What is Math? Why Do I Care What You Think?

By Yevgeniya Rivers @UNHMathZone @UNewHavenMathZone

Some years ago, at a one-on-one meeting with my previous department chair, I was asked "what is math?" My first reaction was not to answer the question but to wonder why I was being asked in the first place. I produced some word vomit to the effect of “the language of the universe; a way to explain nature using commonly agreed upon terms and symbols.” It was a start, but it didn’t feel quite complete.

Despite not knowing the intentions behind the question—the details of which I have yet to uncover—I thought it worthwhile to ask those around me the same question. Would their answers differ from mine? If so, how?

I started by asking a group of undergraduate and graduate peer tutors during a training meeting in the spring of 2019. For background, these are (mostly) STEM major students who (mostly) excelled in math. The peer tutor role requires helping students in introductory math courses in a way that aids them in a) building comprehension, and b) lessening their fear of math.

Peer tutor responses are below.

A way to obtain a numerical values [sic] for physical quantities as well as to obtain expressions to explain simple processes
A set of reasonings and guidelines which apply to the science behind our physical realm
A set of processes that humans have invented in order to understand the physical world around them
Equations and patterns used to determine and measure certain values pertaining to life itself
Math is whatever you want it to be. For me, math is FUN! (heart, smiley face)
A language designed to describe natural phenomena in an efficient way

In the same tutor training session I asked, "what do you love about math?" I figured this question would evoke a different kind of response, a more emotional one, but still a useful point of reflection. The tutors could then think about their responses and how their views might influence their practice. For example, a tutor who likes math because it gives them a toolbox for solving real world problems might help a student with a question by providing more context to show the topic’s relevance. Similarly, a tutor who likes math because they view it as a universal language might help a student by breaking down a seemingly gargantuan application problem into manageable chunks, just as one might do with learning text in a foreign language.

What I like about math is how methodical it is, and that once I understand patterns, I can find a solution. In one word, it's satisfying
The hidden laws of nature (it's like magic)
A diverse field
A universal language that has been the same through time
Ability to solve real-world problems
Can do problems in different ways
Most of the time, one definite answer

You can hear the echo of "universal language" in response to both questions. Since I did not attribute names to responses, I am unaware of whether these were given by the same individual. However, I find striking the statement "...that has been the same through time" which seems to suggest a static nature to the discipline.

For years, I have taught introductory math courses to college students and prior to that, high school students. My experience has taught me that math fears are far more prevalent than I could have imagined and that these fears are borne partly out of prior experiences with the subject (poor teaching, poor curriculum, standardized assessment, attitudes, etc.). I have also found that students' opinions of the subject will determine how they engage with math. Are they keen on "getting it over with" or on satisfying their curiosity? I was convinced that this was initially a binary choice. Moreover, I believed that when they entered the classroom that first week, they had already picked a side, consciously or not.

Yet my optimism suggested that students' mindset—when combined with course activities—could allow for shifting between the two spaces. Perhaps if I curated the activities such that the overarching goal was to pique student curiosity, they would occupy the curiosity space more and venture out of the task-oriented mindset of "getting it over with."

You may be asking yourself, why does their mindset matter? If they are willing to work hard and get the work done, then isn't that enough? Maybe. I'm not here to change anyone's mind. Personally, if I can help a student build intrinsic motivation for math exploration then I will feel more successful as an educator. After all, it is largely their motivation that will help them persevere through academic challenges, including those encountered in my class.

I returned to the question “What is Math?” in August of 2022. It was the first question I asked after our "syllabus quiz show" on the first day of classes in my college math and college algebra sections. Their faces made it clear that they were both shocked that I would ask such a question and anticipating I had an idea of a “right answer” that they were expected to guess.

I asked them twice, once at the beginning of the term and once again about halfway, as a sort of assessment of how the class was going. If I was making any real impact, then their responses—and perhaps mindsets—could shift.

Below are two-word clouds representing the responses of 25-26 students across two classes. Not all students who were present participated.

At first, the images seem similar. Numbers, equation, and problem feature prominently in both. Upon closer inspection you find the figure on the right contains some new terms that were either not present or not prominent in the first figure. For instance, the responses in week six featured words like understanding, real, different, world, can, solve, and find. Overall, it seemed the responses in class one were more noun-oriented while responses in week six were more verb-oriented. While I cannot tell if this implies any mindset changes from week one to week six, it is interesting that there was a shift in the parts of speech used. Perhaps this means they now think about math in a more practical rather than theoretical way. Perhaps the prominence of the word world suggests they recognize the ubiquity of math. It is not clear. And perhaps it does not need to be clear.

In sum (pun intended), I am glad I conducted this exercise with my students and that I started this discussion early in the semester. I learned a lot from their responses. Our whole group discussion revealed their mindsets and provided me with a good starting point for planning my weekly lectures and semester long activities.

I look forward to hearing what my students say next semester. I also want to hear from you. Because I care. So, I ask you: What is Math? Answers should vary!

Yevgeniya Rivers has been an educator since 2007 and a first year math coordinator since 2014. She seeks to empower students by helping them build a growth mindset, overcoming their math (usually test) anxiety, and helping them discover the importance of math in their lives.