# The Calculus of Context

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.)

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.

Context and mathematics have never had an easy relationship.首先,这几乎是不可能达到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 standardCC.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.

## 上下文可以添加到数学学习

Stills from Double Sunglasses Act I video from http://threeacts.mrmeyer.com/ (cc) Dan Meyer

I explained:

• “Tint” means window tint.
• 色调越高，出现较暗的东西。
• P％色调：太阳镜通过P％降低可见光传输。
• Examples:
• With 50% tint glasses, you see 50% of the light headed toward your eyes.
• 含有60％的色调眼镜，你会看到40％的灯向你的眼睛朝向你的眼睛。

With double sunglasses, it is like Dan is wearing 75% tint sunglasses. With double sunglasses, there was suddenly anintellectual need至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.

## 背景和数学学习可以互相构建

Finding a right context for particular mathematics is hard. Deborah Ball (1988) and Liping Ma (2001) famously posed:

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

This task is hard for a各种原因. 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 and then used by Ma in her cross-national study.

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?

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 fromRicardo Martinezabout 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.

TheGerrymandering and Geometry materials由Ari Nieh开发，使用单位方块来模拟地区。我曾经使用过这些材料，结果类似：教师从单位方形几何形状开始，并更加令人着迷，更普遍地迷恋区域度量，以及关于任何公制的政治影响的问题。这些格莱德的问题，尽管处于完全方形土地的不真实，但引发了更多关于替代指标的对话，以及它们的后果，而不是我曾在双曲线或球形几何上教授的任何课程。GerryMandering的背景和其发疯后果激励了一个简单的模型的认真工作，并探索了简单的模型激发了数学好奇心。

## Context can subtract from mathematics learning

Sally gives a hot take on context in math problems

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）。也许术语是常见的。）至于第一个问题，它遭受了多个问题，包括哈密瓜问题。不切实际的背景有可能教导一个人忽视以前的知识，例如房子可能花费多少，或者个人可以合理地购买多少哈拉普斯。结果可以是数学作为一种特殊筒仓，而不是作为美容和描述力的学科。

AsMatt 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 tothat professor, I will use complexification as a metaphor here. When we complexify a real vector spaceV，我们将矢量空间扩展到具有两倍的维度，我们构建复杂的矢量空间。这不仅仅是在更多基础上折腾;它还具有赋予新的空间，具有新的结构，允许以与原始矢量空间结构兼容的方式使用复杂的数字标量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)

At a local Math Teachers Circle a few years ago, teachers worked on theMidge 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?

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呼叫humanizemathematics. 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.

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.

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.

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### 1回应The Calculus of Context

1. Yvonne Lai says:

Hi David, your previous comments were not erased – they just had not been approved yet. Thank you for your work for the Nebraska Correctional Facilities through SCC.