The Calculus of Context

by Yvonne Lai (University of Nebraska-Lincoln)

It is 2020. You are taking a high school mathematics teacher licensure exam. Suppose you see these questions. What do you do? What do you think? (Warning: Your head may spin. These are not licensure exam problems from 2020. Further commentary to come.)

I own a horse and a farm. One fourth the value of the farm is four times the value of the horse. Both taken together are worth $1,700. Find the value of each. Write out a complete analysis.

A merchant gets 500 barrels of flour insured for 75% of its cost, at 2 1/2%, paying $80.85 premium. For how much per barrel must he sell the flour to make 20% upon cost price?

Perhaps you are thinking about proportional reasoning and percentages. You might also be thinking: How quaint. These numbers are unnecessarily contrived; and owning horses, farms, and flour barrels is unrealistic to most students and teachers these days.

It is 1895. You are taking a high school mathematics teacher licensure exam and you see those same questions. What do you do? What do you think?

You might still think the numbers are contrived, but the context may seem more realistic.

Context and mathematics have never had an easy relationship. For one thing, it’s virtually impossible to hit the trifecta of precision, accessibility, and truth to reality. Bring in specific standards to be addressed, and that is a perfecta harder to achieve than predicting what would have happened next in Game of Thrones. (Example: Try writing a set of realistic problems for standard CC.5.MD.2; what are data that make sense to interpret as fractions to the nearest 1/8 that also make sense to add, subtract, multiply, and divide?) And we haven’t even brought in whether the context is actually appealing.

In this post, I present a case that determining whether context can support learning and teaching, through humanizing mathematics, is neither a simple yes-or-no calculation, nor is it value neutral. It is a pedagogical and ethical consideration. Context can motivate mathematics; and it can also extinguish interest. Context can be part of the mathematics; and it can distract. Context can humanize mathematics learning; but humanizing mathematics is not exclusively about context.

Context can add to mathematics learning

In a course for prospective high school teachers, I often teach properties of exponents, with a focus on why definitional conventions make sense and on ways to lead future high school students to discover these conventions (such as defining b^0 = 1 for nonzero b). Year after year, I would reach some teachers but not others. It was hard for some teachers to invest in building up identities that they had been using for years. Then, two years ago, I launched the unit on exponential properties with Dan Meyer’s Three-Act Math Task, Double Sunglasses, which begins with a whimsical clip of Meyer donning one pair of sunglasses, then another pair, then both. In keeping with the Three-Act Math Task structure, I asked teachers after viewing the video, “What questions comes to mind?”

The teachers pounced. “What does he see?” “What’s the tint?” “What does tint mean?” “Would it be 100%?” “No, it’s 0%!” “Or 25%?”

I explained:

• “Tint” means window tint.
• The higher the tint, the darker things appear.
• p% tint: The sunglasses reduce visible light transmission by p%.
• Examples:
  • With 50% tint glasses, you see 50% of the light headed toward your eyes.
  • With 60% tint glasses, you see 40% of the light headed toward your eyes.

With double sunglasses, it is like Dan is wearing 75% tint sunglasses. With double sunglasses, there was suddenly an intellectual need to find the tint, and an intellectual sense for why 100%, 25%, and 0% cannot be right. Teachers in that class, and subsequent classes where I have used this video, speak with gusto to propose, critique, and defend their views. Teachers also almost all come to the conclusion that 75% is correct, even if they originally thought 100%, 25%, or 0% was correct, and they identify various exponential properties along the way. For example, a 0th power corresponds either to wearing 0 sunglasses or wearing sunglasses with 0% tint. In both cases, one sees 100% of the light headed toward your eyes.

Double sunglasses are not realistic. They are not practical. They are ridiculous. And in its concrete absurdity, teachers found a way into the mathematics more deeply than any naked math problem, or any more realistic scenario featuring compound interest.

