[n learning the definitions of sine, cosine, and tangent of an angle in trigonometry', it is necessary for students to coordinate and be able to move fluently amongst four sign vehicles that represent these objects. These sign vehicles belong to the registers of the triangle definitions, the coordinate plane, the unit circle, and the graphs of these trigonometric functions. Data from research projects are used to illustrate issues that arise in the course of students1 learning. These issues include compartmentaSization and use of prototypes, generalization, compression, use of metaphors, use of gestures, use of visual sign vehicles, idiosyncratic notation and sliding interpretants, the power of falsification, and use of a tool as a sign. After presenting data instantiating these categories, some implications for the teaching of trigonometric definitions in these four registers are drawn.
OVERVIEW
In each of two research projects, the central question being investigated was as follows: How may teaching facilitate students' construction of connections amongst registers in learning the basic concepts of trigonometry? In the first investigation, with Susan Brown in her “Advanced Enriched Algebra and Trigonometry" class in a Chicago high school, in addition to observation of her teaching, and collection of students’ work, four students were interviewed six times each over a period of three months. In the second investigation, with Jeffrey Barrett and Sharon McCrone in Jeffs university course mainly for prospective teachers,
“Geometric Reasoning: Geometry as Earth Measures," three students were interviewed twice each to interrogate their knowledge of the trigonometry that Jeff was including in the course. Analysis of more than 300 pages of transcript data from both sets of student interviews provided the categories of issues arising in students’ learning, but the transcripts from the Chicago study are the main focus of this chapter. The specific question addressed here is as follows: What aspects enable or constrain the making of connections amongst signs in learning the basic concepts of trigonometry? Finally, some suggestions are made regarding the teaching of trigonometry. A strong implication of the investigation is that by highlighting structures and patterns across domains, teaching that encourages students to make connections may foster generalizations and help to combat the phenomenon of compartmentalization.
L Radford, G. Schubring. a n d S e e g e r teds.). Semiotics in Mathematics Education: Epistemology.
History. Classroom, and Culture. 103-1/9.
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THEORETICAL LENSES
Three theoretical lenses were used in interpreting the data in this study, namely, Peircean semiotics (Peirce, 1992, 1998), Duval’s (1999) conversions amongst registers, and Hitt’s (1998) levels of understanding of systems of representation.
In analyzing the transcripts for the purpose of investigating the ways that students construct, or fail to construct, connections amongst the triangle definitions of sine, cosine and tangent of an angle, the coordinate plane and unit circle, the sinusoid and tangent graphs, and trigonometric identities, two of Peirce’s (1992, 1998) ten triads were used. These triads are as follows:
- object, representamen, and interpretant;
- signs that are iconic, index tea I, or symbolic according to students’ inter
pretations.
With regard to terminology, because Peirce himself used the word “sign” in different ways, sometimes referring to the representamen, but more often referring to the whole triad of object, representamen, and interpretant, in this paper ! shall use the term “sign vehicle” when referring to the representamen, and “sign” when referring to the whole triad.
According to Peirce (1992),
trichotomic
is the art of making three-fold divisions. By his own admission, he showed a proclivity for the number three in his philosophical thinking. “But it will be asked, why stop at three?” he wrote (Peirce,1992, p. 251), and his reply to the question is as follows:
[Wjhile it is impossible to form a genuine three by any modification of the pair, without introducing something of a different nature from the unit and the pair, four, five, and every higher number can be formed by mere complications of threes, (ibid.)
