The past 30 years have been marked by an increased rate of change in societies around the world. In the span of a single generation, new technologies like personal computers, the internet, smartphones, and social media forced us to reconceptualize the way we engage in social interactions, work, and public life. Now artificial intelligence is challenging the boundaries of what it means to be human.
While these technologies have largely been adopted in positive ways, the systems under which they were created have also caused or exacerbated major world crises. The climate change crisis has shown us the consequences of unsustainable exploitation of nature and little regard for our limited resources. The spread of COVID-19 revealed capacity, coordination, and equity issues in our health, government, and education systems. Finally, most modern societies are also facing an economic crisis as a result of late-stage capitalism,1 which will undoubtedly change the face of our cities, work relations, and public life.
With all of these life-changing scenarios, what can the educational community do to equip students to deal with the challenges facing our world? This kind of question is usually confined to conversations about curriculum policy or reform. Many teachers believe their current curriculum does not address these challenges effectively, making it hard to tackle them in the classroom – especially in mathematics.
Specifically in mathematics, why is it so hard to address matters of public life in class? Perhaps the answer has less to do with curriculum policies and more to do with how we understand math education. The solutions to a quadratic equation, for example, can be found through digital technologies with precision and efficiency, so why is it that we still teach the discriminant technique? If we were to pay less attention to procedures and formulas, what could we do in mathematics classes to support our students to navigate the changes in our societies?
Preparing students to respond to the challenges imposed on our lives in the 21st century is a shared responsibility between policymakers (responsible for designing curriculum and approving pedagogical resources) and practitioners (school boards, schools, and teachers). Many readers might argue that the solution is to change the curriculum. While this is true to some extent, most Canadian provinces and territories have recognized that mathematics teaching in the 21st century must look different from the past.
Provinces already incorporate some of these important issues in mathematics classes, and many others are revising their curricula to increase their importance. They do so in three ways:
- by recognizing cross-disciplinary, real-world problems that must be addressed in all school subjects (e.g. in Quebec, students are supposed to learn about environmental awareness and consumer responsibilities, media literacy, and citizenship and community life in all subjects)
- by incorporating overarching mathematical practices that provide a sense of what it means to think mathematically (e.g. in British Columbia, students learn about reasoning and analyzing, understanding and solving, and communicating and representing while exploring particular concepts in mathematics)
- by introducing new and contemporary areas of mathematics to complement traditional mathematical concepts (e.g. in Ontario, elementary and secondary students now learn coding, financial literacy, and the development of geometry in multiple cultures).
If many provincial curricula already have space to address our generation’s most pressing problems, what is missing? This is an epistemological problem. We, as mathematics educators, have been trained to see mathematics through its scholarly representation. We understand and appreciate theorems, concepts, algorithms, and formulas. But if we take a step back and look at mathematics as a way of thinking and being in the world, we might be able to see how our classes can contribute to a better tomorrow. Teachers often find themselves confused about the values they promote regarding mathematics. What is the purpose of teaching mathematics in the 21st century?
As it turns out, there are three answers to this question. In my research with mathematics teachers, I have identified three main orientations of mathematics education: disciplinary, professional, and citizenship. Depending on their orientation, teachers understand the value of teaching mathematics in unique ways. Consequently, they will respond to social change and tackle contemporary challenges distinctively. Below, I provide a brief description of these epistemologies, along with suggestions for readings and activities teachers can do with their mathematics classes. These are suggestions I’ve been using in my teacher education courses that have proved relevant to mathematics teachers’ visions and values.
This orientation refers to the most traditional – and most common – approach to teaching mathematics. It approaches the subject as the teaching of a scientific discipline, i.e. as an abstract science that is worth knowing for its own sake.
Most secondary mathematics teachers who pursued a specific degree in the subject enjoy mathematics for its own sake. For these teachers, mathematics – just like fine arts – should not serve immediate economic goals. It should instead be appreciated and celebrated as a common heritage of humankind and a way of developing the mind through problem-solving, logic, and rationality.
