At what age should mathematical proofs be taught to students

[mathwonk]

if your dad said he would buy you a car if you made deans list, what would you have to demonstrate in order to get your car? was that so hard?

 

[JyN]

I definitely think people should be learning these things much earlier. I don't think its even necessary to teach proper proofs, or even reasoning and logic in a mathematical context. Why not just teach the kids to play games? That's what kids do right? Games like chess, go, those grid based logic puzzles, or pretty much any games that require logical thinking. Kids could start learning these kinds of things very young i think. Instead of gr 2 math class, gr 2 chess class? (Or maybe a game with simpler rules, just using chess as an example)

 

[Intervenient]

The correct answer is sooner than they do now.

 

[Kindayr]

Just teaching students simply the power of an implication at the beginning of high school or end of elementary school would not only help them in math, but general reasoning, essay writing, etc.

The fact that I had to wait until second semester of my freshman year to have a taste for proofs, let alone wait until second year for a proper introduction to proofs is ridiculous.

 

[mathwonk]

check out my web page for a set of notes on euclid's elements that i taught to 8-10 year olds this past month.

 

[xdrgnh]

I started to see proofs in my Calclulus BC class, but we weren't required to learn them, just to understand them. I think students that show interest in what resembles pure math should be introduced to proofs as early as possible. However the general body of math students should be introduced to proofs but shouldn't be expected to do proofs on there own.

 

[deluks917]

I think 7th grade is a reasonable age for students to understand proofs.

 

[TheEigenvalue]

Other than plane geometry proofs and simple "line-by-line" Middle School Algebra "proofs", I did not encounter rigorous abstract proofs until late in my Freshman Year as an Undergraduate.

I feel like it was adequate for me. I mean, I ended up doing well, and was able to take as many proof-based courses as I could get my hands on as an undergraduate that many undergrads did not encounter because they waited to take "Math Proofs" course.

However, I wish I was taught how to use proofs in abstract mathematics earlier in life- perhaps in Junior Year of High School. I feel like my mind was still more 'plastic' back then, and if I could have started thinking in this way back then, rather than using algorithms for solving routine Calculus problems, I would be able to understand the VERY abstract nature of the proofs I am currently encountering in Graduate School. That is my two cents.

However, it should not be mandatory for all students to do this. But it should be encouraged as a positive option for interested High School-ers.

 

[Strill]

I think college is way tooo late to learn how to write mathematical proofs! Proof writing should begin at least either in elementary school or middle school. Proof writing is just as important to a students education as learning how to write sentences and learn how to combined those sentences to create paragraghs and learn how to combined paragraphs together properly to write a decent term paper, because PRoof writing will improve your deductive and reasoning skills. I think a lot of people hate mathematics because they don't understand how the equations were derived . In high school, math was just memorizing formulas and algorithms . When I got to college, They just threw proofs right at me, and now my system that I have been using all my life to passed mathematics failed because you had to apply systematic methodology for writing proofs and so sadly I dropped my math major.

You assume that just by having a proof, the student will understand intuitively why a given relation is true, which in my experience has been absolutely false. Even if a student understands that dividing both sides of an equation by the same non-zero number preserves the equality, the question is: do they understand what these new quantities are and what they represent? And once they understand what those things represent at each point in the proof, can they relate them all back to one another? I would say that unless the students can relate the equations to the concept at each step in the proof, it's not going to make things any more clear.

I disliked math classes because they kept throwing purely semantic busy-work at me that forced me to do MORE work in order to solve SIMPLER problems. I found that proofs were a perfect example of this, probably because introductory proofs are usually applied to things you already know. I always thought of things in terms of my own mnemonics. I hated semantics and especially jargon-filled "technically correct" definitions that make simple concepts less intuitive and more confusing. For example, in middle school, they were teaching us the absolute value function, which had me confused because that's one of the simplest functions to evaluate. I thought of the absolute value function as "drop the negative sign if present". The definition they forced us to memorize was "the distance of a point from the origin on the real number line". This definition, while accurate, is incredibly convoluted. It introduces a new conceptual universe in which the number line exists, and it replaces a trivial operation (dropping the negative sign) with a non-trivial operation (measuring a distance in 1-dimensional space). This definition is also etymologically sterile. It gives no context as to what situations the absolute value function could apply to, or what the absolute function might represent in those situations. All in all, it does nothing but make things more difficult to understand.

There's also the fact that my proofs teachers would mark off 90% of the credit in a question if you missed so much as a period in a sentence. (How the heck am I supposed to know what part of speech an equation is!?) But that's neither here nor there.

Math is not for everyone. People should get out of the mentality its the schools fault. Schools provide all the proofs and motivations if you actually read the textbook. If not there are excellent resources in public libraries.

Hahahah textbooks what a joke. I haven't been able to understand anything written in a math textbook since middle school apart from the equations in bold. Pretty much all of the content of the text is someone rambling on with derivations for which no context is given. They're incomprehensible to someone not three grade levels higher than the grade the course is intended for.

if your dad said he would buy you a car if you made deans list, what would you have to demonstrate in order to get your car? was that so hard?

Unfortunately mathematical proofs are not that simple, even if they could be, because they're combined with a whole new set of terminology, jargon, and strict rules that turn a conversation with your father into a http://tvtropes.org/pmwiki/pmwiki.php/Main/ptitlei9fyz80ocg6y .

Take this example from the game "Mystery House".

[color=green]>Go North[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>North[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>East[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>West[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>South[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>Go House[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>Go Porch[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>Go Door[/color]
I DON'T UNDERSTAND WHAT YOU MEAN
[color=green]>Go Stairs[/color]
YOU ARE ON THE PORCH. STONE STEPS LEAD DOWN TO THE FRONT YARD.

A lot of the time, that's what proofs feel like. You're forced to use contrived unfamiliar forms of terms and ideas you're already familiar with, which makes it harder to keep track of things and makes you end up lost and frustrated.