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33 votes

Children's counting problems: Is this question phrased correctly?

I don't think there's anything wrong with the wording; it's clear what is being asked. Your example with the three dollars is also not always the way we speak in everyday language. If you ask someone ...
Thierry's user avatar
  • 1,527
32 votes

I want a "true" proof by contradiction of an implication $P \Rightarrow Q$

As you've noticed, there are (at least) three potential ways of proving an implication $p \Rightarrow q$: Assume $p$, and conclude $q$. Assume $\neg q$, and conclude $\neg p$. Assume both $p$ and $\...
Tanner Swett's user avatar
20 votes
Accepted

Can we avoid confusion over using "let" as a quantifier?

Many logicians that I have spoken to have concurred with my assessment that this is an issue of the misleading use of "let". Many teachers use this word in two very different and incompatible ways. ...
user21820's user avatar
  • 2,649
20 votes

Children's counting problems: Is this question phrased correctly?

When we describe counts in natural language, there's almost always an implicit "exactly" when phrasing like this. We use phrases like "at least 4" when we want a more general ...
Barmar's user avatar
  • 369
17 votes

Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?

Your assumption that teaching calculus needs to be backed by the $\varepsilon$-$\delta$ definitions could be challenged, but since it is not your question I won't do that here. My recent experience ...
Benoît Kloeckner's user avatar
17 votes
Accepted

How can I teach intuition why ‘If P then Q’ and ‘P only if Q’ mean the same, to first year undergraduates?

I would totally tune out if you dictated that massive quote to me. I can't even bring myself to properly skim it. You say teaches intuition, but for me, at least, the whole point of intuition is to ...
Justin Skycak's user avatar
16 votes

Children's counting problems: Is this question phrased correctly?

Perhaps "shows" instead of "has". If you asked me to show you 4 apples, I can't think of a logical argument in favor of me grabbing 5 apples and smiling smugly.
Aeryk's user avatar
  • 8,021
15 votes

How to explain (FALSE => TRUE) is TRUE

It makes more intuitive sense if you view an implication as a "promise". The truth value of the expression represents whether the promise can be broken (the promise is considered true unless ...
Justin Skycak's user avatar
13 votes
Accepted

Is it possible to improve logical thinking and problem solving abilities?

First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a ...
Saikat's user avatar
  • 398
13 votes

How can I teach intuition why ‘If P then Q’ and ‘P only if Q’ mean the same, to first year undergraduates?

I'll do a frame challenge answer that is closely related to the others: the phrase "only if" is generally only useful inside the larger phrase "if and only if". It's not hard to ...
user22788's user avatar
  • 854
12 votes
Accepted

Is 'For all $x$' an abuse of language in math?

No, there is no abuse of language here. $x$ and $y$ are placeholders that stand for individual numbers, and your second suggestion captures this: For each number we can insert in place of $x$ and $y$, ...
Natalie Clarius's user avatar
11 votes

Children's counting problems: Is this question phrased correctly?

Nearly every test like this includes instructions to choose the "best answer" to cover exactly this scenario. This looks like it's part of a test of basic counting skills, and in that ...
Jason E's user avatar
  • 111
10 votes

I want a "true" proof by contradiction of an implication $P \Rightarrow Q$

No, I suspect this situation never occurs. Here is why: If $P$ really implies $Q$, then we know logically that $\neg Q$ implies $\neg P$. Thus if you assume $\neg Q$, you will be able to deduce $\neg ...
Chris Cunningham's user avatar
10 votes

How can I teach intuition why ‘If P then Q’ and ‘P only if Q’ mean the same, to first year undergraduates?

To avoid confusion, in examples of implication, it may help to avoid any suggestion of causality, and to have both antecedent and consequent expressed in the present tense. EXAMPLE Consider the ...
Dan Christensen's user avatar
10 votes

Dominance of connectives: Why do we teach this?

