Constructivist Integrated Mathematics and Methods for Middle Grades Teachers Rebecca Walker and Charlene Beckmann Grand Valley State University Department of Mathematics Allendale, Michigan

“Teachers … need a general knowledge of how students think - the approaches that are typical for students of a given age and background, their common conceptions and misconceptions, and the likely sources of those ideas.” Adding It Up (2001, p. 378) “‘mathematical knowledge for teaching’ …allows teachers to assess their students’ work, recognizing both the sources of student errors and their students’ understanding of the mathematics being taught.” The Mathematical Education of Teachers ( 2001, p. 13)

Underlying Concepts  Mathematics teachers need a deep understanding of the mathematics their students are learning.  That understanding should be developed in ways that are substantively connected to teaching and learning activities. Ferrini-Mundy 2001  Teachers need to be familiar with a variety of ways of representing and thinking about mathematical ideas.

Focus on Student Understanding  Middle grades students are often familiar with a variety of representations for ideas.  Teachers need to be introduced to a spectrum of students’ ways of thinking.  Productive and insightful strategies  Misconceived or partially conceived strategies

Integrated Content and Pedagogy  Unit 1: Creating a Learning Community  Reasoning and Patterns in Algebra  Unit 2: The Learning of Mathematics  Rational Numbers and Their Uses  Unit 3: Planning and Instruction  Geometry, Measurement, and Transformations  Unit 4: Assessment  Data Analysis and Probability

Developing Deep Content and Pedagogical Knowledge  Pedagogical and content knowledge are built together in a variety of ways, including  Analyzing student responses,  Considering student uses of problem-solving strategies and representations,  Analyzing student pages.  Teachers engage in middle grades level activities. For each activity, they discuss  Content and process objectives of the activity,  Sample or expected student responses, and  How teachers can help students make sense of the mathematics without giving too much help.

Sample Activities  Learning about how middle grades students think through task-based student interviews:  Reasoning with Patterns: Marcy’s Dots  Thinking about student work and planning for intervention:  Developing Meanings of Operations: Fraction Multiplication and Division  Analyzing middle grades student activities:  Geometry: Pythagorean Tasks

Interviews: A Window on Student Thinking  Teachers learn more about student thinking when they observe students working.  Teachers interview middle grades students as a field experience.  Interview tasks and questions are assigned at the outset of each unit.  At the end of the unit, teachers  Share and analyze the students’ responses, and  Compare student responses to those reported in published articles by researchers

Marcy’s Dots A pattern of dots is shown at right. At each step, more dots are added to the pattern. The number of dots added at each step is more than the number added in the previous step. The pattern continues infinitely. 1 st step2 nd step3 rd step ●● ●●● ●●●● 2 dots6 dots12 dots Marcy has to determine the number of dots in the 20th step, but she does not want to draw all 20 pictures and then count the dots. Explain or show how she could do this and give the answer that Marcy should get for the number of dots. (Kenney, Zawojewski, and Silver, 1998, p. 474)

Shared Student Work

Students’ Work on Marcy’s Dots

Sara’s Dots Sara’s dots make a pattern that grows infinitely in the manner shown. How could Sara determine the number of dots in the 30th step without drawing all 30 pictures? Explain your strategy and find the number of dots in the 30th step. 1 st step2 nd step3 rd step ●● ●●● ● ●●●●● ● ● ● 2 dots5 dots8 dots

Students’ Work on Sara’s Dots

David’s Staircases David's patterns look like staircases. 1 st staircase2 nd staircase3 rd staircase 1 step3 steps6 steps How could David determine the number of steps in the 15th staircase without drawing all 15 staircases? Explain your strategy and find the number of steps in the 15th staircase.

Students’ Work on David’s Staircases

Interviews  Follow-up discussions are rich and varied.  Outcomes:  Teachers are surprised at the variety of students’ ways of thinking about the patterns; several are different from their own.  They observe that students’ misconceptions are not age related.  They learn a great deal about student thinking from a single activity.

Representations for Fraction Multiplication and Division  Activity introduces teachers to a variety of representations for rational numbers through student work.  Teachers are asked to  Analyze how students have begun each problem,  Figure out how to continue each problem using the student’s representation, and  Think about how the representation may connect to a traditional algorithm.

You have 2/3 of a pan of brownies. You give away 4/5 of what you have. What fraction of the whole pan have you given away? I started by drawing a pan of brownies and shading 2/3 of the pan.

