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12

I would like to offer a different way of thinking about the question. The idea that you have to do the proofs for maths students tends to come from people who have a fixed idea of what it means to 'do maths' and what a maths degree is. Instead, I believe the place to start is asking 'what skills do we want the students to learn?' (followed by 'why do we ...


11

I would be careful with the type of result for which one needs a lot of new math to digest the explanation. For example, I would avoid talking about $ 1 + 2+3+.. = -1/12$ because there is basically no way to explain to highschoolers in what sense this could be true. First, one would need a good understanding of the limit and the value of a series (i.e., ...


11

This is an example of what is usually called a flowchart proof (or sometimes a flow proof for short). A quick Google search for "flowchart proof" or "flow proof" shows many, many contemporary examples of the form, including a whole genre of YouTube videos teaching this style of presentation. This style of proof has been promoted at various points since the ...


10

I think there are more like three (or even four) aspects of education here: A. Theoretical understanding (basis, range of applicability, exceptions, etc.) B. Motivation (why care, what's it good for, is it fun, will it make me money, etc.) C. The basic technique (what you do) D. Practice (we are not biological computes--we burn grooves in the brain with ...


10

I always provide the following example whenever a student assumes what they want to prove: Suppose 0=1. Then 1=0 must be true. Then we can add both equations to deduce that 1=1. This is a true statement (1=1) and therefore our original assumption was valid, so 0=1. In our "Intro to Proofs" course for math majors, I show this on the first day of class. Any ...


10

It seems to me pointless and perhaps even irresponsible to teach results/algorithms/methods without explaining them, but between rigorous proof and adequate expanation there is a lot of terrain. Explanation and proof are somehow two extremes of the same thing, the difference being one of level of detail and formalization. For this question there is nothing ...


9

If it's the right amount of "weird", then YES (see Zone of Proximal Development). For example, I often try to show students how $0.\bar{3} = \frac{1}{3}$ implies $0.\bar{9} = 1$. This example alone has so much to unpack, that I often come back to it again and again while teaching rational numbers, series, algorithms, etc. I could see some mathematical ...


5

For the sake of making the question more definite, let's assume we are looking at calculus students. Let's assume we are looking at a generalist class. One with advanced strong high school students or decent college students. Not calculus for business students). But a standard course that meets the needs of physics and engineering students. But also ...


5

Surprise them. Especially in a (mathematical) culture where "getting the right answers" is prized and excellence on such examinations is valued in general, it is reasonable that students will value this above other expressions of mathematics. (I have a little teaching experience in that context, though most of my experience is in the "just get through it" ...


5

Maybe you can motivate the value of proofs by showing seemingly true claims which are in fact false, justifying the need for a proof of even an "obvious" claim. For example, this appears to be a dissection of an $8 \times 8$ square to a $13 \times 5$ rectangle: $64$ vs. $65$ unit squares. It takes some effort to uncover the flaw.        ...


4

I think there are two problems that have to be addressed individually: Wrong usage of implications. Implications just don't describe a relation between terms of the kind that appear here. They can be used between things that have a truth value such as equations or boolean expressions. I'd say there is no way other than fixing this to use only the correct ...


4

This sort of writing is quite common, in my experience, among young mathematicians (in their early years of college) who have had little feedback or training in formal mathematical writing. The first line is simply a statement of the problem, boiled down to its algebraic bones, so to speak. By the arrow, which in mathematical writing is read as an ...


4

Extracurricular math, or math enrichment programs often do introduce proofs in the realm of number theory. At the earliest level, it can include direct proofs such as "even + even is always even", "odd + odd is always even", etc. The geometry proofs are usually taught as very structured -- a build-up from definitions and earlier theorems to the result, ...


4

Math With Bad Drawing has some images that approach an info-graph (and in general is just a great website for math education), for example: https://mathwithbaddrawings.com/2015/07/01/infinity-plus-one-please-check-your-intuitions-at-the-front-desk/ There are some good geometry ones, especially around old compass and straight-edge constructions but that ...


3

I think there's also an issue of how you present the proofs. As an engineering student—knowing that I didn't need to remember proofs, just results—I often sat in class happily following each step of a derivation, and I could see that because all the steps were right the result must be true, but most of the time I didn't have a clear idea of what the proof ...


