Two-Column Proofs that Two-Column Proofs are Terrible

Theorem #1: “Justifying steps” ought to be an opaque, frustrating process.



1. In an argument, all steps must be justified. 1. Definition of Argument
2. In real, adult arguments, such justifications often take the form of cogent explanations, appeals to agreed-upon facts, and clear, explicit reasoning. 2. Definition of Justification
3. High schoolers are too simpleminded for such techniques. 3. Fundamental Axiom of Condescension Towards Young People
4. Besides, it would take too long for instructors to grade such arguments. 4. Overworked Teacher Postulate
5. Instead, high schoolers ought to justify their arguments by reciting the names of theorems and axioms, invoked as if they were not logical statements but magical spells. 5. Property of Nonsensical Schooling
6. Logic ought to be learned through two-column proofs. 6. Theorem of Maximal Damage 

Before sarcasm carries me too far down the rhetorical river, let me plant an oar and explain my stance.

I see the appeal of two-column proofs. They’re clean. They’re easy to grade. They offer a scaffold, a structure, a formal framework for students to lean on. Properly understood, they function almost like diagrams of arguments, and can serve as useful tools.

But in practice, they often obfuscate more than illuminate. A good proof contains not only bare statements of fact, but connective tissue of explanation. In a two-column proof, the organic matter that holds the argument together is flushed away, and replaced with a right-hand column full of terse bullet points that students may use without understanding at all.

Theorem #2: A proof is just an incomprehensible demonstration of a fact you already knew.



1. In a good proof, each individual step is obvious, but the conclusion is surprising. 1. Definition of a Good Proof
2. In many two-column proofs—especially those taught earliest in a geometry course—each individual step is mystifying, while the conclusion is obvious. 2. Definition of a Stupid Proof
3. Experience with such proofs will help students see logic as an alien enterprise, foreign to common sense and deaf to the simplest realities. 3. Definition of Students Not Being Idiots
4. Logic ought to be learned through two-column proofs. 4. Definition of a Bad Idea

My beef here is that a geometry course often begins with totally mystifying two-column “proofs” of elementary facts. For example, take the “Right Angle Congruence Theorem,” which states that all right angles are congruent to one another:



  1. Angle 1 and Angle 2 are right angles.

1. Given

  1. m(Angle 1) = 90o, and m(Angle 2) = 90o

2. Definition of Right Angle

  1. m(Angle 1) = m(Angle 2)

3. Transitive Property of Equality

  1. Angle 1 and Angle 2 are congruent.

4. Definition of Angle Congruence

When I run such arguments by math PhD friends, they look at me dumbfounded. “Why would you prove that?” they ask. There are important lessons here, surely—for example, that even seemingly obvious truths demand justification. But the argument hinges on the fussy technical distinction between the angle (a geometric object) and the measure of the angle (a number describing the size of that object). If you’ve ever been a 9th-grader—or even met one—you know that such a distinction isn’t the most inviting welcome mat to geometry.

Theorem #3: Two-column proofs are great preparation for the future.



1. High school ought to prepare students for their remaining years as scholars, and their future decades as citizens. 1. Definition of Education
2. Real mathematicians employ two-column proofs all the time! 2. Theorem of Lies
3. Two-column proofs are also used in the workplace and the political sphere. They’re everywhere! 3. Theorem of Even More Lies
4. I mean, we wouldn’t be using such an artificial, opaque system for teaching logic if it didn’t have some real-world utility, right? 4. Definition of Wishful Thinking
5. Logic ought to be learned through two-column proofs. 5. Axiom of Systemic Stubbornness in Education

Finally, there’s the fact that no mathematician—indeed, no adult human other than a geometry teacher—uses two-column proofs. Flip through a math journal, and you’ll find only what geometry textbooks call “paragraph proof”—that is, English prose laying out an argument. Sure, the proofs are dense, and often interrupted by equations spanning the width of the page, but they’re written to be understandable, not to adhere to an artificial boxed-in format.

How do we fix the system? Here are some imperfect suggestions:

Begin with better proofs. Start a geometry class with a unit on proof structure, and don’t worry about what you’re proving. Prove that there are infinite primes. Prove the Pythagorean Theorem. Prove that there’s no school on Saturday. Prove that a bear would defeat a lion in combat. The axiomatic development of Euclidean geometry can come later. First, the students need practice playing the game.

Don’t let students give “Definition of A” or “Theorem B” as reasons. At least, not at first. They need to understand that a reason is a truth, not a phrase. The current format of two-column proofs obscures the logical content that underpins arguments. Instead, of letting students invoke “Congruent Supplements Theorem,” make them write out, “If two angles have congruent supplements, then they themselves are congruent.” (You can wean them off of such wordy explanations later.)

Use two-column proofs like spice or seasoning: Sparingly. They can be helpful clarifying agents. But two-column proofs ought to occupy a place in geometry similar to the “flow proofs” that some textbooks like to invoke. They should be a “sometimes” food, not a staple of the diet.

Consider adding a third column. My own geometry teacher started the year with three-column proofs: (1) Statement; (2) Reason; (3) Previous Steps on which This Step Relies. This way, a “reason” does not feel like a secret password, but what it is: a link between statements that have come before and the current statement.

I’d love to hear other geometry teachers’ take on the two-column proof, especially anyone who’s got a passionate (or dispassionate) defense for their pedagogic value.

For related ramblings, check out Black Boxes (or: Just Say No to Voodoo Formulas) and For #9, I got “snake”.

104 thoughts on “Two-Column Proofs that Two-Column Proofs are Terrible

  1. “…and don’t worry about what you’re proving. Prove that there are infinite primes. Prove the Pythagorean Theorem. Prove that there’s no school on Saturday. Prove that a bear would defeat a lion in combat. The axiomatic development of Euclidean geometry can come later. First, the students need practice playing the game.”

