How to Teach Proof Without Two-Column Proofs
Consider the claim: an equilateral triangle has all its angles equal. At one level, this might seem obviously true — you can draw one and measure the angles. But "draw one and measure" is not a proof. It tells you that this particular triangle, drawn with this particular level of accuracy, appears to have equal angles. It tells you nothing about all equilateral triangles.
So you try to prove it. You might drop a median from one vertex, argue that the two resulting triangles are congruent by SSS (three sides equal), and conclude that the base angles are equal. Repeat for another vertex, and you get that all three angles are equal. This feels like a proof, and in most classrooms, it would be accepted as one.
But notice how much this argument leaves unsaid. It does not state that you can always draw a median from a vertex to the opposite side. It does not state that equality of angles is transitive — that if angle A equals angle B and angle B equals angle C, then angle A equals angle C. These are assumptions the proof relies on without making them explicit. A more rigorous version of the same proof would state them. An even more rigorous version would justify them. How far do you go?
This question — when is a proof? — is one of the most important questions in mathematics education, and most mathematics classes never ask it. Instead, they answer it implicitly: a proof is whatever fits the format the teacher expects. In many places, that format is the two-column proof.
The problem with two-column proofs
The two-column proof is a standard fixture of geometry education, particularly in the United States. On the left, you write your statements. On the right, you write your reasons. Each step must be justified by a definition, postulate, or previously proven theorem. The format is tidy. It is easy to grade. And it has very little to do with how mathematicians actually think about proof.
As Weiss and Herbst point out, the high school geometry course is often students' first and sometimes only encounter with proof. And in many cases, it is a caricature of actual mathematics — a course where form triumphs over substance, where there are too many postulates, and where there is a lack of clarity in the meanings of words. The two-column format, as Herbst traces historically, arose from practical considerations in American high school education and gradually became disconnected from the actual process of knowledge construction that proof is meant to serve.
The result is that students come to see proof as a task assigned by teachers — something you do to get marks, not something that helps you understand. Schoenfeld's research shows that after completing their geometry course, students saw empirical methods (trying a few examples) as determining truth, while deductive arguments were just exercises teachers gave them. The proof format was something to endure, not something that revealed anything about mathematics.
This is a significant failure. Proof, at its core, is about understanding why something is true, not just that it is true. It is about seeing the structure of an argument — what rests on what, where the hidden assumptions are, what would need to change if one of those assumptions turned out to be wrong. A format that focuses on lining up statements and reasons in two columns obscures this structure rather than revealing it.
What proof is actually for
I think it helps to distinguish between several things that proof does. One is verification — establishing that a claim is true. Another is explanation — showing why it is true. A third is discovery — revealing connections and structure that you would not have seen without the proof. And a fourth is integration — fitting a claim into a larger body of knowledge by showing how it relates to other claims.
In school, proof is almost always treated as verification. The result is known in advance, and the student's job is to confirm it using acceptable methods. But mathematicians themselves often do not gain conviction about a result through deductive proof. As Weber and others have documented, many mathematicians use authority or empirical testing to become convinced that a result is true before using it, since it would be impossible to work through detailed proofs for every result. The distinctive value of proof is not that it makes you believe something. It is that it shows you the logical landscape — what depends on what, what assumptions are in play, and where the boundaries of the argument lie.
If this is what proof is for, then the format of a proof matters less than its substance. Does the argument make its assumptions explicit? Does it identify the key steps? Does it reveal the structure of the reasoning? These are the questions that matter. Whether the argument is presented in two columns, in a paragraph, or in a tree diagram is a secondary concern.
The tree representation
In my work on theory building, I have used a tree representation of proofs that I think makes their logical structure much more visible than the standard formats.
The idea is simple. The conclusion sits at the top of the tree. Below it are the premises — the claims used to justify the conclusion. Below each of those premises are the claims used to justify them. And so on, until you reach the leaves of the tree. In a complete proof, the leaves are axioms, definitions, or things given in the problem. In an incomplete proof — which is what students are usually working with — the leaves are the places where more work is needed.
Take the claim that two triangles with the same base and between the same pair of parallel lines have the same area. This sits at the top of the tree. Below it are three premises: that the heights of the two triangles are equal (because they are between the same parallel lines), that the area of a triangle is half base times height, and that the bases are the same length. Below the area formula, there are further premises: that the area of a parallelogram is base times height, and that a triangle is half of a parallelogram with the same base and height. Each of these could be broken down further.
The tree makes two things visible that the two-column format does not. The first is the logical structure — you can see at a glance what depends on what. The conclusion at the top rests on the premises below it, which rest on further premises below them. The second is where the gaps are. Any leaf that is not an axiom, a definition, or a given is a claim that has not been justified. These are the hidden assumptions — the things the proof relies on without stating. In the tree, they are visible as leaves that need further development. In a two-column proof, they are invisible, buried in the "reasons" column as things the student is expected to know.