Context can add to mathematics learning, even when the context is not perfectly realistic, because the context gives just enough of a flavor of reality to hold onto lived experience. When Freudenthal (1971) proposed that one should “[teach] mathematics related to reality” (p. 420), he did not mean that the context had to be practical, realistically messy, or even realistic enough to constitute applied mathematics. He meant that students should be able to draw on real experiences to explore ideas and make inferences.

Context and mathematics learning can structure each other

Finding a right context for particular mathematics is hard. Deborah Ball (1988) and Liping Ma (1999) famously posed:

Write a story problem for 1 3/4 ÷ 1/2.

This task is hard for a variety of reasons. Crafting story problems, and evaluating their correctness, can help prospective teachers understand ideas more deeply as well as appreciate student difficulties. This problem was devised by Ball for her dissertation, used by Ma in her cross-national study, and was featured in Kennedy et al.’s (1993) study package.

Sometimes, even when there is a mathematically accurate context, it still may be distractingly unappealing (Nabb, Murawska, Doty, et al., 2020):

(The Condo Problem) In a certain condominium community, 2/3 of all the men are married to 3/5 of all the women. What fraction of the entire community is married?

This Condo Problem was first proposed in Lester’s (2002) Making Sense of Fractions, Ratios, and Proportions. It can be a beautiful mathematics problem that can help students make sense of operating with ratios. In the wake of Obergefell v. Hodges, though, the context is gauche at best, and offensive at worst.

In the changing acceptability of the condo context, Keith Nabb and Jaclyn Murawska, and their students found an opportunity. They asked their students:

How can you rewrite the Condo problem to be more inclusive? (Nabb et al., 2020, p. 696)

Nabb and Murawska report that over time, students in their courses had become increasingly uncomfortable with the problem. And so, their students appreciated the prompt to rewrite, and learned from this experience not only ways to rewrite it (with guinea pigs, penguins, college education rates, and jeans and shirts), but also the important lesson that they have the power to rewrite problems. I would also hazard that writing isomorphic problems also gave students more contexts with which to examine and understand the ratios. Mathematics gives us the power to describe the underlying structure of seemingly disparate situations. Recognizing this power, and wielding it, is part of mathematics learning and teaching. Imagining isomorphic contexts allowed these prospective teachers to confront discomfort about the original context, with mathematical integrity.

Coming up with contexts and then solving the problems with those contexts can be the mathematics. Explaining why contexts do represent a particular mathematical idea is itself mathematical work, and work that poses mathematical challenges beyond the underlying bare mathematics (Ball, Thames, & Phelps, 2008).

Contexts can also inspire mathematical inquiry because of their relevance. I have been struck by a story from Ricardo Martinez about teaching high school students about slope. They saw slope as a bare formula that was not consequential to them.

But when Ricardo presented data of their own school’s population over time, broken into racial and ethnic categories, these students, many of whom identified as Black, Hispanic, or Latinx, wanted to know more. Suddenly, slope was no longer italicized letters on a page but a concept with vivid and personal explanatory power. They asked Ricardo whether they could look up more data and compute more differences. They wanted to use the math to make sense of their life.

The Gerrymandering and Geometry materials, developed by Ari Nieh, use unit squares to model pieces of districts. Across the times I have used these materials, the results are similar: the teachers start with unit square geometry and end with fascination about area metrics more generally and questions about the political implications of any metric. These gerrymandering problems, despite the unrealism of exactly square land, have sparked more conversations about alternative metrics and their consequences than any lesson I have ever taught on hyperbolic or spherical geometry. The context of gerrymandering and its sobering consequences motivate earnest work with a simple model, and exploring the simple model inspires mathematical curiosity.

Context can subtract from mathematics learning

Turning back the opening problems, context can distract from the mathematics. If I were to in fact pose these problems today to students, I would predict that the second problem would be confusing because the language used is (no longer) commonly known jargon. (Footnote: These problems were actual teacher licensure exam problems in 1895 (Hill, Sleep, Lewis, & Ball, 2007). Perhaps the jargon was common then.) As for the first problem, it suffers from multiple problems, including a cantaloupe problem. Unrealistic context has the potential to teach a person to ignore previous knowledge, such as how much a house might cost, or how many cantaloupes an individual can reasonably purchase. The result can be mathematics as an idiosyncratic silo rather than as a discipline of beauty and descriptive power.

