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

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"Use of critical and creative thinking processes." <most under explained line in @OntarioEDU math curriculum! pic.twitter.com/WYIJWbnAwb
— Matthew Oldridge (@MatthewOldridge) May 12, 2015
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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:

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As I said in my #Ignite, what is math for, if not to make critical judgements like this? #OAME2015 pic.twitter.com/uXprdl2kTb
— Matthew Oldridge (@MatthewOldridge) May 12, 2015
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This work came out of our LearnTeachLead project involving proportional reasoning: http://learnteachlead.ca/projects/loving-the-math-living-the-math-part-1/. 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.

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An open space for mathematical thinking. #EngageMath pic.twitter.com/H9no0tJOJy
— Matthew Oldridge (@MatthewOldridge) May 12, 2015
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Critical and creative thinking are a must in math. Even after solving a problem students are critical thinking when they are challenged to evaluate the efficiency of different strategies during a math congress. In order for students to wonder about the math they cannot be taught that math is a set of procedures. A set of procedures kills wonder and kills off creative and critical thinking. When I am following a recipe I am not wondering, but when there is an interesting open ended provocation or question I am. Most teachers in their school experience had a opportunity to wonder about something, to explore and they now see the value in that in other subjects. However math is another story. It is pretty much the only subject where people (even teachers) feel comfortable enough to say publicly that they are not very good at it! I think it has a lot to do with how they were taught math as set of procedures and so a cycle continues of students being taught a sequence of steps to solve math problems and effectively removing any critical or creative thought. Also most math strands seem to taught in isolation meaning that many skills or strategies that students have developed in one area do not seem to students as useful when dealing with different topics in math. The whole idea of inquiry makes some teachers feel uncomfortable because they are not completely in control. You don't know what is going to be said or in some cases discovered. You have to be able as a classroom teacher to let go and work collaboratively with your students. There have been many times where I have said "I don't know" to my kids followed up with a "let's find out". My students and I wonder a lot. They see connections in many things. At this point in the year recognize the value of mistakes and learning from those mistakes. They celebrate different ways to solve a problem. Critical and creative thinking must be a part of all subjects especially math but we need to find ways for teachers to let go of their procedures and allow instead for opportunities for students to become doers of math.

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