We cannot solve our problems with the same level of thinking that created them.
- Albert Einstein
If you're in mathematics education—or any education—it's nearly impossible to avoid task-design conversations. A most recent conversation, coupled with the topic of creativity, had me thinking of relationships between mathematics task-design and art. Specifically, art in the form of painting. Take the following four situations.
Situation A - Paint by Numbers
Situation B - Connect-the-Dots
Situation C - What's the Picture?
Situation D - A Blank Canvas
As with any analogy, don't think too hard on it or it will fall apart. With Situation A and B, no doubt something beautiful can be produced. But is the production process itself beautiful or creative? Is it likely that something novel is created? Or, is it more likely that something from the past is reproduced? With each, the playground for creativity is walled with boundaries well-defined. There is nothing loose (shoutout Jason Page for the share) about Situation A or B.
In contrast, Situations C and D open up a space for ingenuity, creativity, and experimentation. Situation C is admittedly more constrained than Situation D, but each allows the artist a space to create something new. Each provides a space for them to make decisions and express something unique to themselves (even if the end product might resemble or match something of the past). Each, to some extent, removes boundaries and affords looseness. The end product is ill-defined, and that generates the discomfort needed for transformational exploration and learning.
I think mathematical tasks often follow these situations. Tasks can often be closed-ended, either directly stepping a student through a problem or allowing some decision making, but not with respect to the overall mathematical structure or end product. Other tasks can provide a playground, sometimes structured with some initial guidance and other times left with no initial path and limited rules. An example of the latter is an inquiry project in which the individual must propose and pursue a self-directed point of inquiry, where minimal guidance is given with respect to the end product (note for teachers: this means no rubric, no matter how much you want to create one). Shifting such a project to Situation C, one might provide some guidelines as to the topic, a general theme for the point of inquiry, or a general structure fro the end product format (but not content). Modeling projects are also examples that fall into the category of Situation C.
We need more of Situations C and D (and related varieties of these; see Dr. Amy Ellis's Playful Math) in school mathematics. We need more of Situations C and D in our education system, no matter the subject or context. Our job is to promote inquiry, which is applicable to any field. Our job is to promote artistry, which is applicable to any life.