    “In a good proof, each individual step is obvious, but the conclusion is surprising.”

    The first quote is contradictory, but because you also said the second quote, I think I agree with what you’re saying.

    A big part of logic is that the answer is not determined by what you want it to be. I want there to be $1 billion in my bank account, but it I look at the relevant factors, the logic shows that it won’t happen. Logic is more about starting from an agreed position and then using solid manipulations to see where you can get.

    I think that our cultural understanding of debate right now undermines that, and actually makes teaching things like proof very difficult. Kids are surrounded by disregard for logic: extremists pushing unfounded political views, parents making counterintuitive decisions, adults in general behaving badly or even just explaining their actions poorly, etc. A lot of kids get to the “proofs” part of math without having a foundation in their life for a pure understanding of logic.

    I wonder if this is something a teacher can work on before getting to the “proofs” part of the school year. I’m envisioning daily warm-up exercises of asking students “What do you want to happen?” vs. “What are the actually likely outcomes?” …. “If you get to the front of the lunch line and they run out of pizza before you get there…what do you want to happen? What are the things that are likely to happen?” “If you forget your homework…” “If you enter the lottery…”

    Just rambling! 🙂

    1. Good ramblings! I think you’re right that kids are more familiar with “arguments” where the goal is to attain victory for your side, rather than to establish a necessary truth.

      Those warm-up exercises you mention are worthwhile questions, and maybe better deployed WHILE learning proof than beforehand. Anticipating misconceptions is important, but IMO it’s sometimes easier and more natural to deal with them as they arise.

  2. I don’t love the two column method but there is a certain neatness to it by forcing a reason for each statement. If I had a dollar for every conversation I heard about who is better, Kobe or Lebron, -that was never backed up by reason, I’d have at least $20-$25.

    I think your best point is that the reasons don’t need to be ‘definition of ____’ rather just tell me in plain words why you think what you said is true.

    1. Yeah, the opacity of the “reasons” column (with stuff like “definition of ___”) isn’t inherent in the two-column format; it’s just a consequence of how it’s often employed. Good point that two-column proofs “force a reason” for every step, which makes them a useful teaching tool (while not necessarily justifying their ubiquity).

  3. While I agree with you on a lot of this, let me try to give a partial defense.

    For one, paragraph proofs offer at least as much room to replace understanding with magic. Ever bluffed an argument with the word “clearly”? I think that’s a mainstay of every math major, and a lot of professional mathematicians. Two-column proofs (try to) remove this temptation: every line needs a justification, no matter what.

    For another: neither type of proof has a proper normal form, but two-column proofs come a lot closer. This has multiple benefits. Yes, it’s easier and faster to grade, which is of at least some concern. It’s also easier for a student to check his or her work. I’ve certainly written proofs that I doubted were correct, just as students do. I can compare my paragraph to a solution paragraph to see if they’re the “same” proof, but that takes a lot of background knowledge- eg, to recognize when the order of two statements matters, and when it doesn’t.

    Ultimately, a mathematical proof is one thing, and a mathematical argument is a closely related but different thing. If and when you’re teaching the former, two-column proofs can be very useful. And for what it’s worth, the claim that “no adult human other than a geometry teacher uses two-column proofs” isn’t entirely true: formal proofs, especially Fitch-style deductions, look awfully similar.

    1. Thanks. You’ve clipped the corners of my argument, and I think it comes out better for the trimming. Most of the “action items” I list manage to avoid undermining the benefits you point out, but you make a good case that, if you adopt the other changes, then diminishing the role of two-column proofs isn’t necessarily urgent (or even desirable).

      1. This makes me wonder if it is beneficial to have students write paragraph proofs and then check their work with a two column proof key. Does it help scaffold their reflection to see if they’ve attended to each important idea?

  4. Proofs, I feel, are ways of allowing further understandings of certain concepts. While I see and understand your points regarding column proofs, I think people should use a format they find most comfotable with. After all, there isn’t a concrete “template” for writing a proof as far as I know.

    1. That’s fair. Though they should probably experiment with proofs in formats they’re UNcomfortable with, too, to make sure they really know their way around an argument.

  5. Wow, I remember writing 2-column proofs in logic class that I was totally proud of, consisting of tens of steps. I haven’t used them since, and I don’t know if they taught me to reason at all. That would have been 8th or 9th grade, I think.

    1. Yeah. I think my 9th-grade geometry teacher did a great job teaching logic and reasoning, while relying mostly 2-column proofs. But that just shows that experts can do excellent work with pretty much any tool.

  6. I’m having trouble with this right now, because as much as I want my kids to make paragraph proofs, the lack of structure leads to total crap. I’m backing away from proofs for a month or two, see if I can just get one or two steps with justification first.

    1. There are other forms of proof out there. I’ve always liked the tree-style proofs that come up in certain branches (yay, tree puns!) of formal logic, where you start with the thing that you want to prove, and work backwards, spreading out to all of the things that you need to support it, until you get back to some claims that you’re happy to take as axiomatic (or don’t, in which case what you’re proving is probably wrong, or you need to take another approach to getting to it).

  7. So, just to throw a monkey wrench in here – why does proof only have to happen with geometry? Doesn’t proof also happen in algebra every time students solve equations or analyze patterns? What about “proof” in statistics? I think that if we are allowed to expand our concept of what proof is, we will find that it can look very different in different contexts. But fundamental to all kinds of proof is solid reasoning and communication. Personally, I’m not a fan of the two-column proof format, but I’ve seen the structure help some kids develop logical chains that form arguments. Ideally, wouldn’t a two-column proof be a draft on the way to writing some prose?

    1. I’ve wondered before why geometry is the default arena for teaching proof. My take is that it’s as good as any other venue (though not necessarily better). And certainly it makes sense to continue incorporating proof in later classes. (I like teaching very proof-driven version of Trig.)

