25

Do NOT give exam questions that are intentionally more challenging than homework or in-class problems. I would recommend precisely the opposite. The point of the exam is really a spot-check that students know the basics and aren't just faking their way all through the class. If there is a time-limit, then that is already a lot more pressure/high-stakes ...


15

This is a good question, but it is impossible to answer in general terms. It depends upon (at least) the subject, the person, and (perhaps most importantly) what you mean by "learning it". I hope it is no secret that most undergraduate courses aim to give an introduction to the subject they concern. Things in the undergraduate curriculum which need to be ...


13

I wouldn't put any such sentence on your door, just one in the syllabus explaining your make-up exam policy which should be determined at the beginning of the semester. You don't want to come off as mean, intimidating, or otherwise to students who might seriously have an issue where a make-up exam would be justifiable (illness, death of family, etc); not ...


12

As other posters have noted, the answer varies from person to person. There is, however, some knowledge from education research (much of it conducted specifically in mathematics) that you might find helpful. One key insight is that the number of problems isn't usually as important as the timing or spacing of your practice. Once you've "worked to criterion", ...


9

I think your question is good, but a little vague. When you say "how many problems," would ten easy problems be counted the same as ten difficult problems? If exactly the same problem was posed twice, would it count as one problem or two problems? As you can imagine, the answer to your question would depend on the subject to be learned, the types of ...


8

A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically, a monohedral tiling). This can go as deep as you'd like, perhaps in stringing together several mini-sessions. Can every triangle tile the plane? (Yes.) Form parallelograms, then argue that a parallelogram tiles ...


8

I always make homework (from the textbook and online in WeBWorK) and written assignments MORE difficult than exam questions. I tell my students this, with the reason being “if you can run 10 miles in training running 5 miles on race day is easy”. Keep the course learning goals in mind. Your exam questions should be a chance for your students to demonstrate ...


6

I recommend OpenStax open-resource digital books from Rice University for all levels of college math, from prealgebra through calculus and statistics. For the student, they're free, can be shared and copied without restriction, carried on mobile devices, etc. From the instructor's perspective, one can rely on immediate access by all students, show the ...


5

Perhaps logic puzzles would work in this case. Some classic examples are: You're traveling along a road and arrive at a fork. Two guides are posted, but one always lies and the other always tells the truth, but you don't know which one is which. What one question can you ask to find out which path you should take? Three boxes are labeled "Apples", "...


5

I have taught such courses and I don't think that a long list of exercises without solutions is good idea - at least it didn't work for me. It is because 1) students are not yet able to tell a correct proof from an incorrect proof. 2) often students don't know how to get started, but if you give them a hint, they will be able to solve the problem. What ...


4

From my own experiences as a undergrad math major, I would say that good, quality exercises are one of the best tools for teaching math to people. Paul Halmos was right: "The only way to learn mathematics is to do mathematics." I myself, got by for a long time on just showing up to class and paying attention in college math classes (I even aced vector ...


4

If these are exercises from a published textbook, then it's probably self-deluding to imagine that the students don't have access to them already. Chegg et al. probably already have the solutions available to anyone willing to pay the monthly membership fee. Your lead instructor has already chosen a philosophy and a set of rules. They've made the homework ...


4

This sounds vaguely similar to an "inquiry-based learning" course. So although this isn't what you are describing (notably the lectures), you may find resources at the Academy of Inquiry Based Learning helpful. Here is a separate description from a related venture. A most extreme version of this (extreme in a good sense, I think) is the so-called Moore ...


4

Since Benjamin Dickman mentioned The Tokyo Puzzles in the comments, I'll include a couple of the questions from that book here that I thought fit the prompt nicely; neither requires a pen and paper to think about, and a student can just ponder them mentally. Two brothers decided to run a 100-meter race. The older brother won by 3 meters. In other words,...


3

The courses that you mention -- real analysis and topology -- are most often senior-level courses for undergraduate mathematics majors. At this point the program is trying to give you a taste of graduate/professional work, where the overall project is one of generating proofs for as-yet unsolved open problems. It's also pretty universal that textbooks at ...


