Like many math teachers, I have been following Dan Meyer’s discussion on “fake world” math tasks versus “real world” math tasks with interest, especially since one of my early blog posts was on this subject and one of the presentations I do for teachers is on this very topic.
My observation is that it is not the task that defines whether it is fake or real, it is the person doing the task. Our work then should focus on developing criteria on what makes tasks real for children, and then see which tasks support the criteria we establish. Here are some criteria I think we should consider when developing tasks for students, aside from the obvious; the task should engage students in mathematical thinking.
Children do mathematics in a socio-emotional context, and virtually always with the support of a mentor (peer, parent, teacher, etc…). A strong relationship with someone who can support their mathematical reasoning is critical. Tasks which the mentor finds interesting or appealing are more likely to be interesting or appealing to the children they support. There are lots of stories of lone mathematicians working in secret for years on developing mathematics, but I do not know of any stories of children doing the same.
Children are more engaged with tasks that they have questions about. By this, I do not mean the pretend inquiry questions that people sometimes start a unit with (Imagine here a teacher-led discussion that leads to a wall full of questions children made up on the spot to satisfy the “let’s make up our inquiry questions” game…), but actual questions that students have about the world and the objects in it.
Children need to be able to access the task and to do it substantially by themselves. A task where the adult with them has to do most of the work, either physical or otherwise, quickly becomes much less interesting for children. I built (from a kit) a compressed air rocket a couple of months ago with my son. He spent most of the time bored as I fit pieces together and he occasionally got to glue things together. Although he was very interested in the final product (who wouldn’t be interested in something that can shoot paper rockets up 50 metres in the air?), the process of making the rocket was tedious because he only had periphery access to the building process. This is true of mathematical tasks as well. Tasks where children have to rely on a list of “how to” steps provided by someone else are rarely interesting, unless some significant thinking has to occur to make the steps useful.
Children often like to do things because they are challenging so we have to be careful not to make things we ask children to do to be too simple when we are ensuring they have access. I once gave the Seven Bridges of Königsberg problem to my 9th grade class. The problem was accessible because every student felt like they have a possible solution path (ie. draw a picture), but it was challenging so the students kept working on it. Working on the problem became infectious, and soon, most of the 9th grade math classes in the school worked on the problem at least a little. Some of my 9th graders spent three weeks trying to solve the problem before finally coming to ask me to prove that it was impossible.
Children do not have questions about things with which they lack familiarity. If the context you are using is completely unfamiliar to children, then they aren’t going to have questions. If one takes the time to develop context around a situation (ie. story-telling), then it is more likely that children will begin to wonder about it. Every good game as a plot that hooks the player into the game, a good math task should do the same thing.
Note that these criteria all lead to an important conclusion; some tasks will be considered fake by some students, and real by others. It is important to note too that because of our shared society and context, there are some tasks which will be real for almost all students, and there are other tasks which will be fake for almost all students.