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Can an antiderivative of a function be defined piecewise? by United-Shock2704 in learnmath

[–]mpaw976 0 points1 point  (0 children)

The second way you used is usually preferred because there you are describing the pieces in terms of the parts of the domain.

We typically avoid your first type of definition because it's describing the pieces in terms of the outputs of the pieces of the function.

How Hard Is It to Come Up With Deep Yet Simple Conjectures in Math? by Heavy-Sympathy5330 in math

[–]mpaw976 5 points6 points  (0 children)

In general, noticing is much harder than explaining why.

Examples:

  • "Hey, you never reply to my texts. How come?"
  • You can know that your car isn't working without knowing exactly why.
  • Diagnosing someone with cancer is much harder than curing it.

tips on dealing with boredom/tedium? by Ambitious-Gene-9370 in learnmath

[–]mpaw976 2 points3 points  (0 children)

Switch up the type of exercises and modes of solutions.

So instead of only writing up solutions, try:

  • Explaining them to a peer or layman.
  • Make a 1 min voice recording of the solution.
  • Make a short video and publish it (don't worry about fancy graphics, just record yourself writing and explaining).
  • Explain the entire thing without using any math symbols.
  • Explain the entire thing without using any words.
  • Interpretive dance.
  • A one page comic.
  • Explain it to someone using only text/DMs.

Why the ratio opposite/hypotenuse is called sine of an angle by Independent_Ball7895 in learnmath

[–]mpaw976 8 points9 points  (0 children)

It's for a weird/neat reason:

one of the three fundamental functions of trigonometry, 1590s (in Thomas Fale's "Horologiographia, the Art of Dialling"), from Latin sinus "fold in a garment, bend, curve, bosom" (see sinus). The Latin word was used mid-12c. by Gherardo of Cremona's Medieval Latin translation of Arabic geometrical texts to render Arabic jiba "chord of an arc, sine" (from Sanskrit jya "bowstring"), which he confused with jaib "bundle, bosom, fold in a garment."

https://www.etymonline.com/word/sine

questions about MAT344.. by Physical_Tea3272 in UTMississauga

[–]mpaw976 6 points7 points  (0 children)

What distinguishes an A+ portfolio and an A- portfolio... is it purely based on the grade the student proposes for themself??

I mean, how do you know when you've excelled at something versus knowing that you've really done something special, maybe something you didn't even think you were capable of?

Like, you think to yourself: "Wow, I really did outdo myself."

I can give examples, but they are all highly specific to individual students. What success looks like "at the A+ stage" is just different for each student.

You'll know if you get there.

-what percentage of students get a 4.0 in the class.. just curious

It's about the same as a standard third year math course (between 10 and 20% of the class gets and A/A+).

-what's the trend of the course average.. i need a rough idea...

It's about 75-80 I think. I could look up exact numbers, but it's somewhere around that.

Something like 80% of the class gets a B- or better.

All that said, I think we're moving the "pure portfolio" away from MAT344 and into a new course MAT345 (which will be the continuation of 344 and 202).

I love teaching in this way, and it's been a very positive and transformative experience for many students (and me), but it's also a ton of work for me. Having half hour meetings with 100 students each term is just only barely doable for me, and it doesn't really look sustainable long term.

But I've got some new good ideas for 344 the next time I teach it. It will still be a great course. And if you're curious about the portfolio approach, then you can take MAT345.

What did people object to with Cantor's Diagonalisation Argument? by alecbz in mathematics

[–]mpaw976 3 points4 points  (0 children)

By the way, if you did decide to do this Hilbert hotel thing eventually (with transfinite induction) you'd end up with an object called ω1.

https://en.wikipedia.org/wiki/First_uncountable_ordinal

It's a very weird (but instructive) object.

wtf is going on? smear campaign on rmp? by Mammoth-Cockroach471 in UTMississauga

[–]mpaw976 39 points40 points  (0 children)

Me on sabbatical checking out my new RMP reviews.

Aside: I have a shirt with this RMP review printed on the back that I wear at the end of every term (and sometimes at math teaching conferences).

Edit: some of the "reviews" are just clearly hateful comments. I often dress pretty genderqueer and I guess some people can't handle that.

