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 ...
Amy B's user avatar
  • 8,037
39 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 ...
TomKern's user avatar
  • 4,022
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 ...
smernst's user avatar
  • 401
30 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(...
Kevin's user avatar
  • 586
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 ...
fedja's user avatar
  • 3,831
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, ...
Daniel R. Collins's user avatar
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 ...
Daniel R. Collins's user avatar
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 ...
Joseph O'Rourke's user avatar
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, ...
Daniel R. Collins's user avatar
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 ...
Daniel R. Collins's user avatar
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. ...
JRN's user avatar
  • 10.8k
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 ...
Mike Pierce's user avatar
  • 4,737
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 ...
Nicola Ciccoli's user avatar
21 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 ...
Chris Cunningham's user avatar
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? ...
Nick C's user avatar
  • 9,406
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 ...
mweiss's user avatar
  • 17.3k
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. ...
guest's user avatar
  • 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's user avatar
  • 7,672
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 $x$ ...
Gerald Edgar's user avatar
  • 7,499
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 ...
user52817's user avatar
  • 10.5k
17 votes
Accepted

Does there exist a (statistical) topology induced by students on the space of algebraic formulas? :)

Has this "topology", i.e. the organization of student errors in algebra, been studied systematically? Is there any reference in literature that thoroughly deals with the issue of really ...
Justin Skycak's user avatar
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, ...
Aeryk's user avatar
  • 8,025
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... "...
Nick C's user avatar
  • 9,406
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 ...
Fluidized Pigeon Reactor's user avatar
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 ...
Dirk's user avatar
  • 1,308
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 ...
Fareed Abi Farraj's user avatar
14 votes

How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?

It actually depends on exactly what you're asking. Or even what you SHOULD be asking. If you want the average profitability of all the 500+ operators in the Permian, you could just average all the ...
guest's user avatar
  • 141
14 votes

How do you coach students who often make small errors?

I used to have this problem. What helps me more than anything is: Solve it two different ways if you can and make sure they agree If you are finding a general formula, test it on some examples If ...
causative's user avatar
  • 241
14 votes

Is patching up a student's poor solution better than providing a good solution?

A lot of studies in math education have found that mathematical confidence is an important mediator of success - that is, students who aren't confident in their ability to do math are less likely to ...
Reese Johnston's user avatar

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