#math-pedagogy

1 messages ¡ Page 58 of 1

quasi musk
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My HS english teacher had that up on her board. I had her for like 3 years

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I thought about that phrase so much because I just didn't understand it

pastel sundial
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I mean the notion of function generally taught in highschool algebra is pretty far from the proper mathematical definition either

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Only looking at elementary functions from R to R, and conflating a function and its graph

tidal whale
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then blame poor teaching, not poor concept

pastel sundial
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Would functional programming not be a good way to teach functions?

tidal whale
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no

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functional programming might be interesting from a logical sense

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but then you're restricting your idea of function to that

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then there's matters such as currying

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and again

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functional programming and the history of computation is all built on "how do we formalize computation"

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which was entirely mathematical up to that point

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you're replacing the source with its own product

pastel sundial
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Idk maybe I'm just all wrong here

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My idea though was that the concept of a function can be tricky to learn because of how abstract and general it is, and programming could make it hands on to an extent

tidal whale
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but then in an attempt to teach functions, you'd be stuck explaining monads sooner or later

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I think by that point you're hitting the highest form of abstraction

pastel sundial
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I wasn't thinking "replace math with Python 101 and expect the same result" but rather build programming into math courses

tidal whale
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programming might be good as a way to apply math to scenarios (e.g. backprop stuff, regression, etc.) but it's not a substitute for core concepts

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application oriented teaching is one thing, substitution is another

quasi musk
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If you're not planning on being a mathematician/math educator

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Then you shouldn't care that much

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Just like I don't care too much to read a thick literary analysis of shakespeare, although I can appreciate shakespeare

tidal whale
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having applications more apparent in a math class is one thing

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but substituting them with something that hinges on the abstraction is another

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programming courses use math, not the other way around

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also, if said book were actually university level or something, then at that point you're talking about a book for a subset of people

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I feel like your actual problem is something else, being directed at something else

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a simple analogy - automata teaches state machines

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people wonder why study state machines

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because you can effectively model a ton of software as one!

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game development? state machines
software modelling and design? state machines
UI control? state machines

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is the neglect of such applications bad? yes

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but does that equate to "We're now going to teach state machines using the UML convention in Larman"? nope

pastel horizon
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Well, mathematical programming is a thing and there is a genuine use for things like binary variables in engineering

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Here's an example.

Let's say you want to minimise the cost of a process. There are three unit operations to choose from but you can only pick one.

The cost function in index form would look like:
COST = y_i c_i

First constraint would be:
y1 + y2 + y3 <= 1

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Of course normally there'd be further constraints too so it wouldn't be as trivial as just picking the cheapest option

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Should clarify as well in this context y is a binary variable. A value of 1 means true and 0 means false

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So, time for my rant 😅

I wonder if the reason why kids struggle so much with negative numbers is choosing to use phrases like "minus 5"? Imagine if it was consistently taught as either adding "negative 5" or subtracting five.

How commonly do you hear yourself saying "minus minus X"?

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That leads into the next question, how do you teach a rule like

5x - 3(x+2)

Would that be via distributing a negative 3, or subtracting positive 3 times the quantity (x+2)

Think there's a subtle difference between the two pedagogy wise

long pelican
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I'm gonna be honest and say that when I think of programming as it applies to high school or earlier, I don't think of monads, formalizing computation, backpropagation, regression, automata, state machines, or UML. Didn't even know these terms in 8th grade. Yet I think programming was pretty essential to avoid being confused about high school math concepts (for me)

pastel horizon
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Yeah, I think programming at high school is simple things like what is a function, what is a variable, what is a bool, etc. Stuff we take for granted

long pelican
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As for 5x - 3(x+2), the first thing they should be able to do is to read it as written and understand exactly what computation is being shown here

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Pulling a random number out of my left wrist, maybe 75% of people who struggle with things like this in fact aren't aware that this combination of symbols can actually be read

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analogy: teaching someone how to tell the difference between similes and metaphors, having a hard time explaining the difference, and never even finding out that it's because they're illiterate

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It's so easy to take literacy of math expressions for granted

pastel horizon
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I think that would end up being in subtracting the quantity 3(X+2) camp

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But what I think is damaging is when it kind of just gets taught procedurally that you multiply everything by -3 and collect like terms

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If you could read it correctly, of course it doesn't matter since both methods are equivalent anyway

long pelican
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How would you test for math literacy?

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Say you want to know whether they can read 5x - 3(x+2) and you want to avoid false positives as much as possible

pastel horizon
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You can certainly test for illiteracy

long pelican
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e.g. getting a question right because you know the procedure, but don't actually understand, would be a false positive

pastel horizon
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If they get it right they may not have correct understanding but if they get it wrong they probably don't have the right understanding

long pelican
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But I want to test for literacy !

pastel horizon
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I guess it's about being pushy in the classroom and demanding explicit explanations

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Keep being as pedantic as possible until they have no choice but to give exact reasoning

long pelican
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How would an English class test literacy of a passage?

pastel horizon
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I don't necessarily think something like that is much easier as well

long pelican
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An English literacy test shows a sentence or a paragraph then asks questions requiring inference from understanding the sentence or paragraph

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So hmm... you could show a piece of math (a proof, for example, or just a theorem statement) and ask questions about it

pastel horizon
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But in that example, how do you know they're making reasonable inferences, not just pulling something out of nowhere because it's what they were taught

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I think English teachers are catching on to that now

long pelican
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I think with the variety of English passages available it's a very high chance that they aren't going to get something they know how to do completely procedurally

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Also up to the test designers to make good questions

pastel horizon
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I've seen a very good one kind of counter argue with his student that what they thought of the passage wasn't necessarily correct

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That they read too much into a sentence

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That's a good analogy to compare with maths. A student applied a method they were taught procedurally without thinking about whether it's applicable or not

long pelican
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What do you think of using literacy tests in math class?

pastel horizon
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And yeah every UK exam will have a passage of a book students won't have seen before

long pelican
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As opposed to completely procedural

pastel horizon
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Because they know most students just revise essays of the classic books like Of Mice and Men

long pelican
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Mm-hmm

pastel horizon
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I mean arguably literacy is already part of UK exams

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Unless you mean including formal notation

long pelican
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In what way is literacy already part of UK exams?

pastel horizon
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They rarely give a question explicitly like "simplify the ratio 5:10"

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Maybe they'd have to read a recipe for 4 cakes and work out how many eggs are needed for 16 as an example

long pelican
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So that's just word problems

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I'm talking about mathematical expression literacy

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Questions where if they get it wrong, it's because they don't know how to read a mathematical expression (which they should have known)

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Like, questions that uncover a misunderstanding of the entire nature of mathematical expressions

pastel horizon
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So how could you implement something like that with ratio?

long pelican
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I would definitely like to come up with examples of this type of literacy test questions

pastel horizon
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Reason why I say that btw is a whole section of the curriculum is dedicated to ratio

long pelican
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(-1)^n + (-1)^(n+1) always equals 0 because (-1)^n + (-1)^(n+1) = (-1)^n * (1 + (-1)) = (-1)^n * 0. Does the same argument show that (-2)^n + (-2)^(n+1) = 0?

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My first attempt

pastel horizon
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That is a good question and it would definitely throw a lot of students off simply because it's negative numbers

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Plus the wording too. I guarantee some would try to think about factorising (-2)^n + (-2)^n+1 and give the result without realising it's asking if the same argument applies

long pelican
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Ya this is a pretty complex literacy question testing literacy on many levels

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Since you mentioned ratios is a big topic

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Here's a possible test question for literacy

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Does a/b = c/d imply that a/c = b/d? does it imply that a/d = b/c?

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(assuming none of these numbers are 0)

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This question would be very difficult for those who don't read "a/b = c/d" as an equality of two numbers

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I'm noticing being drawn to a type of question format for literacy: the question gives some hypotheses and maybe an argument, and the task is to answer if these hypotheses imply a conclusion and/or whether the argument extends to another setting

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I'm liking this format for math exam questions for literacy

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It's very much in line with being able to infer something from a passage

cosmic ibex
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That's very well for your own teaching and your own exams.
But suppose you're the Board of Education and you want to come up with a way of monitoring that the school system you fund manages to teach mathematical literacy to students. We'll assume that for the sake of the discussion that you, the Board of Education, actually possess mathematical literacy yourself, and are able to come up with test questions. However, now you have to be extremely careful to keep coming up with completely novel questions to test with -- because if you fall into a rut and start reusing question types, teachers will eventually catch on, figure out a mindless procedure for deciding the kind of questions you usually ask, and start teaching that so they'll have good numbers on their evaluation.
The English subcommittee of the Board of Education has it easier, because there seems to be a lot more narrative literature to pick passages from, than there are really interesting "understanding" questions to ask about the subset of mathematical formalism that K12 students are supposed to learn.

long pelican
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That's the eternal problem of mathematical testing I suppose, not just for literacy but for everything in math

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But I don't think it's doomed. Math contests seem to have just enough variety that people can't beat them by just studying question types

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e.g. the AIME (15-question 3 hour test run by MAA) has been going on for many years and every year the average score remains 2 to 5 out of 15

cosmic ibex
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Yes, but there's more leeway for being creative when you're explicitly trying to filter for the top few percent and it's your goal that the average "not stupid" kid should fail.

long pelican
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Most contest math questions can have hints added or be broken down into parts to eliminate the factor of being creative/experienced enough to see the solution

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then what's left is the ability to understand what you're doing

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But anyway, I have no reason to believe that you can't make good standardized tests year by year

wispy slate
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This will probably come across as an incredibly immature and idiotic sexist question but that's fine I'd rather learn to do the right thing and say something idiotic than keep on doing the bad thing.

How do I, as someone who grew up socialized as a man, not alienate women in math? I've been told both to "not assume people know less than you" which is absolutely reasonable, but I also often feel the pressure to correct people, regardless of their gender, to change from an "I hate math" mindset and often that comes with explaining things people might know in a well-digested manner (or go a level of complexity down with the explanation), even if they already know them.
So at the core of what I'm saying is, I think I express myself poorly sometimes and feel different pressures to say different things and sometimes it comes out insensitively independent of intent

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If anyone has any thing they think I could improve the way I communicate and express myself please do let me know

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I'm not a tutor or anything I just thought none of the other channels would be a better place for this kind of question than this one with the crowd it attracts, people who genuinely care about how they affect others in speech and practice

tepid smelt
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The way I communicate with every student is slightly different. I generally find what works by making lots of mistakes and slowly building relationships with them. It takes time to learn how to effectively commuicate. I think its important to not be hard on yourself and realize you will not do the right thing often early on as you develop a relationship with your students. I find being open to your faults helps and be patient as students begin to get comfortable with you.

