# Tagged Questions

Mathematics education consists in the practice of teaching and learning mathematics, along with the associated research. Research in mathematics education concerns the tools, methods and approaches that facilitate the practice of mathematics or the study of this practice.

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### Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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Someone recently asked me why a negative * a negative is positive, and why a negative * a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume $(-x) * (-y) ... 3answers 1k views ### Motivation behind the definition of GCD and LCM According to me, I can find the GCD of two integers (say$a$and$b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the ... 1answer 2k views ### Is Lagrange's theorem the most basic result in finite group theory? Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group ... 152answers 34k views ### What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ... 1answer 612 views ### Alternative construction of the tensor product (or: pass this secret) The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ... 21answers 5k views ### What is your favorite application of the Pigeonhole Principle? The pigeonhole principle states that if$n$items are put into$m$"pigeonholes" with$n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ... 18answers 7k views ### Examples of mathematical induction What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form$\bullet+\cdots\cdots\cdots+\bullet$... 33answers 33k views ### Pedagogy: How to cure students of the “law of universal linearity”? One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$\frac{1}{a+b} ... 4answers 903 views ### What are or where can I find style guidelines for writing math? I am a scientist writing my first manuscript with a substantial amount of mathematical methodological documentation. I am using LaTeX, but this is not my question. I would like to find a list of ... 11answers 5k views ### How do you define functions for non-mathematicians? I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ... 9answers 3k views ### Motivating infinite series What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, ... 9answers 26k views ### Why is \pi equal to 3.14159…? Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any ... 19answers 33k views ### How do I explain 2 to the power of zero equals 1 to a child My daughter is stuck on the concept that$$2^0 = 1,$$having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough. How would you explain the ... 7answers 1k views ### Can this standard calculus result be explained “intuitively” Recently I stumbled upon someone who said he wanted to understand why \arctan x = \int\dfrac{dx}{1+x^2} At first I was confused. This is an easy result in any integral calculus course. But then he ... 4answers 1k views ### Fun math for young, bored kids? For 6 months, I'll be organizing, as part as my volunteer work in an NGO, math classes with small groups (~10 students, aged 16 or 17). These classes are not compulsory, but students willing to stay ... 4answers 891 views ### Linear combinations of sine and cosine If you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift.$$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + ... 17answers 8k views ### Interesting “real life” applications of serious theorems As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ... 11answers 2k views ### Puzzles or short exercises illustrating mathematical problem solving to freshman students At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ... 15answers 5k views ### How can I introduce complex numbers to precalculus students? I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ... 24answers 9k views ### “Negative” versus “Minus” As a math educator, do you think it is appropriate to insist that students say "negative$0.8$" and not "minus$0.8$" to denote$-0.8$? The so called "textbook answer" regarding this question reads: ... 14answers 783 views ### Examples where it is easier to prove more than less Especially (but not only) in the case of induction proofs, it happens that a stronger claim$B$is easier to prove than the intended claim$A$(e.g. since the induction hypothesis gives you more ... 6answers 2k views ### How to tell$i$from$-i$? Suppose now we are trying to explain to students who do not know complex numbers, how do we distinguish$i$and$-i$to them? They will object that they both squared to$-1$and thus they are ... 6answers 2k views ### Why do we need to prove$e^{u+v} = e^ue^v$? In this book I'm using the author seems to feel a need to prove$e^{u+v} = e^ue^v$By$\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$Hence$e^{u+v} = e^u e^v$But we know from basic ... 5answers 938 views ### Alternative set theories This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ... 10answers 514 views ###$2\times2$matrices are not big enough Olga Tausky-Todd had once said that "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." There are, however, assertions about matrices that are true for ... 9answers 804 views ### Motivating implications of the axiom of choice? What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ... 6answers 2k views ### Why Doesn't This Series Converge? I am teaching a Calc II course and came across the following series when finding the interval of convergence for the Taylor series of$f(x)=\sqrt{x}$centered at$x=1$: $$\sum_{n=2}^\infty ... 14answers 10k views ### Identification of a quadrilateral as a trapezoid, rectangle, or square Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to ... 32answers 7k views ### What are some conceptualizations that work in mathematics but are not strictly true? I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ... 17answers 3k views ### What are some good ways to get children excited about math? I'm talking in the range of 10-12 years old, but this question isn't limited to only that range. Do you have any advice on cool things to show kids that might spark their interest in spending more ... 12answers 11k views ### How to convince a math teacher of this simple and obvious fact? I have in my presence a mathematics teacher, who asserts that$$ \frac{a}{b} = \frac{c}{d} $$Implies:$$ a = c, \space b=d $$She has been shown in multiple ways why this is not true:$$ ... 4answers 6k views ### Teaching Introductory Real Analysis I am currently helping teach an introduction to real analysis course at UC Berkeley. The textbook we are using in Rudin's "Principles of Mathematical Analysis" (aka baby rudin). I am trying to find ... 3answers 2k views ### Why the emphasis on Projective Space in Algebraic Geometry? I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ... 19answers 18k views ### How do I convince my students that the choice of variable of integration is irrelevant? I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a ... 2answers 6k views ### Relearning from the basics to Calculus and beyond. Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ... 7answers 2k views ### Quotient geometries known in popular culture, such as “flat torus = Asteroids video game” In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ... 26answers 4k views ### How to teach mathematical induction? Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of ... 12answers 2k views ### explaining the derivative of$x^x$You set the following exercise to your calculus class: Q1. Differentiate$y(x) = x^x$. A student submits the following solution: Let$g(a)=a^x$and$f(x)=x$. Then$y(x) = g(f(x))$, so by ... 16answers 4k views ### Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect. For example, ... 6answers 279 views ### Can one show a beginning student how to use the$p$-adics to solve a problem? I recently had a discussion about how to teach$p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to$p$-adics because no one told them why the ... 9answers 967 views ### Sources of problems for teaching/tutoring young mathematicians I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, ... 3answers 1k views ### Books that develop interest & critical thinking among high school students I heard about Yakov Perelman and his books. I just finished reading his two volumes of Physics for Entertainment. What a delightful read! What a splendid author. This is the exact book I've been ... 7answers 772 views ### Exciting games and material to motivate children to math We are a group of people trying to motivate children, especially living in the countryside, to science and math. We have different activities with children such as doing scientific experiments and ... 15answers 12k views ### What concepts were most difficult for you to understand in Calculus? [closed] I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in ... 3answers 820 views ### Infinite Series: Fibonacci/$2^n\$

I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner) In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... each term ...
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### Differential Geometry of curves and surfaces: bibliography?

Dear all, next year, I will probably teach a one-semester course of Differential Geomtry of curves and surfaces. Its content must be something along the lines of the first four chapters of Do Carmo's ...
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### Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...