33

You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo each other. Their limits -- the integral and the derivative -- still undo each other. Added: The first sentence is a part of what is called “Discrete Calculus” https://en.wikipedia.org/wiki/Discrete_calculus


22

Instructor at University of Washington here - we were one of the early closures, so I feel like we're starting to get the hang of it. Here's what I'm using: Zoom: Zoom is similar to Skype, with better support for many-participant calling and additional features. It has a built-in whiteboard you can write or type on, mechanics for allowing students to "raise ...


17

(this is from my calculus notes, see page 233 of: http://www.supermath.info/OldschoolCalculusII.pdf)


16

Already put two comments but ideas keep coming to me so I'll just package them here. Keep in mind I'm not an educator, I'm just trying to think of practical solutions to the problem as a whole. Another answer recommends YouTube to upload source material but I feel this might be inadequate interaction. Other alternatives: Discord Recently in response to ...


14

I would just say you have a square in 2D, and a cube is the similar shape in 3D, then what is the next shape in 4D. Then show them the cube in cube view and cross like fold out. Don't jump to Schlegel diagrams and the rotating pictures on Wiki so fast...they are confusing. Do like I said instead. I think showing that transition from 2d to 3D, gets ...


13

The case for WeBWorK tl;dr - use WebWork as an easy way of giving students as much practice solving problems as they can handle. Learn by doing lots of problems with a tight feedback loop. It engages the game-playing, obsessive nature in us. Main use case - Homework engine The two biggest features of a VLE/LMS are the presentation of materials and ...


11

I gave a presentation to my department about this today. Like you, as I see in a comment, I am also at a CUNY math department. I haven't done all-online classes before, but I've used Blackboard heavily for ~20 years and have had a hybrid (partly online) class for the last two years. I have access to Blackboard Collaborate. My only cameras at home are ...


10

The Lone Ranger had escaped many dangers, but this time, the situation was hopeless. Back to the wall with 200 wild Indians surrounding him. He turned to his faithful sidekick, Tonto, and said "what will we do now?" Tonto replied, "who's 'we', paleface?" The needs of the administrators, students, teachers are all different. In other words, "who's '...


10

My suggestion would be to demystify the concept and try to disassociate it from spatial interpretations at the first approach. Having $n$ dimensions is just having $n$ variables. As one professor I had liked to repeat, a grocery store owner who is trying to maximize gains by selling oranges, apples, bananas and peaches with some constraint in his storage ...


9

I like You Tube for posting videos. Once you get started it's pretty simple. There are various levels of privacy possible which you can read about. If your school has a convenient way to post videos and you have broadband (we're talking about 1-1.5 Gb files here, do NOT use HD resolution or worse yet the 4k resolution...). Given all that, basically the thing ...


9

Derivative of integral is the original function: Let $F$ be the integral function of $f$, so that $F(x)$ is the area under the graph from zero to $x$. For small $h>0$ the difference $F(x+h)-F(x)$ is the area of a narrow vertical strip. The width is $h$ and height approximately $f(x)$. As $h\to0$, this means $F'(x)=f(x)$. If you like thinking in terms of ...


8

I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet. When I was young, I read the book "Flatland: A Romance of Many Dimensions," probably in high school, and it made me "grok" the fourth dimension.


8

That reminds me of my first college programming course where they drew a square picture of a 2D array, a cube for a 3D array, and then said 4D arrays were very hard to understand. But I'd already made 4D arrays that were fine, since they weren't representing points in 4d space. I'd played a computer dungeon game where you had continent, province, dungeon, ...


7

I am using zoom, and getting better attendance sometimes than in my face-to-face classes. Modified tools for getting interaction: 1. I ask students to put a number from 1 to 5 in the chat to rate their understanding. 2. I ask for a brave volunteer to work with me sometimes, and I walk that one student through, asking them to give me a next step, etc. ...


7

I once read a suggestion to explain the difference between local and global maxima by using the example of Mount Everest (global maximum of the height function) and K2 (local maximum). I consider this a bad example because it gives two erroneous impressions: (1) local maxima almost achieve the global maximum value (because K2 is almost as high as Everest) ...


