Mathematics Is the Art of Giving the Same Name to Different Things

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

In 2012, 100 years after Henri Poincare's death, the magazine for the members of the Dutch Purple Mathematical Society published an "interview" with Poincare for which he "wrote" both the questions and the answers (Verhulst, 2012). When responding to a question about 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 postal service, we consider the perspectives of learners of mathematics by looking at how students may see two uses of the word tangent—with circles and in the context of derivative—equally "giving the same proper name to different things," but, every bit a negative, as impeding their understanding. We also consider the artfulness that Poincare points to and ask about artfulness in mathematics teaching; perhaps one aspect of aesthetic teaching involves helping learners capeesh why mathematicians make the choices that they do.

Our efforts accept been in the context of a engineering science that asks students to requite examples of a mathematical object that has certain characteristics or to use examples they create to support or refuse a claim about such objects.ⁱ The teacher tin can then collect those multiple examples and use them to achieve their goals.

Kayla: Algebra two students often get a super minimalized and overbroad definition of an asymptote. Many leave Algebra 2 saying something similar "a horizontal asymptote is a line the graph gets close to but doesn't affect." In calculus, they become a limit definition for asymptotes. As a teacher, I'm prepared for students to enter calculus with the Algebra 2 definition—information technology's acceptable cognition for Algebra 2—but if a student left calculus with the impression that a horizontal asymptote is a line we get close to but don't touch, I would be mortified.

Willy: I think the purpose of learning nigh asymptotes changes too, right? In Algebra 2, students are getting an overview of a lot of functions and their general behavior. At that bespeak, it seems fine to have such a loose definition. Calculus introduces limits to explicate office behavior at diverse parts of the domain. That includes wrestling with infinity.

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

Willy: Correct! The terms shift meaning a bit. When I took calculus and geometry every bit a educatee, I don't recall whatsoever emphasis or discussion of a shift in the definition of tangent. In geometry, the just use of tangent that I remember was with circles: the tangent is perpendicular to the radius. That's not at all how nosotros talk about tangents in calculus.

Dan: And that'due south Poincaré's "giving the aforementioned 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"—because yous can't unfold the complete complexity of a concept all at once. In different contexts, you recall about particular dimensions of concepts. And so it's natural that when we're but talking about circles, tangent is a special case of a broader concept. It's one that you meet first. Lines whose slopes describe the instantaneous rate of change in graphs of functions are mathematically different, but it tin can brand sense to give them that same name in order to capture some mode in which they're the aforementioned. Kayla, it sounds like yous hadn't thought as much near how differently the word tangent was used in calculus and geometry. What in particular, now strikes you as dissimilar?

Kayla: I believe nearly calculus students acquire the new definition—how to derive a tangent, what information technology looks like, what it tells us about a curve—but I worry they may leave calculus still expecting tangent to mean "touching only at ane betoken" as it did in geometry. I besides worry that the geometric idea that the tangent line must lie on simply one side of the circle causes some students to trip up and struggle in calculus when they encounter a tangent line that crosses the graph either at a betoken of inflection, or just at some other point. I also have students who think it is non possible to accept a vertical tangent; they conflate the derivative being undefined with the tangent line not existing.

Willy: I wonder if that could exist a result of trying to brand sense of the idea that there is no linear role of 10 that volition requite a vertical line.

Dan: Kayla, it sounds like you're maxim that, on the i mitt, there are things that are chosen tangents in calculus that wouldn't accept been called tangents in geometry and also the reverse, that there were tangents in geometry that calculus students would not recollect are tangents.

Kayla: Yes.

Dan: That's really helpful, because it identifies a challenge beyond the curricular discontinuity of changing definitions. When definitions change, people might recognize and remember the changes—a changed concept definition—but the things that come up readily to their minds might non modify, what Tall and Vinner (1981) phone call a "concept image." So really, Kayla, what you were maxim is that only some of the things that come to students' minds every bit tangent lines from a geometry perspective remain useful when they're thinking in a calculus sense. A tangent sharing more than ane indicate with a curve is acceptable in calculus but didn't make sense in geometry; a vertical tangent fabricated sense in geometry but worries the calculus student. The tricky affair is that students might notice that while their concept definition has evolved, their concept images might not have.

