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. 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:
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22 Comments
Michael Paul Goldenberg
3/4/2017 12:16:56 pm
1) Why proofs in geometry in K12 but not in virtually any other topic?
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3/4/2017 03:31:06 pm
Hi Michael,
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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!
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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 axiomdefinitionproof format, served as the standard geometry textbook in the Western world from the time of its writing through the 19'th century.
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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!"
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3/5/2017 09:31:44 pm
That is awesome! :)
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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.
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3/2/2018 12:05:05 pm
Thanks so much, Vasudha!
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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.
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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 noncalculusbased sciences, would have been only memorization and not deep understanding.
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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.
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.”
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Mickey
11/14/2019 10:52:35 am
Hi Alex.
Jenny
2/6/2019 12:44:17 am
I guess this was posted a while ago, but I only just saw it.
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2/6/2019 02:56:12 pm
Hi Jenny,
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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.
Mickey
2/6/2019 08:49:05 pm
Hi Math Giraffe.
Mickey
2/6/2019 08:52:33 pm
Hi Jenny.
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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:
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.
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10/15/2019 11:20:53 am
Hi Pearl,
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