1. Effective Mathematics Teaching Practices The Case of Kelly Polosky
Principles to Actions
Effective Mathematics Teaching Practices
The Case of Kelly Polosky
and the Triangle Task
Fifth Grade
Please Note: Although this lesson took place in a fifth-grade classroom, the content currently aligns most closely with the sixth grade in Common Core and other state standards. We recommend using this module with a middle school audience.
Background
This session focuses on Procedural Fluency from Conceptual Understanding. It is suggested that teachers have some familiarity with the Effective Mathematics Teaching Practices prior to engaging in this session.
There are three sessions available that provide an overview of the Teaching and Learning Principle, with some discussion of each of the effective teaching practices -- The Band Concert (elementary school content), The Candy Jar (middle school content) or the Pay It Forward (high school content).
For this session you will need to provide teachers with copies of the materials highlighted in green below.
1-Slides-MS-Polosky
2-TriangleTask-MS-Polosky
3-EffectiveMathematicsTeachingPractices-MS-Polosky.docx
4-VideoClip-MS-Polosky.mp4
5-Transcript-Clip-MS-Polosky
6-TeacherandStudentActions-MS-Polosky
7-SomeTriangleTasks-MS-Polosky
Developed by Michael D. Steele at the University of Wisconsin-Milwaukee. Video courtesy of the Wilkinsburg School District and the Institute for Learning.
These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Lynn Raith, and Michael Steele.

2. Overview of the Session
Solve and Discuss the Triangle Task
Watch video clips and discuss what the teacher does to support her students’ engagement in and understanding of mathematics
Discuss the effective mathematics teaching practices of procedural fluency from conceptual understanding
Suggested time allotment:
Task: 45 minutes
Video and identification of mathematics teaching practices: 45 minutes
Brainstorming follow-up activities in the unit: 30 minutes
TOTAL TIME - approximately 2 hours

3. The Triangle Task
Using the triangles shown above and the grid paper, construct a formula for the area of a right triangle. You may cut your triangles out and manipulate them in any way that might help you make your argument.
After you explore with your particular triangle, make a general mathematical argument using words, symbols, and diagrams about the formula for the area of any right triangle.
Handout-TriangleTask-MS-Polosky
Engaging teachers in solving the Triangle Task will take approximately 45 minutes. If possible, you should produce the triangles on card stock or posterboard and provide participants with scissors, rulers, tape, and additional grid paper.
There are two primary ways in which teachers might make a mathematical argument about the area of the triangles. One possible path is to create a surrounding rectangle with the same length and width as the triangle, and to argue that the triangle takes up exactly half of that area. A second solution path is to cut the triangle in half either lengthwise or widthwise, and to align the cut portion such that a rectangle is formed with the same length (or width) and half of the width (or length).
It is recommended that you share both solutions. Slides 19 and 20 show two visual representations of the second strategy.
Task adapted from Everyday Mathematics Grade 5, Copyright © 2007 by Wright Group/McGraw-Hill, Chicago.

4. The Triangle Task Video
School District: Wilkinsburg
School: Kelly Elementary School
Coach: Darin Cole
Teacher: Ms. Kelly Polosky
Class: 5th grade
Curriculum: Everyday Mathematics
This small district is adjacent to the City of Pittsburgh and features a diverse student body. A coach, Mr. Cole, is also present in the video interacting with students.
The students have just begun the geometry unit. Prior to the day’s lesson, the students sorted polygons and non-polygons and they identified the characteristics of polygons. They have also found the formulas for the area of rectangles and squares.

5. Mathematics Learning Goals
Ms. Polosky’s
Mathematics Learning Goals
Students will understand that:
The area of a triangle is ½ of its length times its width.
The relationship between area, length and width of a triangle can be generalized to a formula.
There are several equivalent ways of writing the formula for the area of a triangle and each can be related to a physical model.

6. Connections to the CCSS Content Standards
Geometry★ G
Solve real-world and mathematical problems involving area, surface area, and volume.
1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors.