Accordingly, he used triads not only in his semiotic model including object, sign vehicle (or representamen) which stands for the object in some way, and interpretant based on the relationship between sign vehicle and object, but also in the types of each of these components. These types are not inherent in the signs themselves, but depend on the interpretations of their constituent relationships between sign vehicles and objects. In a letter to Lady Welby on December 23,
1908, he wrote as follows. i
i define a Sign as anything which is so determined by something else, called its Object, and so determines an effect upon a person, which effect I call its Interpretant, that the latter is thereby mediately determined by the former. My insertion of “upon a person” is a sop to Cerberus, because I despair of making my own broader conception understood. I recognize three Universes, which are distinguished by three Modalities of Being. (Peirce, 1998, p. 478) For the purpose of using some of Peirce’s triads in examining data on the teaching and learning of trigonometry, I shall embrace his “sop to Cerberus’ and acknowledge that different individuals may construct different interpret ants from the same sign vehicle, thus effectively creating different signs for the same object.
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TRIGONOMETRIC CONNECTIONS
To illustrate the differences among iconic, indexical, and symbolic signs by using some of Peirce’s examples, in an iconic sign, the sign vehicle and the object share a physical resemblance, e.g., a photograph of a person representing the actual person. Signs are indexical if there is some physical connection between sign vehicle and object, e.g., smoke invoking the interpretation that there is fire, or a sign-post pointing to a road. The nature of symbolic signs is that there is an element of convention in relating a particular sign vehicle to its object (e.g., algebraic symbolism). These distinctions in mathematical signs are complicated by the fact that three different people may categorize the “same" relationship between a sign vehicle and its object in such a way that it is iconic, indexical, or symbolic respectively, according to their interpretations. In practice the distinctions are subtle because they depend on the interpretations of the learner—and therefore, viewed in this way, the distinctions may be useful to a researcher or teacher for the purpose of identifying the subtlety of a learner’s mathematical conceptions if differences in interpretation are taken into account.
As an example, let us examine the quadratic formula in terms of this triad. The roots of the equation ax2+bx+c = 0 are given by the well known formula
_ _ -b ± \lb2 ~4ac
x_ —
Because symbols are used, the interpreted relationship of this inscription with its mathematical object may be characterized as symbolic, involving convention.
However, depending on the way the inscription is interpreted, the sign could also be characterized as iconic or indexical. The formula involves spatial shape. In my original research study of visualization in high school mathematics (Presmeg,
1985), many of the students interviewed reported spontaneously that they remembered this formula by an image of its shape, an iconic property. However, the formula is also commonly interpreted as a pointer (cf. a direction sign on a road): it is a directive to perform the action of substituting values for the variables a, b, and c in order to solve the equation. In this sense the formula is indexical.
Thus whether the sign vehicle of the formula is classified as iconic, indexical, or symbolic depends on the inlerpretant of the sign.
Peirce’s model includes the need for expression or communication: “Expression is a kind of representation or signification. A sign is a third mediating between the mind addressed and the object represented” (Peirce, 1992, p. 281). In an act of communication, then—as in teaching—there are three kinds of interpretant, as follows:
- the “Intemional Interpretant, which is a determination of the mind of the utterer”;
- the “Effectual Interpretant, which is a determination of the mind of the interpreter”: and
- the “Commimicational Interpretant, or say the Comintcrprciani, which is a detenuination of that mind into which the minds of utterer and interpreter have
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to be fused in order that any communication should take place1’ (Peirce, 1998.
p. 478, his emphasis).
It is the latter fused mind that Peirce designated the com mens. The commens proved to be an illuminating lens in examining the history of geometry (Presmeg, 2003), and this third triad of interpretants is a useful one to bear in mind when considering implications of the results of the trigonometric investigation for teaching.
These lenses are augmented by Duval’s (1999) notion of registers, taken in this paper to mean types of sign vehicles. There are four distinct registers in the trigonometry investigated in this research, namely, triangle definitions, sign vehicles of trigonometry in the coordinate plane, the unit circle, and in the graphs of trigonometric functions.
Using the theoretical lens of registers rather than Peirce’s signs, in an enlightening analysis that is relevant to the conversions among registers of the trigonometry students, Hitt (1998) identified five levels of understanding of the concept of function, as follows.
Level 3: Imprecise ideas about the concept (incoherent mixture of different representations of the concept).
