I would suggest a distinction: A MOOC really should be massive, that means some 1000 participants or even more. In this case your problems will be about server capacity and technical things. The work ...

I very much welcome connections of research and teaching, although results from research still need or be interpreted in your specific situation. Let me give an example. It might happen that your ...

Imagine a linear mapping $f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$. (As long as $R$ contains the numbers $1.5$ and $0.5$, it could be any ring. The real numbers serve as the ...

I will try to give a research-related answer. There are several suggestions from the literature and you may have to take a deeper look at them. First, a decrease in motivation is also observed in ...

Your question will not have a research-based answer. You assume that there is a determinable "fraction of the population" that "is incapable of learning algebra, even with repeated effort and ...

The literature lacks a clear mechanism, that is why theories on the process-object-duality are criticized sometimes. Anna Sfard's reification, Dubinsky's APOS and Tall's procept may help describing ...

I think from a philosophical point of view, both terms may be used. From an educational point of view, I would generally use the term "the" Cartesian plane in school context as every two Cartesian ...

In order to help you, we should know which country you are from. In Germany, the library of the MPI in Berlin seems to be the only library where you can find these books, see http://gso.gbv.de/DB=2.1/

I would recommend you to specify the term "educational quality". I think there is a study indicating that in German schools, the two educational goal variables of students' motivation and students' ...

Many good arguments have been presented. I would like to add that working with the decimal expansion does require much less understanding of what a real number is. The decimal expansion gives you ...

There will be definitely some material on assessment in school. Since my research focus is on tertiary education, I present some resources from calculus and post-calculus: Abramovitz, B., Berezina, M....

Do you mean something like this: The source is provided in the picture and there is a vast body of literature on modelling. You might want to specifiy your question: Which institution / age? Is your ...

Whenever I want to fill time with kids, simple reverse-problems come to my mind. Imagine a number, say 5. Which computation might lead to "5"? One might start with 2+3. That's right, let's see if they ...

I've discussed this several times with other education researchers and it does not seem easy. A learning style (or maybe thinking style) should scarcely depend on the knowledge someone has or ...

To me it is clearly a definition. Maybe, in addition to Behzad's post have one more idea: What if $a=0$? Then $a^n=0$ for positive values of $n$. So $3^0, 2^0, 1^0$ all equal $1$ wheras $0^3, 0^2, 0^1$...

You can easily make them draw $\aleph$s; however the rest is much more demanding. There is a nice analysis from a researcher group taking a constructivist perspective. They distinguish potential ...

As an alternative to non-mathematical or very complex examples: I like to go back to the very basics: terms. Is $1+2$ the same as $2+1$? As an expression, it's not. The first one begins with a $1$, ...

If the student struggles with this task, what do you think he (or she) thinks of a decimal number? What is .37 for him? Let him try to explain the meaning of .37 in terms of examples or calculations ...

In elementary education, "is" would be the right term since students hardly know what "represent" means. However in teacher education when discussing the rationals, it's different. Once you understood ...

I think one there is a bunch of problems. First, university curriculum tends to treat every topic exactly one time and then assumes people know how to deal with it. Second, as long as students may ...

The problem you describe is well-known in mathematics education research. I cite the paper of De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: ...

I usually give credits for everything that is right, ignoring what ist wrong, iff things are not contradictory (I wouldn't give credits for "answer is right or wrong"). If things are contradictory (e....

I suggest that this is more than a problem in wording. In mathematics, many objects are introduced as a process (e.g. functions as "making a y out of a given x", sets as "taking things together") but ...

Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined. If you have never been in trouble since everything you were ...

In teacher education I had some positive experiences with term papers. Students had to write 10-20 pages where they discussed a topic related to the course and their future teaching. Topics were like "...

Two suggestions: The first is to try clickers. They work like the voting system in "who wants to be a millionaire?". You can activate students and foster discussions on questions. There is no need for ...

I think there is no clear answer, although there has been some research on this topic. I remember one study which focussed on gender differences of university math students: Mischau, A., Blättel-Mink,...

Maybe, one could argue that $f(x)=x^2-2$ has a rational zero using the number line. Just imagine its graph in $\mathbb{Q}^2$. You cannot "see" whether the completeness axiom of $\mathbb{R}$ is given ...