1. Region 11 Math and Science Teacher Center Equality

2. Session Goals:  Identify benchmarks students reach on their way to understanding equality  Identify strategies students might use in solving equations  Understand PLC structure you will use in your school during the coming month

3. Carpenter, Franke, & Levi… (2003) …contend that a “limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra. Virtually all manipulations on equations require understanding that the equal sign represents a relation.” Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School. Portsmouth, NH: Heinemann, 2003, p. 22)

4. National Recommendation

5. The Importance of Equal Sign Understanding in the Middle Grades –NCTM Mathematics Teaching in the Middle School May 2008 Knuth Research

6. Analyze your thinking Explore the True/False equations. As you work, think about your own thinking. Share your answers with a partner.

7. Benchmark 1 BASIC NUMBER SENTENCE SENSE Students begin to understand writing number sentences and can describe their understanding of the equal sign (correctly or incorrectly). 8 + 4 = ____ + 5 Children would answer 7, 12, 17, or 12 & 17

8. Benchmark 2 EXPERIENCE WITH A VARIETY OF EQUATIONS Students explore equations that go beyond the form of a + b = c They understand that equations in these forms might be true: 7 = 3 + 4 2 + 8 = 5 + 5 356 + 42 = 354 + 44

9. Benchmark 3 CALCULATING EQUALITY Students recognize that the equal sign represents a balance of both sides. Students carry out calculations to determine that the two sides of an equation are equal or not equal 8 + 4 = ___ + 5 12 12 Students need to catch both sides to fill in the missing value

10. Benchmark 4 RELATIONAL THINKING Students compare the expressions on each side of the equation and check for balance/equality by identifying relationships among numbers and reasoning instead of actually carrying out the calculations. 8 + 4 = ___ + 5 “7 is the missing number because 5 is one more than 4, so I need a number that is one less than 8.”

11. Our Teaching Goal… Moving students  away from the idea that the equal sign (=) means “the answer comes next”  toward understanding equal as meaning “the same (amount) on both sides” and toward using relationships between numbers to determine equality or inequality

12. Observe student thinking (1.3): How does the teaching sequence move students along in their understanding? How does the teacher use questioning strategies to build from students’ misconceptions? Where do you see examples of the different benchmarks in understanding equality? Where did students start out? Where did they end up?

13. Another look: Use observation sheet to record student thinking Again watch for operational or relational and benchmarks for each student

14. PLC’s: Purpose 1. Apply Teacher Center training ideas to your teaching; 2. Learn what to listen for to better assess student understanding; 3. Form a community to support each other.

15. PLC Structure  Week 1: Share and record baseline assessment student data  Week 2: Discuss teaching strategies  Week 3: Share student interview assessment data  Week 4: Share and record summative assessment student data

16. PLC Structure  Baseline Assessment: How do my students compare to other classes?  Teaching strategies: How can my teaching impact student understanding?  Student interviews: What is the range of understanding in my classroom?  Summative Assessment: What did students learn?

17. Explore the Assessment Tools: Do the Baseline Assessment. With a partner: discuss the related rubric and scoring grid  Review the Student Interview  Review the Summative Assessment

18. Summary Equality is an equivalence relationship Our goal is to help students think relationally about equations Equality is a convention - students cannot discover it; you must teach it explicitly and keep coming back and reinforcing it We can watch and listen for various benchmarks that can help us identify where students are in their thinking about equality Students will use various strategies to solve equations on this journey toward understanding, including calculation, unwinding, and algebra

19. Region 11 Math and Science Teacher Center Math Success: It’s In Our Hands