Why should pre-service math basic school teachers take calculus courses? Should this course be different from calculus courses offered to engineering?
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asked Aug 26, 2022 at 12:59
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7$\begingroup$ Teachers of children of which age? Also, are you interested in or assuming a particular country, as the claim does not hold everywhere. $\endgroup$– TommiCommented Aug 26, 2022 at 14:46
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5$\begingroup$ Since many of these teachers may be teaching future engineers it might be nice for them to have some insight into the way algebra is required for understanding the most basic math course in most engineering degree plans. Also, to be a math teacher, it should be expected you like math, and, if you like math, the problem of taking calculus and other higher math ought to be no problem at all. But, of course, society will take the easy way out and produce less and less qualified teachers with each passing generation. Even now I see my state doubling down on its pattern of ignorance. $\endgroup$– James S. CookCommented Aug 26, 2022 at 23:52
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2$\begingroup$ In North America, requiring pre-service primary teachers to learn calculus would eliminate almost the entire incoming labor pool. bit.ly/3RBSlWV Pre-service primary teachers should focus on playing math games, math stories, rich math experiences, how to manage group vs solo work, how to make paper-and-pencil drills more than rote monotony (e.g. number strings, elaborations, compare/contrast, etc.), having fun with math, and, in terms of content, fractional arithmetic. This would be a colossal victory. $\endgroup$– BravoMathCommented Sep 5, 2022 at 3:34
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2$\begingroup$ @BravoMath It is generally better to avoid URL shorteners. For future readers, the link points to the following article: Elementary teachers’ weak math skills spark mandatory crash courses (archived on the Wayback Machine). $\endgroup$– BrahadeeshCommented Sep 5, 2022 at 7:38
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1$\begingroup$ @BravoMath: I agree, but it's not clear what the OP means by "basic school teacher". Just from the U.S. perspective (to say nothing of that from a hundred or so other countries), the OP could mean primary school (typically grades 1-3), elementary school (typically grades 4-6, but could be 1-5 or 1-6), middle school (grades 6-8 or 7-8), high school (grades 9-12, and sometimes 10-12). In fact, it's not clear to me whether "basic" is intended as an explicitly descriptive adjective or an extra word used to emphasize the distinction between school mathematics and university mathematics. $\endgroup$– Dave L RenfroCommented Sep 5, 2022 at 10:56
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6 Answers
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When the undergraduate curriculum for preservice teachers does not include mathematics beyond middle school (say grades 6-9, algebra 1 or algebra 2 in the US), then what can happen in some cultures is that preservice teachers with negative attitudes about mathematics or with low mathematical self-esteem tend to end up being those who gravitate towards teaching elementary school. This is turn perpetuates negative societal memes about mathematics, because those teaching mathematics to formative young children hold these negative attitudes.
So we can argue that the existence of undergraduate pathways to certification that do not require “college algebra” or precalculus or calculus will only perpetuate negative societal memes about mathematics.
Of course simply requiring preservice teachers to have mathematical competence beyond basic algebra is not in itself sufficient to counter this problem created by self-selection of who teaches elementary school based on their mathematical self-efficacy. Teacher preparation programs must teach precalculus or calculus in such a way that mathematical self-efficacy of the future teachers increases, or is at least not ruined.
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$\begingroup$ "Teacher preparation programs must teach precalculus or calculus in such a way that mathematical self-efficacy of the future teachers increases" Or, rather, in such way that those who cannot master mathematics even at such elementary level (remember, it is 2023 AD, not 212 BC!) would never be able to get their credentials to go into the teaching profession. $\endgroup$– fedjaCommented Jan 4, 2023 at 21:05
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As Tommi commented, the question is a bit imprecise, but attempting to address it:
I don't think lower level teachers (early grades) [IN GENERAL] are required to take calculus. If they are as part of a general ed degree, the only rationale would be for their general knowledge as university students. And yes, a "calculus for non-STEM" would make total sense here.
I do think there is some benefit to going a "step higher" than what you teach. So, it makes sense for high school and intermediate math teachers (say ages 12-18ish) to have had calculus.
