Mathematics Is the Art of Giving Same Name to Different Things

by Daniel Chazan, Academy of Maryland; William Viviani, University of Maryland; Kayla White, Paint Branch High School and University of Maryland

In 2012, 100 years later on Henri Poincare's death, the magazine for the members of the Dutch Royal Mathematical Society published an "interview" with Poincare for which he "wrote" both the questions and the answers (Verhulst, 2012). When responding to a question nearly elegance in mathematics, Poincare makes the famous enigmatic remark attributed to him: "Mathematics is the art of giving the same names to different things" (p. 157).

In this blog mail service, we consider the perspectives of learners of mathematics past looking at how students may see two uses of the word tangent—with circles and in the context of derivative—as "giving the same name to different things," but, equally a negative, as impeding their understanding. We also consider the artfulness that Poincare points to and ask nigh artfulness in mathematics teaching; perhaps one aspect of artful educational activity involves helping learners appreciate why mathematicians make the choices that they do.

Our efforts accept been in the context of a technology that asks students to give examples of a mathematical object that has sure characteristics or to utilize examples they create to support or refuse a merits nigh such objects.^one The teacher tin then collect those multiple examples and use them to achieve their goals.

Kayla: Algebra 2 students often get a super minimalized and overbroad definition of an asymptote. Many go out Algebra 2 saying something like "a horizontal asymptote is a line the graph gets close to but doesn't bear upon." In calculus, they go a limit definition for asymptotes. As a instructor, I'yard prepared for students to enter calculus with the Algebra ii definition—it's acceptable knowledge for Algebra 2—but if a student left calculus with the impression that a horizontal asymptote is a line we get close to just don't touch, I would be mortified.

Willy: I call back the purpose of learning almost asymptotes changes too, right? In Algebra 2, students are getting an overview of a lot of functions and their general behavior. At that point, it seems fine to accept such a loose definition. Calculus introduces limits to explain role behavior at various parts of the domain. That includes wrestling with infinity.

Kayla: Yes, yes, only what I hadn't noticed until recently was that students' understanding even of tangent in calculus might exist influenced by what they retained from geometry.

Willy: Right! The terms shift meaning a bit. When I took calculus and geometry as a pupil, I don't call back any emphasis or give-and-take of a shift in the definition of tangent. In geometry, the but use of tangent that I retrieve was with circles: the tangent is perpendicular to the radius. That'southward non at all how we talk nearly tangents in calculus.

Dan: And that'due south Poincaré'due south "giving the same proper noun to different things." David Tall (2002) argues that evolutions in definitions of mathematical concepts are natural in a curriculum—he calls the phenomenon "curricular discontinuities"—considering you can't unfold the complete complication of a concept all at once. In different contexts, you think almost item dimensions of concepts. So it'southward natural that when we're only talking about circles, tangent is a special case of a broader concept. Information technology'south one that you lot meet outset. Lines whose slopes draw the instantaneous charge per unit of alter in graphs of functions are mathematically different, but it tin can make sense to give them that same name in guild to capture some way in which they're the same. Kayla, it sounds similar you hadn't thought as much about how differently the give-and-take tangent was used in calculus and geometry. What in item, at present strikes you every bit different?

Kayla: I believe most calculus students learn the new definition—how to derive a tangent, what information technology looks like, what it tells us about a curve—merely I worry they may leave calculus still expecting tangent to hateful "touching only at ane point" as it did in geometry. I likewise worry that the geometric idea that the tangent line must prevarication on just ane side of the circumvolve causes some students to trip up and struggle in calculus when they run across a tangent line that crosses the graph either at a point of inflection, or simply at some other point. I also have students who call back it is not possible to have a vertical tangent; they conflate the derivative being undefined with the tangent line not existing.

Willy: I wonder if that could exist a upshot of trying to make sense of the idea that there is no linear office of ten that volition give a vertical line.

Dan: Kayla, information technology sounds like you're proverb that, on the i hand, in that location are things that are chosen tangents in calculus that wouldn't have been chosen tangents in geometry and also the reverse, that at that place were tangents in geometry that calculus students would not recall are tangents.

Kayla: Yes.

Dan: That's really helpful, because information technology identifies a challenge across the curricular aperture of changing definitions. When definitions alter, people might recognize and recall the changes—a changed concept definition—but the things that come readily to their minds might non change, what Tall and Vinner (1981) call a "concept image." So really, Kayla, what you were maxim is that merely some of the things that come to students' minds as tangent lines from a geometry perspective remain useful when they're thinking in a calculus sense. A tangent sharing more than one point with a curve is adequate in calculus just didn't make sense in geometry; a vertical tangent made sense in geometry just worries the calculus educatee. The tricky thing is that students might notice that while their concept definition has evolved, their concept images might not take.

