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 ...

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 ...

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 $b\neq0$, ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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: $... View answer Accepted answer 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. ... View answer Accepted answer 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 ... View answer 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. ... View answer 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 ... View answer Accepted answer 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 ... View answer 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 ... View answer Accepted answer 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 ... View answer 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 ... View answer 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 ... View answer Accepted answer 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 ... View answer 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 ... View answer Accepted answer 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 ... View answer Accepted answer 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-... View answer 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 “... View answer 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} = \...