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At what age should mathematical proofs be taught to students

by pentazoid
Tags: mathematical, proofs, students, taught
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pentazoid
#1
Feb24-09, 10:03 AM
P: 185
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.
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SticksandStones
#2
Feb24-09, 10:19 AM
P: 104
What makes you think that elementary school children have the intellectual maturity necessary to understand mathematical proofs?

Further, given that their mathematical sophistication would most likely extend as far as addition of three digit numbers what precisely would they be seeking out to prove?
pentazoid
#3
Feb24-09, 10:36 AM
P: 185
Quote Quote by SticksandStones View Post
What makes you think that elementary school children have the intellectual maturity necessary to understand mathematical proofs?

Further, given that their mathematical sophistication would most likely extend as far as addition of three digit numbers what precisely would they be seeking out to prove?
I hope 9 year olds aren't just learning how to add 3 digit numbers. IF they are, then the kids who had the luxury of being taught in the public education system, are really screwed.

sure they have the intellectual capacity to write proofs just like they have the intellectual capacity to to write sentences and paragraphs. I think they lacked mathematical sophistication because we are not taught how to write proofs. Students should learned how mathematics equations and diagrams like Pascal's Triangle and N-factorial are derived
because right now they really don't have a strong grasp on understanding what those equations mean.

Tobias Funke
#4
Feb24-09, 05:15 PM
P: 138
At what age should mathematical proofs be taught to students

I can't say what age exactly, but certainly sooner than they are now. Do high schools in the US even do proofs anymore? It doesn't even have to be formal proof, but some kind of "mathematical reasoning", to use a tired phrase, is necessary. Students are so used to being fed rule after rule that they completely shut down when they're asked to think. This happens even in AP calculus with strong students. Why are volumes of revolution so hard for them? Because there's multiple steps and no simple formula to just plug things into. We're really doing a disservice to students.

It happened to me today when trying to explain the exponent rules. Students couldn't understand why I wrote out (5^5)/(5^2) as (5x5x5x5x5)/(5x5), canceled, and got 5^3. They asked why I didn't just tell them that the answer was 5^3 instead of trying to explain why. They also couldn't understand why I crossed out two fives from top and bottom (they're in 9th grade).

Even worse is that it seems to be acceptable and normal for entering high school students to not be able to work with fractions! Why is there no national outrage over this? Just about anything short of proving that every vector space has a basis in the 3rd grade seems worth trying.
physics girl phd
#5
Feb24-09, 08:04 PM
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P: 936
Quote Quote by pentazoid View Post
I hope 9 year olds aren't just learning how to add 3 digit numbers. IF they are, then the kids who had the luxury of being taught in the public education system, are really screwed.
I think the first time I encountered proofs was in 8th grade algebra or 9th grade geometry. I think that was a fine time to do so (I remember really enjoying them in geometry... since my teacher had given us noticed that if you proved something you thought was useful and handed it in as extra credit, then you could also use it later in your proofs on tests). I went through the public system and think I turned out just fine with regards to handling higher mathematics.

I'm not sure if you have a nine year old, but my nine-year old still still can't remember to tuck in his shirt or wash his face, and tying his shoes is hit-and-miss. He can't always get his single digits multiplication tables with reasonable accuracy, especially if we don't quiz him on them. I'm trying to keep him interested in math by having him help me "test" activities I'm designing for high school teachers on probability, etc. (and get practice on addition, subtraction, etc. that way).
pentazoid
#6
Feb24-09, 08:33 PM
P: 185
Quote Quote by physics girl phd View Post
I think the first time I encountered proofs was in 8th grade algebra or 9th grade geometry. I think that was a fine time to do so (I remember really enjoying them in geometry... since my teacher had given us noticed that if you proved something you thought was useful and handed it in as extra credit, then you could also use it later in your proofs on tests). I went through the public system and think I turned out just fine with regards to handling higher mathematics.

I'm not sure if you have a nine year old, but my nine-year old still still can't remember to tuck in his shirt or wash his face, and tying his shoes is hit-and-miss. He can't always get his single digits multiplication tables with reasonable accuracy, especially if we don't quiz him on them. I'm trying to keep him interested in math by having him help me "test" activities I'm designing for high school teachers on probability, etc. (and get practice on addition, subtraction, etc. that way).
Well I didn't encountered proofs until sophomore year of college and I did terrible in my proof class, because up until then , I just wasn't used to proof writing. At least in elementary school, they should teach you what deductive reasoning is and introduce you to logic, and they should demonstrate to the student how proofs are written. It just doesn't make sense to me to have study math for 13 years and just begin encountering proofs in your sophomore year just after you studied calculus. I supposed when you are only studying arithmetic, writing proofs are not necessary, but students should still be taught deductive reasoning. I think you should begin proof writing when the student begins to study geometry and algebra.
csprof2000
#7
Feb24-09, 09:59 PM
P: 287
It should be available at the advanced middle school level for good students, and at the early to intermediate level of high school for everybody.