Kayla: Yes. A couple years agone, when we had students sketch a graph with a vertical tangent, a lot of what nosotros got was graphs like x = abs(y), a 90° clockwise rotation of the accented value graphs students take seen, which doesn't define a role of ten at all. And, they treated the y-axis as the "tangent." I only wonder if, to students, the picture just seems really similar to a circle despite its shape.
Dan: Correct. 1 point of contact with the vertex of the "5" bend, the curve all on i side of the "tangent," just like the tangent to a circumvolve. From a geometry perspective, a pupil could retrieve, well, that's a reasonable example of a tangent. But, from a calculus perspective, information technology's not. In calculus, we want the derivative to be well-divers, determining one specific gradient for the tangent at a point.

Willy: If in that location is an fine art to the fashion mathematics names dissimilar things with the same name, then students should be able to understand why mathematicians over time decided to use the same proper name. It seems like the teacher has to help students appreciate the benefit of having the derivative as a well-defined function, with either one unique tangent line or none at all.

Kayla: I agree, but I don't feel like I accept a great answer to a student who asks why it is of import that at that place not be multiple tangents to a point on the graph of a part. I would probably say something like: "At the vertex of the graph of abs(10), the slope to the left of the vertex and the slope to the right of the vertex are really different (one positive and one negative) creating a drastic change in slope where the two lines meet. And unlike a parabola where the slopes modify from positive to negative beyond, those slopes are both approaching zip—simply one from the negative direction and one from the positive direction. So, when looking at the vertex of the graph of abs(x), when you lot go to draw the tangent line what slope would you choose? The two drastically different slopes is why the derivative does not be at that signal—the slope from the right and left are unlike and the derivative function cannot have on 2 values for ane x.

Willy: This is one of the reasons that asking students to produce examples of concepts has been really idea provoking when I retrieve virtually teaching. Asking students to sketch a function that has a vertical tangent has the possibility of having students stumble upon things that might claiming their conceptions of how mathematics operates across contexts.

Dan: Those sorts of tasks can besides give teachers information about what definitions their students are using, and what kind of concept images they take. But then, Kayla, information technology seems y'all've also been saying that such tasks give you lot a way to influence students' concept definitions and concept images. Is that truthful?

Kayla: Yep, tasks like these help surface students' concept image for me to work on with them. With some tasks, students all basically submit the aforementioned thing, showing how limited their image is. And, this applies not simply to tangents. I especially like asking students to submit multiple examples. When we were doing rational part tasks, we asked them to submit multiple functions that would have a seemingly identical graph to a linear function and students could not think of multiple ways to do so. And from these sorts of tasks, I can also learn about how students call back about related concepts: Practise students think that points of tangency are different from points of intersection or simply special ones? Or, do students retrieve that a horizontal asymptote is a tangent?

Dan: And so, your comments are about non but the lucifer betwixt the concept image and the concept definition, but also the richness and multifariousness of the concept image infinite and connections to nearby concepts. Having surfaced all of those examples from students, in what manner do you feel that those are a resource for your teaching separate from their office in assessing students?

Kayla: For the past couple years, students' submissions have concluded upward being used in hereafter discussions. When you take this bank of submissions that students actually submitted, yous can develop a whole lesson based on what a couple students have submitted. I think the ability to run across all those submissions easily, option ones that are interesting, and utilize those, is great. Sometimes just seeing someone else'due south submission can shift your concept image or support the new definition yous are learning in a way that you lot weren't able to without that extra nudge. I think that part is key. It can be super powerful simply for students to see each other's piece of work.

Willy: I agree! And in the context of teacher preparation I also retrieve about how hard and time consuming it is for teachers to brand up a diverseness of examples. And so using student generated piece of work helps! The work is already washed for you lot, and and so you lot can select the almost appropriate examples for your purpose and accept more time for other things.

Kayla: And I think often we make fake student work to use as teachers, nosotros are proverb these are the mutual submissions nosotros know to expect. But now that nosotros're presenting this job to students, it has been interesting to see examples year after year that I hadn't expected the first time around.

Dan: What'southward an instance of that?

Kayla: Twelvemonth after year, students seem to recall that there is a horizontal tangent on an exponential function where the horizontal asymptote is; they call up the same line is both an asymptote and a tangent.

Dan: And, they aren't thinking about a point at infinity!

Kayla: This comes usually in response to a prompt like "Enter a symbolic expression for a function whose graph is a line parallel to the x-centrality. Then write a function, or sketch its graph, such that the line is tangent to the graph of the function at 2 or more than points."

Willy: To assistance usa acquire how students call up virtually a concept, we tin can pattern assessment tasks that reveal students' concept images or the definitions they're operating from. Students can produce examples that practice not fulfill all or any of the requirements of the task only still reveal possible gaps in agreement or overly broad or narrow concept images. For example, the "sideways absolute value" graph is not a role and does not have a tangent at the vertex. We can also blueprint tasks that push button students in a particular direction to further their learning—to run into a concept in a certain style so that there is no prescribed solution or method and responses will vary. Such tasks could be used to shift student thinking for the purpose of, say, evolving their definition of tangent lines from a geometry sort of definition to one more advisable for calculus. Interestingly, when I spoke with calculus teachers from my old school, i of the teachers idea it was weird that we would care whether a tangent line intersected the graph somewhere else because the curriculum focuses on tangents locally, not more globally. I wonder how extending the tangent line in calculus is helpful.

