What are the chief aims of teaching mathematics in university engineering programs? To what extent should mathematical subjects be presented to students, when, and by whom? What are the most efficient methods of presenting mathematics to engineering students?

A reader of the summer 1908 issues of Science will find a series of five articles—written by a group of distinguished mathematicians and engineers—addressing these very questions. These articles grew from a joint session of the Chicago section of the American Mathematical Society and the American Association for the Advancement of Science held at the University of Chicago in December 1907.

For a modern reader who has experience with mathematics courses for engineering students, the topics discussed in those century-old Science articles may seem surprisingly current. Their scope greatly overlaps with topics central to current global discussions about the role of mathematics in education.

Perhaps the strongest message that emerges from the 1908 articles is that teaching mathematics is about increasing students’ ability to think mathematically as outlined in “The teaching of mathematics to students of engineering” by Ralph Modjeski, J.A.L. Waddell, Gardner Williams, and Arthur Talbot:

The essential thing that we ask of mathematics is that it should develop the quantitative reasoning power, and the student must be able to think mathematically.

But what is mathematical thinking? Does this term have the same meaning for today’s students and instructors as it had even a few decades ago?

## Defining mathematical thinking

Searching for definitions or conceptualizations of mathematical thinking is futile—not because they do not exist, but because there are too many of them. Nevertheless, the definitions have many points in common and agree on an important idea: It is the processes and activities that constitute mathematical thinking that are most important.

In the mid-1980s, Leone Burton wrote “Mathematical thinking: The struggle for meaning,” which described mathematical thinking as “operations, processes, and dynamics.” Operations include enumeration, iteration, relationship (such as symmetry), and transformation, by combination or substitution for example. Processes are specializing, conjecturing, generalizing, and convincing. And the dynamics of mathematical thinking are a purposeful mix of specific, repeated, re-iterated, and applied operations and processes.

This view of mathematical thinking is echoed in the notion of “mathematical habits of mind,” which emerged in the 1990s as a response to the question: “As our society changes, and often in unpredictable ways, what mathematics should we teach?” In their response, the authors of “Habits of Mind: An Organizing Principle for Mathematics Curricula” suggest that the essential question is “how” rather than “what” to teach. They propose that, regardless of its topic, math course curriculum should be organized around the elements of mathematical thinking: experimenting, pattern-sniffing, visualizing, conjecturing, and so on.

What these two conceptualizations have in common is an attempt to describe what mathematicians and those using mathematics actually do, in contrast to passive messages that students get when it is suggested they accept mathematical facts based on their face value.

We believe that, since they contain the essence of how mathematics has been discovered, created, and applied over many centuries, the conceptualizations in “Mathematical thinking” and “Habits of Mind” may be comparable to what mathematical thinking meant to the participants of the 1907 Chicago meeting.

## Computational thinking and experimental mathematics

Since the mid-1990s, modern technology has dramatically changed how mathematics is done and how it is applied; from a research setting, to business and industry, to completing everyday tasks. In our view, these changes have influenced the way modern mathematicians, scientists, teachers, and engineers perceive the notion of mathematical thinking.

The term “computational thinking” was popularized by Jeannette Wing with the aim to describe the educational ecosystem that includes computer programming (“coding”) as one of its species. It has been defined as “the thought processes involved in formulating problems so their solutions can be represented as computational steps and algorithms.” Wing argues that computational thinking complements and combines mathematical and engineering thinking. Shesuggests that, “To reading, writing, and arithmetic, we should add computational thinking to every child’s analytical ability.”

Related to computational thinking is David Bailey and Jonathan Borwein’s conceptualization of experimental mathematics. In “Experimental Mathematics in the Society of the Future,” they describe it as a computationally assisted approach to mathematics that enhances the processes of:

1. Gaining insight and intuition;
2. Visualizing mathematical principles;
3. Discovering new relationships;
4. Testing and especially falsifying conjectures;
5. Exploring a possible result to see if it merits formal proof;
6. Suggesting approaches for formal proof;
7. Replacing lengthy and error-prone hand derivations; and
8. Confirming analytically derived results.

Although Bailey and Borwein refer primarily to research in mathematics, their list can be used to describe modern and future ways of teaching mathematics at any level, including elementary education. It does not take much to realize that this conceptualization of experimental mathematics has many points in common with mathematical habits of mind and with Burton’s description of mathematical thinking.

## Enriching the cognitive toolbox with mathematical sense

There is no doubt that the very act of mathematical thinking is often enhanced through technology-facilitated reasoning. Using technology to predict, gain insights, or check a mathematical fact or to visualize abstract objects has become a routine for many researchers and students of mathematics, as well as for those who, like engineers, need to use mathematics in their jobs (think of modern architecture, for example).

For a mathematician developing a mathematical model whose results might affect people’s lives, computational thinking may be one of the crucial components of their work. But for many mathematicians (as well as other scientists and engineers) and almost all of their students, computational thinking might not be needed to better understand and more quickly solve a given task. To paraphrase Peter Denning in his 2017 paper “Remaining Trouble Spots with Computational Thinking,” the mathematician (be it a student, teacher, or researcher) is judged not for skill in computational thinking, but for their knowledge, ingenuity, esthetics, and vision.

Why do we emphasize computational thinking? Because, as Wing proclaims, it has the capacity to enrich our cognitive toolbox. If computational thinking is to be truly integrated into a mathematics curriculum, and not just added to it as an afterthought, it needs to establish its identity and presence.

Going back a century, we read the advice given in “The teaching of mathematics to students of engineering”:

It is not mathematical skill so much as a mathematical sense, or mathematical common-sense, which is wanted.

And, for us, mathematical common-sense embraces all of the above: mathematical thinking, experimental mathematics, and computational thinking.

Andie Burazin is an assistant professor in the Robert Gillespie Academic Skills Centre and Department of Mathematics and Computational Sciences at the University of Toronto Mississauga. Veselin Jungic is a Teaching Professor in the Department of Mathematic at Simon Fraser University. Miroslav Lovric is a professor in the Department of Mathematics and Statistics at McMaster University.