The tree also makes it natural to fix flaws. If a leaf is not justified, there are three possibilities: it can be justified by a lower layer of premises (in which case you extend the tree), it contradicts other claims in the theory (in which case you look for counterexamples), or it needs to be accepted as an axiom or definition. These are exactly the moves that mathematicians make when constructing proofs, and the tree format makes them concrete and discussable.
What this looks like in a classroom
When I ran the triangle theory building module with students in Pune, the central activity was exactly this: students listed claims they believed to be true about triangles, and then tried to justify some of those claims using others. The justification process naturally produced trees — each proof connected a conclusion to its premises, and the premises to their premises.
What made this different from a standard proof exercise is that students were not working from a known result toward a predetermined argument. They were working from their own beliefs toward whatever structure emerged. A student might start by trying to prove that angles opposite equal sides in a triangle are equal. To do this, they would need the concept of congruence. To justify congruence results, they would need assumptions about what happens when you move a triangle — assumptions about rigid motions that had never been made explicit. Each justification opened up new questions. The proof was not a closed exercise but an open exploration.
I found that students in this setting did not appeal to authority or to empirical arguments. This surprised me, because the literature suggests that students typically do not see value in deductive proof. One possible explanation is that many of these students had worked with me before and knew what I expected. But another explanation, which I find more interesting, is that the structure of the activity made deductive reasoning natural. When you are trying to connect claims to other claims, and you can see the tree developing on the board, the question "why should I believe this?" has a clear meaning — it means "what goes below this node?" The tree gives you a place to put your reasons and a way to see whether they are sufficient.
There were also productive failures. A student named Anya constructed an argument about the angle subtended by a diameter of a circle that applied a theorem about sides and angles of a triangle to two different triangles — using a claim outside its scope. The tree representation made it possible to locate exactly where her reasoning broke down: the step from "this is true within one triangle" to "this is true across two triangles" was a leaf without justification. In a two-column proof, this error might have been harder to spot, because the format encourages listing steps sequentially rather than seeing their logical relationships.
The question of rigor
One thing teachers often worry about is rigor. If you move away from the two-column format, how do you decide what counts as a sufficient proof? The answer, I think, is that you negotiate this with students, and the answer varies from task to task.
Keith Devlin frames the question well. There are two ways of thinking about what a proof is. The first is a logically correct argument that establishes the truth of a proposition. The second is an argument that convinces a typical mathematician. These are very different standards. The first demands formal rigor. The second is contextual and pragmatic.
In a classroom, I think the second standard is more useful. The question is not "have you formally verified every step?" but "have you made your reasoning clear enough that someone else could follow it and identify any gaps?" This shifts the focus from format compliance to intellectual substance. A proof that omits the transitivity of equality but makes every other step explicit is, for most purposes, a better proof than one that includes transitivity but hides three other assumptions.
Of course, stating every assumption explicitly would be impossibly cumbersome. You would never finish a proof if you had to justify modus ponens from scratch each time. The level of rigor expected needs to be negotiated collectively and will vary depending on the goals of the particular task. Sometimes you want students to be very explicit about their assumptions — when the point is to discover that even "obvious" things rest on hidden premises. Other times you want them to focus on the big picture — when the point is to see how different claims connect.
This flexibility is one of the advantages of the tree representation. You can always extend a tree further, adding more levels of justification. But you do not have to. The tree shows you where you stopped, and you can make a conscious choice about whether that is deep enough for your current purpose.
What I am proposing
I am not proposing that two-column proofs be banned. There may be contexts where the format is useful — for very short, highly structured arguments where the main goal is to practise using specific theorems. What I am proposing is that the two-column format should not be the default, and certainly should not be the only format students encounter.
The core of proof education should be about substance: can you identify the assumptions an argument rests on? Can you see where the reasoning has gaps? Can you trace a chain of justification from a conclusion back to its foundations? Can you explain why a claim is true, not just that it is true?
The tree representation is one tool for developing these skills. It makes logical structure visible, highlights hidden assumptions, and gives students a concrete framework for constructing and evaluating arguments. But the tool matters less than the underlying shift in what we are asking students to do. If we ask them to verify known results using a prescribed format, they learn that proof is a bureaucratic exercise. If we ask them to trace their own beliefs back to their foundations, identifying gaps and hidden premises along the way, they learn that proof is a way of understanding the world.
Polya conceived of proof as problem-solving, and I think that is exactly right. Proof is not a genre of writing with rules to be followed. It is an activity — the activity of figuring out why something is true and making your reasoning explicit enough that others can evaluate it. The format in which you express that reasoning should serve the thinking, not the other way around.