Knubs' Teaching Adventures

[Oxymoron]

Just to show ignorant I am on the subject : Do matrices work only with integer values, or can you put floats in there ?

[Agent Bert Macklin]

I have no idea. I just started the course.

[Agent Bert Macklin]

Here you go, Oxy, from my textbook:

[Oxymoron]

I see. So it isn't strictly limited to integers, you could use floats, or even fractions if you wanted to - I think ?

So, matrices are just a "container", a way to represent a set of things ?

[RyanThunder]

You could use other variables or even entire functions as well.

[ 1 2 3 ]

is pretty much exactly the same as 1 + 2x + 3x^2 or a + 2b + 3c.

[Agent Bert Macklin]

I see. So it isn't strictly limited to integers, you could use floats, or even fractions if you wanted to - I think ?

So, matrices are just a "container", a way to represent a set of things ?

I believe so, yes.

My book puts it this way: "The essential information of a linear system can be recorded compactly in a rectangular array called a matrix." As you saw in the example I posted, I simply used the coefficients as the numbers, which is called a coefficient matrix.

[Zod]

I really hate textbook definitions sometimes. They define it without actually telling you why you'd want to use it.

[Oxymoron]

I'd just need the exact definition of "linear system" to completely understand this definition, but otherwise I think I get the idea. I'd need to practice the thing to really understand and assimilate it, though.

And yeah, to go with Zod's commentary : I find that textbooks tend to be useless if you don't take an actual course with an actual teacher in parallel. At least in my own experience.

[Zod]

It's one of the reasons I sucked so bad at math in High School and am still not that great at it. If I don't have someone/thing telling me how it applies to the real world outside of bad examples that are likely to never occur, it just won't stick.

[RogueIce]

You may be overthinking this Politifacts thing just a little.

It's one of the reasons I sucked so bad at math in High School and am still not that great at it. If I don't have someone/thing telling me how it applies to the real world outside of bad examples that are likely to never occur, it just won't stick.

Would you like a wealth redistribution worksheet?

[Oxymoron]

In my own experience, Math are best taught through practical examples and lots of practice, bordering toward or even fully going into "drilling" territory (you know, like army drills, but for something intellectual).

But that's because I learn through "touching" / practice. Other people may have other learning sensibilities (most people are visual, but some are auditive / verbal).

[Oxymoron]

I also know lots of people had difficulties with math problems not because of the math itself, but because they had difficulties understanding the problem itself ; they had problem with the language used, and thus their understanding of the problem's data was hindered.

[Oxymoron]

That's a really practical example of why language is always important, even in the most technical disciplines.

[Agent Bert Macklin]

I'd just need the exact definition of "linear system" to completely understand this definition, but otherwise I think I get the idea. I'd need to practice the thing to really understand and assimilate it, though.

A linear equation, as per my professor's words: an algebraic equation in which each term is either a constant or the product of a constant and (the first power) of a single variable.

You really don't want to see the mathematical notation of the definition. Without seeing those a lot, it's difficult to follow.

Examples? Remember, parentheticals denote subscripts.

x(1) + 2x(2) = 5 -> linear equation

x(1) + 2x(1)x(2) = 5 -> not a linear equation.

The 2x(1)x(2) is basically 2x*x, resulting in a second power x term.

A system of linear equations is just a collection of linear equations involving the same set of variable, like I posted in the example involving tying them in to matrices.

Edit; the subscripts trip people up. here's are two lniear equations:

2x + 3y = 7
4x + 9y = 17

See if you all can determine what the values of x and y are to make both of those equations true,.

[Zod]

[RogueIce]

In my own experience, Math are best taught through practical examples and lots of practice, bordering toward or even fully going into "drilling" territory (you know, like army drills, but for something intellectual).

I think I'm sorta like this. If I don't do math in quite some time, I'll tend to "forget" how to do it. As I learned back in HS with student-teacher day when I tried to grade some papers and discovered I didn't remember how to do those problems after 8 months or so of not, well, doing them. Once the actual teacher showed me a few times I was able to pick it back up again, but going in cold just wasn't working.

2x + 3y = 7
4x + 9y = 17

See if you all can determine what the values of x and y are to make both of those equations true,.

no u

But seriously, this is a good example of the above. Back several years ago when I was taking this stuff I could probably do it. Now?

Well, maybe. I could probably work through it eventually, but not as fast as I could back then. Alternately if I looked up how to do it I'd likely do better even though I'd likely make mistakes. But just presented as "here's the problem, now solve" not so much.

