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Beyond the Equation

February 16, 2016
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By: Dr. Sean Chorney

Years ago, as a young mathematics teacher, I gave the following problem to a grade 11 mathematics class: A theatre with 1000 seats can be filled if tickets are 12 dollars, but for every increase of 25 cents the theatre will lose 5 attendees. What is the cost of the ticket that leads to the maximum profit? Given the necessary information needed to set up a quadratic function which would produce the answer, some students listened, others contributed suggestions, while others made notes.  It seemed like a good lesson. When we had finished, a girl at the back of the classroom raised her hand, asking “What if our goal was to fill the theatre?” My world as a teacher turned upside down. I had been caught advocating a perspective that valued financial profiteering over the consideration of people. “What a great question,” I thought and wondered why I hadn’t anticipated it. Nothing in the teacher’s manual had indicated this might happen. The student had posed a genuine question: why weren’t we considering any of the people in the ‘story’ problem: this made the situation all the more dispiriting.

The messages in mathematics

Ever since that day I’ve tried to remain aware of the implicit messages conveyed in mathematics teaching. In particular, I’ve recognized that mathematics application questions that involve people’s lived experiences need to be thoughtfully posed and considered from different perspectives.  Even more so, I have found it extremely important to frame mathematics content from within a context.  That is, the mathematics I teach has been transformed from a discipline that practices computation and algorithms with a further intention of application, to a study that is only defined and meaningful in the application.

In the example above, the context would be the theatre production, the people, and the ticket price. The content would be the mathematical investigations that draws awareness to what choices there are regarding price accessibility. What I found was that not only did students engage more in discussions, posing their own mathematical questions and observations, but also mathematics was intrinsically tied to world issues and was quite evidently relevant and meaningful for students. This, I would suggest, becomes a social justice approach because it inculcates a sensibility toward others as the impetus for meaning in mathematical teaching.

The inherent relationship between students and their world has since informed my teaching of mathematics. I have adopted a social justice perspective that relates not only to important social issues but also to the differing and unique perspectives of my students. Social justice is not simply an application to be implemented once in a while but has become a way to introduce and make meaning of mathematical thought.  To help students re-conceptualize mathematics as a discipline which supports an engaged citizenship that considers the ‘ways’ and ‘knowings’ of others as an essential step towards a more accessible and democratic mathematics.

Some challenges in mathematics teaching

I have often wondered how I could have been totally unaware of the implicit message conveyed in the problem I had posed to that grade 11 class. I’ve finally come to understand that it’s partly to do with my own mathematics education that had advocated the separation of mathematics from context. Attention in mathematics class was given to the numbers while the words that described details were ignored. This dissociative practice, lauded in school mathematics, supports a dualistic framing. It’s as if there are two storylines existing simultaneously and only one of them is prioritized. When the practice of mathematics is solely about numbers and algorithms, there can be a loss of connection to what numbers mean in everyday activity. I would suggest that there may be profound implications when numbers are retroactively applied to a real life story.

For example, in the case of the theatre scenario described earlier, I’ve already implied (and it is true) that the most profitable ticket price will lessen the number of people in the theatre.  However, it is interesting to note that, by considering how long the show will run and what kinds of reviews the show gets, the more efficient price might be the one that fills the theatre ($12). Basically, if the show is good, and there are more people who will see it then there will be more people to tell others about it.  A mathematical investigation of this word-of-mouth phenomenon can also elicit investigations into, among other ideas, exponential functions. Different mathematical understandings can emerge from different approaches to a situation.  It is worth noting, though, how the second approach considers and appropriates people’s interests.  By considering context, students remain in touch with humanity.

Another example of how numbers become meaningful when they remain in contact with people’s lives could include an investigation into payday loans.  The mathematical concept of compounding interest shows up at different levels of degree throughout the mathematics curriculum.  However, when an investigation of compound interest is introduced through the topic of payday loans, different topics enter into the conversation such as who might be the clientele for payday loans.  When it is realized that payday loans are often used by people who are barely able to live within their economic means, a payday loan compounded at 60% becomes a much more interesting and questionable calculation.  Sixty percent is not just a percentage on the page but a number that is tied to a human who has to survive.

Textbooks often present mathematics problems as a way to practice an algorithm, particularly as a method of manipulating numbers.  The long division method would be considered an algorithm.  What often ends up happening, however, is that the context of the problem is often simplified to such an extent that it becomes forced and meaningless.  In addition, students become familiar with well-defined problems that all have answers, usually at the back of the book, and methods of solution are presented in fluent and efficient ways. Rarely are there problems that point to the act of revising, reformulating or backtracking, yet these are the kinds of practices in which mathematicians are engaged. Simple approaches obscure real life. Mathematics problems are rarely, if ever, straightforward.

Furthermore, this method of ignoring the context produces a vision of teaching mathematics that focuses on functionality. The functional literacy approach ‘reads’ the needs of the world and ‘prepares’ the students for what they will need to do to function. Some social justice scholars have criticized the term ‘functionality’ because it supports the status quo and prepares students to accommodate themselves to the current system. It has been argued that within this model people do not ask questions but only carry on with the way things are. However, critical literacy, or more accurately the term critical numeracy, questions and challenges the status quo. It supports the approach that mathematical application has a lot to say about our world so let’s use it to say something.

