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At what age should mathematical proofs be taught to students

  • Thread starter pentazoid
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Here's another idea. Question to the class: Starting with a square, how many times do you think you can cut the figure in half and then remove a half and then repeat the process on the figure remaining? The students may suggest a finite number (say 15). The teacher gives each student a square sheet of paper and has them actually go through the process. Eventually, they end up with a piece so small that it seems impossible to cut in half. Now, the teacher comes back and says "imagine if you were really really tiny and had a tiny pair of scissors, do you think you would be able to cut the figure in half again." The students may suggest yes, and suggest the idea that as long as they could keep shrinking themselves and their scissors, there will always be a piece to cut in half. The teacher can have the students represent their work with numbers: How much area of the original square remains after each cut? This example introduces students to the idea of infinity, infinite series, and one-to-one correspondence while employing their current knowledge of areas (each piece is some fractional area of the original) and geometric figures (the figures oscillate between square and rectangle).

I remember in third or forth grade they had use cut out a triangle, then cut all the corners off the triangle and put their tips together. No matter what triangle you made when you put all the corners together they made a straight line. I distinctly remember that glimmer of insight I saw. I still didn't understand why, and almost didn't believe it. But it was right there in from of me.

I don't think I saw proofs again till high school pre-cal. I had a very good math/physics/comp sci teacher all through high school and I remember he showed proofs for nearly every formula we used, but only demanded we learn a couple. I think in that in Manitoba, Canada it is expected that all grade 12 pre-cal students be able to prove the law of cosines; it often comes up on provincial exams. Also, I noticed someone here said they didn't do proof till after intro calc? Well I am not sure about anywhere else but at my university on the first day of intro calc they said you will be expected to be able to prove every theorem used in the course, and believe me they didn't shy away from that promise in the final exam! Every university math test I have ever taken is at least 50% proofs.
 

mathwonk

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My friend taught his kids that cookies with alcohol in them were bad for them so they should not eat them. The three year old saw the 5 year old eating a cookie and argued if that cookie was not bad for the 5 year old it must not contain alcohol hence would not be bad for her either. Having proved her case she got her cookie.

Proofs are merely valid logical arguments and can begin very early in life.
 
First time I saw a proof was in Discrete math 1 and then the next time was Real analysis . I think students should be introduced to proofs as soon as they can handle it.
 
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First time I saw a proof was in Discrete math 1 and then the next time was Real analysis . I think students should be introduced to proofs as soon as they can handle it.
Are you for real? I have seen derivations for everything we did in maths starting from 7th grade starting with things like the formula for quadratic equations. Also don't they at least teach proof by induction in the calculus sequence?
 

symbolipoint

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When to teach proofs depends on the student. Grade 9 or grade 10 should be the maximum age-range to begin teaching proofs, regardless of the traits of the student - as long as the student is in something like Introductory Algebra or College Preparatory Geometry. Either they adapt to proofs or they do not, but they need to try. For myself, I struggled badly with proofs in Geometry in high school, but later I improved.
 
Are you for real? I have seen derivations for everything we did in maths starting from 7th grade starting with things like the formula for quadratic equations. Also don't they at least teach proof by induction in the calculus sequence?
I didn't see induction until Stewarts Calculus book in first year.

Thinking about it we sort of did "proofs" in geometry with similar traingles and congruence. However, it was only given eyeservice and the treatment was very quick. Not only do I not remember how to do them but I also haven't used it since highschool.

My highschool teacher was like.... "here this the quadractic equation it gives you roots to 'stuff ' ". :(
 

Tom Mattson

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I teach proofs to my students as soon as they have the tools to do them. For instance in our elementary and intermediate algebra courses, students are taught that parallel lines have the same slope and that perpendicular lines have negative reciprocal slopes. Since these students know nothing of trigonometry, I present these as basic facts. In our college algebra course, they know some trig (at least what the tangent function is and that the tangent of 90 degrees is undefined), so I present the formula for the angle between two lines as a basic fact, and use it to prove the assertions about parallel and perpendicular lines from that. And when we get to precalculus, we do analytic trigonometry and so I prove the formula for the angle between 2 lines.
 
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I don't exactly know what proofs are even now(I had to prove physics derivations, is that what you're talking about) but I'm sure if I was taught that way as a kid rather than just having random seemingly useless facts thrown at me I'd have learned math a lot better and earlier than taking til Junior year of high school before I finally started to get it.
 
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I encountered proofs in 8th grade (geometry) but it was taught so rigidly that it just became another 'plug and chug' crap course. conjugates and propositions were taught like arithmetic operators. the other part was that i had no interest in math, which totally destroyed me, but i can 99% of people don't have any interest in it until much later on anyways. then i finished calc in 10th grade, and stopped doing math altogether until college (we only needed 3 years, so i took stat instead).

horrible choice.

what i would have loved was a course describing the properties of numbers, something along a middle school level, and a bit of logic injected in there (basically discrete math for kids). with today's technology, this seems like it'd be appealing.
 
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Yeah, I like math when I was really young, you know, when it was just adding and subtracting, by 4th grade I started to hate it as I had crappy teachers pretty much every year from that point on until I got sent to a remedial class in 11th grade where I finally got a teacher who taught the reasoning behind why this and that were and what point there was to it, that's when I learned I liked science and math. I think kids can learn a lot more than we give them credit for.
 

mathwonk

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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

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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)
 
The correct answer is sooner than they do now.
 
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Just teaching students simply the power of an implication at the beginning of highschool 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

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

Take this example from the game "Mystery House".
Code:
[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.
 
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