposed into two 6s, and that the answer could be described as 6 + 6 + 6 + 6 + 6 + 6 or six 6s and could be written as 6 x 6. Another student observed that each 6 could also be thought of as two 3s, yielding twelve 3s or 12 x3. Another student realized that "This one is the backwards of our first one, 3 ´ 12." What follows is a description of the extended investigation that occurred.


Activity: The students continued to find ways of grouping numbers that totaled 36. One student looking at the column of 3s, suggested four groups of three 3s, or 4 ´ 9. Another student noted that “we can add another one to the list because if 4 ´ 9 = 36, then 9 ´ 4 = 36, too." One student objected, asking a question the teacher found interesting: "Does that always work? I mean, saying each one backwards will you always get the same answer?" When the teacher asked her what she thought, the student said, “I'm not sure. It seems to, but I can't tell if it would always work. I mean for all numbers." For homework, the teacher asked them to explore ways to prove (or disprove) the student’s question. The next day, the students explained their thinking, noting various number pairs such as 3 ´ 4 and 4 ´ 3 and sometimes using manipulatives to illustrate their examples. Although the original objector was still not convinced that this would work for all numbers, the teacher decided to leave the issue unresolved temporarily and continue exploration of multiplication by introducing arrays. Two weeks later, however, the teacher reintroduced the problem, suggesting students use what they now knew of arrays “to prove that the answer to a multiplication equation would be the same no matter which way it was stated." The class worked on this for a while, alone and with partners. Finally, one student decided she could prove it. Holding up three sticks of 7 Unifix cubes, she said,
See, in this array I have three 7s. Now watch. I take this array [picking up the three 7- sticks] and put it on top of this array [turns them 90 degrees and places them on the seven 3-sticks she has previously arranged]. And look— they fit exactly. So 3 x 7 equals 7 ´ 3, and there's 21 in both. No matter which equation you do it for, it will always fit exactly. At the end of this explanation, another student eagerly explained another way to prove it:


I'll use the same equation as Lauren, but I'll only need one of the sets of sticks. I'll use this one [picks up the three 7-sticks]. When you look at it this way [holding the sticks up vertically], you have three 7s. But this way [turning the sticks sideways], you have seven 3s. See? . . . So this one array shows both 7 x 3 or 3 x 7.
At this explanation, the class objector agreed: Although both students had used a 3 x 7 array to explain their points, the final, simpler representation convinced her of the general claim. As she noted: "That's a really good way to show it . . . It would have to work for all numbers."


In this example, students were attempting to generalize what they saw in a few cases of multiplication to all cases of multiplication and (because they had not yet worked with formal language in mathematics) to articulate their generalization through a variety of notational devices in combination with “natural,” informal language. The basic issue was the range of the generalization—Did it hold for all numbers? The students used cubes and sticks to generate their ideas, to show one another their thinking, and to justify claims that were clearly theirs not their teacher’s. The questions of certainty and justification arose as an integral aspect of the process and were interwoven in their use of notations. Thinking of this activity merely as the children developing the concept of commutativity of multiplication (of natural numbers) trivializes what happened during this extended lesson. The students were actually constructing both the very idea of multiplication (although only two aspects: repeated addition and array models) while beginning to develop the notion of mathematical justification and proof. Although the episode began in a concrete situation, it quickly became a mathematical exploration. Pencils and cases were the stepping-off point that (inadvertently) led the students to the grouping and decomposition of whole numbers and, after some reflection, to the articulation of their newly constructed knowledge (the equivalence of alternative groupings) through use of concrete arrays of cubes. Students found ways to articulate the invariance of the "amount," or total, first under alternate groupings of 21 and then under alternate orientations of the same physical grouping. In the end, despite the fact that they did not have a formal language available, the generality was not only realized, but made explicit: "It would have to work for all numbers." It is easy to imagine that this property might be given a more formal expression later, first perhaps as "box times circle = circle times box," and then later as a x b = b x a.


One other aspect of this situation deserves attention: This is certainly not traditional symbol-manipulation algebra. Although this was clearly an excellent teacher doing a good job in an arithmetic, this extended lesson focusing on generalization rather than computation took place in what many teachers would regard as the normal course of mathematical concept development in an "ordinary" mathematics classroom ("ordinary" in the sense of fitting the NCTM Professional Standards for Teaching Mathematics [1991]). Algebra as Syntactically Guided Manipulation of (Opaque) Formalisms When we deal with formalisms, whether traditional algebraic ones or those more exotic, our attention is on the symbols and syntactical rules for manipulating those formalisms rather than on what they might stand for, with much of their power arising from internally consistent, referent-free operations. The user suspends attention to what the symbols stand for and looks at the symbols themselves, thus freed to operate on relationships far more complex than could be managed if he or she needed at the same time to look through the symbols and transformations to what they stood . To paraphrase Bertrand Russell, (formal) algebra allows the user to think less and less about more and more.


The problem is that our traditional algebra curriculum has concentrated on the "less and less" part, resulting in many students’ inability to see meaning in mathematics and even in their alienation from mathematics. The power of using the form of a mathematical statement as a basis for reasoning is lost when students practice endless rules for symbol manipulation and lose the connection to the quantitative relationships that the symbols might stand for (coming, along the way, to believe that this is what mathematics really is). What happens too often in traditional mathematics classrooms is less learning with understanding than learning with
misunderstanding. Research provides many examples of the difficulties into which students have been led when they do not construct their own knowledge or are not given sufficient time to reflect upon what they have learned.The classroom examples below suggest comparison. The first illustrates what can (and too often does) happen when students do not construct relationships among pieces of mathematical knowledge. The second describes a task taken from a reform curriculum that supports learning with understanding.





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