Making Algebra Realistic Knowledge (MARK) The First Step: 5th & 6th Grade Diana Underwood Marcia Weinhold & Gayle Millsaps Purdue University Calumet Diana Underwood Marcia Weinhold & Gayle Millsaps Purdue University Calumet

What is Algebraic Reasoning? 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? 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? 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) 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 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! 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 Tables Subs 44/5816 Kids

Ratio 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 Reasoning Division Fractions Percents Ratios Rates Units Division Fractions Percents Ratios Rates Units

What did the teachers do with it? Jerry’s lesson: “Al Capone does my Laundry”Al Capone does my Laundry” Donna’s banana problembanana problem Jerry’s lesson: “Al Capone does my Laundry”Al Capone does my Laundry” Donna’s banana problembanana problem

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? 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 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.