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Intermediate Algebra

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Re: Intermediate Algebra

#101

Post by admiral33 »

azalemeth wrote: post #83
alexdowad wrote: post #75

When a book is titled "Algebra", it's always a bit ambiguous whether that means "high school algebra" or "abstract algebra". I clicked on this link thinking it was about group/field/ring/etc theory and wondering what an 'intermediate' treatment of those topics would look like.

I thought this too -- in fact the contents of this site are about the level of what good 16-year-olds learn in my country. I'd love to read an accessible 'intermediate' introduction to abstract algebra. Something I've inferred -- and would love to be corrected on -- is that in the US there's a relatively well-defined course implied by the words "calculus" or "algebra". I can guess what they are, but I'm not certain,…

Before College all math courses in the US 'build up' to Calculus 1 & 2 [1,2] presumably to prepare motivated students for engineering. Calculus 2 is the highest you can go in a public High School before a student will start thinking about starting their college career early or doing an independent study. This one track path is not without its side effects. As you can see by skimming this textbook the information on linear systems is lacking - something that might be more interesting to those who want to study CS.

[1]: https://www.khanacademy.org/math/calculus-1

[2]: https://www.khanacademy.org/math/calculus-2

Re: Intermediate Algebra

#102

Post by ookdatnog »

syops wrote: post #4
Gehinnn wrote: post #3

> Irrational numbers are defined as any numbers that cannot be written as a ratio of two integers. > Finally, the set of real numbers, denoted R, is defined as the set of all rational numbers combined with the set of all irrational numbers. I'm sorry, this is not how math works. Reals are basically defined as number. But what is a number? Also the definition of Q is missing the quotient construction (or any motivatio…

I think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number. My background on this topic is that I’ve taught intermediate algebra for over 20 years.

I don't necessarily disagree with your point that for the given audience it's not appropriate to rigorously define the different sets of numbers.

However, I absolutely detest it when teachers just "sweep it under the rug", when they pretend that they just provided a definition when they evidently did not.

Like the commenter you replied to, this sort of stuff genuinely threw me off in high school and made me feel like I didn't understand mathematics.

Re: Intermediate Algebra

#103

Post by ookdatnog »

zsmi wrote: post #41
Gehinnn wrote: post #6

Earlier quoted context omitted.

You are right, I didn't teach low level math courses, but this brain washing is also precisely why I didn't understand math in high school. You cannot argue with this kind of definitions. Everything feels as if it was randomly defined by the teacher. This "intuition" simplifies teaching, but makes understanding harder. It is like a game where you invent rules as you play. No student can win this game.

> Everything feels as if it was randomly defined by the teacher. I suppose you prefer things randomly defined by Euclid? Just kidding... kinda. Seriously though, randomly defining things and then working through the consequences of that definition is a totally valid way to do math. Those random definitions are called postulates.

Euclid's postulates would now be called axioms, not definitions.

Re: Intermediate Algebra

#104

Post by xphos »

Sorry for the rant but as a math lover this is terrible. In my experience "algebra 2" is taught for multiple years and is in my opinion the most boring soul crushing math I was taught. And I say taught because its forced into you with very few interesting observations and thinking. It focuses on memorizing formula's a procedures, which while these things have there applications I don't consider someone who can say y=mx+b or the a^2+b^2=c^2 knows much about math or more importantly engages with things in a mathematical way. I think there is this focus to produce calculators when you want thinkers and maybe and I hope this has changed in the last 10 years (age 24) but I feel like the American middle school level produces bad math habits and skills wrapped around solving test questions instead of thinking.

Re: Intermediate Algebra

#105

Post by pwdisswordfish8 »

alexdowad wrote: post #75

When a book is titled "Algebra", it's always a bit ambiguous whether that means "high school algebra" or "abstract algebra". I clicked on this link thinking it was about group/field/ring/etc theory and wondering what an 'intermediate' treatment of those topics would look like.

If it were abstract algebra, the title would be ‘Basic Algebra’.

https://golem.ph.utexas.edu/category/2013/12/the_long_grind_...

