# Is there any evidence about the effectiveness of “table proofs” in pre-college mathematics education?

I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol manipulation. We had a big list of axioms and postulates, and proofs were fairly mechanical. They were done by a table, one side was the result, the other was what you invoked to get it

Result                |     Method
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1 Lycanthropy         |     Given
2 A cute bunny        |     Given
3 A werewolf rabbit   |     Flibbergoth's theorem of lycanthropic mammals on (1) and (2)
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∴ Wolf isomorphism    | The harewolf-lupine equivalence axiom on (3)


You can see an actual, non-silly simple example here.

However, I question whether or not this is a good way to introduce proof writing. Certainly the proofs that come out are valid (if done correctly). There's nothing inherently wrong, but even now they feel foreign to me. I don't feel like I really got proofs until I took a discrete math course in college, where proofs were more free, we could choose our own approach, and we tended to write in paragraphs. Certainly we still invoked axioms and theorems and brought up givens, but it was much less structured and, for me, was easier to understand and perform.

(Full disclosure: we did introduce more freeform, paragraph-like proofs by contradiction near the end of the course, but it was introduced as a novelty and never used on any exams or homework).

However, I am not the world. I was wondering if there was evidence to support this method for introducing proofs. Is there any evidence that this improves mathematical reasoning, or is easier for students to learn and execute than more free-form proofs? Certainly it's a lot more similar to the mechanical symbol manipulation most students have been doing up to that point, and I can see an argument that while argumentation and mathematical creativity is still involved, providing the structure to make it more mechanical should make it easier to perform given their likely education up to that point.

Still, I can't shake that I had very little idea what was going on until I got to less structured looking proofs. What does the evidence say? Is introducing proofs in a more structured format helpful? Am I just atypical? I'll grant it's entirely possible I wouldn't have even understood freeform proofs if I hadn't done the structured approach first, but I still have trouble doing the same proofs in table format that I could do easily in a paragraph with sentences interspersed with notation, so I don't know. I've always personally had a more argumentative/wordy approach to doing math (post-arithmetic at least).

Edit: As mweiss indicated, "effectiveness" isn't very well defined. I was trying to get at concepts such as "does the 2-column method help foster an ability to prove mathematical statements, or does it hold back or confuse students more than the more freeform style of proving things?" Overall, I think mweiss attempt to answer the question in terms of whether it provides reasonable structure to guide students or else acts as a straightjacket that stifles their ability to freely think about problems is what I was getting at.

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I'd go with your gut feeling that some (many?) students will see this as meaningless symbol pushing. For me proofs should enlighten and explain –  vonbrand May 1 '14 at 2:02
I've added the "reference-request" tag, since this question asks for specific research and evidence. –  Jim Belk May 1 '14 at 2:20
It might be worth mentioning that people familiar with this kind of proof will be better prepared to use proof assistant software like Coq. I don't really have an opinion about the value of that. –  Neil Strickland May 1 '14 at 7:42
You are not atypical as there is no typicality here. Some people have no problem at all with structure in a proof. The idea of "proving" something is structure enough to them at least for direct proofs. 2CF are a mere formal burden. It can make some steps or methods clearer to them, but not more. But some people, maybe those who don't understand the necessety for proofs, cannot see a way to proving. Giving them a structure like 2CF reduces the task to seeing the next step and some trial-and-error. But I doubt, that it helps all of even those students. –  Toscho May 1 '14 at 9:37
The 2-column format can get said to have variants in abstract algebraic texts and papers, as well as in papers and texts on symbolic logic (where a formal proof gets defined as such a sequence without the reasons... though everyone writes the reasons down also). There do exist things in symbolic logic which almost surely would come as more suspect without such proofs (that Frege's axioms for propositional calculus weren't independent, that certain sets of axioms does form a basis for a known system, that a sole axiom forms a basis for a system). A potential weakness of the method though 1/2 –  Doug Spoonwood May 2 '14 at 23:55

## 1 Answer

The two-column proof form has been the dominant mode of presentation for proofs in secondary geometry in the United States for most of the past century. You ask about its effectiveness; unfortunately, I think that question is ill-posed, because the goal state isn't clearly defined (effective at what?) and so there's no way to measure whatever it is you want to know about.

