Alienum phaedrum torquatos nec eu, vis detraxit periculis ex, nihil expetendis in mei. Mei an pericula euripidis, hinc partem.

Teaching Mathematics for Understanding – Guaranteed!

by Tom Clark

First and foremost, students shouldn’t be ‘taught’ a lesson. The concepts in a lesson should be experienced… the foundational principles must be laid out and explored, before any procedures are developed.

Rhombicuboctahedron

Rhombicuboctahedron drawn with “open faces” (circa 1496), illustration, by Leonardo da Vinci

When students are engaged in the study of mathematics, you will often hear them say rather blandly, “I understand.” That sounds encouraging, but based on the traditional way mathematics is taught, they are really saying, “I know how to do this.” That is not conceptual understanding. It is mechanical, often a mere procedural protocol. Truly understanding mathematics can be a beautiful and rewarding experience, when you know “why” mathematics works the way it does. And that is what the word “understand” really means: to “stand under” the mechanics, to know what supports the procedure.

Yes, that sounds good, but how do we get to that level of comprehension? In short: we have to teach in the Socratic fashion.

Of course, every subject should be taught that way, but more narrative courses, such as history and literature seem to lend themselves to a “conversational” or interactive approach. Mathematics, by its apparent functional nature, seems suitable to a more mechanical approach. At least that is the common view.

For example, you might have heard a teacher comment, “We teach the concept of long division.” But, long division is not a concept. It is a procedure. Pure long division requires an understanding of what division really means. And, if you do understand what division really means, the phrase “goes into” immediately becomes a meaningless and misleading expression. We must focus on the “why” to guarantee meaningful understanding.

So, what does Socratic instruction entail?

First and foremost, students shouldn’t be “taught” a lesson. The concepts in a lesson should be experienced. In other words, the foundational principles must be laid out and explored, before any procedures are developed. Note-taking should be eliminated (or at least minimized) to allow students to concentrate on, participate in, react to, and question, proposed strategies. Follow-up instruction should include, as much as possible, students “teaching back” the fundamental concepts, to demonstrate conceptual understanding. One of the most powerful learning experiences is teaching someone else what you have come to understand. By teaching another, you have to consider your own thought processes, and then communicate them logically and clearly.

Of course, after comprehension some practice is necessary, so a reasonable number of problems will be assigned, with the requirement that the steps of each solution be demonstrated. And, while the answers should be checked by the instructor, errors provide students with the opportunity to reconsider and try again, thus learning from mistakes.

By contrast, most of us have heard an instructor say, “You missed that problem. Go do it again.” Which translates into: “You missed that problem. Go miss it again.” Solution manuals can serve as teaching tools, so that students come to learn from every attempt to solve the problem, step by step. Moreover, in the process of error analysis, students develop the capacity to explain, either verbally or in writing, how the error was made—and, most importantly, how a later solution was uncovered.

This Socratic approach equips students to take more responsibility for their learning, increasing their confidence and motivating them to undertake the further exploration of mathematical reality, just as Socrates did with the young boy in Plato’s Meno.

This instructional method requires students to take an active role in learning, for it assumes that every student should have the thrill of exploring and solving mathematical problems, a liberal art that continues to be passed down through the generations.


Tom Clark is a teacher, trainer, consultant, and the author of VideoText Interactive, a mathematics program (www.videotext.com) sold by Memoria Press.

In the modern era, mathematics has arisen to become the international lingua franca of science, enabling empirical researchers and scientists to symbolically represent all types of patterns.

Doing mathematics properly requires one to think in a certain way about a certain sort of thing (mathematics), and eventually one becomes familiar with it and how it tends to go.

A proof is a movement of logic that unfolds as a narrative. It is incumbent upon the mathematician not only to know the truth, but also to communicate the truth to an audience. Therefore, the very act of proof writing is also an act of rhetoric.

Edna St. Vincent Millay (1892–1950) was one of America’s most talented 20th century poets, whose brilliance as a sonneteer has secured her place in numerous collections of children’s poetry.

This Socratic approach equips students to take more responsibility for their learning, increasing their confidence and motivating them to undertake the further exploration of mathematical reality, just as Socrates did with the young boy in Plato’s Meno.

Practical application often motivates such beautiful abstract mathematics; while, at the same time, abstract thinking often motivates practical mathematics. It’s time to share that side of the coin and how both sides work together.

About Virtue Magazine

Virtue is the flagship publication of the Institute for Classical Education. It disseminates stories, ideas, research and experiences in classical education to readers across the nation, helping them pursue the classical ideals of truth, goodness, and beauty.

Subscribing to Virtue’s mailing list is absolutely free. Sign-up today to receive your first copy!