# An alternative to "two column" geometry proofs

I'm a high school teacher in New York State (US), starting in on my first year of teaching Geometry. One of the things that really intrigues me is that the Regents exam (the state-mandated final exam) accepts a graphic organizer for proofs that is different than the traditional two column Statement-Reason style that I learned. Here, for instance, is a fully correct model response in a recent exam.

My colleagues and I are really intrigued with this style and the potential that it has to make the "flow" of a proof more visually intuitive for students. However, I'm afraid that I don't even know what this style is named, much less research on whether it helps students master skills in proof-writing and strategies for teaching it in a class. Does anyone have any leads that could point me in the right direction?

• It looks awful to me. But that's just me... Sep 1 '19 at 5:33
• @Sue LOL, it's probably not just you. I definitely prefer it to no response, which sounds like a genuine risk with my student population. Sep 1 '19 at 5:44
• Yes, I would prefer this (mess) to no response also. In the course I taught this summer, I tried to model proofs in paragraph form. My proofs kept looking more like 2-column, even though I didn't think I liked that format. (I taught geometry for the first time this summer.) Sep 1 '19 at 6:07
• I bet that's roughly what you write on scrap paper/in your head when you work out the proof. This shows one of the two steps of proving: working out the proof, then writing it up formally. Since many of my students struggle with the first part, and one reason for that is they expect to be able to work out and write down the perfect proof in one straight line in one go, I think this is a great idea. It might help them learn how to solve problems, rather than just run algorithms. Sep 1 '19 at 6:29
• My definition of "proof" is "an argument which can convince your peers". Since a proof is an argument, it should be conveyed in natural language. If you are working in English, that means English sentences supported by some computations and drawings which explain your reasoning. The size of "gaps" in the reasoning which are acceptable vary depending on the intended audience. I think "2 column proofs", or any other sort of rigidly structured approach do a great disservice to mathematics education. Sep 1 '19 at 14:22

This is an example of what is usually called a flowchart proof (or sometimes a flow proof for short). A quick Google search for "flowchart proof" or "flow proof" shows many, many contemporary examples of the form, including a whole genre of YouTube videos teaching this style of presentation.

This style of proof has been promoted at various points since the 1960s. As it happens, way back in 2006 (when I was a grad student) I once prepared an annotated bibliography of articles from The Mathematics Teacher on proof and proving; one section of that bibliography was about "Articles proposing alternatives / modifications to the 2-Column Form", and a subsection of that was "II. Articles about 'Flow Proofs'". I am reproducing the entirety of that section below (with a few minor edits and elisions for clarity). (In what follows, MT is an abbreviation for Mathematics Teacher.)

It seems that the "Flow Proof" gets rediscovered every eight years or so:

Ness, H. (1962). "A method of proof for high school geometry". MT, 55, 567-569.

Thorsen, C. (1963). "Structure diagrams for geometry proofs." MT, 56, 608-609.

Hallerberg, A. (1971). "A form of proof." MT, 64, 203-214.

McMurray, R. (1978). "Flow proofs in geometry." MT, 71, 592-595.

Basinger, D. (1979). "More on flow proofs in geometry." MT, 72, 434-436.

Brandell, J. (1994). "Helping students write paragraph proofs in geometry." MT, 87, 498-502.

A few words about the articles above: Ness (1962) introduces (is this the first reported case of it?) the flow proof. He begins with a concise summary of the drawbacks of the 2CF, and then goes on to say:

"The School Mathematics Study Group, in its geometry course, introduces the paragraph type of proof presumably to eliminate some of the objections listed above. However, in my opinion, the paragraph type of proof would be extremely difficult for sophomore geometry students, and these proofs are often vague and lack precision. I would like to introduce a method of proof that we tried last year as an experiment... that I believe eliminates the difficulties of the two-column proof without introducing the vagueness and lack of precision of the paragraph proof."

Ness then goes on to give an example of the same proof, done both in 2CF and as a flow proof (although he does not name his 'method'). Curiously, his flow proof consists solely of what are normally called "statements"; there is no space in his diagram for the "reasons". Curiously, he seems not to notice this – at least he makes no mention of it.

