Do you hear a lot of grumbling in your home when high schools kids struggle with geometry proofs and can’t see the point?

Are geometry proofs really necessary? Do they have a purpose other than passing the math test and doing well on standardized tests? Is this a life skill your kids will use?

Unless your child majors in math, it’s highly unlikely he’ll ever need to the specific skill required in completing geometry proofs. And chances are he’ll soon forget how to do them.

So what’s the point? Why do we make kids go through this ordeal?

Learning to do geometry proofs is a brain-boosting activity that helps improve children’s brain function, often permanently. In other words, the thinking skills used in doing geometry proofs are important thinking skills your child will use in other academic classes . . . and life in general.

Doing geometry proofs requires the brain to operate in new and complex ways, forming and reinforcing complex brain connections. Once developed, these neural connections remain, ready to “jump into action” in real-life situations, long after how-to-do-geometry-proofs has faded into mental oblivion.

Brain patterns developed by doing geometry proofs include three essential skills.

**Organization:** Doing proofs requires organization, forcing the brain to cultivate and improve neural paths in the executive function area. This involves sorting the given information, making diagrams, labeling, and keeping track of the progress throughout the task. Until they’re doing geometry proofs, your kids probably haven’t done any activity that requires such complex organizing skills.

**Logical thinking:** Doing proofs requires logical thinking, a mental process that is rarely well developed in the younger high school students. The act of doing proofs provides your child with a great opportunity to develop, or improve upon, this valuable higher order thinking process.

**Self-discipline:** The mental and physical tasks required when doing proofs are unnatural, tedious, and difficult for many students. Now they’re being asked to make diagrams from sentences and symbols, and to plan, carry out and coordinate all the required activities. As a result, your kids tend to develop greater self-discipline.

So doing geometry proofs isn’t just for passing tests. It’s good for the brain. Doing proofs help make young brains better, often permanently better. In fact, math in general is good for the brain.

Please leave a comment about geometry proofs or math in general. We love to hear from you.

geometry proofs are not needed. the only answer I’ve ever gotten for “why do we need to learn this?” to geometry proofs is “because for when you’re a teacher, you can teach it.”

don’t kid yourself that it helps the brain, all it did for anyone in my class, and anyone who I’ve ever talked to about proofs is get them frustrated and annoyed that we have to do such pointless work.

The problem is, many kids have trouble accepting that their brains are still developing. They think their brains are perfect the way they are. They are not correct.

It is hard to believe, but it is true. Until 30 years old and even beyond, the well cultivated brain becomes better and better.

I agree that school subjects appear to be useless, but learning them stimulates brain development. The more variety, and the more complex, the more brain connections. Math develops the brain one way, science another, foreign language another, and so forth.

Unfortunately, many teachers act as if their subjects are the end-all. Some even skirt the issue and say, as you describe, learn it because “I am the teacher.” The truth is, and you’ll surely agree, kids will forget the details very quickly. But, don’t stop reading here.

By learning school subjects, not just cramming and getting by, but by really learning the material, the brain becomes “wired” better. And it stays that way, forever, ready to jump into action when needed. And, please believe me, it will be needed.

Being able to think well is good! And a well developed brain is a treasure that remains so long after CPCTC has faded from memory.

As a current high schooler taking geometry and advanced Alg. 2, I realize both that my brain is not perfect and that am an outlier, but I feel that I already understand the concept and don’t need the overly complicated notation. It certainly doesn’t help when I see people like Vi Heart of the Khan Academy mocking the notation, it reinforces my disgust and disappointment with the curriculum.

I agree that some of the notation is excessive, however, much of it is needed so that everyone involved can communicate precisely. If it’s a matter of being too formal, especially when doing proofs, I hope you’ll be permitted to use less formal language, as long as it is still precise.

