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I looked it up. Interesting. I might take a peek at it.
I am not sure what it was, and since it was nearly fifty years ago that I was in ninth grade, I don't recall all the details of what frustrated me. All I know is that it seemed completely illogical, and the fact that so many math heads say it teaches logic is puzzling to me, because I think it's the reverse.I do remember not being able to understand and remember the steps.
Interestingly, I was able to move from the clerical to the management ranks where I worked without a degree by a combination of experience, writing skills, and a minimum score on three General Subject CLEP exams. English and Math were required, and I could choose the third (I chose History).
I aced English, very high score. Got above average on History, expected. I like History. But most amazing was that I got an above-average score on the Math test, too, which included quite a few algebraic problems. It was multiple choice. The answer was often obvious, but I would not have been able to tell you how I arrived at the answer. That was often the problem I had with algebra. The steps. Which, I suppose, is the whole point of algebra, to make you go step by step in order to arrive at an answer that you could get to either in a simpler way or by instinct. With a multiple choice test, you could just pick the correct answer without having to "show your work".
Do you mind going deeper into this? To me algebra came automatically. One of the things I saw with my classmates in school, and with my kids when they took algebra, was the desire to skip steps and jump right to the obvious answer. In fact many teachers encouraged this. I'd had a math teacher who required us to go step by step, even on the easy problems. This is something I required my kids to do as well. Which they fought in school, but thanked me for in college.
The reason is going through all those tedious steps helps to build in the fundamental logic for the more complex problems that aren't obvious.
For example:
4+X=5. Here I think the answer is obvious: 1. But why? If we follow the steps:
4-4+x=5-4 We want all the numbers on one side and letters on the other. We can subtract the same number from both sides.
0+X =1
X=1
Writing down those steps may seem a waste of time, but learning how to do it in more and more complex evolution is the fundamental core of understanding algebra. Skipping steps at this stage creates confusion later on.
I agree with Jim we
The majority of high school diploma welfare recipients here are reading and going math at 6th grade level -according to casas exam -
Do we really want to dumb down more?
We should also stop telling students that everyone should go to college, or college is the only way to achieve economic success. Encourage apprenticeships, vocational schools, etc. Get students thinking about different career options and maybe have them try out different careers so they will have a better idea about what they want to do.
Even the most ardent proponents of education recognize college isn't for everyone. That said, the only opinion which matters is your own, relative to your own education/direction.
Quote:
Originally Posted by katie1215
Abstract algebra shouldn’t be a requirement for high school graduation
It's not; abstract algebra and linear algebra (often referred to as 'Algebra II' or AP/IB Algebra in a high-school setting) are not the same as basic/introductory/high-school algebra (which is a prereq for the aforementioned).
Do you mind going deeper into this? To me algebra came automatically. One of the things I saw with my classmates in school, and with my kids when they took algebra, was the desire to skip steps and jump right to the obvious answer. In fact many teachers encouraged this. I'd had a math teacher who required us to go step by step, even on the easy problems. This is something I required my kids to do as well. Which they fought in school, but thanked me for in college.
The reason is going through all those tedious steps helps to build in the fundamental logic for the more complex problems that aren't obvious.
For example:
4+X=5. Here I think the answer is obvious: 1. But why? If we follow the steps:
4-4+x=5-4 We want all the numbers on one side and letters on the other. We can subtract the same number from both sides.
0+X =1
X=1
Writing down those steps may seem a waste of time, but learning how to do it in more and more complex evolution is the fundamental core of understanding algebra. Skipping steps at this stage creates confusion later on.
THIS IS WHAT I MEAN!!!! How is that the next step? How is it logical to decide that this is what you do? It's not one bit logical to me. It would never in a million years occur to me to write what you wrote on the second line.
THIS IS WHAT I MEAN!!!! How is that the next step? How is it logical to decide that this is what you do? It's not one bit logical to me. It would never in a million years occur to me to write what you wrote on the second line.
You want to "solve for X".
As tnff said, you want your final step to be
X =
Look at what you need to do in order to get X all by itself before that =.
You start with 4 + X = 5
To get X all alone by itself you have to get rid of the 4. "Get rid of" means subtract. The rule you have to remember is that whatever you do on one side of the =, you have to do on the other side, too. Think of the = as the pivot point for a teeter totter: you want it to stay balanced, not one side lower and one side higher.
Therefore:
4 - 4 + X = 5 - 4
The next rule is do calculations in order, so 4 - 4 = 0; 5 - 4 = 1
0 + X = 1
X = 1
Another one:
2X = 12
To get X all by itself we need to get rid of the 2. In this case we have to divide by 2
THIS IS WHAT I MEAN!!!! How is that the next step? How is it logical to decide that this is what you do? It's not one bit logical to me. It would never in a million years occur to me to write what you wrote on the second line.
I think that falls under math education. I volunteer as a math tutor for my local library and quite awhile back they connected me with a low income 2nd grader whose mama thought he had mastered all elementary school math. The kid had zero interest in math enrichment, but it didn't take much goading to realize his mama was right. He attends the worst school pipe line in our metro area, so I've been tutoring him ever since. Skipping all those steps is second nature to him, but now that he's a hormonal teenager and dealing with related brain fog, it is becoming obvious that getting that self discipline to write out those steps to where it is routine is really important. Teaching those steps is a big part of it. His school district only cares about test scores and he is still 2+ grade level ahead, so school education wise, he gets ignored. I have pushed him into participating in Duke TIP, and that has help a lot.
THIS IS WHAT I MEAN!!!! How is that the next step? How is it logical to decide that this is what you do? It's not one bit logical to me. It would never in a million years occur to me to write what you wrote on the second line.
Thank you. Can we explore this a bit? I do think that's where math starts to break down for a lot of people. If you prefer not, that's fine.