Teachers who share the disciplinary orientation of mathematics can respond to contemporary challenges by portraying mathematics problems that have yet to be solved. It is important to show students that mathematics as a discipline is still unfinished. Most students would assume that mathematical knowledge is already established and there’s nothing else to discover or invent. That is the result of the way math is often represented in the curriculum and textbooks, with formulas and concepts that must be memorized. By sharing unsolved mathematical problems, teachers can also show students that mathematical investigations can be done with a variety of technologies, including spreadsheets, coding, software programs, simulators, etc. Unlike what many might think, mathematical work is not isolated, and it certainly uses more than just paper and pencil.
Instead, I would invite teachers to introduce to their students the notion of a conjecture (a proposition that seems to be true but for which we still lack proper proof). Exploring a conjecture provides many opportunities for students to learn about the work of mathematicians and use a variety of technologies to investigate mathematical propositions. Students can also learn about the history of mathematics and how mathematicians pushed the boundary of human knowledge in attempts to prove conjectures.
Suggested book: Fermat’s Last Theorem, by Simon Singh (Fourth Estate, 2017). This book explains the history of more than 300 years of mathematical endeavours to prove a relatively simple proposition. Students can create a book club to discuss the book in parts. They will learn that mathematics is a lively science with lots to explore. The theorem (previously known as a conjecture) was only proved in 1995, 350 years after it was first proposed.
Suggested classroom activity: Explore the Collatz Conjecture2 with students in class. This conjecture can be easily understood by middle and high-school students and can generate many beautiful representations. Use Excel spreadsheets to automatically create a sequence based on a seed number, implement an algorithm (in Python language) that creates the sequence based on the user’s input, and create a concept map (use CmapTools) of multiple sequences.
This orientation is perhaps the most pragmatic of all three; it stems from an economic view of education as training. For teachers (and students) who espouse this perspective, the teaching and learning of mathematics should prepare students for future professional life. Particularly in high school, mathematics classes should develop appropriate skills that students could use in the workplace and/or prepare them for university programs that demand mathematical skills. With the intensification of the use of technology, skills associated with mathematics (counting, estimating, measuring, comparing, reasoning, etc.) have become ubiquitous in virtually all fields of professional life, from life sciences and STEM to literary work and fine arts. Most professionals face some, if not multiple, strands of mathematics daily. These demands intensify as they attempt to get promotions and climb the ranks of their organizations (typically moving toward management positions).
Consequently, mathematics classes should be responsive to these changes and portray the use of mathematics in a range of professions, so that students can see the value of learning mathematics and make informed career decisions in an increasingly precarious job market.
Many teachers see this phenomenon as a way to increase their students’ motivation to study mathematics. However, when faced with the infamous question, “When am I ever gonna use this?” they struggle to bring authentic examples of math in professional life. After all, it is unrealistic to expect mathematics teachers to be aware of how different fields are evolving. Do we expect industrial engineers to solve quadratic equations by hand to optimize costs in a production line? Or do they use software programs to simulate different scenarios under budget and resource constraints?
To tackle this challenge, teachers could provide students with opportunities to explore mathematics in professional life through research and social media. Platforms such as Twitter, LinkedIn, Reddit, Quora, and others provide a much-needed connection between school settings and real-life professionals. Through these platforms, students can reach out to workers from many fields and ask specific questions about the way they engage with mathematics in their daily tasks. Not only can this practice increase motivation, but it also allows teachers to create a portfolio of examples of math used in real-life workplace settings.
Suggested books: For middle school mathematics, On-The-Job Math Mysteries: Real-life math from exciting careers, by Marya Tyler (Prufrock Press, 2008). This book presents a set of mathematical problems faced by real-life professionals in interesting and unique fields. It provides teachers with explicitly mathematical problems that can be used in a class while also portraying mathematics authentically.
