Thursday, May 14, 2015

Surprised By Their Mathematical Thinking

I have been thinking a lot about mathematical surprise these days.  Specifically, I have been thinking about all the different ways we can be surprised in our math classrooms.

We have worked a lot at creating mathematical thinking spaces for our students. The lovely monograph by Dr. Chris Suurtamm Making Space for Students to Think Mathematically nicely formulates how our classrooms can be mathematical thinking spaces.

The above image is some student work on a proportional reasoning problem. (I was going to say unit rates, but leading in that direction might already be taking some of the thinking away, don't you think?) 

The video clip linked here shows 3 of us analyzing the chart paper you see in the picture:  analyzing the thinking. Our students used many surprising strategies. Some were radically different, using completely different thinking tools, or perspectives on the problem.  Some were subtly different. There were huge mathematical implications on the rates they chose to use, for just one example.

I had a chat with a brilliant teacher the other day, and he told me he identified 26 distinct solution paths for one problem (the classic "tug of war") problem. It may be this Marilyn Burns one, or another- it's a classic context for a problem. One this one you may rightly note: we as experienced thinkers might choose to model it with algebra. This is only one type of solution! Don't underestimate the flexible thinking of the novice, with less experience and context to hem in their thinking! (Dr. Brent Davis out of Calgary calls this the mistake of the experienced thinker)

Once we remove our preconceived notions of what the problem "should" be, we can focus on what sort of thinking our students are doing. They will surprise us!  I haven't been involved in one collaborative inquiry where we haven't been surprised by at least some element of one solution! We must be open to surprise, in our classrooms.

One minor caveat: I am not saying here we are "surprised" because we don't know the math at hand. We must know the fundamental big idea in the math, and the connections to curriculum (both content and process). Many of our teachers and coaches are working from the "Five Practices for Orchestrating Productive Mathematical Discussions" book. The key practice for me is anticipation: we must know the math, do the math, and anticipate student responses to the best of our ability.

But, I maintain: the element of surprise will still remain. Sometimes it's a powerful conjecture from a usually quiet student. Other times it's a subtle variation on the math you expected. It could be a question that arises from a student, and spurs them to create more math.  Sometimes it's a lightning flash of insight- an amazing and new solution path that you have never seen before.

Last, what surprises will emerge from the murk and (seeming) mess, when students are allowed open spaces for mathematical thinking?

Tuesday, May 12, 2015

Critical and Creative Thinking in the Math Classroom (Outtake from OAME Ignite)

Critical and creative thinking are both essential to doing math.  Yet both are relatively unexplored areas with our young student mathematicians.

Here is the lone reference to critical and creative thinking in the Ontario curriculum:

The star below is a footnote below the achievement chart explaining that critical and creative thinking are present in some, but not all, math processes. It does not elaborate which! Obviously, this is not helpful- if the math processes are the actions of doing math, it makes sense then that these actions will, at times, encompass critical and creative thinking.

But what is critical and creative thinking in the math classroom?  I'm leaving aside here the debate over "traditional" and "new" methods in math teaching and learning. I am starting from the presumption that all kids are capable of critical and creative thinking. It depressed me to no end when I did my literature review and found that much of the work on these two types of thinking were done with gifted learners.

I also don't buy the false binary that critical and creative thinking are somehow "opposite" or "at odds" with each other. Typically this binary is set up as making versus assessing or judging. But I believe that both are intrinsically tied together.

Here's a nice quotation on the matter:

"These two ways of thinking are complementary and equally important. They need to work together in harmony to address perceived dilemmas, paradoxes, opportunities, challenges, or concerns (Treffinger, Isaksen, & Stead-Dorval, 2006).

Further, Poincare said something to the effect that mathematical creativity is simply discernment, or choice. Doesn't that sound like critical and creative thinking?

I have an intense dislike for overly complicated frameworks and definitions that clutter and obscure important concepts.  So here are my personal working definitions of each:

Creative thinking: making something new.
Critical thinking:  making sound judgements.

How does this happen in the math classrom? How can we harness these two powerful types of thinking?

In the first case, if we don't see math as a generative process, a creative process, then we will not find creative thinking. Look closely at the picture: problem-solving and inquiry are mentioned.  To the former: problem-solving classrooms will always have an element of creativity, unless we force our own methods, techniques and processes on our students.

One of the best parts of really getting to know your students is starting to see inside their idiosyncratic mathematical thinking. For a long time, I felt like creativity was that certain "je ne sais quoi" of the math classroom, a "know it when I see it" type of thing. When I thought this, I probably didn't have a broad enough definition of creative thinking. I was waiting to be bowled over by stunningly divergent solution paths (and that does happen!)

Since, I have been watching for more subtle evidence of creativity.  Students using new thinking tools, or subtly tweaking a solution path or process they may have got from talking with their classmates. Creativity is there to be found in the math classroom.

Inquiry is also hidden in that little line in the picture from the curriculum above. Inquiry to me means: asking good questions. Are our students question askers? There are some astounding numbers floating around about the ratio of students asking, to teachers asking, in a typical math classroom. Question askers are typically critical thinkers.  Once your classroom is an open space for wonder, your students don't stop wondering! Questions lead to answers, leading to more questions (I once called this the "inquiry tumbleweed").

The key thing is that students are becoming more confident in their judgements as young mathematicians.  I want them to be able to use their mathematical thinking tools to decide "what's best", or "what's fair". I want them to justify their thinking. I want them always probing the mathematical world around them with their confident judgements.

This is one of my favourite things to tweet now and again:

This work came out of our LearnTeachLead project involving proportional reasoning:  I found some very precision judgements happening, like students telling me a cup of pop was worth exactly $1.26. Not $1.25, not $1.27- $1.26. The power of their thinking led them to this conclusion.

There a nice quote in this book excerpt about how the "best way to think critically is to think critically". We are risking circular logic there, but think about it: the best way to learn to think, is to think. That is why our classrooms should be open thinking spaces.