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I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math.

What is a good point to start learning proofs? what are the pros and cons? does the whole curriculum have to be geared towards proofs?

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    $\begingroup$ I would say that understanding modus ponens (that is, the intuition behind the notion of inference) is a precondition. There is no point in doing proofs if one does not grasp causality, after all, theorems are in "premise => conclusion" form. However, one should distinguish making inference from formality (both could be named "doing proofs"). There's a difference between teaching explaining cause-effect relationship and forcing boring formal derivations. $\endgroup$
    – dtldarek
    Mar 14 '14 at 0:23
  • $\begingroup$ Some relevant references: Teaching and Learning Proof Across the Grades: A K-16 Perspective, Stylianou et. al. (amazon.com/gp/product/0415887313/…), and Theorems in School, Boero (amazon.com/Theorems-School-Epistemology-Directions-Mathematics/…). $\endgroup$ Mar 14 '14 at 22:05
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    $\begingroup$ Also I think the description in Paul Lockhart's A Mathematician's Lament (maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf) is very relevant. Several answerers have pointed out that it is important not to insist on too rigid a conception of rigor when working with young children; this doesn't mean that you have to avoid proof. $\endgroup$ Mar 14 '14 at 22:08
  • $\begingroup$ For what it's worth: I nearly failed grade 9 and grade 10 math. However, I added a math major to my university degree after I learned about logic and proofs in my philosophy classes. I'm doing well in my math classes now. It seems to me that it would have been useful (to say the least) to have known proofs in high-school, and likely as least as early as grade 7. $\endgroup$
    – Hal
    Mar 31 '15 at 1:11
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Not to be annoying, but what is a proof?

Here's my best take: a proof is an explanation that could convince all conceivable skeptics.

Proof, then, is absolutely crucial at all levels of math instruction, because explanation is crucial to learning.

I figure that you're imagining something different, a moment when you would start asking kids to offer the sorts of formal, rigorous proofs that would satisfy higher level students and working mathematicians. I think that this is a mistake. If we want students to be ready to offer and receive more rigorous proofs in later years, we prepare them by creating environments where offering convincing arguments that make sense to us and everyone we can imagine.

We teachers will push them to be a bit more skeptical of themselves than they're comfortable being, and in doing so push their proofs to an increasingly higher level of rigor. What's crucial is remembering not to rush them. It makes sense that our kids' proofs wouldn't satisfy us. My students aren't me, and their skeptics are different than mine.

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  • $\begingroup$ Although I agree with your sentiment, your answer is not useful in its current format. In particular it does very little to answer the question of the OP. I think that adding more explicit argument behind your opinion would greatly improve it. $\endgroup$
    – dtldarek
    Mar 14 '14 at 0:28
  • $\begingroup$ "What is a good point to start learning proofs?" I thought that I answered that pretty directly, no? $\endgroup$ Mar 14 '14 at 0:33
  • $\begingroup$ Both yes and no. Yes, because you did it literally. No, because there is much more to this question than just its basic meaning. For example you can do "math" with very small children (e.g. the notion of quantity, its conservation etc.), but there would be no proofs as the set of conceivable skeptics is empty (at this age being a skeptic is inconceivable). Also your notion of proof involves words like "explanation", however, explanation why 2+2=4 is not a proof. Proofs accompany theorems, so there is no need for proofs until the child can understand (in some informal way) what a theorem is. $\endgroup$
    – dtldarek
    Mar 14 '14 at 0:49
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    $\begingroup$ "at this age being a skeptic is inconceivable" [citation needed] That's a bit glib, but my experience working with kids as young as 3rd grade is that they are very, very capable of mathematical skepticism. $\endgroup$ Mar 14 '14 at 3:38
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    $\begingroup$ "For there to be a proof, you need axioms and assumptions." Suffice it to say that this is not a universally held conception of proof. For dissent, check out Paul Lockhart's writing, Proofs and Refutations, or the view of educators like Michael Serra. $\endgroup$ Mar 14 '14 at 4:09
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My next door neighbor's daughter is in fifth grade, and she is enrolled in an extracurricular course with advanced math curriculum. I sometimes help her with her homework, which includes proofs. For example, the other day she was asked to prove that the square root of 5 is irrational. I went through the standard argument with her, and even though she has some understanding of what prime factorization is, the notion of proof by contradiction was a bit beyond her, and she quickly lost interest.

I tell this story because I think being too rigorous too soon can be damaging to children's enjoyment of mathematics, which may give them a sour taste and never give them the chance to enjoy math later on. I know that in California high schools, geometry is where the notion of proof is first introduced (albeit, in a very awkward style), and students usually end up hating it.

