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At what age should mathematical proofs be taught to students

  1. Feb 24, 2009 #1
    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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  3. Feb 24, 2009 #2
    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?
  4. Feb 24, 2009 #3
    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.
  5. Feb 24, 2009 #4
    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.
  6. Feb 24, 2009 #5
    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).
  7. Feb 24, 2009 #6
    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.
  8. Feb 24, 2009 #7
    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.

  9. Feb 28, 2009 #8
    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.
  10. Mar 2, 2009 #9
    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.
  11. Mar 2, 2009 #10
    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.
  12. Mar 3, 2009 #11

    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. http://www.maa.org/devlin/LockhartsLament.pdf" [Broken]
    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.
    Last edited by a moderator: May 4, 2017
  13. Mar 3, 2009 #12


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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.
  14. Mar 9, 2009 #13


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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.
  15. Mar 13, 2009 #14
    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.
  16. Mar 13, 2009 #15
    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 [Broken]
    Last edited by a moderator: May 4, 2017
  17. Mar 13, 2009 #16
    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.
    Last edited by a moderator: May 4, 2017
  18. Apr 4, 2009 #17


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    I think it's more like people forgot the equations they memorized the night before, and they don't know how to derive it.
  19. Apr 4, 2009 #18
    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.
  20. Apr 5, 2009 #19
    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.
  21. Apr 5, 2009 #20
    Students should be given proofs as soon as possible. Starting from elementary arithmetic.

    I remember in like 3rd/4th grade my teacher told us


    At the time, that seemed revolutionary to me. I asked her why that works, and she didn't answer me.

    I hate her.
  22. Apr 5, 2009 #21
    Well, most people's math problems start early. Earlier than the age when a student can be expected to read a textbook on his own (this is probably up to about 20 years old now). And in any case, lots of texts don't provide decent proofs or explanations or lots of teachers skip them, and the student isn't at the level to wade through them on his own.

    I don't think a lot of basic things are ever explained, and if they were, the student would realize that they're incredibly clever and beautiful. How many teachers take the time to explain why "carrying", "borrowing", and the long division algorithm work? Not too many I think. How many teachers truly understand them themselves? I don't think I'd like to know. Insufficient teacher knowledge and training are serious problems.

    I'm not conveniently forgetting the fact that even after explanations and attempts to convey what's actually going on to students, many of them forget or just don't care, but this happens at the high school level for me. I wonder how much of that mentality is due to their previous schooling.
  23. Apr 6, 2009 #22
    That's interesting that your teacher provided you with a theorem that contained abstract symbols for numbers in the third/fourth grade. My earliest experience that I can recall was in 7th/8th around the time of algebra.

    I think simple theorems like this can be used as tools for mathematical discovery in the elementary grades. As an example, consider the theorem "If a, b, c are positive numbers such that a > b, then ac > bc." Instead of actually presenting the theorem verbatim to the students, the teacher can provide the scaffolding for the students to start thinking in this direction. For example, the teacher has the students pick any two numbers, where one number is larger than the other (say 7 and 5), and then chooses another third number (say 4) and shows the result: 7*4 = 28 and 5*4 = 20, since 28 > 20 this means 7*4 > 5*4. The teacher may ask the class after a few more examples if this (the idea in the theorem but in words) is always true? The class can then try to find "counter-examples", and while doing so they are reinforcing their multiplication and comparison skills, and likely subtraction skills (28 is 8 more than 20). The class can't seem to find a counterexample, so they suggest that the teacher's idea is always true. The teacher then has the class construct a rule for what they just learned; although it lacks true mathematics rigor, mathematical thinking is still employed. Along the way some students may also realize that this rule applies to addition (a+c > b+c for a>b) and by seeing enough examples that ac - bc = nc where n is a positive number.

    EDIT: 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).
    Last edited: Apr 6, 2009
  24. Aug 15, 2010 #23
    Here's the problem I can see, teaching kids how to solve a problem... teaches them how to solve that problem.

    Teaching kids how to construct a problem, and how to derive a proof from it, teaches them how to solve MANY problems.

    If I showed a 1st grader two big boxes, one holds three balls, and a smaller closed box. The other has five balls in it. I could ask them, "if both boxes have the same amount of balls, how many are in the closed box?", and I'm pretty sure they'd tell me two.


    I could ask them to show me how they got to that answer, and have them explain it. If they guessed, or if they weren't sure, I could demonstrate by having them remove three balls from both sides.

    Pow, basic algebraic reasoning, why can't you go further?

    Why can't you show a child that asking a question about something they see, can be used to form a hypothesis, and that a hypothesis can be tested with an experiment? That the results of an experiment can be gone over with others to see what information can be gathered, and that there is no limit to the number of times you can do this?

    I find it hard to believe that if I basically informed a kid that he now had a basic toolbox with which he could try to understand everything, they wouldn't be excited by the idea, that new power they now possess, not simply having to ask a grown up for an answer, being able to instead try to find an answer on their own is fun!

    Why do elementary school kids need culture? They need the basic tools with which to understand the world around them, not etiquette, or a sense of national identity.
  25. Aug 15, 2010 #24
    That wouldn't work with most first graders, you could demonstrate by adding two balls to the missing side but subtracting from both sides is too abstract reasoning. Many of these kids can barely count you know...
    This isn't true, kids rarely needs help with maths since they rarely needs maths at all except at school. Also maths certainly is not the answer to even a major part of life's questions, it is just exiting if those kinds of things are exiting to you. For example I never asked why the sky was blue or why it was colder in the winter or any other question relating to things like physics or maths, looking through a microscope was the most boring thing ever beaten only by looking through a telescope.

    A certain kind of people find those things fascinating but far from everyone and those are just distant applications of maths and not maths in itself.
    Last edited: Aug 15, 2010
  26. Aug 17, 2010 #25
    Perhaps you're right, perhaps we should keep teaching math and science exactly as we are, it's producing wonderful results.
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