      I’m with you that the two-column format may help some kids build logical chains. It might be useful either as a “draft,” as you say, or as an occasional tool for straightening out misconceptions about proof.

      1. I’m old enough to remember starting Alg II with proofs as well in the old Dolciani texts.
        Ben – you touch on a couple of HUGE issues. The first is our stubborn reliance on distinction between equality of measures and congruence of objects. Good golly, why would a 13/14 year old care a whit about that? The more important one is the structure of a logical argument. Nobody (NOBODY) other than poor old Geom students follow this rigid if-then two column proof model. If we care about this as a life skill, then we have to open this up and have students make cogent, verbal arguments spoken or written in a way that is similar to how they speak.

        I just shared this with my Geometry colleagues. I’m interested in any feedback I get from them.

      2. That geometry is the class to which proofs are attached seems to be a vestige of early twentieth century and late nineteenth century education where it had such a prominent place in the liberal arts. There really is no reason to stubbornly stick to the subject. Personally I would much rather see discrete mathematics (start with the basics: sets, counting, permutations, combinations, leading to some basic graph theory and probability) used instead. For one, discrete math is a bit more relevant in our digital world compared to Euclidean geometry, and secondly, in discrete mathematics one can reach more quickly to some of those good proofs with obvious steps leading to surprising conclusions (something like Hall’s marriage theorem or Arrow’s impossibility theorem).

        1. Yeah, I do like the idea of introducing proof through discrete mathematics. If that’s a year-long class, it raises the question of where geometry fits in. You could do a sequence something like this…

          8th: Algebra 1
          9th: Geometry (w/o proof)
          10th: Discrete Structures / Intro to Proof
          11th: Algebra 2 / Precalc
          12th: Calculus

          And colleges could offer proof-based Euclidean geometry as a fun, low-ish level math elective?

  8. I too HATE two-column proofs and do not see the use since most arguments are not linear but pull from different parts of an argument to get to the conclusion. I only teach two-column proofs when introducing proof structure with solving algebraic equations and writing reasons. As soon as we move into “geometric” proofs I teach them all as flow proofs. I do not teach them to “memorize” the theorem by name but by what the theorem states. Reasons can be shorted to basic vocab or other geometric relationships. Unfortunately I am the only one I know who teaches proofs this way. Also I have to say my students do not dread nor hate proofs and actually enjoy trying to figure out the logically hole in the puzzle. I love proof and see it as the foundation for pre-calc students when they have to work with trig identities.

    1. Yeah, that sounds like a great approach. If they’re (1) Enjoying proofs, and (2) Emerging well-prepared for proving trigonometric facts, then obviously things are going well.

      I find the “three-column proof” (where the third column lists the prior steps being used) can also serve a similar purpose to flow proofs.

      1. Suggestion for a more compact “three-column” format: put the third column in the middle, with no words in it, just arrows linking earlier steps to later ones that use them. Not so easy to do in a word-processor, but easy on paper 🙂

  9. I’ve gone through the proof curriculum once as a kid and once as a teacher, and they make much more sense as an adult. I ask many kids what they think geometry is in general, and I always get the answer of “shapes.” The idea that geometry isn’t about shapes is pretty foreign, and proofs don’t have much to do with shapes at all.

    I always respond that geometry is about a new way of thinking; that everything you say needs to have a reason why. In algebra, I accept right answers; in geometry, I always challenge answers (even correct ones) with…why? Geometry is the first time students really have to come up with a reason for their statements, usually difficult for a high school sophomore that hasn’t been challenged to think in his/her educational career.

    Seriously, though: most of these kids have not been trained to THINK, but to simply rely on others as a crutch. Two-column proofs — I think — are a great, concrete way to show kids what critical thought can look like.

    I had a comment on another post that one day my geometry kids said “Why do we do these proofs at all?” and my response was “We all use these every day, you just don’t know it.” And we proceeded to prove that Given: A bear is chasing you in the woods. Prove: Your survival. Some kids got my point, others didn’t, as is wont to happen with any lesson, but it turned into a pretty funny topic.

    And with kids that can barely put together sentences in the super-technology world, two-column proofs are more concrete and easy to follow. I do, however, LOVE the third column idea of what previous steps went into the one used. Next time I teach geometry, I may use that.

    I also agree that proofs should take place in more high school curricula than geometry; there are proofs in all facets of math, and, while I prefer a paragraph roof, I think students would do better at first with a two-column proof.

    I think proofs are really important at beginning the logical thinking. I love that they frustrate the kids, and I love when the kids keep fighting and want to solve it like a puzzle. It’s to train them to put together a logical argument instead of simply accepting things as they are. When I left that geometry class, I said I hoped they learned one thing from me and geometry in general: don’t believe what your eyes tell you. I believe proofs are instrumental in providing concrete reasoning as to why things aren’t always as they seem.

    (As a kid, I HATED proofs. They seemed so obvious, tedious, and stupid. So I get how they feel. I like the idea of life-related proofs rather than shape-related proofs. But being able to reason with yourself and come up with a logical sequence of events to do anything is a very difficult task, even for most adults. That’s part of the reason why this country is where it is. If you can tackle logic, you can tackle almost anything.)

    1. You’re right about a lot of the obstacles that kids face in making sense of geometric proofs.

      I’m happy to be hearing from you (and others) a defense of the two-column format. It may be that two-column proofs don’t need to be replaced, just modified (with that third column, a more detailed “reason” column, and better proofs to start the year).

  10. Proofs are probably the most difficult topic for any geometry teacher and I would argue most students. I understand that no good mathematician will write a proof in two columns, but it does work as a draft for the prose like Pamela mentioned above and provide a concrete structure for students. Students who are not at the right development level (Van Hiele levels) will struggle with proof no matter how it is taught. I would argue that even the “best” high school students don’t really understand proof but “get” the pattern of proofs like SSS, SAS, etc. I never really understood proof till I was majoring in mathematics in college. I still think, however, there is value in it. The two-column format is often the most structured way to teach proof so it is the default method. It would be similar to teaching a student to write poetry. The students first poems may be fairly formulaic and then progress as they develop. Finally, I do agree that proving obvious results can be confusing to students and definitely clouds the purpose for proof.

