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I have been teaching trigonometry for many years and, surprisingly, did not know of this issue. I read the responses and the references therein, and I am putting my thoughts on this issue here. The question is how to approach trigonometry, right triangles first and then unit circle; or circle first and then triangle. I think the answer depends on the ...


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Think of the naturals instead of the reals. We learn to count around age four. We learn a lot of properties of arithmetic, like addition, subtraction, multiplication, division, prime numbers, unique factorization, etc. without reference to any axioms. This is just how numbers work. Similarly we learn how reals work before we see a careful construction of ...


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You are confusing definitions and models. A definition of real numbers is a set of axioms they obey. Different such sets exist, but they can be shown to be equivalent. A model of real numbers is a construction like Dedekinds, which is a set of sets of sets etc. that is carefully constructed so that they obey one such set of axioms. To work with real ...


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Read Dirk's answer - we take the axioms of rational numbers (which are just fractions of integer numbers, so easy to understand) and get quite a few nice theorems. We even have Cauchy sequences (which are kind of sequences with limits but not quite, but the difference was never mentioned). In my Analysis I, the professor then went to prove the Intermediate ...


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Since real numbers include whole numbers, rational numbers, and irrational numbers (almost the entire landscape) I’m uncertain why you think teaching does not occur. Does it not begin very early with counting and learning numbers? It isn’t until later that these numbers are labeled as real, when needing to distinguish from imaginary as introduced in pre-...


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This answer may seem off-topic, since this is "Math Ed" SE. But a similar question can be asked about many scientific concepts. Based on observations of the world around us animals (including humans) have built up heuristic models of physics, biology and psychology. In order to be a useful creature, one must have some predictive models, even if they are ...


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"In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined." There is no reason, this has to be true. It is just asserted. Two examples have been given where significant content can be learned without strict axiomatic buildup of the subject: arithmetic and calculus. We could come up with others (set theory, ...


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Introductory mathematics is not done with formal definitions. This is because most people at the corresponding age (kids) don't know what the words "formal" or "definition" actually means. This only come to play much later. You start teaching kids mathematics by showing them how to count things up to ten using their fingers. Then, you teach addition with ...


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At German universities, one of the first lectures in mathematics is "Analysis 1" which is a kind of "rigorous calculus" and there one always proceeds more or less like this: We start with an axiomatic approach to the real numbers. In short: The real numbers are a complete, ordered, Archimedean field. In practice we first introduce the axioms of a field, ...


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But as far as real numbers were concerned, they were taught very early without any clear definition was even given. Were they? I would say that usually no teacher really speaks of real numbers to students until the first calculus lesson. In almost all pre-calculus exercises, students only meet rational numbers, in the form of fractions or decimal ...


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It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. 376) calls the Fundamental Assumption of School Mathematics. In terms of the nitty-gritty of classroom instruction, real numbers are handled in K–12 by what ...


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Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of calculus predated formal definitions of the real numbers (Dedekind cuts, Cauchy sequences) and a formal definition of continuity, individuals must first make ...


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I believe that, unlike many other mathematical concepts (especially more abstract ones), even fairly young people have a basic understanding of distance, including fractions to any degree of units of measurement (e..g., inches, feet, miles, etc., in the imperial measurement system and/or centimetres, metres, kilometres, etc., in the metric measurement system)...


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This question is mixed with a lot of speculation and commentary, so I'm going to try to focus on the questions. Should the limits of one system of set theory be the limits of a student's mathematical world? No. The idea that a system of set theory should be a limit to mathematical ideas seems like poor pedagogy (as arbitrary restrictions usually are), ...


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Set theory has no important implications for 99% of normal mathematics, and the difference between, say, ZFC and NF has even less. Mathematics isn't generally written in the language of ZFC, it's written using more basic ideas and notation that are the same in other foundational systems. If I'm teaching freshman calculus and I talk about $\{x|x>0\}$, ...


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The following aren't the same thing: The meaning of an expression An option for computing the value of the expression A rigidly imposed sequence of instructions to be obeyed to compute the value The following example may help you more than your student: Define x = 1 if there are infinitely many twin primes, and x = (- 1) otherwise. If what is inside ...


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Your question relates to a 5-year old child. This should be taken into account (e.g., no point in talking about algebraic manipulations). Parentheses are here to help us; and if they do, there is no reason to remove them. I would still tell a child that we usually prefer expressions that have as fewer symbols are possible, and explain that sometimes, ...


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