7. Connections to the CCSS Standards for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
In this lesson, Ms. Polosky is working on several of the Standards for Mathematical Practice in varying degrees. Her most explicit targets are 2, 3, 4, and 7. Students are asked to model the area situation, make use of the structure of area to develop and justify an area formula for a triangle, construct arguments that justify their formulation of area, and de-contextualize the initial triangle situation into a formula and re-contextualize their formula in justifying why it works for all triangles.

8. The Triangle Task: Video Context
Prior to this clip:
Ms. Polosky launched the task, ensuring that students understood what they were being asked to do
Ms. Polosky provided students with cardboard triangles and other tools such as scissors and grid paper
In this clip, we see Ms. Polosky facilitating a whole- class discussion that focused on comparing two possible formulas for the area of a right triangle generated by students.

9. Lens for Watching the Video Time 1
As you watch the video, make note of what the teacher does to support student learning and engagement as they work on the task. In particular, identify any of the Effective Mathematics Teaching Practices that you notice Ms. Polosky using. Be prepared to give examples and to cite line numbers from the transcript to support your claims.
WILL NEED TO REVISE LINE NUMBERS WHEN FINAL TRANSCRIPT IS MADE
Handouts - EffectiveMathematicsTeachingPractices-MS-Polosky and Transcript-MS-Polosky
Teachers are likely to notice Effective Mathematics Teaching Practice 5 (pose purposeful questions) and 6 (build procedural fluency from conceptual understanding). Specifically:
MTP6: The task chosen and the discussion of the generalizations is an example of laying the groundwork for MTP 6 – this lesson could have easily been executed by giving students the formula and having them practice using it. By generating the formula and making mathematical arguments, students developed a richer conceptual understanding.
WHEN LINE NUMBERS ARE READY, ADD SOME OF THE PROBES ABOUT PUSHING FOR GENERALIZATION IN SMALL GROUPS
MPT5: Ms. Polosky’s questions are purposeful in supporting students in justifying their formulas and determining whether their formulas will work for any triangle.
Some particularly salient examples:
Line 112: How can I prove that this is right? (advancing question, targeting SMP3)
Lines : Do you think there’s another way besides counting [that] you could have come up with this? (advancing question, laying groundwork to connect mathematical ideas)
Lines How are you going to solve for the area of this triangle now? What was your thought process? (assessing student thinking, emphasizing importance of explanations)
Line Do those [formulas] mean the same thing? (Connecting mathematical ideas)
Here are some of additional ideas that teachers might notice:
The teacher moves the students progressively from consideration of the specific example to the generation of a general formula
A wide variety of questions are asked that focus on the mathematical meaning and clarifying students’ ideas
The teacher consistently presses students to justify that their methods and conjectures are correct
Two formulas are presented that can be compared; they are equivalent
The discussion and use of the two formulas are consistently connected back to the examples and mathematical meaning
You may want to chart the things teaching notice (and the practices they identify) so you can keep track of what has been said. If building procedural fluency is is NOT something teachers mentioned, you might want to ask teachers: Why was the teacher working so hard trying to get students to come up with a formula when it would have been so much easier to just give it to them?

10. Mathematics Teaching Practices
Effective
Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.
WILL NEED TO REVISE LINE NUMBERS WHEN FINAL TRANSCRIPT IS MADE
While many of the Effective Mathematics Teaching Practices can be seen in the excerpt from Ms. Polosky’s lesson, today we are going to focus explicitly on build procedural fluency from conceptual understanding.
This is an example of laying the groundwork for MTP 6 – this lesson could have easily been executed by giving students the formula and having them practice. By generating the formula and making mathematical arguments, students developed a richer conceptual understanding.

