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At what age should mathematical proofs be taught to students 
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#1
Feb2409, 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.



#2
Feb2409, 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? 


#3
Feb2409, 10:36 AM

P: 185

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 Nfactorial are derived because right now they really don't have a strong grasp on understanding what those equations mean. 


#4
Feb2409, 05:15 PM

P: 139

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. 


#5
Feb2409, 08:04 PM

P: 936

I'm not sure if you have a nine year old, but my nineyear old still still can't remember to tuck in his shirt or wash his face, and tying his shoes is hitandmiss. 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). 


#6
Feb2409, 08:33 PM

P: 185




#7
Feb2409, 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 10yearold 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? 


#8
Feb2809, 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.



#9
Mar209, 11:21 AM

P: 539

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 K6 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 23 weeks straight. 


#10
Mar209, 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 socalled '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. 


#11
Mar309, 07:09 PM

P: 276

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. 


#12
Mar309, 08:11 PM

Mentor
P: 12,071

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. 


#13
Mar909, 08:42 PM

Admin
P: 21,887

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 79th 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. 


#14
Mar1309, 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 1112 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. 


#15
Mar1309, 08:56 AM

P: 539

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 


#16
Mar1309, 10:46 AM

P: 117

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. 


#17
Apr409, 01:44 AM

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P: 712




#18
Apr409, 10:10 AM

P: 539

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 highstakes 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 K12 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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