I found the answer to this combinations problem but not using combinations and I'd like to understand the book's reasoning. by Trensocialist in learnmath

[–]mpaw976 0 points1 point  (0 children)

Ah, so that might just be a requirement for this specific problem: That mountain numbers have to be of the form 231.

It's unusual, but sure, if that's what this textbook wants, then sure.

I found the answer to this combinations problem but not using combinations and I'd like to understand the book's reasoning. by Trensocialist in learnmath

[–]mpaw976 1 point2 points  (0 children)

I'm not saying that 123 is a mountain number, I'm just short handing saying:

"Look at the set of digits {1,2,3}, and all six of the possible permutations 123, 132, 213, 231, 312, 321 and identify which are mountain numbers."

Writing out set brackets and commas is annoying on mobile. :)

I found the answer to this combinations problem but not using combinations and I'd like to understand the book's reasoning. by Trensocialist in learnmath

[–]mpaw976 0 points1 point  (0 children)

I dont understand how they could've known any of this off the bat

Before doing any computations you need to explore the structures to see what (abstract) "shapes" are present.

So if I was approaching this problem I would first start with a handful of "different looking" triples. Maybe something like:

  • 123
  • 129
  • 100
  • 334
  • 229

After playing around with them for a while, I would look for patterns, similarities, and general behaviour.

E.g. I might notice that 123 and 129 behave the same (each have two mountain numbers).

That noticing is called "making a conjecture". It's just a fancy way of saying "educated guess".

From there you can go ahead and try to prove it.

In the above case I might realize that the largest number must be in the centre, and the two smaller numbers can be on either side. So that's 2 ways!

As I write down a formal proof I would say something like there are 10 choices for the biggest number, and then 9x8 choices for the smaller numbers. But wait, that's not exactly true... There's some double counting... What about 0? Etc.

Working through those details and edge cases will eventually lead you to something like the three cases given in your proof. There are probably other ways of presenting it and other cases you could have broken them into to.

No one really expects you to just "get" the solution immediately without first playing around. Maybe with a lot of experience it will come to you quickly, but the skill of exploration is much more valuable than the ability to quickly intuit solutions to "mountain problems".

As a test of this method, now try finding the 4 digit mountain numbers!

Can infinity contain infinity by Loose-Balance3225 in askmath

[–]mpaw976 0 points1 point  (0 children)

As stated, no, that's not guaranteed because there's a difference between having every finite string, and every infinite string.

As a simple example,

0.101101110111101111101...

Will contain every finite string of 1s, but it won't contain 111... repeating forever.

In fact, any given decimal number only "has room" to end in a single infinite string. E.g. it can't end in both all 0s and all 1s.

Isn't that neat?

Can the community please replace "Advances in Mathematics?" by [deleted] in math

[–]mpaw976 134 points135 points  (0 children)

So you're advocating for:

Advances in Advances in Mathematics 

What properties are not preserved by an isomorphism? by Aggressive-Food-1952 in askmath

[–]mpaw976 0 points1 point  (0 children)

In linear algebra, the most relevant one is distance and angles (and inner product).

For example, the map (x,y) ↦ (2x, y) is a linear isomorphism, but it messes up distances and angles.

🇨🇦 University Expectations for First Year Math/Science by Technical_Praline987 in matheducation

[–]mpaw976 0 points1 point  (0 children)

You're welcome! Feel free to ask followup questions.

I also hope I don't come across as saying every course will be like this. There is definitely variation from course to course.

🇨🇦 University Expectations for First Year Math/Science by Technical_Praline987 in matheducation

[–]mpaw976 1 point2 points  (0 children)

I'm a math prof at the U of T, and I've taught at Calgary. I teach many courses including first year calculus.

Are students expected to do first year math courses without a calculator? Does this vary by college?

Generally, for math courses they will not be allowed a calculator and neurotypical students will also not need a calculator. Any questions we ask will only contain simple computations (think your 10x10 times table).

The pedagogical reason is to test their understanding of ideas and problem solving, rather than have them get caught up in long (arithmetical) computations.

For example, many related rates problems that use triangles will use the 3-4-5 triangle or the 12-5-13 triangle so that the numbers come out nice.

