Yesterday I was doing some literature review for an article I’m writing about my inverted transition-to-proof class, and I got around to reading a paper by Guershon Harel and Larry Sowder¹ about student conceptions of proof. Early in the paper, the authors wrote the following passage about mathematical proof to set up their main research questions. This totally stopped me in my tracks, for reasons I’ll explain below. All emphases are in the original.
An observation can be conceived of by the individual as either a conjecture or as a fact.
A conjecture is an observation made by a person who has doubts about its truth. A person’s observation ceases to be a conjecture and becomes a fact in her or his view once the person becomes certain of its truth.
This is the basis for our definition of the process of proving:
By “proving” we mean the process employed by an individual to remove or create doubts about the truth of an observation.
The process of proving includes two subprocesses: ascertaining and persuading.
Ascertaining is the process an individual employs to remove her or his own doubts about the truth of an observation.
Persuading is the process an individual employs to remove others’ doubts about the truth of an observation.
Central to this paper is the question:
How are conjectures rejected or rendered into facts?
This is a really fascinating way to frame the nature of mathematical proof and what it means to do mathematics in the first place: as a back-and-forth tension between certainty and uncertainty. The clarity of this description helped me reflect on some thoughts I’ve had about proof and teaching students how to prove. Here are some of those thoughts:
- At its heart, the process of proof – and one could extend this to the entire discipline of mathematics – begins with observation. And proof without observation, such as what can take place when we just give students fully worked-out conjectures to prove, is at most half the mathematical story. And yet this construction of proofs given fully worked-out statements is the primary way we teach students proof.
- Using the definition above of “conjecture”, it’s impossible to make conjectures without entertaining doubts about the truth of the observations. This is a powerful thing and very hard to teach. Many students have been socially conditioned not even to make observations unless they are already sure of their truth. Therefore to teach students to prove, we have to teach them first how to conjecture, which means we have to teach them how to doubt.
- At the same time, a person who can’t move beyond doubts will never prove anything. They will remain in a world of conjectures. This is also hard for students – doubt is hard, but so is certainty. Most students are equally uncomfortable with each. A socially acceptable indeterminacy about things, accompanied by an emotionally felt truth, is a much more comfortable place for most students. But mathematics somehow demands we reject both casual uncertainty and warm-fuzzy conviction and replace them, respectively, with principled skepticism and rigorous belief.
- It seems obvious that “ascertaining” has to take place before “persuading”, and I’m sure that those two concepts are given in that order in the paper on purpose. But a key misstep of most students in learning how to prove – really in learning how to solve any problem at all – is that they skip over the ascertaining step and jump straight into trying to persuade other people of the truth of a statement that they themselves don’t believe in. Getting students to go completely through both stages in an intellectually honest and patient way is a key challenge for transitioning to proof.
- If the previous point is true then it’s no wonder that many solutions of problems, including but not limited to proofs, that are given by novice math students look incomprehensible – symbols sprayed indiscriminately on the page, proof rituals attempted but not done right – because the only way to persuade somebody of a truth that you have not accepted yourself is brute force, to shout down the other person’s thought process. Most solutions like this are really a form of a denial of service attack against the intellect of the reader. The reader’s faculties are spammed so vigorously as to be unusable, and hence the argument or solution is accepted by surrender. Many students make it through high school and even freshman calculus with this approach. But then the targets of this attack get more and more hardened and the old non-mathematical ways of “proving” don’t work anymore.
So based on those five thoughts, here’s a short list of what a professor teaching a transition-to-proof class has to accomplish:
- Convince students that mathematics is about observations and truth, not about computations and right answers.
- Put students in a position where they must experiment and observe, and then have enough confidence in their observations to make conjectures – but not so much confidence that they can’t reserve judgment on the truth of those conjectures until the doubt is removed.
- Get students to be intellectually honest, to the point that they can judge objectively and accurately whether they really believe in something or not – and not merely to “believe” in the truth of a conjecture merely by capitulating because an authority figure said so or because credulity is easier than proof.
- Get students to understand that doubt is not only OK but essential to mathematics – but also that the mathematics is not truly finished until that doubt is truly removed both within oneself and in the minds of all others.
- Teach students to see their hard-fought conjectures, which they have “ascertained”, from the viewpoint of others, so that even though they are convinced of the statement’s truth, they can anticipate what objections others may have.
And of course, the prof also has to teach students that yes, they do have to write in professional English, use mathematical notation correctly, be able to recite definitions and perform calculations correctly, and all that other stuff.
So, in case this wasn’t clear already, transition-to-proof is really hard, and it’s hard to teach. In such a course, students not only have to learn new mathematical content and proper professional writing standards but also come to grips with notions of doubt and certainty that are at the center of humanity itself. It’s no wonder students struggle with this! At the same time, it makes a very good case for why students need to learn proof, because emerging from the other side of struggles with belief and doubt makes a person that much more capable of living a full life.
¹Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in collegiate mathematics education III, 7, 234–282.