We had an interesting discussion about mathematical tasks this morning, specifically, rich tasks, and what makes them so.
A principal in the room brought her unique physical education background to the discussion. If you have, say, your baseball unit happening, and you have some elite level players in your class, and some who have never touched a bat and glove before, how do you structure your classes?
I think it's fair to say you can deploy what Boaler, and others, call "low floor, high ceiling" tasks. Check out the YouCubed tasks here and see what you think of them. Further, you can recognize that some students will need more time with thinking tools, or exploring of concepts, where other will need more time on specific practice elements.
This brought to mind my own experiences in grade 9 phys ed class. I was only ever good at one sport, basketball. This was because of endless hours spent shooting baskets at the school down the street. This led to being on the basketball team. That year, the basketball unit in phys ed class happened in the middle of basketball season. The games in class were too easy. Most people didn't know how to play basketball at all. Having only this one single entry point to the game of basketball was probably quite frustrating to those who were just learning how to play. That said, those at lower skills levels would have benefited more from skills drills, than from being in a game they weren't prepared for.
I think we see something similar play out in our math classes. Many of our students come with prior knowledge of the topics that will be taught, especially as math is now more "out there" in the culture than ever. (If you don't believe me, ask your students what they know about pi!) Others will struggle to even enter the mathematical content, having gaps in their skills or conceptual knowledge, or a history of struggling with math.
The challenge for us then is structuring our classes so everything can have something to do that will challenge them, as they work toward understanding the big ideas of the math curriculum, and acquiring the skills they will need to progress.
Here is an example of a task, Four 4s, that I think nicely allows students at all levels to practice their operational skills. This is less a task, and more a class of tasks. In other words, this task could easily generate many more extension or follow up tasks involving operations. "Use all/any operations to make a certain number/integer/fraction", etc. It is possible to enter with just adding and subtracting, then multiplying and dividing, negative integers, squares, exponents, square roots, and so on. In my experience, students who have the comfort with the basic operations will try and come up with more and more elaborate expressions- nested brackets, exponents raised to exponents, fraction bars, and so on.
Another example: change "basketball class" from my example to "integer class".
If our curriculum goal is understanding and being able to add and subtract integers, consider all the actions any given student (or small group of students, for guided explicit instruction) could take:
-working with 2 colour tiles to explore the zero principle
-working with number lines to understand integers visually
-exploring a context for integers, like golf or temperatures
-doing a worksheet on adding and subtracting integers, if they are ready
Many more activities are possible. Having flexible groupings based on readiness, and providing direct instruction as needed are the key things here. The math teacher is more like a coach, providing feedback, guidance, and explicit instruction as needed. "Sage on the stage" and "guide on the side" are outdated descriptions of what we do- perhaps "coach in the middle" is more like it. Here is a piece I wrote on being a guide on the side, versus being a "coach in the middle".