Math is Math! Why would they change Math?!

[Grey_Wolf_c]

I can't beleive I only now remembered this.

Enjoy.

Wait, so this "new math" American keep complaining about is the borrow method of subtraction I was taught in a british-style school decades ago?

Bloody hell, America really is stuck in the 50s and refusing to be dragged kicking and screaming into the century of the fruitbat, isn't it?

I'm told below that it is from the 60s, so that makes sense.

Grey Wolf

[Khedrac]

Wait, so this "new math" American keep complaining about is the borrow method of subtraction I was taught in a british-style school decades ago?

Bloody hell, America really is stuck in the 50s and refusing to be dragged kicking and screaming into the century of the fruitbat, isn't it?

Grey Wolf

That being a Tom Lehrer video I would guess that it was recorded in the 60s (70s at the latest).

[Grey_Wolf_c]

That being a Tom Lehrer video I would guess that it was recorded in the 60s (70s at the latest).

Ah, fair enough. Never heard of the guy. But OK, so the new math is something else - presumably not the borrow method. WOuld make sense, that is a terrible way to do subtraction - counting up makes so much more sense when it comes to understanding what's actually going on.

Grey Wolf

[Tyndmyr]

I suggest that, as a fun way to explore problems, you just plug them into Wolfram Alpha. Now, you still need to know what you're looking for, but it handles a lot of the tedious calculation steps.

Glancing at the graph resulting from graphing the function would suffice to demonstrate if it is always positive or negative. There are shortcuts here, in that some equations are always going to have specific results. The pattern of n² - 6n + 14 always ends up with a parabola, for instance. Therefore, if you're attempting to prove something's always positive, figuring out the minimal value would suffice.

[Grey_Wolf_c]

Therefore, if you're attempting to prove something's always positive, figuring out the minimal value would suffice.

This, however, runs into what I call the Barometer Issue. You certainly can prove that this curve never dips under 0 by calculating its minimum... but this being homework, the purpose is to demonstrate the comprehension and applicability of the day's lesson, not that you have mastered a different lesson.

It ties in with the "change Math" thing - it is not random cruelty on the math teacher's part to insist that math problems be solved by the method they are teaching and not the way the parents are used to. There is often serious thought put into the methods, and teachers are right* to object to alternative methods being used, because the only way you can get through a curriculum when teaching multiple kids at once is for everyone to be on the same page.

Grey Wolf

*I mean, they can also be wrong, certainly, and cruel, definitely, but in this case, clearly the day's lesson is function manipulation into squares, not derivation, or whatever it was that you needed to calculate minimums and maximums.

[Caerulea]

I can't beleive I only now remembered this.

Enjoy.

I posted that upthread a bit ago.

This whole discussion reminds me of a Tom Leher song, titled (appropriately) New Math.

Tangentially related to Math, but it annoys me so much when teachers insist on teaching an outdated model or view of the subject. Yes, you know it's wrong, I know it's wrong, but we are still going to spend a week on the Bohr model of the atom just because it is easier than explaining the quantum mechanics behind the correct model, which we are going to do anyway next week! It is infuriating.

[Radar]

I posted that upthread a bit ago.


Tangentially related to Math, but it annoys me so much when teachers insist on teaching an outdated model or view of the subject. Yes, you know it's wrong, I know it's wrong, but we are still going to spend a week on the Bohr model of the atom just because it is easier than explaining the quantum mechanics behind the correct model, which we are going to do anyway next week! It is infuriating.

Well, there are some merits to talk about Bohr's model first, since it introduces ideas important for quantum mechanics and it might be easier to understand the more accurate description of a hydrogen atom once you know Bohr's framework first. Besides, below university level any talk about quantum mechanics lacks the mathematical language needed to fully explain it, so being able to properly and fully describe an earlier empirical model has some advantage. Another reason is historical: Bohr's model made many other breakthroughs possible and after corrections by Sommerfeld was actually very accurate. As such, it would be good for any educated person to at least know abuot what Bohr did.