6

Well, since I have also been obliged to teach from home due to CoVID-19 these days, I will describe here a possible solution to your problem. As a fast and cheap solution - I have been granted no access to any platform, unfortunately - I use the following: Skype, as a platform to communicate with my students and conduct the major part of the lesson. ...


6

In some sense this question is waaaay too broad, but it is attracting a useful collection of hints, and it's super topical for thousands of college math instructors (likely to be followed by primary/secondary ones), so here are a few things which I don't see mentioned yet, collated from the far too much time spent on this subject today. Web/Doc cam. There ...


6

I am surprised that a school would effectively say "go figure it out". You ask for "brainstorming"... here are my thoughts.. You haven't quite defines your goal, although I did hear, loud and clear, you'd like 'free' or close to it. Still, there are a number of outcomes. Live video - I believe there are many options, but in general, a multiuser ...


5

My answer also comes from physics. Say p(t) is the position of an object in the time t. For concreteness, suppose you are traking a truck and the truck is going forward on a road from A to B. (i.e., all derivatives are positive) It is very natural to graph p(t), and derive it, arriving at v(t), the speed of the truck for each time. After all, the ...


5

The best explanation of the general concept I've encountered so far is the introduction to a 4D game, Miegakure. The idea of extra dimensions is described in the jump from 2D to 3D first, which makes it much easier to visualize and extrapolate. https://www.youtube.com/watch?v=9yW--eQaA2I


5

I strongly recommend the film Dimensions by Jos Leys, Étienne Ghys and Aurélien Alvarez. It's free! The main tools used by the authors to explain the several dimensions are cross sections and steregraphic projections. The animation is very didactic, building the ideas in 2D and 3D as preparation for 4D. There are dubbed versions in Deutsch, American English, ...


4

I gave a similar post to this one touching on this on Math.StackExchange. Basically, the way I would go about it is to say that there is a very easy way by which one can think of at least Riemann integration (the usual definition given in a "most courses" calculus course) as an inverse of differentiation by construction: that is, the relationship between ...


4

When I first began teaching Calculus, I realized that I really didn't understand the Fundamental Theorem. So I looked for something that would help me have a deep understanding, that would also help me help students to see it. I found a lovely project, which I have modified over the years. Here are links to the pdf and to a .doc version (in which the ...


4

My university (I am in Spain) has Microsoft Teams integrated with student accounts, so I use this. My classes are two hours. I open a chat (there are 60-70 students). (Zoom for free only allows 40 minutes and I am not sure it can handle the 60-70 students; a colleague is using the free video mixing software OBS and broadcasting class via Youtube, but ...


4

I'm a full-time faculty member at a community college in NJ. I've been teaching for 12 years but have never taught an online course. When our college left for spring break I was left scrambling for how to make use of the tools that I had at that moment: my course notes and textbooks, my laptop, and my iPhone. With these tools I write, by hand, a short ...


4

There are deep mathematical questions everywhere. I am fairly certain that, given enough time to prepare, I could take any topic in mathematics, no matter how elementary, and present a 60 minute talk that is both (1) accessible to a lay-audience and (2) concludes with several fairly deep—possibly even research-level—questions. By way of example:...


4

Many logicians that I have spoken to have concurred with my assessment that this is an issue of the misleading use of "let". Many teachers use this word in two very different and incompatible ways. The first is universal quantification, as in your example. The second is existential instantiation, as in "Let $z = \exp(x+y)$. Then [blah blah] about $z$.". The ...


3

To enhance my last comment, this is what I was talking about with Blackboard Collaborate Ultra. I did a trial session with some of the other faculty yesterday and it seemed to work alright. We come "back" from spring break next week, so I haven't had a chance to try it with students yet. It might be worthwhile to either do a training session with students ...


3

Intuitively, the fundamental theorem of calculus states that "the total change is the sum of all the little changes". $f'(x) dx$ is a tiny change in the value of $f$. We sum up all these little changes to get the total change $f(b) - f(a)$. I elaborated on this explanation here: https://math.stackexchange.com/a/1537836/40119.


3

For more than four dimensions, I would consider treating each pixel in a gray scale image as a "dimension" and its brightness as the value of the corresponding "coordinate". Then, ℝm x n is just the set of all images (including photos) of m pixels by n pixels. There is a video on YouTube which explores this approach (and a few geometric interesting aspects ...


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