Kayla: Yeah. A couple years ago, when nosotros had students sketch a graph with a vertical tangent, a lot of what we got was graphs similar ten = abs(y), a xc° clockwise rotation of the absolute value graphs students have seen, which doesn't define a function of ten at all. And, they treated the y-axis as the "tangent." I merely wonder if, to students, the picture just seems actually like to a circle despite its shape.
Dan: Right. One betoken of contact with the vertex of the "five" curve, the curve all on one side of the "tangent," just like the tangent to a circle. From a geometry perspective, a student could think, well, that's a reasonable example of a tangent. Simply, from a calculus perspective, it's not. In calculus, nosotros want the derivative to exist well-defined, determining ane specific slope for the tangent at a point.

Willy: If at that place is an art to the manner mathematics names different things with the same name, then students should be able to empathise why mathematicians over time decided to use the same name. It seems like the teacher has to help students capeesh the do good of having the derivative as a well-defined role, 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 educatee who asks why it is important that there not exist multiple tangents to a point on the graph of a function. I would probably say something like: "At the vertex of the graph of abs(x), the slope to the left of the vertex and the slope to the right of the vertex are really dissimilar (one positive and i negative) creating a drastic change in slope where the 2 lines see. And unlike a parabola where the slopes modify from positive to negative across, those slopes are both approaching null—merely ane from the negative direction and one from the positive direction. And so, when looking at the vertex of the graph of abs(10), when you go to draw the tangent line what slope would y'all choose? The two drastically different slopes is why the derivative does not exist at that point—the slope from the right and left are different and the derivative function cannot have on two values for one x.

Willy: This is i of the reasons that asking students to produce examples of concepts has been actually thought provoking when I call up about pedagogy. Asking students to sketch a office that has a vertical tangent has the possibility of having students stumble upon things that might challenge their conceptions of how mathematics operates across contexts.

Dan: Those sorts of tasks tin also give teachers information about what definitions their students are using, and what kind of concept images they have. But and then, Kayla, it seems you've too been saying that such tasks give you a way to influence students' concept definitions and concept images. Is that truthful?

Kayla: Yeah, tasks like these help surface students' concept paradigm for me to work on with them. With some tasks, students all basically submit the same affair, showing how limited their image is. And, this applies not just to tangents. I especially like request students to submit multiple examples. When we were doing rational role tasks, we asked them to submit multiple functions that would take a seemingly identical graph to a linear role and students could non think of multiple ways to practice so. And from these sorts of tasks, I can also learn about how students think most related concepts: Practice students think that points of tangency are different from points of intersection or simply special ones? Or, do students think that a horizontal asymptote is a tangent?

Dan: So, your comments are almost non just the match between the concept image and the concept definition, but also the richness and multifariousness of the concept epitome infinite and connections to nearby concepts. Having surfaced all of those examples from students, in what fashion do you feel that those are a resource for your teaching separate from their part in assessing students?

Kayla: For the by couple years, students' submissions accept ended upward beingness used in future discussions. When you lot have this depository financial institution of submissions that students really submitted, you lot can develop a whole lesson based on what a couple students have submitted. I recall the ability to run across all those submissions hands, pick ones that are interesting, and use those, is nifty. Sometimes just seeing someone else's submission tin shift your concept image or support the new definition you lot are learning in a manner that y'all weren't able to without that extra nudge. I think that part is key. It tin be super powerful just for students to see each other's work.

Willy: I agree! And in the context of instructor preparation I too think about how difficult and time consuming it is for teachers to make up a variety of examples. And so using student generated work helps! The work is already washed for yous, and then you lot tin select the well-nigh appropriate examples for your purpose and have more time for other things.

Kayla: And I remember ofttimes we make fake student work to utilise as teachers, nosotros are saying these are the common submissions we know to expect. But now that we're presenting this task to students, information technology has been interesting to see examples year after year that I hadn't expected the commencement time around.

Dan: What's an example of that?

Kayla: Year later yr, students seem to think that there is a horizontal tangent on an exponential function where the horizontal asymptote is; they retrieve the same line is both an asymptote and a tangent.

