#change 11: Leveraging my learning
This morning as I was looking for images to use in a new math course I'm developing for students in a country I've never visited, I came across the work of a math educator named Ben Chun. Chun seems to be one of those brilliant people who, fortunately for his students and those of us who read his blog, gravitated into teaching. His "Facebook Visualization" in Flickr Creative Commons
... bears an interesting resemblance to the Fibonnaci seqence (from Algeblog9)
... and will fit into my unit on Patterns in math, nature, music, art, and more.
Conceptualising this course has been a bear of a task. Many of the students drop out when they are young teens, and a one-to-one laptop program with individualised learning packages has been instituted as a way to keep them engaged in school work. The students will have no connection to the internet so the wonderful world of online interactives cannot be accessed.The courses have to be completely self-contained and self-managed so if the students go into school only occasionally, they can still be successful.
Interestingly, depriving me of the ability to use what I've spent 5 years becoming good at has forced me to think more deeply about the relationship of curriculum structure to student learning. How do I build in interactivity? What will be its purpose? How can I make math into a 'do with' rather than a 'get through' subject? How can I give the students more choice and control and at the same time ensure they take on the responsibility for ensuring they have the skills needed to meet the requirements of standardised tests? I want them to look forward to opening their laptops to do math. I want them to be interested enough to take on the hard stuff rather than have the course reconfirm that math and staying in school are not for them.
Chun's reflection on the benefits of teaching programming to math students is a nice metaphor for my underlying big idea:
"School teaches that errors are bad; the last thing one wants to do is pore over them, dwell on them, or think about them. The child is glad to take advantage of the computer’s ability to erase it all without any trace for anyone to see. The debugging philosophy suggests an opposite attitude. Errors benefit us because they lead us to study what happened, to understand what went wrong, and, through understanding, to fix it." (http://goo.gl/Mdqgu)
You can't convince a kid on the verge of dropping off that math is good for him because it will help him get something he wants some day in the future. The work either has to be intrinsically interesting or have a real-world usefulness. One has to get the kids staying with the work and enjoying the results so they have an opportunity to engage in what Chun calls 'computational thinking':
"When we analyze the effects of computation, we take note and measure how data is transformed. We look at how information is processed and what is accomplished by that processing. We can think about what we might do if such computational power wasn’t available. That can also help us start to imagine new things we can strive to accomplish using computation." (http://www.ctillustrated.com/)
Another math educator, David Tall, comes at mathematical thinking in a different way (http://www.lsri.nottingham.ac.uk/mtw/onlinevideos/Tall.html). His writing reminds me that there is also still a place for solid procedural learning in my course. Part of math learning requires that students give up replicating real or natural processes over and over and do the work of commiting facts to memory and coming to recognise relationships automatically.
Most of us don't have to examine and mentally aggregate the parts of a table to know we're looking at table. We've been there and done that enough times that we can instantaneously recognise table-like shapes and objects. Yet many children, especially those who struggle in math, cling to primitive routines such as counting (instead of learning math facts). This makes such processes as long division and calculating fractions into Herculean tasks. If students don't encapulsate mathematical processes into "thinkable objects", all their attention and brain power is sucke up be by the lowest level tasks. These students end up with huge 'no go' mathematical zones that at first slow them down and eventually cripple their ability to enage in the the kind of nimble, flexible thinking needed to use math more powerfully.
So today's conclusion is that crystallized intelligence + fluid intelligence = a more "global capacity to learn" (http://psychology.about.com/od/cognitivepsychology/a/fluid-crystal.htm). People aren't either left or right brained. The 2 hemispheres are in continouous communication with each other giving us our ability to engage in complex behaviour and higher order thinking (http://goo.gl/Qdfm and http://goo.gl/R6GqH).
The problem of what will make a good course structure then becomes a sort of 'chicken vs. egg' problem of where to start. How to keep more students learning more math more successfully comes down to letting the students choose where to begin --i.e. with learning prodedures if that makes them more comfortable or exploring problems and projects if that's what will draw them in. It doesn't really matter where a learner starts in the mathematical landscape, but the couse must also develop the understanding that it's not good enough to just get better at what you already does well or like.
Becoming a powerful thinker involves leveraging what you do well to learn something new or different. In math it means growing and expanding your talents rather than using them as a boundary or circumference to delineate what you will (safe; inside the circle) and won't (too far outside the compfort zone) try. It also means being able to later look back, engage in a little of Chun's computational thinking, and savour how far you've come.







