Blending of Conceptual Physics and Mathematical Signs∗

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Blending of Conceptual Physics and Mathematical Signs∗ Blending of Conceptual Physics and Mathematical Signs∗ Tra Huynh† and Eleanor C. Sayre‡ Kansas State University, Physics Department, 116 Cardwell Hall, 1288 N. 17th St. Manhattan, KS 66506-2601 (Dated: September 26, 2019) Mathematics is the language of science. Fluent and productive use of mathematics requires one to understand the meaning embodied in mathematical symbols, operators, syntax, etc., which can be a difficult task. For instance, in algebraic symbolization, the negative and positive signs carry multiple meanings depending on contexts. In the context of electromagnetism, we use conceptual blending theory to demonstrate that different physical meanings, such as directionality and location, could associate to the positive and negative signs. With these blends, we analyze the struggles of upper-division students as they work with an introductory level problem where the students must employ multiple signs with different meanings in one mathematical expression. We attribute their struggles to the complexity of choosing blends with an appropriate meaning for each sign, which gives us insight into students' algebraic thinking and reasoning. I. INTRODUCTION ics literature and a lack of a strong theoretical framework that accounts for a homographic property of signs, which Mathematics is important in understanding physics, would give us better insights to students' reasoning and but many students struggle with integrating mathemat- struggles. ical formalism. The language of mathematics, like other We pull out two conceptual spaces which might be affil- languages, includes homographs and homonyms: a sym- iated with sign in physics: location and direction. Using bol in mathematics can bear multiple meanings, possi- the machinery of conceptual blending, we show how those bly causing trouble for language learners. This can be two spaces may blend with sign to produce five different especially prevalent (or problematic) when mathematics blends. We illustrate the blends in a one-dimensional is used in other scientific contexts, such as physics, when problem from electrostatics, first with a principled solu- the meaning of a symbol can be more specific, yet diverge tion and then with two student solutions. Our goal is from the standard mathematical conventions at the same to present substantial theoretical discussion grounded in time. a simple physical scenario, to show how even a simple Signs, both negative and positive, are important in problem and the relatively limited space of signs have both math and physics. Building on Vlassis' work in el- rich conceptual complexity undergirding them. This is ementary algebra [28], a map of how the negative sign not an exhaustive accounting of all that students can do is used includes three natures: unary (structural signi- with sign, or even with this problem. fier), binary (operating signifier), and symmetric func- Our work is in contrast to prior work which suggested tions. For example, the signs in the metaphor of floors in that two input spaces have only one resulting blend, and a building can play different roles, such as moving from that difficulty in blending arises from difficulty choosing one floor to another (binary - `going up two') and estab- appropriate input spaces or running a blend. lishing the location of a floor (unary - `the second floor'). Successful sense-making of a mathematical expression re- quires students to flexibly and suitably ascribe meaning II. SIGNS IN EDUCATION RESEARCH to signs. Yet, algebra students often fail to consider ex- plicitly that the minus sign could have a double status or Historically, the concept of negativity was attached to meaning [28]. the operation of subtraction. In mathematics history, In physics, signs can signify a direction (\towards"), the minus sign is first introduced to describe a quantity a flavor (\negative charge"), a location (\−3^x"), or an yet to be subtracted. A precise difference between the operation between two terms (\−kx−cv"). The complex- minus sign that assigns a negative number and an op- ity of reasoning with signs in physics, especially negative eration (subtraction) was not clearly defined until the arXiv:1909.11618v1 [physics.ed-ph] 25 Sep 2019 signs and negative quantities, has focused primarily on development of algebra, when negative numbers gained the results that reasoning with signs is hard, especially the status of mathematic objects [14]. at the introductory level. However, mechanisms for why Significant research on how we make sense of negative reasoning with signs is hard have been elusive. More- numbers has been conducted within the context of math- over, there is both a lack of connection with mathemat- ematics education since the 1980s. Lakoff and N´u~nez [15] suggest that negative number sense could be ac- quired by extending and stretching their set of ground- ∗ Portions of this paper previously appeared in preliminary work ing metaphors for counting-number arithmetic to the set [13]. of integers. Various instructional models have been de- † [email protected] veloped that employ metaphor to better teach negative ‡ [email protected] numbers and the operations on them to children [1, 14]. 2 In contrast, Vlassis [28] argues that it is not the nature count for multiple meanings, the authors used the Vlas- of the negative number itself but the negative sign itself sis' categorization scheme to ask students to explicitly that causes difficulties. explain what physical meaning is embodied in individ- In physics, negative and positive signs embody addi- ual negative and positive quantities in different mechan- tional characteristics for quantities. The meaning of such ics and electromagnetism contexts. The results show signs varies specifically with context. For example, nega- that even though students' flexibility in interpreting the tivity in reasoning about energy can be particularly chal- meanings of signs is highly context-dependent, they gen- lenging [16, 24]. Two common ontologies for energy are erally struggle with the symmetrical meaning more than substance (energy contained in objects) and vertical lo- unary and binary functions. Moreover, combining more cation (higher or lower energy level) [7, 23]. Each ontol- than one nature of negativity in a single calculation ap- ogy has both advantages and drawbacks in making sense pears to present significant challenges to introductory of different aspects of negativity. For example, the sub- students. stance ontology is helpful in thinking of the transference Altogether, these disparate accounts of signs in math- of energy but appears to be a hindrance in the case of neg- ematics and physics suggest that students struggle with ative substance or negative energy, where the vertical lo- flexibly choosing and applying specific meanings of cation ontology is more appropriate and convenient[6, 7]. signs, from children to intermediate-level undergraduate Therefore, accounting for the sign contexts while flexi- physics students. This is not surprising: there are multi- bly choosing the signs' meaning is significant for making ple meanings of the negative sign which may vary among sense in physics. problems and within a given problem or expression. As a scalar, energy itself does not have a direction. However, when it comes to vector quantities, positive and negative signs carry the additional meaning of direction- III. THEORETICAL FRAMEWORK ality, which has been shown to cause additional trouble for students. Middle school physics teachers inappropri- We turn conceptual blending theory to help us make ately interpret the signs of acceleration using the speed sense of the different meanings of signs in physics. model [25]: accelerating is always in the positive direc- tion and decelerating is always in the negative direction. The speed model makes sense from a person-centric mo- A. Conceptual blending theory tion perspective: we usually move forwards and think of that as the positive direction. Walking backwards is rare; Conceptual blending theory was developed by Faucon- if we need to go back, we usually turn around and walk nier and Turner [9] to account for how people create forwards. meaning. The theory posits a mental network model Reasoning about sign for both scalar and vector that processes and forms new meaning (Figure 1). The quantities is complicated by mathematical formalism in mental network model is composed of at least two input physics; in one-dimensional problems, we often treat spaces containing information from discrete domains, a quantities flexibly as either vectors or scalars. For exam- generic space containing common information and struc- ple, consider the kinematics equation for final velocity as tures, and a blended space where new meaning emerges. a function of initial velocity, acceleration, and time: The input elements in one space connect to their coun- terparts in the other spaces via vital relations, such as v⃗f = v⃗i + at⃗ (1) time, space, change, cause-effect, identity, etc. When If an object is slowing down, there must be something projected into the blended space, these vital relations subtracted from v⃗i [18]; the consequent minus sign can could be compressed into the same types, or more often, be interpreted to be an operation (remove at⃗ from v⃗i) different types of relations in the blended space. or as a comparison between relative directions (of v⃗i and The process of generating new meaning in the blended a⃗) or simply as a negative acceleration (as in the speed space involves three operations: composition, comple- model). Students may be unaware of the multiple
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