Hot answers tagged

41

I'm slightly concerned that Is there a mathematical reason (like a proof) of why this happens? is a purely mathematical question, but since you write "we just warn students" I will assume that this question is purposefully asked here on Math Educators StackExchange. As to a proof: Given $a>b$, subtract $a$ from both sides: $0 > b-a$. Next, ...


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

Depending on the context and the previous curriculum, the following might work: "less than" means "to the left of" on the number line. Multiplying by a negative number flips numbers around 0. Thus, "left of" becomes "right of", or "greater than".


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

Had the asker/OP written the following, their proof would be legitimate. They left out a preface on the first equation, and inappropriately used only left-to-right implication between steps. To prove: $$ x\oplus(y\oplus z) = (x\oplus y)\oplus z.$$ Proof: \begin{align} x \oplus (y \oplus z) &= x \oplus\sqrt[3]{y^3 + z^3}\\ \\ &= \sqrt[3]{x^3 + (\...


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 ...


9

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

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 ...


6

Perhaps one way to see (and explain intuitively to children) the "multiply by $-1$ part" is the following. Imagine your two numbers, $a$ and $b$, lying on the numberline. Multiplying by $-1$ is like 'rotating the numberline through 180°': imagine it's a straight metal pole, lying flat on the ground; pick it up by the middle, and rotate it 'long ways' (ie not ...


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

For multiplying or dividing by -1... $$\begin{align} a&>b\\ a-b&>0\\ \\-(a-b)&<0\\ -a&<-b \end{align} $$ (You can then extended to arbitrary negative numbers by multiplying or dividing by the [positive] magnitude.) For taking reciprocals... assuming $ab>0$ $$\begin{align} a&>b\\ \left(\frac{1}{ab}\right)a&>\left(...


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

In practice, one does have to explain (even to university engineering students) that although A implies B, it need not be the case that B implies A. However, there is a confounding factor, which is that mathematical logic is not what is used in scientific reasoning. As is explained by Polya in one of his books (the one on plausible reasoning) and by V. I. ...


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 ...


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

In a more general case this happens if you deal with a strictly monotonic decreasing function like f(x) = -x or f(x) = 1/x (in the positive or negative numbers).


3

Edit - Following the edit made to the OP, I must admit it is a bit unclear what you are looking for exactly. Here is what I understand. The proof must: Pertain or lead to algebra Algebra must be harder than algebra I and II Be college level Be hard Not be 'related to any subject' First, I would ask your professor for an example of something she'd find ...


3

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, ...


3

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 ...


2

I found this post as a student trying to figure out why induction proofs are so difficult to understand. There are good answers here, but I think many are not specific enough to why induction proofs (rather than proofs in general) are so challenging. The metaphor and principle of induction is straightforward. For me, the real issues arise in following along ...


2

You could take a look at basic set theoretic proofs, like for example showing that $$A \cap (B \cup C) = (A \cap B) \cup (A \cup C).$$ You can draw a diagram here and you can show and emphasize formal proofs. This property is not hard to prove, but it teaches the students how to do formal proofs, step by step. Another such sort of proofs would be the ...


2

The binomial theorem is a great way to introduce proofs, since it provides a "shortcut" for high school algebra problems using Pascal's triangle (which students enjoy drawing). You can ask why it keeps working and lead them to proof by induction. You can also pivot to a question about choosing from sets, and let students realize they're the same numbers. ...


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

I can't help but feel this question highlights the self defeating nature of teaching maths via "cheat tricks". In my opinion it is probably best not to teach students ever to just "switch the sign" of the inequality in an arbitrary manner, which can be confusing. As elegantly illustrated by @Benjamin_Dickman's proof, the alternative notation $$a>b \...


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

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

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 ...


Only top voted, non community-wiki answers of a minimum length are eligible