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Infinitesimals-Based Registers for Reasoning with Definite Integrals

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Infinitesimals-Based Registers for Reasoning with Definite Integrals Infinitesimals-based registers for reasoning with definite integrals Rob Ely University of Idaho Abstract: Two representation registers are described that support student reasoning with definite integral notation: adding up pieces (AUP) and multiplicatively-based summation (MBS). These registers were developed in a Calculus I class that used an informal infinitesimals approach, through which differentials like dx directly represent infinitesimal quantities rather than serving as notational finesses or vestiges. Student reasoning reveals how the AUP register supports modeling with integral notation and how the MBS register supports sense-making with and evaluation of integrals. Keywords: calculus, integral, register, semiotics, infinitesimal, differential My goal is to examine and illustrate two registers for interpreting and working with definite integral notation, registers that are particularly useful for supporting student modeling and sense- making with integrals. These registers—adding-up-pieces (AUP) and multiplicatively-based summation (MBS)—are situated in a Calculus I course that uses an “informal infinitesimal” approach to calculus. I briefly summarize this general approach to calculus before describing these registers and how students reason with them. Informal Infinitesimals Approach to Calculus For nearly two centuries, Calculus was “the infinitesimal calculus.” For its inventors, G. W. Leibniz and Isaac Newton, it was a set of techniques for systematically comparing infinitesimal quantities in order to determine relationships between the finite quantities that they comprised (and vice versa). In the 19th century, calculus was reformulated in terms of limits rather than infinitesimals, and 20th century calculus textbooks have followed suit. Yet calculus textbooks and courses still use Leibniz’ notation, dx and ∫ , but without the meanings Leibniz assigned to these: dx is an infinitesimal increment and the big S is a sum (“summa”). The notations are now vestiges, and in particular, differentials no longer directly represent quantities that students can manipulate and reason with. The guiding principle of the informal infinitesimals approach is to restore this direct referential meaning to calculus notation. The approach is supported by the work in nonstandard analysis in the 1960s showing that calculus can be founded upon infinitesimals with equal rigor and power, but it uses the informality of Leibniz’ reasoning rather than the formal development of the hyperreal numbers (e.g. Keisler, 1986). For instance, the derivative at a point dy/dx really ! !!! !!(!) is a ratio of two infinitesimal quantities, not code language for lim . The chain !→! ! rule is canceling fractions. And an integral really is a sum of infinitesimal bits, each of which in MBS is given by the product f(x)·dx. By taking differentials seriously, students can develop formulas for volumes of rotation, arclength, work, and many other ideas, formulas, and applications in first-year calculus (Dray & Manogue, 2010), and in vector calculus and physics (Dray & Manogue, 2003). Registers and Signs For Duval (2006), a representation register is a collection of signs and a set of transformations by which some of these signs can be substituted for others. Transformations within the same register are treatments; transformations of signs from one register to another are conversions. Barthes defines a sign as a combination of a signifier and a signified (1957/1972). For instance, a bunch of roses (signifier), together with the concept of passion it is representing (signified), comprise a sign. In our case, when a mathematical representation (e.g., “dx”) signifies a concept (e.g., an infinitesimal increment), the combination of the representation “dx” (signifier) and infinitesimal increment (signified) is a sign. An interpretation is thus a signified concept. So if a representation stays the same but its interpretation changes, it becomes a different sign, since it signifies a different thing or concept. This points to a conversion to a different register, because within a given register interpretation should remain relatively stable. Such a conversion is often accompanied by a new lexicon of signs, interpretations, and treatments that which might support the new purpose or apply to the new context. Consider the following example: We can use infinitesimals to develop the formula for the arclength of a curve in the plane between x = 0 and 1. We imagine a curve to be comprised of infinitesimal segments, each of which is the hypotenuse of a right triangle with legs dx and dy. Then the arclength of the curve would be the sum of these hypotenuse lengths: ! ��! + ��! !!! . So far we have performed a modeling step, treating the dx and dy as quantities representing magnitudes, and have used the adding up pieces (AUP) register (which I detail soon). But this