2

Ask them to give you a number between $0$ and $100$. The number they give you must be as close as possible to $2/3$ of the average of all the class' numbers. We could argue that this is more psychology than math, but it fits the no wrong answers. See Wikipedia for some history of this.


2

Matt Enlow recently posted a wonderful collection of problems on twitter in his tweet here. He links to the "More Questions than Answers" pdf on dropbox here; it is a compilation of 100 math problems, which run the gamut in difficulty and range (for me) from well-known to never-before-seen.


2

It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I really like it. https://www.wolframalpha.com/pro-for-educators/ Also, Math.com has a worksheet generator, which allows some specification of fraction use and ...


2

At first, I thought that my exam is too hard. However, I usually, definitely, give problems that need the understanding of materials I give in the class to solve, but add some "spices" to them. Part of the problem is the "extra spice" you're giving them during the test. Why are you surprising them with extra spice for the first time during the test? That's ...


2

If you ask me in how many ways I can arrange 4 letters or digits, this is what I will do: 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321 Then, I'll count them and answer your question: "Sir, I know, it's 24!!!". If you ask me how many there are for a general n letters, I don't have ...


2

I don't think the two "answers" you consider ("on the one hand", "on the other hand" ...) require an "either/or" answer. I think a "happy medium" exists, in which you do not offer pdf solutions to all exercises at each exercise session. What you describe during "in class exercise sessions" is spot on, in my experience. Allowing them to ask questions, ...


2

Typical constrained optimization (useful to "get it") is asking for the rectangle of largest area that can be enclosed in a fence of given length. Sure, it can be reduced to one-dimensional, but leave that option out. Or ask for the largest volume box with given surface area.


1

Something like this? $$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$ $$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \left({y \over x} + {x \over y}\right)$$ $$c. \left({n \over m-n} + {m \over {m + n}} \right) \times \left({m^2 \over n^2} + {n^2 \over m^2} - 2\right)$$ $$d. \...


1

The Fahrenheit and Celsius temperature scales are linearly related. A change of one degree Celsius is a change of 1.8 degrees Fahrenheit. The freezing point of water is 0° Celsius or 32° Fahrenheit. Find the formula for computing the Fahrenheit temperature $F$ given the Celcius temperature $C$.


1

I found another one I like. It's the example in this question on MathSE. There are lots of other good ones in this post too. Three friends Albert, Betty, and Chadwick ask the Game-Master to play a game with them. The Game-Master agrees, and proceeds to paint two colored dots on each of their foreheads. The dots are either blue or ...


1

There are some great puzzles here on Andrej Cherkaev's website, most of which he attributes to Vlad Mitlin. I'll list paraphrased versions of a few of them here which I thought most fit the requirement that they may be pondered over without any suggestion that they require calculations to answer. Note that I've reworded these questions rather dramatically. ...


1

Here's another one that I like. Suppose you and a friend are sitting at a circular table and you each have a large stack of pennies, so you start to play this game where you take turns placing one penny flat on the table. At each turn you may not place a penny over any other penny (because the pennies must be flat on the table), no pennies may lay over ...


1

I keep rather copious statistics on student performance in my classes, but these statistics are unpublished and not peer-reviewed. In my community college remedial algebra courses, doing the assigned homework exercises is, among 40 factors assessed, the single highest-correlated item with the test average, final exam score, and end-of-semester grade ($R^2$ = ...


1

I have created a website of public domain mathematics teaching materials. Most of the material is in the K-12 range. (I have been developing these materials since the late 1970’s.) The address is: http://www.public-domain-materials.com/ One of the items in my website is autocorrected exercises in proving theorems: http://www.public-domain-materials.com/...


1

Word? Yuck. Ultimately you need a document management system, each exercise has to become it's own file, although you can cheat and put multiple similar exercises in the same file, then each exercise is assigned keywords. You could do this with screen snapshot software and a good photo cataloging package. Adjust word settings so that the full path to the ...


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