the focus on “looking for patterns” also resulted in specific skills-based revelations from students as they worked towards solutions including:
Here is a glimpse into some of their observations and conversations.
It is important to note that though the work of some of these students in attempting to develop algebraic representations of factorial (!) extended well beyond the scope of the Grade 7 math curriculum, the intended outcome of this work was simply to have all students organizing their thinking systematically such that they could be certain in each case, that they had found all the possible ways of buttoning a given number of buttons. We encouraged students not to suppose that there were 6 solutions to 3 buttons, but to know, because… The nature of the problem also encouraged learners working at a higher level to identify that writing out all 120 ways for five buttons (for example) might not be necessary and to look for ways in which their work could be structured in order to simplify the problem solving process. All students were encouraged to look for generalizations or make conjectures as they worked through the problem, some of which were expanded upon or disproven as they progressed. Some of these conjectures have been included below.
In the Galileo Educational Network’s recent digital publication: “Focus on Inquiry”, they elaborate on the difference between a strictly “thin or superficial knowledge that comes from a general, surface acquaintance with important concepts or ideas”, and the “deep knowledge that engages [these concepts] within the central ideas of a topic or discipline” (see Chapter 1: Knowledge Building within Discipline Based Inquiry). Textbooks, and curriculum, for the sake of organization and accessibility, are generally structured in very linear, segmented formats. Teaching is a practice that engages the teacher as a liaison between these individual concepts and the depth and breadth of the discipline within which they exist. Teaching in the inquiry-based classroom requires that students solve problems, construct explanations, and work with simple concepts to construct more complex understandings.
One of the biggest challenges of this work can be communicating what this work looks like, and how it connects to those individual concepts. This year, as a teaching team, Jocelyn Monteith and I are focused on trying to tell that story.