Why We Need to Keep Teaching Formal Proof in Geometry

2/16/2017 43 Comments

Proof That Proofs Belong in Geometry

WHY formal proofs should not be watered down or cut back, plus links for HOW to teach them in a better way

You probably already know how much I LOVE proofs. It's my absolute favorite thing to teach.

But this post is focused more on why I argue that we should never "water down" or cut back on explicitly teaching formal proofs in Geometry class.

The formal proof is a staple of the geometry curriculum. It has also been the center of debate among educators for quite some time.

Some educationalists believe that the proof should be abandoned for less formal ways of understanding geometric ideas, while others believe that the emphasis of the formal proof is an integral part of learning geometry.

However, any decrease in proof based lessons is an extreme disservice to our students.

What Is a Proof in Geometry?

If you’re jumping into your first round of teaching geometry, a quick refresher – Yes, we are talking about the two column proofs that we learned while we were in school. But do you remember exactly why or what they were for?

Basically, a proof is an argument that begins with a known fact or a “Given.” From there, logical deductions are made through a series of conclusions based on facts, theorems and axioms. This will finally prove the proposition at hand, for example, the sum of the angle measures in a triangle equals 180˚. By writing out a proof, the answer is undeniable.

Why Are They So Important?

Well, logical reasoning and deduction are central to understanding not only geometry, but mathematics as a whole. Being able to tell the difference between obvious mathematical concepts and ones that need to be justified is a new level of understanding in math. It shows comprehension of deductive logic and the ability to structure arguments to make mathematical conclusions. All of these skills are paramount to reaching a more mature and complete knowledge of geometry and arithmetic.

As powerful as our brains are, they can miss key facts and be fooled. There are times where things seem perfectly reasonable and they turn out to be wrong. That’s why we need to learn how to PROVE things. When you go through step by step, with the deductions laid out, you know what you’ve done is absolutely correct.

When mathematicians first began to form rules to prove valid mathematical statements, they did so through trial and error. This allowed congruence in learning. One person could show another person a mathematical rule and prove it through reproduction, which in turn made it valid.

However, proofs aren’t just ways to show that statements are true or valid. They help to confirm a student’s true understanding of axioms, rules, theorems, givens and hypotheses. And they confirm how and why geometry helps explain our world and how it works.

Why formal two - column proofs must be kept in Geometry courses

What Harm Does It Do When Proofs Are Removed?

Limiting the amount of substantial and challenging proofs in a geometry curriculum pretty much defeats the purpose of the course. Now, that may sound a little exacting, but it is true. The reality is that geometry is different than other math courses.

All mathematics are rooted in problem sets, however the problems in geometry that require proofs of propositions do more than apply a theory. They are a part of it. When students learn how to postulate and prove concepts, they are tapping into a deeper stage of mathematics.

Geometrical proofs offer students a clear introduction to logical arguments, which is central to all mathematics. They show the exact relationship between reason and equations.

More so, since geometry deals with shapes and figures, it opens the student’s brains to visualizing what must be proven. It rounds out their knowledge, building upon the concepts of basic algebra.

Students have to combine all of their acquired knowledge. They have to develop a mental list of steps that will lead from the given to the conclusion. Then, they have to find ways to show algebraically that it all works out while simultaneously following along in a diagram. They must combine two lines of logic to create a new one and flow from one step to another. It can take some deep planning and thinking for a more challenging proof.

It’s a whole new way of thinking that develops entire new brain connections for them!

Benefits Beyond the Classroom

Reasoning is a skill that has a multitude of applications. Whether you’re proving a geometric postulate, working through a detailed word problem, navigating facts in a debate, or even making a monthly budget, you will need reasoning. While we do learn reasoning outside of geometry, students that practice proofs strengthen that skill even more. You learn how to reason carefully and find links between facts. This is something that is important for everyone, not just mathematicians.

Basically, proofs do have a very important role in the geometry classroom. They offer:

a means of communicating your reasoning with others,
a justification for the way things work,
and a basis of developing other applications of logical reasoning.

Sound Overwhelming? Read This for Help:

If proofs intimidate you as a new Geometry teacher, or even if you’re a veteran and your kids always struggle through the first weeks of proofs, you are not alone! It’s such a tricky new way of thinking for them.

But don’t skip them! Instead, go about it in a better way. Check out this twist: It’s a key step that I added into my introductory unit on proofs for a much smoother transition into teaching proof writing. It made a world of difference for my students.

I wrote up this post to guide you through a smoother way to set students up for success with proof writing. Click the image to read how to teach it and to download the free files to get you started!

Looking for even more support?

