# Questions tagged [mathematical-pedagogy]

for questions on general considerations and problems of teaching mathematics, i.e., issues specific to teaching mathematics yet relevant to various contexts and courses.

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### Whence the "everything is linear" phenomenon, and what can we do about it?

$$\color{red}{(a+b)^2 = a^2+b^2}$$ $$\color{red}{\sqrt{x^4+y^4} = x^2+y^2}$$ $$\color{red}{e^{t^2+C} = e^{t^2}+e^C}$$ I've observed this phenomenon -- wherein, implicitly, students say, "...
An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=... 44 votes 28 answers 11k views ### Good, simple examples of induction? Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ... 33 votes 14 answers 2k views ### Revisiting topics from previous courses [closed] I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.) ... 26 votes 9 answers 2k views ### How can mathematics educators encourage innovation and creativity? Almost by definition, innovation requires that things be done differently than established custom has it, and comes from the young more often than from the old. In a field as old and established as ... 19 votes 12 answers 2k views ### Mathematical problems for preschoolers What are some mathematical problems that are feasible for preschool children to stimulate their intellectual development? There are multiple natural laws that are not apparent to them, for example: ... 15 votes 4 answers 1k views ### When should I say "nothing is as it seems"? "Intuition" is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true ... 36 votes 24 answers 6k views ### Imbuing a six year old with a sense of mathematical wonder My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home ... 26 votes 4 answers 2k views ### Lesson plan to self-teach real analysis to student with comp-sci background For my background, I'm a software engineer currently studying for his master's degree in information security. But when that's all done, I plan on going back to mathematics to keep me busy. But with ... 21 votes 2 answers 4k views ### Example "bad proofs"? As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the ... 51 votes 3 answers 10k views ### How do blind people learn mathematics? I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often taught using a lot of visualization; how does this work with blind people? My ... 27 votes 8 answers 6k views ### When should we first teach variables in school math? And how? From a pedagogical point of view, when is the "right" moment to introduce letters and variables to school children? Let's say we taught arithmetic, the four operations, did computation exercises, or ... 9 votes 3 answers 569 views ### How many problems do we have to do as undergraduate mathematicians in order to learn a subject? I'm wondering how many problems are needed in order to learn a subject, let's say Calculus of Several Variables. We know that the professors often assign us a list of problems to solve as homework, ... 41 votes 2 answers 2k views ### What does math education research know about difficulty vs. effectiveness? I've asked basically the same question previously on on math.SE, then cogsci.SE without much response, surely here is the place to ask this. As anecdotal evidence is plentiful, but unfortunately ... 37 votes 7 answers 5k views ### A Lexicon of Math Mistakes Neil Postman wrote an interesting (and freely available) article called "The Educationist as Painkiller." I highly recommend you read the article for your own enjoyment and as a background to this ... 21 votes 6 answers 2k views ### Is there a good way to explain determinants in an elementary linear algebra class? Many colleges offer an an elementary linear algebra class for sophomore math, science, and economics majors. Such a class typically covers a chapter on determinants, including the following aspects: ... 19 votes 6 answers 12k views ### Good examples of proof by contradiction? In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ... 16 votes 5 answers 1k views ### Is Peer Instruction suited to mathematics classroom? Peer Instruction is a method developed by Eric Mazur in Harvard, designed with a student-centered approach in mind. In a nutshell, the core of the method is that when presented with a problem, ... 13 votes 3 answers 1k views ### Moving From Rote Learning To Creative Thinking My mathematics education was essentially rote, you learned the formulas and applied them almost algorithmically to the problems you were presented with; the teacher dictated a method and you followed ... 12 votes 3 answers 755 views ### How to teach a student algebra who misses too much previous knowledge? I am now tutoring a student in Grade 9, who falls behind in math study. He lacks the basic understanding of operations and inverse operations, and have trouble dealing with negative numbers and ... 6 votes 1 answer 682 views ### Propositional and predicate logic, with quantifiers: Is there any research when it is ideal to explicitly teach in mathematics education? In terms of helping students to understand propositional and predicate logic, with quantifiers, is there any research regarding when it is most advantageous for students studying mathematics, to first ... 95 votes 18 answers 19k views ### Unique candidate that fails In the comments to David Speyer's answer here, he points out that "the distinction between 'if there is a formula, it is this one' and 'this formula works' is subtle." Does anyone have any simple, ... 64 votes 13 answers 9k views ### How to get past the "mystique" of Maths This question is primarily discussing maths education for adult learners, on courses teaching engineering mathematics at an undergraduate level. These students generally never set out specifically to ... 46 votes 9 answers 3k views ### Knowing mathematics does not translate to knowing to teach mathematics. Why? Many brilliant mathematicians seem to make average or even poor classroom teachers. Is this an accurate assessment? Has there been any research to explain the phenomena? What is the difference ... 37 votes 4 answers 3k views ### Taxonomy of bad proofs I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in ... 