How I talk to freshman is way different also than seniors or college students also.

I don't have a good answer for this but I do know some teachers have a gift of charisma where students just respond better to them. In the end your goal should be to present the material clearly and try to help get as many students as possible to understand it using lots of different techniques.

tepid smelt
# long pelican That's the eternal problem of mathematical testing I suppose, not just for liter...

Testing emphasis in math is so damaging. I have found well thought out projects have been a better way to demonstrate student understanding and allows higher students to really push themselves.

For example I gave a simple project this week for my math 1 students after going through graphing inequalities. They had to create some type of picture where they define at least 6 of the inequalities correctly and have at least one correct shaded region. Lower students did very simple designs. Where my higher students had very complicated figures with correct restricted domains for segments multiple shaded regions along with other functions like circles/seni circles etc.

I wish there was a way to develop a culminating project that we could use instead of traditional tests. Though grading would be a nightmare. I think the integration of math and computer science could lead to interesting coding projects that could be checked easily.

pastel horizon
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Testing is damaging you're right. We all agree making mistakes is important for learning and yet tests focus only on what you got correct

earnest trail
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yeah teachers need to learn how to use technology lmao

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then we can do cool shit w math

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anyway what do u guys think about the SAT starting to be administered online in 2023 or 2024 or whatever

real mauve
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supposedly it will allow more flexible schedules instead of the traditional small number of sessions a year

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that should be positive

quasi musk
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I'm a huge fan of it for rural students

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As it increases accessibility

earnest trail
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wdym

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what if the rural areas have poor connection

quasi musk
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Yeah, then they can go into the suburban/urban areas

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Nothing has changed

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But if they do have internet connection

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Then it's a good thing

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So it just increases accessibility

flat cargo
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I think cheating can be sidestepped by recording the students then having someone skim the tape, this is how https://schoolhouse.world does it that is

earnest trail
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it's not feasible for college board to skim through every tape

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and im sure there will be a lot of patents who don't want their children being recorded

quasi musk
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Yeah, in every case that testing went online

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It happened to coincide with a massive spike in cheating

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This is true both in the k-12 system and the higher ed system

earnest trail
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yeah

quasi musk
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So we would be naive to think that skimming videos would do anything really

cosmic ibex
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Unless the consequences of being caught cheating are significantly worse than scoring too low in the first place.

earnest trail
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reported to colleges

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not allowed to retake the sat

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those are really detrimental

cosmic ibex
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If you believe you can't pass anyway, that's not much worse than failing honestly.

analog token
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There are companies which specializes in online exams
They use AI for analysing the videos

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If the algorithm detects anything suspicious it's notified to someone

wise onyx
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Do you think lecturing is easy?

quasi musk
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In what sense ultra?

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Not the real ultra but I'll play along

wise onyx
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Is it easier than actually teaching

long pelican
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I mean, if the goal of teaching is to induce as much understanding as possible, what's the goal of lecturing?

pastel horizon
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I'd say the difference is with lecturing the student is expected to take ownership of their own understanding. The lecturer just provides them the knowledge.

Is that easier? Not necessarily, I think it comes with its own challenges. The content you lecture is likely to overlap with your own research. Technical terms that are obvious to people working within your field won't be obvious to students

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Plus you also ideally want to convince students that your research is interesting and they should hopefully study it further after passing your course.

wispy jolt
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lecturing is okay when the students have the resources to make use of it

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i think sometimes people lean too close to the "teaching is when i shout the words from the book and they memorize it" side of things

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it's not so simple in america anyways, unless you are in a certain position or fine with holding strong on standards

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i had a lecturer who seemed real "shouting the words" teaching style but it turned out he wasnt, and actually supplied a lot of resources to succeed

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like an online platform that would get people ready for his weekly quizzes

pastel horizon
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Yup. The difference is lecturers aren't just teaching from the book they probably wrote the book themselves

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I had one lecturer who used his own book as a resource

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Now that I think of it another main difference is assessment. In teaching you're expected to use assessment to work out what they're struggling with and adapt accordingly. In lecturing that's more on the students to use their surgery hours if they feel like they need it. Of course it doesn't mean they shouldn't address common misunderstandings

wispy jolt
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a lot of the people with title "lecturer" at my uni have been very good at structuring stuff

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of course they are generally teaching linear algebra tops at my school

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i have been shocked by a professor before, how little effort they put into it

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i actually think lower of it the further i get out, he was also a super harsh grader

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i just didnt get why he was teaching intro analysis

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well, it was a second section opened so he wasnt first pick i guess

tepid smelt
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Has anyone here tried writing an exam for a particular topic your teaching before actually teaching it? I wonder If i just write my exam first and focus my lectures on helping them to get to the level of my test. I worry I might be teaching to the test though.

Yet I find at the other end I write a test at the end based on what I think they can do. I am wondering when you create your exams if its at the start or end of a topic?

long pelican
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I do it informally, not actually writing the exam but I sort of think about what it means to "know" the topic. Rather than skills, I kind of aim to get students from "Level n" to "Level n+1" for that topic, where "level" is in the same sense as the Wired 5 Levels video series (if anyone 's seen that)

earnest trail
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is it fun to write an exam?

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bc to me it sounds fun

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but idk it could be like a "ugh I guess I have to do this" to a prof for all I know

tepid smelt
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I like writing exams and coming up with questions. It is time consuming though. I kind of like my tests to build to a interesting problem that uses several techniques that were covered in previous problems. I try to have the first page be a little easier with straight forward conceptual/skill problems I would hope for all students to reach(C level) and second page trickier conceptual problems or problems that involve using techniques learned applied to problem solving(kind of like contest math flavor). These are problems needed to reach B or A. I only get about 45 minutes for an exam though and teach high school basic math so I keep them fairly short.

Writing good tests is a art though and I be no means have it down. I have noticed most tests I see at the secondary level are garbage. I really like the questions Icy has shared here though and am trying to get to that level. The best tests I took in undergrad were from a professor who was an Olympiad competitor and he came up with interesting problems. The worst tests were just too easy or similar to HW

tawdry venture
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anyone else massively peeved by this type of problem ive seen a few times that goes like
"the volume of a box is [cubic in x]. find the dimensions of the box" and nothing else

real mauve
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is that really all there is? thonk

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no mention of at least regular box or somesuch

tawdry venture
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yes

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ive seen it many times

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i know the intended answer is to factor the cubic but just

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ive seen probelms where yes, that was it, there was literally nothing else

molten urchin
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As a part of my work, I've had to look into a lot of American middle school math curricula. In contrast to the education here (India), I was impressed by some of the problems that gradually "build up" in steps, but at the same time, I felt that a ton of problems (almost all the ones I had to deal with, but this may be a selection bias) are phrased in an applied, "real world" context. We do have word problems here but still a proportional number of "symbolic" or abstract problems. I'd like to know if phrasing problems in the former manner actually reinforces the mathematical idea, or just makes the students miss the forest for the trees.

long pelican
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The real world problems that average teachers often pick (i.e. the ones they can understand themselves) are too homogeneous, forced, and test-like. The nice real world problems are often the more difficult ones to grasp, because many mathematical ideas are used in a deep way for those problems. Also, average teachers may be scared off of using them in class or exams because after all, in their thoughts, if they struggle to understand it, there is no way their students can succeed in understanding it. Textbooks also tend push the nice real world problems to the very end of the exercises section for that reason.

molten urchin
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Ah, I see. How do students usually respond to these elaborate word problems? One point of concern for me is that the amount of words/information can itself get overwhelming and become a barrier in understanding the essence of the problem.

real mauve
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my usual experience with engineering students is exactly as you described, manan

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through no fault of their own of course, but rather how engineering math courses tend to be structured

molten urchin
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I agree. But is it possible to not allude to real-world phrasing of the problems to such an extent? Or at least give some feel for the abstract setting, and then apply it in context?

real mauve
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certainly, and that would be for the best

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but it has to be done early on

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i unfortunately mostly deal with masters students

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and at that point it is rather deeply ingrained, so increasing the level of abstraction to the required level is met with a response like "everything is super difficult all of a sudden"

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and it's very difficult to work around that when students are multicultural

molten urchin
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Oooof, that makes sense. Do they eventually get a hang of things, or do most of them just get-by with things somehow?

real mauve
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most just get by and decide to end their studies after the masters program

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of the ones that adapt well, about half and half go into academia (phd+) and research positions in companies

molten urchin
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Hmm, I see.

real mauve
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part of how these programs try to deal with the issue is to speedrun undergrad in 1 semester

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the core topics are covered again very quickly

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this is good for people that are doing ok. those with a poor background are immediately turned away, so the first semester has a very high dropout rate

molten urchin
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This must be very uneasy for international students I guess

real mauve
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it's one of the main concerns and complaints, yes. universities in germany have state-regulated indicators that include things like number of new students per semester and number of students that graduate on time

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so to some extent these gung-ho first semester approaches serve the double role of helping students make up for deficiencies and also weed out many of them

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because drop out rate is usually not as heavily weighted as an indicator

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not if it happens early in the program

cosmic ibex
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The major downside of having too inane (and yet too predictable) word problems is that students will internalize an approach to them that is not "understand what's happening and then make an informed decision about how to model it mathematically", but rather "consider the symbolic procedures we've learned recently. Pick the procedure whose motivating example in the book sounds most vaguely similar to this problem, and crank the handle on some numbers from the problem". That's how we end up with kids who will cheerfully calculate the age of the shepherd.

earnest trail
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I think the word problems serve as a way for teachers to combat the infamous "when will I ever use this" question

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but I think (middle and high school) teachers should just emphasize that if they go into stem, they will definitely use the concepts discussed in class, but for everyone else, it is wonderful for problem-solving practice

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especially as topics become abstract

cosmic ibex
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Well, word problems are also an attempt to respond to the criticism that "students just learn to execute symbolic procedures without ever understanding what those procedures achieve".

