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17

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 numerators to sum of denominators is $\frac12$. However, we can also write $a=\frac03$ and $b=\frac22$. Now the ratio is $\frac25$, which is not equal to $\...


14

Agreeing with comments and other posts: If you want more conceptual answers, give them less details in the set-up. Using your velocity problem, here are a couple of examples of making it more conceptual: Suppose that a truck's distance from you in meters at a time $t$ seconds after the big bang is given by the function $p(t)$. What does $p'(19)$ tell you (...


14

This is not really a math problem, it's a social problem. Some schools, such as West Point and Cal Tech, have their own honor code systems for this sort of thing. From what I understand, they work very well. However, most schools do not have any such system. Social scientists and psychologists have studied what factors promote or prevent cheating. Cheating ...


13

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 profit margin percentages. This is taking the ratios (profit/revenue) for each company and averaging them. It corresponds to your expected (mean) profit margin ...


12

I like guest's answer. To elaborate, here is a possible question to ask them. You take two trips in your car: Trip 1 is a 100 mile drive that takes you 2 hours. Trip 2 is a 200 mile drive that takes you 1 hour. (a) What is the average speed of your car? (b) What is the average speed on an average trip? The answer to (a) is $\frac{...


7

Asking students to explain why something happens can be useful for assessing understanding, although it is often harder to grade and works best with many demonstrations before the exam. (Students need to know what your expectations for a thorough explanation are.) I have found that asking students to critique a process will sometimes help me assess their ...


6

Comment-answer, but too long for a comment: I think you are thinking about this wrong. Tests are some of the MOST valuable hours in a course. They are high stakes performances (like in music or sports). Preparation for them drives a lot of learning. Then the actual execution and subsequent feedback is often much more valuable training than routine ...


5

I agree with @BrendanW.Sullivan's comment. That is, when teaching an undergraduate course, like calculus, students need more than procedural knowledge. For a deeper understanding, and efforts to evaluate such, students should be asked on exams to answer a few "free form" questions, like the one Brendan suggested. A good question to ask following any ...


4

As stated in other comments, try not to "invite" your students to merely apply rules they do know. In the specific example you mention, I would prefer a multiple choice question of the form: A truck's distance in metres from you as a function of time $t$ (in seconds) is given by a smooth function $p:[0,+\infty)\to\mathbb{R}$. Knowing that $p(19)=12$, $p'(...


2

(Too long for a comment and it is kind of a soft question anyways) I'm not so sure that your assumption of an imbalance is valid. Maybe it is. But would be better if demonstrated (or at least explored) first. Otherwise, we end up finding an explanation for a phenomenon that doesn't actually exist. Plus the exploration would probably inform the answer, ...


2

Analysis is useful Physics uses analysis. Engineering uses analysis. Continuous models are widespread everywhere, and typically they look like differential equations, if there is time involved. Finance uses lots of analysis. Note that a lot of modern geometry is differential geometry, which builds on a foundation of analysis. Certainly, other parts of ...


2

The problem is with the question, not with the students' answers. The question is ambiguous and I think the students' answer is actually much better than yours. Suppose I drive a thousand miles at 25mpg and you drive one mile at 35mpg. What's the average fuel efficiency? Your answer is 30mpg but I honestly can't think of any situation in which that is a ...


2

Think of an example with two ratios: 1/3 and 4/5. When you add the numerators, and divide this by the sum of the denominators, you get (1 + 4)/(3 + 5) = 5/8. Now, think about what is happening with the denominators - the denominator of the first ratio should only act on the first numerator. But instead, when you add the ratios in this way, the denominator ...


2

Allow me to offer another example: Imagine you and your best friend both want to buy a new smart phone. The phone you have chosen will cost you 300€ but your friend chooses a phone that will cost as much as 600€! Luckily, you have two vouchers that will give you a discount: The first voucher will give you the cheaper phone for free, if you buy two phones. ...


2

I can only help with (3). A. This behavior is not unusual and not just with intuitionists. It's good to be able to do things in your head, but you need to "know your head" and when you will have issues. B. In general, when doing pen and paper you should try to write down all the steps prone to an error. Of course there's a balancing point. But ...


2

This might go over differently at a liberal arts school, but we've had some success inviting students to an optional, uncredited program where they work in groups on fun problems from various topics in math (ideally associated with upper level courses, so we can say, "and if you liked this, you should take..."), supervised by two older undergrads. (There ...


1

There is quite a bit of literature under the banner "humanizing mathematics" (or cognate phrases). I am just beginning to read this literature, so I can do no more than point to a few references. This seems a slightly different emphasis than your first bullet on "a culture of diversity." Luis A. Leyva. "Toward humanizing undergraduate mathematics ...


1

This is a fairly minor issue, but probably worth a mention. At a technical university there was (and probably still is) a question about how to attract more female students. One aspect was the diversity in promotional material. The outcome was that male students reacted equally well to diverse promotional material than to material with mostly white male ...


1

My advice is to emphasize the "get a job" aspect. The university is filled with the dreamy beauty of knowledge stuff. (And that's fine. But you need to distinguish yourself and provide additional info that kids might not have.) Do you even know what percent of graduating kids go into what fields? How easy/hard it is to get jobs (like some sort of ...


1

What one has to do to test for conceptual understanding is hard to state in terms of general principles (although Polya's books on Plausible Reasoning do a pretty good job of addressing the issue) and maybe is best addressed via examples. Here is one example. Consider a cubic polynomial in one variable that is increasing as a function of its argument. ...


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