Constructivist Integrated

Mathematics and Methods for Middle

Grades TeachersRebecca Walkerand

Charlene Beckmann

Grand Valley State UniversityDepartment 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 strategiesMisconceived or partially conceived strategies

Integrated Content and Pedagogy

Unit 1: Creating a Learning CommunityReasoning and Patterns in Algebra

Unit 2: The Learning of MathematicsRational Numbers and Their Uses

Unit 3: Planning and InstructionGeometry, Measurement, and

TransformationsUnit 4: Assessment

Data Analysis and Probability

Developing Deep Content and Pedagogical

KnowledgePedagogical and content knowledge are built

together in a variety of ways, includingAnalyzing 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 discussContent and process objectives of the activity,Sample or expected student responses, andHow 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 DivisionAnalyzing 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 DotsA 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.

1st step 2nd step 3rd step

●●●●●●●●

●●●●●●●●●●●●

2 dots 6 dots 12 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.

1st step 2nd step 3rd step

●●●●●● ●

●●●●●● ● ●

2 dots 5 dots 8 dots

Students’ Work on Sara’s Dots

David’s StaircasesDavid's patterns look like staircases.

1st staircase 2nd staircase 3rd staircase

1 step 3 steps 6 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 2 Recipe 3

Recipe 1 Recipe 3

Recipe 1 Recipe 2

Recipe 1 Recipe 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 considerThe 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 ofLengths 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

a2 4ab2

b2 4

ab2

c2

so a2 b2 c2

Analyzing Student Activities

Benefits to teachersThey 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, walkerre@gvsu.edu

Charlene Beckmann, beckmannc21@aol.com