1. Recap
Summarize (on an index card) a key idea or insight you have gained from our work together so far...
Understanding equality (e.g., equal sign, levels, relational thinking)
Mathematical representations
OA  EE  Algebra progression
Other
Then take turns sharing and discussing.

2. HL-TP3. Use & connect mathematical representations
Using concrete objects to act upon or manipulate the idea (cubes, paper strips).
Illustrate or show the math idea with diagrams, pictures, number lines,
graphs, and other math drawings.
Recording the idea with numerals, variables, tables, and other symbols.
Situating the
math idea
in real-world, imaginary, or mathematical situations or contexts.
Marshall, Anne Marie, Alison Castro Superfine, and Reality S. Canty. “Star Students Make Connections.” Teaching Children Mathematics 17, no. 1 (2010): 39–47.
To develop students’ representational competence: (1) Discuss the explicit connections among representations; (2) Alternate directionality in making connections; and (3) Encourage purposeful selection of representations.
(Marshall, Superfine, & Canty, 2010)
Effective mathematics teaching includes a strong focus on developing students’ “representational competence” through making important connections among contextual, visual, verbal, physical, and symbolic representational forms.
Using language (words) to interpret, state, define, or describe the math idea.

3. Homework Due Wednesday, July 2, 2014
Design a sequence of open number sentences or true/false statements that you might use to engage your students in thinking about the equal sign.
Describe why you selected the problems you did and the rationale for the sequence of tasks.
Equation-structure items (p. 331) Matthews article

4. Course Assignment
by Sunday night: Sequence of T/F or Open Number Sentences and Rationale (5% of grade)
Equations
Rationale z

5. Pathways to Teacher Leadership in Mathematics Wednesday, July 2, 2014
Equality and Relational Thinking: Abstracting from Computation
Common Core State Standards for Mathematics
Pathways to Teacher Leadership in Mathematics
Wednesday, July 2, 2014
Session Description:
Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning,
to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

6. Learning Intentions & Success Criteria
We are learning to understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value.
Success Criteria
We will be successful when we can recognize the difference between computational and relational thinking
We will be successful when we can use relational thinking to build, express, and justify mathematical relationships.

7. Shifting Perspective Session Description:
Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning,
to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

8. Math Practice 7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure…(CCSSM). They can also step back for an overview and shift perspective. What shifts in perspective supported your thinking as you engaged in relational thinking during yesterday’s class?
Read MP 7 out of Think Math to add to their knowledge of the what it means to develop a habit of mind of looking for and making use of structure.
The CCSSM suggest mathematical habits of mind that are essential for mathematical proficiency – one of these is MP7.
Structures can be: place value, the properties of the operations, or generalizations about the behavior of the operations
***Developing Algebraic Habits of Mind by Mark Driscoll is very helpful to facilitating and structuring this discussion.

9. Supporting Relational Thinking
True or False? = 100 – – 28 = 86 – = What shifts in thinking did you have to make to view these true/false sentences through a relational lens?
Chart the shifts in thinking that participants made to think relationally
Structures can be: using place value, the properties of the operations, or generalizations about the behavior of the operations

10. When mathematical ways of thinking begin to become automatic – not just ways one can use, but ways one is likely to use – it is reasonable to call them habits: mathematical habits of mind. --EDC Transitions to Algebra

11. Developing a Habit of Mind
Fill in the box to make these statements true: = x = 46 + ¾ + x = ¼ +
Meaning of the equal sign and thinking about the relationship between the quantities
Equation needs to balance
Structure of the base 10 system
Also MP 8.

12. What happens when…
How would students typically approach solving for x?
11 –
3x – 2
= 6 50
Read MP 7 out of Think Math to add to their knowledge of the what it means to develop a habit of mind of looking for and making use of structure.
The CCSSM suggest mathematical habits of mind that are essential for mathematical proficiency – one of these is MP7.
Structures can be: place value, the properties of the operations, or generalizations about the behavior of the operations
***Developing Algebraic Habits of Mind by Mark Driscoll is very helpful to facilitating and structuring this discussion.

13. 1.OA.7
Work with addition and subtraction equations.
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false
6=6, 7=8-1, =2+5, =5+2
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.

