# Tag Info

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 ...

8

It might work better, with this age level, not to be concerned about how large n is when the apparent pattern falls apart. My favorite example is the problem of making all possible straight-line segments between n points on a circle, and then counting the regions. 2 points makes 2 regions, 3 makes 4, 4 makes 8, and 5 makes 16. It sure looks like doubling...

7

Here is another one. Prove that the power of $13$ can be writen as a sum of two squares. I will give two proofs of it. First one is more involved and includes lemma $$(a^2+b^2)(x^2+y^2)= (ax+by)^2+(bx-ay)^2$$ yet second takes step 2 and it is much more elegant. 1.st proof Base: $n=1$, then $13 = 2^2+3^2$ and we are done. We know that $13^n = a^2+b^2$ ...

7

I have found this site to be an invaluable online resource for commentary. Long story short, nobody in history ever wrote an authoritative second edition of Elements, so we're still "stuck" with Euclid's original thoughts. Undeniably a work of genius that deserves its place in history, but it does have a few rough patches. Here are David Joyce's thoughts ...

7

I second Sue Van Hattum's suggestion that you should not be so concerned with how large the $n$ is where the pattern eventually fails. I'll go one step further and recommend an example where that $n$ is not only fairly "small" but also such a situation where the students can see why that $n$ makes the pattern fail. Consider the function $f(n) = n^2+n+41$. ...

5

Perhaps related to "The Tangle" is what I call "Wishful Thinking". This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, but makes great (or incorrect) leaps to get from one to the other, e.g. in proofs by induction. There is also the "Using What You Are Proving" (maybe similar ...

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

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

A simple consequnce of: Postage Stamp Problem, which states that for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$, is that every natural number greater or equal to $mn-m-n+1$ can be writen in a form $am+bn$ for some $a,b$. And for some ...

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

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 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 ...

3

This happened to me once on a topic that should be accessible to middle-school students. As an undergraduate, I formed the following conjecture: An antiprime (also called a highly composite number) is a positive integer that has more divisors than any number less than it. The first few antiprimes are $$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360,..... 3 Putnam 1963 Let \mathbb N be the set of positive integers, and let f:\mathbb N\to\mathbb N be a strictly increasing function such that f(2)=2 and f(m)f(n)=f(mn) for all positive integers m,n such that \gcd(m,n)=1. Find all such functions f. Watered down version: Let \mathbb N be the set of positive integers, and let f:\mathbb N\to\... 3 For what natural n does there exist a square composed of n squares? Example: 1, 4, and 6 are valid, but one cannot construct a square from 2, 3, or 5 squares. Proof: 2 Hmm, I'm surprised than no one mentioned de Moivre's formula:$$(\cos \phi +i\cdot \sin \phi)^n= \cos (n\phi) + i\cdot\sin (n\phi) It is an excellent theorem to prove where we must also use the addition theorem for sin and cos.

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

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

Perhaps it is a good idea to start with truly simple examples. The simplest example accessible to children is the notion that every odd number is prime. Another is to calculate $\cos(\pi/n)$. For $n = 1, 2, 3, 4$ it works. It is simple in principle to construct a (degree $N-1$) polynomial that takes "nice" values at the first $N$ integers by solving an ...

1

Let the two original lines be $AB$, $AC$, with the angle to recreate as $\angle BAC$. Draw a line parallel to $AC$ that passes through $AB$ and is not $AC$. Call this line $DE$. Where $DE$ and $AB$ intersect ($F$), drop the perpendicular of $DE$ to $AC$, and call this point $G$. $AG$ and $GF$ can be used to recreate the angle.

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

As an undergraduate math major, there's a radical change in structure around late-sophomore to early-junior year (U.S. experience here). Prior to that point, most of the work is to calculate things with given formulas or algorithms. After that point, it switches to mostly proof-writing exercises. The big picture here is that the junior-senior years of the ...

1

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (p.190ff), explain the paradox in a way I hadn't seen before. Pearl imagines changing the rules to "Let's Fake a Deal," where "Monty opens one of the two doors you didn't choose, but his choice is completely at random." Of course he could open the door containing the car/prize, ruining ...

1

Prove that, for all x in N, we have x+1 =/= x. (Requires use of proof by contradiction.) Use only the following properties of addition on N: Addition is closed on N. x+1 =/= 1 for all x in N. If x+1 = y+1, then x=y for all x, y in N. The Principle of Mathematical Induction. OUTLINE Base case 1+1 =/= 1 follows directly from (2). Inductive step ...

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