45
votes
Accepted
How does a teacher come up with plausible wrong answers for multiple choice tests?
I freelance as an item writer, someone who writes questions for standardized tests. When making up alternate choices, I always have to justify my reasons for the "wrong answers" or distractors.
Here ...
- 7,939
38
votes
Accepted
Students confusing "object types" in introductory proofs class
I personally use terms like "type disagreement" and "type error". This agrees with the notion of types within computer science (https://en.wikipedia.org/wiki/Type_system).
When I ...
- 3,882
30
votes
Misuse of parentheses for multiplication
It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis ...
- 401
29
votes
Accepted
Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$
While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, this is the limits unit. When doing limits questions, most ...
- 4,960
29
votes
Is patching up a student's poor solution better than providing a good solution?
As Besicovich once said, the reputation of a mathematician is determined by the number of their badly written papers (the pioneering work is often clumsy and follows the route far from the shortest ...
- 3,439
28
votes
Accepted
Grating mathematical phrases---How to correct?
Personally, I don't think we attend to this sufficiently in lower-level mathematics (where it's actually needed most). Students need that vocabulary to interface with books, future teachers, tutors, ...
- 23.6k
27
votes
How should a student's inefficient calculation be pointed out?
Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just ...
- 23.6k
27
votes
How to explain what's wrong with this application of the chain rule?
The root of the difficulty is that $x$ appears free in $f(z)$, but we are trying to "capture" it with $g(x)$, which is illegal. When we substitute $g(x)$ into $f(g(x))$, we have a variable clash:
$$
f(...
- 556
26
votes
Accepted
Quote to show students don't have to fear making mistakes
Johann Wolfgang von Goethe: "By seeking and blundering we learn."
Original German, 1825.
Albert Einstein: "Anyone who has never made a mistake has never tried anything new." (However, attribution to ...
- 29k
22
votes
A Series of Unfortunate Examples!
Personally, I refer to this phenomenon as students "submarining" a broken understanding on a particular kind of problem.
Example #1: Our in-house elementary algebra textbook, in its first edition, ...
- 23.6k
22
votes
Accepted
Misuse of parentheses for multiplication
To answer the ultimate question ("Can anybody explain where this writing tradition comes from?"): It's explicitly taught that way by many U.S. instructors and textbooks.
Examples: From the otherwise ...
- 23.6k
22
votes
How does a teacher come up with plausible wrong answers for multiple choice tests?
If a teacher has taught the course before, and has asked questions that are free-response (not multiple-choice), then the teacher can look at the incorrect answers previously given by the students.
...
- 10.7k
22
votes
Accepted
How do you coach students who often make small errors?
Ask the student to "talk through" their calculations
Having a student verbalize their calculation may force them to pay more attention (or a different kind of attention) to their work that ...
- 4,506
21
votes
teach that $\frac10$ not defined properly
What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever ...
- 1,705
20
votes
Accepted
How should a student's inefficient calculation be pointed out?
I like your second option the best:
...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?
...
- 8,856
20
votes
Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ -- why?
When a student writes incorrect notation, ask them to read it out loud. I would say something like:
Something here doesn't look right, but we can fix it. Could you read this work out loud? I think ...
- 20.8k
19
votes
Accepted
Students problems with reasoning, not exactly math
You can say that this is "just reasoning", but the truth is that this is a specific application of basic logic, in particular the implication (if/then) relation. I have a colleague with a ...
- 23.6k
19
votes
Why do students only see the last term of a sum abbreviated with an ellipsis?
I suspect that the issue is not so much the ellipsis per se but a problem with notation in general, and in particular with the correct use of the equals sign. At the risk of repeating what I wrote in ...
- 17.2k
19
votes
How can a teacher help a student who has internalized mistakes?
Coach them through doing it the right way. Have them repeat it the right way, several times. In front of you. And go very easy, including repeats. Gradually relax the guardrails and keep drilling. ...
- 207
18
votes
Why does the widespread erroneous definition of "irrational number" persist without being taught?
I can think of two related reasons:
The characterization via the decimal expansion might be perceived more strongly like a property of the number: "This number is irrational, because this number's ...
quid♦
- 7,632
18
votes
Misuse of parentheses for multiplication
I disagree that it is "terribly harmful".
Do not prevent them from writing $(2)(5)$. Instead prevent them from writing things that are actually wrong.
Thinking that $\sin x$ is $\sin$ times $x$ ...
- 7,273
18
votes
How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?
One observation is that (sum of numerators) divided by (sum of denominators) is not well defined.
For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$.
The ratio of the sum of ...
- 9,483
16
votes
Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$
First of all, I think that the problem statement can be confusing.
use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method
It can be ...
- 261
16
votes
Is this just a mistake or a more serious misconception?
It seems clear that there is a certain conceptual gap in the student's understanding. My suspicion is that the student is essentially running the following program in his mind:
Initialize factorial =...
- 2,581
16
votes
Taxonomy of bad proofs
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, ...
- 7,252
15
votes
Mnemonics for some properties in mathematics
Recently, a student in my beginning algebra course offered the following to the class, regarding signed number multiplication:
Assuming positivity is like love, and negativity is like hate, then...
"...
- 8,856
14
votes
Grating mathematical phrases---How to correct?
I wish to give a slightly different answer compared to the others.
Strict and Standardized Notations is Very Important
They not only help us communicate better, they also help us think. They prime us ...
14
votes
Accepted
How much credit to give a short exam question with one error?
I'll try to make this answer a little more general than just telling how many points I would give for this particular error (if interested: I'd give 5/10 at most, most likely less).
For that, let's ...
- 1,298
14
votes
Quote to show students don't have to fear making mistakes
"I have not failed. I've just found 10,000 ways that won't work."
"Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time."
"Many of life's ...
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