Thorsen (1963) is a response to Ness (1962). Apparently Thorsen had also been using an essentially identical format, the only difference being that Thorsen's statements are enclosed in boxes ("balloons"). Like Ness, Thorsen does not include reasons in his flow proofs, which he calls "structure diagrams". Thorsen cautions against using structure diagrams instead of a 2CF; rather, he uses them as a planning tool, to help understand the logic of an argument, which still needs to be written as a 2CF "which includes 'authorities'" (i.e. reasons). The structure diagram thus serves as a scaffolding device only.

Hallerberg (1971) re-discovers the flow proof independently (?) eight years later. His flow proofs are extremely formal in nature; nodes in the proof correspond to implications (of the form P -> Q), with incoming edges representing known assertions (P) and outgoing edges representing deduced conclusions (Q). Thus Hallerberg, like Berger, Schacht, and Shields, makes the law of modus ponens the central feature of his format. Hallerberg's flow proofs lack the readability of Thorsen's and Ness's, largely because his diagrams consist solely of code labels ($$S_n$$, $$G_k$$, etc.) with a "legend" alongside the diagram decoding the symbols. The result might please a logician but is manifestly classroom –unfriendly.

Two remarkable features of Hallerberg's article are that he makes no claim that this has worked well in the classroom (I believe his article is unique in not making this claim), and that he offers examples of non-geometric proofs as well. He also addresses using the flow proof with indirect proofs.

Flow proofs are rediscovered once again in McMurray (1978). It is here that the name "flow proof" is first (?) used. McMurray's flow proofs consist of statements, joined by arrows; each arrow is labelled with a number, and under the flow proof the numbers are associated with "reasons". As in Hallerberg, McMurray stresses that the law of modus ponens (which he calls the "law of detachment") is made manifest through this representation.

Basinger (1979) is a response to McMurray (1978). Basinger reports that he has also used flow proofs, with a few key superficial differences in notation. Interestingly, Basinger writes "The flow proof format does seem to show graphically the 'flow of logic' in complicated proofs. But overemphasis of this proof format may be just as bad as over-emphasis on the traditional formats... It may be better to use several proof formats, with near-equal emphasis on each of them... And certainly the paragraph proof should not be left out."

My final paper in this section is Brandell (1994). Despite its title ("Helping students write paragraph proofs in geometry") the paper has much more to say about flow proofs than about paragraph proofs. In fact, Brandell's approach to teaching paragraph proofs seems to be: (1) First, make a flow proof. (2) Then, turn the flow proof into sentences and paragraphs. This approach seems amazingly backwards to me. The whole point of paragraph proof is that it's less rigorously formal, more like natural language. (Isn't it?) To teach paragraph proof as just a highly-condensed form of flow proof is just weird. Brandell's flow proofs have statements in boxes, linked by arrows, with the reasons labelling the arrows. Interestingly, he suggests as a learning tool that unnecessary statements could be included in the flow diagram; when the paragraph is finally written, those extraneous statements are jettisoned.

• And a comment: coincidentally, the last author in that list (Joe Brandell) was a teacher at West Bloomfield High School in suburban Detroit, which happens to be the same high school I graduated from in 1989 -- although he was not my teacher, and at the time did not teach math; I knew of him as "Coach Brandell", and only later did he become a math teacher, and eventually department chair. Sep 1 '19 at 15:57
• I disagree with your last paragraph, that writing a flow-proof before a paragraph-proof is backwards. I would say it is very sensible. Paragraphs force any argument to be written as linear, regardless of whether it is or not. That makes it harder for students to understand, partly because it becomes less algorithmic. They may well worry about which bit to do first, even though in reality it doesn't matter. A flow-proof deals more naturally with the non-linearity. Indeed, I would say a flow-proof is actually closer to what a proof is, and the paragraphs are enforced by tradition. Sep 2 '19 at 7:19
• @JessicaB I don't entirely disagree; I wrote that paragraph 13 years ago and I'm not sure I know what I was thinking about at the time. What I would say now is that a well-made flow proof does not really need to be rewritten in paragraph form, as (if it is complete and clear) the flow proof itself should be adequate, and may even be preferable, for precisely the reasons you mention. Sep 2 '19 at 16:35