Calculation versus writing in mathematics

Question

Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas along with the results of calculation.

In requiring students to write mathematics in full sentences, we often require that they provide justification for what they do. We get a more complete picture of what the students are thinking, and check blind and mindless computation.

I have recently told my students that mathematical notation is a very powerful shorthand that one must earn the use of by demonstrating that one knows why one is doing what one is doing.

Of course, in reality we all compute when working on a problem: we all make formal substitutions according to rules appropriate to a symbolic system. We are not always mindful, in language, of the validity of these substitutions as we are doing them. Arguably, a different part of our brain is engaged when we are carrying out computations than when we are writing them up. In short, we all (I imagine rather often) behave like formalists.

A more serious objection to writing while one works is that mathematical notation is a very powerful shorthand. Using symbolic substitutions provides a great deal of leverage that writing everything down misses. Just think of basic arithmetic with place value!!!

Of course, attempts to reduce mathematics to formal logic a la Whitehead and Russell surround the issue.

Question: Has there been any study of the dichotomy between the mental processes of writing mathematics and calculation?

Writing and calculating just feel so different, I wonder if someone has studied this.

Answer 1

What do we understand as mental processes? All of us (Math teachers) dream of entering the brain of our students, see what's happening and adjusting some connections...

However, the thing is that their brains are a kind of black box for us. So the only way we actually have to be sure they have catched some mathematical concept (or argument, or property, etc.) is to check if they manage to use all of the possible semiotic representations of such objects. What you call writing is just our verbal language, a semiotic register. On the other hand, numerical calculations, algebra, graphic representations, etc. are different semiotic registers. The more registers used, the better. All of this is one of the principles explained in The onto-semiotic approach to research in mathematics education (Godino, Batanero & Font, 2007)

So, answering to your question, I'd say that instead of a dicotomy, writing and calculation are complementary.

It's also interesting F. Spagnolo's approach (Semiotic and hermeneutic can help us to interpret teaching/learning?). In that article, he shows how to analyze a didactic situation from the semiotic point of view. Which registers are employed and when to express the ideas? The natural tongue? A symbolic representation? Algebraic language?

Both references will lead you to more bibliography.

Answer 2

For all of the community college algebra classes I teach, I certainly make proper mathematical writing the number one priority; which is not to say that I have students compose everything in English writing. It's already an overwhelming challenge for students to get the algebraic grammar and syntax right, such that I already feel like there's not enough time to properly exercise even that skill. So my focus is to say that our algebra class is equivalent to a foreign language class; I assume they know English (or something), and now we're interested in translating those ideas into properly-formed mathematical statements (algebraic equations), with short side-justifications as appropriate.

I'd say the fact that the mathematical transformations are more concise and convenient isn't something that should be resisted; it should be embraced as the whole raison d'être. Here's one of my favorite quotes on the subject by Alfred North Whitehead, in An Introduction to Mathematics (1911):

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that … a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility … Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

Answer 3

Another thought occurred to me regarding this, after having read Reuben Hersh's collection of essays. There is a quote of Bill Thurston, which I paraphrase as "thinking is the same as seeing". In a sense, having the logic of one's proof "at the tip of the tongue" produces a sense of unity in the proof that is analogous to an object one can mentally manipulate. This sense is like the reification phenomenon observed by mathematics educators: the proof process is "reified" into an object (the proof) that can be thought about as a coherent whole. The process of "reifying" a proof can be greatly aided by mathematical writing that shines light on potentially obscure details that threaten the coherence or consistency of the object...how it fits together.

From this viewpoint, mathematical writing and calculation stand on an equal footing. Both serve as processes to be reified into mental objects. Oddly, our mathematical activity wants to change verbs into nouns. We've all experienced this phenomenon when we sit to calculate something in order to gain intuition or feelings for a mathematical object or theory. The calculation is incorporated in our memory as an aspect or vantage point of the theory. We are looking at a new part of the theory. The attitude here is that whenever we DO something in order to understand a mathematical object, we are LOOKING at another facet of the object. These facets fit together as our mental picture of the object as a whole. We experience this feeling of the object as a whole as mathematical intuition, and get a sense of "what's going on".