Modern Learning with Flexible Structures

“Modern learning” is a phrase I am encountering in more and more places, but still I just can’t seem to find a definition of what exactly it is. The general idea seems to be that a changing 21^st century world demands collaborative creative critical thinkers, and that modern learning involves whatever tools and approaches are needed to help students develop those traits. Discussion about modern learning typically emphasizes the role of the student in learning, flexibility in approaches to the curriculum content, and use of technology to enhance the learning experience and facilitate collaboration.

To experiment with one modern learning idea, I used a flexible structure this semester for the Linear Systems unit with my Grade 10 Applied Mathematics class. I divided the unit’s learning goals into two learning cycles of about five days each. At the end of each day students completed a Google Form to self-assess their progress towards each of the cycle’s learning goals, identify homework, and choose what and how they would like to learn the next day. Content choices aligned with learning goals, and method choices included watching a video, trying an investigation, reading from the textbook, researching online, or participating in a mini-lesson with me. Several times in each cycle they were also asked to choose between writing a quiz and creating a video with Lensoo Create as a method for formative assessment. Each student had a Nexus 7 tablet to use in class to access learning and assessment resources.

OAME Top Three

Year after year the OAME conference gets me thinking about mathematical things in a new or renewed way.  Here are three ideas that have stuck with me from today's speakers at OAME 2015

1.Spiraling through curriculum
The structure of most mathematics courses (or even I suspect most courses) mirrors its curriculum document.  The curriculum divides the course into three or four strands, each with three or four overall expectations that are elaborated on with a half dozen or so specific expectations.  Similarly, long range plans usually divide the course into a half dozen units derived from the overall expectations which are subdivided into lessons to sequentially address each of the skills required.  A linear document (the curriculum) is translated to a linear teaching plan, and most of the time the system works to deliver prepared students to the next grade so long as they don’t fall off the linear skills assembly line (... or if they do so long as we can pick them up and put them back on quickly enough...)

In their tiny five minute long IGNITE talks both Mary Bourassa and Alex Overwijk referred to an alternative approach.  When “spiraling” through the curriculum the linear planning model is gone.  Instead, students wind their way through activities and lessons that span the curriculum.  All of the overall expectations might be addressed within the first six weeks of a course, and then revisited in more depth as the course circles back again and again. 

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.