 Your carpet measures 2 2/3 yd by 1 4/5 yd.  What is the area of the carpet? I drew a rectangle measuring 2 and 2/3 by 1 and 4/5. I see 2 full square yards. Now I need to figure out the other pieces.

You have 2 cups of milk. A recipe requires 3/4 of a cup of milk. a.How many full recipes can you make? b.What is left over? c.How many recipes can you make if no milk is left over? I tried to show 2 cups. I divided them into quarters. I know I can make more than 2 recipes but I'm not sure how much more. Recipe 2Recipe 3 Recipe 1Recipe 3 Recipe 1Recipe 2 Recipe 1Recipe 2

You have 2 cups of milk. This amount is 3/4 of what you need for one full recipe. How many cups are needed for a full recipe? I drew 2 cups. This is supposed to be 3/4 of what I need. I need more than this to make a full recipe.

Analyzing and Extending Student Work  Teachers are asked to  Make sense of possibly unfamiliar solution strategies,  Implement those strategies, and  Make connections to more traditional and algorithmic ways of solving the problems.  Through this work, teachers  Extend their repertoire of problem solving strategies,  Enhance their understanding of student solution processes, and  Enhance their understanding of traditional algorithms.

Analyzing Student Activities  Teachers are asked to consider middle grades activities. They consider  The mathematical intent of the activity,  The mathematical prerequisites students need,  What questions students might have as they work through the activity,  Questions to ask students to help them move forward when they get stuck,  Extensions for activities, etc.

Pythagorean Tasks  Directions for teachers:  Complete one of the tasks with your team. Find at least two mathematical observations you want students to make.  Write responses you could give to support students’exploration and conjecturing without preempting their thinking.  Imagine that a group of students has discovered something significant. What would it be? How would you encourage students to think beyond what they have found?  What relationships exist between tasks?  Should students complete 1, 2, or all 3 of the tasks?  If students should complete more than one task, in what order should they do so? Explain.

Tangrams and Right Triangles 1. Using tangram pieces, create a square on each side of the smallest isosceles triangle. 2. Assume the length of a leg of the small isosceles triangle is 1. Find the areas of the squares you have created. 3. Repeat steps 1 and 2 with the largest isosceles right triangle. 4. Find a relationship that works in both cases among the measures you have found for the areas of the triangles.

One Solution

Dot Paper and Right Triangles On square dot paper the distance between two adjacent horizontal or vertical dots is 1 unit. 1. Make a right triangle with legs 3 and 4 units long. Make a square on each side of the right triangle. Explain how you know your shape is definitely a square. 2. Find the areas of the squares you have created. Explain how you found the areas. Record your findings. 3. Repeat steps 1 and 2 for a right triangle with legs 6 and 8 units long. 4. Find a relationship that works in both cases among the measures you have found for the triangles.

Triangle with Legs of Lengths 3 and 4

The Right Triangle Puzzle 1. What is the relationship between the right triangle puzzle pieces? 2. What is the relationship between the side lengths of one of the right triangles and the side lengths of the squares?

A Right Triangle Puzzle Continued 3. Find a way to arrange all 11 puzzle pieces into two congruent squares without gaps or overlaps and using four right triangles in each square. Draw your solution. 4. Use the puzzle solution to find a relationship among the areas of the three square puzzle pieces. Explain your thinking. 5. Consider one of the right triangle puzzle pieces. Let a and b represent the lengths of the legs of the right triangle. Express the relationship in problem 4 using these variables. What does this tell you about the relationship between the lengths of the sides of a right triangle.

Right Triangle Puzzle Solution a b c

Analyzing Student Activities Benefits to teachers  They begin to think more deeply about using activities.  What mathematical purpose does the activity serve?  How will students respond to the questions asked?  How can I facilitate student learning?  They experience a variety of ways that mathematical ideas might be developed in a classroom.

Conclusions Teachers benefit from studying student work that they and others collect and from thinking about how students might interact with a mathematical task. Studying and analyzing student work helps teachers  Become better listeners,  Become familiar with a wide spectrum of ways of thinking,  Take students’ thinking more seriously  Think more deeply about the nature of students’ reasoning,  Think more closely and specifically about subsequent teaching steps, and  Work collegially with others.

Questions? Comments? Contact us at:  Rebecca Walker,  Charlene Beckmann,