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Algebra vs. Calculus pedagogy

  1. Dec 16, 2013 #1
    Out of curiousity,

    Why did it take roughly 5-6 years to teach Algebra in the K-12 system, then 1.5-2 for calculus

    6th grade prealgebra
    7th grade more prealgebra
    8th grade even more prealgebra
    9th grade algebra I
    10th grade algebra II
    11th grade basically precalculus stuff, which isn't very far removed for algebra II

    I swear 6th- 8th grade was a repeat of the same material. some geometry, probability and trigonometry is thrown into the mix. This sums up my experience of K-12 math education. then calculus is taught in 3 semesters (1.5 years).

    I mean it seems like I literally spent 6 years to learn algebra and 1.5 to learn calculus, yet calculus is far more encompassing and complex. I don't know how it took so long to teach us how to do logarithms, exponentials, and algebraic equations..this is a quandary.
     
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  3. Dec 16, 2013 #2
    I think that most of the problems that students have with calculus stem from a weak grasp of the fundamentals. Often, it's not the calculus that is the problem but a lack of basic mathematical fluency. The algebra is what gets in the way.

    I look at it like this. You spend a lot of time learning English to a high standard and then you can learn quickly to write essays on nearly anything. I think algebra is to maths as English is to History, Law, Sociology etc. I did a postgrad in social science. I felt that it was quite easy because although some of the ideas were subtle, half the challenge was writing stuff down such that it can be understood. If you can already do this then you're half way finished.
     
  4. Dec 16, 2013 #3

    jgens

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    I imagine this is partly due to cognitive development. At a young age it just takes time to really learn some of the abstract concepts in algebra. By the time you are finishing high school and taking calculus, the brain has developed bunches, and most people can probably pick up on the concepts quicker. No citations here as this is just a hunch.
     
  5. Dec 27, 2013 #4

    Redbelly98

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    My own experience was to spend 4 years on algebra (in grades 7,8,10,11, with a "break" in grade 9 for geometry) and 2 years on calculus.

    But to answer your question, here are a couple possible reasons -- strictly my opinion:

    1. Those algebra courses have kids who will never take calculus, even the advanced-track algebra courses. Plus a 7th/8th grader is a lot younger than a 12th grader or college freshman. The repetition of material through those years, at increasingly advanced levels, may be necessary based on the level and maturity of the students in those courses.

    2. Doesn't calculus have just two basic concepts? Differentiation and integration -- which turn out to be inverse operations of each, so you might even argue that there is just one single concept. So perhaps it is not so surprising that less time is spent on calculus than algebra.

    Just my 2 cents.
     
  6. Dec 27, 2013 #5

    QuantumCurt

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    I agree with what others have said. It's not like algebra just goes away once you hit calculus. If anything, there's even more of it. I've seen a lot of people really struggling in my calculus classes, and it isn't because of the calculus itself, it's because their algebra skills aren't up to par. The actual concepts of differentiation and integration aren't really that difficult right off the bat. There's a huge mess of algebra involved in both at the introductory level though, before you learn things like the power rule, chain rule, or fundamental theorem of calculus. When you're learning these concepts by the limit definitions and such though, the algebra involved can be formidable.

    I think it's necessary in a lot of ways to spend more time on algebra, just because it's so important. Once you know algebra like the back of your hand, calculus is going to be significantly easier. If you're fluent in algebra, you aren't going to have to work through the process by using 'the rules of algebra' as such. You just intuitively understand what it is, and how it works. Learning algebra from a formal standpoint is important though, because without learning it rigorously, you aren't going to learn it to the point of fluency.
     
  7. Dec 27, 2013 #6

    D H

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    … and limits and sums. Those side excursions into developing the concept of limits, sequences, and sums reveal infinitesimal calculus for what really is: It's algebra.

    Woopydalan, you didn't stop learning algebra when you started calculus. The infinitesimal calculus is a side branch of algebra. If you continue studying math after those first two college level calculus classes, you are not done studying algebra. Not even close. You'll take analysis (real and complex), introduction to linear algebra, and introduction to modern (or abstract) algebra. That these upper level mathematics courses are still called introductory courses means that there is a lot more algebra to come.
     
