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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 ...


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

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 ...


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

Should we impose that $(a^m)^n=a^{mn}$ only when $a \gt 0$? Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule $(a^m)^n=a^{mn}$ cannot be true with negative bases and rational exponents, and this is a great achievement. Try to explain that he/she proved that If the said rule is true for negative ...


9

That gives a different result for the cube root of -8. It doesn't give a different result, it just gives two additional roots that are complex numbers, for a total of three roots. A physics or engineering student in the US probably first learns about complex numbers in high school, but never sees any interesting applications. Then in college classes they ...


7

Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work. So please don't accept this answer. Anyway, the educational answer is to see that student is using the fact that $1^2 = (-1)^2$ in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand ...


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 ...


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

When I asked the What are the Laws of Rational Exponents? question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. While it wasn't the top-voted, my answer there represents my best thinking about the status of this issue in classes at that level. As I wrote: Regarding the ...


6

In the UK, what you describe lies somewhere in between GCSE and A-Level Maths. Both of these exams are intended for secondary school students. It is unusual for these topics to be taught at the university level, and if they are taught, it is often not by the Maths department. For example, the Economics department might have such a course for their first ...


6

This is a complicated question, and there are a number of articles written on the topic in the math education literature. Here are some of the entries that I would recommend (taken from the bottom of this answer): Goel, Sudhir K., and Michael S. Robillard. "The Equation: $-2 = (-8)^\frac{1}{3} = (-8)^\frac{2}{6} = [(-8)^2]^\frac{1}{6} = 2$." ...


6

There are two levels at which someone can understand algebra. (1) They can do stylized tasks using a set algorithm, such as multiplying out $(a+b)(c+d)$. (2) They understand what it means and can apply it to real-life problems. In principle it is possible for someone to master #2 while still not being competent at #1. The reality is that this never happens. ...


5

It's important to clarify the difference between the way mathematicians do things, and the way scientists do things. In math, we typically reserve $x$ for the independent variable (aka input) and the horizontal axis, while $y$ is reserved for the dependent variable (aka output) and the vertical axis. With these variable choices, we need to swap variables in ...


4

I usually introduce the idea of inverse functions by linking it to the idea of basic composite functions For your example, I'd ask: ``What are the (intermediary) functions done to $x$ to get $2x+2$?" Given $x$, we'd multiply by $2$ and then we'd add $2$ So it's like we have $y=f\left(g\left(x\right)\right)=2x+2$, where $f(x)=x+2$ and $g(x)=2x$ Then to get ...


4

Since $y=2x+2$ and $x=0.5y-1$ have the same graph on the x-y plane, I am hesitant to call them inverse functions. Clearly, it's just a matter of making a distinction between independent variables and dependent variables. It is both traditional and sensible to treat $x$ as the default independent variable and $y$ as the default dependent variable. Based ...


4

I can't spot in any of the other answers what I think is the main point. Neither teaching students to swap variables or teaching them not to switch variables is really the solution, as both of these simply train students to carry out a mechanical process without understanding what is going on. Either way, for the student the 'inverse' of a function remains '...


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

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 am going to list the topics from the Russian Algebra/PreCalc school program, and you decide for yourself. The material below is mandatory for all students. I did not include optional material. Russia has 11-year grade school; it used to have 10-year until late 1990s. No K class. The age to enroll 1st grade is between 6.5 and 8 years old. There is no "...


4

Here's one that students always enjoy. A bottle and a cork cost 1 dollar and ten cents. The bottle costs $1.00 more than the cork. How much does each cost? Student often think that it is one dollar for the bottle and 10 cents for the cork. That's incorrect because \$1.00 isn't \$1.00 more than 10 cents but the answer is easily discovered with algebra. ...


4

Pedagogically speaking, factoring is a lot less intuitive than 'simple' rearrangement. For your example we have that, $$ 2x +4 =10. $$ When first teaching Algebra, there are many nice and neat tricks/visualizations to understand the process of unraveling the equation to solve for $x$. A classic analogy is to see the equation as a kind of seesaw that's ...


4

That is a great question that I struggled with the one year I taught Algebra 1. I came to the same conclusion that factoring by grouping problems are inauthentic because they only work when the expression is built to work. The story that got me through it is that you can use factoring by grouping to factor a quadratic expression if you manage to come up ...


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

My home teaching setup. Still to be tested.


3

Here I broach only the case in which students have consistent internet access; the more general case is clearly much more difficult. Moreover, this answer is written for the present context, i.e., a sudden shift to remote learning (perhaps disaster distance learning is a more apt name) due to a pandemic. This is not an answer about teaching mathematics ...


3

Algebra is a like a hammer that always works unlike arithmetics that may require inventive tricks, different for each problem. You may want to read a pertinent short story Tutor by Anton Chekhov. The problem posed in the story can be solved arithmetically, algebraically and arithmetically with tools like abacus. Do you do word problems with your students? ...


3

I should disclaim I haven't managed to solve this equation yet, even with your description. What were your failed attempts like? What was your own approach to this problem, and can't this approach help your son? Overall my approach for a problem like this would be: in general, to solve for N variables you need N independent equations. Here the problem gives ...


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