# Describe, verify, justify

I have just finished a series of patterns investigations with my math 10 class. In this specific series, students explored the connections between quadratic equations (in both vertex and factored form) in order to connections between the equation and the features of the graph (like the position of the vertex, and intercepts).

These kinds of activities are a regular feature of math in the IB Middle Years Program (MYP) in which students are required to consider a series of specific situations to identify a pattern, describe it as a general rule, verify that their rule works and then justify why it works.

I’ve had a love-hate relationship with this kind of task since I started teaching MYP math almost 10 years ago.

I hate investigations because…

• Students find it difficult to approach novel situations.
• I find it difficult to prepare students for novel situations.

However, I love investigations because…

• Investigating is what mathematicians do. While math students typically spend more time doing tests (i.e. using specific examples to demonstrate that they understand a general rule that someone else discovered), mathematicians are on the frontier of searching for new pattern, generalization and rules. By engaging students in this process (even if they are discovering patterns that are new to them, but well-known to others), they are engaging in the authentic work of mathematics.
• Investigations support inquiry skills. In looking at a variety of specific situations in order to find trends, patterns and generalization, students develop strategies of problem-solving, visualization, hypothesizing and generalizing. These skills support similar processes in other disciplines, like finding trends in a data set, identifying themes in a work of literature or cause-and-effect relationships in history.
• Investigations foster independence. By looking for patterns and general rules, students develop the confidence to use what they know as the basis to discover new things, making them less reliant on the teacher as the source of knowledge.
• Investigations cultivate persistence. Not all problem-solving strategies will work in every situation. The more experience students have investigating patterns, the more comfortable they will be with trying an approach and switching to a different strategy when necessary. Rather than seeing this as a set-back, they will accept it as a normal part of the process.

As you can see, the benefits of investigations outweigh the challenges. Promoting a classroom culture in which students are willing to take a risk on an unfamiliar problem and persist with challenging work requires on-going effort, but I believe the benefits are on-going as well.