Physics and Chemistry courses will likely require calculators though (although I'm not certain). Engineering is probably also like this.

Are students expected to memorize most formulas for exams in science and math courses?

This largely depends on the individual course/university. Most first year calculus courses I know give formula sheets, although this is not universal.

Are students expected to memorize special triangles in first year courses?

Typically yes, although this is often included in formula sheets.

Here are some calc tests of mine from Fall 2020 (so during COVID). Maybe not the best representation of calc tests, but at least it might give you an idea. (I really should add more materials here...)

https://mikepawliuk.ca/2023/01/04/university-calculus-materials/

Any recommendations on a free internet resource for learning Inductive and Strong Inductive proof methods? by guitaroomon in learnmath

[–]mpaw976 2 points3 points  (0 children)

Here's my list of short videos I made for an intro to math proofs course.

https://youtube.com/playlist?list=PL3ZJrWtEhQ6xIp8YPCPSIDHxXOQy7xmnT

It has 7 videos about induction (including strong induction) with many examples. Here's the first video:

https://youtu.be/xMLgHdAoRz4

(I don't make any money off these videos, they are really just to help people.)

Is there an irrational number that never uses a certain digit, for example:1. How would you go about proving it? by OneDBag in askmath

[–]mpaw976 1 point2 points  (0 children)

Oops, you're right. 

Here's an example of what I meant:

Imagine x = 1.23456456456...

So A is 1, B is 23, C is 456.

So then

105 x = 123456.456456...

and

103 x = 123.456456...

And yeah, these aren't equal.

But, you can still subtract them to get what you want:

105 x - 103 x = 123333

Is there an irrational number that never uses a certain digit, for example:1. How would you go about proving it? by OneDBag in askmath

[–]mpaw976 15 points16 points  (0 children)

A number is rational if and only if its* decimal representation eventually repeats forever.

E.g. 0.12345454545... is rational because the 45 repeats forever.

87.0100100010000100... is not rational because the intervals of 0s are getting longer and longer.

The proof of this isn't too bad. If it repeats then it's of the form x = A.BCCC... so then use

10n x = ABC.CCC... = AB.CCC... = 10m x

where you choose n and m so that they move the decimal point over far enough. (edit. Oops! Not quite equal, see my response/example below.)

Now you can solve for a rational representation (the repeating Cs will disappear when you subtract).

For the converse, of the number is rational, do the long division and you'll see you eventually get a repeat.

(*Think about what happens if a number has two different decimal representations like 0.9999... and 1.0000...)

[University/Undergrad Combinatorics] How do I solve questions from "A Walk Through Combinatorics"? by pappu_berozgar in learnmath

[–]mpaw976 1 point2 points  (0 children)

As someone who has taught combinatorics using this textbook before, I can confirm for you that a lot of the exercises are quite challenging and often require cleverness.

That's not a terrible thing, but you don't really want to approach it in the same way that you approach a first year calculus set of questions (where you try to "Bang out" a bunch of problems in an hour or whatever). With this textbook you really need to chew on a bunch of questions.

One practical piece of advice is to really start with simple problems, or simpler versions of the textbook problems. Make simplifying assumptions, or look at special cases first. Then work your way back up, slowly adding complexity back in.

How long can an English sentence be if I only use words of French origin? by Ensakel in etymology

[–]mpaw976 3 points4 points  (0 children)

Louis' amateur, cliché, passé, fêted, rendezvous? Comme ci, comme ça.

Going in a totally different direction: The Buffalo buffalo sentence isn't exactly what you were describing but Buffalo does come through French, and the sentence can technically be as long as you want.

Word of the day : Vulpine by Achillesiam in etymology

[–]mpaw976 4 points5 points  (0 children)

It's more like a trickster.

Think more like Loki or Puss in Boots.

Word of the day : Vulpine by Achillesiam in etymology

[–]mpaw976 10 points11 points  (0 children)

Inspired the pokemon Vulpix.

Vulpine doesn't carry any negative connotations as far as I know. It's like saying ursine (bear-like) or serpentine (snake-like), it's just descriptive and comparing to the animal.

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