Habits of Mind: Experience Before Formalization

January, 2011

In the CME Project curriculum, every investigation begins with a Getting Started lesson. These special introductory lessons include exercises that are to be tried, but not necessarity completed, before instruction. The exercise sets in a Getting Started lesson preview main ideas, suggest experiments that may lead to conjectures about mathematical content, and which activate prior knowledge that students will need during the investigation.

Here's an example of one exercise from the Getting Started lesson that kicks off Algebra 1's Investigation 5A: Functions---The Basics. It follows two exercises that ask students to play the Guess My Rule game, in which players try to guess a function rule from input-output pairs.

3. In Round 1, Sasha is the Rule Maker.

Tony says, "That rule's not fair!" What makes

Tony say that?

In responding to this question, students will have to think about what constitutes a "fair rule." They may or may not remember a formal definition of function, but they can see that in the context of the game, if you get different outputs for the same input there's no way to use earlier responses to predict future responses. They will develop informal ways of describing this property of functions, such as "For any input, you always have to give the same output. You can't change it."

The idea for the Getting Started lesson came out of our early field-test experience and feedback from our Teacher Advisory Board. Teachers wanted a curriculum that balanced two seemingly opposing goals:

Students need a chance to grapple with mathematical situations and discover key ideas for themselves. Students need a textbook that can act as a reference, with formal definitions, theorems, and worked-out examples. By using Getting Started lessons to introduce an investigation, students gain experience in mathematical situations related to the concepts they will be studying. They struggle, experiment, conjecture, and discover. Then, as the investigation continues, they will draw on these experiences as they make sense of the mathematics. The formal definitions in the text will give names to concepts that students have already talked about in their own words, and theorems will act as capstones, summarizing students' conjectures and experience.It's important that teachers step back during a Getting Started lesson and let students do the work themselves. It can be tempting to step in and provide answers to their questions or cover topics that students don't remember, and students may find the lack of resolution uncomfortable at times. It can be tempting to get stuck in a Getting Started lesson, trying to make sure that students master the material and complete all the exercises, rather than trusting the rest of the investigation to take care of lingering questions and confusions. However, these struggles will motivate students to learn the formal results. Their experience will give them a context in which they can understand definitions and theorems, and as they see their own conjectures become proven theorems, they will feel ownership of and connection to the mathematics they study.