As Matt Felton-Koestler has written, “real-ish” problems, such as the Double Sunglasses task, or Gerrymandering tasks, have a place as stepping stones, but not as the only kind of context.
Exclusive use of real-ish problems, no matter how good they are, can teach students that in math, problems should always be well-posed: students should never have to seek out information, and the answer is always the operation-of-the-day with the numbers featured.

Moreover, adding real world context can hurt a mathematics task. Consider the problem

How many different three-digit numbers can you make using the digits 1, 2, and 3, and using each digit only once?
Show all the three-digit numbers that you found.
How do you know that you found them all? (Ball & Bass, 2014, p. 302)

Perhaps there could be a reason that someone needed to arrange the digits 1, 2, and 3; or to arrange a particular three other numbers or objects. But this task needs no real world context to make it work. Ball has used this task for multiple years with rising 5th graders in a summer “turn around” program for students who have not been successful in mathematics at school (Ball & Bass, 2014). As her teaching demonstrates, so long as students understand the conditions of the problem (using each digit exactly once), it is a task that is highly accessible and engaging.

There is another way to think of this problem’s apparent lack of context and its appeal to students nonetheless. Sometimes, mathematics itself can be a context for other mathematical ideas, and adding more context than that would take away from the task. Curious phenomena themselves can be context for discovery and explanation.

Context can complexify mathematics teaching

One of my favorite classes in grad school was the day we learned about complexifying vector spaces. With apologies to that professor, I will use complexification as a metaphor here. When we complexify a real vector space V, we extend the vector space to have twice as many dimensions and we construct a complex vector space. It’s not just about tossing in more basis vectors; it is also about endowing the new space with a new structure that allows for working with complex number scalars in a way that is compatible with the original vector space structure in V.

When we add context to mathematics presented to students, we are not just adding a dimension to the mathematics. We are also adding potential interactions between the context, the mathematics, and the class community. These potential interactions can enrich learning while also complicating our work as mathematics instructors.

Suppose that you use a context with statistics about race, such as income (Casey, Ross, Maddox, & Wilson, 2018) or honors class enrollment (Berry, Conway, Lawler, & Staley, 2020). Discussing race can be uncomfortable, because it is so charged. But increasingly, Black scholars are calling for explicit discussion of race in the classroom (e.g., Berry et al., 2020; Love, 2019; Milner, 2017; Tatum, 2016). As Casey, Ross, Maddox, & Wilson (2018) wrote, “Race … is a reality in our cultural moment, and it too important to be ignored when discussing issues of equity in education.” (p. 84)

It is often said that one power of mathematics is to understand the world around us. There are powerful and important statistics and mathematics to be done with real world data. Avoiding charged contexts can also mean losing opportunities for our students and ourselves to understand the world around us.

At a local Math Teachers Circle a few years ago, teachers worked on the Midge Problem, which asked them to find ways to differentiate two kinds of insects based on antenna and wing length:

[Given existing data] How would you go about classifying specifies Af or Apf?

Suppose that species Af is a valuable pollinator and species Apf is a carrier of a debilitating disease. Would you modify your classification scheme and if so, how?

This context gave teachers a concrete entry into determining metrics, and spurred a lively discussion. When it came to defending points of view, teachers drew on data, and made precise connections to others’ ideas. There was also a sense, though, that this kind of sorting was all in fun and an academic’s hobby – until it was mentioned that cutoffs are something that happens with standardized exams. Context can change how mathematics is taken up. In retrospect, the midge context may have allowed for more play, which was important for exploring the math. But the standardized exam context was more convincing, to this audience of teachers at least, of real life consequences of mathematical decisions.

With issues of race, students may inevitably begin hypothesizing potential reasons for disparity, which raises the danger of accidentally reinforcing stereotypes. Whether you disagree or agree with these hypotheses, and whether other students do or not, it is in the spirit of mathematics and statistics, or any other disciplined inquiry, to consider counterfactuals. This opening of explanations is part of what makes teaching problems with charged context so difficult: How do you approach the explanations with integrity and sensitivity?