Quote:
Originally Posted by suzy_q2010
...
To get X all alone by itself you have to get rid of the 4. "Get rid of" means subtract. The rule you have to remember is that whatever you do on one side of the =, you have to do on the other side, too. Think of the = as the pivot point for a teeter totter: you want it to stay balanced, not one side lower and one side higher.
...
That's a great way to explain it. Most everyone automatically knows how a teeter totter works and what happens if the kid on one side gets off before the other.
THIS IS WHAT I MEAN!!!! How is that the next step? How is it logical to decide that this is what you do? It's not one bit logical to me. It would never in a million years occur to me to write what you wrote on the second line.
Well, it's not intuitively obvious that the next step would be to subtract 4 from both sides. That's why it took 50,000 years after humans first learned how to count and add and subtract, for the Arabs to invent algebra in about 1000 AD.
What it is, is a technique or tool, to solve for an unknown. There are a whole set of these techniques or tools, which you are taught in algebra class. You can multiply both sides of an equation by the same thing, you can add or subtract the same thing from both sides, you can raise both sides to a power. The quadratic equation allows you to solve for X when presented in the form ax^2 + bx + c = 0. And so on, and so on.
The purpose of algebra class is to teach you a) how to set up a mathematical problem in a format that can be solved by certain tools; and b) how to use that set of tools to solve the problem.
Unfortunately this underlying concept is not often communicated, and teachers launch into instruction on the use of the tools without explaining what they are or why you're going to use them. Honestly, in my experience, math teachers especially in grade school were amongst the least effective teachers at explaining WHAT we are doing and WHY we are doing it, at least in a way that I could grasp. The first sentence above is why the "discovery method" of teaching math doesn't work for most people. Do they really expect that an 8th grader, who's not overly interested in the subject in the first place, will discover in his Thursday night homework session, what it took generations of Arab mathematicians to figure out? Just teach us the damn tools.
Do you mind going deeper into this? To me algebra came automatically. One of the things I saw with my classmates in school, and with my kids when they took algebra, was the desire to skip steps and jump right to the obvious answer. In fact many teachers encouraged this. I'd had a math teacher who required us to go step by step, even on the easy problems. This is something I required my kids to do as well. Which they fought in school, but thanked me for in college.
The reason is going through all those tedious steps helps to build in the fundamental logic for the more complex problems that aren't obvious.
For example:
4+X=5. Here I think the answer is obvious: 1. But why? If we follow the steps:
4-4+x=5-4 We want all the numbers on one side and letters on the other. We can subtract the same number from both sides.
0+X =1
X=1
Writing down those steps may seem a waste of time, but learning how to do it in more and more complex evolution is the fundamental core of understanding algebra. Skipping steps at this stage creates confusion later on.
This "shorthand" form of writing algebra confuses some students.
Also, if you aren't able to color in the -4 on both sides as the same color but different color than everything else in the equation, some people have a hard time understanding the teeter totter.
It's important to show a "top-down" approach instead.
Look at what you need to do in order to get X all by itself before that =.
You start with 4 + X = 5
To get X all alone by itself you have to get rid of the 4. "Get rid of" means subtract. The rule you have to remember is that whatever you do on one side of the =, you have to do on the other side, too. Think of the = as the pivot point for a teeter totter: you want it to stay balanced, not one side lower and one side higher.
Therefore:
4 - 4 + X = 5 - 4
The next rule is do calculations in order, so 4 - 4 = 0; 5 - 4 = 1
0 + X = 1
X = 1
Another one:
2X = 12
To get X all by itself we need to get rid of the 2. In this case we have to divide by 2
2X / 2 = X
Do the same on the other side of the =
12 / 2 = 6
X = 6
OK, and I can see where I would have run into problems.
The first:
Quote:
To get X all alone by itself you have to get rid of the 4.
That would not occur to me naturally. I would not think "I have to get rid of the 4" on my own as a first step. This is an example of the mechanics, if you will, that would have to be specifically taught to someone like me to whom this not come naturally, yet I don't recall anyone ever teaching anything that specific about algebra. It is likely assumed by math teachers that it is obvious, but it is not obvious to me.
This:
Quote:
The rule you have to remember is that whatever you do on one side of the =, you have to do on the other side, too.
is a little more obvious because by the time we hit ninth grade, we certainly know what "=" means, but I think it is still something that should be specifically pointed out (as you just did) in the early stages of learning algebra.
I would have had to have someone point out to me, starting from the most elementary basics of algebra--the stuff that those of you to whom math comes easily--these basic steps very early on without the assumption that I could "see" them, and maybe go over and over the basics until I had it down because my brain just does not see these things as obvious.
Look, I'm not a complete idiot. I could read at a fifth-grade level in second grade. I used to be puzzled by how someone could not remember how to spell a word because once you read it, you can call it up on the little screen inside your forehead (and then I discovered that not everyone has that little screen inside their forehead, lol.) Historical dates stick with me. I have a photographic memory to some extent and can remember approximately where in a book and where on the page I read something. Even with numbers, as part of my job I used to have to calculate individual scores then averaged and multiplied by percentages to come up with total weighted average scores for purposes of evaluating bids and proposals, and I would amuse myself by doing it on paper to see if I could beat the guy with the calculator, and I did.
But for algebra, I would have needed more repetitive instruction on the very basics early in the game, and I think that's where I got lost. Everyone else moved on, and I was stuck on Step One. Even later, when I tried to take the remedial class at the college level going to night school, not enough time was spent on the mechanics of how this type of work is done. I am missing a certain type of thinking ability in my brain, and I suspect it is related to the same thing that makes me unable to figure out how to put something back together once it is taken apart, even when I was the one who took it apart. I need very detailed, very precise, step-by-step "idiot" instructions, or I won't get it.
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