    1. Agreed that a deep grasp on proof develops slowly, and that two-column proofs may be more useful when first learning proofs than they are later. Maybe just emphasizing “this 2-column format is only one of many ways to write a proof” would help, rather than letting the two-column style shape students’ entire view of proof as an intellectual enterprise.

  11. Ben — first of all, the 2-column format for your 2-column critique was laugh-out-loud hilarious, with the “Definition of a Bad Idea” and so forth. 🙂

    Second, let’s not forget to tighten up the logic on why 90 degree angle #1 is congruent to 90 degree angle #2. You wrote that a = 90 and b = 90, from which a = b by the transitive property. Not so fast, my friend! First we need the following theorem:

    a = 90. Given
    b = 90. Given
    90 = b. Symmetric Property of Equality
    a = b. Transitive Property of Equality.

    i.e. just because “a = b and b = c” implies that a = c, does not mean that “a = b and c = b” implies a = c. To reach that conclusion, we need my argument above, which I call the theorem of WHOCARESABOUTSUCHNONSENSE. aka “Definition of Pedantry.”

    And third, I’m sure you realize that there are a lot of geometry teachers out there who are not going to part with their 2-column proofs readily. Therefore, it is incumbent on you, me, and other like-minded teachers to show the world an alternative. Again, to SHOW the world, not just tell them “you should write paragraph proofs instead.” So do us a favor and post your students’ work product!

    Great blog. I’m subscribed.

    1. Ah, how could I forget the symmetric property of equality! What a sloppy proof I gave. 😉

      Alas, I’m not teaching geometry this year, just mouthing off about a class I’ve taught in the past. But if you’ve got examples of student work I’d be happy to add a link at the bottom fo the page.

      Reading over comments has heightened my sense that parting with two-column proofs altogether may not be necessary; all that’s needed is some sensible reform.

  12. We teach them this way too! Love love LOVE the flow proofs with justifications in students’ own words.

  13. I would have thought keeping track of the horse and the jockey was simple and straightforward before reading this piece, now it appears the jockey has his own life to lead…..

  14. Thanks for couching something so important and so unnecessarily debilitating to so many students in such delightful humor. This should be required reading as discussion fodder for everyone one of the 30-40,000 HS geometry teachers in the US.

  15. Ben, one more thought. My district’s geometry curriculum starts the treat of proofs with the “Overlapping Segments Theorem” and the “Overlapping Angles Theorem.” i.e. if A-B-C-D are collinear with AC = BD, then AB = CD. When you use numbers, say AB = 5, BC = 10, and CD = x, then it’s super clear that if AC = BD = 15, then x = 5. But try proving that with arbitrary numbers, and all of a sudden BAM! the students are hopelessly confused. And in the end you’ve proved a result that gets used………………………..NEVER AGAIN! What a way to start the year…

  16. Yet another thought: the two-column proof may be a decent tool for organizing your argument AFTER you’ve already thought things through 90% of the way. (I think you make a similar point somewhere above.) It’s a bad tool (IMO) to *start* thinking about how to prove something, but it might be a decent place to *end up.*

    Example: proof of vertical angles theorem, with x next to y next to w.
    1. Since x and y make a line, I know that x + y = 180.
    2. Since y and w make a line, I know that y + w = 180.
    3. Since both expression are equal to 180, I know that x + y = y + w.
    4. Now I can subtract y from both expressions to get x = w.
    5. This proves that vertical angles are congruent. Yay!

    Now for the 2-column stage:
    Hmm…how did I reach the conclusion in step 1? Answer: I used what I know about straight lines. What’s the name of that postulate again? Oh yeah, the “Linear Pair Postulate.” Let me write that down:
    1. x+y = 180 Reason: Linear Pair Postulate

    Last point (for now): further evidence that “all is not well” is this — point to a picture with a linear pair of angles, and ask a student what they know. 99% of the time they’ll tell you them make 180 degrees. Ask the same student to “State the Linear Pair Postulate,” and 93.42% of the time they will…………..stare back at you.

    This just goes to show that WORDS CAN BE HARMFUL. Take something simple and try to express it in WORDS in a precise manner, and you might just kill everything right there.

    Students can easily form the impression that all ideas have nicknames. What is the name of the theorem that says, “If a parallelogram has a right angle, then it is a rectangle?” Foolish endeavor.

    1. Well said. I’ve definitely seen that “all ideas have nicknames” error.

      In particular, I think that words are harmful when the label/jargon precedes the understanding. This is a very broad problem in math. If you grow familiar with symbols (be they words or mathematical notation), before you understand WHAT they symbolize, then math becomes about manipulating and combining meaningless symbols, rather than manipulating and combining ideas.

  17. I have seen something like 2-column proofs used in a higher level class. In a course on automatic theorem proving, a very formal detailed definition of a proof was given, in which each “line” consisted of a statement and a justification, but the justification was more than just a definition, axiom, or previously proved statement. The justification consisted of all the parameter setting (substitutions) needed to create the new statement from the previously known one(s). Writing proofs like this by hand is incredibly tedious, and the 2-column approximation to it loses what is often the most important part (what the substitutions are).

    1. Interesting. I’ve got the Wikipedia page for automated theorem proving open in another tab now. Not surprising, I guess, that something akin to the two-column proof would arise in a computer setting, since the main advantage of the 2-column format is standardization and lack of ambiguity.

  18. I tutor geometry often, and speaking from that particular perspective, I agree strongly on a few points. First off, I also don’t see why geometry is the only math subject that includes proofs. There are plenty of math students getting good grades in other courses by memorizing steps and algorithms, but who can’t tell you the reason why the just did what they did. This further stresses the uselessness of the two-column, geometry-only proofs as they are currently taught in that students are not often then encouraged to justify their steps/reasoning in later math courses.