Dan: I was asking myself that question with a focus on the mathematics. I don't have anything conclusive, just I accept an observation to offer. On the interval between a signal of tangency and a point of intersection farther down the line, even if that signal of intersection is non another point of tangency, I call back the boilerplate value of the derivative function is equal to the derivative at the point of tangency or the slope of the tangent line. For example, consider Red(x) = (x-i)(10-ii)(x-iii), and Green(x) = ii(10-one). The point of tangency is (one,0) and Cherry'(i) = 2. The point of intersection is (4,6).

Think about the interval [1, 4]. This interval reminds me of Algebra One where nosotros oft work with average rates of change and linear functions, rather than more complex curves. As long as we know the values at two points, in social club to interpolate or extrapolate, we imagine a hypothetical situation where the change is distributed evenly, rather than the messy reality of change that is non evenly distributed. This observation about the interval between the point of tangency and intersection seems like it might suggest a mathematical value for because when the continuation of a tangent line intersects with a function.

Kayla: I see the mathematical hope in that direction only wonder how many teachers would see that equally standard calculus fabric. I wonder what information technology might take to have my colleagues consider using these tasks. I know I am a bit of an outlier. At the get-go of the year, I generally move through content with my BC Calculus class at a slower footstep than other teachers in my district. From what I've heard from other teachers, many either skip the limits unit (assuming students understand the content from precalculus) or simply do a quick review (a week or and so of course time). Similarly, with tangent lines, the concept of tangent line is pretty much skimmed over (pun intended!). The introduction to derivatives usually begins with defining derivative and then a quick transition into derivative rules, the relationship betwixt functions and their derivative graphs, and applications of derivatives (related rates, optimization, linearization, etc.). Our district'due south curriculum materials frequently enquire questions about calculating derivatives and writing the equation of tangent lines at specific betoken, but there's little digging into what the definition of a tangent line is and how information technology might have inverse from geometry. Personally, I recall it's important to spend fourth dimension on the issues almost tangents that we've been discussing, but I worry many teachers may find these tasks a distraction that would take time away from other topics and skills in the curriculum that they see every bit more of import/relevant to the AP exam.

Willy: Does that influence what you lot are going to exercise side by side yr?

Kayla: No, not actually. Using these tasks over the last few years has surfaced important areas of student confusion, even across the ones we've talked nearly here. I want students to think hard virtually definition and how definitions change. These "give-an-instance" tasks help. They engage students with something interesting and challenging, and help them to pay conscientious attention to mathematical definitions and to be precise in using them.

Endnote

1. For the last two years, nosotros have been using the STEP platform developed by Shai Olsher and Michal Yerushalmy at the MERI Middle at the University of Haifa (Olsher, Yerushalmy, & Chazan, 2016). The ideas represented in this chat were spurred by utilise of this program with activities developed in Israel (Yerushalmy, Nagari-Haddif, & Olsher, 2017; Nagari-Haddif, Yerushalmy, 2018) and adapted for apply in the United states of america.

References

Verhulst, F. (2012). Mathematics is the art of giving the same proper noun to different things: An interview with Henri Poincaré. Nieuw Archief Voor Wiskunde. Serie five, thirteen(3), 154–158.

Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of engineering in formative cess support changes in mathematics didactics? For the Learning of Mathematics, 36(3), eleven–eighteen. https://www.jstor.org/stable/44382716

Yerushalmy, Thousand., Nagari-Haddif, G., & Olsher, S. (2017). Design of tasks for online assessment that supports agreement of students' conceptions. ZDM, 49(5), 701–716. https://doi.org/10.1007/s11858-017-0871-vii

Nagari-Haddif, G., & Yerushalmy, K. (2018). Supporting Online E-Assessment of Problem Solving: Resource and Constraints. In D. R. Thompson, M. Burton, A. Cusi, & D. Wright (Eds.), Classroom Assessment in Mathematics: Perspectives from Around the Globe (pp. 93–105). Springer International Publishing. https://doi.org/10.1007/978-3-319-73748-5_7

Alpine, D. (2002). Continuities and discontinuities in long-term learning schemas. In David Tall & Chiliad. Thomas (Eds.), Intelligence, learning and agreement—A tribute to Richard Skemp (pp. 151–177). PostPressed. http://homepages.warwick.air-conditioning.uk/staff/David.Alpine/pdfs/dot2002c-long-term-learning.pdf

Tall, D., & Vinner, S. (1981). Concept paradigm and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

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Source: https://blogs.ams.org/matheducation/2020/07/15/pedagogical-implications-of-mathematics-as-the-art-of-giving-the-same-name-to-different-things/

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