What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and where by effective i mean "putting someone in the best position to use a mathematical result as a tool for solving problems"? Please note that the answer "it should come with a better understanding of modular arithmetic" is not a valid answer (even if i support this point of view actually), because that is something that is going come with undergraduate studies (or a really smart high school student). I'm not asking about a magic wand, just some suggestion about what you consider the most insightful examples or effective ideas. Please note that while being an obvious result for those that are used to it, this theorem requires some examples and exercises to be grasped by a beginner. Thanks!
[Math] Best way to introduce the Chinese Remainder Theorem (to a high school student)
congruencesnt.number-theorysoft-questionteaching
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My view is that there are essentially two strands in a first calculus course.
The first is not really concerned with a rigorous presentation; rather it tries to get the main ideas, their interrelations, and uses across.
The second is concerned with the technicalities, showing how abstract mathematics can lead to very useful, interesting, and important results.
This means that we are really working with two different definitions of the derivative. The first is to draw the tangent line and measure its slope. The second is to compute a certain limit. To be sure, the limit is motivated by the tangent approach, but no attempt is ever made to show that the two approaches give the same answer (indeed this can't be proved using the usual definitions, since ultimately the definition of tangent line is based on the derivative).
The MVT is the basis for all proofs that geometric intuition about slopes of tangent lines holds for the limit definition. That is, the metamathematical content of the MVT is that the intuition definition matches the formal definition.
When I realized this, I decided that this point was so subtle that I either have to make a big point of explaining the question or else drop it. This choice varies from class to class.
EDIT: You use the MVT to show that positive derivative corresponds to increasing function. This is obvious from the intuitive point of view, but not from the formal point of view.
This is a good question, given the way calculus is currently taught, which for me says more about the sad state of math education, rather than the material itself. All calculus textbooks and teachers claim that they are trying to teach what calculus is and how to use it. However, in the end most exams test mostly for the students' ability to turn a word problem into a formula and find the symbolic derivative for that formula. So it is not surprising that virtually all students and not a few teachers believe that calculus means symbolic differentiation and integration.
My view is almost exactly the opposite. I would like to see symbolic manipulation banished from, say, the first semester of calculus. Instead, I would like to see the first semester focused purely on what the derivative and definite integral (not the indefinite integral) are and what they are useful for. If you're not sure how this is possible without all the rules of differentiation and antidifferentiation, I suggest you take a look at the infamous "Harvard Calculus" textbook by Hughes-Hallett et al. This for me and despite all the furor it created is by far the best modern calculus textbook out there, because it actually tries to teach students calculus as a useful tool rather than a set of mysterious rules that miraculously solve a canned set of problems.
I also dislike introducing the definition of a derivative using standard mathematical terminology such as "limit" and notation such as $h\rightarrow 0$. Another achievement of the Harvard Calculus book was to write a math textbook in plain English. Of course, this led to severe criticism that it was too "warm and fuzzy", but I totally disagree.
Perhaps the most important insight that the Harvard Calculus team had was that the key reason students don't understand calculus is because they don't really know what a function is. Most students believe a function is a formula and nothing more. I now tell my students to forget everything they were ever told about functions and tell them just to remember that a function is a box, where if you feed it an input (in calculus it will be a single number), it will spit out an output (in calculus it will be a single number).
Finally, (I could write on this topic for a long time. If for some reason you want to read me, just google my name with "calculus") I dislike the word "derivative", which provides no hint of what a derivative is. My suggested replacement name is "sensitivity". The derivative measures the sensitivity of a function. In particular, it measures how sensitive the output is to small changes in the input. It is given by the ratio, where the denominator is the change in the input and the numerator is the induced change in the output. With this definition, it is not hard to show students why knowing the derivative can be very useful in many different contexts.
Defining the definite integral is even easier. With these definitions, explaining what the Fundamental Theorem of Calculus is and why you need it is also easy.
Only after I have made sure that students really understand what functions, derivatives, and definite integrals are would I broach the subject of symbolic computation. What everybody should try to remember is that symbolic computation is only one and not necessarily the most important tool in the discipline of calculus, which itself is also merely a useful mathematical tool.
ADDED: What I think most mathematicians overlook is how large a conceptual leap it is to start studying functions (which is really a process) as mathematical objects, rather than just numbers. Until you give this its due respect and take the time to guide your students carefully through this conceptual leap, your students will never really appreciate how powerful calculus really is.
ADDED: I see that the function $\theta\mapsto \sin\theta$ is being mentioned. I would like to point out a simple question that very few calculus students and even teachers can answer correctly: Is the derivative of the sine function, where the angle is measured in degrees, the same as the derivative of the sine function, where the angle is measured in radians. In my department we audition all candidates for teaching calculus and often ask this question. So many people, including some with Ph.D.'s from good schools, couldn't answer this properly that I even tried it on a few really famous mathematicians. Again, the difficulty we all have with this question is for me a sign of how badly we ourselves learn calculus. Note, however, that if you use the definitions of function and derivative I give above, the answer is rather easy.
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What things like partial fractions decompositions (in resolving rational functions) are really all about is the algebra surrounding the Chinese remainder theorem. Often when I've taught this sort of thing to calculus students, I get them to try to write, say, 5/21 as a sum of integral multiples of 1/3 and 1/7, or something like that, and algebraically we are doing something similar with partial fractions. It's sort of fun thinking about all this methodically, from algebraic first principles: how to do it in a principled way will eventually get one into idempotents of rings $\mathbb{Z}/n$ or $\mathbb{R}[x]/(q(x))$.