    As a tutor it is often my job to figure out some slight or tangential deeper meaning in a mundane topic or assignment, because often a student’s lack of understanding in the first place comes from feeling absolutely no connection, intellectually or personally, to the material. Usually with two-column proofs, the route I go is like what someone mentioned above: they force you to justify every step and encourage students to leave no room for error. BUT there are plenty of other ways to do that without the inanity! And also, the mindset of justifying each step often stems from the same attitude that results in checking one’s work completely before handing it in, being interested in making (school)work to be proud of, and other such habits. I’ve been trying to put a finger on what this attitude is and recently I’ve been considering it a propensity for utter completeness, which is different from being “detail-oriented” or “following all the steps,” and which is a very meaningful quality that can easily carry over into other fields of learning.

    1. I like the idea that an affinity for rigorous proof is linked somehow to an intellectual tendency for completeness. I’d be curious how general that tendency is. When I was a student, I found striving for mathematical completeness (where we’re working within a well-defined axiomatic system) more comfortable than striving for “completeness” in science or history (where the potential sprawl of data and fact to consider is virtually infinite). But that’s my personal n = 1 experience.

  19. Ben, your description of two-columns proofs is hilarious. But you back away from the implications too fast! I am all for students explaining their reasoning — and as a geometry teacher for several years, I know students can often guess the right answer: “Sure, those lines look parallel!”; “But of course that angle is right! Even a fool could see it” — without really understanding why. You, Ben, have used the phrase “conceptually motivated solution” on this blog before, which is exactly right. I do not care that my students can formulate a two-column proof (or any other proof); all I care is that they can explain their thinking clearly and correctly, providing a real, conceptually motivated solution. Yeah, if my students tried to write a paragraph proof, they’d probably skip steps, but if a kid shows he gets the idea — “oh, these lines are parallel, therefore the alternate interior angles are congruent, so I can show the triangle are congruent” — what is the harm? They know how to actually do some geometry, to think through novel problems, and that’s pretty darn important.

    The key comment made so far is, proofs are the residue after the kids have the idea already. A proof does not teach the students new geometry knowledge. If you find that you have the time to see that they know the material to an exactitude appropriate for a computer, not a human being, then sure, tack on proofs at the end of the year. But I always find that my kids need the time to really get the ideas of geometry conceptually, and I’m not sad proofs don’t fit in. And I think the truth is, for 90% of the students, they are a black box. The top 10% of kids might really benefit a bit from doing a proof, but the overwhelming majority are mystified and fake their way through.

    You ought to read Mathematician’s Lament, by Dr. Paul Lockhart. I wrote a post about it here on my blog: He absolutely lays into proofs for pretty much exactly the reasons you outline. He’s extreme, but in decrying proofs as a staple of Geometry classes, he’s right on the money.

    1. Russel: ‘I know students can often guess the right answer: “Sure, those lines look parallel!”; “But of course that angle is right! Even a fool could see it” — without really understanding why.’

      I think one of the things missing from most mathematical education is “trickster proofs” – which often rely on (judiciously) bad drawings, by the way – in which the diagram used to illustrate the “reasoning” is slightly off in some subtle way that makes certain claims seem “obviously” reasonable (that actually aren’t true, if you draw the diagram accurately); or that (though drawn exactly) rely on the quite particular angles or ratios of lengths you used, while some part of the diagram doesn’t vary with those angles or ratios in quite the way one might simplistically intuit. Exposing students to such “proofs” might help them to understand why the real thing is worth mastering.

      This is closely related to one of the families of courses I think every education system needs (and I’ve never heard of one that has): the “How to lie with X” courses, for various X – e.g. generalisations, syllogisms and, of course, statistics. They’re mainly about building the student’s intellectual immune system but also about teaching why it’s perfectly proper and reasonable to demand that “arguments” should actually hold water. They’re also an occasion for teaching how to demand that *civilly*, with a readiness to help fix up (or fill in gaps in) the argument thus challenged; folk do make correctible mistakes in reasoning through to valid conclusions, after all, so (politely) quibbling a step on their reasoning can be a way to help them tighten up their logic rather than “opposing” them.

  20. HI Ben,
    Thanks for shedding light on the elephant and opening the discussion.
    The most success I have had in transference of proof is with proof by contradiction. I ask the students to argue a point (some get sticky like abortion or immigration, some silly like why we should eat pizza everyday) and use arguments with solid reasons to convince us their side is correct, however, use the ” If we don’t allow…” to reason to a ridiculous conclusion, and go back to conclude, therefore “we should allow…”

    This is the first year in about a hundred years that I have taught both Advanced Algebra and Geometry. At this point in the year, I kick myself and all my colleagues who didn’t drill the Algebraic properties into the kids heads. Yes m<K + 60 =180 means m<K = 120. That is obvious, but when putting f(x) = 4x^2-12x + 5 in vertex form, the adding and then subtracting for completing the square is completely lost on them and I want to pull my hair out!

    In Algebra 1 (I teach high school) I have N.E.V.E.R. had a mature enough group to spend time in Algebraic proofs and memorizing properties of Algebra, but by the time I teach Advanced Algebra, I sure wish they had those suckers closer by. Suggestions?

    1. Amy, I don’t understand your statement “Yes m<K + 60 =180 means m<K = 120."
      "m<K+60=180" means "m<K+60 and K+60=180", which in turn means "m<180 and K=120", but "m<K=120" means "m<120 and K=120", so your two statements have different meanings.

      1. I think she’s using the “<" sign to mean "angle," and "m" to mean "measure."

        So "m<K + 60 = 180" means "[measure of angle K] + 60 = 180."

