Venn-style thinkingAn example of mathematical thinking beyond the narrow world of numbers.Dave C, 2013.
Beneath the numbers that make up mathematics lies a philosophy that goesfar beyond simple counting, and teaches us (if we let it) how to measure asocial situation.Let me give you a few examples, starting with this one: the Venn Diagram.
The Venn diagram protects us (if we let it) from falling into the trap ofseeing a debate as simply having two sides. Take any controversial issue thatyou hear discussed on TV; there are champions for one side and championsfor the other, neither party allowing for the possibility that there could be anysolution other than we-win you-lose.
The issue could be Charter Schools versus Public Schools. One or the other,we would think.Maybe the solution is Charter Schools, end of story.
The issue could be Charter Schools versus Public Schools. One or the other,we would think.Maybe the solution is Charter Schools, end of story.Maybe the solution is Public Schools, end of story.
Maybe the solution is BOTH Public schools and Charter schools.Maybe these two systems serve different groupswithin a given society and allow more kids tosucceed despite their differences. The BOTHsolution allows for diversity.
Maybe the solution is EITHER Public Schools or Charter Schools.Maybe it doesn’t matter who pays the teachers andsets the rules; any kind of schooling is beneficial.
Or maybe the solution is NEITHER Public Schools or Charter Schools.Maybe both options fail to address the basic issues.Arguably, this could be that institutionalisingeducation is wrong, and so debating which kind ofwrong is therefore pointless.
That’s five different ways of answering what looks like a YES-NO question.But there is another solution suggested by mathematics: the NULL solution.The NULL solution is not the same as the NEITHER solution. TheNEITHER solution lies outside the two circles, whereas the NULL solutionlies nowhere and everywhere, existing as a region so small it covers noterritory.
The NULL solution says ‚The question makes no sense‛. Like looking forthe Inverse Sine of 2 or the number of times zero divides into 2, the NULLsolution says, ‚The question was poorly stated; it has no answer.‛
In the case of the Charter School/Public School debate, the NULL solutioncould mean that ‚It’s not whether these schools exist but rather how youchoose to manage them‛. Maybe Charter Schools and Public Schools areonly as good or bad as people choose to make them. Maybe both systems ofeducation lend themselves easily to mismanagement, deliberate oraccidental.
Perhaps the question should not be ‚Do we have Charter Schools at all?‛,but rather ‚How do we ensure that Charter Schools are run well?‛And what do we mean by saying ‚run well‛? Does it mean exam scores arehigh or that the kids are happy, or that the school is making a profit?
And if we can ask these questions of one side, should we not ask thesequestions of the other side too? That is, shouldn’t we ask the question, ‚Howdo we ensure that PUBLIC schools are also run well?‛
And what does it mean to run a public school ‘well’? Is it sufficient to coverall your expenses and make a profit? Or are there subtle agendae thatlobbyists on each side of the school systems debate want to promote at theexpense of clear rational thinking?
I’ve latched onto the debate of Charter Schools versus Public Schoolsbecause it is topical in my community at the moment. But the tool that I amillustrating with this example could be – and should be – applied to allsituations where two sides of a debate emerge, and seem to be sinking into aswamp of vitriol.
What about the debate over nationwide standardised assessment ineducation? Maybe it’s not the existence of standardised exams, but ratherhow these exams can be implemented in a way that does no damage. That’sthe NULL solution.
On the other hand, maybe it’s BOTH, EITHER or NEITHER. Maybeschools should decide for themselves whether they want to adopt ‘nationalstandards examinations’, in which case families that like exams aresupported as much as families that don’t. That’s respecting diversity. That’sthe BOTH solution.
The EITHER solution implies that it doesn’t really matter whether kids areexamined or not: an education is an education and will work for a kid, or failhim, regardless of the exam process.
The NEITHER solution implies once again that schooling of any kind is bad,and while I don’t believe this no matter how long I consider the idea, at leastconsidering the idea at all makes me a bit wiser.People dislike exams because of the damage itdoes to some kids. Okay, but is it the exam thatdamages the kids or the forced education that isdamaging the kids? Maybe it’s the entire schoolsystem that is damaging kids, in which case theforced end-of-year exam is just another brick inthe Pink Floyd wall?
Wherever you stand on these issues, and other issues too, I want to suggestto you that Venn-style thinking will widen the scope of your focus to includemore possibilities and perhaps arrive at better solutions. But moreimportantly, I want to show through this example that there is value tolearning the philosophy that accompanies mathematical thinking, and that itcan be applied in situations that have no smell of mathematics to them.