I had no idea that so many people hate geometry proofs . I personally was a great fan of them in high school . But after seeing so many geometry haters I think there can be a problem with teaching methods or the proofs are a bit difficult for majority . But beyond any doubt the proofs are a great way of intellectual exercise but may be ineffective for mass . Studies should be done to measure its effectiveness in comparison to alternative activities . It can be made optional after a trial . Proofs requires abstract thinking , bit of creative imagination, analytical reasoning in a disciplined and organized step by step manner . So it is like a multi gym . To people who think proofs have no practical application , I would like to ask what are the practical uses of push ups and sit ups ? Often students are not properly taught how to think and prove these and start memorizing and hating them simultaneously. So its ok if you hate proofs ,you may not have been taught properly or you have a different personality with another form of intelligence or talent . But if you think proofs are stupid or pointless then it really gets geometrically proved that you are a stupid . People who does not like a subject will always find it useless .

You made some good points. Thanks.

I had no idea that so many people hate geometry proofs . I personally was a great fan of them in high school . But after seeing so many geometry haters I think there can be a problem with teaching methods or the proofs are a bit difficult for majority . But beyond any doubt the proofs are a great way of intellectual exercise but may be ineffective for mass . Studies should be done to measure its effectiveness in comparison to alternative activities . It can be made optional after a trial . Proofs requires abstract thinking , bit of creative imagination, analytical reasoning in a disciplined and organized step by step manner . So it is like a multi gym . To people who think proofs have no practical application , I would like to ask what are the practical uses of push ups and sit ups ? Often students are not properly taught how to think and prove these and start memorizing and hating them simultaneously. So its ok if you hate proofs ,you may not have been taught properly or you have a different personality with another form of intelligence or talent . But if you think proofs are stupid or pointless then it really gets geometrically proved that you are a stupid . Many people who does not like a subject will find it useless .

I see the point you are trying to make here, but why such a bizarre technique? If they want to tease our brain why not just do it with less complex EVERY DAY problems. Not “Why is this equal to this?” even though we all know it to be true and not needing to be proved?

Geometry proofs are NOT neccessary, there are dozens of ways and activities that “boost” the brain in several different matter. Proving that triangle ABC is congruent to triangle DEF is pointless. Why not do problems that boost the brain everyday a little bit, instead of huge gallops through proofs that just make everyone angry? Why not learn in a double fashion meaning that they will both use the skill and it boosts brain power?

is there even medical evidence to suggest this activity boosts the teenage brain more than any other geometric related activity?

OK Linda, I'm a 50 year old PhD student. I hated geometry when I was in high school, and HATE it now as I review math in prep for a graduate level stats course. I'm a medical professional that has well-developed cognitive function. I can truly say that after a half-century of life geometry is useless. I'd love to see the research that supports your contention that geometry forms those magical synapses in the brain. I'd rather use some Dopamine. Geometry and those that support it can go pound sand. I'm just glad that Mr. Oliver, the useless geometry teacher from 1976, is probably dead by now….good riddance.

Positively no connection. Notice how no 'math' was used to answer the question. Just a selling point. Really? How to prove that a right angle is perpendicular to a straight edge? Like inventing a new thought process? It's sickening if it weren't so lame. Why geometry isn't being taught at all!

Positively no connection. Notice how no 'math' was used to answer the question. Just a selling point. Really? How to prove that a right angle is perpendicular to a straight edge? Like inventing a new thought process? It's sickening if it weren't so lame. Why geometry isn't being taught at all!

im a sophmore. i hate geometry. it's fifty five minutes of torture.

a prooof is nt logical thinking, there is different logic for different things like i can say programming is logical thinking and a geometry teacher that logically thinks wouldnt b able to figure it out. there is different logic in different things in order to do proofs u need to know the math reasons on why the peices are s=cingruent or similar, its un neccessary now in programming u need to know the code which is logical thinking for that there is a big difference. proofs r an extreme waste of time because they rlly have nothing to do w/ math. people can learn organization and self discipline doing other tasks. proofs u need to know the reason to prove ur statement true and u will never use these reason ever again in ur life, if u can actually think of a decent argument please e mail me

[This comment answers the other comments with which this commenter strongly disagrees.]

