Sense Making, Seeing Structure, And Making Generalizations Breakout Report for Sense Making, Seeing Structure, And Making Generalizations

Classroom Scenario: Comparing Two Fractions Problem: Which of the following fractions is greater? 29 / 9 35 / 11

Classroom Scenario: Comparing Two Fractions 29 / 9 35 / 11 Suppose a student gives the following solution: 29 divided by 9 = 3 R 2 35 divided by 11 = 3 R 2 So 29 / 9 = 35 / 11.

Classroom Scenario: Comparing Two Fractions Questions: What might a teacher do next in this situation? What mathematical understandings or practices might help a teacher navigate this situation?

Next Steps and Practices What to do next Try to get another student to put up a different solution – elicit disagreement Use concrete representations of fractions or a context to understand the problem Use an extreme example Be willing to say “I don't know – let's explore this”

Next Steps and Practices Teacher skills and practices: Understanding what the fractions represent (MP.1, MP.2) Understanding that “3 R 2” is not a precise expression of a number (MP.6) Understanding of appropriate use of the = sign Ability to connect various representations and student approaches (MP.3, MP.5) Belief that we can discuss this question and make progress, even if we don't know the answer right now

Classroom Scenario: The Border Squares Problem Consider an n x n square, divided into unit squares. If we shade all of the unit squares that are on the boundary of the figure, how many squares get shaded?

Classroom Scenario: The Border Squares Problem Questions: How can a teacher structure this task so that it is accessible to students and invites mathematical practices to occur? What mathematical practices seem likely to occur?

Aiming for Mathematical Practices Suggestions for Task Structure: Encourage students to work on specific cases first: 3 x 3, 4 x 4, 5 x 5, … Ask students to solve a specific case in different ways Have students represent the relationship between n and the number of border squares in different ways (table, graph, …) Use a large case (e.g., 100 x 100) to push students to generalize Have students write their reasoning

Aiming for Mathematical Practices BUT: All of this must be situated within the mathematical content teachers intend to teach. Need to think about learning progressions when deciding what to do next.

Thinking About Structure A possible framework for getting preservice teachers to think about structure: Part I: Solve a set of problems Part II: What “mathematical structure” do these problems have in common?

Thinking About Structure Example: (a) How many three-digit numbers can we make using each of the digits 1, 2, 3 exactly once? (b) If three people run a race, how many different possible outcomes are there (first, second, third)? (c) How many symmetries does an equilateral triangle have?