103
My suggestion is that you stop thinking of homework as an assessment tool, and instead find the true meaning of homework, which is a method for students to gain proficiency with the ideas on their own. The key is to design your other assessment tools so as to motivate the students to see that doing homework is a good idea.
My practice, for example, is to ...
48
TL;DR 2-semester course is not enough.
Disclaimer: I write this as a computer-scientist that uses math a lot in his work (I'm a research assistant at a university).
Introduction:
There are three (overlapping) aspects of math in computer science:
Math that is actually useful.
Math that you can run into, and is generally good to know.
Math that lets you ...
45
The underlying issue is something that is hinted at in JDH's answer, but not explicitly stated. So I'd like to state it.
The key is to decide what your homework is for. The big words are Formative and Summative. In short, formative assessment is designed to help you learn, summative assessment is designed to figure out what you've learnt.
The ...
42
First, a little background. I switched to slides a few years ago when teaching a 3rd year course at my university. Because of how teaching works at my university, this course is one of the first where as a lecturer I can assume that the students taking it are interested in mathematics as a subject in its own right. That is, they aren't students from ...
42
I'm an old-school biologist (animal physiology) who works with mostly cell biologists. I sent out an email to a bunch of grad students and postdocs I work with. Here is the data so far:
Senior undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her.
Grad student: Undergrad biophysics course used ...
28
This is not an answer to the posed question, but only an
anecdote. This semester, teaching US college students (Discrete & Computational Geometry), I prepared all my assignments in LaTeX, and made available a .zip file of the .tex, .bbl, .bib, Figure/ directory constituting
the assignment. Students could submit assignment answers in any form—from ...
27
I think that actually trying to get students at this age to contemplate infinities in a rigorous way is probably ill advised.
I do think that exploring counting from both an "ordinal" and a "cardinal" point of view is probably a good idea.
Example for a 5 year old: Something you could do is have 20 stuffed animals, only 18 of them wearing hats. You can ...
26
The Moore Method is alive and well, and so are a great many variants. These days the community is more likely to use the term Inquiry-Based Learning (IBL), because the Moore Method can be seen as a restrictive set of practices, and people use the same underlying ideas in a lot of different class structures.
If you want to learn more and meet people who are ...
26
My favorite textbook for an undergraduate course in Abstract Algebra, Ted Shifrin's Abstract Algebra: A Geometric Approach, uses a rings-first approach. The primary pro is that students are much more familiar with examples of rings (integers, polynomials) than they are with the standard examples of groups (symmetries of simple shapes, permutations). Indeed,...
24
In my experience, students in sophomore or junior level math courses usually have very little trouble picking up LaTeX on their own. They typically require the following assistance:
Some guidance in downloading and installing it, e.g. links to user-friendly distributions for both the Mac and the PC. I post links to my course webpage.
A sample LaTeX source ...
22
The Moore method is used at the University of Chicago in some sections of "Honors Calculus", which is really an introductory real analysis course for top incoming freshmen. I assisted with it a couple of times and taught it on my own once. It absolutely depends on having a well-constructed sequence of notes to use; we started with a truly excellent set of ...
21
I always organize my courses with the totality of the course set from the outset whenever I can. I see this as being closely tied to your question concerning homeworks assigned.
The benefits I see:
Keeps me and my students on track. The semester invariably gets busy, it's nice to have a go-to place where everything is set from the outset. I can always ...
20
I happen to have revised our calculus syllabus for first year biology majors about one year ago (in a French university, for that matter). I benefited a lot from my wife's experience as a math-friendly biologist.
The main point of the course is to get students able to deal with quantitative models. For example, my wife studied the movement of cells under ...
19
What are the most important considerations when writing your own lecture notes for a course?
I would say that the single most important consideration is to make life easy for yourself. It is easy to fill your time with writing lecture notes but that isn't always the best use of that time. Nevertheless, it can be useful for the students so here are my tips....
