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Awareness and Cavalieri's Principle ALGEBRAIC FORMULAS, GEOMETRIC AWARENESS AND CAVALIERI’S PRINCIPLE JÉRÔME PROULX, DAVID PIMM It seems appropriate in a new journal of mathematics deployed, geometry has vanished (despite the etymological education [For the Learning of Mathematics] that there origins of the word itself). should be an attempt to tackle a perennial problem [the We argue that offering and focusing upon formulas in this loss of geometry] with new terms. (Tahta, 1980, p. 9) context transports the mathematical work from the geomet- ric realm of area and volume to that of substitution of One of the fundamental tensions at work in both learning numerical values into formulas – a quite different mathe- and teaching mathematics is that between fluency and matical arena of potential technique, fluency and skill. (One understanding. [1] Formulas are frequently offered as an of us, for instance, remembers classroom work involving aid to fluency, as a concise and manipulable summary of an ‘changing the subject of the formula’ – algebraic tasks that invariance, an indifference to change and to the contingent. were quite independent of the meaning and referents of the Formulas contain a tacit ‘always’ (with ‘under suitable con- particular formula in question and, indeed, whose unavowed ditions’, sometimes said sotto voce). One unfortunate result purpose was precisely such semantic suppression.) Signifi- can be students at various levels seeing mathematics as cantly, however, this arena has nothing to do with the being ‘all about formulas’. More than twenty-five years ago, geometric notions of area and volume themselves: when stu- Hart (1981) observed: dents work with area or volume formulas, they are not mathematics teaching is often seen as an initiation into working on understanding concepts of area or volume, they the rules and procedures which, though very powerful are working within a particular sub-domain of algebra, (and therefore attractive to teachers), are often seen by namely substitution. children as meaningless. (p. 118) Underlying understanding is awareness, a term much The presence of formulas is particularly prominent in the drawn upon by Caleb Gattegno who offered the challenging teaching of geometric topics such as area and volume. This, proposition ‘Only awareness is educable’. In this journal, Janvier (1994) explains, has the consequence of making Hewitt (1999, 2001a, 2001b) has published a series of arti- people think of area and volume as easy topics to teach, cles concerned with teaching and the educating of mathe- where it “simply consists of leading students to memorise matical awareness, in relation to a distinction between the some formulas” (p. ix, our translation). In these instances, arbitrary and the necessary, as well as examining the chal- learning about area and volume is reduced to knowing the lenge of memory for mathematics. Hence, lying behind this formulas, recognizing which one to use and applying it in a article’s more surface concerns is a desire to find ways to problem to obtain an answer. work more geometrically (which often means more directly) The starting point for this article is an exploration of the with much of school mathematics. We would argue pedagog- complex set of connections between algebraic formulas and ically for approaches that attempt to educate geometrical geometric awareness, in particular the limited extent to which awareness of the structure of some area and volume concepts. the latter is either embedded in or necessary for the former. To do this, we bring forth a powerful historical mathematical We do so within the core school mathematical topic of area resource, an early seventeenth-century mathematical theorem, (and, to some extent, of volume), but we wish to go beyond the curiously named Cavalieri’s principle. [2] what is now explicit in curriculum documents of various countries, namely that such formulas should be motivated or Cavalieri’s principle even proven using geometric arguments (e.g., NCTM, 2000). Bonaventura Cavalieri (1598–1647) was an Italian mathe- One complaint in relation to this minor move (as we see it) matician whose work is often mentioned in accounts of the is that it leaves unchallenged the prominent place taken by emergence in Europe of infinitesimal calculus and infini- formulas themselves as the primary or even sole goal of tesimal calculations in the early part of the seventeenth teaching and learning about area and volume. We wish to century. One of his theorems reads: examine a possibly unfamiliar way (namely Cavalieri’s prin- If between the same parallels any two plane figures are ciple, our mediating third of the title) of staying with the constructed, and if in them, any straight lines being geometry on its own terms. In doing so, we take seriously drawn equidistant from the parallels, the included por- Tahta’s (1980, p. 7) formulation that “the geometry that can tions of any one of these lines are