How to work with complex social contexts is an open question for educators, and a hard and important one at that. For the issue of race, Casey et al. (2018) propose one way that has worked for instructors using their materials. Students begin by discussing explanations for disparity in small groups. Then, students are asked to consider causes and separate them into causes from within the group and causes from outside the group. Then from readings, videos, and data investigations done throughout the semester, students are asked to constantly reflect on their lists of causes, adding and removing items from the list based on what they’ve learned, with encouragement to add notes to themself with respect to why they are making an edit to the list when they make it. The materials ask students to examine data sets related to various within-group causes, including ones that debunk likely potential explanations from within the group.

Organizing proposed explanations seems outside of mathematics, yet upon reflection is central to both the discipline and its teaching. As mathematicians, when we come up with potential explanations or theoretical counterarguments, we don’t stop at the proposal – we see the proposal through with proof and evidence. But doing this also takes time and energy that we may not have before our 9am class. I am optimistic that in coming years, there will be more teaching practices and materials that will make teaching with charged contexts more possible.

Context as one variable in humanizing mathematics

There have been calls to humanize mathematics. Humanizing mathematics has come to encompass many meanings, including finding joy in mathematics; finding utility in mathematics to explain and understand real world phenomena; seeing the role of mathematics in making the world a better place; and using mathematics as a resource for social justice. Context in mathematics problems is then integral to some definitions of humanizing mathematics, especially those with social agendas.

Context is one variable in humanizing mathematics because of pedagogical and ethical considerations. Pedagogically, context may or may not enhance a task. Ill-chosen context can turn people off from the mathematics. At the same time, carefully considered context can give students a foothold to discovery and joy, though context is not always necessary for this purpose.

Federico Ardila-Mantilla, in a Notices article, urged mathematicians to consider the question of what it means to “do math ethically” (Ardila-Mantilla, 2020, p. 986). Context is one variable in addressing this question. Context can challenge narratives about what “is” mathematics, who does or needs mathematics, and the consequences of using mathematics. As Ardila-Mantilla observed, mathematics can be used to improve lives, and mathematics may be used for harmful purposes. At the same time, there is a place for mathematics for mathematics’ sake, as emotional responses to the work of teachers like Jo Boaler or Francis Su suggest. Mathematics–like song, poetry, and art–can and should be a place to play, find beauty, and experience joy.

The calculation for whether, when, or what context should be incorporated is not simple. I do not believe that we should prescribe context as a cure for all educational disease, and I do not believe that for all instances of mathematics learning, there should be a fitting contexts. Improving mathematics teaching and learning, through humanizing mathematics, can have to do with context, but is not exclusively about context.

Acknowledgments. I am grateful for encouragement from Erin Baldinger and Younhee Lee to pursue this topic, and for critical feedback on this essay from Stephanie Casey, Andrew Ross, Paul Goldenberg, and Mark Saul. I also appreciate editing from Cathy Kessel.

Errata. Hyman Bass has pointed out that it was Deborah Ball who devised the “1 3/4 ÷ 1/2” problem in her dissertation, and Liping Ma who then used this problem in her cross-national study. The post has been corrected with this information.

References

• Ardila-Mantilla, F. (2020). CAT(0) geometry, robots, and society. Notices of the American Mathematical Society, 67(7), 977-987.
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• Nabb, K., Murawska, J., Doty, J., Fredlund, A., Hofer, S., McAllister, C., … & Welch, E. (2020). The Condo Problem: Is this culturally responsive teaching?. Mathematics Teacher: Learning and Teaching PK-12, 113(9), 692-701.
• Tatum, B. (2016). Defining racism: “Can we talk?” In P. S. Rothenberg & S. Munshi (Eds.), Race, class, and gender in the United States: An integrated study (Tenth edition). New York: Worth Publishers/Macmillan Learning.