20 votes

How to answer a three-year-old the question "Why is $2+6$ the same as $4+4$"?

I'm nearly sure I did this with my child when she was young. First, establish that she understands that a number, like three, is equal to $1+1+1$. Hold three fingers up and ask her "how many is this"?...
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  • 7,713
15 votes

Why don’t all professors let students use notes, books, etc. on exams?

Making a course easier to pass is not the same as making a good course. There is a certain corpus of knowledge with which practitioners are expected to have instant recall (a.k.a: "automaticity"). If ...
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13 votes

Why don’t all professors let students use notes, books, etc. on exams?

I think that allowing students to use resources on exams can work well in certain situations, such as when the knowledge they can pull from those resources is peripheral (not of core importance to the ...
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  • 593
12 votes

Why don’t all professors let students use notes, books, etc. on exams?

I think it is usually counterproductive. Instead of building skills, it rewards students who are good at information retrieval. And I say that as one of those advantaged. Still remember a P-chem ...
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  • 636
11 votes

Why don’t all professors let students use notes, books, etc. on exams?

I allow students to have one index card of notes during their exams. I believe this allows me to test for understanding and skills that are more important than the ability to memorize. I also ...
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  • 10.2k
11 votes
Accepted

What are questions from students that improved the teachers understanding of mathematics?

At my university, the usual multivariable calculus class is split into two semesters: the first does geometry in $\mathbb{R}^3$ and then derivatives (gradients, Lagrange multipliers, etc.) while the ...
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10 votes

Ten options for multiple choices questions

Presumably, you are trying to reduce the chance that someone will get a correct answer by guessing. In that case, I think that 10 options are more than you need. Suppose that we have an $N$ question ...
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  • 4,784
8 votes
Accepted

Falling into the calculus trap

Echoing @AndreasBlass' remark, and having experienced somewhat similar episodes, it is already precarious enough to make such choices _for_oneself_. So, to directly answer your question: I think "no, ...
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  • 13.6k
7 votes

Why teaching undergraduate-level mathematics is so complicated?

There are many reasons, but the classic explanation is that professors (especially at research universities) are picked (and compensated) for research ability versus teaching efficiency. (In general.....
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  • 79
7 votes

What are questions from students that improved the teachers understanding of mathematics?

I am not a research-level mathematician, but this absolutely happens to me. There is so much in math that we can think more deeply about. Just last week, a student came into my office to get help on ...
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  • 17.5k
7 votes
Accepted

Funny things students say when learning mathematics

From John Stembridge's web site at http://www.math.lsa.umich.edu/~jrs/plans.html : "Are we going to have to think today, or is it going to be all math?" --a student in Phil Hanlon's Math 115 class
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6 votes

Ten options for multiple choices questions

There's a fair amount of research about this, mainly from psychology and medical education as well es test development research for high stakes tests. In practical development, I usually resort to ...
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  • 803
6 votes

How to answer a three-year-old the question "Why is $2+6$ the same as $4+4$"?

Blocks work well for thinking about addition. Have her count out 8 blocks, and then ask her about all the addition problems that have 8 blocks as the answer. A lovely children's book which looks at ...
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  • 17.5k
5 votes

How to answer a three-year-old the question "Why is $2+6$ the same as $4+4$"?

A good answer to this question is one that (is correct, and) she finds convincing. As kids are growing up and making sense of the world around them, experimentation is often one of their key sources ...
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  • 51
5 votes

How cheating on homework affects student questions

Imagine a scenario where you don't have a hypothesis about your students' unwillingness to ask questions in class or during office hours. Your question may just boil down to: "How do you get students ...
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  • 7,713
5 votes

Ten options for multiple choices questions

I do think that 10 options for a multiple-choice question is excessive. A few things to consider: This will be outside the range that standard automation tools can handle (TestGen application, ...
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4 votes
Accepted

Ten options for multiple choices questions

I would say it completely depends on what is being asked. For example, something simple like "What is the derivative of $\cos(x)$ with respect to $x$?" could have as many as 10 options, because no ...
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  • 7,713
4 votes

Falling into the calculus trap

[VERY LONG ANSWER, needs patience to read through] I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school ...
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3 votes
Accepted

Why don’t all professors let students use notes, books, etc. on exams?

An exam is a totally artificial scenario. In real life you will never be in a scenario with all these fact at same time: a) you can not consult external resources or colleagues; b) you have limited ...
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3 votes

Ten options for multiple choices questions

Like others, I feel this is too much. It turns the question into almost a mini research project (assuming there is some reasonable amount of checking needed). It's one thing to just scan through ten ...
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  • 61
3 votes

Math Scholarships for the highly advanced

I don't know about scholarships in the USA, but here's an alternative: Consult a maths department at some university (maybe college is more appropriate in USA?), preferably one that is close enough ...
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  • 4,626
3 votes

Funny things students say when learning mathematics

I once had a colleague when I was teaching in Asia who was new to teaching high school, but was a very accomplished Mathematician. In one of his first lessons he was teaching simultaneous equations, ...
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2 votes

How can a Q&A contest help a math teacher to improve his/her students' understanding?

You can organize it as a low-tech version of StackExchange! Collect questions (anonymously) and present them to the class, letting them upvote the questions they feel are relevant. Then collect ...
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2 votes

Funny things students say when learning mathematics

From a student learning real analysis: The sequence diverges because the Cauchy criterion is dissatisfied.
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  • 6,367
2 votes

Why don’t all professors let students use notes, books, etc. on exams?

I believe the choice is based on the expectations: what does the teacher expect the student to be able to do with the course content? If the student needs to show that (s)he is able to use the course ...
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  • 832
2 votes

How to answer a three-year-old the question "Why is $2+6$ the same as $4+4$"?

None of the answers so far seem to give a name for the kind of concept in the inquiry. What you're asking is "how to make 8" or more generally "making numbers". See here and here. "Making numbers" is ...
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