# Research on the use of outlined / structured proofs in instruction

Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs?

For the purpose of this question:

• A "structured proof" is something akin to what Lamport described in How to write a 21st century proof. I don't necessarily want something strictly as what he defines, but you should keep in mind a proof format where a proof is broken down to smaller steps, each step individually justified, with the whole proof presented in a way that the hierarchical nature of the statements is made clear. (Traditional proofs often do invoke lemmata and propositions, but the presentation is not as organized.)
• An "outlined proof" is a proof given in several iterations. At the first iteration a sketch of the main proof is given. At the second the sketch of the proofs of the various steps using in the main proof is given. And so on. So if you think of a proof sort of as a tree-like data structure, a structured proof presents it "depth first" while a outlined proof presents it "breadth first".
• A "prose proof" is what we see in most textbooks or journal articles.
• Higher level mathematics education I refer to "Intro to proofs" and beyond in a US system. This would be proof-based courses that often leads to the completion of an undergraduate degree in mathematics.

Motivation: anecdotally I have observed that students often struggle when doing proof writing because they have no grasp of the "big picture". This either manifests in a written proof that gets derailed (the student losing track of what he or she is trying to prove) or is circular (the student losing track of where they are in the proof). The students often cannot concisely explain what they have learned from reading a proof.

Part of this I think can be safely blamed on the non-transparent nature of traditional proof writing and reading. If you give me a random sample of three consecutive sentences from a published mathematical proof in a journal, out of context, most likely I will not be able to tell you whether there are any logical relation between the three of them (is one justifying another? Are they three independent conditions?) One may imagine making the proofs available to students in a more structured format, and making the students practice writing some proofs in the same format, can make them better at the mathematical craft.

Have people seen anecdotal/empirical justification for this? Are there published research concerning this aspect of mathematical teaching?

• I am very interested in answers for this question. Seriously considering using the format. Commented Feb 20, 2016 at 1:38
• Not answering the question, but I have an article discussing about taking the prose proof to its maximum potential: Making concrete analogies and big pictures Commented Jul 28, 2018 at 18:00
• I only have anecdotal evidence, but it is extremely strong: Every student I have seen who actually learns to use a good Fitch-style deductive system for FOL such as this one always reaches a point of 100% crystal-clear understanding of the logical reasoning needed in mathematics. The system is crucial; other deductive systems are almost all useless for mathematical pedagogy. Feel free to ask me more if you're interested. Commented Aug 29, 2022 at 16:20
• This is slightly off topic, but IMHO you can write either clearly or incomprehensibly regardless of the particular style you choose (for very formal and structured but extremely hard to follow text, see Federer's book on Measure Theory). What matters is not the formalization itself, but the ability to convert from one representation to another (both ways). It requires training in both styles. I personally think in pictures but my adviser thought in symbols and it was a good training for me to learn to translate my proofs into his format (though I still think in pictures and won't change that). Commented Oct 19, 2022 at 13:33
• I don't know if this will answer your question, but definitely related to this is Lakatos's lovely little book: "Proofs and refutations" you can find freely online: dl1.cuni.cz/pluginfile.php/730446/mod_resource/content/2/… Clearly, it reflects strongly the views of the writer. But I think he does a decently balanced job of exploring the various aspects of this problem.
– Amit
Commented Feb 7, 2023 at 15:32