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Sitting in a doctor’s waiting room (as you do!) the other day, I read one of those waiting room magazines with the crossword puzzles and jokes.  One of these jokes was: The mother asked the birthday boy: ‘Do you want your birthday cake cut into twelve pieces?’  ‘No,’ the boy replied. ‘We won’t be able to eat twelve pieces.  Can you cut it into eight pieces instead?’

Funny, yes.  But it also got me thinking about the real concepts we are teaching kids in schools.  This boy was obviously taught that “The smaller number implies ‘it’ (whatever the smaller number refers to) is smaller.”  This makes sense, doesn’t it?  But what about if the smaller number has to do with fractions or parts of an area?  If it is the same area cut into less pieces, the smaller number indicates a bigger piece of the area (i.e. a bigger fraction.)  Do we ever make those connections and concepts clear to the kids?  The smaller number is NOT necessarily smaller!  What about negative two and negative ten?  Negative two means you are closer to the zero on the number line, so it is actually a ‘bigger’ number than negative ten.  What about two to the power of negative two against two to the power of negative ten?  Once again, two to the power of negative ten indicates a (much) smaller fraction that two to the power of negative two…fractions again! 

But then, what about one to the power of negative two against one to the power of negative ten?  These two numbers are equal…..both are equal to one!  Same thing with zero to the power of a small (positive) number, against zero to the power of a big (positive) number: it remains equal to zero.  And zero to the power of any negative number does not even exist!

Yes, we need to be careful when teaching blanket rules in Maths.  We need to be astute when we teach kids about concepts.  Wherever possible, we need to relate concepts back to concrete materials and digital manipulatives (ICT in Mathematics).  It is clear that digital manipulatives can be seen as the scaffold between moving from concrete, hands on materials to the more formal concepts.  This is one area where computers can play a vital role in the understanding of Maths concepts.  Teachers should also be aware of explicitly asking students to visualise what we teach them, i.e. ask kids to “make a picture in your head”.  It is much easier to remember things we have actually ‘seen’, and, where possible, if we played with them or experimented with them, rather than just remembering numbers and cold facts.

If you would like to play with some virtual manipulatives in your Mathematics/Numeracy classes, visit this site and click into the year level and Maths area you want to explore: http://nlvm.usu.edu/en/nav/vLibrary.html

Looking specifically at manipulatives that may address the problem that started this discussion (cutting a cake into twelve or eight pieces), look at these manipulatives: