1. Semiotics as a Theoretical Framework for Research in Mathematics Education Norma Presmeg Illinois State University July, 2009

2. Semiotics, language, and mathematics Both in the use of language, and in the doing of mathematics, and its teaching and learning, signs are taken to stand for mental concepts. Semiotic theories, including the dyadic models of Saussure and Lacan, and the triadic model of Charles Sanders Peirce, provide useful lenses for examining issues pertaining to mathematics and language.

3. Ferdinand de Saussure Saussures ideas were worked out in the context of language. His dyadic model consisted of concept and acoustic image (roughly called signified and signifier).

4. Charles Sanders Peirce Peirces all-embracing triadic model takes the act of interpretation into account. A Peircean nested model of signs, each involving object, representamen, and interpretant, casts light on issues of representation – including metaphors and metonymies – in mathematics education.

5. Terminology Peirce: a sign entails an object, a representamen, and an interpretant. In this presentation I shall take a sign to be the interpreted relationship between some representamen (signifier), called the sign vehicle, and an object (signified) that it represents or stands for in some way.

6. A Peircean example of the three components of a sign Object (signified): Its going to rain! Sign vehicle (representamen or signifier): The barometric pressure is falling. Interpretant: Take an umbrella!

7. Peirces trichotomies Trichotomic is the art of making three-fold divisions. Peirce showed a proclivity for the number three in his philosophical thinking. But it will be asked, why stop at three? [W]hile 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 (Peirce, 1992, p. 251).

8. Types of signs: icon, index, and symbol Just one of Peirces ten trichotomies. Not inherent in the signs themselves, but depend on the interpretation of the relation between sign vehicle and object. Icon: physical resemblance, e.g., photo. Index: physical connection, e.g., smoke and fire. Symbol: conventional, e.g., algebraic symbolism.

9. The sign vehicle and its object partake of different levels of generality. Seeing an A as a B (Otte, 2006). More than one sign vehicle may refer to a mathematical object. Conversions amongst registers (Duval, 1999). Visual imagery and inscriptions refer to internal and external representations respectively.

10. Abstract notion: this particular function - object Algebraic notation: y=x 2 Sign 1: interpreted sign vehicle 1 (parabola) and object Sign 2: interpreted sign vehicle 2 (equation) and object Two registers illustrated by two signs with the same object

11. A significant issue is the investigation of ways that teaching may facilitate learners building of connections amongst mathematical signs. By highlighting structures and patterns across domains, such connections may foster generalizations and help to combat the phenomenon of compartmentalization.

12. Some Early Research Results Mr Blue obtained a score of 3 out of a possible 36 on my test for preference for visuality (MV) in mathematical problem solving. Yet his teaching visuality (TV) score was 7 of a possible 12 classroom aspects. Mathematical visuality (MV) and teaching visuality (TV) were only weakly correlated in this sample of 13 teachers. (Spearmans rho = 0.404 n.s.)

13. Mr. Blue [with excitement]: Youve got to be careful sometimes I think … bringing things from the abstract, to too concrete. Then its that way forever, then everything is like that. Youve got to be careful with that because sometimes you must remember that our abstractness carries us to flights of imagination of where we can go with it. And thats what I would like them as often to see here, when we do something, this is another possible way of doing this problem; more algebraically, what you can do with it. For Mr. Blue, algebra was often the vehicle of abstraction.

14. Beauty is truth, truth beauty: That is all ye know on earth - And all ye need to know (John Keats)

15. In his teaching, Mr. Blue frequently expressed his own pleasure in the beauty of mathematics. In a trigonometry lesson he spoke with his students about errors that some of them were making. Mr. Blue: Dont just square things, and suddenly they disappear into space. … And then of course I was really saddened by this: now let me say this to you. Dont do this any more. Now you know better than that in this room. You cannot take the square root of individual what? Boys:Terms.

16. Mr. Blue (continued): Terms. … Dont force it! Maths just wont be forced. Thats the beauty of it, thats its beauty: where it stands strong against this forcing things into it that dont have any place for it at all. It must go on the way it always has gone on. The way that Mr. Blue encouraged metacognition was also apparent in an algebra lesson on change of base of logarithms. In a test, the boys had solved a quadratic equation in logarithms: (log 3 x) 2 – 10log 3 x + 9 = 0

17. Mr. Blue: So this would be the fastest way: factorise. You can do the change of base with tens, you can get it, it will be right, when youve finished [but it would be slow]. … We could put a y in for log to the base 3 of x, couldnt we? Then factorise. … The whole thing in higher grade is to think in patterns, and relate to patterns of the former work received. And you get bigger and bigger problems.

18. Mr. Blue (continued): If you look at this one now, how many ideas were in this problem? This idea was a log idea, this turns into a quadratic idea, this turns into factorisation, this turns into exponentials to get the answer. All in one problem. Thats what you must start getting used to. The connections between domains that Mr. Blue was helping his boys to identify are a central topic that I wish to highlight. The theoretical lens that I shall use is that of Peircean semiotics.

