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.
