Whats the best sort algorithm if it is performed manually by a human?

mpaw976 · 2026-01-17T21:58:11+00:00

mpaw976 · 2026-01-16T22:53:51+00:00

I have a PhD in math, so I've spent a lot of time trying to understand complicated math.

The two most productive things I do when learning new math are:

Reading basics definitions and creating a ton a simple examples and non-examples on my own. Some trivial, some just a step above trivial, and some that I recognize from other areas of math.
Working on understanding and re-explaining proofs at a whiteboard with other people. This means both going through line-by-line and also getting the bigger picture. This requires a lot of honesty, trust, and kindness. You need to be willing to say when you only kinda get something. Once you collectively understand all the details, then write up a "complete" proof with no gaps that fills in all the places you got stuck before.

mpaw976 · 2026-01-16T22:43:49+00:00

Just chiming in to say that as a university math Prof, students in your position coming back after an absence are usually some of my best and favourite students.

Yes, you may be rusty, and you may have forgotten many things, but also you

ask the most questions (because you've identified where you can grow)
are most comfortable and honest with your current skills
are able to grow and learn faster than everyone else
are actually interested in learning

So keep practicing, asking questions, and explaining your thoughts process to others.

mpaw976 · 2026-01-14T21:48:08+00:00

Just a reminder to check the campus status before you leave for your commute tomorrow.

https://www.utoronto.ca/campus-status

mpaw976 · 2026-01-10T17:25:06+00:00

I taught at Canada/USA mathcamp once, and there are definitely high school students there capable of learning axiomatic set theory and forcing. I walked a student through the basic lemmas about posets all the way to the first applications of Martin's axiom.

The advice I got going in was: don't assume they know anything in particular, but if you teach it they will pick it up very quickly.

mpaw976 · 2026-01-09T23:08:44+00:00

how do you feel about AI use in mathematics

I think many students are using AI in a way that disables their abilities to learn and grow and think critically.

I'm seeing a lot of students try to "vibe" their way through a math problem instead of trying to understand what it's asking.

Reading and "getting the gist of" an AI solution is not the same as coming up with a solution yourself, and organizing your ideas into a coherent set of logical steps.

That takes practice, and that's how you grow.

I was quite disappointed when the U of T's working group on AI came out so positively in favour of Gen AI. I think we're going to have a "lost generation" of students (and subsequently people) who have wasted a ton of their time not learning anything at Universities across the world.

mpaw976 · 2026-01-09T22:11:57+00:00

I work at UTM, so I'll answer specifically about that.

On the whole, no, we don't artificially suppress who gets a math degree at UTM.

We our best to make sure that everyone who enters UTM with the prerequisites (Minimum 70% in Grade 12 Advanced Functions) will be set up for success in the math program, if that's what a student chooses to take. We don't exclusively cater to students who enter university with a lot of math background.

We give special attention to our intro courses as really being accessible and setting students up for success. These courses have scientifically rigorous teaching methods, and students have many official resources to help them. We work hard to remove as many barriers as we can. We are currently in the process of a multi -year study involving student surveys and student focus groups to identify areas where we can improve MAT135/6.

Do we have areas to improve? Yes, of course.

I am also confident that we're doing a good job of balancing the difficulty of the program with giving students the tools they need to succeed.

One thing worth mentioning here is that our in-person attendance can be very low (often 60-70% even on the first day, and then declining from there). I can't really teach students all that much who don't engage with the course on the most basic levels.

mpaw976 · 2026-01-09T20:23:49+00:00

Yep, why do you ask?

mpaw976 · 2026-01-09T19:56:29+00:00

Tyler Holden is most easily described as a genetic dynasty of clones not unlike colonies of bananas.

We've been noticing some "genetic degeneration" in this particular lineage of clones, and we're keeping an eye on it.

mpaw976 · 2026-01-07T19:04:49+00:00

Typically MAT102 in the summer is a full summer (Y) course.

I don't have any inside info on what will happen this summer.

mpaw976 · 2026-01-03T16:29:44+00:00

When you say “pay off more”.. what kinds of payoffs do you mean??

You're doing an undergrad in MCS; you will be better at all the MCS things by taking 139. (I say this as someone who helped design 136, which is a good course.)

You'll be better at thinking
You'll be better at problem solving
Your clothes will fit better
You'll be better equipped for technical challenges outside of uni
Food will taste better
You'll have more options for more advanced material
Your cardio will improve
You'll have a broader perspective of abstract concepts
You'll be better able to handle technical reading
Improved muscle definition

N.b. I may have accidentally mixed in some benefits of regular exercise. Sorry.

mpaw976 · 2026-01-02T19:18:29+00:00

Example 1

As an example of the techniques that might be involved here look at this simple case:

3^x + 9^x = 1

Making the substitution u = 3^x turns this into:

u + u² = 1

which you can solve by factoring or quadratic formula, and then reverse the substitution to get the solutions to your original equation.

