Question about my mathematics education, how to strengthen my foundation, etc

[DrStrangelove]

Hey, I'm a first time poster here and I was feeling a bit uneasy about my mathematics preparation and my foundation in math. I'll be entering my senior year in high school in about a month, and I plan on majoring in physics at a decent, rigorous university after high school.

As a junior, I took Calculus AB AP as I could not enroll in BC (because I had not taken honors math courses previously) and I simply did not feel comfortable about my math skills. I was mistaken. I had always been a strong student, and in junior year I really focused on my schoolwork. I ended up taking 6 weighted classes (4 APs and 2 honors) and earned a solid to high A in all of them. In calculus I kept a steady 100% (over 100 towards second semester) in the class and received a 5 on the AP test. This surprised me as I had received a B- in algebra 2 freshman year and I felt that my math skills were terrible (I guess one can attribute it to the teacher or my lack of maturity towards math back then). The fact that I was taking algebra 2 as a freshman was an indication that I was a slightly "advanced" student compared to the average people at the school, but it wasn't too exemplary, especially since most advanced students were taking honors algebra 2.

The next step was precalculus (regular), and in this class I received an A+. What puzzles me is that I went from a b- in a fundamental class to an A+ in a class that draws heavily on algebra skills. Perhaps it was simply a change in my studying habits as I actually went home, sat down, and worked through all of the assigned homework problems? Maybe I had become more mature towards math?

Junior year came and I applied the same studying tactics and genuinely tried in Calculus AB. I ended up with a grade of over 100% due to setting the curve on every test and not missing a single homework assignment.

Here's where my uneasy feelings about my mathematics background begin to reaffirm themselves. I took the SAT (2/3 of the way into junior year) and I scored terribly on the math section (~600). I took the SAT Mathematics Level 2 subject test (at the end of junior year) and scored a perfect 800. The tests cover different mathematics material, and I feel that my lack of adequate preparation in mathematics prior to pre-calculus will come back to hurt me in the long run. How accurate do you think this assumption is? How is it possible that I even did so well in Calculus I without fully knowing all the trig identities, random things from algebra 2, and theorems from geometry perfectly and without skipping a beat? What would you recommend I review in order to feel comfortable about my foundations in math? I was actually considering a physics and math double major as I seem to be able to handle a strenuous courseload, and I have developed a new found respect and admiration for mathematics as a result of taking Calculus AB. I have been trying to get into a Calculus II class at the local community college, but I was not able to enter one during the summer. I am trying again this fall. Anyways, do you think I am overreacting in terms of my lack of confidence in my mathematics foundation, or do I have legitimate concern?

 

[ArcanaNoir]

Study the trig as much as you can stand. Never forget synthetic division (it comes up in other math courses and then is not the time to relearn it). Memorize the formulas to factor AND expand polynomials up to three degrees. (x^3). That will save you so much time. Get really comfortable with absolute values in equations and how to make then come and go according to the rules. The rest of the skills you will hone as you go. My algebra has drastically improved since I've been through calc I-III.
[EDIT] oh, and get a ti-89 or casio class pad (I strongly recommend the class pad). They have symbolic computation (the ability to work with multiple variables). I have found that you will either be allowed to use such a high-function calculator, or none at all. Having symbolic computation will eliminate the chance of algebra errors because you can check each step as you go along. Huge tool.

 

[micromass]

Arcana has pretty good advice. Practise and drill it as long as you need to. Buy a decent algebra book (make it college level) and make all the exercises! You'll only feel comfortable once you made a lot of exercises on the subject, so I suggest you do that.

Here is a site with decent resources: http://hbpms.blogspot.com/2008/05/stage-1-elementary-stuff.html
Take an algebra and trig course and practise hard!

 

[ArcanaNoir]

I forgot to mention! It is imperative to know how to graph functions without asking a graphing calculator!

 

[DrStrangelove]

Thanks for all the advice guys, I really appreciate it.
I've got a Ti-84 Plus (silver ed.), so I'll pick up one of the more advanced ones maybe this year or once I get to undergrad. Would you guys recommend me practicing the basics before moving on to anything else past calc 2? If i take calc 2 this fall I will have the opportunity to take multivariable in the spring and then possibly diff. eq/linear algebra in the summer before college. I've heard that this is unwise, and getting too far ahead isn't a good idea.

 

[ArcanaNoir]

Don't upgrade your 84 unless you feel you need to. Lots of peeps don't have an 89, it's not necessary, just a luxury. Although for less than the cost of one textbook, not a bad buy either. I would move right along and take whatever you can as the opportunity arises. Just try to use any extra time to keep the basic skills fresh. Linear is a good summer course. I just took it this summer with a bunch of high schoolers who came to the university for summer camp. Totally doable. DiffEq I haven't had yet, but I guess how you feel about DiffEq should determine if it's a good summer course. There's bound to be a section on diffEq in your calculus book, or any calc book. I struggled with that section. I would not be inclined to take it in the summer. But, again it's about what you feel comfortable with.

