While usually C.S. Lewis is treated as an authority in classical education (and for good reason), his experience with math unveils a familiar struggle. He wrote, “I could never have gone far in any science because on the path of every science the lion Mathematics lies in wait for you.” Students as they enter my classroom at the beginning of the year display the same fearful sentiment. Many teenagers think of math not only as an intimidating enemy but also, judging from the pain expressed on their faces, as a means of physical harm. Sounds dramatic, but it’s an appropriate exaggeration.

How could I combat this misunderstanding and very visceral feeling so many students had? Why should I combat it in the first place? I could just take the easier route and seek to produce a path to right answers for my students; however, my classical conviction screamed otherwise, my conviction that the greater goal of learning must revolve less around the right answer and more around right thinking through articulate verbal expression. And in my experience, verbal expression was consistently the key! I could blunder through this (and believe me I did, especially at the beginning), and still trust that submitting to a bigger, more beautiful picture of math education was worth my time and effort.

One particular student comes to mind. He was so quick to say statements like, “I hate math!” or “When will I ever use this?” Now, I’d be lying if I said there wasn’t a part of me that was very offended by these statements. So much so that my petty yet humorous side would come out and say things like, “Well I spoke with math the other day, and it’s not too fond of you either!”Joking aside, these moments of honesty from this student would springboard us into deeper discussion. And the more I conversed with him about math and the more I brought in the eternal truths and lessons about it, the more I would see him surrender to the beauty of it. With that surrender always came the fruit of understanding and articulation…what a gift! These experiences in the classroom led me to desire a deeper study of and commitment to Socratic discussion.

In addition to my work in the classroom the last ten years, in 2021 I completed my Master’s program at the New Mexico Institute of Mining and Technology while writing my thesis, *The Socratic Method: How Using This Classical Pedagogy in Math Education Can Lead to Progress*. While so much of my time and study for the thesis went into finding the practical benefit of Socratic discussions in math education, it is very important to me that I first say that Socratic discussions are propitious and worthwhile because math is a medium to more clearly behold what is good, true, and beautiful. Mathematics is a language. Like any language, it is a collection of symbols that represent truth that precedes its symbols.

My husband and I spent two years in China studying Mandarin. It seemed I could not possibly learn the new language without the use of English to translate. At first, studying a new language was frustrating and uncomfortable. Learning it in light of the language I already knew so fluently seemed impossible. But the harder I worked, the more something beautiful began to happen. I was still mostly thinking of Mandarin in terms of English, but was less and less the more I practiced and learned. This new language was communicating truth and beauty apart from my mother tongue. I have found math capable of this as well.

Many of our modern-day math students (and most people in general) do not think of math as a language. They tend to think of it for its utilitarian purpose solely, its only goal being usefulness or practicality, and too often students are expected to simply mimic their teacher and their textbook. Our job as classical math educators is to help our students approach math as a language to both think and speak. Then perhaps we can see success in mathematics that moves beyond just computation and utility.

Really, there are two types of language: number and word. Through the use of word, we can come to a realization of mathematical language. This means that verbal expression acts as a bridge between a student and math—at least at first, and that Socratic dialogue in its use of word is not only useful for teaching math, but the best way to do so.

Consider the way that Socrates teaches the slave boy a lesson in the *Meno*. Having no prior mathematical training, the boy showcases his ability to reason mathematically. He does this not through Socrates teaching him, but because Socrates patiently uses a connecting point between known language and what is seemingly unknown to expose truth already within the boy. The longer the discussion between Socrates and the boy continues, the more the boy becomes confident in his inherent ability to reason mathematically. This new language is communicating truth and beauty apart from the boy’s mother tongue.

From this example in the *Meno,* I understand the Socratic method as a patient, inquisitive, question-driven discussion that births recollection. Most new educators in the renewal community think of a Socratic discussion as a tool in the humanities, but as we see in the *Meno*, it is wise and good to use this tool in our math classes as well. Our students know these truths that math seeks to display. It’s written on their hearts. A teacher’s job is to guide them in their recollection. Too often in our classrooms, students are expected to simply mimic their teachers or their textbooks instead of being given the tools through language to recollect mathematics rightly.

Teachers who seek to produce right answers devoid of right thinking and eloquent expression engage in a transient endeavor. Simply getting right answers will not stick with a student, but going through the process of wrestling with math and recollecting mathematical thinking will. If teachers teach students formulas to right answers, these formulas will be despised by some students and quickly forgotten by most following graduation. Besides, a person can always look up a formula, but the skill of thinking mathematically has to be taught. This skill will be retained by the student because showing them the beauty of math engages them as a full human being—a thinking, wondering creature. Students will want to remember what they are taught in a math classroom which engages their verbal reasoning skills and their aesthetic sense.