I think we should always prioritize enjoyment/curiosity/intrigue/wonder over rigor when teaching children mathematics. Rigor can come later, when they thirst for it. If they don't care about the foundations of mathematics right away, and just enjoy looking at nice pictures and doing interesting activities, great, run with that. Of course, there will be a few who at an early age desire more, but we shouldn't force students to get there until they are ready.

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    $\begingroup$ I understand your point, and while I agree that it can be detrimental to introduce rigorous proofs too early, I think that students could (should) be given problems or asked questions that help them develop a sense for what proof is and why it is important from a very early age. By the time they are introduced to formal proof in Geometry, they should already have a very solid basis on which to build their understanding. Sadly, many students lack this today. $\endgroup$
    – mkasberg
    Mar 14 '14 at 1:15
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    $\begingroup$ To lump all proofs into "foundations" (as your post suggests) is an unusual practice, to say the least. $\endgroup$
    – Shay
    Apr 4 '14 at 4:39
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There is really no age where you can learn mathematics, but where you cannot learn mathematical proofs if you do not confuse rigor with technical notation and language.

Obviously, the younger the child and the smaller the mathematical knowledge base, the less technical the results need to be.

The types of results that I am thinking of:

The sum of two even numbers is even. Why?

Can I find out the sex and the home planet of the aliens in this puzzle? If not, why not?

And no, you do not need to introduce set theory and information theory to give a "rigorous" proof. It is just a matter of using an "axiom system" that is adapted to the child's level of knowledge. Hardly any research paper that is not about mathematics foundation refers to foundations to give a rigorous proof. Teaching students a myth as introduction to proofs is not helpful. I have also noticed that first-semester courses on the use of quantifiers usually tell the students the lie that $\Rightarrow$ is used only in the logical sense ($0=1$ implies all beer is free) and then go on to mark down student papers because their implication arrows do not correspond to an actually reproducible step in the proof.

If the question actually is: "Is there a good age/level to start forcing everyone to learn proofs?" then the answer is probably "never".

I am biased here, because my first introduction to "rigorous proofs" included months of Aristotle's classification of syllogisms (see here ), which is about as useful as learning different names for problems of the type $a-b=?$ and $b+?=a$.

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Very young children can prove that two odd numbers will always add up to an even number. They can explain with pictures. There are other good questions young children can address with proofs, the style of which will be more juvenile, but which can still be "rigorous".

Math circles are a great way to introduce young children to logically figuring out a mathematical idea. This post at Talking Stick Math Blog might help you imagine what it looks like: http://talkingsticklearningcenter.org/logic-for-the-very-young/

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The concept of 'proof' in mathematics (education) is overloaded with importance/fear/magnificence/awe. What is 'proof' if not just an explanation to that or another claim by a sequence of logical steps? How is it different then 'proofs' in history? Few educators of history would suggest teaching historical facts without understanding and arguing about their cause, effect or importance. Yet, in mathematics education there seems to be more room for debate.

What is a good point to start learning proofs? Well, whenever the student is showing an interest would be fine. Showing an interest means that, at least on some level, they are ready to tackle it (whatever it may be). Then, of course, some care needs to be taken as to the level of maturity the student has in order to decide on the approach to take.

What are pros and cons? Mathematics without proof is, at best, a cook-book recipe for long and boring computations. Not understanding what one does may lead one to mistake mathematics for magical rules to be followed just because somebody said so. It robs the students from the ability to understand what they are doing and how to check their answers make sense. Of course, introducing long and tedious proofs is also dangerous. Before a typical student can appreciate the intricacies of, say, proving the reals exist, one must first come to appreciate the merits of such a proof. Not all things should be proven, it's quite alright to take some things on faith, as long as we are not oversimplifying anything.

Does the whole curriculum needs to be geared towards proofs? Well, it should be geared towards understanding, and that means that it should include explanation. Some people call it proof.

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With regard to Math Education literature on proofs: a person to look to is Eric Knuth (Google Scholar).

However, it may be more fruitful to shift from talking about writing (formal) mathematical proofs to discussions around incorporating sense making into the mathematics classroom.

Sue VanHattum comments in another response here that:

Very young children can prove that two odd numbers will always add up to an even number.

For an example of this, consider the excerpt below, which mentions Deborah Ball, and comes from an article by Alan Schoenfeld (citation: Schoenfeld, A. Mathematical Modeling, Sense Making, and the Common Core State Standards. The Journal of Mathematics Education at Teachers College, 4(2).)