18
Theoretically, there's no way to determine the next term in the sequence
$$
1,\quad 2,\quad 4,\quad 8,\quad 16,\quad\ldots
$$
It literally could be anything.
At the same time, it is a vitally important skill to be able to look at this sequence and say "it looks like the powers of 2". This answer is correct in the sense that any mathematician looking at ...
17
Of course there are many differences between undergraduates and graduates, including:
Mathematical independence. In a graduate class, one may generally presume a much greater level of mathematical independence and self-motivation than in an undergraduate class.
Mathematical sophistication. Graduate students generally are more knowledgeable than ...
17
After reading the previous answers, I'd add logic and proofs. To reason about a program's (in)correctness is proofs, even if you don't go to the length of proving correctness formally.
Many students struggle with the idea of recursion, recursively defined functions/data structures, and the related proofs by induction.
More on the soft side, being able to ...
17
Your assumption that teaching calculus needs to be backed by the $\varepsilon$-$\delta$ definitions could be challenged, but since it is not your question I won't do that here.
My recent experience about a few logic classes first has been disappointing. It took much hard work, and the outcome seemed good at first, but vanished as soon as we got to the main ...
16
There are a few ways to do this. I recently did it in Carnegie Mellon's introductory proofs course. I'll try to outline here a few different solutions that various places have tried.
Carnegie Mellon (15-151): We alternated introducing new topics (usually via lecture) with what I called workshops. A workshop is during normal lecture time. Students are ...
16
I agree that this is a problem in teaching calculus and other classes where solutions manuals and online forums are readily available and I don't know that there exists any one thing that you can do. No matter what, a student can always find "help" somewhere.
First some remarks
If a student gets hold of a solutions manual, then it is of course easy for the ...
16
I took a number theory class at University of Cincinnati that was taught using a modified Moore method. The class size was pretty small, and credit-wise it was an "upper level mathematics elective" (i.e. not a core class), so our professor had some room to be experimental.
Like with the Moore method, the students presented all course material. The ...
16
I hand out notes in most of my courses. These notes parallel my lectures, sometimes precisely, sometimes just roughly. Some of them are typed, some of them are pdf-scans of my handwritten work. I usually post a day-by-day lecture schedule detailing where I'll be in the notes as best as I can forecast. I suppose, in principle, there is a possibility that ...
16
I'm just finishing up a graduate course in computational topology which could be adapted very effectively for this purpose. We're focusing on topological data analysis and computational homology. All the topology in the course has been self-contained, meaning that essentially no previous experience in topology was required.
The book we're using is ...
16
Here's a silly example:
Give students collections of the same type of thing, where each collection contains "good" objects and "bad" objects -- for example, a stack of Pokemon cards with both rare and common cards. We might assume common cards are worth $1$ and rare cards are worth $5$. Have ready another stack of cards that are all common -- call this the "...
16
Here's my advice.
I have no teaching experience.
Remedy that first before you lay out plans for a 6-month course of study. Find some way where you can teach just for a single day in some way at the high school that you're targeting. After that, find some way where you can teach for a week (i.e., say, 4 interlinked days). I'd suggest that you restart your ...
15
Here's another idea: Give a completely different type of homework assignment.
For example, this year, collect all of your homework as usual. Look through the homework for examples of solutions that are clearly done by students, and which have mistakes in them. Choose representative samples of the student homework which have evidence of common misconceptions ...
15
I think you will definitely want to check out the The Journal of Inquiry-Based Learning in Mathematics. It is dedicated to being a "resource for designing undergraduate mathematics classes that are not lecture-based."
Another place to start that is overwhelmingly huge and mostly aimed at the K-12 level, but definitely has a lot of good ideas that are ...
15
One technique which is fairly obvious, but (at least for some of us) surprisingly difficult to implement consistently, is to just model for them in class what you expect them to write on their own. When I solve a problem in class, I try to show the same work and write the same explanations that I expect them to show. I also try to talk about it as I do it, ...
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