        Turns out WordPress comments aren't great for detailed mathematical writing…

        1. Ah. That reading had not occurred to me. Too many years since I last had geometry. I’d forgotten some of the cryptic notation that isn’t used anywhere else. “m∠K” might have been clearer, but “∠K+60°= 180°” would have been clearer still.

    2. You’re definitely right that proof by contradiction is a good gateway to proof. I’ve found kids can get a pretty quick handle on that “Well, suppose you’re right. If so, then this awful or contradictory conclusion must follow” fashion of argument.

      As for algebraic properties, that’s tough. I’m not sure memorization is quite the right paradigm, but a lesson discussing each property in turn, and why they must hold true, seems on point. (“We can always add the same thing to both sides of an equation, because then the two sides will still be equal.” I like to demonstrate by having an equals sign connecting two happy faces, and then adding a nose to each. “See? Still equal.”)

      Others might have better ideas.

  21. Of course it’s impossible as a practical matter to write out complete formal proofs. On the other hand sometimes when I have tried to formalize an argument somewhat the attempt to do so has caused me to see something which I had overlooked.

    1. Formalization definitely has its merits. (All branches of non-Euclidean geometry emerged from fussing about the formalization of parallel lines.) The problem with 2-column proofs isn’t that they’re insufficiently formal (or even “too formal”), but that they often lead to specific misconceptions among kids first learning about mathematical proof.

  22. An amusing example of the practical limits of formalization is given in a footnote on page 234 of Potter – Set Theory and Its Philosophy. He quotes a calculation by Mathias that the term for the number 1 in Bourbaki’s formal system written out in full has 2,409,875,496,393,137,472,149,767,527,877,436,912,979,508,338,752,092,897 characters.
    A Fermi question – If the term for the number 1 were written out in full how would the mass of the resulting inscription compare with the mass of the universe?

    1. I like that!

      My quick calculation gave me about 1/10,000th the mass of the universe (excluding dark matter/dark energy). Is that about where you wound up?

  23. Ben – I was too lazy to go through a calculation but your number seems about right. Mathias number is about 10^54, a mole of atoms is about 10^23 and I vaguely recall reading somewhere that there are about 10^80 atoms in the universe. I’m not sure whether my question includes the paper as well as the ink or graphite for the inscription. Roughly I think this is about the mass of 10 million galaxies.

  24. I’ve never taught high-school geometry but If I did I don’t think I would bother much with highly formal proofs like the two column format unless it were a requirement from higher up.
    I would think that doing so would distract students from actually understanding the geometry.

    1. There’s actually a trend towards that, lately. In NY, for example, the state test doesn’t ask much about proof, so lots of teachers don’t bother covering it.

      I think it’s possible to teach an engaging, intellectually stimulating class without proof. But I also think proof is important, and worth teaching as early and often as possible in math.

      1. So for the sake of argument (get it?), how does one teach an “engaging and intellectually stimulating class without proof” and without creating Black Boxes?

        The possible recourses seem to be

        (a) presenting interesting ideas that are intuitively plausible. These are Black Boxes, but the students don’t see the blackness of them until they hit a corner case.

        (b) proving things at some level of rigor.

        Is there another way? It would seem that you must either provide reasons or else not provide reasons.

        1. That’s a good question. You probably do wind up creating some black boxes. I have to confess, I’ve never tried teaching such a geometry class, but my hunch is that there are lots of interesting lessons you could do (classifying quadrilaterals, exploring slopes of parallel and perpendicular lines) relying mostly on careful inductive reasoning, rather than deductive reasoning. The approach would have its pitfalls, but you could probably hit more interesting geometric ideas than if you’re ALSO teaching logic.

          That said… your comment is right on, and makes me want to see a class do this effectively before I’m confident it can be done at all.

  25. I think that you can provide reasons without using the rather artificial and distracting formality of two column proofs. If you want to prove theorems in proof theory then yes you need to be more formal about how you describe proofs. But if the goal is to understand geometry than focusing on the formal structure of proofs is a confusing distraction. The goal is to study geometry not meta-geometry.

    1. I guess it’s a matter of what you call a proof. A true, well-reasoned deductive justification of a general fact is probably a “proof” in some sense, even if it lacks formality. But I certainly see the argument for dispensing with formality.

      1. There different ways to communicate proofs, such as flow-chart proofs, paragraph and so forth… Use what works for you and your students, yes, thinking through a problem logically and sequentially, while having solid justification for each step is the goal… the two column proof is just the iPad of proofs… its not the only one

  26. Ben: Would it be ok if I use your “2-Column Proof that 2 Column Proofs Are Terrible” for a workshop I’m doing this weekend at the CMC-South conference in Palm Springs? The title of my workshop is “Proof Doesn’t Begin with Geometry” so this would be perfect. And of course, there would be a big shout out to your work and blog.

  27. I would try for a presentation designed to convey understanding of geometry at a level apprpriate for the students. I think trying to dissect the formal structure of the arguments would not contribute much and might well be an impediment to that goal.

  28. I really like this post… Couple of things I notice as well.

    1. I think this may be a great opportunity for us to go across the aisle to the English teachers and tie proofs in with the technical writing in English. I remember and enjoy the lesson on making a peanut butter and jelly sandwich. This could turn into a win on both sides.
    2. I am teaching Geometry for the first time since I took it about 20 years ago. Since I am rusty on the proofs, I have had to redo them myself before going to class. I then go to textbooks, proof wiki or another resource I find on the internet and realize that many proofs are justified the same way but with completely different language. Not that this is necessarily bad, but it leaves me with the idea that I could be missing a step or putting in a step that is not necessary. I have no issue with creating a path for success in my class for my students that is based on sound reasoning. I just have a concern that other broad assessments may not be on my wavelength. This is just unsettling.
    3. I know that many of us do not have the authority to create a course in logic, but I would love to see students getting at least a semester in logic before Geometry. This would serve them in the long run and us as well.
    3. I like the progression in the development of proof. This year, I have made some missteps in the rollout and am lucky to have a class culture where the class is still in good shape. Thanks for sharing the ideas.