Elementary school? Maybe some simple proof system... like propositional calculus. I bet a 10-year-old could prove "If A then B means NOT(A) or B", "If A then B and if B then C then if A then C", etc. This would have the added benefit of giving them exposure to things like set and logic notation. Proofs are also easy for logic.

Thoughts?
Dr.D
#8
Feb28-09, 08:40 PM
P: 619
Long, long time ago, I first encountered formal proofs as a high school junior in plane geometry, and I thought it was just wonderful. It was the thing that has been missing in all my math classes up to that point. It could have come earlier.
buffordboy23
#9
Mar2-09, 11:21 AM
P: 540
The first time I saw a proof was in my 8th grade algebra course. I don't recall the teacher going over them or assigning them as homework, but the text, Algebra I by Dolciani, offered numerous examples. I really didn't understand what a proof actually is and why they are necessary until college, which is really sad.

What's worse is that the middle school algebra texts of recent publication appear to offer no example proofs and are not very rigorous from what I have seen in the school district I used to work at. Worse still is the fact that mathematical rigor is being replaced by "gadget tricks", that are supposed to aid in student understanding but is probably detrimental over the long hall.

I think formal proofs can begin as early as the 6th or 7th grade. However, many students will need heavy scaffolding on the teacher's part to be successful at first. I think activities can be designed for K-6 that nurtures mathematical discovery and teaches one to think mathematically, but when it comes down to it, it's a lot easier for a teacher to run a game of "Around the World" to teach students their times tables for 2-3 weeks straight.
Bourbaki1123
#10
Mar2-09, 02:57 PM
P: 326
I know that when I was 7 or 8 I could probably have handled some simple proofs. I did math quite a bit and learned basic algebra skills by figuring out how to play the math games you got to play if you finished your work early. I really enjoyed algebra and worked on it(with my teachers and parents and books) until 6th grade when I had to repeat the same material in a so-called 'advanced' math class. I got bored and stopped until college only going up to algebra 2 in high school, and now I will be taking my first graduate course next year as a junior.

That being said, I know that not everyone has an affinity for math and that some people are even put off by proofs after they learn how to do them and drop the major; so, how certain are you that younger kids who have questionable enthusiasm for mathematics would latch onto proofs? I think that if we could have more extracurricular math besides competitions, where interested students can learn a more rigorous version of what is presented in class and not be scared away by the competition aspect(anyone should be allowed to join, no classroom competitions, save that for the math team), then we could certainly bring in those with inclination to see what mathematics is really about.
Wellesley
#11
Mar3-09, 07:09 PM
P: 276
Quote Quote by Tobias Funke View Post
Do high schools in the US even do proofs anymore? It doesn't even have to be formal proof, but some kind of "mathematical reasoning", to use a tired phrase, is necessary. Students are so used to being fed rule after rule that they completely shut down when they're asked to think. This happens even in AP calculus with strong students. Why are volumes of revolution so hard for them? Because there's multiple steps and no simple formula to just plug things into. We're really doing a disservice to students.
It happened to me today when trying to explain the exponent rules. Students couldn't understand why I wrote out (5^5)/(5^2) as (5x5x5x5x5)/(5x5), canceled, and got 5^3. They asked why I didn't just tell them that the answer was 5^3 instead of trying to explain why. They also couldn't understand why I crossed out two fives from top and bottom (they're in 9th grade).
Even worse is that it seems to be acceptable and normal for entering high school students to not be able to work with fractions! Why is there no national outrage over this? Just about anything short of proving that every vector space has a basis in the 3rd grade seems worth trying.

My first experience with formal proofs was 7th grade geometry....I hated it. Maybe it was how it was taught, or maybe it was the book, but when I got to high school AP Calculus (BC), proofs took on a whole new meaning. I couldn't necessarily write them, but I could interpret them. I went from no grasp of proofs, to applying them in one year. This probably had something to do with me being motivated and wanting to learn. But my point is that it can be done.
I am saddened that middle school's have "Advanced Algebra" that is really only doing the kids a disservice. Not being able to work with fractions?! In High School?!

I found this article on the Web. I'm sure some people have already read this, but I think it fits well with what is being talked about, so I'm going to post the link. Lockhart's Lament
It's kind of long, but it is the best article I've ever read that describes the current condition of the American Math system in public schools.
Redbelly98
#12
Mar3-09, 08:11 PM
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It's somewhat standard to get proofs in h.s. geometry (9th or 10th grade). However, 2 years ago I tutored a kid in this subject and his teacher never had them do proofs.

So I guess it depends on what school system you're in, and maybe on whether you are in the "honors track" for math.
Astronuc
#13
Mar9-09, 08:42 PM
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IIRC, my first encounter with proofs was probably 7th grade in introductory algebra.