For high-school mathematics, 101 Careers in Mathematics – Fourth Edition, by Andrew Sterrett Jr. (Maa Pr, 2019). This book can be used by students, teachers, or counsellors to explore a wide range of careers for those who enjoy mathematics in high school. The book features real people in different fields and how mathematics was part of their professional trajectory.
Suggested classroom activity: Although most “traditionally mathematical” professions now use software programs for mathematical tasks, there is a lot of value in knowing how particular mathematical concepts were developed within those fields. One good example is geometric instruments and constructions, both of which were developed in the context of architecture. Notable angles, parallel and perpendicular lines, and triangle centres can all be constructed with high precision simply through a compass and a ruler. Students can learn a great deal of math by exploring why such constructions work. Here is a guide for a variety of constructions: www.mathsisfun.com/geometry/constructions.html
This orientation perceives the teaching of mathematics as a way of facilitating active participation in social life. Mathematical knowledge is one lens through which students can understand the world. When we discuss issues of public policies, the planning of our cities, the distribution of resources, or the electoral system, it is important to understand how mathematical information is produced, used, and communicated. It is therefore paramount that our students learn to decode different discourses through mathematics.
Most teachers present this orientation in implicit or explicit ways. They agree that mathematics is required to become a well-rounded individual in our societies, but sometimes struggle to identify proper opportunities to discuss important issues in the classroom. How much time should be spent discussing the context before diving into the “actual” mathematics? How much preparation does a math teacher need to approach sensitive topics? How do we identify the underlying mathematics concepts that can be explored in such topics as city planning or government budgets?
It is true that mathematics teachers need to go above and beyond their original training to make connections between mathematics and citizenship. However, once this connection becomes clear, it can save time in the classroom by interweaving different math strands into one unit. Also, the most recent curriculum revisions have introduced topics that facilitate these connections explicitly. Financial literacy, coding, and data literacy are just some examples of new mathematics strands that can easily be implemented with a citizenship epistemology. These concepts are unequivocally connected to social situations.
Suggested book: How Not to Be Wrong: The power of mathematical thinking, by Jordan Ellenberg (Penguin, 2015). Each chapter of this book explores a different mathematical concept or principle and how it has been used to shape our daily lives. It is a great resource to find deep and authentic connections between mathematics and social life.
Suggested classroom activity: One of the biggest debates in Canada over the last decade has been electoral reform. Currently, Canada uses the so-called first-pass-the-post system: each of 338 districts elects a member of Parliament to represent its interests, and the party with the most seats then forms the government. Through publicly available data,3 students can organize a spreadsheet according to each district and the votes received by each party.
A range of questions can be explored: What is the percentage of votes received by your MP? In which riding does a vote have the most/least percentage impact? Which riding elected an MP with the highest/lowest number of votes? Which riding had the closest race or largest landslide victory? Which non-elected candidate received the greatest number of votes? Has any MP been elected with less than this number? These questions elevate the debate about Canada’s voting system without promoting any specific position about electoral reform.
Similar to art, which can be valued for its aesthetic contribution as well as its depiction of social issues, mathematics is multi-faceted in its contributions to our world. The orientations described above are present in curriculum expectations, textbooks, teaching practices, and students’ rapport with the subject. They are certainly not mutually exclusive and can emerge in the classroom at different moments. Mathematics educators can benefit from a deeper look at their own values related to mathematics in order to recognize the biases and ideas guiding their instructional choices. In doing so, they might also be able to recognize the orientations their own students bring to the classroom and express in mathematics.
Which of these orientations is most closely aligned with your values? How do they inform your practices in the classroom?
First published in Education Canada, September 2022
1 Commodification of housing and health, widespread industry monopolies, precariousness of workers’ rights.
2 See The Simplest Math Problem No One Can Solve – Collatz Conjecture. Veritasium. www.youtube.com/watch?v=094y1Z2wpJg
3 2021 results: www.elections.ca/content.aspx?section=res&dir=rep/off/44gedata&document=index&lang=e