  29. I chuckled through the proofs to disparage proofs. Similar to the emperor’s lack of clothes, we do have to admit to weaknesses, before we can thoughtfully analyze the topics. Keyjames said above, “yet another thought: the two-column proof may be a decent tool for organizing your argument AFTER you’ve already thought things through 90% of the way”. I agree and believe that developing a “plan” for a proof is often the most important part of a proof for younger students. That plan becomes a “paragraph” proof or can lead to a more structured two-column. I like class discussions on a plan as they try to find a way to attack a problem, including working backwards from their conclusions. I agree with several above that there need to be more conversations outside of just geometry on “why” something might be true. I agree it takes time to develop the sense of logic demanded in larger measure in the traditional geometry course and prefer not to begin more formal proofs (paragraph, flowchart, two-column) until the students have grappled/played/visualized the components/objects used for several weeks.

  30. I tried your three column idea (I call it “What?”, “Why?” and “Glue”) and have them state the theorem in its entirety — I like it, but I still think students are missing the boat. Students feel that they are “stating the obvious”. The picture “looks like” the altitude of the triangle is an angle bisector, etc. Therefore, they often will not be “surprised” by the result. Any ideas on how to help the reasons be clearly understood?

    1. Hey Zach, that ‘looks like it’s true’ problem is a really tricky one. I have three suggestions, which may or may not prove useful.

      First, maybe you could spring a trap on them – let them prove something that ‘looks’ at first, and then demonstrate afterwards that the statement is false. A non-geometric example: “n^2 – n + 41 is always prime.” This turns out to be true for n = 1, 2, 3, … 40, but not n = 41.

      Another option: Have them prove “the altitude of a triangle is ALWAYS the angle bisector,” accompanied by a picture that happens to be of an isosceles triangle. Then, when they’ve proved it, point to a scalene triangle, to make them realize they were going off of a specific picture and not thinking about the actual relationships.

      Or: Use a computer program to create several triangles that are either isosceles or very close to it (side lengths 4, 8, and 8.01, for example). For each, ask them if the altitude bisects the angle. That might shake their confidence, since it’s so hard to tell a perfect bisector from a 50.01/49.99 split.

      Anyway, I hope something in there is vaguely useful!

    2. I converted to the three column format several years ago and like it. I am glad to hear of your three new words (What, Why, Glue). Fresh words help some students to connect with the structure.

  31. I barely managed a C in both semesters of my 9th grade geometry class thanks to these kinds of proofs, despite liking math and having been fairly good at it through elementary and middle school. I’m terrible at rote memorization- I need some level of understanding first, either of how a thing works or what it’s used for, before I can memorize. And even then, I might not be able to. Proofs like this feel like being handed the table of contents for a textbook of a subject I know nothing about and being told to memorize it. If I couldn’t say what any of the chapters are about just from knowing the title, and have no clue why they’re in the order that they’re in, it’s going to take a lot of time and effort rereading and writing out that table of contents and when tested I’ll still probably put several chapter titles in the wrong parts of the list, get titles wrong, and forget at least a couple. And then somehow memorization of the table of contents is equated with understanding of the text.
    On the exams that I often failed, if I couldn’t remember then instead of listing the names of the steps of the proof, I’d just show the proof. Like I’d draw and label a series of diagrams, or write out an explanation. But because I couldn’t list the names of the proof steps, I’d get a zero for the question. The teacher would tell me “I said you need to write the proof, not show the proof. When you take notes in this class you need to write down the steps of a proof when I’m teaching, not just copy the pictures and explanation. You’re making this hard for yourself by doing it the long way, what the test is asking for is much easier!”
    I’m also pretty terrible at remembering names of formulas. If I know how and/or why the formula works, with a bit of practice using it I can memorize it. But if I’m asked “what formula did you use to solve this problem?” I might be able to repeat the formula. But it’s a toss up if I can remember the name of the formula.
    I also had a similar problem in French class. Test question “list the verb endings for the conjugation of the imperfect tense” or “What is the imperfect tense?” and I’d probably guess or leave it blank. Meanwhile, I’d confuse my teacher like crazy because I’d earn full points on the essay portion of the test, “You have just moved to France for a study abroad program. Your host family is curious about your life. They want to know what you usually did on weekends when you lived in America. Write a paragraph or two using the appropriate tense.”

    Now, in the Calculus 2 class I’m taking in college, the professor is very big on doing a proof involving showing how/why it works and that we understand it. He says if we want to list steps for a proof, and we use the name of a theorem as a step of the proof…we need to back it up. For example we can’t just write “because Pythagorean Theorem”, but “because a^2 + b^2 = c^2” next to a little drawing of a triangle with the right angle marked and sides labeled a/b/c, is fine.
    “When I ask you to prove something on a test, I’m asking you to show me how and why the thing works. I don’t want to know a list of reasons why someone else claims it works, I want to know that you understand the formulas and concepts I’m teaching you. Not parroting back what you crammed into your head last night. Give me the summary, not the table of contents.”

    1. Sounds like your school-teacher was too preoccupied with nomenclature and using the jargon of geometers. There is (some) value to learning those things, but they aren’t what’s most important about geometry or about its proofs – they’re just the language conventionally used to describe it (which’ll make it easier for you to talk to geometers about it). That language is commonly alien to the vernacular (for example: why is the other part of a straight line’s half-turn at a vertex on the line, where another cuts it, called a “supplement” rather than, say, a “complement” ?) which tends to get in the way of remembering it. An artificial insistence on using conventional wordings to express something both obstructs to the student’s ability to do what’s asked of them and distracts them from seeing what they’re actually doing; encouraging them to express the same things in their own words would remove the obstacle and enable them to see more clearly what they’re doing and why proof is useful (and beautiful).

