Reflecting on mathematics as the art of giving the same name to different things (Part 2): Averages finite and continuous


by Bill Rosenthal, Queens, NY; Whitney Johnson, Morgan State University; Daniel Chazan, University of Maryland

The July 15 blog post by Dan Chazan and two colleagues referred to Poincaré’s enigmatic remark: “Mathematics is the art of giving the same name to different things.” Poincaré called “giving the same name” an “art,” no doubt referring to the beauty and depth of showing mathematical relatedness and also the care with which that must be done. The post explored how the word tangent in the contexts of circles and graphs of functions connotes different things for learners and argues that, for learners, “the same name” can add depth, or confusion, and that teachers must be alert. In this post, we return to Poincaré’s remark and consider how reuse of names may be seen differently by students who come to the mathematics classroom with disparate experiences.

Our conversation began with a conjecture about a line tangent to the graph of a function—specifically, about the interval between the point of tangency and the point at which that tangent line intersects the graph a second time—then moved to averages in finite and continuous contexts. These two types of averages are the “different things” we study here.

Poincaré’s “different things” are generally understood as objects such as vector spaces and groups, not constructs such as averages. Perhaps he would have considered our appropriation of his aphorism to be illegitimate. For the purposes of teaching and learning mathematics, however, “the same name for different things” fits finite and continuous averages. We (authors and perhaps readers) conceive of these averages as different instances of the same thing. It’s unrealistic to expect students to think this way. School mathematics remains procedural and formula-laden. Vis-à-vis finite and continuous averages, addition and integration are different computations, and the formulas look different. It’s likely that most students perceive finite and continuous averages as different things.

The mathematics of continuity and the infinite are barely present in school mathematics, yet many students wonder about, even puzzle over, continuity and the mathematical infinite. How might being unable to express their puzzlements and pose their questions affect these students’ mathematical learning? We take up this question later in this post.

Our work began with Dan’s conjecture to Whitney and Bill about a tangent to the graph of a function: “[o]n the interval between a point of tangency and a point of intersection farther down the line…I think the average value of the derivative function is equal to the derivative at the point of tangency.”

Bill thought of the Mean Value Theorem (MVT). Dan’s observation felt intriguingly strange to him, especially when he labeled the point of tangency as (a, f(a))  and the second intersection point as (b, f(b)), then expressed the conclusion as $f'(a)=\frac{f(b)-f(a)}{b-a}$.  The MVT guarantees a c between a and b such that f'(c)=\frac{f(b)-f(a)}{b-a}.

That is, it vouchsafes that somewhere between the two points of intersection, the tangent line is parallel to the secant line connecting these points. Dan appeared to have done the MVT one better by discovering that the point of tangency is always a “somewhere.” Closer attention to the MVT brought back the nuance that its conclusion insures a c strictly between a and b, meaning that

f'(a)=\frac{f(b)-f(a)}{b-a}

doesn’t improve on the MVT in the case of a re-intersecting tangent.

Whitney approached Dan’s observation differently, beginning with a generic picture.
sketch of graph with tangent line and several lines parallel to tangent

She drew lines parallel to the line through the point of tangency  (a, f(a)) and the point of intersection (b, f(b)), whose slope is

\frac{f(b)-f(a)}{b-a} (call this line LT), to see if it made sense that there could exist a line parallel to LT that would be tangent to the graph of f(x) at some point c in (a,b). Intuitively it made sense; however, it was not clear why LT’s slope is referred to as the average rate of change (ARC) of f from a to b.  This name did not seem to fit. It made more sense that the ARC would be the average of all of the derivatives over (a, b). Since there was no easy way of computing this average—we can’t sum the derivative over all points in (a, b) and divide by how many derivatives there are—Whitney accepted that \frac{f(b)-f(a)}{b-a} is a reasonable estimate to what she thinks of as the ARC.

Whitney’s thinking made a strong impression on Dan. The computation gives the value of the slope of the secant line, LT, connecting (a, f(a)) and (b, f(b)). Dan ventured that many students, having been taught that the derivative computes the instantaneous rate of change, might well wonder, “Why should this value be the way to compute the average of all the instantaneous rates of change on the interval,” if that’s what the “average rate of change” might mean.

