Friday, November 11, 2011

Is this real?

Today, part 2 of Harry Potter and the Deathly Hallows is available in stores. When the movie opened in theaters this past July, I used it as an opportunity to write about teaching as storytelling. In this post, I want to take a look at a particular exchange between two main characters and consider what it might mean for teaching and learning mathematics. Warning - spoilers ahead.

The following happens near the end of the story. Harry has sacrificed himself to save his friends and finds himself in an ethereal version of King's Cross Station talking with his deceased mentor, Albus Dumbledore. The conversation is nearly over when this brief exchange occurs.
"Tell me one last thing," said Harry. "Is this real? Or has this been happening inside my head?"
Dumbledore beamed at him, and his voice sounded loud and strong in Harry's ears even though the bright mist was descending again, obscuring his figure.
"Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real?"
There seems to be a strong push in mathematics education to attach some real-world significance to nearly every topic in the curriculum. I appreciate the effort and agree that content a learner can connect to is more likely to be engaging. For me, however, the search for context to wrap around content can sometimes be distracting and inauthentic.

How can real-world examples be inauthentic? Let me provide a personal example. As a middle school mathematics teacher, I often extolled the practicality of learning to add and subtract fractions with unlike denominators. I would say, "You'll need this skill in cooking and building." Now maybe someone else can pull this off in an authentic way, but I cannot remember ever adding or subtracting fractions outside of an educational setting. Granted, I do not cook or build anything from scratch so I may be missing something. This only reinforces that this purpose was inauthentic to me and a distraction to my learners.

My problem was that I was confusing context with purpose. I now explain to learners that the rationale behind our lessons on fractions are related to the NCTM Process Standards and not some potential future use. I do not know if my learners will ever need to add or subtract fractions while cooking. I am sure that their future success will depend on their ability to problem solve, reason, communicate their thinking, use representations, and make connections. Yes, even connections to the real-world.

Let me be clear that I am not suggesting context is not important. In order to learn something new, we need to connect to something known. Looking back, I am not sure how many 8th-graders have experience with cooking and building. They are familiar with time, however, which is why I like using the clock model. They also have worked with whole numbers, and I often try to connect to these computational experiences. Sometimes the mathematics itself is the context.

Essentially, it is about finding what is real for my learners. Certainly, there will be times when their reality is at odds with that of the mathematical community - that is when the fun begins. I can only hope that my learners, like Harry, will begin by asking, "Is this real?"

1. When I cook, I am really lazy about washing measuring cups, so I am forever wondering, "how many times do I have to fill a 2/3 of a cup measure to measure 3/4 of a cup?" (which is dividing fractions). I also often think about doubling, tripling, or halving fractions when I change the size of a recipe. I rarely add or subtract fractions when cooking, except that I somtimes use repeated subtraction or repeated addition to do the dividing or multiplying.

2. Popularity of Sudoku means that that puzzle became a reality for so many people. Their reality did not include Sudoku until the puzzle had been invented. This should serve a model for math education or any other kind of education. Students' reality is important and must be taken into account by a teacher who molds their reality to include what s/he teachers. But mold s/he has to do.

3. Thank you for your comments. I see teachers as cultural ambassadors for their disciplines. Being able to share my authentic interests and the accepted norms of the scholarly community are important aspects of my role as a teacher. Some of these ideas (measurement division) and activities (Sudoku) may be new to my learners and I will need to support their adoption into the learners' schema.

I also want learners to add their own unique perspectives to our exploration of the discipline. Who knows what new ideas will be created, challenged, and consolidated as a result? Maybe we are talking about different levels, but as a teacher I would rather be a model than a molder.

4. I think context is very important, more than function(utility) - but that's just my opinion. This might help clarify my point.

Stories provide great contexts.

5. I think there is always a tension between real-world applications of what students learn in school and the need for simple thought play, as I call it.
At one end of the spectrum you have this push for making the context "authentic" and rooted in adult-like life, and at the other our innate need for imaginative play.

While you refer here to mathematics only, I see it as a broader issue with education today. The focus on STEM, on project-based learning and the general dismiss of humanities as being "non-productive" subjects makes me think how much we need to recover from this abyss. I don't believe school should be a replica of the world of work to this extent. Students need to play, experiment, create, wonder beyond it.