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]]>Even though Wikipedia is not the final authority on all things, it does give a nice history of the concept:

http://en.wikipedia.org/wiki/History_of_the_function_concept

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]]>Computers and internet have diminished importance of human memory. We need people who possess enough understanding to search, process and use new information more than we need people with total recall. So, for students to achieve such understanding (deep learning) we need teachers who know their subjects and are able to engage students in discussions that reveal what is “inside their minds” so that these minds can be properly “formed” (“formation” is the word used in many languages to mean “education”). An ability to use group dynamics and emotional devices remains of utter importance. Ability to use technology is helpful but, in my view, its importance is sometimes exaggerated.

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]]>Here is my take on things:

Division by zero is undefined whether you work with real numbers or complex numbers. Khan’s video on the subject is a bit confusing. The consistent exposition of algebra is given in Euler’s “Elements of algebra”. There division is defined by saying that a/b=x, a unique number, such that bx=a. I am sure you can define it otherwise, making certain that there are no circular arguments involved, just as in Euler’s approach. Do not know why would one want to do it, but this is a separate issue. Using the commonly used Euler’s definition of multiplication (and multiplication rule x.0=0, for any x) and his definition of division we can show that division by zero is undefined:

The proof is carried out by *reductio ad absurdum*. Assume that a/0 IS defined, that is, there exists an unique number x, such that x.o=a. First let a^=0. Then, by definition, a=0.x. The number on the left of the = sign is 0 and the number on the right is not. Hence we arrived at a contradiction. Hence the assumption is wrong. Now let a=0. Then by definition, 0=x.0. Since all numbers satisfy this identity the assumption proves wrong again.

You can, of course, make an artificial distinction between the two cases of zero and non-zero numerator as Khan does and introduce two different words to distinguish between the two cases. This is not advisable for two reasons:

1. the real alternative is between having and not having an unique answer (known in advanced mathematics as the Fredholm alternative, would be a good idea to prepare students for dealing with it)

2. the word “indeterminate” is applied to a symbolic “solution” when seeking limits and it means “the formal solution does not produce an answer, some tricks are required to ‘resolve’ the indeterminancy, that is, determine (establish) the unique answer or else that there is more than one answer or no answer at all.”

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]]>1. STEM students with weak backgrounds do not understand mathematical symbols. Therefore, they cannot read the simplest mathematical texts or follow the simplest mathematical explanations. To get real education they need a teacher who will patiently teach them the mathematical language and mathematical reasoning first. Simple lecturing will not do, Socratic dialogue – the continuous two-way feedback – is a must. Most students will not be able to follow internet lecturers who have material suitable for students with strong background.

2. Automated tests have a value but this value is limited. To make sure that students get education and not simply meaningless “degrees” they have to be encouraged to learn the material and not just train themselves to exams by studying previous exam papers.

3. When I was giving my students home assignments, many (too many) simply copied from others. I came to conclusion that assignment marks were quite useless. Only once did I have a student use a substitute coming to sit an exam, but even that happened.

4. As a student at an elite University I learned most not from lectures or tutorials but from other students or aside remarks made by teachers. Internet forums will hardly be a substitute. Look at all the trolls frequenting internet forumsand stifling most interesting debates.

Having said all that, I hope MOOCS will encourage better teaching.

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]]>extolling the virtues of teaching arithmetics as music. I

always feel sad when teachers put competency over

comprehension. Of course, at work, competency is king,

what matters is only whether you accomplish your tasks and

how quickly. But when learning, comprehension should be

paramount, eventually it will lead to greater competency.

Yes, one can learn the times table by heart without any

understanding. It takes a lot effort though and has

limited value. Yet, if one is explained how numbers work,

the times table can be learned in 2 weeks, working 10

minutes a day. And it is real fun!

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