19. Thirteen significant questions 1.What aspects of pedagogy are significant in promoting the strengths and obviating the difficulties of use of visualization? 2.What aspects of classroom cultures promote the active use of effective visual thinking in mathematics? 3.What aspects of the use of different types of imagery and inscriptions are effective in mathematical problem solving? 4.What are the roles of gestures in mathematical visualization?

20. 5.What conversion processes are involved in moving flexibly amongst various mathematical registers, thus combating compartmentalization? 6.What is the role of metaphors in connecting different registers? 7.How can teachers help learners to make connections between visual and symbolic inscriptions? 8.How can teachers facilitate connections between idiosyncratic and conventional inscriptions (and/or imagery)?

21. 12.How do visual aspects of computer technology, including dynamic geometry software, change the processes of learning mathematics? 13.What is the structure, and what are the components, of an overarching theory of visualization for mathematics education?

22. Recent research: Conversions amongst trigonometric signs Compartmentalization in interviews with preservice elementary teachers (with Jeff Barrett & Sharon McCrone). Teaching that encourages conversions amongst signs in high school trigonometry (with Susan Brown).

23. Research question in both studies How may teaching facilitate students construction of connections amongst registers in learning the basic concepts of trigonometry?

24. Sams case: Compartmentalization in trigonometry Sam was chosen as one of three students (from 27 in Jeffs class) to be interviewed, because of his strong abductive thinking in class discourse. Sam was trying to recall trig. principles that he learned several years earlier. The facilitative teaching of my colleague Jeff Barrett had not yet taken place.

25. Third question in the preliminary interview This graph shows an angle. Give the approximate value of the sine of the angle. (Brown, 2005)

26. Right triangle, unit circle, and sinusoid curve Sam knew the right triangle definitions of the trig. ratios (using SOH CAH TOA). He called the rotation angle theta, and marked its supplement, the reference angle in the second quadrant. He dropped perpendiculars to the x and y axes, and joined P to the circles intersection with the y axis. He identified the sine of the reference angle as having a value of 0.8.

27. Sams inscriptions on the unit circle The value of the sine of the reference angle is point eight.

28. Sam: Thats point 8, yeah. But over here, lets see, […] the sine of the whole angle is one. Interviewer: The sine now of the obtuse angle? Sam: So this would be point 2 [pointing to the arc between P and the y axis]. Sam appears to have an image something like this: 0.8 0.2 1.0

29. Sam: So Id say negative one point two … I dunno. Interviewer: How did you get one point two? Sam: So, the sine of this angle is one. Interviewer: The ninety degree angle? Sam: Yes, so this is one. And then this is … lets see, this point is … negative point six, point eight. Interviewer: Oh, I see. Youre figuring out the coordinates? Sam: I was just thinking of a unit circle. And with coordinates …cause now like, the sine of this angle here [indicating point of intersection of circle and y axis] is, the cosine zero, the sine one. […] And then it goes, thats 90, which, it still stays positive though, so … one point two, because this is point two.

30. Sam then explained negative and positive values in the quadrants using coordinates of points. After the interviewer told him that the correct answer for the value of the sine of theta is point eight, he persisted: Sam: But if this … if the sine of this angle here is one, how can a bigger angle be less? Interviewer (I): Ah, thats a good question. Do you know what a sine graph looks like? Sam: Yeah. I: Can you draw me one? Can you put values in there? Sam drew the sine graph for one revolution and inserted appropriate radian measures on the θ axis, and one and negative one on the y axis.

31. I: There you go! Now you just said, how can it be less, if its [the angle is] bigger that 90? Sam: Yeah, it, its not … [following the curve with his finger]. I: So it goes down again. Sam: So that spot is the same here. Yeah! [He marks symmetrical points on the sine curve on either side of π/2.] Sam seemed elated to make this connection between sine in the unit circle and the sinusoid.

32. Teaching that encourages conversions amongst signs Laura and Jim were two of the four students (from 30 in Sues class) who were interviewed six times each. In the second interview of the series, all four students completed without difficulty the task given to Sam. Lauras solution is typical (although two of the four students felt no need to invoke the Pythagorean triples, as Laura did).

33. I: So first of all, wheres the angle that youre looking at? Laura: Its the … It goes through the first into the second quadrant. […] The y value over the radius. And Id say the y value is approximately point eight. I: Point eight. Where are you looking? On the y axis? Laura: Yes. Hm … Im not quite sure but I assume it [the radius] would be … about one. It has to be greater than … [5 seconds] I: Is there a way that you can see what the radius is? … Have a look at other points on the circle. Laura: Oh! Yeah … So that would be point six, minus point six. … Draw a triangle.

