Than Two-Column Proofs

Flow Proofs

Are More Logical

Than Two-Column Proofs

By

Lanis Lenker

Wesclin High School

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" computer-based 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 dàx 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

dàx

zàq

eàp

này

jàm

pàf

kàt

aàj

kàz

fàh

xàe

qàk

càg

màd

Given: m is true

Prove: n is true

bàx

aàg

tàc

aàw

bàs

tàh

xàr

màr

uàn

wàb

tàa

ràa

aày

bàr

càu

xàt

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.

Given: t is true

Prove: b is true

yàe

aàw

pàb

yàg

dàm

càm

wàa

yàa

pàd

mày

yàd

tàd

tàz

hàb

wàc

pàh

xàb

càp

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

tàm

hày

màg

pàc

zàd

pàw

kàd

zàm

pàh

gàa

màz

aàw

hàp

càt

tàg

aàh

càa

màp

hàd

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 two-column 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-gramà ______? 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?

______à 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…