The complete unit I developed includes a presentation and printables to lead your students from the basics (properties, postulates, etc.), through a special revamped Algebra proof that scaffolds their learning all the way up through writing their first batch of Geometry proofs!

This method is the smoothest way to introduce this challenging unit. Check out what's included in the full unit.

Teaching Unit - Formal 2 Column Proofs in Geometry

Free Proof Writing Downloads:

I send out some free resources for two-column proof writing to my email subscribers as part of a welcome kit! Enter your email here to get those downloads delivered right to your inbox:

To Read Next:

43 Comments

Michael Paul Goldenberg

3/4/2017 12:16:56 pm

1) Why proofs in geometry in K-12 but not in virtually any other topic?

2) Do you REALLY believe that doing geometric proofs contributes in any meaningful way to reasoning and/or argumentation about real-world issues outside of math class? Do you imagine that the widespread illogic that informs so much of contemporary American political discourse is somehow due to lack of training in classroom geometric proving?

3) Why "two-column" proofs? Have you EVER seen a mathematician use the two-column proof format in ANY published paper whatsoever?

4) Do you seriously suggest that the sorts of proofs done in K-12 geometry comprise "formal proving"?

Math Giraffe link

3/4/2017 03:31:06 pm

Hi Michael,
Even better if you can do them in other classes K-12! I do like to, but it can be easiest to incorporate in Geometry since that is more standard. Any time you can have students prove new properties in Algebra or other courses, it can be so helpful for them. I absolutely do think that teaching logic in math class can help develop thinking skills, for the future and the real world, yes. The logical structure really does develop different thinking patterns in their brains. It does not have to take a 2-column structure. You can expose them to trying it in paragraphs as well, but for high schoolers, the two colum structure can help them to organize it and have a consistent structure to help them jump in without getting as overwhelmed. It's a good first stepping stone for teens. It's as formal as they get at this age and is a good way to build the concepts. The goal is not for them to be able to achieve what mathematicians writing papers can at this point, but to develop the logic and steps in their mind to get from A to B. Thanks so much! Have a great weekend.
-Brigid

Lindsay

11/17/2017 01:18:41 am

As an geometry and AP calculus teacher, students absolutely use proofs in other courses besides Geometry - calculus students are expected to grapple with logical reasoning and justification everyday.

Big Lenny Jr

9/3/2018 05:17:39 pm

Stop with that cookie cutter conformist BS. Teach them proofs like a true freak would do!

Joseph N Rushton

2/16/2019 10:48:32 pm

1) The reason proofs (as well as definitions and axioms) are emphasized geometry is historical rather than logical: it is because Euclid's _Elements_, which had a rigorous axiom-definition-proof format, served as the standard geometry textbook in the Western world from the time of its writing through the 19'th century.

2) I think the article author really thinks this; I agree with him, and so did Abraham Lincoln. In the Sept 1, 1964 issue of _The Independent_, Lincoln was quoted as follows:

\begin{quote}
In the course of my law reading I constantly came upon the word ``demonstrate''. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?...

At last I said Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.
\end{quote}

3) While most published mathematical proofs are written in prose, yes I have seen published proofs written in two-column format by, among others, Turing Award Laureate Leslie Lamport. In my own experience teaching college students to write proofs (in computer science at Texas Tech), they write *much* better proofs if they are encouraged or required to use a two-column (assertion - justification) format.

4) The term "formal proof" is used in two senses. In the strong sense it means a proof in a format checkable by an algorithm, such as the ACL2 or Coq formats. In the weak sense, as it is more usually used, it means a rigorous proof in the style normally appearing in published proofs. The two-column format is of course less formal than Coq or ACL2, but actually more formal than most published proofs. So, though I would not use the term `formal' this way myself (I prefer "rigorous" for the two-column format), I think the author's use is obviously reasonable.

John P.

10/8/2020 08:59:19 pm

I majored in business. My late mother was a high school math teacher before I was born. She had a BSE and an MS in Mathematics by in the 1960s at age 19! She really worked hard in her classes back when we did not have pocket calculators. Slide rules were the norm, and she never learned how to use the slide rule. She later returned to teaching when I was in high school. When I reached college, she got the opportunity to teach college-level math. She even went back and earned a PhD in Math Education. Cancer stuck her rather young. She died a few years after I finished college.

I came to hate mathematics during high school during the 10th grade because my geometry teacher was using an at least 20-year old Geometry book that emphasized two-column proofs. How bad was it? My Mom was completely appalled that the teacher had the gall to justify giving the bulk of the points on eight-to-nine two column proof questions for a hour long test. It was awful.