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It makes sense, in theory, that if the students are able to solve word problems using a certain procedure, that certifies that the students have understood what the procedure is for.

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Unfortunately this reasoning underestimates the ability (and motivation) of students to meta-game homework and tests. "The thing we have just learned is vertex form of quadratics, so I can ignore all the flavor text and just plug the numbers I can see into my formula for vertex form".

earnest trail
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yeah I agree

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in order for students to have more motivation, we need to make sure that:

  • teachers should be knowledgeable of the content at a deep lever so they can teach more precisely why things work
  • teachers should be motivated to work, which means that school districts should treat their staff better
  • the curricula of the courses should be stripped of the less necessary ideas so that teachers can spend more time on more important and applicable ones
tepid smelt
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Yeah whats actually happening is the opposite of everything you said

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Math teachers are becoming less knowledgeable as districts scramble to lower the bar to fill those positions. Teachers are treated worse every year by admin/parents/students and the curriculums are getting worse each year while we continue to get more work and overcrowded classes. The school system in America is barley holding on and it's only going to get worse based on my few years in the system. It is quite depressing that politicians and the general public could care less also.

pastel horizon
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Less knowledgeable usually implies less passionate/enthusiastic as well. Not always true in general though

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I get the impression a few of the teachers in the US don't have enthusiasm and the kids end up not caring resulting in a vicious cycle

earnest trail
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exactly, it's really not feasible

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and yeah encouragement of learning would be another outcome of the things I mentioned

earnest trail
turbid zenith
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Working on my first day activity in Calculus I, and wanted to share it here

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And skipping to the end...

real mauve
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pretty nice

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also a little sus around round 2

turbid zenith
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Anyway the idea is to start getting them to review vocabulary that they should remember from precalc

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And preview some of the things we'll be able to describe more precisely with limits

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What do y'all recommend for polling students using cell phones or laptops?

real mauve
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i usually use doodle

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it's kind of limited though, i think it's only for plain text and dates (or at least that is what i've used it for)

wispy slate
tawdry venture
pastel sundial
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Sus

next relic
cosmic ibex
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Perhaps instead of a single student in each of the "viewer" and "sketcher" roles there should be two or three on each side, with the sketchers allowed to ask clarifying questions (that is, the viewer doesn't just speak into a void and hope for the best). Then if just one student in each group remembers the math function of the symbolism they can bring it up in the conversation.

turbid zenith
wispy slate
pastel horizon
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Ironically I genuinely think algebra is what kids generally find to be the easiest

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Where they struggle fundamentally is when negative numbers, fractions and decimals are involved.

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Put it this way, how often have you seen introducing negatives as a way to make a question more challenging?

ancient vapor
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English isn't my first language and I went searching what the channel name means.....things went very wrong

IMG_6845.png
real mauve
ancient vapor
tepid smelt
# pastel horizon Put it this way, how often have you seen introducing negatives as a way to make ...

Not really they generally do the procedure the same but negatives do cause lots of mistakes with arithmetic. Often they don't distribute negatives properly or they make mistakes adding/multiplying with negative numbers. I would say making a problem more challenging is presenting a question in a different way trying to stress there conceptual understanding or multi step problems using several techniques together.

pastel horizon
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Yup. It's just a general trend I noticed.

Algebraic fractions are a whole other beast to be fair

long pelican
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The best tool a student can have to fix their algebra mistakes is the habit of checking their work by another method

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Self-feedback instead of relying on the teacher or answer key or pure memory

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it’s much faster to learn a song on the piano and fix mistakes if you can hear what you are playing on the piano than if you can’t

pastel horizon
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This is actually the most practical thing you can ever teach someone. How to check their own work

earnest trail
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the only issue with this is if the students are not motivated to check their work

long pelican
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That’s like saying some programmers are not motivated to debug their programs before they deploy them

shadow basalt
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some programmers are not motivated to debug their programs before they deploy them is true

long pelican
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Are there any programmers who can write 10k lines of code without running any part of it to test it once, and have zero bugs (or even compile errors)?

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Somehow many people think algebra is expecting that of algebra students when it comes to math

shadow basalt
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I think the point is that these programmers work does have bugs, as do these students' work have errors

long pelican
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Ok yeah

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The saying “it’s okay to make mistakes” should be fine-tuned to: “it’s okay to make mistakes, as long as you catch most of them pre-production”

real mauve
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the analogy breaks down because "early access" is now largely "open beta"

shadow basalt
#

Yeah, my interpretation of gmod's statement was that many students for some reason (myself included when I was younger) do not seem sufficiently motivated to check work, even when it is a simple process

#

So one has to help them find that motivation as well

long pelican
#

I wonder why simply “I want to be right” isn’t sufficient motivation

shadow basalt
#

Well, for me it was neurodivergence, ADHD can have a big impact on these things

#

Others suffer from time pressure or other forms of paralyzing anxiety

real mauve
#

that component can be there, but there are many that will simply immediately think "oh i saw this in class, it's easy" while being completely wrong in their procedure/answers

#

not even being able to detect that they need to double check

shadow basalt
#

A lot of people also like, don't view math in the same way mathematicians do, and they see the idea of being wrong vs right as like, incomprehensibly mysterious

long pelican
#

Dam but they don’t then say “it’s easy so I better make sure my answer is correct so I don’t look like a fool in case it’s completely wrong”

shadow basalt
#

Not always

#

also, many times the consequences of being wrong are minimal

real mauve
#

this is similar to asking how students did after an exam

shadow basalt
#

Like, a lot of calc students know they can always just differentiate their antiderivative

earnest trail
real mauve
#

you will find many answer with something like "i don't know" or "i answered everything"

shadow basalt
#

yet most don;t

#

don't*

real mauve
#

not "i know roughly what i have right and wrong"

#

so you get a combo of lack of interest but also not even knowing what needs to be checked in the first place

long pelican
#

Self debugging is a pretty universal life skill tbh

#

Make it an explicit part of the curriculum!

real mauve
#

i do agree that it should be so

#

but it simply isn't at the moment

cosmic ibex
#

OTOH plenty of students seem to be motivated enough not to be wrong that they ask in places like here to have their work checked, and still don't seem to know how to sanity check the result themself.

long pelican
#

Yeah I wanted to mention that aspect too

real mauve
#

that's certainly true, yeah

#

in that scenario, i do wonder whether the motivation itself makes a difference, then

#

e.g. wanting to pass/get a good grade/understand/etc.

earnest trail
#

100% extrinsic

pastel horizon
#

Reason why engineering teams always scrutinise each others calculations

pastel horizon
#

In fact that refinement of the statement is destroying them further. You're kind of just saying make a mistake as long as you fix it in time. The whole point is to make mistakes so you don't make them again or at least less often

long pelican
#

Making a mistake is okay, like I said

#

But on problems of a type where you can see if the answer satisfies the constraints

#

Eventually there should be no excuse for the teacher finding the answer doesn't satisfy the constraints which the student could have found himself as well

pastel horizon
#

Yes. In fact maybe problems where the constraint isn't explicitly said but is common sense.

For example, using trigonometry to find missing side lengths. Some students just lack the common sense to realize it's impossible for the opposite to be larger than the hypotenuse

#

By definition the hypotenuse is the longest side

left vault
#

When I was in high-school, the main reason I wouldn't double check all my answers is because I wouldn't have enough time in an exam

earnest trail
#

true

left vault
#

I would rather lose 2-3 marks from silly mistakes in return for like 5-6 mins of extra working time on a harder problem

#

Plus, when you make a silly mistake, it's not because you don't understand the procedure, it's just because you fumbled in the moment. So I cared much less when I made a silly mistake as opposed to a conceptual one

#

Most of the students I tutor have said similar things

#

Obviously it's different when they're just doing textbook exercises (under no time constraint)

earnest trail
#

when I took algebra, I didn't check answers because it was extra work
now I don't check answers because I trust myself enough with my ability and im paranoid ill change my correct answer to an incorrect one; now, if I was struggling on a question I will check it but for the most part I don't

left vault
#

Yeah, the less confident I am with an answer the more willing I am to check my answer

earnest trail
#

but yeah I tend to lose a few points on exams for a dumb mistake so idk

left vault
#

Do those mistakes really bother you though?

earnest trail
#

not really

left vault
#

I get annoyed, but I realise that I don't care too much since I "know" the answers

earnest trail
#

I just don't pay attention sometimes

earnest trail
left vault
#

I think the problem is more that: Students don't have an accurate gauge of when they should check their answers and when they don't need to

earnest trail
#

if I don't get a 5 on the ap calc exam im blaming my lack of attention

earnest trail
left vault
#

So maybe teaching people to recognise a lack of confidence in an answer

earnest trail
#

how would you do that

left vault
#

Cringe, kinda-impractial, proto-solution:

have students rate each of their answers on a likert scale for their confidence

#

Then they get feedback on that

earnest trail
#

eh maybe

tepid smelt
#

On difficult exams in undergrad I didn't have enough time to check a solution. Professors seemed to design exams to take the full time and most were pressed for time. I have made the mistake of having exams go too long.

In general based on how long you take to finish an exam how much time do you add for your students? Like if you can finish your exam showing full work in 10 minutes how long do you expect your students to take?

tawdry venture
#

is there any pedagogical value in having students explicitly verify that a particular map is linear, over and over again?

#

(talking about linalg specifically)

long pelican
#

This will be relevant to me in a few weeks, but I think the answer to your question is no!

#

Exception is if you were able to come up with a large variety of presentations of linear maps; once per variety is okay

tawdry venture
#

i mean yeah just

#

yknow

#

like it pays to see the various ways linear maps can look like but ive seen people come here with like 10 exercises that are all the same just with different numbers and are like "show that this map is linear" etc

pearl gate
#

does anybody else have the problem where you can prove something, but its hard to zoom out to really get the essence of what it is u are proving?