In contrast, here’s a sense making example that comes from Deborah Ball’s third grade class. Deborah had a bunch of kids who were playing with numbers—adding, subtracting, doing various things. They noticed that every time they added two odd numbers, they got an even number. One of the kids said, "Is that always going to happen?" She then stopped and said, "But the odd numbers go on forever. We can never test them all, so we can never know." That’s pretty damned good for a third grader.

Another kid says, "Well, I was thinking about seven plus nine, and actually I knew that the sum was going to be even before I added them up. If you look at seven, it’s a bunch of pairs with one left over, and if you look at the nine, it’s a bunch of pairs with one left over (Figure 1). So when you put them together, the pairs are going to stay the same—but the two leftovers become a pair, so everything together is in pairs, so it’s going to be even" (Figure 2). The girl stopped at that point and she said, "But wait! It doesn’t have to be seven and nine. No matter what that first odd number is, it's going to be a bunch of pairs with one left over. The second odd number is also going to be a bunch of pairs with one left over. And when you put them together, you have all the pairs you started with plus the pair you made, so it’s going to be even."

Now that’s a third grader. I guarantee you that every mathematician I know would say that the student produced a completely rigorous mathematical proof. That’s what I call mathematical sense making, and that’s what I’d like to see in our classrooms. The real challenge we face is to support sense making in our classrooms.

The topic of sense making comes up a fair amount in Math Education; it also features prominently in the Common Core State Standards for Mathematics (PDF) as a Standard for Mathematical Practice:

Mathematics | Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Of course, sense making is something that can (should!) be taught from the earliest years.

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One difficulty that I have seen is that many mathematics teachers present a course as if all their students were going to become mathematicians, and forget that many, even most, will prefer to become builders, restaurant cooks, nurses, elementary educators, musicians, or advertising copywriters instead. They are studying mathematics because they believe, or have been told, that it will someday be useful for something they really want to do.

It's also impossible to tell in advance, who will take an interest in mathematics for its own sake and enjoy rigor. In either case, if the proofs are difficult, tedious, or incomprehensible, it will indeed put them off learning any mathematics at all.

It seems that the time and place to introduce proofs (and disproofs) is when students ask "How do we know this for sure?", In that case, it's not so much a question of whether to teach proofs, but how.

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    $\begingroup$ Well, your argument could be extended to "we need only to teach children enough history so that they would know, as adults, when the national holidays are, so that the can plan their leave accordingly". I know, this is extreme (and kind of silly), but I wanted to show why your argument is not good at all (even if the conclusion might have a lot of sense). $\endgroup$
    – dtldarek
    Mar 14 '14 at 0:35
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    $\begingroup$ One of the first questions students commonly ask is "Why should I study this?". If the teacher doesn't have a good reason why they should study it now, one that makes sense at their level of understanding, they are likely to tune the teacher out. "Someday you might find this useful" is usually not a good enough answer. The teacher should have a good reason for why the proof, or technique of proof, is necessary.. $\endgroup$
    – Confutus
    Mar 14 '14 at 0:45
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    $\begingroup$ You missed my point - many people would not find knowing history (anything above the dates of free national holidays) useful. Yet, not knowing history is detrimental to the community. In this case I don't know of any arguments which could be understood as intended by children. This reminds me of an conversation between a child and her mother "Why do I need to be vaccinated? Because you could get sick. But I'm not sick. But you could, in the future. But I'm not..." $\endgroup$
    – dtldarek
    Mar 14 '14 at 0:58
  • $\begingroup$ If a subject is sufficiently important, there is usually more than one reason for it: that applies to vaccinations as well as history, and to proofs as well as those. Part of the art of teaching is knowing why you are teaching something. The why is often just as important as the what. $\endgroup$
    – Confutus
    Mar 14 '14 at 1:31
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    $\begingroup$ Yeah, I have to agree with you on the last comment. Perhaps this is why I can handle university students, but not so much the younger bunch. Yet, I'm still not convinced. For example, some most useful methods for teaching children involve tricking them into learning important stuff besides the thing they actually think they are doing. It's like making them play some game to let them learn teamwork. You could argue that it's a way of explaining why they need teamwork, but "to play a game" is just a lie. $\endgroup$
    – dtldarek
    Mar 14 '14 at 1:41
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I think proofs can be/should be taught as early as possible. I am not sure at what age exactly but as soon as they can understand the concepts maybe around 12-13 years old. So I keep giving them bits and pieces of it and if they are able to pickup the concepts then I teach them fully. One of the things that we need to do is to coherently represent all the moving parts in a Mathematical Proof. I always struggled with "the big picture" when I was studying. We also need to explain the underlying concepts in easy way. For example what is Logic? I tell my kids, Logic is a vehicle that takes your ordinary day today language and converts it into mathematics. Towards the big picture, I created an infographic which you might find useful. Have a look here.

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