is no form to be evaluated for any particular curve, however. To be evaluated, the integral must first be converted by imagining all the dx’s as uniform in size and then being ! !" factored from the integrand, to get 1 + ( )! �� (if y is a function of x, say g(x)). Now the !!! !" ! �(�) �� 1 + �′(�)! integral is of the form !!! (where f(x) is ). So the integral can be evaluated by F(1) – F(0), for some F as an antiderivative of f. Thus using algebraic manipulations we have converted to a new register that is suited for evaluating integrals. The [ ]! + [ ]! structure lost ! �(�) �� significance and the ! structure gained salience, and it became important for the original curve to be seen with y as a function of x. The Adding Up Pieces (AUP) and Multiplicatively-Based Summation (MBS) Registers The elements of the AUP and MBS registers are based on the work of Jones (2013, 2015a, 2015b). The interpretations involved in both registers are summarized in Figures 1 and 2, and the strange numbering on these (I2, etc.) draws from the learning progression in my class through which they emerge, which I reference elsewhere in detail (Ely, in review). The interpretations in the AUP register (Figure 1) entail the idea that a definite integral measures how much of some quantity A is accumulated over an interval of a domain, say from t = a to b. This domain is partitioned into infinitely many infinitesimal increments of uniform size dt. For each infinitesimal increment dt there corresponds an infinitesimal increment of A, dA. The integral ! ! adds all of these up to give the total accumulation of A over the interval from t = a to b. In order for this to make sense, one must appeal to conception C5: The sum of infinitely many infinitesimal bits is a finite accumulation of A. The AUP register allows one to transparently represent a general bit of the sought quantity and of the whole quantity as an accumulation of these bits. The treatments in the register include (a) writing a symbolic expression for a generic bit dA and (b) rewriting this expression for dA in terms of other infinitesimal quantities that specify the expression for the domain at hand, usually in terms of its corresponding domain increment dt. These treatments require viewing infinitesimals as legitimate quantities that behave Figure 1 – Interpretive elements normally under algebraic operations (including possibly of integral notation in the adding some additional Leibnizian rules for operating with up pieces (AUP) register infinitesimal quantities). The treatments also rely on two other grounding conceptions: (1) equivalent expressions can be substituted for the same quantity (conception C6), and (2) a foundational understanding of covariation, which in turn relies on the basic understanding that variables vary (Thompson & Carlson, in press). The student must be able to coordinate changes of A with changes of t in order to reason that for each increment dt there is a corresponding increment dA. The MBS register includes many of the same notational interpretations as AUP, but it also adds to these the expression of each piece dA as a product r(t)·dt. This introduces the integrand, which is necessary for the Fundamental Theorem of Calculus (FTC) to apply. We follow Thompson, Byerley, & Hatfield’s (2013) approach to treat the integrand r(t) as a rate at which A accumulates over the increment dt. This ultimately allows us to recognize and use that an accumulation function f for A will have r(t) as its rate-of-change function. Thus, one supporting conception in this interpretation of the product r(t)·dt is the idea that A accumulates at a constant rate r(t) over the dt increment, and that this constant rate is determined by the value of t closest to that dt increment. This relies on conception C4: A concept image for rate of change at a moment (Thompson, Ashbrook, & Musgrave, 2015). This momentary rate of change can vary constantly as t varies. I1 . dA: a generic The second supporting I4i. Sum of all the bits of A, i as t hops along by dt-sized “little bit” of the conception is the increments from a to b quantity A multiplicative structure associated with rate, expressed in conception b I2i. An infinitesimal C1i (see Figure 2). The ∫ r(t)·dt increment of t a Figure 2 – Interpretive I3i. “Rate” at which A accumulates elements of over the dt-sized increment of t C1i. The multiplicative integral notation structure: If A accumulates at in the a rate of r(t) A-units per t-unit, multiplicatively- over an increment of dt t- C5i. The sum of infinitely many units, the product r(t)·dt is the based summation infinitesimal bits is a finite accumulation. accumulated bit of A. (MBS) register word “rate” is used here broadly. It does not necessarily mean that r(t) is measured in a compound unit like miles-per-hour. Rather it means that as t changes, A changes by a proportional amount, and r(t) provides that rate of proportionality (Lobato & Ellis, 2010). Most broadly, r(t) serves as a factor allowing conversion from an increment of t to an increment of A, by dA = r(t)·dt.
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