I hated the subject to the bitter end after that. I was completely turned off. I even had the teacher criticize me on a geometry problem in front of the class that my Mom and I discussed. I was a construction crane problem. When did not round up the answer to the nearest whole number--the answer had several decimal places. The teacher was unforgiving and said no to my answer. If the crane's limit in degrees came to a certain limit out to several decimals places, you could round up because you would risk breaking or damaging the crane. The teacher would not reason with me. My Mom later told the principal that my sister would not have that teacher. My Mom did not agree with that teacher's teaching philosophy after having a parent-teacher conference. I never learned the details of that conference. My Mom knew her math. She knew it well. She was appalled to say the least.

My memories of that math teacher were burned in my mind forever. I found it interesting that all of the standardized tests I would take would ask questions about the geometric theory but no two-column proofs. The emphasis should not be on just proofs.

I never found a use or any situation where a two-column proof was used anywhere else in any subject. I majored in business with a BBA and MBA. I think to two-column proof can be interesting to explain. I will say that I hated doing them.

My Mom later told me that one or two problems in proof format should be the maximum on a geometry test, and they should not be the bulk of the points to be earned. The concepts and formulas are more important. Putting the bulk of the test points on two-column proofs is ridiculous. To this day, I still feel that way about it. By the way, my sister did not get that geometry teacher that I got. Ironically, the math department at my high school scrapped that geometry book the next year when I was taking Algebra II. When my sister got to high school for her 10th grade year while I was in 12th grade, she got the new book that did not overly-emphasis two-column proofs like my old, antiquated geometry textbook did.

My Mom also thought it was stupid to put Geometry between Algebra I and Algebra II. Unless you could have tested to be allowed to take Algebra I in eighth grade, you would not take it until ninth grade. The state were I lived had this stupid eleventh grade test standardized test--that could force 11th graders to repeat the grade if they did poorly on it--that needed geometry on it. As a result, this forced the stupidity of putting Geometry before Algebra II. My Mom said that students did not have a firm grasp of algebra when the state mandate sandwiched Geometry between the two algebra classes. The sequence used to be Algebra I, Algebra II, and then Geometry. Back then, some education folks in state government got this crazy idea to put in these education revisions that screwed up a lot of stuff. Now, there is this common core stuff. I am glad that I am no longer in grade school to put up that foolishness.

I told a college math teacher a few years ago that another thing that I thought was stupid was memorizing all of the formulas for the sake of memorizing them. They did not teach us how to memorize the formulas. The teachers would go through the motions of how to solve problems. It was suggested that we had to memorize the formulas because we could not look at our books or any notes. I was told, "Times have changed."

I found out years later that I had a learning disability. My Mom did not want to believe I had a problem. Unfortunately if I had been tested, the alternative learning classrooms were abysmal in how they taught those students. That is a whole other issue altogether. I am lucky that I was able to adapt in some ways to get through school. I had some really good teachers when I was in school. I also had some folks that could be total jerks.

Laura

10/9/2020 11:55:34 am

I have noticed with my own kids that teachers' (a) comfort level with math and (b) emphasis on form and process are inversely related. The ones who know less about math tend to overemphasize things like proofs. I absolutely think proofs should be thoroughly taught, but not to the degree you describe. They are a necessary first step in teaching the unnatural sort of abstract thinking that higher mathematics (and logic in general) requires. But they are only the first step. Unfortunately, they are easy to grade and a convenient crutch for someone who doesn't know or care about the endgame.

I'm interested in what textbooks you are referring to because I actually use a couple of old textbooks from the late 60s/early 70s with my kids (Dolciani algebra, Jurgensen geometry). There is a lot of emphasis on proof in the early chapters of both - partly because computer programming was the new Huge Thing but also partly to get students into the habit of thinking about math as a tool - a way to communicate, make an argument, or discover/derive scientific knowledge. Nowadays other subjects emphasize that, too: "tell me how you know this, how do you justify your claim." But until maybe 10-15 years ago, most kids only ever got that in math class. In any case, these books move to looser methodologies pretty soon.

I wasn't overly fond of geometry, but it did force me to learn how to approach, analyze, solve and use problems outside my comfort zone (incomplete info, presented from a weird perspective, etc.) That's an invaluable skill that permeates nearly all professional careers and is becoming increasingly necessary for all of us.

Karen Gold

3/5/2017 12:13:38 pm

I taught high school (not math) for 25 years. Now I'm retired and I teach basic math and reading to adults. About 25 years ago a girl in my homeroom came in and told me, gloomily, that she had a geometry test that day. I asked her what the problem was, since she was doing great in geometry, and she said, "But this test doesn't have any proofs!"

