The scene has shifted away from the harshness of a 'lesson' toward more student-action-centred words like project, problem, prompt, or task. I like these words because they accurately describe what I am trying to do as a teacher--make the students think.
My personal favourite remains the "math task", or more desirably, the "rich math task".
The phrase "rich task" crept its way into my planning, blogging, and collegial conversations quickly. Maybe it was the way it rolled off the tongue. Maybe it was how refreshingly different it seemed when juxtaposed to 'lesson' or 'example'. "Tasks" seems free, and "rich" just seemed to fit.
The more I read, the more I realized that tasks have strong roots in the beginning of math reform. The 1991 Professional Standards for Teaching Mathematics say tasks "frame and focus students' opportunities for learning mathematics in school." (p. 24). There isn't a higher pedestal that they can be put on. They are the basis for a solid mathematical education.
Even more confusing to me, was the fact that in the same publication, the NCTM seemed to lump a bunch of words together under the heading of "task" which included the likes of "projects, problems, constructions, applications, exercises, and so on" (p. 24). I was soon losing my long-sought distinction; I wanted "tasks" to surface as a sort of distinctive champion of effective teaching, but it seemed far more inclusive than the barriers I had set up in my mind. The word had taken on a certain amount of "semantic inflation" (Piaget, 1969). It became used by many different people to mean many different things. It was rapidly approaching buzzword territory.
Along with the supposed interchangeability of the word "task" with many others, the adjective "rich" soon lost its lustre when I went looking. The keystone article in Rich and Engaging Math Tasks: Grades 5-9 (an NCTM publication) didn't refer to tasks as rich at all. Instead, the word "good" was used. Could such a primitive descriptor really be synonymous? It seemed like the word "rich" was only used to create a smooth title, and past that it held little unique consequence.
In an effort to pick up the pieces, I turned to colleagues and asked them what they felt constituted a "rich task". I got several responses, and distilled the requirements into categories. What fell out was a collective--albeit primitive--definition of a "rich" task. Those characteristics mentioned most appear at the top of the list.
A "rich task" must have:
- Multiple entry points
- The opportunity for divergent thought was a constant theme in the responses. The availability of a low-level entry option was important for engagement of all involved.
- Multiple solution paths
- Much like the first attribute, the divergence of thought and method was an important feature. This leads to valuable connections and communications throughout the process.
- A curious, captivating, or surprising element
- This was, by far, the most vague of the requirements but several people mentioned that student intrigue played a large role. What makes a task "curious" is variable from student to student, and very hard to determine.
- My favourite of the requirements, and often the hardest for classroom teachers to implement. A rich task provides natural extensions to students as they work toward solutions.
There you have it. A muddy, crowd-sourced look at what "rich" really is. Notice that it is taken for granted that the task involves meaningful mathematics. I think the constant framework of curriculum (especially at the high school level) makes these tasks infinitely harder to find, develop, and execute.
I'd love thoughts on tweaks, flaws, or downright blaspheme.
Here are some of my thoughts thus far:
1. Rich tasks do not need a "real-world" context or consequence.
There is no question that context does help in some instances. Context can help students frame the task in familiarity. The danger in focusing on the context alone (and not the attributes listed above) is that the real-world quickly becomes skewed into what Jo Boaler calls a psuedocontext. It transports the students to math land where anything can happen no matter how trivial or abstract. Context can be helpful if it is realistic and aids in the student's mathematical modelling.
2. Rich tasks should be paired with discourse.
This is where a skilled teacher separates themselves from the pack. The multiplicity of thought sets the stage for an exhibition of student learning. Rich tasks need to be taught within a rich ecology where students are sharing ideas, methods, and connections. Margaret Schwan Smith and Mary Kay Stein--who authored the keystone article mentioned earlier--have written an excellent book on teaching using productive discourse. It is a must read for any teacher hoping to use rich tasks in class.
3. Rich tasks are often shockingly pedestrian.
If there is one thing that the explosion of web 2.0 math resources has shown us, it's the ability to create a curious modelling context out of the most mundane of circumstances. An ounce of wonderment can go a long way. There are powerful mathematical moments waiting for students who explore the relationship between area and perimeter. Students can find it very empowering to be able to predict patterns minutes in advance. Something as simple as different arithmetic strategies can create a buzz.
NCTM, Professional standards for teaching mathematics, 1991.
NCTM, Rich and engaging mathematical tasks: grades 5-9, 2012.
Piaget, Science of education and the psychology of the child, 1969.
Boaler, What's math go to do with it?, 2008.