In 10th grade, geometry and trigonometry included many proofs, but the methods were based on what was studied in 7-9th grades.

I would have like to learn more about analytical geometry and linear algebra early. I was introduced to matrices as early as 6th grade.

I found the flow of mathematics and science was sporatic and disjoint. I would have preferred to be allowed to learn when I was ready, but the educational system wasn't structured for me.
khemix
#14
Mar13-09, 04:29 AM
P: 117
I believe they used to encounter proofs in grade 10, in a geometry course that lasted a year. Most schools don't do that anymore, and proofs are usually only briefly seen in a grade 11-12 geometry course that lasts a semester.

I'm not so much for exposing young students to proofs as I am for exposing them to logic. Instead of teaching say, history and science in elementary school (where most students are too young to appreciate it), a basic course in logic should supersede. Kids should be taught about double negatives, conditional statements, etc. Then again, its difficult to say whether they would understand it. I Know most schools actually teach the proof of pythogaras' theorem, but virtually no one understands it.
buffordboy23
#15
Mar13-09, 08:56 AM
P: 540
Quote Quote by khemix View Post
I'm not so much for exposing young students to proofs as I am for exposing them to logic. Instead of teaching say, history and science in elementary school (where most students are too young to appreciate it), a basic course in logic should supersede. Kids should be taught about double negatives, conditional statements, etc. Then again, its difficult to say whether they would understand it. I Know most schools actually teach the proof of pythogaras' theorem, but virtually no one understands it.
I disagree with the idea of not teaching science to elementary students. Kids have a natural curiosity about the world around them and like to ask questions. This can easily lead to scientific inquiry in the classroom. In my experiences as a former science teacher, I see that students tend to dislike science more as they get older.

Students in elementary school have the ability to think logically, but usually in concrete terms, so a course in logic may be too abstract depending on the content and target audience. Here is a link to a general overview of Piaget's learning theory and stages of cognitive development: http://coe.sdsu.edu/eet/Articles/piaget/index.htm
khemix
#16
Mar13-09, 10:46 AM
P: 117
Quote Quote by buffordboy23 View Post
I disagree with the idea of not teaching science to elementary students. Kids have a natural curiosity about the world around them and like to ask questions. This can easily lead to scientific inquiry in the classroom. In my experiences as a former science teacher, I see that students tend to dislike science more as they get older.

Students in elementary school have the ability to think logically, but usually in concrete terms, so a course in logic may be too abstract depending on the content and target audience. Here is a link to a general overview of Piaget's learning theory and stages of cognitive development: http://coe.sdsu.edu/eet/Articles/piaget/index.htm
I'm surprised you'd defend science and not history, as I was expecting more criticism for removing a history course which at least gives young students culture.

A lot of this is from personal experience, but any science I did in elementary school hardly inspired curiosity. What I remember doing was some very basic chemistry and physics, which was both too shallow to be of any taste, and the teacher was not qualified to answer our questions. Astronomy on the other hand I really enjoyed, and the study of rocks, but that was only grades 3 and 4. Highschool grade 9 is when I got a real taste for science.
thrill3rnit3
#17
Apr4-09, 01:44 AM
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Quote Quote by pentazoid View Post
I think a lot of people hate mathematics because they don't understand how the equations were derived .
I think it's more like people forgot the equations they memorized the night before, and they don't know how to derive it.
buffordboy23
#18
Apr4-09, 10:10 AM
P: 540
Quote Quote by khemix View Post
I'm surprised you'd defend science and not history, as I was expecting more criticism for removing a history course which at least gives young students culture.
I just now had a chance to read your response. Don't get me wrong. I think history is important too, because of the reasons you mentioned and more. My window of perspective in the educational world was science, so I know now how important the structuring of the K-12 science curriculum is rather than the history curriculum.

The OP's post falls under the general category of how we can improve mathematics education. It's funny that you mentioned the removal of certain elementary school subjects to achieve this end, since I know that some school districts in the U.S. have previously done so to focus on mathematics. Why did they do this? Because of their students' poor performance on high-stakes standardized tests and the negative consequences that would follow.

In my opinion, this is not the answer. The thinking is somewhat analogous to throwing more and more money into education in the hopes that achievement scores will rise. Elementary subjects are the foundation of higher learning, so we must not sacrifice them. It's unfortunate that your science teacher was not qualified to teach science (although they may or may not have their teaching certificate), but this is one part of the problem, which also pertains to mathematics education. From my experience as a student and as a teacher, I see that much of mathematics education is based upon memorization. This does not lead to true understanding of the subject. Moreover, not exposing students to proofs and similar methods of mathematical thinking in their K-12 education is a great disservice to our students, because this is the central force that drives our discoveries in mathematics and students don't recognize and appreciate that.


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