As a sidenote, I would like to say, that one can understand principles of quantum mechanics practically without any mathematics, since they can be derived from experiment (if you can get it somewhere, Dirac wrote a book on quantum mechanics with a fantastic introduction). Still, making any predictions with it requires university level knowledge.

[factotum]

Well, there are some merits to talk about Bohr's model first, since it introduces ideas important for quantum mechanics and it might be easier to understand the more accurate description of a hydrogen atom once you know Bohr's framework first.

Yeah. Complaining about that seems to me to be like complaining that they still teach Newtonian mechanics before (or even instead of) Einsteinian ones, and the simple answer there is that Newtonian mechanics are perfectly adequate for 99% of situations you'll ever encounter in real life. Similarly, the Bohr model of the atom answers most questions--you rarely need the quantum model.

[crayzz]

This, however, runs into what I call the Barometer Issue. You certainly can prove that this curve never dips under 0 by calculating its minimum... but this being homework, the purpose is to demonstrate the comprehension and applicability of the day's lesson, not that you have mastered a different lesson.

It ties in with the "change Math" thing - it is not random cruelty on the math teacher's part to insist that math problems be solved by the method they are teaching and not the way the parents are used to. There is often serious thought put into the methods, and teachers are right* to object to alternative methods being used, because the only way you can get through a curriculum when teaching multiple kids at once is for everyone to be on the same page.

Grey Wolf

*I mean, they can also be wrong, certainly, and cruel, definitely, but in this case, clearly the day's lesson is function manipulation into squares, not derivation, or whatever it was that you needed to calculate minimums and maximums.

Aye, exactly.

"Everyone on the same page" is one concern; the other being that, if you can only solve a problem one particular way, you aren't very good at those kinds of problems. You're passably adequate, certainly, but if you only have one technique under your belt, you're going to be very limited at what you can do. If you want to learn a subject in depth, you should learn to approach it in a variety of ways.

In my experience, multiple methods are usually taught, and when it comes to evaluation, you'll be asked both "solve this problem" and "solve this problem using this particular method" type questions. So you'll have ample opportunity to solve problems with your preferred technique, but for perfect marks you'll need to learn a broader understanding of the subject.

Well, there are some merits to talk about Bohr's model first, since it introduces ideas important for quantum mechanics and it might be easier to understand the more accurate description of a hydrogen atom once you know Bohr's framework first. Besides, below university level any talk about quantum mechanics lacks the mathematical language needed to fully explain it, so being able to properly and fully describe an earlier empirical model has some advantage. Another reason is historical: Bohr's model made many other breakthroughs possible and after corrections by Sommerfeld was actually very accurate. As such, it would be good for any educated person to at least know abuot what Bohr did.

As a sidenote, I would like to say, that one can understand principles of quantum mechanics practically without any mathematics, since they can be derived from experiment (if you can get it somewhere, Dirac wrote a book on quantum mechanics with a fantastic introduction). Still, making any predictions with it requires university level knowledge.

The Bohr model is great because it's simple but still widely applicable; you can make a career out of chemistry and never stray out of using the Bohr model. Models are tools as much as they are theories, and not teaching the Bohr model because the quantum model is better is like not teaching someone how to use a hammer because nail guns exist.

Incidentally, the quantum models aren't "correct" in any strict sense. They're better models than the Bohr model, usually, but there's plenty of fudging and simplifying assumptions that go into those models. At the end of the day, they're just models, and you pick your model to match the level of precision you need; quite often, you just need the Bohr model. Like how you often just need a simple hammer, because hanging family photo's with a nail gun is just plain overkill.

[Radar]

Incidentally, the quantum models aren't "correct" in any strict sense. They're better models than the Bohr model, usually, but there's plenty of fudging and simplifying assumptions that go into those models. At the end of the day, they're just models, and you pick your model to match the level of precision you need; quite often, you just need the Bohr model. Like how you often just need a simple hammer, because hanging family photo's with a nail gun is just plain overkill.

