Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning

Jo BoalerCathy Humphreys

About the Book• Eight cases of teaching and an

accompanying set of student interviews• Year-long project videotaping two seventh-

grade math classes (higher tracks)• Filmed everyday for entire year• Connections between:

• students’ ideas and between student and teacher ideas

• areas of math – encourages reflections on ideas, representations, and different domains such as algebra and geometry

Eight Cases

1. Building on Student Ideas – connections between algebra & geometry

2. Building Understanding of Algebraic Representation – algebraic expressions, variables, and functional dependency

3. Defending Reasonableness 4. Introducing the Notion of Proof – algebraic equivalence5. Class Participation – students share feelings on talking

publicly and different methods of eliciting participation6. Continued discussion of Proof7. Volume of Prisms and Cylinders 8. Surface Area – deriving a formula for SA of a cylinder

The Case for PCK• The demands teachers face when

questioning a student are complex!• First listen to student, understand his

reasoning, and locate his ideas in the broader mathematical terrain.

• Consider what the student is capable of and willing to do plus consider a mathematically productive direction for the student (and class).

• Post a good question – accessible, challenging, and useful.

Instrumental vs. Relational Understanding

• Cathy Humphrey’s Teaching Journey – select segments p. 7-9

• “Being able to perform the appropriate algorithmic procedures… does not indicate any depth of understanding.”

• “But for me as my colleagues, mathematics had always been about following directions.”

• “I learned that I had developed an “instrumental” understanding of algebra (what to do) rather than a “relational” understanding (what to do and why).”

Developing Mathematical Proficiency• The cases were selected based upon interesting,

unexpected, and sometimes difficult moments that occurred during ordinary lessons.

• Theme emerged… students grappling with making connections among math ideas, representations or models of real-world contexts… which is at the heart of math proficiency.

• As teachers, if we are able to more profoundly connect our math understanding to our growing understanding of how children learn math, the fabric of our own teaching can become stronger.

Learning Mathematics• Means making sense of mathematical relationships.• Means getting better at the action verbs used to describe

the thinking habits mathematicians routinely use: looking for patterns, conjecturing, justifying, analyzing, wondering, and so on • Relates to Mathematical Practices• edTPA – developing a Language function for central

focus of a learning segment

Teaching Mathematics• Means helping all students learn to think mathematically.

Students need to understand that:• Math is more than arithmetic• There is no direct route to understanding• There are many ways to be good at math• Many ways to approach most problems, even those

with only one answer.• We can get better through practice, but talking and

listening to each other (Not just the teacher!) about mathematical ideas help us understand the ideas in different ways. We really understand what we can EXPLAIN.

How to use the book• Read Cathy’s notes about designing the

lesson• Watch the case• Ready Cathy’s reflections• Read Jo’s Case Commentary• Use CD discussion questions as source

for reflection and discussion as well.