However, when you return to placing sample tasks into your grid, it seems you’ve removed the student’s experiencing of the task from the decision making again

I’m admitting several times throughout the post that the horizontal axis is really, really subjective, but not so subjective it isn’t worth talking about broadly. I place Shaughnessy’s partitioned square task in the “abstract context” category, for example, even though for most people here it’s fairly concrete. It’s a useful frame for me but I try to keep its limits in mind.

Do, Draw, Dream (aka concrete, 2D, abstract).

To be able to do abstract maths students need to experience it, before they will be able to represent it on paper and then as a process/formula. Of course they can go straight to abstract but I think we can safely say that most of our students won’t ‘get it’ this way. This means that if I really want to build a bridge for my students to get to me, I have to meet them where they are and bring them across – not yell from the other side ‘you should be here!’. We do lots of ‘do’ activities before giving them any recipe for something (ie this is how you multiply fractions). Another example with rename 327 at least 4 different ways allows them to show me they know: 327 ones, 32 tens & 7 ones, or 3 hundreds, 2 tens and 7 ones and then they must think and learn to rename eg 7 ones is 70 tenths because this is important when dividing or multiplying (not just carrying the number to the next column!) This is to build a robustness of their knowledge and understanding and allows them to be able to maths at a higher level later on with confidence.

This realization opens up a more interesting discussion, I think….one that you have started here on your blog. If it isn’t real-world context, what is it about a task that engages students in the type of rich mathematics/thinking that we know to be the true benefit of math education? My action research this year has led me to hypothesize that students enjoy tasks in which imposition from the teacher (in terms of expected content outcomes) is most minimal. They enjoy exploring, looking for patterns, making and testing conjectures, and pursuing individualized paths from an open/rich task. I continue to think that the less I dictate the path of their work, the more they enjoy it. The less I hold them accountable to some concept/idea, the more they feel a sense of agency and capability as a mathematician. The curriculum must be the product of THEIR work, not some plan of MINE. For me, this doesn’t mean we necessarily need to abandon a general outline of suggested standards but it does mean that we need to abandon this sense of accountability and “mastery” that (to me) are all false indicators of success and security.

I appreciate that you have created an ongoing dialogue for all of us about designing interesting tasks for students. You empower teachers to free themselves from crappy textbook problems (read: exercises) and create an opportunity for their students to become participants in doing mathematics again.

I think so… In comparison to these Learning Trajectories I hear Jere Confrey insert into the CCSS conversations.

What I like about this framework is this: I’m fearful of saying “it’s ok if your tasks aren’t concrete” because that tends to let teachers off the hook. I say “hey! let’s use math creatively in fun and interesting ways of exploring shapes!” and colleagues hear “it’s ok if I keep having kids solve 25 equations”. This better parses the difference between alleged authenticity in a task versus authenticity in the student behavior.