… Mathematical reasoning skills are vital for modern living. To Nate… computer science is based on discrete mathematics, which, you guessed it, is based on proofs.

Being able to logically reason through geometric proofs is a very useful skill. Being able to take two ideas, and equate them is vital to modern life. Geometric proofs help create a structure of logical equivalence in your head that applies unilaterally to almost any situation.

Outside of all the practical applications of proofs, quite frankly I think that you need to consider that the only thing that pushes us forward as a society is scientific discovery, which requires an educated population. Even if you’re not going to be a scientist you need to be able to understand basic coloration/causation differences and the one-way properties of the logical implication operator, and a multitude of other skills taught by hated geometric proofs.

Yeah, having to think to complete your homework, inside your nice home in your nice first world country, built and provided to you by the ‘evil’ engineers who thought it was important to teach thinking skills, is difficult. Get over it.

GEOMETRY IS STUPID! we are NEVER going to need to know geometry proofs!!! you DON’T open a child’s mind for life with proofs!

there’s no such thing as teaching useless things in school to “open a child’s mind” because it doesn’t work!

all it does is make kids go to school twice as long as what they would normally have to if they could just learn stuff they actually need to know! i’m a sophomore (10th grd) and i’m in geometry and i know that i will NEVER use ANY of this! my dad builds houses and etc. and he doesn’t use geometry! he uses inches for measuring! that’s all! he doesn’t know that vertical angles are equal and what 4x-3=23x+9 and all that mumbo jumbo for figuring out a proof! he uses common sense! in fact, my mom was a banker and both my parents NEVER even took PRE-algebra in high school! if you need to know how much liquid to pour in a container with a circumfrence of 7 and a height if 9, then instead of doing that math, just fill up the container until it’s full! DUH!!!!! why do you have to make everything so hard on yourself and people! just do things with what you were born with…..common sense and you’ll be perfectly fine! if you are going into some weird math major then learn math in college! don’t make EVERYONE learn it for the less than 1 percent who will ever need to know it!!!!!!!!!!! that’s what college is for isn’t it???!!!!!

Jessica,

I hope your parents don’t read what you just wrote.

I think your father would take offense to your statement that he doesn’t know geometry. I’ll agree he probably doesn’t use proofs, but he regularly uses geometry – and you’ll likely do so as well.

Inches are part of geometry. I bet your dad knows and uses the 3, 4, 5 rule. If you’re building a square or rectangle and you need to make sure that you’ve created a right angle (so books don’t slide off your shelves), you can use this simple geometry rule.

Measure 3 inches down one piece of wood and draw a dot. Measure 4 inches down the other piece of wood and draw a dot. Then using your tape measure make sure the two dots are 5 inches apart – and the ends of the wood come together at the corner. Now you have a right angle – and you can finish building your house, shelves, picture frame, etc. . .

He knows that vertical angles are equal because he builds roofs. The back has to have the same angle as the front.

He cuts molding so he understands how to cut and measure angles, how to calculate square footage and much, much more.

If you spoke only of geometry proofs, I’d have agreed. But your parents use geometry, ratios, and algebra more than you know.

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Jessica I can tell you as a Master Student in Applied Mathematics I really don’t like proofs (I feel like I should have been an engineer) and I do think in lots of cases they are a waste of time (e.g. prove 1+1=2 or prove their are infinite prime numbers). However geometry (e.g. pythagorean theorem) I have used in my summer job, and I have used algebra many times when I have been cooking (e.g. making a serving size that isn’t one of the default sizes and figuring out how much of each thing I need), so there are practical reasons for it. When people say “what are you going to do with math” I laugh back at them and say “anything” if you have the computer skills to go along with it. It is ridiculous how people think degrees like history and English are more applicable then math, almost every technical field (c.s., engineering, operations research, economics, etc…) uses calculus. And if you don’t believe me ask a professor in any of those fields.

Hi, I’m a freshman, and like most geometry students i dislike proofs greatly. My teacher is making my class write an essay (yes, in geometry i have to write an essay) about the benefits of them and why they are put into the curriculum. But now, looking over my research i am starting to realize their importance. Still, I’m sure Euclidean geometry could of adapted to some other method of building organization, logical thinking and self-discipline besides PROOFS!

