The Importance of Non-Technical Questions to Successful Mathematics Education

Students who are hard-working and otherwise successful, but whose peers, mentors, and home environment are mostly non-technical and disengaged from the ideas behind science and technology, are at substantially higher risk of disorientation, dissatisfaction, and disillusionment with mathematics and science.

In this article, I’ll develop this conjecture and suggest an approach that incorporates philosophical and humanistic elements into technical subjects. To reach and engage a broader popluation of students is critical if mathematics education is to directly contribute to the technical (& technological) literacy of a broader population of students.


Some Observations

In students who are otherwise quite successful, difficulties with mathematics and science sometimes boil down to what are essentially meta-questions, i.e. questions that are about mathematics as a subject itself rather than about its content. These questions are often of more philosophical interest and don’t fit within the syllabus of most courses offered within a typical mathematics department. For example: “What is mathematics really about?” “Why does mathematics seem to work?” “Why bother to learn mathematics at all?” “Is there any use for mathematics?” “What’s the point of this?”

In these otherwise good students, unaddressed meta-questions such as these seem to cause greater lasting damage than one might expect. In the mildest cases, we have good students able to perform specific computations adequately but who have missed the essential ideas underpinning the material at hand; in the severest cases, otherwise good students dismiss mathematics as inaccessible, irrelevant, or both. The resulting departure from mathematics and science subjects by students who are otherwise capable, engaged and hard-working, seems particularly tragic. (Though I have less evidence for the situation in the hard sciences, I suspect the same to be true, especially in physics and chemistry.)

Why I Believe this to be True

This suspicion has strengthened over time as I mentored secondary and post-secondary students as well as recent graduates. All too often when I encountered a good student who was turned off to mathematics, it seemed that if they could be satisfied on what were essentially their particular meta questions about mathematics, their sense of alienation from the goals and methods of mathematics seemed to diminish, and in more than a few cases, after a time they came to appreciate the kinds of questions mathematicians ask, and showed a renewed willingness to resume studying the technical content.

Why is this? It seemed that their questions were more philosophical, the kinds of questions that people ask before they engage in something for pleasure or for their own personal benefit — the very reasonable questions that guide the free direction of effort. But when these questions were unexpressed and therefore unanswered, they seemed to have created a hurdle that was hampering the students’ ability to freely absorb new mathematical ideas.

Encounters such as these prompted me to reflect on my own pathway through secondary and higher education. Certainly, I can trace my continuation in mathematics and science to positive encounters with key mentors and structured experiences allowing the active pursuit of engaging mathematics. Curious to see how others who stayed in the field came to be there, I followed up these informal investigations with industry and academic colleagues. The results of the conversations suggest that there is perhaps a common thread that runs through a specific class of difficult educational experiences.

The Conjecture

Here’s what I currently believe:

It seems to me that students who are hard-working and otherwise successful, but whose milieu (peers, mentors, and home environment) is both non-technical and disengaged from the ideas behind science and technology, are at a higher risk of disorientation, dissatisfaction, or disillusionment with mathematics and science.

But this disadvantage can be addressed.

The Why’s and Wherefore’s

The rather sophisticated perspectives that are required in order for students to satisfactorily resolve for themselves the most common meta questions in mathematics are frequently absorbed, almost by osmosis, when the student’s milieu has within it mentors with this so-called “cultural knowledge”.

The important exchanges typically happen informally during general conversation with mathematically or technologically literate parents, uncles, aunts, or mentors within the student’s community. Or they happen silently if the student has had the good fortune to observe or participate first-hand in how mathematics or mathematically related concepts are used in areas that they find interesting, be it physics, technology, computer games, artificial intelligence, robotics, digital graphics, social research, monetary policy, or business.

For students whose environments lack the opportunity for such interactions, the typical school or even university curriculum is not well suited to compensate for the lack. For such students, there is little further opportunity for meta questions to be addressed, putting them disproportionately at risk, with the expected disengagement and lack of success in science and technology education.

This disadvantage can be partially addressed by providing a structured opportunity to philosophically examine the meta aspects of mathematical activity. Addressing this “cultural omission” in a structured fashion can guiding the student toward greater personal intellectual engagement with the subject and can turn unhappy potential departures into continuing, successful science and technology students.

Several national initiatives are being explored to reverse this difficult situation.

In the UK, the recent Mathematical Needs Conference held by the Royal Society’s Advisory Committee on Mathematics Education (ACME) led to some fruitful discussion, recognizing as they do that improvement in this area is crucial both for individual student success and for the success of science and technology education.

In particular, the working group on “The mathematical needs of higher education” explored obstacles to student engagement with mathematics in the UK and how these could be reduced. (The results from this working group and the findings from an extensive research project sponsored by the Mathematical Needs committee of ACME and led by Huw Kyffin will be contained in a forthcoming report, to be available from the ACME website.)

Further Action

The Course Reading Materials provided here as part of a Course on the Philosophy and Foundations of Mathematics, are intended to address that portion of students who are attracted to Science, Technology, Engineering, and Mathematics (STEM) subjects, but whose background may not have included the cultural pre-requisites required to resolve common meta questions without supplementary sources. Making these cultural assumptions and perspectives explicit for these students will hopefully lead them to be able to study further and with greater urgency in both theoretical mathematics and its applications than they might otherwise do, and by doing so, enable them to continue more confidently and comfortably with their science and technology education.

In each of these contexts, the primary intent is to head-off the otherwise inefficient blind search for this cultural perspective early enough in the students’ technical education so that they better understand the origins and meaning of what they are doing, why mathematics has evolved in the way it has, and have a useful sense of perspective to ensure that the forest is not missed for the many, many trees they will encounter. The secondary intent is to help catalyze the development of that elusive “mathematical maturity”, a characteristic often desired by instructors but unfortunately less frequently found.

It is my opinion that a course such as this would be fruitfully offered initially as an elective, but then possibly required for undergraduate mathematics majors as well as beginning graduate students in mathematics, preferably during the first year of graduate school. With appropriate modifications, an approach such as this may be useful for talented secondary students in an abbreviated, colloquium setting run in tandem with faculty from a university mathematics department interested in school partnership and outreach.

Where the humanistic questions around mathematics are addressed, implicitly or explicitly, one often sees energy and effort in mathematics. When this is the result of an intervention, this frequently causes a step change in attitude, engagement, and performance.


>> Continue reading: A Course in the Philosophy and Foundations of Mathematics


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