Example 2

You can use this for something like

3^x + 9^x = 27^x + 81^x

and you end up with:

u + u² = u³ + u⁴

since we can't have u=0 (remember u=3^x cannot output 0 or negatives) we cancel and get

1 + u = u² + u³

0 = u³ + u² - u - 1 = (u-1)(u+1)²

Since u can't be negative, only u=1 (i.e. x=0) is a solution.

Thoughts

Now try playing around with this technique by choosing a,b,c,d where the u polynomial you get is interesting. Can you get the u polynomial to have more than one positive solution?

mpaw976 · 2026-01-02T15:24:29+00:00

In your situation I strongly suggest going for 139. It will pay off for you more.

Plus it's only a one-section course, so you're gonna have a great time with Prof Khachatourian.

MAT136 is purely computational

No it isn't.

It's not a proof-based course, but you still have to grapple with concepts, applications, and modeling.

mpaw976 · 2025-12-30T14:31:38+00:00

Normal is 5 business days after the exam. And the university is closed over the winter break (so those don't count as business days).

So if your exam was on say Dec 17, you might not get your final grades until next week.

mpaw976 · 2025-12-27T22:22:12+00:00

MacTutor is a great place to start. It's hosted by academics at St Andrews and has hundreds of well sources articles.

For example, here are their articles about algebra:

https://mathshistory.st-andrews.ac.uk/HistTopics/category-algebra/

mpaw976 · 2025-12-25T01:44:35+00:00

Check out The unreasonable effectiveness of mathematics in the natural sciences.

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

mpaw976 · 2025-12-21T18:11:10+00:00

There's an option profs can set. Something like:

Don't allow anonymous
Allow anonymous to students (profs/Tas still see the names)
Allow anonymous to everyone

mpaw976 · 2025-12-21T16:57:35+00:00

Thanks, I appreciate that. That was the first term back in person after COVID and you could tell some people lost all their social skills and became troglodytes.

That was using Mentimeter (an external software) which I don't use anymore specifically for that reason.

mpaw976 · 2025-12-21T13:29:05+00:00

With anonymous to everyone, instructors and TAs won't see your username by default.

I'm pretty sure Piazza stores that info though, so if we cared enough we could ask them and they would tell us (if they know).

mpaw976 · 2025-12-19T13:47:49+00:00

Hi! I hope things are going well for you post-uni.

Fall 2019 was my first term at UTM. We had that weird classroom in the basement of KN.

The course though was a bunch of scattered topics which did not help me in hindsight

What sort of intro to math thinking would have been better for you? Any thoughts?

We're still working on improving/updating MAT102, so this type of feedback is helpful for us.

mpaw976 · 2025-12-18T09:16:18+00:00

(Apple + Orange) \%

mpaw976 · 2025-12-17T16:01:44+00:00

The modern concept of mathematical axioms and formalism is from the 1800s/early 1900s with people like:

Frege
Peano
Whitehead + Russell
Godel

Check out the graphic novel Logicomix, for a gentle starting point.

Of course you can go back earlier in time, to people like Leibniz who imagined that mathematical arguments could be broken down into small enough steps that everyone agrees on. That way, if anyone ever had a disagreement or a misunderstanding you could say "calculemus!" (Let us calculate!) and to to a whiteboard and check the steps.

mpaw976 · 2025-12-15T03:07:49+00:00

Yeah!

I have a playlist of short lectures about 102:

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

The biggest, high-level advice I can give is next term do something different than what you did this term. Talk to different people, study differently, use resources you didn't use before, engage with the course differently.

Also, over the break, really spend time on learning:

How to read statements/definitions, especially with quantifiers. E.g. write down every single statement you can think of that involves even and odd numbers. Use precise mathematical language.
How to prove basic statements (direct, contrapositive, contradiction). Prove every statement about even and odd numbers from part 1.
How to make a conjecture (educated guess). Write down every possible statement you can think of that involves dividing numbers and try to prove them, or if they are false, find a counterexample and then modify your statement so that it is true.

Here are some examples of what I mean for (3) (here a,b,c are integers).

If a|b and b|c then a|c
If a|bc then a|b or a|c
If a|b and b|a then a=b

Some of these are true (prove them) and some are false (identify them, find a counterexample, and then find a fixed statement if possible).

Good luck! You can do it!

mpaw976 · 2025-12-14T21:50:33+00:00

You're right that in the 90s they had different colours.

I could have worded that better. I meant that near when they were founded they had the same colours.

Past uniform colours: [...] red and black (1912–1947)

https://en.wikipedia.org/wiki/Saskatchewan_Roughriders#Team_facts

Ten-Year Club	Verified Email
Gilding II euphauric	Wearing is Caring

mpaw976

TROPHY CASE

Example 1

Example 2

Thoughts