Until Whitney brought it up, Bill hadn’t conceived of “the average value of the derivative function” as the mythical arithmetic average of the derivative’s values at all points in the interval. Average rate of change caused him to think of how “average” is used in other parts of analysis. For him, the phrase “average value of the … function” evoked only the Calculus II sense of defining the average value of any continuous (or just integrable) function f on [a, b] to be \frac{\int\limits_a^b f(x) dx}{(b-a)}. The semantic logic behind Whitney’s conceptualization of the ARC enthralled him.

Whitney turned to a calculus textbook (Simon, 1992, p. 149) for a curricular depiction of the ARC. She found herself lingering on a passage from the MVT section.

passage from calculus textbookThe statement that “the instantaneous rate of change of f equals the average rate of change of f at least once in the interval (a, b)” unsettled her. In students’ prior experience, when they compute an average of a set of values, the result is not necessarily a member of the set. The average rate of change isn’t a familiar “average.” First, it is not the average of all of the pointwise rates of change of f  on (a,b); now it is being equated to one of these instantaneous rates. Averages of a finite set of numbers do not behave in this way, and Whitney’s curiosity was piqued. When comparing these ways of taking an average, the salient difference seemed to be that one set is continuous and one is finite. What is it about continuity that would cause an average to be a member of the set of values that one is averaging? Although the MVT guarantees this outcome for rate-of-change functions — as does the MVT for Integrals for continuous functions in general — we’re unaware of a proof that explains it.

A second way in which finite and continuous averages are “different things” surfaced when Whitney revealed her dissatisfaction with the denominator of \frac{\int\limits_a^b f(x) dx}{(b-a)}, the expression for the average value of any continuous function. Even though (b-a) is also the denominator of the average rate of change \frac{(f(b) - f(a))}{(b - a)}, how is it in any way analogous to the number of numbers that we use in the denominator of an average of a set of values? Absent a satisfactory explanation, it was difficult for Whitney to have complete confidence in the definition of a continuous average.

Out of the tangle of ideas spawned by Dan’s conjecture (most of which we spare the reader), the two questions italicized were most prominent in our consideration of finite and continuous averages as different things with the same name. The following dialogue is a pared-down version of our Zoom conversation about these questions and more.

Dan: My Algebra 1 students helped me with the  (b – a) in the denominator of the average rate of change between two points. In thinking about how to compute the average fuel efficiency of a car, one student emphasized that the number of gallons is like the number of people when one computes an average bonus per person. It’s “how many ones you have,” to which you are equally distributing the total.

Whitney: My concern is about the average value of any continuous function, derivative or not. I don’t see how to use your student’s insight in the continuous case!

Bill: Perhaps I do. Almost 45 years since first teaching calculus, until our conversations, I remained uncomfortable with thinking of b – a as the continuous analogue of the number of data points in a finite set.

The discomfort was strongest with the average value of a continuous function, and starkest when I placed the expressions \frac{\sum_{i=1}^n f(x_i)}{n} and  \frac{\int\limits_a^b f(x) dx}{(b-a)} side by side. Thinking of integration as the continuous analogue of addition made it make sense for the numerators to correspond. But the denominators? I could conceive of both n and b – a as measures of the size of the set over which numbers are being averaged. This association was too vague and weak to assuage my discomfort.

Dan, it took your Algebra 1 student’s wording, interpreting b – a as “how many ones you have,” to help me with b – a in the denominator of a continuous function’s average value. This interpretation set off a chain of realizations: The number of elements n in a finite set is calculated by adding 1s, one 1 per element. This is  \sum_{i=1}^n 1. The continuous analogue of this expression is \int \limits_a^b 1 dx. And the value of this integral is  b – a. Finally, it made sense for   b – a to correspond to n.

