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 ...
- 1,527
32
votes
I want a "true" proof by contradiction of an implication P => 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 $\...
- 631
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. ...
- 2,567
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 ...
- 359
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 ...
- 9,079
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 ...
- 6,342
16
votes
Can students tell the difference between the "definition if" and the "theorem if"?
Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such:
It appears that even many serious professional ...
- 14.2k
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.
- 7,500
15
votes
What is a variable?
This is a very difficult question to answer; I recommend as a first place to look:
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra, K-12, 8, 19. Link (no ...
- 18.4k
15
votes
Accepted
Are logic textbooks wrong to teach ‘unless’ = ‘or’? How to motivate?
There is nothing misleading or wrong with the textbook passages you cite. How would you represent the sentence "P unless Q" in propositional logic, especially after reading the beautifully ...
- 10.3k
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 ...
- 854
12
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 ...
- 378
12
votes
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 $y$, ...
- 229
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 ...
- 111
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 ...
- 1,074
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 ...
- 17.3k
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 ...
9
votes
I want a "true" proof by contradiction of an implication P => 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 ...
- 10.3k
8
votes
What is a variable?
I teach people (informally) how to make iOS apps. A lot of the people I teach are not people who were good math students. Of course in programming variables are important and anyone with a basic ...
- 181
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 ...
- 81
8
votes
I want a "true" proof by contradiction of an implication P => 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 ...
- 21.2k
7
votes
What does maths teach you that logic does not?
I think the only sense in which the quote is accurate is if you interpret "maths" broadly and "Intro to Logic" narrowly. Intro to Logic would only introduce limited proof techniques tailored to ...
- 29.5k
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 ...
- 10.7k
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 ...
- 5,768
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 ...
- 11.6k
6
votes
Can students tell the difference between the "definition if" and the "theorem if"?
iff and if
In my experience, students who have a solid grasp of first-order logic have absolutely no problem with the inconsistent use of "if" in definitions. The problem is that most ...
- 2,567
6
votes
Accepted
What does maths teach you that logic does not?
In the United States, some universities offer "Introduction to Logic" courses. These courses are often offered to undergraduates who are not majoring in mathematics, as a way that the undergraduates ...
- 3,158
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) \...
- 2,021
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 ...
- 2,102
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 ...
- 809
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