by Daniel Chazan, University of Maryland; William Viviani, University of Maryland; Kayla White, Paint Branch High School and University of Maryland
In 2012, 100 years after 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 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 post, 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—as “giving the same name to different things,” but, as 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 artful teaching involves helping learners appreciate why mathematicians make the choices that they do.
Our efforts have been in the context of a technology that asks students to give examples of a mathematical object that has certain characteristics or to use examples they create to support or reject a claim about such objects.1 The teacher can 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 leave Algebra 2 saying something like “a horizontal asymptote is a line the graph gets close to but doesn’t touch.” In calculus, they get a limit definition for asymptotes. As a teacher, I’m prepared for students to enter calculus with the Algebra 2 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 but don’t touch, I would be mortified.
Willy: I think the purpose of learning about 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 have such a loose definition. Calculus introduces limits to explain function behavior at various parts of the domain. That includes wrestling with infinity.
Kayla: Yes, yes, but what I hadn’t noticed until recently was that students’ understanding even of tangent in calculus might be influenced by what they retained from geometry.
Willy: Right! The terms shift meaning a bit. When I took calculus and geometry as a student, I don’t recall any emphasis or discussion of a shift in the definition of tangent. In geometry, the only use of tangent that I remember was with circles: the tangent is perpendicular to the radius. That’s not at all how we talk about tangents in calculus.
Dan: And that’s Poincaré’s “giving the same name 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 you can’t unfold the complete complexity of a concept all at once. In different contexts, you think about particular dimensions of concepts. So it’s natural that when we’re just 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 can make sense to give them that same name in order to capture some way in which they’re the same. Kayla, it sounds like you hadn’t thought as much about how differently the word tangent was used in calculus and geometry. What in particular, now strikes you as different?
Kayla: I believe most calculus students learn the new definition—how to derive a tangent, what it looks like, what it tells us about a curve—but I worry they may leave calculus still expecting tangent to mean “touching only at one point” as it did in geometry. I also worry that the geometric idea that the tangent line must lie on just 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 point of inflection, or just at some other point. I also have students who think 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 be a result of trying to make sense of the idea that there is no linear function of x that will give a vertical line.
Dan: Kayla, it sounds like you’re saying that, on the one hand, there are things that are called tangents in calculus that wouldn’t have been called tangents in geometry and also the reverse, that there were tangents in geometry that calculus students would not think are tangents.
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 readily to their minds might not change, what Tall and Vinner (1981) call a “concept image.” So really, Kayla, what you were saying is that only 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 acceptable in calculus but didn’t make sense in geometry; a vertical tangent made sense in geometry but worries the calculus student. The tricky thing 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 we had students sketch a graph with a vertical tangent, a lot of what we got was graphs like x = abs(y), a 90° clockwise rotation of the absolute value graphs students have seen, which doesn’t define a function of x at all. And, they treated the y-axis as the “tangent.” I just wonder if, to students, the picture just seems really similar to a circle despite its shape.
Dan: Right. One point of contact with the vertex of the “v” 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. But, from a calculus perspective, it’s not. In calculus, we want the derivative to be well-defined, determining one specific slope for the tangent at a point.
Willy: If there is an art to the way mathematics names different things with the same name, then students should be able to understand why mathematicians over time decided to use the same 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 have a great answer to a student who asks why it is important that there not be 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 different (one positive and one negative) creating a drastic change in slope where the two lines meet. And unlike a parabola where the slopes change from positive to negative across, those slopes are both approaching zero—just 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 go to draw the tangent line what slope would you 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 take on two values for one x.
Willy: This is one of the reasons that asking students to produce examples of concepts has been really thought provoking when I think about teaching. Asking students to sketch a function 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 can also give teachers information about what definitions their students are using, and what kind of concept images they have. But then, Kayla, it seems you’ve also been saying that such tasks give you a way to influence students’ concept definitions and concept images. Is that true?
Kayla: Yes, tasks like these help surface students’ concept image for me to work on with them. With some tasks, students all basically submit the same thing, showing how limited their image is. And, this applies not just to tangents. I especially like asking students to submit multiple examples. When we were doing rational function 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 think about related concepts: Do students think that points of tangency are different from points of intersection or just special ones? Or, do students think that a horizontal asymptote is a tangent?
