Meta-Questions and Educational Success

Abstract

“It seems that students who are hard-working and otherwise successful, but whose milieu (peers, mentors, and home environment) are at once 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.”

In this article, I will explore this conjecture, and suggest a potential course of action, rooted in the methods of inquiry of philosophy and the humanities, that may help reverse an otherwise tough pedagogical situation.

Some Observations

In students who are otherwise quite successful, difficulties with mathematics and science often seem to boil down to what are essentially meta-questions — of more philosophical or human interest than technical problems. In these otherwise good students, unaddressed meta-questions seem to cause greater lasting damage than one might expect. In the mildest cases, we have good students well able to perform computations 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. Though I have less evidence for the situation in the hard sciences, I suspect the same to be true, especially in physics and chemistry. The unfortunate departure from mathematics and science subjects by students who are otherwise capable, engaged and hard-working, seems particularly tragic.

The Origins of a Conjecture

The ideas behind this conjecture have taken shape over some time now as I have mentored an increasing number of secondary and post-secondary students. The ideas were strengthened the more I encountered good students who were turned off to mathematics, but who, after being satisfied on what were essentially meta questions about mathematics, seemed to feel less alienated from the goals and methods of mathematics. The questions themselves were often more of a philosophical or humanistic nature. Nonetheless, it was these questions that seemed to have created a hurdle that was hampering the students’ ability to freely absorb new mathematical ideas. For some, there was renewed energy and effort in mathematics, often accompanied, in these cases, by a step change in attitude, engagement, and performance.

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 that students who are hard-working and otherwise successful, but whose milieu (peers, mentors, and home environment) are at once 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. This disadvantage can be partially addressed by providing an opportunity to philosophically examine the meta aspects of mathematical activity. Addressing this “cultural omission” in a structured fashion, and guiding the student toward greater personal intellectual engagement with the subject can turn potential unhappy departures into continuing and successful science and technology students.

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 in 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. Though reversing this situation is difficult, there are initiatives underway¹ to improve the situation, 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.

Suggestion 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.

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

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