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In the documented lessons of Fran Dickinson (5th/ 6th grade) and Cathy Humphreys (9th/10th grade), several of the mathematical practice standards are evident within one lesson/problem. Here, we invite you to explore their classrooms to learn how they engage their students in mathematical habits of mind.
Holistic Approach to the Standards of Mathematical Practice
Although the Common Core identifies eight standards of mathematical practice and Inside Mathematics highlights the ways in which all eight are evident in many different classrooms, Dickinson and Humphreys commonly engage their students in multiple practices simultaneously. These single clips demonstrate the students’ capacities in several practices.
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Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he harvests students’ observations about the underlying operation that yields specified outputs. His students demonstrate that they can:
- CCMP 3: “justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.”
- CCMP 6: “communicate precisely to others..the meaning of symbols.”
- CCMP 7: “look closely to discern a pattern or structure.”
- CCMP 8: “notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity…might lead them to the general formula…”
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Cathy Humphreys engages her students in exploring properties of quadrilaterals. In this clip, her students into small groups to focus on specific quadrilaterals and use materials to test their observations. Her students demonstrate that they can:
- CCMP 1: “start by explaining to themselves the meaning of a problem and looking for entry points to its solution.”
- CCMP 3: “understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.”
- CCMP 6: “try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning... By the time they reach high school they have learned to examine claims and make ... use of definitions.”
- CCMP 7: “look closely to discern a pattern or structure.”
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Classroom Environments to Support Content Learning
To support their students’ content learning, Dickinson and Humphreys have established a rapport with their students and a set of norms for students to turn and talk to each other as they work together. We don’t expect teachers or students to immediately establish this climate on the first day of school, but even from the first day teachers can establish a context for students to engage in mathematical thinking. Dickinson and Humphreys serve as facilitators in their lessons—they lay out the task, and then invite the students to dig into the problems.
In these classrooms, students have mathematical conversations and are able to state their thoughts, and are also comfortable with making uncertainty explicit. In Dickinson’s classroom, for example, he and his students use a hand signal to make any dissent, disagreement, consensus, and confusion explicit. In Humphreys’ classroom, a poster reminds students that the problem solving process is a circuitous “tinkering” cycle, not a vector from problem → solution. In their classrooms, it’s okay to say what you think, and it’s also okay to disagree and change your mind. Too often, students’ reaction to mathematical confusion is silence. We believe these classrooms demonstrate a way to honor the process of developing mathematical understanding in students. The students’ independence in their math work rests on a climate of acceptance and mathematical risk-taking.
Learn More
Click below to see how 5th/6th graders and 9th/10th graders demonstrate most of the standards of mathematical practice in Dickson’s and Humphreys’ documented lessons; you can also see all classroom examples of specific standards here.
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
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Exemplary Lessons Integrating the Standards for Mathematical Practice