Five ideas from OAME

Recently I attended the OAME 2014 conference at Humber College.  Here are five speakers who got me thinking.

1. Jo Boaler: With a Growth Mindset, anyone can learn mathematics
Performance praise like “You’re smart” actually encourages a fixed mindset, a belief that ability is inherent rather than developed.  In contrast, growth mindsets are cultivated by practices of: persistence, learning from mistakes, determination to keep going, and being encouraged by other’s success (Carol Dweck).  Those with fixed mindsets underachieve when compared with those with growth mindsets, regardless of high or low present ability.  She believes that with a growth mindset, all people can learn mathematics, except perhaps those few limited by cognitive disabilities.   I think I agree with her point of view, with the great challenge being that some students have established habits of behaviour and belief that reinforce fixed mindset behaviour.

Describing Graph Transformations with Input / Output diagrams

I regularly stress to my grade 10 and 11 students that when transforming a function it is imperative to dilate before translating in any direction. In other words, if a function is both stretched and translated horizontally the stretch must happen first. Ron Watkins’ demonstration of Input / Output diagrams at OAME2014 has helped to see how rearranging the equation that describes a function can translate to rearranging the kind and order of transformations

An Input / Output diagram shows the sequence of operations applied on a function’s argument, x, to arrive at its value g(x). Naturally this sequence follows order of operations. What is interesting is that transformations are made clear through the diagram. Look first at the parent function indicated by the red arrow, which in this case is f(x) = x². Moving towards the right are the vertical transformations: stretch by factor 3 then translate 5 up. Following the arrows towards the left are the horizontal transformations: compress by factor one half, translate 4 right.

 Often students don’t see why horizontal transformations on the parent function are related to the inverses of operations seen in the function’s equation.  The diagram shows that to go from the input of the parent function (red arrow) to the horizontal coordinate x requires inverse operations, while going from the output of the parent function to g(x) does not.