2.1. For one thing, you solidify a lot of algebra skills by doing calculus! Feynman noted this in his famous remarks.
2.2. Also, there is sort of a track of pre algebra to algebra to second year algebra to trig to pre-calc to calc. (US basis, and the exact names vary, but my sister went to school in Europe and was very similar.) So, at least you've seen the track you're heading kids on. [Geometry is sort of off the track, but this is irrelevant to my point.]
- There is probably a minor benefit in ed research in knowing calculus given the use of stats in papers and the like. But really, even for masters in ed, I think a baby algebra-based stats course is fine. So, wouldn't justify the pain of making kindergarten teachers do Thomas...but just including the rationale, to be comprehensive.
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4$\begingroup$ Concerning knowing (or at least working through) material a step beyond what you teach, an analogy I like to make is that people probably would not want their young children to be taught how to read by someone with the reading comprehension level of an elementary school student. $\endgroup$– KCdCommented Aug 28, 2022 at 12:52
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$\begingroup$ @KCd, "reading comprehension level of an elementary school student" is a construct invented in English-language countries by "educators" who use "look and say" method of teaching reading instead of phonics. In other countries, you either can read or you cannot, there are no levels to it (maybe there are in the countries that use logograms). The litmus test is an ability to read a word you haven't seen yet in written form. So yes, anyone who is able to read can teach reading using a decent textbook. $\endgroup$ Commented Aug 28, 2022 at 16:50
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7$\begingroup$ @RustyCore "reading comprehension" is not just about figuring out which combination of letters represents which word. It is about understanding the meaning of text. This develops over a lifetime. $\endgroup$ Commented Aug 29, 2022 at 9:18
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$\begingroup$ @StevenGubkin Reading comprehension is an artificial construct that has nothing to do with reading. On the most basic level, reading is converting of written symbols into sounds, which one recognizes as words. Then one comprehends these sounded out words. If one does not know what, say, "apogee" means, it does not matter whether he read it or heard it. Comprehension has nothing to do with reading. Reading is purely the process of converting symbols to sounds, period. It is a different story that after one masters sounding out he starts reading silently. $\endgroup$ Commented Sep 2, 2022 at 15:15
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3$\begingroup$ @RustyCore: Could you link any research which confirms your claim that the ability to read something is not related to comprehension and meaning? I think most people will find "Charles the off airplane at de took airport Gaulle" much more difficult to read than "The airplane took off at Charles de Gaulle airport". Obviously this is due to the messed up syntax of the first "sentence". But if you don't know any of those words, you can't even know that the syntax is messed up - so it seems very likely that our knowledge of words does influence how easily we are able to read something. $\endgroup$ Commented Sep 5, 2022 at 23:06
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You’ve got to understand where Calculus sits in the history of Math Education.
What we know as the math curriculum was essentially set up by the Committee of Ten, a group from 1892. It’s an oversimplification, but essentially they set Calculus as the capstone course and arranged Algebra, Geometry, everything really, to focus on the skills one needs to learn Calculus in college. Think about it: why would students need to learn how to manipulate and verify trigonometric identities if it weren’t for Calculus? Why so much on sequences and series? There are tons of examples of topics that are essentially unjustifiable to students because they’ll never see them again, never use them in real life, &c., except if you think about what you need to take two years of Calculus. If that’s your goal (and, again, it was) then pretty much anything in the curriculum has a reason for being there. That’s the history.
So if you’re a math teacher, even today with things like Statistics crufted into the curriculum, you’re basically teaching Pre-Calculus. Algebra I is Pre-Calculus. Geometry is Pre-Calculus. It’s all pre-Calculus. And guess what—take serious statistics in college and you’ll discover it’s pretty much Calculus at its core.
Knowing that, is there a course that should be taught to all future math teachers? A branch of math that all math teachers need to experience? Even if you never think you’re going to teach anything higher than Pre-Calculus Algebra I, what course should be unquestionably in your preparation? I mean, there are math courses that you can take that have no calculus in them like this one, this one (both excellent bases for courses) or things like this blah book or especially this travesty, but you should think about those in addition to a calculus sequence.