  8. Dec 28, 2013 #7

    mathwonk

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    because precalculus teachers are clueless?
     
  9. Dec 30, 2013 #8
    From perspective of students I can add that maybe calculus is not necessary to be learned for all of those six years. If you want students to learn calculus then at this point there should a concrete problem that requires the effort of learning it. In high school and beyond you definitely need it to understand physics and math itself.

    However most of the day to day mathematics can already be solved by using ordinary arithmetic and algebra learned in elementary schools and it's fair that most students should have good grasp of it as they will be using it all the time. Moreover, I think the amount of time spent in school and lesson is shorter in the elementary years, which explain why it seemed to take longer overall.
     
  10. Dec 31, 2013 #9

    mathwonk

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    there was in my day very little demand on kids in 6th grade and 7th grade. i skipped 6th grade and learned all the math they did in that grade in about 5 minutes in one afternoon. i.e. in those days they spent 6th grade teaching how to multiply and divide fractions. hello.... (a/b)(c/d) = ac/bd, and (a/b)/(c/d) = (a/b)(d/c) = ad/bc. that's it. i suspect it ain't a lot different today.
     
  11. Jan 1, 2014 #10
    When I was in sixth grade the most advanced thing that I remembered was geometrical problem related to circle, so we had to find the area given half circle, or the radius given the area and etc. There were also other simple pythagorean construction but the circle was probably the heaviest one in calculation because of that magnificent ##\pi##.
     
  12. Jan 9, 2014 #11
    I think age is by far the biggest factor. Consider learning a second language, for example. I spent about 8 years learning basic French (starting early in elementary), but I'm pretty sure that an adult who took 1 year of French at a college level would be more proficient in French than I ever was. It's exactly the same material, but being older (more mature, fully developed) makes a huge difference.

    There's also the fact that some kids just hate math. I (and a few others in my class) could have handled a much harder math sequence throughout elementary school and high school. But if we had done that, most of the class probably would have been overwhelmed and they wouldn't have learned anything at all.
     
  13. Jan 9, 2014 #12
    Some of it surely comes down to the difference between high school and college students, the latter choosing to study calculus (for longer than a single required semester at least) because it's necessary for their major, instead of to boost their GPA to look competitive for colleges.

    As for taking six years to learn algebra, a lot of the posts have been correct that students struggle with calculus because of weak algebra skills, but this can be taken a step further; students struggle with algebra because they don't know arithmetic. Someone who can't add fractions taking six years to learn algebra (and many or most never learn) isn't surprising. Age comes into play as well, but I think moreso because algebra is being taught earlier and earlier, so the students' mathematical foundation is less and less developed.
     
  14. Jan 9, 2014 #13

    QuantumCurt

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    That's very true. I work in my schools tutoring lab, and I tutor a lot of people in algebra. Some of the things I see them struggle with most are basic things like the properties of fractions and basic properties of multiplication and division. They get into things like long division of polynomials, and they struggle with it horrendously because they can barely even use long division to divide two numbers. I blame a great deal of the issues that people have with math on calculators. Calculators are surely an awesome thing...I love my TI-84+, but you've gotta understand how to do basic arithmetic without them. People have to use calculators for basic multiplication that they should have had memorized for years. Memorization certainly isn't everything in math, but it's still important. Things like memorizing the basic multiplication table should be done very early on. People struggle like crazy with things like negative signs, and the basic commutative/associative properties of addition/multiplication. That's going to create problems that are just going to keep following.
     
  15. Jan 9, 2014 #14
    Because algebra is a much larger field of math (and vague-er term).

    Taken literally, if someone says they are "doing algebra" I would have no idea what they were doing.

    If they were doing calculus, I'd have an idea, I think.
     
  16. Jan 24, 2014 #15

    mathwonk

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    yes, the 3 fields of pure math are algebra, geometry, analysis, so the "calculus" term analogous to algebra is analysis. these deal with finite operations and symmetry, shapes, and infinite processes, i.e. sequences of convergent approximations. (calculus on the other hand, in the sense of finding tangents and areas, can be done largely without limits.)
     
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