Math Giraffe link

3/5/2017 09:31:44 pm

That is awesome! :)
I love when they get to that point after dreading/hating them. It can feel really successful to finally get the hang of it and enjoy thinking them through. Thanks so much for sharing, Karen!

Vasudha Uddavan link

3/2/2018 07:19:08 am

Awesome write up. You explained why we need proofs beautifully. I use this argument with my students as well. I always tell them to focus on the logical reasoning part of it. Once the kids get the hang of it they love it as well.

Math Giraffe link

3/2/2018 12:05:05 pm

Thanks so much, Vasudha!
I appreciate your comment. I'm glad you are teaching this to your classes! :) Have a great weekend,
-Brigid

Mickey

10/13/2018 04:25:45 pm

Sorry, I just can't buy into this. I'll concede that one will need this to be an engineer or an architect, but I absolutely hated this stuff in high school. Frankly, knowing that alternate interior angles are equal never, I repeat...NEVER helped me ANYWHERE in real life. For me, mathematical proofs were worthless whether I worked in a supermarket, did taxes, or took the bar exam, or anything in between. And my high school's "learn it or else" only added to the worthlessness.

Laura

10/16/2018 10:29:38 am

I presume you are a lawyer? I have found that logical reasoning (being able to parse something out and evaluate each point) helps enormously in negotiating and in trying to convince people to do/believe something without them figuring out my game. It also helps with not getting tricked or sidetracked myself. Aside from mathematical or career benefits, geometric proofs are a vehicle for learning the reasoning process, which is applicable in many fields, whether you are designing bridges and roof trusses, calculating astronomical trajectories, analyzing statistical claims, constructing a usable website, auditing corporate finances, or deciding whether you should buy alkaline water and refuse to vaccinate your kids. We do something similar with literature. Nobody needs to analyze the symbolism of the white whale, and most people hate doing it. We make kids do it anyway so they can practice grammar, argumentation and expressing themselves in a coherent manner. You may have hated doing proofs. (So did I, BTW. Trigonometry was like a breath of fresh air!) But I guarantee that unless you bombed the class you learned something worthwhile that has enabled you to be successful in things like law school. You could not have learned it any other way, IMO. Anything else, including logic classes and non-calculus-based sciences, would have been only memorization and not deep understanding.

Mickey

10/21/2018 09:42:48 pm

Laura: With all due respect, I cannot disagree more. There is absolutely nothing...repeat...NOTHING that a mathematical proof ever did for me in everyday life, logically or otherwise. Recalling that vertical angles are equal somehow didn't help me figure out the marital deduction on my Estate Planning final exam while I was in law school. Nor did knowing the area of a triangle ever come up in any job interview I never had. Admittedly, if someone had ever taken the time to explain the real life relevance, I could at least see the point of it all. But to have it shoved down my throat and told to just "LEARN IT OR ELSE." is forever unconscionable. Now that it's been almost 40 years since my escape from high school, the fact that proving two triangles are "Side Angle Side" congruent is just worthless to me...now as then.

Jalal

5/26/2021 09:04:40 am

@ Mickey -

I would say that an advantage of using geometry to teach children to reason [somewhat] formally [or rigorously, if you prefer that term], is that you have a pictorial connection that allows you to visualize what is being "proven" ["proof" here of course is in a loose sense, not the formal mathematical one which is impossible at this stage to teach to children]. Rather than using abstract concepts or algebra, the rules and application of which require a lot of getting used to, you have something there IN PLAIN SIGHT that you can refer to, when devising a plan of attack. For children, to introduce logic using more abstract material, in my opinion, would be pretty pointless. It is not natural for humans who have not become comfortable with abstract concepts and ideas already. Which is why algebra seems to cause so much confusion. And in particular, it is extremely difficult for children to come up with scenarios where they could use it in real life [even for adults, to be honest!].

As for whether it is actually USEFUL in daily life, I sympathize with both viewpoints. I agree that if you are reasoning in any one SPECIFIC area, be it biological process or business law, logic [whether symbolic or through basic school geometry "proofing"] is not really necessary. You spend so much time learning the material, and you have so much information in mind, that you have probably already made a set of connections between various areas, and have developed [albeit looser] semi-structured arguments as to why something is the way it is.

Having said that, when you do symbolic/geometric/formal logic, you are essentially GENERALIZING your argumentation form, since it isn't necessarily the particular facts of the argument that are important, but HOW you have come to certain conclusions and accept a given argument as VALID [not sound, since that would require true premises]. In geometry, you would use certain "tricks" like similarity/matching, transformation, or "flipping", to come to a conclusion, or else guess your way to a conclusion. But the general problem-solving scheme[s] you are using can be re-iterated in more recent or novel examples [which is why you tend to get better at all].

