One of the aspects of our current research with a colleague in Mathematics Education requires the definition of what should/could be called a "non-standard" mathematics problem. One of the ways to go around this is to define non-standard problems as those which are not standard, in which case we need to define what is meant by a "standard" mathematics problem.
Your suggestions on either of definitions (standard or non-standard problems) are most welcome.
Thank you in advance for your contribution!
My suggestion is that a non-standard problem is one that does not typically fall within the purview of an applied mathematician in any field.
The most sensible approximation of a definition - as I do not think there is one - is to view text books from a few decades ago (let's say the 60s), and let the standard problems presented therein be the definition of "standard".
The reason why you should not look at books published these days is the fact that more modern books have been expanded with lots of other things than "problems", and therefore it will be more difficult to pinpoint the good ones.
Also, modern books tend to "talk" a lot around the subject and problems, while older books are "to the point" - that is, the standard problems will be easier to locate in those older books.
This is an interesting approach, Michael, but I doubt this would work as we no longer use the texts from 60s. I know well the difference as I used them myself as a student but I can't imagine using them now for my regular teaching. Many problems from the textbooks published 60s that used to be routine could be easily termed now as challenging, and I do not like this tendency at all.
Search the net for this article:
Development of Children’s Solutions of Non-Standard Arithmetic Word Problem Solving
by: Laura Jiménez and Lieven Verschaffel
This article and its references may help you formulate your own definitions of standard and nonstandard mathematics problem.
In Russian, the best answer to the question posed is the article by I.V. Arnold (the father of Academician V.I. Arnol'd) "The principles of selection and compilation of arithmetic problems." To intrigue the reader, I will inform you that in this article, among other things, a selection of 20 non-standard problems is given, each of which is solving by the same action: 3 -- 1 = 2.
Dear Aslanbek Naziev ,
Could you give us the reference about I. V. Arnold's article? I was looking for it through the Net, but I couldn't find it. Maybe it was written in russian.
Thank you in advance
'Standard' sounds relative. I have never come across the standard/non-standard problem term in mathematics. In what context this standarization is asked about?
Dear Sona, most problems in traditional mathematics textbooks are usually routine or drill tasks which help to develop mainly procedural skills whereas non-standard problems require better conceptual understanding and a more careful analysis. In my humble opinion, by posing non-standard questions, the lecturer stimulates better understanding and deeper learning.
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