#

if so, how do u get around it?

pastel horizon
#

Straight line graphs using function machines. An idea I had. Seems kind of obvious but it emphasises the fact that the equation of a line maps an x coordinate to a y coordinate

shadow basalt
wispy slate
#

Also getting used to definitions and negations of definitions, how some things are characterized usually and then also alternative characterizations of those same things

#

Basically the more you look at something from different angles the more familiar you are with it

#

Even the symbol-pushing has a certain beauty to it sometimes

proper trout
#

Sorry if this is the wrong place, but how do you balance TA workload with your own stuff 😓

shadow basalt
#

Could you describe your personal issues with it? Could be a lot of things

#

Ie your prof is overworking you vs you are overworking yourself

pearl gate
pearl gate
#

it can be hard to do that, but sometimes i do a problem and the next day i have a completely different way of looking at it

wispy slate
#

that's good! yeah time off def helps, at least for me

wispy slate
#

A very experienced research mathematician could probably interpret a simple statement in dozens of ways relating deeply to various contexts
Meanwhile for a fresh student in that subject, it might be the first time they see that subject, so their toolbox only has general non-specific problem solving skills

pearl gate
#

i see so its just a matter of exploring and trying things

#

and learning more content of course

wispy slate
#

I would agree with that

#

but I would emphasize again for proofs specifically: alternative characterizations, knowing one's way around definitions, and relating what you see on paper with something more vivid

pearl gate
pearl gate
wispy slate
#

There are a lot of reasons it can be a good thing sure!

pearl gate
#

especially for things like analysis

#

so hopefully i will get better at this over time and i can grow my skills for problem solving and mathematical thinking more in college as well

#

thanks

shadow basalt
wispy slate
pearl gate
#

thanks

tepid smelt
pearl gate
#

lol itd be great if we had problem solving in school

real mauve
#

this is one of the points that many people here argue toward, yeah

#

taking a different approach to math teaching that values problem solving skills and abstract thinking, instead of procedure grinding

#

the matter of restructuring curricula and availability of (human) resources is unfortunately highly political and difficult, as well as ethical when considering new structures have to be somehow "tested" on students

#

this is met more positively in undergrad+, where there is more flexibility in teaching, but very challenging before that

long pelican
#

It’s be great if whatever is being used to “test” new structures on students was a good test of general mathematical ability rather than specific procedures

molten urchin
#

One of the younger students I've been tutoring (they're in sixth grade) seems to be struggling with keeping a track of important definitions. For instance, despite repeating the definition of prime numbers throughout my interaction, elaborating it with examples over and over, they still seem to have difficulty in recalling it correctly, affecting their ability to apply it properly in the context of problems. I'd like to know if there are any measures I can take to make them more comfortable with the definitions/ideas they have been explained already, other than just plain repetition and practice.

#

tl;dr how to make younger students comfortable with precise definitions in math?

real mauve
#

it might be a lack of intrinsic motivation, or maybe they are struggling to really understand the definitions? do they seem responsive if you ask them questions about the definitions?

molten urchin
#

They often sound hesitant and confused

#

I'm not sure if I can pin down the problem precisely

real mauve
#

all right. that is probably it, the struggling to grasp the concept, but that alone shouldn't be a problem. they might have a negative learning environment at school where this difficulty in understanding gets them chastised in some sense (grades, reprimand, etc.). maybe you can start by getting them more comfortable with making mistakes during your study sessions and they'll stop being so hesitant, even when making mistakes

molten urchin
#

That's a good suggestion, I'll keep that in mind. Also, they've been in the online-school environment for 2 years now, and I'm tempted to think this has scarred the learning experience for many.

real mauve
#

aha

molten urchin
#

But yeah, I'll keep in mind to bring them at ease. Thanks Edd! catthumbsup

real mauve
#

you bring up a good point because multiplication and division are practiced extensively in 3~5th grade, i think? so the online experience might have left them lacking in the understanding and overall confidence doing these operations

#

so prime numbers would certainly be challenging

molten urchin
#

They actually seem to have some difficulty with basic arithmetic

real mauve
#

aha

#

yeah, maybe some arithmetic games so that they practice and stop being scared

molten urchin
#

Like, in calculating correctly. They'll often trip and say things like "13 times 5 is 165"

#

I started teaching fractions to them and it was going fine, till we got to the point where we had to compare fractions

#

Now I had to explain the idea of considering a common denominator for fractions, but turns out they were completely oblivious to the idea of factorisation, etc.

#

So I had to move back several units in their curriculum

#

And hit this new roadblock of failing to get definitions across

molten urchin
#

So many students will just finish their mandatory math education and probably not even learn how to add fractions properly, even less why they're added the way they are.

real mauve
#

mhm, i see. also do keep in mind these are things the average adult also struggles with 😛 if you grab a random person on the street and have them compare fractions, oof . the "weird operations" don't seem to ever have a physical meaning. the goal ultimately is to reach some level of abstraction, but since numeracy is involved, maybe considering practical scenarios helps. idk, putting apples into boxes

molten urchin
#

Cakes were my go-to example for fractions KEK

#

I understand though, I also need to develop a more realistic outlook for sixth graders who haven't received the best education over the last couple of years.

real mauve
#

yeah, this is a very challenging situation. depending on how invested you are, you could spend a session talking with the student about how they feel about their math, or even asking the parents about it

#

since the whole deal also has an effect on mental well-being

#

very rough :x

molten urchin
#

I should be able to talk to them, if not their parents. I've been put in touch with them through a non-profit organisation that I'm volunteering for, so I've had to interact with them directly so far.

long pelican
#

I can relate to the original situation about the definition of primes not sticking if I pretend that instead of trying to remember what a prime number is, I'm trying to remember what, say, a Lagrangian subspace is

#

Without some ideas about what's important about the background stuff, i.e. symplectic vector spaces, the idea of Lagrangian subspaces would seem pretty arbitrary and it's harder to remember

molten urchin
#

This is true, but in this case the discussion for primes and composite numbers was preceded by the idea of factors/divisors of a number. The student seemed to be fairly comfortable with that bit, but somehow the leap to classifying as prime/composite didn't go as smoothly.

long pelican
#

What definition(s) of prime did you work with

molten urchin
#

The state textbook goes with "has exactly two factors"

#

I've tried phrasing that in one or the other way

long pelican
#

That's a pretty steep "why do I care" curve

#

The "why do I care" element affects memory too

#

Can they say what the fundamental theorem of arithmetic says?

molten urchin
molten urchin
real mauve
#

i struggle to come up with any "down to earth" applications of primes for this school level of the top of my head, though. especially for a student in this situation

#

i don't think providing abstract motivation here will be very helpful

long pelican
#

Real world applications of primes is not a route you want to go to, yes. Don't really think that will help. Properly done abstract motivation is the way to go here

#

Some ideas: Demonstrate Sieve of Eratosthenes, demonstrate arranging numbers into boxes, demonstrate complete factorization of some numbers "until you can't do any more factoring"

real mauve
#

i just don't know that that can be done at a pace where they will catch up with the curriculum, which sadly must also be considered. since this is a student struggling with many basic arithmetic concepts, like comparing fractions, this will require a lot of backtracking and filling in a lot of gaps. i do agree that ultimately this is ideal, but i don't know if at present it is practical

long pelican
#

The abstract motivation here can be as simple as: "As you can see, numbers like 12 can be written as 3 times 4 -- a product of two smaller numbers. Can every number be written as a product of two smaller numbers?"

#

As an aside, here's my re-imagining of a short and effective comparing fractions lesson
Act I: Which is bigger, 4/7 or 3/5?
Act II: Showing 4/7 and 3/5 precisely on a number line using computer software. Students will be convinced 3/5 is bigger
Act III: Ask students how they could have worked out on paper that 3/5 was bigger. Suggest to them to look at the space between 4/7 and 3/5 on the number line and how small it is. How small is the space? (1/15, which is very small indeed)

Acts I and II set up the abstract motivation which I believe is very important and people don't do it enough justice. Without abstract motivation the thing itself risks being seen as a skill to pass a test. In my experience, this doesn't change even when real life motivation is used to introduce the subject

#

Ok rereading the context, @molten urchin It seems that comparing fractions is the larger goal here. The student seems to be lacking the motivation for the common denominator method, is that right? Did the subject of factorizations come up because the method was to find the least common denominator? The "idea" of common denominators doesn't require finding the lcm, just multiplying the two denominators is enough. Finding lcm is just a way to get a more reduced fraction with less work

molten urchin
molten urchin
#

But I guess I also wanted to use this opportunity to fill gaping holes in their understanding of a rather important topic

#

Another thing I've noticed is a frequent reluctance to "take action": I often prompt them to answer a question, or explain why they think something should be the answer and I'm hit back with a "I don't know" rather quickly.

molten urchin
tawdry venture
#

Act III: Ask students how they could have worked out on paper that 3/5 was bigger. Suggest to them to look at the space between 4/7 and 3/5 on the number line and how small it is. How small is the space? (1/15, which is very small indeed)
not 1/35? :p

long pelican
real mauve
#

icy needs to take manan's tutoring too 😌

#

ah btw icy, i'll DM you about something really quick, if that's ok

long pelican
#

😮 I enabled direct messages then

real mauve
#

that was all, feel free to disable them again 🙂

cosmic ibex
#

"Has exactly two factors" sounds like a pretty horrible way to introduce the concept of primes to children. It may be the most succinct way to avoid 1 being a prime, but that's about the only thing it has going for it. Just saying "a number that's not the product of any two smaller numbers, except that 1 doesn't count as a prime anyway" would feel less like it's being pulled out of a hat for no interesting reason.

wispy slate
#

what's a good way to prove the binomial theorem combinatorically with lemmas

shadow basalt
#

You can phrase it as a counting problem in the expononents, i.e., given the expansion how many ways can I make a term x^ky^n or whatever

inland wren
#

if you're teaching a topic, how much beyond that topic do you need to understand to be able to teach it well?