Can these particular techniques ever be useful in the "real world"? I suppose you would have to hunt for examples of that. One technique I found useful when I took a sales stint quite a few moons ago, was something that Euclid employed in his Element's over and over again - reductio ad absurdum. If you read his proofs, many of them employ this technique where he shows that if you make particular assumptions, the logical deduction shows you would come to a false conclusion. Therefore, the assumption must be flawed, and he goes on to show that this means his posit [which was the opposite of the assumption] must be correct. I used just such a ploy with my customers, loading up assumptions [you use basic information about them, their job, social and marital status, particular locality etc.] referring to them, and showing how NOT buying the product/service was incongruent with their lifestyle and social standing. Curiously, this worked better than simply listing why my product/service was even good in the first place. Perhaps because it was more personal and negative [I've read consumers are more haunted by loss than they are thrilled by gains].

In any case, that sort of thing is what I have in mind when I talk about GENERALIZATION. You can't do specific and general at the same time. So if you are asking what SPECIFICALLY learning the SAS conjecture is good for [area of a triangle might actually come in handy if you have to paint an awkward spot and need to estimate how much paint goes in the tray, and whether you'll need to buy more etc.], it's going to have to be something related to actual space [which dramatically cuts possibilities]. But practice with the GENERAL FORM[s] of reasoning may have lots of applications beyond the immediate goals.

Alex

11/11/2019 02:57:58 am

“Frankly, knowing that alternate interior angles are equal never, I repeat...NEVER helped me ANYWHERE in real life. For me, mathematical proofs were worthless whether I worked in a supermarket, did taxes, or took the bar exam, or anything in between. And my high school's "learn it or else" only added to the worthlessness.”

It sounds like you had a really negative high school experience, at least when it came to geometry class, and that you harbor quite a bit of resentment over it.

If you'll indulge me for a moment, let's take a step back to untangle the "proof" itself from the thing being proved by the proof.

The author isn't saying that the specific mathematical facts that you had to learn in geometry class are going to be useful in general life. As you've stated, the alternate interior angles theorem has not been useful to you outside of geometry class.

Rather, the author is saying that the practice of formally proving something provides exercise in the skill of logical reasoning, and that is useful in general life.

So, maybe you had to formally prove the alternate interior angles theorem in high school geometry class. And, you say that knowing how to prove the alternate angles theorem has been absolutely useless to you outside of that class.

That's fine. I'll give you that. You've never had to prove the alternate angles theorem ever again after high school. That's probably true for almost all of us.

But that doesn't mean that logical reasoning isn't useful outside of geometry class. I think most would agree that general logical reasoning skills are widely useful in real life.

The author is just saying that doing formal proofs in geometry class helps us develop our logical reasoning skills in general.

Mickey

11/14/2019 10:52:35 am

Hi Alex.

My old high school, to be charitable, was just a s**thole, and geometry was just a symptom of the problem. That said, I completely understand the need to learn how to think through things differently. The bigger issue is when schools have a systemic teaching mentality that says LEARN THIS OR ELSE, that can cause a student to shut down mentally, even forever. That's what happened to me in high school, and I'm forever turned off from STEM courses in general, and especially geometry and trigonometry. THANK YOU, BROOKLYN TECH!!! 🤬🤬🤬🤬

Jenny

2/6/2019 12:44:17 am

I guess this was posted a while ago, but I only just saw it.

I'm so frustrated with my daughter's geometry class. They did a couple of proofs, but now in 2nd semester it is all algebra. They are given the theorems, and they solve ratios with them, and stuff like that. Like me, my daughter has a head for geometry and proofs, and was looking forward to sharpening her skills and learning some formal logic. I remembered having fun doing that too. But now it is just another algebra class.

As you say, the reason for learning proofs is that it requires a different way of thinking, so you get better at all kinds of math. It forces you to consider why you believe something is true, and whether it is. The techniques of working from basic principles to useful theorems can apply in other math classes and get you through problems that are extremely challenging.

I get that most students don't like proofs, and they won't use them. So what? I had to take tennis and gymnastics and I have never had any use for that. The Love Song of J. Alfred Prufrock has never been something I needed. That doesn't mean someone who genuinely wants to learn how should be deprived of that opportunity. My daughter asked me if a later class will cover the material, and I had to tell her no. Here's a kid who genuinely wants to take on the challenge and learn, who seems headed for some STEM field, and she is being denied the education. If we have time, we can do some over the summer, I guess.