Finite average: \frac{\sum^n_{i=1}f(x_i)}{ \sum^n_{i=1}1}

Continuous average: \frac{\int\limits_a^b f(x) dx}{\int\limits_a^b  1 dx}

Whitney: If we are going to use the word average in the phrase “average value of a function,” then we should compare how this “average” is the same and how it differs from the average of a finite set of numbers. In providing some intuitive thinking on the MVT, the excerpt from my calculus book states, “… the instantaneous rate of change of f equals the average rate of change of f at least once … .” This is not the case for finite averages. In many instances the average of a finite set of numbers is not also a member of the set of numbers. This is a fact that we emphasize to students when they compute averages. It creates a probable point of confusion for students. Why must a function’s ARC over an interval be assumed in the interval? In order to make sense of it I would look to the differences between conditions in the MVT and an average of a finite set. Specifically, the MVT presupposes continuity on the closed interval (and differentiability on the open interval). Changing a constant function’s value at an endpoint shows that a discontinuity can prevent the ARC from being assumed. So it seems that continuity should play a key role in the different ways we employ the term “average.”

Bill: When I ponder which differences account for this discrepancy between finite and continuous averages, the conjunction of continuous functions’ properties and the real numbers’ completeness satisfies me. In particular, the theorem that a continuous function on a closed interval has a minimum and a maximum, and takes on all values in between, carries much explanatory power. When you couple this property with completeness, there is no other value that exists to occupy a space in the function’s range between its minimum and maximum.

Whitney: Although I see how you are describing the two situations as parallel to each other, I still think this is an unsettling example of using the same name, average, for different things (the average of a finite set of numbers; the average value of a continuous function). I think this a particularly extreme case; it is a big move from the finite and the summation to the continuous and the integral. The art in using the same name for different things is being able to communicate what similarity warrants using the same name, while not overlooking differences, and, in the context of teaching, addressing the similarity and differences with students.

It was obvious to me as a student that there are counterintuitive and ambiguous ideas that professors never addressed but needed addressing. I always wondered why, and I still do. As a learner who is an African American woman, I often felt as though my instructors did not think it was worth their time or effort to explain things to me because to them, in some way, I was unworthy or unable to understand what they had to say.

Dan: It seems that you are using Poincaré’s statement to reflect on your experiences as a student; this reflection not only allows you to articulate your questions around averages, but also to address your experience as a Black female learner of mathematics. How does Poincaré’s statement help you understand your experience as a Black female mathematics learner?

Whitney: Yes, many might be surprised at how my experiences as a Black female have impacted my experiences as a learner of mathematics. I think I stuck with math for so long first, because I really like it, and second because I never wanted someone else to determine for me if I was able or willing to learn mathematics. As a Black female I have learned through my family teachings and through life experiences that it is not wise to take things at face value. It is never obvious what is trustworthy and reliable and when people may be leading you astray. I naturally brought this skepticism to mathematics classrooms and learning the subject. Thus my questions were essential to my understanding, but just as essential for me to be able to comfortably rely on what I was learning. If I could not find reasonable responses to my questions and justifications for the mathematical ideas I was being taught, then there was no reason that I had to accept them as true. Although I could still work with them in most cases, I was unable to totally embrace my learning or be completely comfortable with the ideas.

When my professors dismissed my questions, they were also dismissing me as a person. This made me skeptical about their commitment to my learning and it made me skeptical about the mathematical ideas I was learning.

Bill: This White male’s mathematical coming-of-age was quite different. Two high-school teachers took an interest in me. One gave me two SMSG books I’ve kept through floods and fire for 50 years. The “New Math” introduced me to axiomatics, and eagerly I bought in. I don’t doubt that feeling safe in my melanin-poor skin contributed to being able to keep faith with the axioms and resulting theorems. Whether it was New Math, old math, borrowed math, or blue math, I cannot recall doubting the truth of any true mathematical statement made by an author or teacher.

Whitney: If my calculus instructors had been willing, or maybe able, to let me know that the foundations of the real numbers are vital to what we were studying, it would have impacted how I oriented myself as a learner to the course material. We aren’t giving people—not only people like me—a full enough story about continuity, let alone completeness, to appreciate the character of the real numbers.

Bill: No question. We have no choice but to keep our own counsel on some of the full story’s characters and plot twists. None of us believes that calculus syllabi should include a unit on the real numbers as a complete metric space or equivalence classes of Cauchy sequences of rational numbers.