#

like, would someone whose teaching Calc 1 need to be comfortable with analysis to teach it well?

earnest trail
#

not necessarily

#

for me, teaching what im learning is a great way for me to learn

#

I make youtube videos on things im learning about

#

it helps enforce the ideas for me

long pelican
#

Do you have a youtube channel?

earnest trail
#

yea

long pelican
#

Dam

quasi musk
#

And give an overview of what's going on, but it's not required to teach the calculus sequence

#

I just proved the mean value theorem for a second time for my students

#

I think it clicked more this time

#

that you can take a difference function, and that difference function will have a minima or maxima

long pelican
#

I'd say learn enough (not necessarily analysis) so you can equip students with the purpose and/or history and/or larger abstract context of every day's lesson/lecture and not just "This is what we're learning today, practice it for the test"

quasi musk
#

Which is by no means an easy task for a subject as intricate as Calculus

#

There's a lot of moving parts and pieces that interlock in unexpected ways

#

It's a beautiful subject, literally changed human history

earnest trail
#

exactly, it can help a lot but is not required

molten urchin
junior tapir
#

it's not too difficult to explain

#

if you only say that if 1/x is still a whole number, then x is a unit

pastel horizon
#

Has exactly two unique positive integer factors, wouldn't this definition solve the problem while still being concise?

#

1 fails to meet this definition since while it has two positive integer factors they aren't unique

#

I edited in the positive integer part since while it's implicit when we talk about factors it might nevertheless open up the whole can of worms of "but -2 times -1 = 2"

cosmic ibex
#

The problem isn't to make the definition precise, but to get through why one would even care.

pastel horizon
#

That's usually why you follow up with questions like "is one a prime?" "What about zero?"

#

Tbh asking kids the question of whether it makes sense to have a factor of 0 can give interesting discussions

#

Even negative numbers can be an interesting discussion. When you relax the restriction that positive factors are needed it turns out you won't have 2 unique factors anyway for something like -2

cosmic ibex
#

None of that feels like it will make the definition feel less completely random and arbitrarily pulled out of a hat.

pastel horizon
#

Maybe you can link it to cryptography?

#

Algorithms are based on prime numbers because the fact that there are only two unique factors means that it's harder to brute force

cosmic ibex
#

Which problem are you trying to solve here?

pastel horizon
#

You want prime numbers to have a meaning right?

#

I mean at the end of the day it's just a set of numbers generated from a definition. You could say the same thing about square numbers why do we care about numbers multiplied by themselves

cosmic ibex
#

I want prime numbers to be explained to children so they agree it's a cool and natural concept that it makes sense to care about.

#

To start with the "exactly two factors" definition hides the cool things you can use primes for, and as far as I can see with the sole purpose of avoiding the need to explain why 1 is excluded later on. You won't need to explain it because kids who are told this-is-the-definition-because-I-say-it-is won't care about whether 1 is a prime in the first place.

#

Instead, just start with "a number that you can't get to by multiplying smaller numbers together" already.

#

Don't worry overly about whether 1 is prime or not until you're at a point where you can talk about prime factorizations and the fact that they're unique, which is when we get a motivation for treating 1 as a special case.

long pelican
#

Another motivation that just occurred to me is that multiplication by 1 is the same as doing nothing, and we agree that “nothing” does not count as a building block of a system

cosmic ibex
#

Right.

#

I think the "building block" language goes naturally together with the idea of prime factorization.

#

I suppose my point depends on just how early children are to be taught about to prime numbers. You can speak about primes in and of themselves while only mentioning quite small numbers -- I don't think it should be done that early, though.

#

If it's postponed until the kids are comfortable with three-digit division, one can show a convincing amount of examples of prime factorizations being independent of the order one discovers the prime factors in, and then it makes sense to talk about the subtlety of 1 simultaneously. Then it could even make sense also to discuss the "exactly two factors" concept and make a teachable moment out of "definitions are something we choose for reasons", if the kids are not struggling too much to be ready to appreciate that.

shadow basalt
#

whereas when you just assert this

#

people think there is like, some deeper meaning

#

idk how to explain it but this sort of thing comes up a lot

#

it's similar to those horrible PEMDAS questions on facebooks where people think that mathematics is about enigmatic and poorly written order-of-operations questions

#

maybe its like, "1 is not prime is a Truth of the universe" vs what is actually true, "1 being prime would make some theorems more annoying to state"

shadow basalt
faint yarrow
#

I think if Z^+ was the only thing we thought about factorization in we would call 1 prime but it would be a weird prime. It's certainly not composite, and inventing a whole new bucket to put it in seems a bit silly. But as soon as you think instead about Z[i] you realize the thing 1 is shows up in a lot of places and although you can skirt it in the case of Z^+ you will never avoid it in general. But it is somewhat unsatisfying to tell people that the reason this number that is obviously prime isn't prime is something they'll learn about in grad school is quite unsatisfying. My standby answer that sounds less condescending is that "The fundamental theorem of arithmetic becomes especially tedious to state if 1 is prime."

shadow basalt
#

+1

left gust
#

One way to make it clear that 1 isn't a prime would be to draw the divisibility lattice for the natural numbers

354px-Infinite_lattice_of_divisors.png
#

Then draw a circle around the primes, and define them as the numbers on the second layer

#

I think that makes it clear why 1 doesn't belong with them

shadow basalt
#

I am not sure this makes sense

#

It is clear that 1 is distinct (initial)

#

but it isn't any less arbitrary than the other defns

#

in fact the same argument could be used to say 1 is the most prime number lol

cosmic ibex
#

I don't think one wants to draw divisibility lattices (with infinite fanouts and dotdotdots going off in all directions!) at the point in elementary school where prime numbers are first introduced, anyway.

verbal bloom
#

Excuse me, my sister is struggling with her homework... she is trying to solve one step problems involving fractions and apparently I am bad at explaining it... is anyone here a teacher that can help? (Don't worry I am old enough to be here, I have a young sister)

distant fractal
#

can you provide an example of a problem

verbal bloom
verbal bloom
burnt vesselBOT
long pelican
#

First ask if she has a crystal clear understanding of what the goal of these problems is

verbal bloom
verbal bloom
cosmic ibex
#

Does she get to 9/10 - 2/3?

verbal bloom
cosmic ibex
#

That sounds like she has an opinion about what she does need to do?

verbal bloom
cosmic ibex
#

Well, what is the explanation of yours that doesn't work? We're pretty much guessing in the dark here.

long pelican
verbal bloom
long pelican
#

So basically, the goal she needs to understand is to "find the number which when added to 2/3 yields 9/10"

#

Needs to be those exact words or close

verbal bloom
long pelican
#

Does she know by instinct how to prove a number isn't a solution, by demonstrating that it doesn't give 9/10 when added to 2/3

verbal bloom
long pelican
#

Try getting her to the point she can instinctively do that

verbal bloom
#

Sorry about the ambiguity

verbal bloom
cosmic ibex
#

It seems unclear so far whether she is comfortable with adding fractions in the first place.

verbal bloom
cosmic ibex
#

Yes to it's unclear, or yes she's comfortable?

verbal bloom
#

Wait, she came up, (I came upstairs to talk to you all) and she appears to be done with her homework.

#

I think my mom helped her

#

Thanks for your support and help!

tepid smelt
#

I am doing transformations with my freshman and can come up with interesting problems using reflection. Like typical problems involving shortest path and using Pythagorean theorem or bouncing balls on billard table etc. Anyone have interesting ways to use rotation or translation in ways to solve harder problems easier? I can make interesting problems using dilation and reflection but unsure how to show how the other transformations can help us solve difficult problems.

My students seem to enjoy seeing techniques we learn applied to non typical problems. I think this is helping them to really see the value of the techniques. Pythagorean theorem is a great one because it can show up in so many ways. If anyone has some fun problems on transformations or ideas for this topic I am interested to hear.

#

The curriculum is quite dull just moving shapes a bunch of ways so trying to get my students to see the value of these techniques. We just finished systems which has so many immediate application that it wasn't difficult to get them to want to learn it

earnest trail
#

that's amazing that you're engaging your students with interesting and applicable problems, props to you

#

what kinds of questions for example? im curious

#

also are these college freshman? if so, what course?

tepid smelt
#

"I own a cabin in the woods thats 4 miles directly north of a river that run east-west. I am currently out exploring in the woods. I am 2 miles north of the river and 3 miles west and 2 miles south of my cabin. I want to walk to the river then to my cabin. What is the length of the shortest path I can take? " The idea is shortest path is a straight line and you can use reflection to do that and build a right triangle. Also some with out calculations playing billiards or minatiure golf and where should you hit the ball etc. Simple solving and its for HS freshman.

I have learned students respond better to things that might be harder rather then simple drill problems.

earnest trail
#

oh alright that's awesome

#

that question is super interesting too

#

better than a stupid typical word problem

tepid smelt
#

Yeah I like word problems that require drawing it out and it leads to what to do

#

Rather then just words for the sake of it

earnest trail
#

yeah exactly

cosmic ibex
#

What do you mean you can't make everything relevant by smearing some words out around it?

William wants to sell his bicycle. The price William is willing to accept can be described by a differential equation of the following form: ...

tepid smelt
#

Lol exactly

Somehow physics classes have figured this out but its still a problem with math at least at the secondary level. I was always impressed with my physics class that problems would often become trivial with a good picture. I think this should be encouraged at a younger age.

charred silo
#

When I tutor 1-on-1, I find the most success in just going through their homework for the week, or past papers. And then fill in any gaps they have for questions involved. Is this normal?

#

And if I try to set them questions to do for the next session, they almost never bother.

real mauve
#

it's common because the main motivation in many education systems is simply getting a good/passing grade

charred silo
#

Mm. Makes sense. In fact, I usually have the opposite to this:

I have learned students respond better to things that might be harder rather then simple drill problems.
Like, they just want the methods to past the exams. Preferably drilled into them within the same hour session, so they need no revision.

real mauve
#

it's really a person by person thing, but yeah, this approach is reinforced heavily by the standard teaching and grading approaches. and sadly, as a tutor, you are also expected to help the student survive through this

distant fractal
#

I have an idea for getting my students to engage more with the content of the course. (Essentially it is college algebra and a little bit of precal and linear algebra)

I give weekly quizzes.

I was thinking about giving weekly "challenge questions" which would be entire conceptual. Essentially, I would take the material taught that week, and ask "why" about various things. For example, this week I taught how to test if a given equation is a circle, and the conclusion was that after putting in center-radius form, if r^2 is negative, it isn't a circle. So a challenge problem would be "Why is it not a circle? Could we represent it is a circle in the complex plane? What would be the challenges in doing so? What if r^2=0?"

I would offer them half of the points they missed back on their quiz if they give a cogent response which appears to have been well thought out.

#

My high-school teacher did this sort of thing, but I never experienced this in undergrad. Is there a reason? Has anyone tried this in undergrad before?