Math Giraffe link

2/6/2019 02:56:12 pm

Hi Jenny,
That is so frustrating! :(
I am so sorry to hear this. I think in some courses, they are resisting the challenge, or maybe losing sight of the reasons we teach proof. Sadly, some people think that we do not need to be teaching this anymore. I am so bummed that she is not doing much proof in high school geometry. I guess she will get more if she majors in math. Is that her plan? For many students who do not take much college math, this is still so valuable as a thought process in high school. There are fun resources out there if she wants to dive in over the summer! :) Thanks for reading and commenting!
-Brigid

Jenny

2/6/2019 05:26:34 pm

My daughter is a freshman, so it is hard to say what she will study, but for years she has wanted to be a bryologist. On the other hand, this year she discovered engineering class. So who know? I suspect there will be a lot of math, but it doesn't seem likely to be her major.

Really, it is a bummer, but this is hardly the first shortcoming of her schooling that we have had to overcome.

Mickey

2/6/2019 08:49:05 pm

Hi Math Giraffe.

My problem has always been that my old high school never took the time to explain WHY I had to learn this. If someone had at least said it gives one a different way of thinking about and analyzing a problem, I could at least see a rational basis for it, even if I didn't necessarily agree. That said, since my old high school pretty shoved this down my throat with a culture of LEARN THIS OR ELSE, it turned me off to mathematical proofs forever. As I'm an undergraduate law professor now, I can definitely tell you that proving two triangles "Side Angle Side" congruent did nothing for me when I was in college, when I became an accountant, when I went to law school, or in my current teaching career (21 years in). Sorry, but I just cannot buy this.

Mickey

2/6/2019 08:52:33 pm

Hi Jenny.

I'm terribly sorry to hear about your and your daughter's frustration. Have you guys considered the possibility of taking an advanced math course elsewhere? I know geometry sucked for me, but I hate to see someone who obviously likes it to be denied the opportunity to learn it further. God bless you both.

Jenny

2/7/2019 01:28:31 am

Thanks for the kind words, Mickey. I get what you mean about not seeing the point of proofs. I mentioned a couple of things that I certainly didn’t see the point of, and I still feel that way. I like mathematician Paul Lockhat’s explanation for the purpose of math:

“At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement; that works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?”

Our schools act approach math as one thing that students are either good at or not, which seems silly to me, like thinking history is one thing. Not liking geometry should in no way preclude possibly liking calculus, or discrete math, or probability. I think we do students a disservice when we force them to follow a prescribed set of exercises in math and take away what makes it special for each student. You didn’t have to learn geometry, but you were owed a real chance to see if you liked it, and it doesn’t sound like you got that. But you did find other things you liked, such as law, and that’s a very fine thing.

Pearl

10/13/2019 01:23:31 am

I am designing a lesson on quadrilaterals and triangles for 9th graders in India. Most of the questions are proofs and a lot of proofs require students to construct (a line segment, extend a side etc) to solve the problem. How can I help students identify the need for a construction and the kind of construction? Any tips will be deeply appreciated.

Math Giraffe link

10/15/2019 11:20:53 am

Hi Pearl,
It sounds like a great lesson! I have not taught much as far as proofs that require construction - I'm sorry. I wish I could be more help on this.
We generally teach constructions but not integrated with proofs. Best of luck! Have a great day,
-Brigid

Bean

9/23/2020 11:20:55 am

Hi, I am in HS right now and we are "learning" geometric proofs, and right now I can not be more frustrated at this article. I can not see why we are made to learn this, in any job that uses geometry, architecture and fabrication for example. If it is for logic and reasoning then it should be in a english class, it would make later english classes more useful instead of writing multiple page essays and trying to figure out what a quote from a book means, and then forcing one persons take of it down our throats.

Math Giraffe link

9/23/2020 01:33:49 pm

Hi,
Great to hear from a student! :) I definitely get that. They are really frustrating at first as a student. I had the same struggle, but then later on do find myself using the logic even when thinking through political arguments, life decisions, etc. You may be surprised (I hope!) someday. It's really handy to have the skills to lead another person step by step through logical reasoning that makes it impossible for them to dispute. Thanks for commenting, and hang in there! It's usually only a couple of weeks of frustration.

Michael P Goldenberg

9/23/2020 02:42:59 pm

I suspect that this discussion would work better if people all recognized that geometry and algebra are two major flavor groups in high school (and higher) mathematics and that math education researchers probably would tell you that a LOT of people preferred one or the other in school. Assuming they haven't become professional research mathematicians, it's not terribly likely that they've changed their preferences over time.