      Fortunately, your professor seems to grasp the importance of understanding: the substance matters more than the form !

  32. I’m totally using the 1st proof in the first professional development I’m leading with a geometry team! I’m an instructional/master coach in Texas and one of the “fun” things the geometry teachers run into here is that our state standards specify that students have to use two column, flowchart and paragraph proofs. The issue I’ve had with most teachers in the past is that they are mathematical (obviously, they teach geometry) and they think in the form of two column proofs most of the time. Students don’t! So I’m going to challenge the teachers to teach proofs by starting with a discussion and having the students justify to them verbally a statement. I think students nowadays tend to think more holistically and do better with flowchart or the tree proofs you talked about. They do need to know how to move to a more concise form (aka 2 column) but the idea and the thought processes that are going on are essential for their ability to problem solve.

    And when I taught 8th grade 10 years ago I had my students do proofs but they just didn’t realize it. I have always emphasized justifying their steps and their work so that they don’t get into too much of a rut later in school and to help when they totally get sidetracked.

  33. “2. In many two-column proofs—especially those taught earliest in a geometry course—each individual step is mystifying, while the conclusion is obvious.”
    This is exactly how I feel when I do proofs.

  34. Wow, lots of comments on this one! I haven’t read them all yet, but I’d like to throw in my two cents. As someone who both teaches math at the HS level, and someone who graduated with a degree (and love for) pure mathematics, proofs are very near and dear to my heart. They are the single most important thing (I think) we do with our students, and if they could learn one thing from my class I would be very happy if it was the ability to write a solid and structured logical argument. I write proofs most every week, having nothing to do with my job, just for the betterment of my own mathematical understandings. As would any other mathematician, I write my proofs with no set form or pattern, mixing diagrams, prose, equations and (when possible) a little wit. I would LOVE for my students to be able to write proofs in this manner (and I do show them some of my own proofs so they understand where this is all going someday), but I find that the lack of structure in a “normal” proof is a significant barrier to dealing with the concepts honestly and rigorously for HS students. The format of a statement, and a reason, makes it very easy for students to see when they make a statement for which they have no clear reason. It directs attention to this and forces them to consider whether they do in fact have a reason to make that statement. When writing a paragraph proof there is very little opportunity to notice the lack of a reason (or even a missing statement or two) because as soon as you begin writing a proof in normal verse, you begin to rely on the same level of rigor and specificity that you normally apply to written (and spoken) words every day. That is to say, with very little regard for rigor or specificity. When you read something written in prose, it can sound very convincing, especially to the uninitiated, but if you convert it into a two column proof you will immediately notice some glaring omissions. I think this is important in the early stages of proof writing. As the year goes on, I slowly try to wean them from two column proofs, and begin to explore the less structured (but more normal) style of proof when possible. BUT, I see two column proofs as training wheels for the untrained mind. Training wheels are terrible. Do you remember riding with them? They are noisy, get in the way, restrict your movement, and eventually, prohibit you from any further development in bike riding skill. At some point you NEED to take them off to progress further. In the beginning however, they give you necessary feedback (does that statement have a reason?) and prevent you from falling over completely (did you leave out an entire statement that needs to be made to justify that last one?) when you are learning.

    I apologize for the excessive rant, but this is very much on my mind today, and I came across this post googling two column proofs : )

    I’d love to hear your thoughts…

    1. One more thought… Everything can be done well, or done poorly. Teachers who use reasons such as “definition of …” or give reasons without any reference to previous steps of the proof upon which they rely, or who teach steps as a sequence that needs to be done the same way every time rather than develop understanding so students can see there are many ways to write any particular proof, but some may be preferable to others because they “flow” better. In my class, I insist that any statement which relies upon previous statements have a reason that references those statements (i.e. addition steps 1 and 4, or substitution steps 3 and 7). I also encourage students to always consider the “reader,” of their proof. I have them think about the order in which they choose to make their statements and reasons so as to best facilitate the reader being able to follow their argument. We write proofs multiple ways, observing that they are both valid, but one is much kinder to the reader, and therefor preferable as you develop in your ability to write a good proof. Another pet peeve of mine is teachers who have the students write all the “givens” in the first step of the proof. That’s the real world equivalent of walking into a trial (as a lawyer) and dropping a file with all your evidence on the judge’s desk in a huge pile. In my class, as you would in any good argument, we introduce our evidence (givens) one at a time, and at the point in the proof where they are needed. We don’t throw everything at the reader at once, we think about the optimal time and the optimal sequence to introduce each given to the reader. By the time a student can write a solid two column proof, with well formed and correct reasons for each statement, with each statement flowing nicely from the one before it, sequenced in the best way for the reader of the proof, they are ready for the training wheels to come off. Once those basic skills have been adequately developed, removing the restrictions of the two column proof allows them the freedom to develop their own unique style, but with a sufficient background in rigor and completeness to continue producing good work.

      1. That’s got some good resonances with one of the things the software industry has learned: code is read more often than written so, as well as being formally correct (the compiler’s got to turn it into code that does the job it’s meant to do), it has to make clear to the reader what it is doing, why and how the reader can be sure this produces the right answer. When I’m hunting for a bug, all code implicated is suspect as the bug’s locus; so I have to understand that code and clear it of suspicion; code that’s hard to read wastes a lot of time in bug-hunts. Likewise, formally equivalent mathematical proofs may exhibit diversity in ease of comprehension, depending on how they’re written and presented.

        I like your metaphor of “training wheels” – the two-column format (which I have no memory of having ever been taught; perhaps I forgot, or UK teaching doesn’t use it) sounds like a good way to make clear whether all the steps are present and supported, but not necessarily the clearest way of communicating it once you’ve established that. Once you’ve learned the habit of checking every assertion’s reasons are established, it’s time to learn how to turn such reasoning into fluent prose.

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