Background: most of my students are freshmen

quasi musk
#

It'll most likely work ~ I've had profs give a quiz every class

#

That works like magic to get students engaged

shadow basalt
#

but its a great idea if you can pull it off

#

I think a lot of college stuff is a compromise between peoples various time commitments

verbal dirge
#

tfw you are the grader

distant fractal
tepid smelt
#

I give an assessment every other week just because grading can be time consuming. If you can pull it off and give good feedback you will keep your students on their toes and get more data on how they are doing.

I always put a challenge question on my hws and tests fully expecting only a few to do them. It makes a big difference for your higher students who want to be pushed and are bored with the regular material.

long pelican
#

Do you encourage everyone to attempt the challenge questions? There will be good data from that— not expecting everyone to solve it of course, but everyone should be able to make some progress and write reasonable things on their paper. Most importantly, if anyone writes nonsense/false statements and doesn’t make any progress or know what they’re writing, you can see the red flag

tepid smelt
#

Oh yeah definitely I give credit for any attempt. It's not a way to crush a students grade at all. What is surprising is many times a student I don't expect to make much progress on it will. They might not use the most efficient method but they are developing persistence in solving problems which is actually a state standard.

I find most classes have a small minority who are really low or really high but that middle group can be pushed to the higher end by giving more challenging work. So trying to cater to the lowest students hurts that middle group more. It might not be perfect but those that choose to engage are getting more out of the class when you don't baby them or assume they are not capable of a certain level.

tawdry venture
#

does anyone else hate the term "simultaneous equations"

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and how it is very prone to being put in the singular despite that not making any sense

real mauve
#

what is your beef with the term? the latter point, absolutely, though i don't know if that's a language problem instead

#

reminds me of a calculus book which had a section on "diverse substitutions", which had a handful of different miscellaneous cool substitutions. upon using these, some students would justify their procedure as it being a "diverse substitution"

cosmic ibex
#

Recent posts in #geometry-and-trigonometry sound like some people are using the term "simultaneous equations" to mean that the equations in question are linear.

real mauve
#

hmmm

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i agree it's funny, but rather than make fun of them, the point was to highlight the disconnection between the actual meaning of terms and just seeing them occur together often and assuming that is what they mean

tawdry venture
#

but also just like

real mauve
#

that's certainly not good

tawdry venture
#

"simultaneous equations" feels much less like a thing-in-itself than "a system of equations"

real mauve
#

i do think they refer to the same thing. for whatever reason, "system" appears to be preferred over "set of simult." in grad school

tawdry venture
#

yes they do refer to the same thing

#

but "system" is easier to like

#

conceptualize as one object

#

if that makes sense

real mauve
#

i can see that

#

this is circumvented by adding in "set of" in front of the simultaneous equations, but at that point it's wordy. but yeah, gives more intuition on it being a collective thing

pastel horizon
#

Simultaneous equations usually refers to a pair of equations, while a system of equations is more general

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I wonder why Gauss-Jordan elimination isn't taught earlier on? You could introduce systems of equations this way

tepid smelt
#

I do show the method to my freshman but honestly most don't respond well to it which is surprising. I think because they are taught substitution method so early that they tend to want to go to that even when it can make the problem harder. I find in general breaking bad habits is much harder then teaching something they have no prior knowledge of.

At the freshman level I am just happy if they can conceptually understand what a solution to a system means then ideally to take a real life context and use systems to solve it. I finished my systems unit with linear programming and my students really enjoyed the idea that we could use systems to maximize/minimize an objective function in various contexts.

pastel horizon
#

The reason is them not being strong enough to evaluate methods (reasoning) and decide on the most efficient approach (fluency)

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There are some cases where substitution can be efficient, depending on how it's structured. Generally Gauss-Jordan is better as an algorithm

wanton walrus
#

I'm grading an assignment and notice 2 assignments have extremely similar or identical wording on most of the assignment. It's a HW that students are allowed to work together on, is this a red flag or nah?

#

The 2 assignments actually diverge towards the end, but the 1st 2 out of 4 problems seem about identical

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My main concern is if 1 student straight up copied or if they just did the assignment together

tepid smelt
#

I think there is no harm in mentioning what you noticed to the student to ensure they understand and are not just copying before they get to an exam and bomb it.

If students are encouraged to work together your going to see similar arguments I would imagine.

wanton walrus
#

Sounds good, I'll at least leave a not on the assignments then

pastel horizon
#

For the future, if you want to encourage group work it might be best to tell them that they have to form a group explicitly and write the members of that group on the assignment. If you see two different groups with similar work you know they copied

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If you want to see individual efforts within a group it might be good to explicitly mark some questions as individual tasks that need to be split up. It doesn't rule out them colluding on that one question but it does discourage them if only one person gets credit for it

tidal mica
#

am i allowed to ask for opinions on how i should grade a students hw?

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obv wont disclose anything about student i just

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ok it's borderline illegible, how harsh do yall punish when you have to make like an effort to even see their answer

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prof said to be lenient but i cant even tell if they submitted the right thing so my opinion is to give 0 for the question

pastel horizon
#

Yeah I'd say just grade what you can make out. If that's nothing, it's a 0

tidal mica
#

does sending a picture of their submission violate anything or can i

inland wren
shadow basalt
tidal mica
shadow basalt
#

Ferpa**

tepid smelt
#

If it was my student I would give them a chance to explain it to me in office hours. I have had students scribble nonsense then magically write the correct solution on a test question because they likely got the final answer from a friend in a class before. I don't give credit in this situation.

tawdry venture
tidal mica
tawdry venture
#

go against the professor and zero the entire assignment with the comment "LaTeX this shit next time" opencry

tidal mica
#

it's CS students :/

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well im grading a math class but it's a linear algebra class that CS students have to take so they can say "i did linear algebra"

charred silo
#

Happens quite often at my old university. I was always told to give highest I could.

cosmic ibex
#

CS students ought to know TeX too. (Not that I'm champoining Ann's position, though).

tidal mica
#

these definitely do not know TeX

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i took the class by accident two years ago when i was still a pretty ignorant math major, unless you take math major classes here youre never gonna see TeX

tawdry venture
#

i'm not even presenting a position

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well

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not MY position

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that's not a position i hold unironically, as signified by the opencry at the end

cosmic ibex
#

Hmm, CS undergraduates might be excused for not having seen any TeX...

long pelican
#

I introduced latex to my honors linear algebra students and as far as I know, none of them had seen it before. I even typed 4 proofs in real time on overleaf in class. All but 2 students were interested by it and are writing their math homework in latex

shadow basalt
shadow basalt
#

Like you should make every effort to give students every possible point imo

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That said if you really can’t read it you really can’t read it

grim spindle
pastel horizon
shadow basalt
#

like either its actually impossible or it just takes some small amount of effort in 99% of cases

tidal mica
#

in my case i was asking cuz the person submitted pictures of their work in such a way that i could only see the very top of their work and the very bottom

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i couldn't find any conclusion and the work in the picture was minimal so i gave .5/2 points

#

sometimes i have to force myself to be nice tho, not bc theyre CS students but bc theyre students

shadow basalt
#

well yeah ofc

#

being a student is hard

tidal mica
#

idk why but sometimes i find myself defaulting to harsh grading even though i wouldnt wish it upon myself

shadow basalt
#

that is a good thing to know about yourself and avoid

#

Being a lenient grader is also like

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objectively easier lol

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so everyone wins

tidal mica
#

yeah i literally give myself more work lol

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what i do now is before i finish grading i usually consult with other graders and change my scaling if they think im being too harsh

shadow basalt
#

it might also help to change how you make a rubric

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like

tidal mica
#

or just friends who've taken the class, i have a biased pov from doing well in it

shadow basalt
#

I normally decide on my rubric before i start and try to craft it to be generous

#

in particular I normally try to think about counting up, rather than subtracting down

tidal mica
#

hmm i hadnt thought about that

shadow basalt
#

and I only every take away like 1 point for minor algebra errors or anything irrelevant to the material

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like i grade calc

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if students fuck up their algebra but get the integral right

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as far as i am concerned they learned the relevant material

#

so i will only remove some small amount of credit

tidal mica
#

something i realized too that might be problematic(?) is that if a lot of people make the same mistake i take less points off for it

#

idk do you do that too

shadow basalt
#

I try not to change my grading retroactively

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in part bc its way more work

tidal mica
#

im grading online submissions so it's not as hard

shadow basalt
#

fwiw I grade like

#

300 students

tidal mica
#

254 here KEK

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but im not the only grader most weeks

shadow basalt
#

Oh 300 is the number that I grade lol

#

there are 1000 students

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But I only grade part of the exams

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so normally ~300 problems

tidal mica
#

bro your profs devastation

shadow basalt
#

i mean they dont do that much work really

#

that much more work

#

they give the same number of lectures and write the same number of exams

tidal mica
#

ig that's true

shadow basalt
#

and they use piazza for questions instead of emails, which the TAs do more work for

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but they probably have a lot of admin BS

tidal mica
#

i wish my math classes used piazza

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been tryna advocate for it but no luck

#

at least the big ones like calc

shadow basalt
#

i prefer discord

#

but piazza is good

#

piazza has the benefit of having fewer repeat questions

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discord has the benefit of not being garbage

tidal mica
#

i mean yeah but a surprising amount of people are unwilling to learn discord UI devastation

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i was in a class that used discord and only like 3 people really used it

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that's my experience at least but in general i agree

austere inlet
earnest trail
#

discord isn't too hard to understand

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if it's just for a class

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imo

#

it would be fun to be in a class that uses discord

tepid smelt
# shadow basalt Oh 300 is the number that I grade lol

Wow and I thought us secondary teachers had it bad with grading. I couldn't imagine sitting down to grade 300 tests. A 110 already feels like too much. I would just give multiple choice tests if I had a 1000 kids grading a real test would take all semester.