Level 2; Identification of different representations of the concept. Identification of systems of representation.
Level 3; Translation with preservation of meaning from one system of representation to another.
Level 4: Coherent articulation between two systems of representation.
Level 5: Coherent articulation of different systems of representation in the solution of a problem.
These levels of understanding cast further light in interpreting students’ conversions amongst the four registers of trigonometry'.
EMPIRICAL SETTING AND DATA COLLECTION
In spring semester of 2006, informed by Sue Brown’s knowledge of the students in her year-long class in a Chicago high school, “Advanced Enriched Algebra and Trigonometry'”, four students were chosen from the 30 in the class to provide a range of cognitive styles and abilities. These students, Laura, Raj, Jim, and Brian (pseudonyms), were interviewed six times each over the period from January' to April, addressing the time in which they were learning trigonometry'- The audio- recorded task-based interviews were of 15-35 minutes’ duration. The tasks were chosen by Sue and the author in collaboration, and included some tasks from Sue’s dissertation study (Brown, 2005) for the purpose of later comparison. An example of an elementary' task from the first interview is presented in figure 1.
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TRIGONOMETRIC CONNECTIONS
The diagram shows a right triangle for which A is an angle. On the grid, draw a triangle with an Z Q for which tan ZQ is twice as large as tan Z.4.
Figure i. A task in the first interview (Brown, 2005).
The discrepancy between the students1 interpretants in response to the iconic sign vehicle in the diagram of figure 1, in Sue’s class and in the investigation with prospective elementary school teachers (in collaboration with Jeff Barrett and Sharon McCrone), is illustrated starkly by the fact that most of the prospective teachers doubled both the opposite side and the adjacent side, in attempting to double the tangent of angle A. Of the four students’ interviewed in Sue’s class, only Raj made this error. He saw his own mistake when questioned, and modified his interpretant by considering the special case in which tan ZA = ~l would need
4
?
to become _ in order to double the tangent. Most of the students in Sue's class 5
provided a correct new sign vehicle spontaneously when they encountered this item m a test. The prospective teachers’ project provides interesting comparisons but will not be addressed in this chapter.
In addition to the interviews, Chicago data sources included notes from seven observations in Sue’s class, documents consisting of students’ work on tests and quizzes, and Sue’s reflections on ways that she attempted to facilitate students’
construction of connections between registers in trigonometry. The methodology of this part of the research could be regarded as a teaching experiment, which includes cycles of joint reflection based on interviews with students, followed by
107
further teaching, in some ways comparable to developmental research (Gravemeijer, 1994). It is beyond the scope of this chapter to describe the leaching experiment in detail. However, some aspects of Sue’s pedagogy are described in drawing implications for teaching in the final section.
Some of the transcribed interview data from the Chicago project and that with prospective teachers—consisting of a total of more than 300 pages of transcriptions from the 30 interviews—will provide a source For the results reported in this chapter. Analysis of the transcripts consisted of coding the data according to themes that appeared to be relevant to aspects that enable or constrain the making of connections amongst registers in trigonometry'. These themes and relevant examples are provided in the next section.
RESULTS AND ANALYSIS
Themes identified in the data analysis were as follows. (Letters in parentheses refer to the students, Laura, Raj, Jim. and Brian, in whose interviews, 1-6, these themes were identified.)
- Need of students for specificity in a sign vehicle (L4, R4).
- Compartmentalization and prototypes (L2&4, R2, J2, B2).
- Generalization and logic (R2, J4&5, B2): a “mathematical cast of mind”
(Krutetskii, I976)(B1,4,5&6).
- The power of falsification (R4&5, J4, B4&6).
- Compression, reification, encapsulation (Bl,2,4,5&6).
- Spontaneous metaphors (R2, J2,3,4,5&6, Bl,3&4) and taught metaphors (L6.
Jl, B1&6).
- Notation and sliding interpretants (L1&5, R5).