There's nothing wrong with getting ahead if:
1- your grades stay high. Any sign of a b- and you'd be better off if you slow down.
2- you don't have gaps. Don't take all this stuff now and then wait two years to take another math course. Keep it fresh.

 

[blade123]

Study the trig as much as you can stand. Never forget synthetic division (it comes up in other math courses and then is not the time to relearn it). Memorize the formulas to factor AND expand polynomials up to three degrees. (x^3). That will save you so much time. Get really comfortable with absolute values in equations and how to make then come and go according to the rules. The rest of the skills you will hone as you go. My algebra has drastically improved since I've been through calc I-III.
[EDIT] oh, and get a ti-89 or casio class pad (I strongly recommend the class pad). They have symbolic computation (the ability to work with multiple variables). I have found that you will either be allowed to use such a high-function calculator, or none at all. Having symbolic computation will eliminate the chance of algebra errors because you can check each step as you go along. Huge tool.

Hahaha, I actually just relearned synthetic division. I'm applying for a tutoring position at my school and I needed to revisit algebra 2. I wish I would have remembered it, it is fairly useful.

All great advice.

 

[SteveL27]

Thanks for all the advice guys, I really appreciate it.
I've got a Ti-84 Plus (silver ed.), so I'll pick up one of the more advanced ones maybe this year or once I get to undergrad. Would you guys recommend me practicing the basics before moving on to anything else past calc 2? If i take calc 2 this fall I will have the opportunity to take multivariable in the spring and then possibly diff. eq/linear algebra in the summer before college. I've heard that this is unwise, and getting too far ahead isn't a good idea.

If you have the choice, take linear algebra before multivariable calculus. Just as the derivative involves finding a straight line -- in other words, a one-dimensional linear space -- that's a good approximation to a function at a given point; in multivariable calculus, you find an n-dimensional linear space that's a good approximation to a function at a point.

The more you know about linear spaces, the maps between them, and the matrix representations of those maps, the better off you will be in multivariable calculus.

When I took the second-year calc sequence, linear algebra was automatically taken before multivariable calc. You need to know matrices, if nothing else.

As far as the calculators ... I have not been in classrooms since the calculator revolution, but I have to think that if I were teaching calculus I would certainly ban the use of calculators. Oh I'm such a dreamer! Am I being too idealistic? Why not bring Mathematica with you to class and let it solve all the problems? But don't listen to me, just get whatever calculator everyone else tells you to get. I'm sure they're right.

 

[ArcanaNoir]

As far as the calculators ... I have not been in classrooms since the calculator revolution, but I have to think that if I were teaching calculus I would certainly ban the use of calculators. Oh I'm such a dreamer! Am I being too idealistic? Why not bring Mathematica with you to class and let it solve all the problems? But don't listen to me, just get whatever calculator everyone else tells you to get. I'm sure they're right.

I did not use a calculator in Calc I, and although it would have prevented me from accidentally saying 2*3=10 on one unfortunate test, it would have probably defeated the purpose of calc I. However, by the time calc III rolled around, my calculator was a valuable tool. I could set up a triple integral or a polar integral conversion, and in a few seconds know if I had the right set up by asking the calculator what answer my set up gave. The calculator won't set it up for you. I was able to study more problems in less time by not having to spend ten minutes grinding out a long complex integral that may turn out wrong in the end. Also saved time doing cross products of vectors by calculator instead of by hand, since it's an easy thing to understand but tedious, and only a single step in longer processes.

 

[blade123]

If you have the choice, take linear algebra before multivariable calculus. Just as the derivative involves finding a straight line -- in other words, a one-dimensional linear space -- that's a good approximation to a function at a given point; in multivariable calculus, you find an n-dimensional linear space that's a good approximation to a function at a point.

The more you know about linear spaces, the maps between them, and the matrix representations of those maps, the better off you will be in multivariable calculus.

When I took the second-year calc sequence, linear algebra was automatically taken before multivariable calc. You need to know matrices, if nothing else.

As far as the calculators ... I have not been in classrooms since the calculator revolution, but I have to think that if I were teaching calculus I would certainly ban the use of calculators. Oh I'm such a dreamer! Am I being too idealistic? Why not bring Mathematica with you to class and let it solve all the problems? But don't listen to me, just get whatever calculator everyone else tells you to get. I'm sure they're right.

Honestly, even though I had a fancy ti-84 calculator, I really had no use for it in calculus 1. Some tests I wouldn't touch it at all, not even for arithmetic, because there simply wasn't a need.

The only time I really used it was for arithmetic and to check my work. Use calculus to graph a function, then graph it on my calculator to make sure I got everything right.

 

[DrStrangelove]

My teacher simply required a ti-83 or ti-84 for the class as the AP test requires one of these for the calculator section of the test. As a result he split tests up into two days, one day allowing calculator and the next day without one.
In addition, it was cool having one for when we went over polar coordinates, and it was useful for graphing in general rather than arithmetic. Teachers should take this cautiously, however, as constantly using the calculator for graphing can turn into a crutch.