There are multiple parts to each practice. The parts help students develop the habit of mind that is the main practice. Remember that the practices are defined as ways to help students become mathematically proficient. As we look at each practice, think of ways we can help students to take ownership of these practices.
In the second video, students are learning how to determine whether or not a situation is fair. The Essential Question asks: “How can proportions help you decide if things are fair?”
Observe how the teacher begins with a demonstration so that students can develop their understanding about fairness. What questions does she ask? Students are seeing the relationships between different quantities and are able to discover the meaning of those relationships. When they begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.
Mathematical Practice #2: Reason abstractly and quantitatively.
• Mathematically proficient students make sense of quantities and their relationships in problem situations.
• Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships:
– the ability to decontextualize – to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents
– the ability to contextualize – to pause as needed during the manipulation process in order to probe into the referents for the symbols involved
• Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
As you look at your classroom, you probably see students with varying degrees of expertise in this practice. Our job, as educators, is to help students develop a habit of mind that helps them naturally think before they begin to reason abstractly and quantitatively. Ask yourself:
Do you give students enough time to understand the relationship between the different quantities in the problem?
Are students able to visually represent the relationship between the two quantities?
Can the students explain the visual representation to demonstrate an understanding of the problem?
As students take ownership of their learning and develop expertise using the mathematical practices, the content standards (knowledge, skills and understandings, procedural skills and fluency, and application and problem solving) will make sense, allowing students to achieve success in mathematics.