I’m a Sophomore and all I can say is that proofs suck. Waste of my time and all the engineers and stuff mentioned above can better society while I’m getting rich on math that makes the world go round. Thats right, making money in the stock market. So when I am done with geometry it can burn in hell for all I care. End of my rant.

I am of the general opinion that Geometry proofs are not necessary. I have a degree in Engineering and a Master’s degree in liberal arts.

there are many other ways to exercise and train the brain without using geometry proofs, including syllogisms, logic and rules of evidence. debate, and expository writing.

I excelled in math…until I hit proofs, and once I was allowed to move on to Calculus, Differential Equations, Laplace transforms….I never, ever needed to use proofs to excel in mathmatics again.

Since it is not required to do well on the ACT or SAT test, I did not require my daughter or son to do them. And they are doing very will in all of their subjects, including science and theology, thank you very much…

Thanks so much for your comment. Your comment inspired me to write a new blog post. You’ll see that I agree with you in principle, and look forward to seeing changes in education that result in the same benefits with less of a down side.

Here’s the link.

http://drlindasblog.com/geometry-proofs-problems/

I liked math until I had to do proofs in high school. The only thing doing proofs did was make me afraid of any math because I felt inadequate with trying to understand them. I went in before and after school for tutoring, and still didn’t understand them. I took some college again in college and it finally made sense to me, but the teacher used “real world” examples and made it much less intimidating.

I so agree with you that “real world” examples always make learning easier and not just for geometry proofs. Glad you finally found doing proofs less intimidating.

The three headings,

Organization:

Logical thinking:

Self-discipline:

Are really good choices i thought. I believe that doing proofs helps you in life because it gives you insight on how to think through problems in a logical way. “Once developed, these neural connections remain, ready to “jump into action” in real-life situations, long after how-to-do-geometry-proofs has faded into mental oblivion.” this quote describes my feeling for proofs. I dont like them but they are useful later in life.

I’m currently a sophomore in high school studying geometry. I searched for the reason why proofs were necessary, and stumbled upon this article as well as the other one you’ve written. I agree wholeheartedly with your statements.

Proofs still annoy me massively as I can easily solve geometry math problems when I don’t have to present every little modicum of evidence that I come across and memorize every single rule out there – but as it’s all for the sake of building myself a better brain, so to speak, in my mind it really is worth it in the end.

Truthfully, I’m grateful that I even get this chance and look forward to cultivating myself further in the future, in both body and mind.

Bob,

Your last sentence speaks volumes. I hope you reap the benefits of your hard work and good attitude.

Bob,

Your last sentence speaks volumes. I hope you reap the benefits of your hard work and good attitude.

John,

I’m impressed that you searched for the reason for studying geometry. It’s easier to simply gripe about things that are challenging than to learn about possible benefits.

I hope you reap the benefits of your hard work and good attitude.

I’m a high school geometry teacher. I sold my students on learning proofs is about being able to logically construct and support an argument. Half the classwork is math related and half is not.

I flunked geometry twice (the second time with a 64! What an A-hole teacher, right?) That was over 40 years ago. I’ve aquired two graduate degrees since that nightmare and completed seminars at Harvard and Cornell (not in math, of course).

Of course I did not study in high school. That was one reason I was a lousy student. Reason two was because only my high school history teacher told me I had no talent in any subject at all. Encouraging, huh? Why work hard when you’re told you have no academic talent?

What I’ve learned is the following. I subscribe to Howard Gardner’s view (Harvard) that there are different “intelligences” that all people have to one degree or another. Mathematical and logical thinking is one type of intelligence—abilities in language, music, spatial dimensions, etc., are other abilities (eight in all).

Finding out what your most “developed” intelligences are (this is physical inside the brain) is the most important thing you can do to develop what you are good at. If you fail mathematical subjects even though you study to grasp them, the reason is because your brain does not function well in that particular type of intelligence. A tutor won’t help you much.