On the other hand, there has been a ton of scholarship on the two-column form. Rather than attempt to address "effectiveness", instead these studies address the qualitative question, "How is the two-column form used?" For example, here is a paper I wrote with Patricio Herbst and Chia-ling Chen:

Weiss, M., Herbst, P., and Chen, C. (2009) Teachers' perspectives on 'Authentic Mathematics' and the two-column proof form. Educational Studies in Mathematics, 70(3), 275-293.

Here are a few excerpts from that paper, which include citations to other relevant studies for this question:

This paper explores geometry teachers’ perspectives on how the two-column proof form can be both a resource and a constraint in engaging students in proving. The two-column form has been used in the USA to teach students how to do proofs since early in the twentieth century (Herbst 2002a). But the two-column form has been criticized for perpetrating a formalistic image of proving, and calls for reform in the 1980s (NCTM 1989) had recommended that teachers decrease attention to two-column proving. In this article we examine a counterproposal that emerges from listening to teachers: namely, that the two- column form can actually be productive when engaging students in proving, by acting not only a constraint, but also a resource enabling complex and non-linear reasoning.

and from later:

American geometry students often learn to write proofs by way of a numbered sequence of “Statements” and “Reasons” in parallel columns (Sekiguchi 1991). This form of writing proofs is quite different from how mathematicians ordinarily write their proofs (Lamport 1995; Schoenfeld 1988; Wu 1996). The two-column form of “statements” and “reasons” has been the standard for the writing￼of proofs since early in the twentieth century. Herbst (2002a, b; Herbst et al. 2008) has argued that this form, as it is deployed in the high school geometry course, is not merely a special way of recording a finished proof; in common classroom practice the two-column form is both a symbol of, and a tool for, a customary way of engaging students in proving. In most classrooms, a proof in two-column form is written out in a chronological sequence that correlates with the numbered assertions: line 1 is written first, line 2 is written second, and so on. Moreover, the two columns are written in alternating sequences: first the statement of line 1, then the reason for line 1; then the statement for line 2, then the reason for line 2; and so on. This turn-taking structure facilitates the teacher in engaging many students in the construction of a single proof. Thus it is not the case that a proof argument is developed first, and only later recorded in the two-column form; rather, the two-column form becomes a structure that scaffolds the crafting of the argument itself. More than a final summary of the reasoning, it often amounts to a chronological narrative of the development of the argument. For this reason, dispositions important for the working mathematician, such as working backwards, mapping a strategy, or structuring a proof, are poorly accommodated in the way geometry classes tend to use the two-column format (see Anderson 1982; Greeno 1980; Herbst 2002b; Leron 1983; Thurston 1994).

As you may be able to gather from those excerpts, most people who look at the teaching of proof are pretty critical of the two-column form, arguing that it puts too much emphasis on form rather than substance, and that it is an inauthentic representation of what mathematicians actually do when they prove. Given this, one might wonder why the two-column form has stuck around as long as it has. Patricio Herbst traced the evolution of this form and its gradual establishment as a norm over time, and argued that the 2CF made it possible for teachers to hold students accountable for learning to write proofs -- in effect, it solved a problem of teaching. His paper is:

Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283-312.

The main argument we made in our 2009 paper was that, yes, some teachers can and do use the 2CF in a way that inhibits flexible reasoning in the classroom; but others use the 2CF in a way that enables flexible reasoning. The form can function as either a straightjacket or as a safety net, depending on how it is deployed.

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+1 The last paragraph holds the wisdom. –  Toscho May 1 '14 at 9:32