Making Algebra Realistic Knowledge

(MARK)The First Step: 5th & 6th

Grade

Making Algebra Realistic Knowledge

(MARK)The First Step: 5th & 6th

GradeDiana Underwood

Marcia Weinhold &Gayle Millsaps

Purdue University Calumet

Diana UnderwoodMarcia Weinhold &

Gayle MillsapsPurdue University Calumet

What is AlgebraicReasoning?What is AlgebraicReasoning?• Algebraic reasoning is a process in which students generalize mathematical ideas from a set of particular instances, establish those generalizations through discourse, and express them in increasingly formal and age-appropriate ways (Kaput, 1995;1999)

• Algebraic reasoning is a process in which students generalize mathematical ideas from a set of particular instances, establish those generalizations through discourse, and express them in increasingly formal and age-appropriate ways (Kaput, 1995;1999)

What does REALISTIC mean to us?What does REALISTIC mean to us?• Realistic Mathematics Education (RME) is an instructional design theory and an instructional approach

• Philosophy: Mathematics instruction should guide students toward reinventing mathematics and students should experience mathematics as a sense-making activity

• Realistic Mathematics Education (RME) is an instructional design theory and an instructional approach

• Philosophy: Mathematics instruction should guide students toward reinventing mathematics and students should experience mathematics as a sense-making activity

What does REALISTIC mean to children and their teachers?

What does REALISTIC mean to children and their teachers?• Tasks are engaging to students because they are grounded in real world contexts

• Accessible to all students• Encourages students to construct models of/for reasoning

• Tasks are engaging to students because they are grounded in real world contexts

• Accessible to all students• Encourages students to construct models of/for reasoning

How can submarine sandwiches make math real? (an example from CFLM)

How can submarine sandwiches make math real? (an example from CFLM)• Sharing 3 subs among 4 kids, 4 among 5 kids, 3 among 5, and 7 among 8 kids– Was it fair?– How can the subs be shared?– How much will each child in each group get if the subs are shared equally?

– How can these quantities be compared?

• Sharing 3 subs among 4 kids, 4 among 5 kids, 3 among 5, and 7 among 8 kids– Was it fair?– How can the subs be shared?– How much will each child in each group get if the subs are shared equally?

– How can these quantities be compared?

Making ConnectionsMaking Connections

• 36÷4• 72÷8• 144÷16

– What do you notice?– All have the same answer– You did the same thing to both sides

– The ratio is 9 to 1!

Ratio TablesRatio Tables

Subs 4 4/5 8 16

Kids 5 1 10 20

9 36 72 144

1 4 8 16

Ratio Tables as a model of/for reasoningRatio Tables as a model of/for reasoning

• Initially was a tool to record students’ mathematical activity

• Evolved into a tool for mathematical reasoning for the teachers

• ALSO Evolved into a pedagogical tool that had many other applications for the teachers

• Initially was a tool to record students’ mathematical activity

• Evolved into a tool for mathematical reasoning for the teachers

• ALSO Evolved into a pedagogical tool that had many other applications for the teachers

Proportional ReasoningProportional Reasoning

• Division• Fractions• Percents• Ratios• Rates• Units

• Division• Fractions• Percents• Ratios• Rates• Units

What did the teachers do with it?What did the teachers do with it?

• Jerry’s lesson: “Al Capone does my Laundry”

• Donna’s banana problem

• Jerry’s lesson: “Al Capone does my Laundry”

• Donna’s banana problem

Where can it go??Where can it go??

• Linear relations• Slopes• Rates• Rates of change• Related rates??

• Linear relations• Slopes• Rates• Rates of change• Related rates??

What can be learned?What can be learned?

• When given opportunities to engage in realistic contexts AND pedagogical tools that support students’ learning, teachers will engage their students in mathematical inquiry

• When given opportunities to engage in realistic contexts AND pedagogical tools that support students’ learning, teachers will engage their students in mathematical inquiry

Questions for us?Questions for us?

• Questions for Discussion:– What are you using to deepen teachers’ mathematical understanding for teaching?

– What challenges have you faced?– What lessons have you learned?

• Questions for Discussion:– What are you using to deepen teachers’ mathematical understanding for teaching?

– What challenges have you faced?– What lessons have you learned?

The instructional practices and

assessments discussed or shown in this

presentation are not intended as an

endorsement by the U.S. Department of

Education.

The instructional practices and

assessments discussed or shown in this

presentation are not intended as an

endorsement by the U.S. Department of

Education.