The Alberta Math Curriculum is extensive, and for the sake of organization and accessibility it is structured in a very linear, segmented format. Considering how we might connect key skills and concepts with the broader discipline of mathematics in ways that engage diverse groups of learners and breathe life into the Mathematics Curriculum requires not only that we engage with the central ideas of the discipline but that we become intimately familiar with the curriculum document. Thankfully, this work is incredibly engaging. Both the “Introduction” and the “Conceptual Framework for K-9 Mathematics” contain such gems as:
“A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.” (p. 14)
“A true sense of number goes well beyond the skills of simply counting, memorizing facts and the situational rote use of algorithms. Mastery of number facts is expected to be attained by students as they develop their number sense. This mastery allows for facility with more complex computations but should not be attained at the expense of an understanding” (p. 17)
The “Conceptual Framework” outlined by the Alberta Mathematics Program of Studies aligns perfectly with our inquiry-based approach to pedagogy at Connect. It also reinforces the importance of coaching students to develop both computational fluency as well as mathematical habits of mind through problem-solving.
With these “mathematical process” outcomes in mind, we introduced the “button” problem (adapted from nRich) in order to encourage students to consider how a systematic approach to problem solving might help them recognize patterns or connections that could lead to generalizations. As is often the case with this inclusive approach to mathematics, 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.