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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. ...


0

The "deeper" or at least off-beaten path solution should be much quicker than a standard but boring approach to create a system of equations or calculate a derivative or something like that. Give two-three problems and limit time so that they can solve them in time if they apply some sort of a trick. You'll get a lot of angry students :) Here is an middle-...


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'(...


6

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 ...


0

One way is to use simple, atypical mathematical objects, fringe cases, non-examples and counterexamples, etc.


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 (...


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 ...


0

If your assumption on how the students may get the right answer, maybe rephrase the question in a way where doing that is the wrong answer. For instance, you could ask for acceleration rather than velocity. Still, I don't think many students will get the right answer by following the logic you mentiond without having an understanding on what's behind!


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 ...


0

Rankings are imprecise (e.g. may differ by source). In addition, gradation of number 2 versus number 4 (or the like) is rather a nuance, when you consider the restriction of range. More meaningful is top 5 verus top 50 or 500 or the like. Also, any ranking of schools from one country to another is problematic. In addition, prestige of a school seems to ...


0

I think that the most interesting application of (sum of numerators/sum of denominators) “addition” is to continued fractions. For example, if you want to calculate the continued fraction expansion of the square root of 2,start with 1/0 and 0/1 and “add” them in the following way: In the top row you put the results of the “adding” that are greater then the ...


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. ...


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 $\...


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{...


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 ...


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 ...


1

You are interchangeable, like peas in a pod. Cauchy and Weierstrass were usually saying "$f(x)$ becomes arbitrarily close to $L$", with the qualifier "as $x$ approaches $a$", sometimes before, sometimes after, sometimes implied. They also followed Leibnitz and Lagrange to talked about a quantity $f(x)$ becoming infinely close to $L$, when $|x-a|$ is ...


0

Difficult for us to research it for you. But you presumably have access to Norwegian libraries colleagues booksellers and know the language. I would try to research it yourself. One other idea for how to skin the cat is just to pick a less verbose English text. Maybe Schaums Outline. Edit. Reading your question again it sounds like they could use ...


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