# How do I “learn” more difficult algebra?

I do not really understand where I was suppose to learn this kind of stuff. I am always told that my algebra knowledge is the biggest reason I am so bad at math. But I do not understand where I was suppose to learn it. I started math with pre-algebra and I learned basically no algebra in that, just arithmetic. After that was college algebra and I was told constantly that the biggest reason people fail that class is because of a weakness in algebra.

That doesn't make any sense to me, isn't that where I am suppose to learn algebra? Anyways I will have a semester off of math, where is it that I can actually learn algebra? All the math books before pre-algebra are incredibly basic and problems I can mostly do in my head. Where do I learn those incredibly complex trick, formulas to memorize (pascal's triangle stuff, quadratic formula and all that other stuff?

It just seems like people expect you to either be good at algebra or be bad at it.

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Let me begin by saying that people don't expect you to either be good at algebra or not. That's a false dichotomy. I also presume that you decided not to continue with your Calc II class? There are many ways to proceed, but have you considered getting a tutor, asking a professor for help, using some likely-free resource on campus? I tutor this material frequently, for example. Although perhaps the first step is to stop thinking of it as 'incredibly complex tricks' and instead just think of learning bits one step at a time. –  mixedmath Jun 13 '12 at 12:21
I am going to continue with Calc 2, I do not have to drop out for a few more weeks still. –  user138246 Jun 13 '12 at 12:25
@jordan There is a big gap in most high school's preparation and what is expected in college. I really sympathize with a lot of students in your situation, where high school pretty much dropped the ball on preparing you. Nevertheless, if you keep at it, you will catch up! –  rschwieb Jun 13 '12 at 12:45
@Jordan: It seems to me that your problem (assuming what you say is true) is in taking college algebra immediately after prealgebra (doesn't seem possible to me). In the U.S. prealgebra is typically a 7th or 8th grade course (roughly ages 12-14). After this comes a year of algebra 1 (8th or 9th grade), then a year of geometry, then a year of algebra 2, then a year of precalculus. College algebra consists of the more advanced parts of algebra 2 along with some (but not all) precalculus, and it is a course students take in college if their math background isn't strong enough to take precalculus. –  Dave L. Renfro Jun 13 '12 at 14:26
You can have a look at the problems here: artofproblemsolving.com/Alcumus –  Phira Jun 13 '12 at 21:17

Key to learning algebra? I think it depends on what you mean by algebra. To me, algebra is just arithmetic where we're allowed to use symbols that aren't exactly numbers (variables), or represent numbers that are difficult to write (e.g. $\sqrt{2}$ and $\pi$.) I say "just arithmetic" because all of the old operations and distribution works exactly the same way. I have found many students struggle because they never even mastered the basic arithmetic rules. I think if they did, doing it with $x$'s would make no difference.

Incredibly basic exercises I obviously have no idea what's going on in your head about these exercises, maybe you're exactly right in all or some of your thinking about algebra problems. The problem with this is that this isn't useful to anyone unless you can write it down.

I'm not saying you are this type of student, but I have seen some students fall to a sort of overconfidence where they think they understand something, but it's clear they don't, because their work and answers are completely wrong. The only antidote for this is practice and validation of your answers. The clearer your written solutions are, the clearer it will be in your mind.

Memorization Memorization is an extremely blunt tool, and yet many students treat it as their main weapon. Memorization is too often used to avoid or delay thinking about things that really ought to be understood. My wife would say I have an awful memory, so I can probably attest that math is not so much about memory.

Certainly there are some things that need to be memorized, and there is no sense in spending time "understanding" them, like adding and multiplying single digit numbers. The quadratic formula on the other hand is a borderline case that is good to have memorized, but it's also good to understand where it comes from (completing the square on a quadratic equation!).

Contrary to popular belief, we use mathematics to simplify problems. I would casually argue that a core tenet of math is: "You just find the right way to look at it (or represent it) so it becomes simple".