equal, the plane be told is not geometry”, noting that area and volume for- figures are also equal to one another; and if between the mulas are unreservedly about such tellings. The moment the same parallel planes any solid figures are constructed, word ‘measure’ is uttered, the instant that variables are and if in them, any planes being drawn equidistant from For the Learning of Mathematics 28, 2 (July, 2008) 17 FLM Publishing Association, Edmonton, Alberta, Canada the parallel planes, the included plane figures out of Figure 3 shows members of the family of equivalent par- any one of the planes so drawn are equal, the solid fig- allelograms obtainable from a given one (there is a unique ures are likewise equal to one another. […] The figures rectangle in this family). Working geometrically with Cav- so compared let us call analogues, the solid as well as alieri allows us to know directly that these figures have the the plane [...] (Cavalieri, 1635) same area, without any computation. Gray (1987) glosses the two-dimensional version of this claim as follows: “The principle asserts that two plane fig- ures have the same area if they are between the same parallels, and any line drawn parallel to the two given lines cuts off equal chords in each figure” (p. 13). Figure 1 pre- sents two plane figures illustrating Cavalieri’s principle. Figure 3: An infinite family of Cavalieri-equivalent parallelograms A rhombus is a parallelogram with all four sides of the same length. In fact, in the particular infinite family of asso- ciated parallelograms indicated in Figure 3 (that all lie within the two given parallel lines, with the same base), there are two equivalent rhombuses (one left-facing and one right-facing) that could be drawn. So, any rhombus belongs Figure 1: An illustration of Cavalieri’s principle in two to a unique family of parallelograms and so can be associ- dimensions ated with a single rectangle of the same area, that is, the The principle seems to allow qualitative assertions of rectangle that has the same base and the same height as the same or different only (although other circumstances might rhombus. But not all families of parallelograms contain a allow greater or lesser to be argued). Questions of how much rhombus. For example, if the shorter side of a parallelogram bigger or smaller lead to quantification of the relation, is selected as the base of the parallelogram and the height is though not necessarily quantification of the area itself. [3] larger than this length, then no rhombus can be found in this family. Thus, all rhombuses are associated with a family of Plane geometry through Cavalierian eyes parallelograms, but not all families of parallelograms con- tain a rhombus. In addition, no rhombus family (other than As a very straightforward application of Cavalieri’s princi- the one associated with a square rhombus) contains a square. ple, here is a parallelogram and a corresponding rectangle: as One thing Cavalieri provides, then, is a reason to attend to both figures lie between the same parallels (resulting in the related figures drawn between the same pair of given para- fact that each figure has the same height, a consequence of llels. It also invites attention to the notion of an infinite Euclid’s Fifth Postulate [4]) and for every cross-section the family of figures that resemble one another in terms of some segments have equal length, then by Cavalieri’s principle, attribute (in Cavalieri’s terms, the family of analogues). In they have the same area (see Fig. 2). Notice in particular that addition, the principle points to the freedom this equivalence for all Cavalieri figures this condition applies at the two allows us to be able to select any member of this family to be extremes, the top and the bottom of the shape. used as a referent to which to relate other figures. For pur- poses of area (and of volume as we will shortly see), when dealing with categories of shapes for which there are general formulas, these reference figures are often chosen to be rec- tangular. These features illustrate examples of awarenesses that Cavalieri’s principle stresses and brings to the fore. The realm of quadrilaterals is complex, with various fig- Figure 2: Cavalieri’s principle at work with a rectangle and ures named (and some not) as a consequence of particular a parallelogram schemes (e.g., classification by number of pairs of parallel Consequently, particular pairs of parallelograms can be sides, exclusive or inclusive; classification by means of associated: any parallelogram has a single associated rectan- number of equal side lengths; etc. – see de Villiers, 1994). gle (seen as a rectangular parallelogram) relative to a given The result is a set of distinguished, named shapes, but one base (choose the ‘other’ side as the base and there is a dif- which does not necessarily reflect a single coherent scheme ferent family of analogues) or relative to a given pair of with respect to area.
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