Habits of Mind: Experience before Formalization
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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 necessarily completed, before instruction, giving students informal experience with the ideas of 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.
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.
The exercise shown above is typical for a Getting Started lesson. It asks students to use their experience in playing the Guess My Rule game to develop informal ways of describing properties of functions such as, "For any input, you always have to give the same output. You can't change it." Getting Started exercises are designed to preview main ideas, suggest experiments that may lead to conjectures about mathematical content, and activate prior knowledge that students will need during the investigation.
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-outexamples.
As the investigation continues, formal definitions 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 questions or cover topics that students don't remember, and everyone may find the lack of resolution uncomfortable at times. It can be tempting to get bogged down 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 confusion. However, early struggles will motivate students to learn the formal results. Their experience will give them a context in which they can understand definitions and theorems. As students see their own conjectures become proven theorems, they will feel ownership of and connection to the mathematics they study.