The key difference is, Bohr made an empirical model without having knowledge why electrons have to obey such specific laws. Quantum mechanics is a general theory, which can be in particular used to predict behaviour and qualities of an hydrogen atom.

Also: for chemistry, you really need more then Bohr's model, since you cannot describe molecular bonds with it. Even for things like polarizability you Bohr's model does not give any clue - it was built empiricaly to do one thing and only one thing.

[Aran nu tasar]

In this case, clearly the day's lesson is function manipulation into squares, not derivation, or whatever it was that you needed to calculate minimums and maximums.

Interestingly, the two approaches are one and the same. The minimum of an upwards-pointing parabola is the vertex. You find the vertex by manipulating the equation of the parabola into the form y = a(x-h)²+k; the vertex is (h,k) when the equation is in this form. You do that manipulation by, essentially, completing the square.

If somebody starting trying to compute the minimum of the parabola using calculus, of course, that's serious overkill for the problem and not the approach the teacher wants the students to practice/demonstrate understanding of.

[Knaight]

If somebody starting trying to compute the minimum of the parabola using calculus, of course, that's serious overkill for the problem and not the approach the teacher wants the students to practice/demonstrate understanding of.

It's not the approach, but I'm not sure I'd call it overkill. Taking derivatives of polynomials is arguably a lot easier than the algebra involved, and you can also do it once and call it a day. Take the generic case:
ax²+bx+c

For calculus, take a derivative and set it to 0. That's just
d/dx(ax²+bx+c)=0
2ax+b=0
x = -b/2a

The generic case of taking the squares was solved upthread, and it was a whole lot more complicated than that.

[Kato]

Re outdated models :
It is absolutely sensible to use (partially wrong) models in school or even lower university courses. Bohr is a prime case, since understanding the quantum model requires math most people will never come across. I had a pretty good grasp of chemistry in school and we touched on the existence of orbitals without anything resembling quantum physics. It needs some handwaving but...
Even if you look at the better model right after, it's sensible to see what came before to understand where it comes from.

[Algeh]

Interestingly, the two approaches are one and the same. The minimum of an upwards-pointing parabola is the vertex. You find the vertex by manipulating the equation of the parabola into the form y = a(x-h)²+k; the vertex is (h,k) when the equation is in this form. You do that manipulation by, essentially, completing the square.

If somebody starting trying to compute the minimum of the parabola using calculus, of course, that's serious overkill for the problem and not the approach the teacher wants the students to practice/demonstrate understanding of.

It's not the approach, but I'm not sure I'd call it overkill. Taking derivatives of polynomials is arguably a lot easier than the algebra involved, and you can also do it once and call it a day. Take the generic case:
ax²+bx+c

For calculus, take a derivative and set it to 0. That's just
d/dx(ax²+bx+c)=0
2ax+b=0
x = -b/2a

The generic case of taking the squares was solved upthread, and it was a whole lot more complicated than that.

Well, sure, but if a student has a completing the square problem for homework they probably haven't seen calculus yet and won't for several years (at least in the system I teach in).

It reminds me of checking geometry homework about a decade ago. This was using a "spiraling" curriculum where homework problems would be a combination of stuff from a while ago, stuff we're doing right now, and stuff that will help you build intuition about things we'll study formally later in the year.

Anyway, one of the homework problems was, and this is not the exact wording, asking about the different possible number of intersections between a line and a circle. Since we haven't hit circles yet, students wouldn't have formal ideas about things like tangent and secant lines to work from, but it's perfectly reasonable to ask a kid to sit there with a picture of a circle and try to empirically see the different ways a line might or might not touch it. (Phrased the right way, I could probably get some reasonable conclusions out of elementary students about that, since it's a visual question that only relies on definitions of common things that most students "get".) The idea was to be building their intuition for a much later lesson about tangent and secant lines because they would have at least have thought about the visual relationship before hitting a bunch of formal math stuff.

