Anschewski
  • Member for 7 years, 10 months
  • Last seen more than 1 year ago
  • Hanover, Germany
Whence the "everything is linear" phenomenon, and what can we do about it?
Accepted answer
45 votes

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: ...

View answer
Why do students have problems with showing that something is well-defined? How can this be improved?
Accepted answer
28 votes

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 ...

View answer
How to encourage women to study mathematics?
16 votes

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,...

View answer
Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?
16 votes

In Germany, there recently was a Project at two universities which tried to strengthen historical connections in analysis courses, especially for teacher education. For them, history can be an ...

View answer
What is a good motivation/showcase for a student for the study of eigenvalues?
14 votes

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 ...

View answer
What Math(s) Ed literature is accessible to the working math(s) educator?
12 votes

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 ...

View answer
Motivation in School
12 votes

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 ...

View answer
How can I familiarize elementary school students with infinities larger than $\aleph_0$?
12 votes

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 ...

View answer
What was the problem with New Math? Why did it end?
12 votes

I think very roughly speaking: "new math" (also known in europe as a fearful period of time, especially for parents) followed a mathematical construction of mathematical knowledge rather than a ...

View answer
How can I teach my students the difference between a sequence and a series?
11 votes

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 ...

View answer
Experiences with online courses, specially MOOC?
10 votes

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 ...

View answer
What are argument one can give to students on the definition $0^0$?
10 votes

Since there is much choice in Terms of limits as you said, a definition should be reasonable in terms of convenience. If you define the exponential function via the series $e^x:=\sum_{n=0}^{\infty} \...

View answer
Impressive examples where a "proof by picture" goes wrong
8 votes

The missing square puzzle is a nice one. Maybe it doesn's fit your question perfectly, since no one would use this for a proof. But you might take it as a proof for the theorem, that the area of a ...

View answer
Pressure vs. Laissez-faire: Literature dealing with balance in university-classes
Accepted answer
8 votes

Well, I think the answer is "there is no literature on your question"; however, this question is somehow unanswerable. Let me explain: For my PhD, I spent the last 3 years working with literature on ...

View answer
Revisiting topics from previous courses
8 votes

I suggest to have a look at your students' concept of a function. Typically, they see a function as a process but not as an object that can be part of a larger object. (They will identify terms as ...

View answer
What are effective alternatives to a written math exam to evaluate knowledge?
7 votes

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 "...

View answer
How to make calculus lecture time more interactive?
7 votes

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 ...

View answer
Impressive examples where a "proof by picture" goes wrong
7 votes

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 ...

View answer
Should we say that fractions "are" or "represent" numbers?
6 votes

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 ...

View answer
"We already passed that course!" How to overcome this?
6 votes

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 ...

View answer
"A" or "The" Cartesian plane?
Accepted answer
5 votes

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 ...

View answer
How to motivate equivalence classes
5 votes

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 $1$, ...

View answer
Redundant zeros
5 votes

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 ...

View answer
Good definition for introducing real numbers?
5 votes

There are different ways to introduce the real numbers. Even if you choose an axiomatic introduction, you have to choose an appropriate formulation of the completeness theorem. Since you are german, ...

View answer
What is the correlation between students' contentment and educational quality?
4 votes

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' ...

View answer
How to deal with answers containing completely off-topic/random/very wrong arguments?
4 votes

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....

View answer
How to teach logical implication?
4 votes

To supplement Brendan's idea: I like to connect quantified statements to unquantified in the following way: Assume a statement like "If X is a dog, then X has a head". Now, once you have found this to ...

View answer
Books/(auto)biographies/references on how mathematicians study/studied (as students)?
4 votes

For paedagogical use, the book "Amongst mathematicians" by Elena Nardi might serve your intentions.

View answer
Summary of the mechanism of reification
3 votes

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 ...

View answer
Obtaining printed copies of the textbook series Unified Modern Mathematics
3 votes

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/

View answer