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 ...
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 ...
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 ...
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", ...
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 ...
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 ...
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 ...
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 ...
One way to address this issue is with more open-ended problems that have a "low floor" (easy entry) and "high ceiling" (hard aspects that can be addressed but don't need to be).
Sometimes this comes in a series of related problems. See the Julia Robinson Mathematics Festival activities for many great examples (click on 'activities library' for a pdf).
I don't think that "hard" and "easy" are the right categories when it comes to answer the question: "What kind of problems do you learn most from?" Indeed "hard" and "easy" are really subjective and many people might find "hard" the "easy" problems I learn most from and vice versa.
Since I guess that purpose of the question is giving students problems from ...
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 ...
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", "...
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.
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 ...
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,...
WebWork maintains an Open Problem Library. It is not necessarily "easy" to construct your own problems on WebWork, but the system is constructed with purposes like these in mind. In particular, if you have set up a WebWork server, you can maintain a local problem database, with a file structure, course structure, tags, difficulty levels, and so on. In ...
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 ...
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, ...
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.
Also, Math.com has a worksheet generator, which allows some specification of fraction use and ...
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 ...
I think a lot of people here and in research mathematics overemphasize the value of harder problems, discovery learning, etc. It appeals to them since they are the tip of the spear in intellect. And it's more intrinsically interesting to someone who already knows the stuff (them).
Also many of them have never done sports coaching and learning the benefit ...
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.
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.
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 ...
If you ask me in how many ways I can arrange 4 letters or digits, this is what I will do:
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 ...
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.
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 $C$.
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 ...
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.