If you love(d) geometry in high school, chances are that you're a better than average visual thinker and that various elements of geometry appealed to your TASTES. If proofs are part of that, great (for you), though I've known people who generally enjoyed geometric problems but weren't high on doing h.s. geometry proofs.

Similarly, if you're less of a visual thinker, or if proofs really don't appeal to you at all, unless you had an unusual K-12 math education you weren't encountering much proving outside of geometry class. And again, it's unlikely that you had experiences afterward, unless you pursued something very math-based, that would have pushed you one way or another in terms of your sense of what math is, the role of proof, or your proclivities for or against doing algebra vs. geometry.

Of course, all of that is quite artificial, far more to do with the nature of American math education and schooling in general than with the actual subjects. Most of us never get a hint of how big areas of math interconnect repeatedly the more you learn.

I could discuss the impact of delta-epsilon proofs on many students who take calculus that barely touches on such things until the student gets to "advanced calculus/introduction to analysis" when they finally wind up in a course that starts looking at such proofs of limits on day 1, but that's only going to mean anything to you if you've gotten that far in undergraduate math. I'm fairly sure that things would be different for a lot of students had proof been part of every aspect of their math learning from the beginning (naturally, what would comprise a proof of why, say, and odd number plus and odd number must be an even number for a primary grades student isn't likely to be anything like even those secondary school geometry proofs. But mathematical maturity is real and we generally do NOT nurture it for K-12 students, but we easily could.

If these differences were more explicitly recognized, we could stop arguing pointlessly as to what the role of proof is in geometry or any other part of math, students would see proof as an essential component of the entire field of math, and there'd be a lot less silly whining. As things stand, however, we create two major viewpoints, neither one of which is "the truth," and can't understand where people in the other camp get their outlandish ideas. And I suspect that a lot of what strikes me as rigidity and narrow-mindedness from people in both camps would vanish or be significantly reduced.

Jarom

7/7/2021 08:32:56 pm

This. Oh my gosh, this. Thank you. I hate that we have somehow gotten the idea that at least until college, geometry is the only branch of mathematics that uses proof. Mathematics, as an axiomatic system, relies on proof to avoid reinventing the wheel for every problem. Many don't realize that all of the formulae we use in mathematics (e.g., the quadratic formula) have all been incontrovertibly proven to yield consistent solutions when properly applied. Otherwise we wouldn't teach them.

an anonymous geometry student

9/29/2020 05:26:04 pm

I'm a geometry student and its kind of insulting when you say that proofs teach us how to reason. I know how to reason and I'm pretty sure that everyone else in my class does too. Teaching us to memorize theorems doesn't help us to understand math. instead, it gives us shortcuts so that we don't have to understand it. This works in the short term, but in the long term this will make us hate math because if we don't understand it in geometry we're not going to understand it in the future either. Instead of teaching proofs we should teach kids to try to find the answer before teaching it to them. This helps them to find their own way of doing it that they understand more. Of coarse, they shouldn't be expected to be able to find out how to do it. Although, they should be given the opportunity to look for patterns that lead to the answer. I have done this in almost every math lesson in stead of listening to the teacher and have been given the math award for the advanced math class 3 years out of the 3 years i could have received one. I've also come up with a new formula for almost every math lesson. One of the most useful is when i found out that when looking for a missing endpoint and you have an endpoint (e) and a midpoint (m) you can use my formula 2m-e for the x coordinates and the y coordinates individually or you can use an extremely long formula using the Pythagorean theorem.

an anonymous geometry student

9/29/2020 06:52:48 pm

No matter how long I look for people that agree with me, I still have to go back to lesson 3-2 in geometry and write my proof about angles made by a transversal. :(

Bailey

10/2/2020 09:34:08 am

I think that proofs are important because you need to prove that things and equations can be true. Without proofs we would never know if a lot of things are true.

It improves yor comm arts skills and your thinking skills. If you find out how to prove things it will help your smarts and your thinking.

an anonymous math student

10/2/2020 03:36:07 pm

There are other ways to prove things than to do a two column proofs and it seems like more of an English lesson than a math lesson to teach how to prove things.

Another Annoyed Math Student :) link

10/26/2020 09:23:12 pm

I completely agree with this. How the heck does this crap teach us “logical reasoning”. My sense of logical reasoning tells me that we shouldn’t have to waste our time by explaining why we need “reflexive property” to solve certain equations. It seems like Math Giraffe just doesn’t want to hear us though, or just doesn’t care.