Edit Dejan Govc inspired another thought with his comment about a student being uncomfortable with what exactly a variable is. This is true: it makes people uncomfortable when something new/ambiguous is introduced, like this. This is because the student hasn't "made the jump" to that new way of thinking yet. Being able to make these jumps is important, because as they learn to abstract further, they will encounter this feeling of disorientation all the time. The best thing to do is simply admit to yourself that you don't get it totally, but trust that through practice you will finally gain a feel for the idea.

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+1, very well written. I'd just like to add that in my experience some people do have problems with understanding what exactly $x$ represents. Is it a concrete number? Is it an arbitrary number? Half of both? Surely $3x+1$ must be an even number, since $3\cdot 5+1$ is? And so on. The abstraction here is already on a higher level than working with concrete numbers, but this is rarely explicitly mentioned. –  Dejan Govc Jun 13 '12 at 12:59
@all Thanks for bringing that up. It inspired me to add another comment at the end, because you made me realize I didn't write anything about the (perfectly natural) state of confusion we are all sometimes in while learning. –  rschwieb Jun 13 '12 at 13:11

I think the best thing you could do is practice a lot. Try solving lots of really simple problems in algebra and try to completely understand them.

(By "completely understanding", I mean understanding every single step of your solution and how those steps interact. After every step try writing an explanation of what exactly you did in it and why you did it. This will help you think about it more deeply. Also: try asking yourself as many questions about the problem as you can possibly think of. Try answering those questions. Take as much time as you need. When you understand those, write down explanations of them. Having a notebook for that purpose might be helpful.)

When you feel you completely understand them, proceed to increasingly more difficult problems and try to understand those completely. Most importantly, do it at your own pace. If a problem just seems overwhelming, try understanding a simpler (but preferably related) one first. This way you will slowly learn to break complex problems into small little pieces that you have already mastered and your algebra skills will gradually increase.

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this is something crucially important that many students don't get. Solving a question without knowing the motivation for each step in the solution is a recipe for disaster. –  Robert Mastragostino Jun 13 '12 at 15:35
@RobertMastragostino Yeah... unfortunately a lot of them are under the impression that there isn't any motivation: they think that math is a bunch of random tricks and processes to be memorized. –  rschwieb Jun 14 '12 at 0:53
Just wrote an exam for a 2nd year ODEs class that was presented exactly like that, so I don't think it's always the student's fault for thinking that way. But it's still definitely one of the biggest problems in the curriculum, and I'd say that it's most of the reason that math gets all the "when will I ever use this" comments. –  Robert Mastragostino Jun 14 '12 at 2:48

You are absolutely right to be confused by concrete (as opposed to "abstract") algebra. That is a sign of intelligence, not absence of "talent." You should be confused because we are not told what algebra is or what it is about. Algebra is NOT arithmetic with symbols instead of numbers. Algebra is the unrevealing name given to the study of FUNCTIONS.

A function is a QUANTITATIVE RELATIONSHIP BETWEEN UNSPECIFIED QUANTITIES. The quantities HAVE to be unspecified in order for there to BE a relationship between them. For example, the circumference of a circle varies with its diameter and vice versa. The area of a circle varies with its diameter and vice versa. The area of a circle varies with its circumference, and vice versa. The area of a triangle varies with (is determined by) the lengths of its sides. That area is NOT determined by the sizes of its angles. We can ask, "Is the area determined by the sizes of its medians? It's altitudes? The lengths of its angle bisectors? The diameter of its incircle or circumcircle or both?" and so on.

In mathematics we use the word "function" to MEAN a device that shows how one quantity varies with another. Stated in somewhat different words, a FUNCTION is a mathematical object that specifies/shows/evinces the one-to-one RELATIONSHIP between QUANTITIES. Unless this is perfectly clear, nothing that follows will make sense.

The most basic question of algebra is the the problem of how we can REPRESENT, CHARACTERIZE, REALIZE, OBJECTIFY, SPECIFY, such a relationship? It turns out there are a NUMBER of DIFFERENT ways, each of them wonderfully sensible, incredibly useful, and the basis of many further developments.

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please don't shout... ;) –  apnorton Feb 18 '14 at 16:41