At my high school the advanced math teachers have had good response with getting students on discord to help each other.

earnest trail
#

At my high school the advanced math teachers have had good response with getting students on discord to help each other.

damn that's awesome, are these younger teachers?

tepid smelt
earnest trail
#

oh yeah that sounds awesome

tawdry venture
#

does anyone else think we should teach kids to write sin(x) always with parentheses, and never sin x

pastel horizon
#

There's people that don't?

earnest trail
#

I don't write the parentheses with trig functions or logarithms unless it's something other than one variable or number

#

but I do understand how it could be confusing

#

personally if I am multiplying by something, I will put it in front of the function, or any function for that matter

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for me it is always bsinx but never sinxb

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or bf(x) but never f(x)b

tawdry venture
#

the only time i don't write parentheses is if the argument of sin or log or whatever is enclosed in absolute value bars

shadow basalt
#

I think students should just be taught good judgement about when their notation could cause confusion

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like teaching notational "rules" is often misleading imo

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it makes students think that PEMDAS is like, the Truth of mathematics rather than a notationally useful tool

quasi musk
#

I agree with Max here, there's no hard rule for how to write something that conveys information

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The only issue is when it becomes ambiguous

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I currently have some 6th graders that are trying to make "their own" notation

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And it's horrendously ambiguous and confusing

tawdry venture
shadow basalt
#

You don’t teach good judgement by prescribing the rules for them

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You can explain why you think sin(x) is better notation

#

But it should be explained from the POV that all that matters is clear communication

earnest trail
#

I agree with this

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sinx is not ambiguous at all but sinπx can be misinterpreted

earnest trail
quasi musk
#

Oh when solving a system of two equations with two unknowns

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like x + y = 100

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5x + 3y = 300

#

they'll solve

#

5(x+y=100) + 3y = 300

#

@earnest trail

#

Instead of writing x = 100 - then plugging that in

earnest trail
#

bruh lmfao

#

that's terrible

quasi musk
#

He says "but it makes sense to me"

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and I say "Yeah but that doesn't make sense to me"

earnest trail
#

exactly

#

seems like a creative little fucker but the creativity is being used in the wrong way

wispy slate
#

he's just 🍛 -ing respect the drip 😩

#

or traversing an operation tree

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that involves substitution

cosmic ibex
#

It makes some sense if you expect that mathematical symbolism works like English.
"Five times x (which by the way makes a hundred when you add y to it) plus three times y is three hundred."

#

Which I suppose is an easy mistake to make if the first symbolic expressions you see are so simple that they do make sense as word-for-word renderings of English, and the teacher never bothers to explain how mathematics actually works on rather different principles.

gentle talon
#

Why is Stewart considered the gold standard for calculus?

#

In the US so much time is spent preparing for calculus and teaching from Stewart but it seems like a waste of time unless you are in a gifted program or AP out.

#

Why not introduce profs early and often and then all undergrads could start with advanced calculus and linear algebra first year.

#

That would require a lot of change tho.

quasi musk
#

Because to introduce proofs early you have to hold students to standards

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and have to teach the teachers the material

#

Both of which will cost a lot of money for no real gains

gentle talon
#

I'm pretty sure Britian did it this way but recently changed to follow the US system.

quasi musk
#

As to why Stewart is the default, it's because it doesn't require you to think that hard

#

It's just a plug & chugg book

gentle talon
#

No real gain?

quasi musk
#

Yes, are you familiar at all with how schools are funded, what "success" metrics are, and how they optimize for them?

#

If you're familiar with that, then you will realize according to these standards, that intro to proofs earlier for everyone will amount to no real gain in education

gentle talon
#

Plug & chug doesn't teach you how to think but I guess education isn't supposed to do that anymore.

quasi musk
#

Anymore?

gentle talon
#

As far as economic output I think we'd be better.

quasi musk
#

How would teaching kids to prove geometry theorems improve economic output

#

Let's say I grant you $1 Trillion to revamp the education system in the US. What is the best way to go ab

gentle talon
#

teaching them how to write, argue, think and do research

quasi musk
#

about that in a cost effective way

gentle talon
#

There's no reason why quantifiers shouldn't be taught right along inequalities.

#

And you build from there.

quasi musk
#

Have you spoken to your average HS math teacher?

#

It's not so easy to run a classroom or a school

#

Let alone change curriculum across the country

#

If students are already disillusioned with math

#

How will introducing quantifiers magically make them like math

#

Because it makes more sense for you?

gentle talon
#

Becuase it's less of push these symbols around and follow these algorithms which you will never be better than a computer at and more of developing ones thinking.

quasi musk
#

That's the issue though: once you make the standard "develop one's thinking"

#

You have to have a rubric that says this is what you mean by "thinking"

gentle talon
#

I never said get rid of all computation either, but sticking with hey class do 1-29 odd and if you need to do more do the evens.

#

Isn't helping your case either.

quasi musk
#

And if you make it difficult, or introduce abstraction then there will be more students failing

#

Which means less funding

gentle talon
#

So your left with a population that's less educated and those who have the capability be held back. Makes sense.

quasi musk
#

In order to fix the system you have to rework the financial rewards that schools get for lousy teaching

#

I never said what we currently have makes sense

#

I'm saying your changes are ultimately futile

#

Because they've been tried & tried again

#

Just look at the history of math curriculum over the past 100 years

#

Especially new math in the 60s-70s

#

The change in curriculum isn't going to do much

#

It never has

#

If you want to truly change how math education is run, you have to change the way that schools are funded

#

And rework the incentives that schools have

#

Then give local control to teachers on curriculum that they teach with some state oversight

gentle talon
#

Or let people specialize early on if they choose.

quasi musk
#

That can easily backfire hard

#

I'd rather late specialization with flexibility, than early specialization

#

You can already specialize early if you choose by picking electives

long pelican
#

What if we deleted the stuff that doesn’t make sense without adding anything, just giving teachers freedom to do what they want to replace the deleted material

#

Couldn’t make math education worse?

quasi musk
#

I think that's probably the best way to move forward at this point Icy

#

Giving more control back to teachers, students, & parents

#

And away from state bureaucrats

#

But the issue that arises is what do your grades mean? What does your education mean? How do colleges evaluate that?

#

etc.

#

But I think that's more of a minor issue than what we currently have

#

Since grades from region to region, school to school, and class to class are pretty non-uniform

gentle talon
#

If you choose to specalize late you will have a harder time catching up.

quasi musk
#

Catching up to whom?

#

The problem with specialization earlier in school is that it "boxes you in"

#

And there's little room for felxibility

gentle talon
#

Your European peers.

quasi musk
#

You should read about the origin of the US college major

#

And why it was designed the way it was

quasi musk
#

The french university system along with the italian system

#

Are not so great

#

The UK and Germany got their shit together

#

(If the UK counts)

#

I'm ignorant of all the other systems within the EU

#

But I can also tell you that East Asian universities aren't much better

#

The US dominates higher ed because of our flexibility and the fact that professors are held accountable by their students & administration

#

But professors still have a lot of freedom in their curriculum, they just can't make arbitrary grading schemes

#

Also, for a long time the US has dominated pre-k thru 5th grade

#

Education in most categories

#

It's starting in middle school thru HS that the US education system floudners hard

long pelican
#

Those grades are also getting the most attention in education research

quasi musk
long pelican
#

Pre-K to 5th grade

quasi musk
#

I think it's a combination of things: too many teachers (each student having 7 teachers is not a great way to educate kids)

#

Puberty, dumbing down of the curriculum

#

State control, etc.

gentle talon
#

I think US graduates are getting outclassed by their European counterparts.

quasi musk
#

From university?

#

From HS, sure I can agree with that ~ but the US is dominating higher ed

#

We have so many international students

gentle talon
#

Yes, I talk to a lot of US math majors doing masters/phd in statistics and they simply haven't taken enough math and the uni they attend won't catch them up.

quasi musk
#

It's like not even close

#

Well that's an issue with the statistics field

gentle talon
#

Now is it like that in every graduate program, I don't know.

#

mismanaged prerequisites?

quasi musk
#

That's department to department

#

But just the fact that we have so many systems

#

Like in California the CC system, CSU, and UC + world class privates

gentle talon
#

So if you could specialize early that wouldn't be an issue.

quasi musk
#

Just the amount of pipelines and the ways in which you can go through the higher ed in CA alone

#

Is astounding

gentle talon
#

So do we keep all these systems or try to unify them?

quasi musk
#

You want to unify the CC, CSU, and UC system along with privates?

gentle talon
#

Abstraction like computation isn't daunting when you build up to it. And I'm arguing if done at an early age it would be better for the development of the pupil.

quasi musk
#

I'm not arguing against that

#

I'm arguing against "let's make that national curriculum mandated"

muted oriole
#

Is there any well-known readings (papers, books, etc) available regarding encoding and pre-study for undergraduate maths?

tepid smelt
# gentle talon Abstraction like computation isn't daunting when you build up to it. And I'm arg...

The only math programs I have been impressed at the elementary level are those that are willing to present difficult problems to students.

My daughter just did the online math contest through RSM and I was really impressed with the old test questions and my daughter enjoyed working on them despite the challenge. She is now prepping for the math kangaroo test and again enjoys working on difficult problems because they are not boring.

I have noticed my own students respond better to interesting problems even if they might be harder. I think we underestimate how curious kids can be and force them to do mind numbing tasks hurting their growth and interest in math early.

Its much harder to get a kid interested in math once they have convinced themselves its not for them. Young kids don't have this prejudice yet so they should be encouraged to tackle weird problems sinilar to math circles or elementary school math contests.

tepid smelt
#

Its easy to teach a kid who is interested. Its easier for a kid to be interested when they work on interesting problems. I don't know why so many curriculums don't offer any problems worth doing. The math circles have shown me you can have interesting problems at any age. Math is full of rich problems its a shame they are not highlighted more

quasi musk
#

RSM is great ~ from an RSM teacher

wispy slate
#

What are some useful/effective things you have told students or peers that empowered them or made them realize they are capable of verifying their solutions on their own?