34. [Laura draws on Her paper.] I: So youre thinking of a right triangle? Is that … how did you know that? The 6, 8, 10? Laura: Its just a, one of the triples that we learned. I: The Pythagorean triples? Okay. So if that was 6, 8, that would be 10, and now youve got … Laura: Point 6, point 8, and one. So the, it was one. Point 8 over one … point 8. 8 6 10

35. Both Laura and Jim were also asked to describe how they would work with angles in the third quadrant. They both drew right triangles by dropping a perpendicular to the y axis rather than the x axis. Jim in the discussion that ensued expressed resistance to working from the x axis.

36. Jim:Well Id subtract the rotation angle from 270, to get the angle, and then Id use, um, this up here. […] And you can do the same thing: multiply it by the radius. [In working with the second quadrant he had written sinθ.r =opposite, and cosθ.r =adjacent indicating multiplication by the length of the radius.] I: Just be careful. Because if you now say the sine of that angle … which one is it going to give you? […] Your cosine gave you the x here [in the second quadrant], and the opposite gave you the y. Now is it going to be that the cosine gives you the x again? Jim:No. It will still be the opposite and the adjacent legs, but it will switch from x to y. Jim explained later that he did not like drawing the triangle backwards on itself, because then it would block the rotation angle.

37. Some of Sues facilitative principles for converting among registers Connect old knowledge with new, starting with the big ideas of trigonometry. Connect visual and nonvisual registers. Supplementing problems with templates for students to draw and use sketches. Providing memorable summaries in diagram form. Providing contextual or real world metaphors, e.g., boom crane; bow tie.

38. Lauras computer inscription of the bow tie

39. Analysis of the episodes: The power of semiotics * All the mathematical thinking portrayed in these episodes involves activity with signs, i.e., semiosis. * There is an internal logic in the students interpretations (correct or not). * This logic is revealed in the imagery imputed to the students as they interpret the relationships.

40. Icon, index, or symbol? Sams image that results in his claim that the sine of the rotation angle is 1.2 is a sign vehicle that is connected iconically with the way he is seeing the relationships in the mathematical object (the sine ratio defined in the unit circle). When he constructs a different sign, based on his inscription of the sinusoid graph, then his previous icon is no longer viable: the value of the sine of an angle cannot exceed 1.

41. There is also a sense in which the signs are indexical, because they point to what Sam sees as the structure of the relationships involved (as smoke points to fire). In standard mathematics there is an element of convention associated with the principles governing the trigonometric ratios defined in the coordinate plane (e.g., that the radius vector rotates counter clockwise from the positive x-axis), and with the way the sinusoid is organized. Thus the correct interpretation of the relations between sign vehicles and mathematical objects is also symbolic.

42. When Laura and Jim want to subtract the rotation angle from 270 o to find the reference angle in the third quadrant, the iconic sign they have constructed gives way to a conventional symbolic sign under the influence of the bow tie metaphor. Metaphors are particularly memorable if they are iconic, even if the result is a symbolic sign. The change in the form of signs is not arbitrary; it partakes of necessity according to the consistency of mathematical principles. As Mr. Blue claimed in the opening vignette, that is the beauty of mathematicsthat it just wont be forced.

43. Objectification, compression, and prototypes When Laura constructed a right triangle in the second quadrant, with lengths of legs that reminded her of values she had encountered previously (6, 8, and hence 10), a Pythagorean prototype appeared to be invoked. This prototype prevented her from seeing that she could have read off the value of the radius as one directly from the points of intersection of the circle with the x axis. Objectification as a semiosic process is powerful in the learning and doing of mathematics. However, the flexibility of having the ability to convert freely back and forth amongst different signs for the same mathematical objects is paramount.

44. Coming home to the beauty of mathematics as it connects. Only by the form, the pattern, can words or music reach the stillness, as a Chinese jar still moves perpetually in its stillness. T.S. Eliot: Burnt Norton T.S. Eliot: Burnt Norton

45. Thirty years of PME research on visualization Affordances and constraints of using imagery and inscriptions as sign vehicles. Affordances and constraints of using imagery and inscriptions as sign vehicles. Students seeming reluctance to visualize. Students seeming reluctance to visualize. Objectification, compression, encapsulation, and reification. Objectification, compression, encapsulation, and reification. Gesture and embodiment. Gesture and embodiment.

46. Kosslyn and Bruner: the representational development hypothesis [I]ndividual differences in types of imagery, quality and quantity, preference for and skill in using, persist through the school years and possibly through lifetimes, without evidence of general developmental trends in forms of imagery or in their personal use. …

47. The representational development hypothesis... Bruners (1964) well known enactive, iconic, and symbolic modes of cognition should therefore be taken as metaphors for types of thinking rather than as a developmental hierarchy. (Presmeg, 2006, p. 223)