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

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

Willy: To aid us learn how students think nearly a concept, we tin pattern assessment tasks that reveal students' concept images or the definitions they're operating from. Students can produce examples that do not fulfill all or any of the requirements of the chore only nevertheless reveal possible gaps in understanding or overly broad or narrow concept images. For case, the "sideways accented value" graph is not a role and does non accept a tangent at the vertex. We can also design tasks that push students in a detail direction to further their learning—to encounter a concept in a certain way so that there is no prescribed solution or method and responses will vary. Such tasks could exist 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, one of the teachers thought information technology was weird that we would care whether a tangent line intersected the graph somewhere else because the curriculum focuses on tangents locally, not more than 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, but I have an ascertainment to offering. On the interval between a bespeak of tangency and a signal of intersection farther down the line, even if that point of intersection is not another betoken of tangency, I think the average value of the derivative function is equal to the derivative at the indicate of tangency or the slope of the tangent line. For instance, consider Red(ten) = (x-1)(x-2)(x-3), and Green(10) = 2(x-1). The point of tangency is (1,0) and Crimson'(1) = ii. The signal of intersection is (4,6).

Think about the interval [one, 4]. This interval reminds me of Algebra One where we oft work with average rates of alter and linear functions, rather than more than complex curves. As long as nosotros know the values at two points, in society to interpolate or extrapolate, nosotros imagine a hypothetical state of affairs where the modify is distributed evenly, rather than the messy reality of change that is not evenly distributed. This observation near the interval between the point of tangency and intersection seems similar information technology might advise a mathematical value for because when the continuation of a tangent line intersects with a function.

Kayla: I see the mathematical promise in that direction but wonder how many teachers would see that as standard calculus material. I wonder what information technology might take to have my colleagues consider using these tasks. I know I am a flake of an outlier. At the beginning of the year, I mostly motion 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 (bold students understand the content from precalculus) or just do a quick review (a week or so of class 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 and so a quick transition into derivative rules, the human relationship betwixt functions and their derivative graphs, and applications of derivatives (related rates, optimization, linearization, etc.). Our district's curriculum materials oftentimes ask questions about computing derivatives and writing the equation of tangent lines at specific indicate, but in that location's little digging into what the definition of a tangent line is and how it might have inverse from geometry. Personally, I call back it'due south important to spend fourth dimension on the bug about tangents that we've been discussing, just 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 as more important/relevant to the AP exam.

Willy: Does that influence what you are going to do next year?

Kayla: No, not really. Using these tasks over the last few years has surfaced of import areas of student defoliation, even beyond the ones we've talked about here. I desire students to think difficult near definition and how definitions alter. These "requite-an-example" tasks help. They engage students with something interesting and challenging, and assistance them to pay careful attention to mathematical definitions and to be precise in using them.

Endnote

i. For the terminal two years, we have been using the STEP platform developed by Shai Olsher and Michal Yerushalmy at the MERI Heart at the Academy of Haifa (Olsher, Yerushalmy, & Chazan, 2016). The ideas represented in this conversation were spurred by utilise of this plan with activities developed in Israel (Yerushalmy, Nagari-Haddif, & Olsher, 2017; Nagari-Haddif, Yerushalmy, 2018) and adapted for use in the United states of america.

References

Verhulst, F. (2012). Mathematics is the fine art of giving the aforementioned name to different things: An interview with Henri Poincaré. Nieuw Archief Voor Wiskunde. Serie 5, 13(3), 154–158.

Olsher, Due south., Yerushalmy, Yard., & Chazan, D. (2016). How might the utilize of engineering in formative assessment support changes in mathematics teaching? For the Learning of Mathematics, 36(3), 11–eighteen. https://www.jstor.org/stable/44382716

Yerushalmy, M., Nagari-Haddif, G., & Olsher, S. (2017). Pattern of tasks for online cess that supports understanding of students' conceptions. ZDM, 49(5), 701–716. https://doi.org/ten.1007/s11858-017-0871-7

Nagari-Haddif, M., & 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/ten.1007/978-3-319-73748-5_7

Tall, D. (2002). Continuities and discontinuities in long-term learning schemas. In David Tall & Yard. Thomas (Eds.), Intelligence, learning and understanding—A tribute to Richard Skemp (pp. 151–177). PostPressed. http://homepages.warwick.ac.uk/staff/David.Tall/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(two), 151–169. https://doi.org/x.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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