On Target?
The Mathematics
The first set of bullseyes concerns the mathematics at the heart of a lesson.
We ask three big sets of questions:
What is the main mathematical idea? How does it develop? How is it connected to what students know? How is it connected to the grade level content and practice standards?
In what ways do students engage with the mathematical content? What connections are built between procedures, underlying concepts, and meaningful contexts of application?
In what ways do students engage in mathematical practices or other activities that build productive mathematical habits of mind?
Begin an interactive "target practice" by clicking on a target ABOVE.
There are many ways to open up the mathematics to enrich students’ experience of it. As illustrated in the introduction, problems can be re-framed so that students have opportunities to do more of the mathematical work – e.g., by conjecturing likely results and determining their validity, rather than being instructed to validate results they are told are true.
Problems can be posed such that multiple approaches and strategies are possible. If students approach a task in different ways, then comparing, contrasting, and connecting the mathematics involved provides a way of making mathematical connections and deepening the treatment of the mathematical content. Multiple approaches provide ways to engage with the content at different levels of cognitive demand, providing access to the mathematics to more students. (See, e.g., Lotan, 2003, who discusses “group worthy” problems.) Making connections can help students who have adopted more rudimentary approaches see how they link to more sophisticated mathematics, while those who adopted the more sophisticated approaches may need to “stretch” to see the connections to earlier mathematics. (For example, take a classical word problem that results in two linear equations in two unknowns. Some students may graph the equations, and some may solve them algebraically. You can see why and where the two graphs will intersect. Can you see that information when you look at the equations?)
Asking for justifications or applications provides opportunities to delve into the mathematical practices, and asking “is there another way we can see this?” provides opportunities to make connections. Asking if there are advantages to taking one approach over another helps students become more strategic thinkers. (For example, students may have learned a range of techniques for solving simultaneous linear equations. Given a particular pair of linear equations, would it be more advantageous to use substitution, or elimination? Why? Are there easier or harder ways to combine the equations? Operating at this level provides practice in being strategic and resourceful, above and beyond learning the relevant techniques.)
Making a habit of expanding mathematical opportunities does a number of things. It enhances students’ experience with mathematics as a deeply connected discipline. It sets the stage for their developing the full range of mathematical proficiencies – content knowledge, the ability to implement problem solving strategies, to think strategically and metacognitively, and to develop the kinds of beliefs and habits of mind that are mathematically productive (see, e.g., Schoenfeld, 1985; Kilpatrick, Swafford, & Findell, 2001).
The bullseyes on the following five pages correspond to the questions highlighted above. As you think about a mathematical task or activity you plan to use, think about its attributes. What does it do for students? In which ring of the bulls-eye does it land? Could you modify the task or activity so that it lands more toward the center? What would you add?
The Mathematics
The extent to which classroom activity structures provide opportunities for students to become knowledgeable, flexible, and resourceful mathematical thinkers. Discussions are focused and coherent, providing opportunities to learn mathematical ideas, techniques, and perspectives, make connections, and develop productive mathematical habits of mind.