Region 11 Math and Science Teacher Center

Region 11 Math and Science Teacher Center

Equality

Session Goals: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

Carpenter, Franke, & Levi… (2003)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)

National RecommendationNational Recommendation

The Importance of Equal Sign Understanding in the Middle Grades

–NCTM Mathematics Teaching in the Middle School May 2008

Knuth Research Knuth Research

Analyze your thinkingAnalyze your thinking

Explore the True/False equations.

As you work, think about your own thinking.

Share your answers with a partner.

Benchmark 1Benchmark 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

Benchmark 2Benchmark 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

Benchmark 3Benchmark 3

CALCULATING EQUALITYStudents 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

Benchmark 4Benchmark 4

RELATIONAL THINKINGStudents 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.”

Our Teaching Goal…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

Observe student thinking (1.3):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?

Another look:Another look:

Use observation sheet to record

student thinking

Again watch for operational or

relational and benchmarks for each

student

PLC’s: PurposePLC’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.

PLC StructurePLC 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

PLC StructurePLC 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?

Explore the Assessment Tools: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

SummarySummary

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

Region 11 Math and Science Teacher Center

Region 11 Math and Science Teacher Center

Math Success: It’s In Our Hands