I think it's something people tend to forget about others. That the knowledge is in fact perishable, and while it might not be 'gone forever' if you don't use it you'll probably forget how to without needing to go refresh your memory first, at the least.

Of course in some places such as a certain sci-fi related discussion board I will not name, they think you're "stupid" for not being 100% on the ball no matter how long ago you last did anything related to, say, the quadratic formula. Those people are dickheads, generally.

[Oxymoron]

Code: Select all

(1) : 2x + 3y = 7
(2) : 4x + 9y = 17

See if you all can determine what the values of x and y are to make both of those equations true,.

Piece of cake.

Code: Select all

(1) : 2x + 3y = 7
(2) : 4x + 9y = 17

Code: Select all

(1) : 2x = 7 - 3y
(1) : x = (7 - 3y)/2

(2) : 4x + 9y = 17
(2) : 4*((7 - 3y)/2) + 9y = 17
(2) : 2*(7 - 3y) + 9y = 17
(2) : 14 - 6y + 9y = 17
(2) : 9y - 6y = 17 - 14
(2) : 3y = 3
(2) : y = 1

(1) : x = (7 - 3y)/2
(1) : x = (7 - 3)/2
(1) : x = 4/2
(1) : x = 2

x = 2
y = 1

Drill, drill, drill, drill.

The only way I can ever manage to pass math classes.

[Crazedwraith]

i have totally forgot simultaneous equations since i did them at a-level. :'(

[Agent Bert Macklin]

In my own experience, Math are best taught through practical examples and lots of practice, bordering toward or even fully going into "drilling" territory (you know, like army drills, but for something intellectual).

Of all the math classes I've taken (College Algebra, Pre-Calculus, Calculus I, Calculus II, Probability and Statistics, and Differential Equations), I've rarely had the material taught to connect to practical, real world examples. I attend a school predominately in the engineering and engineering technology departments, so any practical applications are completely irrelevant and foreign to me.

I learned more about calculus in the community college I attended and how it pertains to real world problems than all of the other courses. It's a sad commentary on the university system in my state, to be honest. I hate the "this is why it is, just trust it" attitude many professors have. I hope to never have that attitude when I start teaching mathematics. I want to know why it's this way. That helps me understand it more.

[Oxymoron]

First, some theory and history of the thing to explain why people felt the need to invent that particular mathematical tool.

Then some theory to explain the basics of the tool.

Then a first batch of practice in class on the most basic usages of the tool. Trying to bring everyone at the same basic level.

Then a bit more theory, and some explanations on the more advanced uses of the tools in the real world.

Then a second batch of practice in class. Still trying to put everyone on the same page.

A relatively short home assignment, non graded (people learn through mistake, grading would be punishing mistakes), though people who would have done the assignment would get a bonus on their "participation" grade.

After that, if need be, another round of theory & practical examples, followed by practice.

Rinse & repeat as many times as necessary.

[Zod]

I can't help but feel that if more classes put extra emphasis on the why you'd have a much higher math literacy in this country.

[Oxymoron]

Like I always say, if teaching was more "integrated", it's to say if every course wasn't segregated to its own subject but if they worked together on some sort of common project, I think more people would understand the use of what they are working on and be motivated to learn the stuff.

But, of course, it would mean that the teachers would have to actually work as a team and cooperate with each other, and to have knowledge of what their colleagues teach.

[Agent Bert Macklin]

If someone asks me why algebra requires performing the same task on both sides, that can be easily explained. "=" implies they have the same value, thus to keep the integrity of the "=", you have to do the same thing to every side.

But when you ask "why" about something in the higher level courses, it can be more complicated. I asked my discrete mathematics professor why a truth table (more complicated than what I posted) had those specific values, and he just said that's the way it is. Way to go? I learned nothing other than memorization. Thankfully, the book explained it better than him, which is quite rare in my experience.

[Agent Bert Macklin]

I can't help but feel that if more classes put extra emphasis on the why you'd have a much higher math literacy in this country.

I think the problem is that to know why, you have to understand the proof and some teachers just don't want ot show it. In my Calculus I course, my teacher did the proof for the quotient rule when using derivatives. It was awesome because it tied into everything we did up to that point in the course and it made logical sense. He is partly the reason why I want to teach calculus in high school and in community colleges.

[Oxymoron]

There's also a time constraint : you don't have that much hours to teach on a given subject, and some teacher may not want to take time to introduce it in depth, preferring to allocate what time they have to something else, like more practice, or things like that.