Flow Proofs
Are More Logical
Than Two-Column Proofs
By
Lanis Lenker
Wesclin High School
LenkerL@wesclin.k12.il.us
lanislenker@hamiltoncomwb.net
Proof in Geometry
By Lanis Lenker
Should proof be dropped from Geometry courses? Or minimized? I’m from the old school. I vote NO on both
questions.
I understand that proof is not easy for students, but is that sufficient reason to drop or minimize it? Should we
drop or minimize all topics (in any course) that students find difficult? I prefer to stand on the side of the debate
that emphasizes intellectual growth. Algebra II shouldn’t be a re-teaching of Algebra I; it should extend
algebraic knowledge and skills into new territories.
Bloom’s Taxonomy highlights knowledge, comprehension, application, analysis, synthesis and evaluation. Any
fair week in a Geometry classroom will blend all six of these into the fabric of the course. A good week will
highlight and balance all six.
I concede that the majority of my Geometry students will not end up doing geometry weekly, or using
geometric formulas weekly, but almost all of them should benefit from the experience of creating a meaningful
argument in favor of (or opposition to) some proposal at work or at city hall.
I don’t think the issue should be about the inclusion/exclusion of proof in Geometry. The more appropriate
discussion should be focused on making proof more accessible to more students. I don’t like to call it proof
writing; I prefer to call it proof making.
My dislike of two-column proofs comes mostly from watching students struggle with those proofs that start on
one thread of knowledge, and then inexplicably step 6 has nothing to do with step 5. This disconnectedness is
disconcerting to students, and difficult to explain.
I stumbled into Flow Proofs in the early 1990’s in an article in the Mathematics Teacher. It explained an
educational experiment at Carnegie-Mellon University that was field-tested with control groups in the
Pittsburgh public schools. It involved an artificial intelligence-based software package that allowed students to
create flow proofs on a computer screen while the machine monitored their progress. Geometrically legal steps
were allowed, even when they were not helpful to the problem at hand. At the successful conclusion of a
problem, the correct proof-path was highlighted on the screen to show the student the useful and non-useful
work on the problem. When I read that the program had been transported to early Macintosh computers, I
snapped at the chance to try it. (Wertheimer, R. 1990. The geometry proof tutor: An "intelligent" computerbased tutor in the classroom. Mathematics Teacher 83:308–317) I now use the program, the Geometry Proof
Tutor, as an extra credit tool in my Geometry classes, but I like the style of proof so well that I do all of my
classroom work in flow proof style.
Today we are going to investigate the logic of flow proofs versus two-column proofs, with hands-on experience
and an open discussion of the pros and cons of this approach.
I have seen flow proofs in three major styles. The ‘bottom up’ style I use corresponds with the Geometry Proof
Tutor. Some people prefer a ‘top down’ look. I have seen flow proofs written left to right; in order to get them
to fit on a page, the writers usually must use a number for each theorem/definition/postulate they use, and then
make a numbered list of those theorems below the flow proof.
Geometry
Proofs using symbols and the Law of Detachment
In each column the statements have been scrambled. Your job is to find the correct statement order to prove
each problem.
Every column has unnecessary statements, so you are looking for the shortest path to complete the proof.
Use only the statements in each column for that column. No cross-using statements in other columns.
Write the correct set of statements under each column or beside each column. Be sure to start with the given.
Geometry students take note! Statements like dx represent our ‘If… then’ theorems in our book. The
purpose of this exercise is to help you see that putting together of ideas in a correct sequence is the way to
organize your proofs in this class.
Given: a is true
Prove: h is true
Given: m is true
Prove: n is true
Given: t is true
Prove: b is true
dx
zq
ep
ny
jm
pf
kt
aj
kz
fh
xe
qk
cg
md
bx
ag
tc
aw
bs
th
xr
mr
un
wb
ta
ra
ay
br
cu
xt
ye
aw
pb
yg
dm
cm
wa
ya
pd
my
yd
td
tz
hb
wc
ph
xb
cp
This proof is an
example of using the
best theorem when
more than one
theorem has the same
hypothesis. The
wrong theorem leads
to the wrong
conclusion.
More difficulty with
repeated hypotheses.
Sometimes you have
to keep trying other
ideas until you find
the way that connects.
Given: g is true
Prove: d is true
tm
hy
mg
pc
zd
pw
kd
zm
ph
ga
mz
aw
hp
ct
tg
ah
ca
mp
hd
In this packet are several flowproofs, some of them finished, some for us to finish. This first problem is
finished except for keeping the tick marks up to date on the diagram. We will do that as we survey the proof.
This style of flowproof builds the proof from the bottom (the Givens) to the top (the Conclusion). The arrow
tips show the direction of flow. All the theorems are in boxes; the result of each theorem box is above the box.
In problem 2, we will talk strategy and tick mark in advance; then we will fill in all the details to complete it.
I want to try two different strategies on this proof. Again, we will talk and tick mark ahead, planning our
strategy. Then we will fill in the details to make each proof work in detail.
This is a longer, fairly complicated proof. It will work well to show how to ‘convert’ a flowproof into a twocolumn proof. For example, you quickly get the picture that SAS can’t happen first. Other details must be
completed before SAS is ready.
Note that all theorems have an input (I call them feeders) and an output (I call them results or outputs). Some
theorems have only one input; others have two inputs (like Transitive); others have three inputs. The wording
of the theorem dictates the inputs and the output.
In problem 5, we want to prove a proportion. If you study the four segments in the diagram, you notice that
segment SR is ‘out of place’ compared to the others. That provides a strategy: get the ‘normal’ proportion on
the page, and then prepare to make a 1-for-1 switch.
In problem 6, we can take advantage of two semicircles.
Can you ‘see’ why the triangles are congruent? How many feeders does it take to make it work?
What are some of the Pluses?
Train of Thought:
Suppose you are given a perpendicular. The train of thought here could be
Perpendicular leads to right angles leads to 90 degrees leads to complementary angles.
What if you are given a parallelogram. How many theorems do we have that start with
p-gram ___________________________________? Quite a few, I’ll bet.
Suppose you are trying to prove that two lines are parallel. How many theorems did we study that ended with
parallel?
______________________________  parallel
The blank side of that generic theorem gives us some choices of inputs (feeders) we need to look for to make
parallel happen.
Ease of Grading
Did you happen to notice that a good flowproof alternates between theorem boxes and results? Any time
you see two (or more) theorem boxes in a row, you automatically know that the writer left out a result (or more).
Likewise, a student who has two results in a row must have left out a theorem box.
Occasionally you see a theorem box that should have two feeders, but it only has one feeder. You can comment
about the lack of a second feeder and assign partial credit accordingly.
Occasionally you see a theorem box that has nothing to do with the result.
Bad theorem wordings stand out in those theorem boxes.
Minuses?
Most texts use two-column proofs; they don’t teach this style.
Students won’t have any models or samples to learn from in their text.
It looks so different… I don’t think I could adapt to it…