This homework problem was mostly memorable because it taught me that pretty much my whole class was copying homework answers they didn't understand from the internet, because their answers all were the same verbatim explanation involving tangent and secant lines from the website they were all using to "check their work". Sure, that answer was technically correct, but it served no pedagogical purpose for them to copy it into their homework that night rather than try and think about what circles and lines look like. Fun times.

[Radar]

This homework problem was mostly memorable because it taught me that pretty much my whole class was copying homework answers they didn't understand from the internet, because their answers all were the same verbatim explanation involving tangent and secant lines from the website they were all using to "check their work". Sure, that answer was technically correct, but it served no pedagogical purpose for them to copy it into their homework that night rather than try and think about what circles and lines look like. Fun times.

This is one of the most difficult things to eradicate. I only ever tought at university level, but it was a huge problem with any kind of projects or even lab reports. Granted, I did have many more options to make students do their work, but still.

It also contributes to a bigger issue of remembering instead of understanding, but this is a different topic.

[Caerulea]

Re outdated models :
It is absolutely sensible to use (partially wrong) models in school or even lower university courses. Bohr is a prime case, since understanding the quantum model requires math most people will never come across. I had a pretty good grasp of chemistry in school and we touched on the existence of orbitals without anything resembling quantum physics. It needs some handwaving but...
Even if you look at the better model right after, it's sensible to see what came before to understand where it comes from.

I would agree, and perhaps overstated my frustration. It came mostly from spending far too long on something that should have been a half hour review of stuff that was done two years ago. Also, the issue with the quantum mechanical model is not math—if the term "Hamiltonian" was mentioned, 3/4ths of the class would immediately fall asleep or lose all interest—we covered it this week with a very basic overview.

With regards to people copying answers of the internet, is that a big problem? It seems to me that those doing that are wasting their education. Either figure it out yourself, or ask a friend to explain it in a way that you understand. I don't see why anyone would do different. Perhaps I am supremely naive.

[OutOfThyme]

With regards to people copying answers of the internet, is that a big problem? It seems to me that those doing that are wasting their education. Either figure it out yourself, or ask a friend to explain it in a way that you understand. I don't see why anyone would do different. Perhaps I am supremely naive.

Yeah, it is. I'm still in college, but the amount of copying people do on engineering work is astounding. Fortunately, we're at the point where the online sources have about a 50/50 shot of being right (even the official solutions are occasionally wrong). It's not ethically sound, but people who copy their homework tend to generally get screwed on tests. Some people have figured out that the best way to study for the test is to actually do the homework without taking any corners.

It's extremely wasteful, but people are lazy.

[snowblizz]

Perhaps I am supremely naive.

Unfortunately that is the correct answer.

It's extremely wasteful, but people are lazy.

The amount of time people do not spend studying their chosen degree is astounding at times.

I remember one university class where it really struck me. It was a short course with a visiting professor who was trying to engage the class and literally noone was responding. I was struck by the insight that we are all sitting here voluntarily purposefully avoiding to interact with someone who came here specifically to teach us something "special". Mind blowing stuff.

A symptom of chasing credits not knowledge in a world where having a degree is more important than actually knowing.

You see the same thing in grade school in the "when am I ever going to use XXX in real life?" question.

[Chen]

You see the same thing in grade school in the "when am I ever going to use XXX in real life?" question.

To be fair, a huge amount of university education that you get tested on, closed book, during exams will also never be used in your job. Either there will be a shortcut or you'll have programs/reference sheets to do almost all the difficult math work.

[Kato]

So, as has been pointed out, the problem with people who copy answers from online is that (usually) those are the people who think they can just cheat their way through the class. (of course there are other reasons, like terrible educators who give tasks that require you to use other sources because they suck at teaching you what you need for your homework)
And these people waste resources. They take away seats in classes, they take away time from their teachers (even more so because they tend to disrupt class) they have them check homework they didn't do. And all that effort is wasted and cannot go towards students who actually want to learn.

Also, this tangent is going pretty far away from the initial question.