Not to mention that Calculus is a human achievement that allowed the modern world to happen. If you think art and music and Literature are worthwhile for everyone (and I so do) then Calculus should clearly be on that list. I realize it takes a lot of preparation, but it’s a beautiful thing, as is lots of the math leading up to it, especially if taught well. And a prerequisite to teaching it well is having it in your teaching toolbox.
Note I’m not contending for the standard two years of Calculus like math majors or engineers take. You don’t need that much detail. But two semesters of applied calculus (especially if your teacher uses this book) feels to me like exactly what’s needed. Because it’s all Pre-Calculus, see?
answered Aug 29, 2022 at 20:09
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Others have already addressed the first half of the question, and complained about its imprecision.
The second half is this: "Should this course be different from calculus courses offered to engineering?"
Well, more broadly, too many different types of students are currently thrown together in the same one-year calculus sequence in the US. It's absurd to expect people to learn this much calculus in order to become dentists, for example, but that is effectively what is required in places like California. E.g., a community college student who says they want to go to dental school will be steered into calculus and calc-based physics, so that they can apply to UC's in the biology major.
The question has tags for both primary and secondary education. If someone is going to become a primary educator in the US, then they are not required to take calculus. If they're going to become secondary-school math teachers, then certainly they should take a solid one-year calculus course of the type that currently exists in the US. They'll need that background because they may be called upon to teach AP calculus. But realistically, may US high schools are drafting their PE teachers to teach algebra.
In an ideal world, I would suggest two or three different calculus tracks for freshmen in the US, with only the highest-level track doing epsilon-delta proofs and methods of integration. Preservice high school math teachers would take this track. The lowest track would be a one-semester survey of calculus.
I think the reality is that this will never happen. Math departments don't want to lose the enrollment that they would lose if lots of students only had to take one semester of calculus. Biology departments started instituting a calculus and calc-based physics requirement back in the 90's precisely because they wanted the hardest possible requirements in order to filter prospective majors.
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When teaching, one needs to know what one is hiding.
A topic often taught in elementary schools is finding square roots of integers. Beyond guessing above and below, the simplest method is that of Newton, and this can't be understood without calculus (even if in the particular case of square roots it can be reduced to an algorithm that doesn't use calculus).
In middle school algebra one works with polynomials with real coefficients. Many of the properties of such polynomials are best understood using calculus rather than algebra. Even someone who is not teaching such properties to students needs to know them and understand their origin. Some basic exercises in a calculus class should include: prove an odd degree real polynomial has a root and show that a degree $n$ real polynomial with $n+1$ distinct roots is identically zero. I'd be wary of someone teaching elementary algebra who can't prove those things.
Many secondary school curricula include some elementary probability and statistics. Almost nothing meaningful can be said about a Gaussian without using some basic calculus.
The basic properties of trigonometric and exponential functions and logarithms are all proved using calculus (more specifically the existence and uniqueness theorem for ordinary differential equations and convergence results for power series). These are central objects of study in high school math. Who teaches them should certainly know how to justify all their basic properties. For example, it's important to realize that the different definitions of the exponential function - as solution to an ODE, as a power series, as a limit - do not at all obviously yield the same function - or, more relevantly, that the property $e^{a + b} = e^{a}e^{b}$ is not obvious ... Students taught by those who know no calculus might reasonably guess that $\ln(ab) = \ln(a)\ln(b)$ ...
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Well, I found this scientific reference today:
Here is another study:
eAspects of Calculus for Preservice Teachers BY Lee Fothergill
answered Sep 4, 2022 at 21:29
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$\begingroup$ If you're going to go back 100 years, then many articles can be found in U.S. journals such as Mathematics Teacher and American Mathematical Monthly and Mathematics Magazine and Mathematical Gazette and several other such journals in other countries. And somewhat related is Rosenberger's 1921 Ph.D. dissertation The Place of the Elementary Calculus in the Senior High-School Mathematics and Suggestions for a Modern Presentation of the Subject (Columbia University). $\endgroup$ Commented Sep 5, 2022 at 10:46