The answer? Stay away from math and try subjects that are more attuned to the intelligences that are more developed for your particular brain. The term “don’t strain your brain” is true–don’t keep taking mathematical subjects if you can’t get past a C.

Good luck with the school system. It is not based on finding your best intelligences, it is based on a set of standards you must complete to succeed. You must do it yourself.

The Sage of Wake Forest

Thanks for a well written response. Schooling exposes all of us to a little of everything. It doesn’t take us long to find out where our strengths and weaknesses lie. If one area proves to be out of easy reach, we need to do the best you can to get through it or work around it.

Naturally, as you say, go toward a career track that doesn’t require those abilities. However, if you can not avoid a required course, do what you need to to get by. Seek help from a parent, sibling, relative, friend, teacher, or tutor. With luck you’ll find someone who can present the material in a way that WORKS FOR YOU. If the material makes sense to you, it’ll help you succeed.

One final point. Reading between the lines, while the teacher you describe in the opening line may very well deserve your description, I’m inclined to think he or she was delivering a message. This is based on what you wrote, “Of course I did not study in high school…” I’ll bet, had you showed that teacher even a modest effort, you’d have gotten a passing grade, or higher.

Again, thanks for a thoughtful response.

I’m a big fan of Howard Gardner. I agree with the eight intelligences of learning. It has made a major impact in my life. I have been able to impart his teaching onto my daughter.

Hi Caroline,

Thank you for commenting. That’s great that Howard Gardner’s work has had such a positive influence on your life and that now your daughter is enjoying it too.

As a high school geometry teacher, I understand why my students do not like proofs. I see why they think they are pointless. I see why many of these replies (adult and adolescent) are so full of anger and resentment. Even the engineers and doctors who claim they never used them. But what you are saying is exactly right. The content is a vehicle to the growth of the brain’s functions.

I find it absolutely terrifying that many of my students, regardless of writing a proof in its entirety, cannot even connect a statement to the reason it can be said. Proofs to me are the ability to form a leak-proof argument, a skill that everyone needs to have. When I hear grown people fight about the economy, politics, how to raise their children, and have no reasons to back up their statements, I think…wow, did they never learn the point of proofs?

Maybe a stretch. But I still remember my high school geometry teacher telling us, whenever you state something as a fact, you better be able to prove it. I wish more people took that idea away from geometry class.

You’ve nailed it. I think if teachers would allow a more casual approach to proofs, while still being precise, that would help. I’ll bet you concur. Thanks for your comment.

I loved algebra but geometry had to grow on me. I think that’s largely because my teacher didn’t start with an explanation of the practical value of the calculations. If I’d understood the point, learning to do the calculations would have been easier.

It is interesting to me that so many people who love Algebra hate Geometry. I was the opposite. I could understand every last “why” of geometrical problems. Proofs were magical to me. For the first time, I actually understood why formulas are true and what they mean. Algebra was never explained to me adequately. I had difficulty knowing when or why to apply this or that formula. I had to memorize as many different types of word problems as I could in order to know if I should use the quadratic equation and how to plug in the information. I never understood it. Geometry made sense. I loved that. It took awhile for me to get it right, but once it clicked, I could write pages of valid proofs. I never understood why others hated it so much.

Speaking of the developing brain, I took an algebra class at 30 and understood a lot more by the end of the class than I ever did in high school or college. The teacher was horrible, but something started to click for me. So it is true that our brains are still developing well into adulthood.

You’re right, it’s amazing how much easier those challenging subjects would be if we were able to take them after our brains reach the necessary level of development. Thanks for your comment.

I have such a hard time with geometry proofs. I just can’t wrap my head around it. You can give me Algebra problems and I will happily solve it but geometry proofs make me frustrated and upset I get so angy with myself for not undetstanding it. Is there some type of book that could help me? I

Ask your teacher for help? If your teacher can’t help you, ask who in your school helps kids with math. I’m not sure another book other than your geometry book would be very useful.