Benjamin Dickman
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Redundant zeros
14 votes

Check out Berkeley mathematician H.H. Wu's homepage. In particular, see his textbook drafts for Pre-Algebra (pdf) and Introduction to School Algebra (pdf). For example, see p. 20 and the discussion ...

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Is using different notations in one course a good idea?
14 votes

There is a certain benefit to "confusing" students; I alluded to the ideas of disequilibrium and the resulting equilibration in an earlier MESE post. More comments about Piaget can be found on this ...

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How to motivate equivalence classes
14 votes

I would think a good first example is the rational numbers. (Note the "quotient" terminology here, too.) In particular, the rationals can be written as the set of integer pairs $(a,b)$ with $b\neq0$, ...

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How to react to students saying that they are allergic to applied mathematics?
14 votes

When in doubt, I often decide simply to quote others! A nice choice, in this case, would be someone who started as a pure mathematician, then worked in applied mathematics, and ultimately moved into ...

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Graphing functions from a finite field to itself
13 votes

One of the approaches taken in some areas of mathematics (e.g., in arithmetic dynamics and considerations of preperiodic points, etc) is to create these graphs by drawing discrete points and then ...

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Factoring quadratics where the coefficient on the $x^2$ term does not equal 1
13 votes

Factoring non-monic quadratic polynomials can be done by factoring with respect to a particular constraint. More precisely, DL Renfro points to the ac Method of Factoring which can be summarized ...

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How can mathematics educators encourage innovation and creativity?
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13 votes

Edit (5/24/14): For the reader interested in a somewhat longer answer, I am including the literature review (and all references) from my thesis on conceptions of creativity with regard to problems ...

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Exam philosophy
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13 votes

Since you remark that your question is "deliberately non-specific," here is a (necessarily) incomplete response: First are two links to documents about assessment that might be of interest, and then ...

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Why aren't logarithms introduced earlier?
13 votes

Historical comments. Early on, the study of logarithms and logarithmic tables was incorporated into trigonometry. For more on this background from the perspective of the history of trigonometry ...

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Adult Mathematical Literacy
13 votes

I don't see any studies of this sort on prime numbers, though I'm sure you could conduct an informal one and get a good estimate relatively quickly. Instead, I tackle your final note: A good answer ...

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Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III?
13 votes

Your students might find it useful to see this "visual approach" to proving the FTA: Velleman, D. J. (2007). The Fundamental Theorem of Algebra: A Visual Approach. Link. For a more rigorous ...

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Ideas for high school pure maths projects
13 votes

Joseph Malkevitch, based out of CUNY York College but also a visiting professor at Columbia University Teachers College, has a fair bit on his website about (high school) student research. Depending ...

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Algebra 2 textbooks that incorrectly claim that all solutions of polynomial equations can be found
12 votes

At the moment, I can answer bullet point two: Are there any high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree ...

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How are students messing up in this Khan Academy surface area problem? (right triangular prism: 3-4-5 triangular base, height of 11)
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12 votes

One quick way to guess at wrong answers is just to forget a face. Omitting units, the correct answer is $144$ and the faces have surface areas of: $6$, $6$, $33$, $44$, and $55$. Missing one face: $...

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How to think mathematically?
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12 votes

I believe the classic reference from the mathematics education literature is: Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. ...

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Uninsulting way to say "this will eventually be easy"
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12 votes

One thing that you might want to do early on in your course is think about the classroom norms that you wish to establish. From your post, it seems like an example of a norm in your class is that it ...

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Mathematical Knowledge for Teaching
12 votes

(Perhaps this should be a comment, but it's pretty much an answer.) Mathematical Knowledge for Teaching (MKT) is based on the more general term Pedagogical Content Knowledge (PCK) due to Lee Shulman. ...

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What are some great books for exploring mathematics? (not kids' books and not textbooks)
12 votes

For a recent suggestion, check How Not to Be Wrong by Jordan Ellenberg. Lying in the "simple and profound" quadrant, the book also gives deserved attention to Condorcet, in addition to providing a ...

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Requiring students to know all the proofs on an oral exam
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12 votes

This will depend in some part on the course you are taking. If it is an introduction to proofs class, then probably it is reasonable to hold such an expectation. For higher level courses, some of the ...

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Is there a good age/level to start learning mathematical proofs?
11 votes

With regard to Math Education literature on proofs: a person to look to is Eric Knuth (Google Scholar). However, it may be more fruitful to shift from talking about writing (formal) mathematical ...

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Appropriate journal for an article on practical mathematical teaching
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11 votes

Edit: See also the PDF linked here: Wikipedia has a list of mathematics education journals; I happen to prefer the list of mathematics education journals compiled here, as they are provided along ...

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Creativity in mathematics
11 votes

Yes, there is a growing literature at the nexus of mathematics education and creativity. The main name to know is Bharath Sriraman (google scholar) though the classic pieces to read for mathematical ...

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Name the heuristic: exploiting the legitimacy of the questioner
11 votes

The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as trying a special case. I do not know of a more specific term for the context that you have put ...

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What is a healthy and effective way for a math educator to evaluate his or her performance?
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10 votes

One place to look is in the following source: National Research Council. Evaluating and Improving Undergraduate Teaching in Science, Technology, Engineering, and Mathematics. Washington, DC: The ...

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How do blind people learn mathematics?
10 votes

Update for JMM 2020: See the potentially relevant blogpost here. The post begins: I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often ...

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Is there a toy example of an axiomatically defined system/ structure?
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10 votes

This may go beyond what you are asking for, but there is a wonderful book called Introduction to the Foundations of Mathematics by Raymond L. Wilder. I provided its axioms and an example of how they ...

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request for evidence about class perspectives in math word problems
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10 votes

Yes, there is evidence for the claim; for example, consult the following: Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14(3), 219-...

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Learning counting and addition: fingers or in their head?
10 votes

Edit (May 2016): From The Atlantic is: Boaler, J. & Chen, L. "Why Kids Should Use Their Fingers in Math Class." Apr 2016. Link. "Evidence from brain science suggests that far from being “...

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A Series of Unfortunate Examples!
10 votes

One mathematical example that has been explored is the somewhat pathological nature of "anomalous fractions" where digit cancellation produces correct simplification. For instance: $$\frac{16}{64} = \...

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What are the historical reasons for the hostility against standardized testing in the US?
10 votes

One possible reference (mentioned here specifically for its introduction) is: van den Heuvel-Panhuizen, M., & Becker, J. (2003). Towards a didactic model for assessment design in mathematics ...

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