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!