35 votes 15 answers 9k views ### Justifications for: Why learn mathematics? I wonder how you teachers walk the line between justifying mathematics because of its many—and sometimes surprising—applications, and justifying it as the study of one of the great ... 29 votes 6 answers 937 views ### How can we help students who are very anxious about math? In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier ... 27 votes 2 answers 1k views ### What methods successfully identify and eliminate severe math anxiety? What methods are effective in identifying and eliminating severe math anxiety, this most terrible and unfortunate part of modern mathematics education? This question is not about ordinary math anxiety ... 26 votes 7 answers 4k views ### Why are we so careful in saying that dy/dx is not a fraction? Calculus instructors are mostly very careful to explain that$\frac{\mathrm{d}y}{\mathrm{d}x}$is not a fraction, and multiplying both sides of an equation by$\mathrm{d}x$is nonsense, wrong, or evil.... 23 votes 2 answers 1k views ### Is Knuth's suggestion on teaching calculus a good idea? Note: I myself am not a math educator, though I plan to be one someday. In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A ... 19 votes 3 answers 2k views ### Should I teach Laplace Transforms? How much? My question is in the title. Let me elaborate and give some context: I'm teaching a first differential equations course, essentially for engineers, at the university. I'm developing the syllabus ... 18 votes 1 answer 2k views ### Is metacognition ever bad? Metacognition seems pretty universally positive. I'm wary of viewing it as such. Aside from the obvious criticisms like "you can't learn to ride a bicycle by thinking about and writing a 200 page ... 18 votes 5 answers 1k views ### What is a variable? There are two kinds of answers I'm looking for: What do students think a variable is? What do YOU, the teacher, think a variable is? I'm also interested in why you think a variable is what you think ... 17 votes 11 answers 14k views ### Looking for simple "interesting" math problems that cannot be easily solved without algebra I often find students who dislike algebra. They prefer to work with numbers in solving problems. I believe there are many problems that are hard to solve without algebra. For example: Finding the ... 15 votes 5 answers 1k views ### Cost and benefits of compartmentalization in k-12 curriculum This is a soft question perhaps not well suited for the format of the site but I'm interested to hear opinions from this community on this topic. K-12 mathematics textbooks (understandably) divide ... 15 votes 3 answers 514 views ### Calculation versus writing in mathematics Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas ... 15 votes 6 answers 3k views ### Is there any difference between teaching calculus for math and engineering students? In our university both math and engineering students attend in the same calculus classes. There are arguments in our department about the possible influences of this approach on students. It seems ... 13 votes 14 answers 3k views ### What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance? How can we break down the complexity of eigenvalues/vectors to something that is more intuitive for students. I feel like the proofy way isn't a good intuitive representation of the mechanism that ... 12 votes 2 answers 904 views ### Definitions/proofs that allow "useless" cases? I often see students confused/mystified by definitions (and proofs) that allow/consider "useless" cases. A case in point is the definition of a DFA (deterministic finite automaton), which allows ... 11 votes 6 answers 1k views ### How can I convince students that Fourier series are useful? Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ... 11 votes 4 answers 2k views ### Mathematical concepts and techniques that **pay off the most**? [closed] There is a smart way of learning, and it consists in first finding out what are the most valuable pieces of knowledge to acquire. The ones that will give you the highest value for your investment in ... 8 votes 2 answers 3k views ### Test Correction Analysis In hopes of helping my students practice and deepen their understanding of math knowledge, I wanted to bring up the topic of a test correction analysis - I didn't find much on this topic when I ... 6 votes 3 answers 348 views ### Should one justify formulae in middle school? Consider two possible lesson outlines: Check homework. Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279 Calculate the area of a circle as an example. ... 5 votes 7 answers 453 views ### Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion) A friend of mine that teaches math has made many times the following experiment : drawing two circles on the blackboard representing two sets A and B such that A and B are disjoint writing on the ... 5 votes 8 answers 693 views ### How to justify single digit answers to long problems? I have been teaching my student BODMAS and giving him problems like $$4\times4-6+8\div4+10\div2+5-4\times2$$ While solving the problems he asked me that, "So small answers for such big questions?" ... 2 votes 3 answers 2k views ### Why are most math textbooks so uninspiring; unless you already like Math? [closed] Why are math textbooks so often boring? It requires some mental discipline to do math; cartoons can help make the principles easier to digest. The Mathematical Association of America has many FUN ... 47 votes 24 answers 20k views ### How to explain Monty Hall problem when they just don't get it Talking to some friends, I was asked to explain the answer to the Monty Hall problem (see also here;) .... they were having some trouble because whoever explained it to them didn't do a very good job. ... 42 votes 12 answers 7k views ### Is it advisable to avoid teaching "multiplication as repeated addition"? I've had this discussion with a couple of friends. I argued that teaching multiplication as repeated addition isn't a good idea because it doesn't help children differentiate between the two ... 36 votes 6 answers 9k views ### How can teachers warn students about common mistakes without causing the student to make the mistake? For example, if you're teaching integration of$\int \frac{dx}{1+x^2}$, would you mention the common wrong answer of$\ln\left(1+x^2\right)+C\$? -- For myself, I very rarely mention common mistakes ... 