Embracing context

The newly revised British Columbia curriculum promotes new insights into mathematics teaching. The new curriculum contains positive changes that support a connection between mathematics and social justice. With the inclusion of social responsibility as well as the critical thinking competency, the curriculum offers the opportunity for students to develop a sensibility towards themselves and others. Social responsibility as well as a critical perspective open up new opportunities for a mathematics which embraces social justice. The critical thinking competency, as written in the current curriculum, indicates that students “examine their own thinking, and that of others, about information that they receive through…various forms of communication.” This critical approach provides an important step in mathematics education. Within this framework, mathematics is not a discipline to apply but rather an examination of an ethical and moral approach to the world which inculcates a sensibility of care and compassion.

Evidence supports a strong correlation between social justice topics and mathematics in a number of ways: quantification, alignment between mathematical concepts and political ideas and a mathematical expansion which includes multiple modalities of knowing.

QUANTIFICATION

In a world overrun with data, the development of a critical eye is imperative. Every opinion and product seeks to draw attention to its cause. The quantification of phenomena in our world has become a phenomenon in itself. However, quantifying phenomena has its limitations and too often results in suppressing and obscuring information.  How, for example, can a single number represent an idea or an entire population within a context or activity? From measures of popularity and intelligence to differing interpretations of unemployment and poverty figures, the numbers rarely mean what they intend to convey. We are finally coming to terms with this idea when assigning letter grades for scholastic achievement. Negative consequences can too often result when quantifying students’ academic achievements. It is not simply the “number” or its numerical equivalent that is the problem, but how this “measurement” is interpreted, how it is used and what message it conveys. Good mathematics education strives to develop a critical eye able to discern an act of quantifying that goes beyond the use of numbers. Being able to critically approach how numbers are meant to represent societal measures, determine that consequences of different interpretations are more important than being able to do what the calculator can do.

CONCEPTS OF MATHEMATICS ALIGN WITH POLITICAL IDEAS

More than just numbers and calculations, mathematics conceptually deals with notions of equality, of difference, of power. These are powerful mathematical concepts in addition to being political terms. The notion of the equal sign can itself initiate vigorous discussions about equality insofar as it shifts focus from the symbolic to the meaningful. Different conceptions of equality, such as equal opportunity or equitable outcomes, can lead to a productive discussion that encompasses the mathematical idea of equality possibly more than solving an equation. When the political and mathematical are brought together in discussion, students can relate and express their voice. They are provided the opportunity to ‘feel’ others’ experiences through the consideration of numbers and mathematical ideas.

MATHEMATICS EXPANDS TO INCLUDE MULTIPLE MODALITIES OF KNOWING

Mathematics and social justice becomes less about calculations and performing algorithms and more inclusive of different modes of engagement making mathematics a more inclusive and accessible discipline. In the process of researching, and discussing, as well as challenging pre-conceived notions and grappling with differing perspectives, there is a direct communication and an immediacy that connects the student to the actuality of the world around them. Mathematics and social justice directly impact  how students see the world and they act in accordance with their new perspectives.  Social justice in mathematics is not an application. To suggest this would be to shift social justice to a position that exists outside the realm of mathematics. When students and teachers engage in “applying” mathematics to a social justice issue, they themselves feel like outsiders. That is, they are imposing a frame of reference upon a context without first contending with the method of quantifying the problem or by considering mathematical approaches. When approaching social justice issues in the classroom, mathematics instruction should be initially framed within a context. In this way, mathematics learning begins with a focus upon the meanings of the numbers and the outcomes of the operations. When numbers are treated as things to manipulate, students can get the wrong impression as to what mathematics is and what it is intended to achieve.

In the classroom, discussions emerge, reasoning happens within the group and collaborating occurs on how to interpret the situation. In presenting a social issue as an impetus for inquiry, teachers can offer opportunities for students for discussion, provide opinions, receive perspectives, and, with thoughtful nudging from the mathematics teacher, foster the ability to quantify, analyze and question. This process offers students the space to engage in difficult issues, to recognize the decisions that have to be made, and to understand that any decision has consequences.

Conclusion

Mathematics as a discipline is associated with words such as right, wrong, exact, precise, logical or rational. While these terms seem reasonable, they don’t often apply to the real world. Suppose instead that mathematics was attached to words like compassion, care, and empathy. Although these may be odd terms within the current framework, when engagement with mathematics in the classroom is about people, their lives and experiences in the world, students participate in more meaningful way than with symbols and numbers. Mathematics is not an acontextual practice. For students to care and concern themselves in mathematics class, they need to connect to lived practices of people in their community, in their family and also with themselves.

For our students to become personally engaged with mathematics,  I suggest we move beyond calculations and symbolic manipulations and start relating to the world in a less symbolic and more connected way. The common refrain “Why are we learning this?” usually emerges from a mathematics class in which students are performing algorithms with real world issues being ignored. A mathematical sensibility is more than calculating. It’s engaging in an approach that supports people. The newly revised curriculum now supports a social element in mathematics instruction and practice.

An excerpt of this article was also featured in the March 2016 issue of Teacher Magazine