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

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

3. Classroom Scenario: Comparing Two Fractions
29 / / 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.

4. 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?

5. 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”

6. 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

7. 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?

8. 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?

9. 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

10. 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.

11. 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?

12. Thinking About Structure
Example:
(a) How many three-digit numbers can we make using each of the digits 1, 2, 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?