#

Of course looking for a second eye in case of mistakes can be nice, but feeling paranoid that you are not capable of independently producing coherent results seems to be a common ailment

#

I think it's hard if the person has never tried anything independent or creative wrt math stuff, because then their only image of what math is is "something done to you" rather than something you do

#

So it's only natural that the thought "I'm not capable of checking my own work" would be prominent

shadow basalt
#

You can start by listing things you know the answer has to satisfy

#

An example might be like

#

If I ask what 3626 times 8698 is

#

And you give me a prime number

#

Or if your antiderivative differentiates to the wrong thing

wispy slate
#

Like telling a student "list things you know the answer has to satisfy"?

long pelican
#

I think so. A student saying "Is my answer right?" while it blatantly doesn't satisfy the constraints is pretty frustrating

shadow basalt
#

Students can “Sanity check” their work

#

By checking whether they satisfy things they have to

#

This can be as simple as plugging an answer back in

tepid smelt
#

Check the results in the original problem. Interpret the solution in terms of the original problem. Determine if there is another method of finding the solution. If possible determine other related or more general problems which the techniques will work.

turbid zenith
#

Does this look reasonable difficulty-wise for a first-time undergrad abstract algebra course?

shadow basalt
#

It’s an exam?

#

Or homework

turbid zenith
#

Sorta more like a take-home exam I guess? It's a Problem Set.

shadow basalt
#

Oh problem set lol

#

I’m dumb

#

Yeah this is fine

turbid zenith
#

I give weekly "Checkups" that are more like quizzes, and they're more focused on computations and examples/nonexamples and explanations. For Problem Sets I make them more proof-based.

#

This is my first time actually teaching Abstract Algebra so ... I'm learning to calibrate my difficulty

shadow basalt
#

I would say it’s on the easier side, if you want a more accurate description but also I think early abstract algebra problems just tend to be

#

Like it’s hard to come up with an interesting difficult problem without more theiry

#

Theory

turbid zenith
#

That's fine if they're starting off easier yeah. Topics we've covered so far:

  • Algebraic systems in general
  • Groups
  • Proving group properties
  • Subgroups
  • Cyclic groups
#

We're hitting symmetric, dihedral, and more general transformation groups (orthogonal etc) this week.

shadow basalt
#

Oh fun, the last checkup problem has a formal group law 🙂

turbid zenith
#

Formal group law? 😮

shadow basalt
#

x+y+xy is an example, it is basically a power series in two variables that acts like a product in a Lie group

turbid zenith
#

Whoa. I didn't know that's a Thing-In-General. LOL

#

I just knew that's an example of a funky group operation that still works.

shadow basalt
#

Yeah it’s one of very few that are easy to write down

wispy slate
#

I remember when my algebra prof did this one thing it's hilarious in hindsight, I think he assumed we all had him previously for discrete (many of us had someone different lol) and he said "hey you guys remember Chinese remainder theorem? that's basically the idea" and then started doing a bunch of computations that we couldn't grasp the purpose of

tawdry venture
pearl gate
#

i recently came across the idea of teaching geometric algebra alongside linear algebra. ive heard of geometric algebra only recently and purchased two texts that discuss the subject (so i dont know much about it)

#

what are your thoughts on integrating geometric algebra alongside linear algebra in a standard curriculum?

#

seems that it gives a good interpretation of a lot of things. ive heard that 3b1b's interpretation of the determinant was similar to the interpretation in geometric algebra, and that it also gives tools for exploring manifolds and generalizing ideas in vector calculus

cosmic ibex
#

Is there a even good way to define the exterior product in general without depending either on constructing a wildly large space and then quotienting most of it away again, or on axiomatically asserting a universal property? Either of those would be at a rather more demanding level of abstraction than typical freshman linear algebra (which already seems to present plenty of struggles to many beginning students).

pearl gate
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the determinant example was only scratching the surface. https://en.wikipedia.org/wiki/Geometric_algebra is what i am talking about, because i dont know enough to explain it myself

Geometric algebra

In mathematics, the geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which contains both the scalars

    F
  

{\displaystyle F}

and the vector space

    V
...
pearl gate
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but yeah i see what u mean here, somewhat

long pelican
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I’d love to see torque and angular momentum in classical mechanics become taught as a bivector one day

pearl gate
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well yeah i was about to say the first point myself lol

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i am looking more for opinions of more experienced people

wispy slate
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So actually that sounds amazing

quasi musk
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Welcome to physics, how may I help you

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Do as little math as possible to get the right answer is the attitude of many physicists

long pelican
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#physics-pedagogy: why are we not teaching Lagrangian mechanics in high school

earnest trail
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I hate when im tutoring students and I ask them what they're doing in their class and they say "chapter 3" like thanks that helps a lot

quasi musk
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lmao

cosmic stag
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whats pedagogy

earnest trail
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the tutoring company I work for tells students to convert improper fractions to mixed numbers. what are the advantages and disadvantages of this?

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I have some ideas but I want your opinions

shadow basalt
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I think the advantages are close to none and the disadvantages are many

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The one advantage is that some teachers take off points if you do not convert them

pastel horizon
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Maybe in some cases 2½ is more practical than 5/2?

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I'd argue a decimal would be even better though

shadow basalt
earnest trail
long pelican
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A bit cheesy but writing 35907325972314/1239713425 as a mixed number tells you as at a glance how big it is

earnest trail
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yeah but again decimals work too

long pelican
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OK. It tells you simultaneously how big it is and a number which when multiplied by it yields an integer

earnest trail
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and I mean I guess it's the same thing as (x+1)/x vs 1+1/x and shit like that which can be advantageous in different ways

long pelican
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Can't do that with decimals

earnest trail
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true

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but improper fractions make the latter advantage clearer

shadow basalt
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Clearly none of these are good reasons to make students do it lol

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Ofc they should know how

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But a correct rational number is a correct rational number

earnest trail
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exactly

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I don't see why mixed numbers are even taught in the first place

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and the notation is terrible

long pelican
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Off topic, do sully reacts feel like... hostile... to anyone else

earnest trail
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wdym

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I love the feeling of getting sullied
it means my work here is done

shadow basalt
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They are

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It’s meant to be hostile lol

earnest trail
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lmao

long pelican
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It's a very unique emoji among emojis in this server

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I'm in like 4 math servers, only the ones with anonymous people have sully in their culture

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I couldn't dream of someone in AGS reacting with sully

shadow basalt
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It is much easier to be hostile anonymously

long pelican
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I think a decent person would be decent anonymously too

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According to a good definition of decent

shadow basalt
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I think you should avoid conflating indecent with hostile lol

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A great many posts on this server deserve to be sullied

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The vibe here is also intentionally different, as AGS and similar servers are meant to be less casual and more professional

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Whereas this place has more banter

long pelican
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It's just weird how sully evolved from something funny to something hostile over time

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In 2017 sully was funny

shadow basalt
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It was always hostile

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It was invented to be hostile

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The original meme it comes from was an insult lol

long pelican
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I was here for a bit in 2017, it meant the same thing but didn't carry the same weight

shadow basalt
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Idk what weight you’re giving it

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It’s still a pretty lighthearted thing

long pelican
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It's just a perception but it's gotten heavier

shadow basalt
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I can’t say I’ve ever taken a sully seriously either way

long pelican
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Hmmm

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I'll just say getting sullied is very far from a good feeling

shadow basalt
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Did u say something sully worthy

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Be honest

long pelican
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Well, getting sullied by a random guy who just joined wouldn't feel the same way

shadow basalt
long pelican
shadow basalt
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That wasn’t the question lol

long pelican
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Well how do you judge something sully worthy except by your own judgment

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Am I predicting someone else might find something I write sully worthy?

shadow basalt
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You can believe something and still understand whether it deserves a sully

earnest trail
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as I said earlier

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I love getting sullied

shadow basalt
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I believe that

earnest trail
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I sully messages based on one criterion:
is message dumb?
if yes ----> sully
if no -----> probably still sully but less likely

long pelican
# shadow basalt Oh are you referring to this

This is true as far as I said, I said it's cheesy and not a reason to make students do it, but there's some mathematical connection too: In number theory, the real absolute value measures size while the p-adic absolute values measure degree of divisibility by p

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So writing it as a mixed number tells you the interval [n,n+1) the number falls in

earnest trail
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I don't think 5th graders care about p adics

earnest trail
shadow basalt
earnest trail
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lol

long pelican
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I don't really care explicitly about the sully

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but it does have that connotation

earnest trail
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I don't think sullies should be taken seriously

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I mean look at the emoji

shadow basalt
earnest trail
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it is so laughable

long pelican
shadow basalt
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Uh

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No?

long pelican
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Yes?

shadow basalt
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Unless you are requiring they be finite

long pelican
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Yes

shadow basalt
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Why?

long pelican
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A rational number can be stored in finite space

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You can't store a generic real number in finite space

earnest trail
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that is a valid point actually

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you could use the repeating bar thingy for decimals if necessary

shadow basalt
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So can a decimal

long pelican
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I want to emphasize that this isn't a reason overweighing reasons to use improper fractions but just something to think about

earnest trail
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yeah

shadow basalt
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I mean again a decimal can also be stored in finite space, though

long pelican
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A decimal with a bar can be thought of as a mixed number with extra steps

shadow basalt
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This also doesn’t like

earnest trail
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tri

#

tru

shadow basalt
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Differ between mixed fractions and fractions

shadow basalt
long pelican
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I also offer the practical and historical reason for mixed numbers: so students can read them in newspapers and other things where mixed numbers appear

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Chess match results ♟️

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I just had a long argument with a friend whether 11/2 is an acceptable way to write 1 and a half

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or whether 11/2 should be read as 5.5

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And it all came from the match result of 1 1/2 - 1/2 in a chess match

earnest trail
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that's why I hate mixed numbers

long pelican
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According to the practical and historical reason, students should just learn how to read them but don't need to write them when they are taking tests

earnest trail
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yeah I agree with that

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and they should know that 1 1/2 = 1 + 1/2

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oh shit I just realized mixed numbers are a form of concatenation

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:o

long pelican
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Hmmm

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Concatenation talk is your talk?

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I saw that in events

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Ok I have some more thoughts on mixed numbers vs. improper fractions
When I did MathCounts the only correct form of a fraction was an improper fraction. This is for ease of grading, but the emphasis on improper fractions is very good for mathematical development. It makes people more comfortable with rational numbers

For example, I can imagine a student being worried about whether (x^2+1)/(x-1) represents a number in all cases where x is not 1 because "in some cases, it's an improper fraction"

Another advantage of improper fractions: you can see at a glance the numerator of 128/81 is a power of 2 which can suggest some combinatorial interpretation, whereas 1 and 47/81 is much more opaque

pastel horizon
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I think there's no argument that improper fractions are easier to use in calculations but they are less readable as final answers