11. Procedural Fluency from Conceptual Understanding
Mathematics classrooms should:
Provide students opportunities to use their own reasoning strategies and methods to solve problems;
Press students to explain and discuss why the procedures they are using work for particular problems;
Use visual models to support students’ understandings of general methods
To use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. (Martin, 2009, p. 165)

12. Lens for Watching the Video: Time 2
As you watch the video a second time, make note specifically of how you see Ms. Polosky supporting students in developing procedural fluency from conceptual understanding. Be prepared to give examples and to cite line numbers from the transcript to support your claims.
Watch the video again with a specific focus on procedural fluency from conceptual understanding. The next slide provides examples of some of the specific evidence that teachers might identify related to this effective mathematics teaching practice.

13. Mathematics Teaching Practices
Effective
Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.
WILL ADD IN SOME SPECIFIC IDEAS HERE WITH THE FINAL TRANSCRIPT

14. Connecting Conceptual and Procedural Knowledge
Imagine that following the this lesson, Ms. Polosky engages her students in solving real-world tasks using area. For Tasks A-D, consider:
Which tasks might support Ms. Polosky’s students in advancing from a generalization for right triangles to a generalization for all triangles?
What aspects of the tasks provided might be challenging for her students to solve, and why?
This is an extension activity to engage teachers in if you have time to do so.
Of the four tasks provided, tasks A and C directly move students towards generalizing the formula developed in the lesson to any triangle. Task A is similar to the task Ms. Polosky enacted, but does not explicitly address right triangles. Task C begins with a right triangle and asks both for multiple right triangles and non-right triangles.
Tasks B and D would be tasks we would want students to be fluent in by the close of the unit, but does not directly support students in thinking about right triangles. Task D is framed in a way that invites multiple procedures, but may not give students enough scaffolding to move from the right triangle formula to a more general formula based on where her students were at. Students who learned one of the methods for finding the area of the right triangle in Ms. Polosky’s class could solve the first part of the task, but might need some additional support to be able to solve the second part of the task with the acute triangle.

15. Connecting Conceptual and Procedural Knowledge
Consider Task B. If Task B were the type of task that Ms. Polosky would want students to be procedurally fluent in solving as a part of an assessment, what aspects of conceptual understanding would help support that fluency? How might a teacher foster those aspects of understanding?
This is an extension activity to engage teachers in if you have time to do so.
Aspects of conceptual understanding to support finding the area of a triangle:
-Identifying that the base and height (or length and width) are the two quantities that determine the area of any triangle
-Understanding that other variances (such as side length or angle measure) don’t change area as long as base and height remain the same
-Knowing that area is a two-dimensional covering of a shape
-Understanding that the base and the height have a right angle relationship, and that you can choose any side as the base as long as the height is defined properly
-Knowing that changing the orientation of a triangle does not change the base and height of the triangle
A teacher might foster these understandings by giving students experience with:
-finding the area of a wide variety of triangles (with respect to angle category and orientation)
-asking students to find the area of a triangle, and then create another non-congruent triangle with the same area
-asking students to construct two triangles with the same angle measures and different areas
-asking students to identify the base and the height on a wide variety of triangles
-asking students to find the area of the same triangle using two different base-height pairs

16. Consider the Teacher and Student Actions Related to Developing Procedural Fluency from Conceptual Understanding
What will you need to work on in order to develop procedural fluency from conceptual understanding in your own classroom?
Where will you start?
Handout - TeacherandStudentActions-MS-Polosky
You could end with these as rhetorical questions for teachers to think about, have teachers write an exit slip where they address the questions, or have teachers talk with their peers in small groups about the questions.
If you choose the later, you might ask for volunteers to share how they will get started.

17. Build Procedural Fluency from Conceptual Understanding Teacher and Student Actions
What are teachers doing?
 What are students doing?
Using visual models to support students’ understanding of general methods.
Providing students with opportunities for distributed practice of procedures.
Striving to use procedures appropriately and efficiently.

19. Strategy 2: Example 1 (Why 1/2 x w x l works

20. Strategy 2: Example 2 (Why ½ x l x w works)