You don't just learn by solving questions...

... rather you can also learn - perhaps better - by 'posing (or developing) questions'. According to a research article in School Science and Mathematics, "Problem posing, with its emphasis on having students generate and develop their own mathematical problems from particular situations or on the basis of their prior problem solving experiences, may be an important activity that in fact helps students develop as inquiry-based problem solvers."*

Well, I have a friend who does the same with his O/A level students in Pakistan, helping them make DISTRACTORs: questions that only 1 out of 1000 can answer. One of his student said that this is the way of real learning and growth. How does it happen that by posing questions a student may better solve a problem? the article mentions two instance:

1. ... he or she may look to break down or reduce the original problem into smaller problems, the solution of which lead to a solution of the original problem.

2. ... students engaged in the solution of a problem may generate a result that challenges or calls into question their prior goals and actions. In these situations the ways that the student acts to resolve the new question often leads to a reformulation of the original problem which may in turn lead to progress in finding a solution.

* Problem posing and problem solving: a dynamic connection. mathematics education Charlene Sheets; Victor Cifarelli. Full Text: COPYRIGHT 2009 School Science and Mathematics Association, Inc.

'Terrified, Mortified, Petrified and Stupified'

I have always been terrorized by mathematics. I ain't any simpleton, though. i like beauty. Yes I can even talk about it. I stretch matters to no-useful extent everyone may not be capable of doing. I fought wars in my backyard. Build sedentary morchas (bunkers) of mud and brick, bombed them through distant catapults - *boom*. I did calligraphy; compiled huge registers filled with extracts on various subjects, as a sort of a haphazard, passionate encyclopedias; I even led many a times a hard-ball cricket team when I was not even about 14 (a feat my physical manifestations definitely contradict, this shows how many brains I have)!

But mathematics. I had no clue if I could ever relate it to the norms and mentalities of a certain species known as "humans" the way I knew, saw and sensed them. How could a person commit the mistake of initiating into mathematical madness? Where did it come from, in the first place? i had no clue, I still lack a point. To talk about its practitioners: Mathematicians to me have always been like those mythical creatures who are willing to suffer so many dilemmas of their own systems. That's why I revere them, even if they are of the worst sort of characters. I'm in an inferiority complex, generated by maths, to plainly tell you the 'sordid state of affairs'. But I don't want to get into a superiority complex either, the worst form of the former.

To settle the dust and to lift myself above it. i have made my plans. Want to listen to them? I'll stick to it, I'll stick to it, until everything sticks to my mind. Good-bye. 'Where has gone that God Created the Integers?'

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Please email your kind suggestions as to how to handle this demon safely without my major causalities at: final.human.battle.against.complexnumbers@extremesurvivalseries.com. Pray for my safe journey, as well. Thank you.

Causality in Economics

Causality (root word 'cause') determines what cause of an effect (or consequence) is. And in order to make a rational decision, the economist must know the independent variable and the dependent variable, parts of a choice leading to a utility, where dependent variable is a consequence of independent variable.

Another way of viewing causality is finding the existence of causality in a given relation or function of two variables, or parameters. It happens, as we are told, that there may be existent an association between two things/events/choices yet no causality is to be found in the ‘relation’. Word ‘relation’ is important here. Because, here we’re not discussing the matter from the view of classical, philosophical cause-effect theory, rather we are strictly concerned with the mathematical relationship between different variables.

In the following paragraph, I’ll show you two different cases where both variables are associated with each other on the graph. You may notice at the end that intuition works better at the basic level. But the importance of rigorous analysis and empirical research shall also be highlighted by the demonstration of idea.

Consider this case: ‘Sugar’ is a dependent variable. ‘Sugarcane’, on the other hand is an independent variable. There is also a natural (that is to assume quite intuitively) association or relation between these two variables. And sugar is dependent to sugarcane (independent), because we cannot think of sugar, without any sugarcane. Therefore, existence of sugar depends on that of sugarcane. Or in other words, sugarcane is the cause of sugar (effect of sugarcane).

Sugar = f(sugarcane, etc) [Sugar is a function of Sugarcane]

Consider the second case where an association between two things is present with the absence of causality given their mode of function

If I draw a graph and show on x-axis 'number of police constables' recruited in a given period, in which the number of crimes taking place in the city had increased. Crimes data shown on y-axis, while police data shown on x-axis. Meaning by, that crime is a cause of increase in the number of police constables. Police and crime have a relation and association no human being can deny. Yet the idea seems absurd. Because police helps diminish the crime, not vice verse. So intuitively we conclude that although there is an association between the idea of police-crime, there is no causality (that police is a cause of increase in crime) established between them.

Here comes the interesting part of the game. Rigorous mathematical and statistical analysis of these two ideas, so that it could be made ascertain where does valid causality lies. Regression models ar built, in-depth research is carried out, many reports are made to establish the truth. Intuition is just the beginning and in the middle, sometimes it is not the end in itself. Sometimes.

(Warning note: This post had no intention towards public benefit. It was a personal matter. And was addressed to my own self. If you have questions to ask please contact the economist-teacher Mr. N. Gregory Mankiw of Harvard University, U.S.A. But if you've answers and something to educate and share, can't you just go ahead and do it, please?)