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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