Thursday, August 20, 2015

Zooming in and Zooming Out: Considerations for Math Curriculum Planning

Use the scale of the universe website. Zoom in, all the way to the Planck Length. Zoom out, to the edge of the observable universe. Now consider your own perspective on curriculum (a very personal thing). How much detail is too much? Do we see more, or less, when we zoom all the way in  on the curriculum particulars, or do we see more when we zoom all the way out and try to find the big ideas underpinning the topic or subject matter? 

I've zoomed in often enough. My 2005 Ontario math curriculum is missing the cover, and it's spattered with coffee and highlighting, scraps of sticky notes, random notes, and scribbles. "Know thy curriculum", should be our mantra.

It takes a long time to understand how it works, longer still to figure out how to make the language of of overall and specific expectations come alive. It takes years to know it well enough to get past the idea of specific curriculum expectations as a kind of shopping list for your year or course.

Here are some planning considerations that I think make the job of working with curriculum easier. 

My first key point:  know the front matter. In the case of the math curriculum, the mathematical processes come first, and as it says:

-the mathematical processes cannot be separated from the knowledge and skills that students acquire throughout the year.

These five specific processes (reasoning and proving, reflecting, connecting, selecting computational tools and strategies, and representing) are tied in to the larger processes, which are: problem-solving and communicating. The curriculum characterizes these as the actions of math. We need to think about what sorts of things we want our students to be doing in the math classroom.



Your second concern is the overall expectations for each grade. We know from our assessment and evaluation document, Growing Success, that we must base our evaluations on the overall expectations. Using the specific expectations (many that they are) as a checklist for teaching won't do-this is a recipe for frustration. Been there, and done that, when I didn't know any better.

Consider any connections you can make between the overall expectations from each strand. Any connections you can find can only help you. Try and bust out of the "strand by strand" thinking. We do have to report on each strand, but cross strand tasks or problems will help students to see the connective tissue of math, and the underlying structure underneath.

Here is an attempt at using proportional reasoning to tie together a few strands and concepts:

Students will use multiplicative relationships to compare rates, ratios, fractions, and patterns in expressions, graphs.

Here is another bigger idea that I believe fits grades 4-8:


Percent, rate, ratio, and fractions are connected.

Try and consider the bigger ideas behind the curriculum expectations. This is difficult work, but it is the "'zooming out" part.

One version of the mathematical big ideas (by Randall Charles), are here:
http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigIdeas_NCSM_Spr05v7.pdf

Many people prefer the very user friendly "Big Ideas from Dr. Small:  http://www.nelsonschoolcentral.com/cgi-bin/lansaweb?webapp=WBOOKSITE+webrtn=booksite+F(LW3ITEMCD)=9780176110789

Another consideration is our new "Paying Attention to Math" series, which includes proportional, spatial, algebraic reasoning, and fractions. Each of these monographs allows us to focus on the reasoning underneath the math curriculum. In so doing, we find that proportional and spatial reasoning, for example, can be found "underneath" a lot of our curriculum expectations.

Here is a successful planning sequence I have seen in a number of collaborative inquiries:

-use the proportional, algebraic, or spatial reasoning monograph to frame the initial discussion
-find a matching mathematical big idea (e.g. from Dr. Small)
-consider what actions students will be taking (sometimes you want to target a specific math process(es)
-match your thoughts to an overall expectation from the curriculum (and specific ones, if  you are designing something across more than one strand)

The concept web from Paying Attention to Proportional Reasoning:


Here is the concept web showing the  surprisingly large number of things that can be linked to spatial reasoning:



Here is some work on a version of the classic "tables and chairs" problem:


A possible planning sequence for using this problem:
-look at Paying Attention to Algebraic Reasoning, and see what concepts are at play (will your students just make a table, or can they generalize to the nth term?)
-consider the Representation process (you could us a table, graph, expression, or picture here)
-a big idea could be: Algebra helps us to model situations efficiently. An extension concerns a hypothetical dinner party with 1000 tables. Does algebra help us as a "shortcut"?
-grade 7 overall expectation:  "represent linear growing patterns...using concrete materials, graphs, and algebraic expressions.

I want to be totally clear here that know thy curriculum is the number one principle at work here. You must know the specifics, and be clear on the particulars, before you can zoom all the way out and focus on the big ideas. But don't be afraid to "play", and find what works for you!



Wednesday, August 19, 2015

Critical and Creative Thinking in the Math Classroom (Updated)

I believe critical and creative thinking are both essential to doing math.  Yet I believe 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 the achievement chart is a footnote 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. Further compounding the problem, critical and creative thinking are, at best, ill-defined. The role of teachers in teaching critical thinking is debated- see Daniel Willingham’s piece here.


What is critical and creative thinking in the math classroom? What does it look like in the math classroom? I am starting from the presumption that all kids are capable of critical and creative thinking. My second presumption is that mathematical knowledge and skill gained as children go older allows them to think creatively and critically. Third, I don’t buy the typical (and somewhat ill-defined) notion that creativity and critical thinking are only typical of “higher order thinkers”. It depressed me to no end when I did my literature review on these two topics and found that much of the work on these two types of thinking were done with gifted learners.


The other common line of thinking is that critical and creative thinking are somehow opposite, or at odds or competing with each other. I don't buy this false binary. Typically this binary is set up as “making” versus “assessing” or “judging”. I believe that both are intrinsically tied together.

Here is an example I like to come back to. A student came up with his own method for predicting the career points scored of several hockey players. In his judgement, here they are:
Here's a nice quotation on critical and creative thinking:


"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. Our young mathematicians will make judgements as they are solving problems, deciding which path to follow, and when. They will pick the best representations for their mathematical work, and their own idiosyncratic mathematical voice will come out. (Given a classroom culture of math talk, our students will find their voices. “Voice” is not just for the English classroom) Doesn't that sound like critical and creative thinking, combined in one neat mathematical package?


I have a dislike for overly complicated frameworks and definitions that clutter and obscure important concepts.  Einstein may have said something about how if you understand something, you can explain it to a child. If we can explain the quantum world without jargon, we can explain educational concepts without jargon, so here goes. Here are my personal working definitions of each:


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


Yes, these are deliberately economical. Yes, you could add to these definitions if you wanted to. But if you are a student, and you are doing a mathematical problem or task, you are making something new every single time. There will be patterns and trends in the strategies and tools that individual students use that further differentiate more “unique” or “divergent” work which will perhaps “more” creative.  I also maintain that, provided we don’t oversimplify our mathematical tasks to take students’ judgements away, they will be constantly hypothesizing, choosing, testing, and revising their work.  


How does this happen in the math classroom? 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 I started this post with: both 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. It will always be our job to consolidate purposefully, and to offer suggestions as to more efficient or effective solutions. The range and variety of the student work, with all its understandings and misunderstandings will lead us to that point. A balanced math program with strong foundations and and a spirit of questioning will always lead to interesting lines of inquiry-questions leading to more questions.

The beautiful diversity of student work:
Here is a video where we analyze the student work in our LearnTeachLead project, "Loving the Math, Living the Math." 


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. That does happen, but not always.


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.

Here is an example of a student finding a new use for Minecraft as a thinking tool to represent data:



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 questions, to teachers asking questions, 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: 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. If they are, our students will be constantly making judgments, testing them, revising them, and drawing meaningful conclusions about the important mathematical work of the classroom.
See:  

Treffinger, D. J., Isaksen, S. G., & Stead-Dorval, K. B. (2006). Creative problem solving: An introduction (4th ed.). Waco, TX: Prufrock Press.

"Loving the Math, Living the Math" at LearnTeachLead is here.