3

I can only help with (3). A. This behavior is not unusual and not just with intuitionists. It's good to be able to do things in your head, but you need to "know your head" and when you will have issues. B. In general, when doing pen and paper you should try to write down all the steps prone to an error. Of course there's a balancing point. But ...


3

"Kids usually struggle with every one of these concepts, let alone all of them together. It is difficult to get the whole picture and all the moving parts. So this place (proof) seems like a good place to show all of this in action." I'm afraid there is no "complete picture" of all the facets of logic and proof that you mention, interrelating in one ...


2

At any level, I'd ask myself the question "how many new things does the student need to know in order to understand this even at a surface level?". If that number gets beyond 2, I would think that it is a definite no. (So anything involving measure theory is right out.) The $\sum_{n=1}^{\infty} n = -\frac{1}{12}$ result is going to require zeta functions, ...


2

Some lovely answers are here already. I would like to look at an example below the level of calculus. In beginning and intermediate algebra (at a community college), I love working through the proof of the quadratic formula with my students. Here's why: They are highly likely to think of it as a complete black box. It is the nastiest looking formula they ...


2

We cannot reduce the assessment of mathematics education to knowledge of how to prove a specific theorem. To be sure, the inability of so many students to provide a proof for such a fairly simple mathematical proposition (in the context of asking the problem of senior math majors) is troubling, and reflects the fact that we don't give students enough time to ...


2

How big a sin is this? It's flat-out wrong and nonsensical for multiple reasons (as you point out here). The person writing those things is in need of correction. Note that similarly, the person in question also has serious problems writing standard-grammar English sentences. E.g.: Inconsistent use of a capital for the personal pronoun "I", inconsistent ...


2

As usual, my advice is contrarian, but you need to consider alternate viewpoints. I think you would be better off not pushing so hard. It's like pestering a girl who doesn't like you. Not a good approach. Slightly pathetic. Make proofs the spice or garnish or relish or what have you to the main course. Don't try to convince people they are better than ...


2

You will never know everything in mathematics. One place you could read about this is Bill Thurston's 'On Proof and Progress in Mathematics'. He gives a (slightly fictitious) list of many, many ways of understanding the concept of derivative, some of which will almost certainly be beyond you at this point. It sounds like you need to recognise that your ...


2

Concentrate on doing the homework problems. They are most important for developing your skills. Your approach is not time efficient. You need to learn things at a surface level and then come back and learn it deeper. The human brain is not a computer. We are imperfect animals and learn by starting a groove and deepening it over time AND making ...


2

There are two separate things you want to take away from a proof, and they should be the focus of your study. The first important takeaway from a proof is the driving idea (or ideas) that make the proof work. You should generally be able to express these ideas using as much ordinary language as possible. As an example, I will consider the theorem that the ...


1

The latter of your two choices. You don't know it if you can't recreate it. Just reading stuff and saying "yeah I get it" is the lazy way and you don't really learn from it. This was true in technique based math courses as well or chem or physics or languages. (Bio/history/lit/psych/etc. can be read a bit more passively with large gains of knowledge. ...


1

Prove it, or derive it? I had to derive it as my final task in 9th-grade Algebra I (1982). That is, I had to show If $ax^2+bx+c =0$, then $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I memorized it, forgot it, then re-learned it years later as a high school math teacher just to say I could do it, which many of my colleagues couldn’t. Algebra and Calculus are ...


1

That’s not a yes or no question. The proper way to teach mathematics is to give all (or all main) derivations, sometimes even different ways of getting those derivations and formulas. The students must understand why the formulas are the way they are. They must have a feel of the logic behind. They should be able to prove and derive formulas with confidence. ...


1

I like this comment: @DanFox: "between rigorous proof and adequate explanation there is a lot of terrain" (1) I would try to sprinkle in a few very clear proofs of non-obvious theorems. So I would avoid the intermediate value theorem, because it seems so obvious. But proving that the geometric mean is at most the arithmetic mean, $$ \sqrt[n]{ a_1 a_2 \...


1

There's no point in showing people something weird to illustrate how their knowledge is valid and yet their expectations are not, if they don't have the knowledge in the first place. Anything that requires university-level topics should immediately be discarded. A straightforward example that can be approached in a number of ways is the Monty Hall question, ...


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