Dan: So, your comments are about not just the match between the concept image and the concept definition, but also the richness and variety of the concept image space and connections to nearby concepts. Having surfaced all of those examples from students, in what way do you feel that those are a resource for your teaching separate from their role in assessing students?
Kayla: For the past couple years, students’ submissions have ended up being used in future discussions. When you have this bank of submissions that students actually submitted, you can develop a whole lesson based on what a couple students have submitted. I think the ability to see all those submissions easily, pick ones that are interesting, and use those, is great. Sometimes just seeing someone else’s submission can shift your concept image or support the new definition you are learning in a way that you weren’t able to without that extra nudge. I think that part is key. It can be super powerful just for students to see each other’s work.
Willy: I agree! And in the context of teacher preparation I also think about how difficult and time consuming it is for teachers to make up a variety of examples. So using student generated work helps! The work is already done for you, and then you can select the most appropriate examples for your purpose and have more time for other things.
Kayla: And I think often we make fake student work to use as teachers, we are saying these are the common submissions we know to expect. But now that we’re presenting this task to students, it has been interesting to see examples year after year that I hadn’t expected the first time around.
Dan: What’s an example of that?
Kayla: Year after year, students seem to think that there is a horizontal tangent on an exponential function where the horizontal asymptote is; they think 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-axis. Then write a function, or sketch its graph, such that the line is tangent to the graph of the function at two or more points.”
Willy: To help us learn how students think about a concept, we can design 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 task but still reveal possible gaps in understanding or overly broad or narrow concept images. For example, the “sideways absolute value” graph is not a function and does not have a tangent at the vertex. We can also design tasks that push students in a particular 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 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 appropriate for calculus. Interestingly, when I spoke with calculus teachers from my old school, one of the teachers thought 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, but I have an observation to offer. On the interval between a point of tangency and a point of intersection farther down the line, even if that point of intersection is not another point of tangency, I think the average 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-1)(x-2)(x-3), and Green(x) = 2(x-1). The point of tangency is (1,0) and Red'(1) = 2. The point of intersection is (4,6).
Think about the interval [1, 4]. This interval reminds me of Algebra One where we often 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 order to interpolate or extrapolate, we imagine a hypothetical situation where the change is distributed evenly, rather than the messy reality of change that is not evenly distributed. This observation about the interval between the point of tangency and intersection seems like it might suggest a mathematical value for considering 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 it might take to have my colleagues consider using these tasks. I know I am a bit of an outlier. At the beginning of the year, I generally move through content with my BC Calculus class at a slower pace 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 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 then a quick transition into derivative rules, the relationship between functions and their derivative graphs, and applications of derivatives (related rates, optimization, linearization, etc.). Our district’s curriculum materials frequently ask questions about calculating derivatives and writing the equation of tangent lines at specific point, but there’s little digging into what the definition of a tangent line is and how it might have changed from geometry. Personally, I think it’s important to spend time on the issues about 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 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 important areas of student confusion, even beyond the ones we’ve talked about here. I want students to think hard about definition and how definitions change. These “give-an-example” tasks help. They engage students with something interesting and challenging, and help them to pay careful attention to mathematical definitions and to be precise in using them.
Endnote1. For the last two years, we have been using the STEP platform developed by Shai Olsher and Michal Yerushalmy at the MERI Center at the University of Haifa (Olsher, Yerushalmy, & Chazan, 2016). The ideas represented in this conversation were spurred by use of this program with activities developed in Israel (Yerushalmy, Nagari-Haddif, & Olsher, 2017; Nagari-Haddif, Yerushalmy, 2018) and adapted for use in the US.
Verhulst, F. (2012). Mathematics is the art of giving the same name to different things: An interview with Henri Poincaré. Nieuw Archief Voor Wiskunde. Serie 5, 13(3), 154–158.
Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of technology in formative assessment support changes in mathematics teaching? For the Learning of Mathematics, 36(3), 11–18. https://www.jstor.org/stable/44382716
Yerushalmy, M., Nagari-Haddif, G., & Olsher, S. (2017). Design of tasks for online assessment that supports understanding of students’ conceptions. ZDM, 49(5), 701–716. https://doi.org/10.1007/s11858-017-0871-7
Nagari-Haddif, G., & Yerushalmy, M. (2018). Supporting Online E-Assessment of Problem Solving: Resources 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
Tall, D. (2002). Continuities and discontinuities in long-term learning schemas. In David Tall & M. 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 image 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