While I do not have an exact methodology that produces a perfect Socratic discussion, I do have some principles I have followed that help guide me when preparing dialectic discussions for my students, fully understanding that Socratic discussion is much more than these principles, but not less. The first principle is to commit to a question-driven discussion that involves posing inquiries to students, prompting them to think about what they already know. This type of questioning encourages students to deduce logical answers with their own thinking instead of being fed the answer by the teacher. The second principle is to take every opportunity to help students uncover problems with their own reasoning. The teacher should plan every concept to be covered in class, but he or she ought to encourage independent discovery whenever possible. This process strengthens student understanding and transforms misunderstandings into an opportunity to persevere and persist in difficulty. The third principle is to promote inductive reasoning, helping students form general conclusions from specific instances.

These three principles fit into the two major stages of a Socratic discussion: the ironic stage and the maieutic stage. The ironic stage happens when the teacher probes a student’s understanding of a concept and shows he or she has inadequacies of thought. The student may feel lost or discouraged, but as the conversation continues the maieutic stage begins to take shape. “To give birth” is a rough translation of “maieutic” in Greek. This stage is when the student gains, or births, a more accurate recollection of the concept through question and dialogue.

For my thesis research, I chose to focus on Socratic discussion during a test review, an opportune time to encourage students to correct any inadequacies of thought and recollect rightly. One common theme I found in my qualitative data—feedback from participating students—was the comradery students experienced with their classmates during a Socratic discussion. Many times, their questions were answered (questions many of them did not even know they had) because they were present when a classmate was able to verbally express what he or she was thinking. Students also shared that the dynamic of a Socratic discussion makes them less afraid to be wrong when they answer questions. This is because they knew there was a process to trust that leads to right conclusions. Many noted how this verbal expression actually makes them more confident in math. Students also spoke about how a Socratic discussion forces one to engage and pay attention, which naturally leads to improved understanding. They spoke of how it provides an opportunity to understand the “why” behind concepts instead of just the process of a concept, and of how a discussion with the teacher provides students with helpful nuances regarding a concept that reading a textbook or answer key cannot provide.

Not only can the benefit of my research be seen in the qualitative data of student feedback, but it can also be discerned quantitatively. My research showed that the test scores of students who received a lesson review using the Socratic Method were 19.6 percentage points greater than the test scores of students who received only a written review. Almost twenty percentage points higher!

It’s hard to quantify something as delicate and linguistic as a Socratic discussion. It may not even be necessary, but I am still pleasantly surprised by my results. Our students want to verbalize the truths they hold dear. They want to be guided in their understanding, rebuked for their misunderstanding, and come closer and closer to grasping and articulating the truth they are recollecting. I appeal to all of you wonderful teachers tasked with teaching math in our classical schools: push through the difficulty of Socratic discussions in the classroom. I have no doubt you will benefit from the sweet communion these discussions bring, as well as the fruit they bear, both measurable in assessment and immeasurable in the forming of their souls.

## 2 thoughts on “Apart from Our Mother Tongue: Math as a Language and the Results of Socratic Dialogue”

William CareyThanks Michelle for writing this. Thinking about how we talk with students about math is super important! Would you feel comfortable sharing some of the sorts of questions they ask that the sort of dialogue they produce?

Michelle StevensHi William. Thank you for your question. One lesson comes to mind to help answer your question. When first teaching students how to solve a quadratic equation, I also made sure to speak about the Fundamental Theorem of Algebra. I didn’t necessarily teach these lessons at the same time, but in very close proximity to each other. In an introductory Algebra class, many students were solving quadratics and getting one solution or no solution at all. This contradicts the Fundamental Theorem of Algebra. Often, one or a few students would notice the contradiction and ask how this could be possible. A quadratic must have two solutions. I loved these moments when individual students had that lightbulb moment and other students, who were perfectly fine just getting the right answer, noticed that their “right answers” were missing a fundamental truth about Algebra. This would springboard us into deep discussion about why these “right answers” were only partial truths and what was missing. It even led us to discussion about how important semantics are, even in math, and how a term like “imaginary numbers” can cause confusion and misunderstanding.

I hope this example helps answer how questions from students can produce dialogue. Thanks William!