But neither should we stand aside while more generations of students endure a school mathematics inattentive to the real numbers’ intricacies. We consider factoring sums of cubes and arithmetic with radical expressions precalculus necessities, yet we ignore the properties of real numbers upon which limits, continuity, differentiation, and integration all rely as developed in elementary calculus. Doesn’t seem right.

Whitney: First, I agree with you to an extent. I would not advocate for teaching those ideas in their entirety in a calculus class. However, there are ways in which instructors can set a larger mathematical context for the ideas that they are teaching. Teachers could also address using the same name for different mathematical objects or processes.

Second, what you say connects to what I said earlier. It feels that continuity is a very deep concept that many people may not understand. Those who you may think understand it (i.e., professional mathematicians) apparently have much difficulty helping others to understand or appreciate it.

Dan: Whitney, you’re clearly placing learners at the forefront of your considerations. The earlier post confronted challenges students might have with shifts in the meaning of “tangent.” In addition to your point about learners who are highly skeptical of the mathematical authorities, you seem to be arguing that, at this time, mathematical differences between the finite and the continuous may not be well conveyed to learners.

Whitney: For me, many things have been uncovered in this discussion. I think we all want all of our students to see the wonder of mathematical ideas and to experience the joy of coming to understand difficult concepts. Also, we all knew going into this conversation that teaching mathematics is a complex art. Highly effective mathematics teachers are adept practitioners with a deeply woven and intricate knowledge of mathematics and various types of human experience. It is not sufficient to show students how to solve a variety of mathematics problems. We mathematics teacher educators should stress the importance of respecting students’ capacities for learning and making mathematics accessible by opening up the world of mathematics without watering the ideas down.

There is much room for what Poincaré has said about mathematics. Teachers can more accurately present mathematics by taking students’ questions seriously and helping them make connections to seemingly unrelated mathematics that students have previously studied.

Of particular importance is helping instructors to be mindful of the move from instances in finite contexts to those in infinite contexts. This transition is at the crux for much of mathematics and historically has been the site for much debate and the development of new ideas. Instructors should have some responses to questions like, “How have mathematicians in the past set out to mathematize the infinite? Which attempts were more successful than others and why? Why did the discipline set upon the path that it did and not on other paths?” These are questions that can give clarity to students—clarity in the sense that they can know more about the contexts for the ideas, which helps them understand the ideas themselves. Attending to these questions also helps students to see which questions are worth an investment of their time and energy and which are not (which decisions were made for the art of mathematics and which were made out of necessity). It may or may not help in understanding particular mathematical processes or solution methods. But we can remove mysticism and dogma and allow students to perceive mathematics as a human-made body of knowledge that they can make sense of if given a fair opportunity to do so.

And it’s further complicated by the fact that mathematics teachers are often teaching students who are different from themselves. Teachers are continually confronted with the ways in which their students make sense of the world—ways that may be alien or clash with their own. They can try to rope off the mathematics classroom or the learning of mathematics from the rest of the world, but this pedagogical strategy is unlikely to work.

Many students choose to study mathematics because, at the early stages of learning the content, everything appears to be unquestionable and reliable. However, as one continues to study mathematics, the complexity and vulnerability of the subject is revealed.

When students experience the world as always in flux, and always having to decide whether the people they are interacting with, the contextual clues, and the knowledge they are being given are sound and trustworthy they develop a healthy skepticism to navigate daily life. Chances are that these students will bring this level of skepticism to the learning of mathematics. Based on my own experience, students may have questions that challenge basic ideas, facts, axioms, theorems, etc., and they may need a reliable, sound response to their questions before they can believe or accept what they are learning.

By dismissing the integrity and core being of students, insufficient answers to these questions may inhibit their mastery of skills and understanding of mathematical ideas. A part of the art of teaching is understanding and respecting this skepticism in students and preparing to offer answers to students that honor who they are as human beings in a complex, confusing world. If the questions that I posed to my mathematics professors had been treated with respect and as valuable, my experiences in their classes and as a math major overall would have been profoundly different and better. I might even have a Ph.D. in mathematics instead of in mathematics education.

Reference

Simon, A. B. (1992). Calculus with analytic geometry. Glenview, IL: Scott, Foresman and Company.

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