14. 5.OA.2
Write and interpret numerical expressions. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × ( ) is three times as large as , without having to calculate the indicated sum or product.
Where does the idea of understanding the meaning of the equal sign come up. Consider students in first grade and the expectation set as they are working to understand this fundamental idea for developing algebraic reasoning.

15. 6.EE.2b
Apply and extend previous understandings of arithmetic to algebraic expressions.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient);
view one or more parts of an expression as a single entity.
For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

16. 6.EE.4
Apply and extend previous understandings of arithmetic to algebraic expressions.
4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).
For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. .

17. Furthering Relational Thinking
Session Description:
Take a journey into the “Core” to inspect progressions of mathematical ideas and student learning,
to surface shifts from current practice, and to consider implications for instruction, curriculum, and assessment.

18. Learning Intention & Success Criteria
We are learning to…
Examine and further strengthen “relational thinking.”
We will be successful when we can…
Use structure to identify relations between expressions on either side of the equal sign.
Use structure and repeated reasoning to reason relationally.

19. True or False ?? 13  9 = 90 + 27 13  9 = 130 – 13 Individually
 13  9 = 130 – 13
Individually
Decide if the statement is true or false.
Make notes to keep track of your thinking.
Group
Share your thinking.

20. 13  9 =
13  9 = 130 – 13
What approaches were used?
computational strategy
relational thinking strategy
Be prepared to share a computational approach and a relational thinking approach for each equation.

21. Thinking back to 8 + 4 =  + 5 Computational approach
Compute the “answer” of 12 and reason from there.
Relational thinking approach
Reason using relationships and properties and do not need to find the “answer” of 12. “5 is 1 more than 4, so the number in the box must be 1 less than 8.”

22. 1. Select a facilitator. 2. Facilitator pulls out an equation strip and shares it with group. 3. Individually decide if the statement is true or false. Keep track of your thinking. 4. Facilitator asks each person to share their decision and the reason behind it.

23. True or False ???
6  7 = 6   8 = 2   6 = 8    1 = 7  8 6  9 = 5   9 3   8 = 21 8 8  6 = 8  8 – 8  2 13  7 =

24. Seeking and Using Structure
1st viewing: Listen for the relational thinking demonstrated by the students. 2nd viewing: Listen and note how the students use the distributive property to explain their relational thinking. Finally, using the script as a reference, write an expression that corresponds to each student’s reasoning and that highlights the distributive property.

25. MP 7 & MP 8
Read MP7 (p. 18) and MP8 (p. 19) from K-5 Elaborations of the Practice Standards. Use the phrase below to make connections from what you have read to the thinking in the video. …they (students) have identified and described these structures through repeated reasoning (MP8). What structures are students employing?

26. Seeking and Using Structure
Articulate the underlying structure(s) that helped you decide if these statements are true or false.
Structures may include:
Place value
Properties of the operations
Meaning of the operations

27. “Children have a great deal of implicit knowledge about fundamental properties in mathematics, but it usually is not a regular part of mathematics class to make that knowledge explicit.” ---Carpenter, Franke, & Levi, 2000, p. 47 Key Questions • Will this work with all numbers? • Will this work with other operations? • Why is this working?

28. Using Relational Thinking to Support the Learning of Arithmetic
How do we engage all students in developing this habit of mind? The development of students’ mathematical thinking should not be perceived as one more topic to teach. Ideally, it should be an integral part of the teaching of arithmetic concepts and skills.

29. Distributive Property: A Structure to Support Algebraic Reasoning
Distributive Property of Multiplication over Addition and Subtraction a x (b + c) = (a x b) + (a x c) (b + c) x a = (b x a) + (c x a) a x (b – c) = (a x b) – (a x c) How do these statements make the meaning of the distributive property explicit?
Pick one or two of the number sentences. How do these statements make explicit the meaning of the distrubutive property.

30. Learning Intentions & Success Criteria
We are learning to understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value.
Success Criteria
We will be successful when we can recognize the difference between computational and relational thinking
We will be successful when we can use relational thinking to build, express, and justify mathematical relationships.

31. Disclaimer Pathways to Teacher Leadership in Mathematics Project
University of Wisconsin-Milwaukee,
This material was developed for the Pathways to Teacher Leadership in Mathematics project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Any other use of this work—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors—without prior written permission is prohibited.
This project was supported through a grant from the Wisconsin ESEA Title II Improving Teacher Quality Program.