When I see exercises like this, I often find that it teaches students to make assumptions about symbolic statements that may not be there - in a real world situation, if a statement is ambiguous, I ...
Leo Ell's user avatar
  • 101
9 votes
Accepted

Why are proofs by contradiction counterintuitive?

One reason why proof by contradiction is difficult for students is because mathematical notation (and other written language) does not allow for a subjunctive mood. Let me elaborate on this: In ...
mweiss's user avatar
  • 17.4k
9 votes

Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?

I had the same thought this year. My suspicion was that many students get anxious about suddenly dealing with quantifiers and they also struggle with understanding how the ordering of them can affect ...
cocoahomology's user avatar
9 votes

I want a "true" proof by contradiction of an implication $P \Rightarrow Q$

I think you are overlooking the fact that proof by contradiction must invoke the tautology $(P\ \hbox{or}\ \neg P)$, called the law of excluded middle. To prove $P\Rightarrow Q$ by contradiction, we ...
user52817's user avatar
  • 11k
9 votes

How to explain (FALSE => TRUE) is TRUE

I use the example If your flight is on time, then I will pick you up from the airport. If your friend tells you this, the only time they have lied is if the premise is true and the conclusion false. ...
David Steinberg's user avatar
8 votes

Children's counting problems: Is this question phrased correctly?

I showed this question to my three-year old son. His response - because he counted the apples one by one in each picture, passing "4" each time - was B, C and D. Hence, we need to take into ...
Nardya's user avatar
  • 81
8 votes
Accepted

Dominance of connectives: Why do we teach this?

This shouldn't be taught, and those exercises are pointless. There is clearly no intrinsic value in introducing and memorizing precedence of operations. If there is a point, it either is that we would ...
Arno's user avatar
  • 966
7 votes

Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?

Generally speaking, it would be nice to have a foundations class at the initiation of the Math major. Some of my colleagues envision this course centered around teaching college algebra. Well, to be ...
James S. Cook's user avatar
7 votes
Accepted

Teaching logic through "high school algebra"?

Obviously, one place to look is in the huge amount of “new math” curriculum material that was written during the late 1950s to early 1970s, but I’ll leave that for you or someone else to search ...
Dave L Renfro's user avatar
7 votes
Accepted

"Always/Sometimes/Never" vs. "True/False" questions for mathematical reasoning

With extensive anecdotal experience, by now I scrupulously avoid such questions (and analogous ones that exactly hit at the incompatibilities between "ordinary" language and mathematical language), at ...
paul garrett's user avatar
  • 14.7k
6 votes

Determining sets to show sufficiency of a condition?

I would not recommend to teach this method since there are some downsides. Take $A(x) \iff x \text{ is divisible by } 2$ $B(x) \iff x \text{ is divisible by } 42$ Is $A(x) \implies B(x)$ or $B(x) \...
Stephan Kulla's user avatar
6 votes

Determining sets to show sufficiency of a condition?

First, although you talk a bunch about cardinality, I don't see how that makes sense, so I'm going to assume you mean that you have them determine if the set corresponding to p is a subset of the set ...
Henry Towsner's user avatar
6 votes

Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?

It is well known that learning epsilon-delta definitions is difficult and is the intellectual equivalent of jumping over a tall wall in order to join the enlighted ones on the other side, a feat never ...
Mikhail Katz's user avatar
  • 2,240
6 votes

Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?

No. This is the same kind of pedagogical fallacy that led to the "new math" of the 1960s, when they tried to teach elementary school students deep concepts of abstract algebra as an introduction to ...
Ben Kovitz's user avatar
6 votes

Book request: teaching proving and reasoning at an American university

Some things we're currently considering for a similar course at a large urban community college: Epp, Discrete Mathematics with Applications Artin, Algebra Gilbert, Elements of Modern Algebra Lay, ...
Daniel R. Collins's user avatar
6 votes

Book request: teaching proving and reasoning at an American university

Another free option is Lehman, Leighton, and Meyer's Mathematics for Computer Science. It's written for an MIT introductory discrete math course that emphasizes training students in proof-writing.
perigon's user avatar
  • 161

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