Sunday, August 25, 2013

What Makes a Task "Rich"?

In my short career, I have seen the death of the lesson. I remember creating 'lesson plans' to the exact standards of my college of education, and then never looking at them when I began to teach. I was never really in tune with the rigidity of the plan, but knew that there were certain learning goals I needed to get to by the end on an hour. 

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.
  • Depth
    • 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. 

Nat Banting

Lazy references:

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.
Stein and Smith, 5 practices for orchestrating productive mathematics discussions, 2011.

Tuesday, August 6, 2013

Animating Patterns

There is a very strong emphasis on linear relations and functions in the junior maths in my province. In Grade 9, students begin by analyzing patterns and making sense of bivariate situations. The unit--which I love--concludes with writing rules to describe patterns and working with these equations to interpolate and extrapolate.

Grade 10 students continue along this path in the light of functions. There is a large degree of abstraction that occurs in a short amount of time, and droves of students abandon the conceptual background (pattern making) in favour of memorizing numerous formulas. (Slope formula, slope-point, 2-point-slope, slope-intercept, etc.)

For the record, I do not think that the transition between pattern sense making and formal function work should be made between grade levels. It is an unfortunate result of our curriculum structure.

Last semester, I worked a lot with visual patterns and group tasks. I began numerous classes with Fawn Nguyen's "Visual Patterns". We started with ten minutes, and by the end of the unit, we were making sense of several patterns within five. It served as a great warm-up or wind-down to a lesson.

We also undertook several #3Act pattern activities. The class favourite was Dan Meyer's "Toothpicks".  After they got their head around the fact that someone would "waste" all that time arranging toothpicks, there were several ingenious ways of making sense of the situation.

In fact, I noticed a distinct increase of strategies with the 3Acts approach. I started putting visual patterns into a stations format, and class participation increased. Students were creating patterns in novel ways, but still using very similar strategies to decipher and model them.

I was stuck--until a student made a comment to me during a station activity:

Me: "We know where the pattern starts, but how does it grow?"

Student: "It doesn't grow at all; it just repeats itself"

Right then I realized why it was so hard for students to describe growth, because the patterns were not growing. They were presented in stages on paper; paper--a static medium--has an incredibly difficult time representing growth.

Paper had reached its natural limitation. 

Some students just couldn't visualize the representation as a single, growing pattern. They wanted to see them as separate, and who could blame them? It makes much more sense conceptually to create a model that describes growth when you can see the growth occurring fluidly.

Here is my modification:

I began with the two problems from my textbook that I used with the inaugural relation stations (linked above).

 and used Vine to animate the growth that the textbook is trying to symbolize.

The short clips make use of colour and movement to animate the qualities of a linear relation. This is especially true for the rate of change. 

Here are three of Fawn's patterns animated with Vine:

The growth movement that these videos show may not be the pattern that the student was envisioning. This can result in larger misconceptions.

In cases where the pattern may be complex enough to entertain multiple patterns of growth, the teacher should first show a static pattern and ask students how they see the pattern growing as the stages progress. This activity holds the potential for rich classroom discourse. Students could use colour, shading, or prose to describe and illustrate their thinking. Only after the students have had time to visualize the movement, should the teacher show some animated patterns and begin working toward an equation or function to describe the growth.

In this case, several patterns may be anticipated and animated to show several avenues of thought. This takes careful anticipation on the part of the teacher. Here are two anticipated growth patterns of another visual pattern:

Great lessons meet students at their current level and provide the milieu to move them forward into deeper understanding. Animating the growth of patterns enables students to "see" the growth and make better sense of patterns when they are presented in stages, words, or in the world around them.