- Gestures as sign vehicles (R5, J3).
- Visual sign vehicles (L6, J2A5&6, B4&6).
- Use of a toot as sign (R3).
Examples of the limiting effects of coinparlmentalization (lack of the perception of possible conversions amongst registers) and certain prototypes in trigonometry were illustrated in a PME-30 plenary' presentation (Presmeg, 2006) and will not be repeated in detail here. However, it should be noted that prototypes may affect conversions amongst registers in both negative and positive ways (Doerfler, 1991):
the mnemonic advantages of memorable qualities of a prototype such as a right triangle with “horizontal” and “vertical” legs may be offset by the negative aspect that the prototype may prevent students from recognizing properties when the triangle is in a different orientation—and thus hinder conversions (Presmeg, 1992).
In this chapter l shall illustrate examples of the following categories: Need for specificity, Generalization and logic: a "mathematical cast of mind” (Krutetskii, 1976), The power of falsification, Visual sign vehicles. Notation and sliding
interpretants,
and Use of a tool as a sign.108
TRIGONOMETRIC CONNEC TIONS
Need of students for specificity in a sign vehicle (Laura 4, Raj 4)
Tasks for interview 4 were as follows: “Check true or false. Illustrate with an example on the unit circle, sinusoid graph, or triangle.
(1) cos 0 * cos(-0)?
(2) sin (20) = 2 sin 0?
(3) sin 0 = sin (0+180°)?”
Laura originally said that the first statement is false. Then she drew a diagram of an angle labeled 25° in the first quadrant; “Say it’s 25° for the angle. Well, if it was negative 25 ... it would be 25 off of zero, as well ... so the cosine would be the same." Thus her interpretant for this sign was that the statement is true. She had difficulty connecting this interpretant with the cosine graph, in which she did not give the angle a specific value: “Well, it’s kinda hard to explain.”
Raj also worked with specific sign vehicles, namely, 30" and 330" on the unit circle: “It shares the same side so it has to be true” was his correct interpretant.
There is a sense in which the iconic signs Laura and Raj constructed for specific values of 0 in the unit circle were interpreted generally. However, this generality is more apparent in the interpretants of Jim and Brian, as follows.
Generalization and logic (Jim 4, Brian 4, Laura 4) and falsification (Raj 4)
Jim preferred visual sign vehicles in all cases. However, these iconic signs were much more overtly general in his interpretants than in those of Laura or Raj. For the first task he, too, used 0=30° and hence concluded that-B is the same as 330" in a circle: “The cosine will stay the same, still cover the same distance on the x.” His general interpretant was one of symmelry. “The special case doesn’t matter. And then it’ll be the same thing on the other side of the second quadrant.” He also connected this symmetry with points reflected vertically on the cosine graph, which he drew.
For the second and third tasks in the fourth interview, Raj worked with particular sign vehicles again, which quickly led to the correct interpretant, by the falsification principie. because both of these statements are false. The power of falsification was evident in many of the transcripts: a specific example in this case leads to the general result that the statement is false.
Laura, by contrast, constructed a general sign vehicle for sin (20): “it refers to the period—two cycles in 360"''; and for 2 sin 0, making a strong connection between the radius of a circle, and the amplitude of the corresponding graph. The discrepancy between the two graphs, which she drew separately, resulted in her interpretant that sin 0 does not equal sin (20), and thus the statement is false. For sin 0 = sin (0+180°) she again used the particular value, 0 = 25", and drew a circle diagram to conclude from positive and negative values of y that the statement is false. Although she used falsification, she did not refer to this principle.
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The most general task in all six interviews came in the form of an “extra credit”
item on a test the class had done, which was discussed in interview 4 when time permitted (figure 2). Of the four students interviewed, only Brian addressed this task. His thinking during aspects of trigonometric problem solving in all six interviews could be characterized by what Krutetskii (1976) called a mathematical cast of mind.
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