Re: Intermediate Algebra

#106

Post by xphos »

throaway46546 wrote: post #82
Miiko wrote: post #78

Not sure what's the point of section 4.3 Factoring Trinomials introducing "trial and error (or guess and check) method", when this task can be solved easily without any trial and errors, by using 6.2 Quadratic Formula. Shouldn't section 4.3 at least mention that possibility? Is there something I do not understand here?

Guess and check pissed me off so much back when I was in school.

I think guess and check pissed you off because its taught as guess a random number and check they don't really teach you how to guess smartly and then you waste your time guessing in the wrong direction mindlessly. Its like teaching stands they should checkout every array slot when there are bits of knowledge that you can teach to do a binary search and get to an answer faster. And the math intuition to build the pattern of guessing is much less especially for high school level problems.

Re: Intermediate Algebra

#107

Post by xphos »

Miiko wrote: post #78

Not sure what's the point of section 4.3 Factoring Trinomials introducing "trial and error (or guess and check) method", when this task can be solved easily without any trial and errors, by using 6.2 Quadratic Formula. Shouldn't section 4.3 at least mention that possibility? Is there something I do not understand here?

Try factoring this with the quadratic formula x^5+5x^4+10x^3+10x^2+5x+1. Or try guess and check with -1,0,1. The issue with formula's is that they constrain the space of the problems and ones mind especially. the problem above is factoring is (x+1)^5 but there exists no (and there cannot exist) formula to that can tell you that from the equation. Algebra is taught poorly not because kids don't learn the steps but because it stunts people into viewing math as if you only had the formula its easier. I'm not saying you should know that there cannot exist a 5'th or greater degree integer formula for factoring equations because to say that is very hard. But when it leaves students with very little scaffolding to use when things go wrong / don't fit the formula neatly

Re: Intermediate Algebra

#108

Post by dqv »

xphos wrote: post #104

Sorry for the rant but as a math lover this is terrible. In my experience "algebra 2" is taught for multiple years and is in my opinion the most boring soul crushing math I was taught. And I say taught because its forced into you with very few interesting observations and thinking. It focuses on memorizing formula's a procedures, which while these things have there applications I don't consider someone who can say y=…

I'm definitely feeling the effects of this now trying to get back into math. It's frustrating because I can tell exactly what's happening: I get the same kinds of problems wrong over and over again and exhaust the question bank. The next time around I recognize the problem and basically just solve it from memory.

Do you have any recommendations for learning math the right way? Or a better way at least.

Re: Intermediate Algebra

#109

Post by nicklaf »

gxs wrote: post #77
alexdowad wrote: post #75

When a book is titled "Algebra", it's always a bit ambiguous whether that means "high school algebra" or "abstract algebra". I clicked on this link thinking it was about group/field/ring/etc theory and wondering what an 'intermediate' treatment of those topics would look like.

I have Artin’s Algebra on my bookshelf, most people who see it always tell me they took it in high school. I’m always impressed.

You know for a few confused seconds I didn't see the joke about the title and took your statement literally, assuming you were talking about some precocious student, who perhaps had been lucky enough to be tutored or concurrently enrolled while still in high school.

Funny story, though: this is the upper division algebra text we used at UC Davis, and one amusing anecdote that our professor relayed was that this book was in fact written for MIT freshmen. Supposedly, this is why the book introduces linear algebra from scratch rather than assuming it as a prerequisite....

Re: Intermediate Algebra

#110

Post by [deleted] »

adrian_b wrote: post #73
whatshisface wrote: post #65

Earlier quoted context omitted.

Assuming you already know what a rational number is, the next step is to tell you what a real number is. A real number is defined as the equivalence class of all sequences of rational numbers that converge to the same value. For example, every sequence of rational numbers that gets arbitrarily close to the square root of two as you go to higher terms is considered "the square root of two." If you don't know what a ra…

Instead of assuming an understanding of what natural numbers are, you could have continued to define all of them as equivalence classes, as that is what they are. The integers are the equivalence classes of differences of natural numbers, while the natural numbers are the equivalence classes of finite sets having the same number of elements (i.e. which may have a bijection between themselves), including the empty set…

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