Another Annoyed Math Student :) link

10/26/2020 09:24:30 pm

I meant that I agree with “anonymous Math Student”

Math Giraffe link

10/27/2020 10:53:04 am

Some Geometry teachers do not go as deep the logic and deductive reasoning skills. Sounds like you may be missing some of that, and it's not feeling like much logical reasoning. That can be frustrating, and I can see how it can feel like a waste of time when the math is disconnected from the logic. I would definitely encourage you to explore some more logic. Check out symbolic logic. You may like that better since you can see more clearly why it's so important to know, and how it applies to real life. (less silly and pointless feeling than the old Reflexive Property)... If p implies q, and q implies r, then p implies r... etc. That stuff is really interesting, and then you can look into logical fallacies. Just pull out a few news articles and hunt for those logical fallacies! :) You'll spot a lot right away once you get familiar with them. Two column proofs are just a structure that helps some students to get started with this, but don't give up! Move on and dive deeper into some more applicable logic if math is not your favorite path for it. Reason is still critical! :) Good luck.

Herbert

12/7/2022 01:56:11 am

The fact you call it ‘crap’ tells me you were very poorly taught at school. It’s a shame to see people later on in life reject certain parts of their education because of their experiences as a child. As an example I was taught history badly, why were we learning this ‘crap’, who cares what the romans did? Later in life I revisited history and saw something different. In fact the collapse of the Roman Empire turns out to have so many lessons that apply today. There are a lot of things we learn in school that appear to have no practical value, geography, history, mathematics, science (the vast majority of people won’t become scientists so why learn it?), music, Shakespeare, sports (I don’t care about sports at all, yet I was forced to take part in it), the list goes on. Part of the reason for teaching this range of subjects is to create, hopefully an informed and educated society. Republics and democracies tend not to survive long when the population is uneducated.

teacher teaching link

7/5/2021 01:21:13 pm

Irresistible! Thank you so much for this kind and good service.your services is better than better.

Irresistible! Thank you so much for this kind and good service.your services is better than better.

Joel Campbell

2/3/2022 12:53:39 pm

The big problem with this is it’s a required course and many kids that age haven’t yet developed the ability to think abstractly. As a result it’s nothing but pure torture for many of these kids. Even at the university level the only people concerned with this are pure mathematicians - not scientists and engineers, business majors, etc. So why subject kids to something like this when most of them will neither need more want it? If this is such an important topic offer an analysis class as an elective for those who are interested in it and leave everybody else alone.

Math Giraffe link

2/14/2022 09:48:47 am

Hi Joel,
Thanks so much for your comment. This is exactly why it's so needed. The fact that kids that age are not strong in abstract thinking shows why they need this type of learning. It can be tricky, but is essential. Thinking with logic is needed for everyone, not just mathematicians. :)
Have a great week, and thanks for chiming in,
-Brigid

Mickey

10/14/2022 07:16:07 pm

Hi Math Giraffe.

Regarding your response to Joel, I submit that unless one can convincingly prove WHY this has relevance to non-mathematical, non-engineering life, it is an exercise in learned hopelessness. I think one is going to need a more compelling answer than the one-size-fits-all response of "it helps your logical thinking skills." Fine...LET ME SEE IT.

Mickey

10/14/2022 07:02:07 pm

Jalal: The problem, as I saw it in high school is that no one ever took the time to explain WHY the process is the way it is. Alternate interior angles are equal? Vertical angles are equal? Great. Why is this relevant??? Why do I need to know this??? I believe that if teachers took the time to really connect the dots and show WHY something is relevant to the process instead of just showing how to it, maybe so much niformation would not go to waste.

Herbert

12/7/2022 01:27:31 am

I’ve been rediscovering Euclid and I’m loving it. I like the idea of deducing ‘truths’ based on reason and a set of elementary definitions and axioms. I think geometry is great for learning this mode of thinking because it’s so visual and for those more practically mined it has immediate practical applications. Whether one needs to use the two column format is another matter. I think there is some truth however to say that learning how to reason using say geometry, does seep into one’s subconscious so that there is a tendency to start reasoning automatically in other areas of your life. As for learning things in school that one will never need, imagine we only taught basic arithmetic, basic reading skills and perhaps some elementary civics. Imagine what our society would be like within a generation or two?

YOUR CART

Proof That Proofs Belong in Geometry

What Is a Proof in Geometry?

Why Are They So Important?

What Harm Does It Do When Proofs Are Removed?

Benefits Beyond the Classroom

Sound Overwhelming? Read This for Help:

Free Proof Writing Downloads:

To Read Next:

Leave a Reply.

Archives