© 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Students’ Solution Paths to Maximize Student Learning Supporting Rigorous Mathematics Teaching

© 2013 UNIVERSITY OF PITTSBURGH

Selecting and Sequencing Students’ Solution Paths to Maximize Student Learning

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of EducationHigh School MathematicsGeometry

Rationale

Orchestrating discussions that build on students’ thinking places significant pedagogical demands on teachers and requires an extensive and interwoven network of knowledge. Teachers often feel that they should avoid telling students anything, but are not sure what they can do to encourage rigorous mathematical thinking and reasoning.

(Stein, M.K., Engle, R., Smith, M., Hughes, E. 2008. Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell)

In this session, we will focus on monitoring, selecting, and sequencing student work so you can assess and advance student learning during the Share, Discuss, and Analyze Phase of the lesson.


Session Goals

Participants will:

• learn what to monitor in student work when circulating during the Explore Phase of the lesson;

• learn about guidelines or “rules of thumb” for selecting and sequencing student work that target essential understandings of the lesson; and

• learn about focus questions that target the essential understandings.


Overview of Activities

Participants will:

• discuss the content standards and identify the related essential understandings of a lesson;

• analyze samples of student work;

• select and sequence student work for the Share, Discuss, and Analyze Phase of the lesson;

• identify “rules of thumb” for selecting and sequencing student work; and

• write focus questions that target essential understandings.


Midsegments Task

A midsegment is a segment that connects the midpoints of two sides of a triangle.

1. Draw a triangle on the coordinate plane and label the coordinates of the vertices. Draw and label the coordinates of the midpoints of the sides and then draw the three midsegments.


Midsegments Task (continued)2. Analyze the relationship between the midsegments and the sides of the

triangles. What conjectures can you make? Support your conjectures with mathematical evidence and compare your findings with the findings of partners.

3. Marco’s group made the five conjectures listed below.• The three midsegments of a triangle always have the same

length.• A midsegment is parallel to the side of the triangle that it does not

intersect.• The three midsegments of a triangle form an acute triangle.• The length of the midsegment is half the length of the side of the

triangle that it does not intersect.• The three midsegments create four congruent triangles.

a. Determine which conjectures are incorrect. For these conjectures, describe a triangle that Marco may have drawn for which this statement holds true. Then describe a counterexample for which the statement is false.

b. Determine which conjectures are true. Describe using words, diagrams, or symbols why the conjecture is a true statement.


The Task: Discussing Solution Paths

• Solve the task in as many ways as you can.

• Discuss the solution paths with colleagues at your table.

• If only one solution path has been used, work together to create others.

• Consider possible misconceptions or errors that we might see from students.


Linking the Standards to Student Solution Paths

The task has been selected with specific Standards for Mathematical Content and Practice in mind.

Where do you see the potential to work on these standards in the written task or the solution paths?

The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry

Common Core State Standards, 2010

Expressing Geometric Properties with Equations (G-GPE)

Use coordinates to prove simple geometric theorems algebraically

G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

Common Core Standards for Mathematical Practice

What must happen in order for students to have opportunities to make use of the Standards for Mathematical Practice?

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.


Five Representations of Mathematical Ideas

Pictures

Written Symbols

ManipulativeModels

Real-world Situations

Oral & WrittenLanguage

Adapted from Van De Walle, 2004, p. 30

Five Different Representations of a Function Language

TableContext

Graph Equation

Van De Walle, 2004, p. 440


The Structures and Routines of a Lesson

The Explore Phase/Private Work Time

Generate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions

2. Assess and Advance Student Learning

MONITOR: Teacher selects examples for the Share,

Discuss, and Analyze Phase based on:

• Different solution paths to the

same task

• Different representations

• Errors

• Misconceptions

SHARE: Students explain their methods, repeat others’

ideas, put ideas into their own words, add on to ideas

and ask for clarification.

REPEAT THE CYCLE FOR EACH

SOLUTION PATH

COMPARE: Students discuss similarities and difference

between solution paths.

FOCUS: Discuss the meaning of mathematical ideas in

each representation

REFLECT by engaging students in a quick write or a

discussion of the process.

Set Up of the Task

Share, Discuss, and Analyze Phase of the Lesson

1. Share and Model

2. Compare Solutions

3. Focus the Discussion on

Key Mathematical Ideas

4. Engage in a Quick Write


Analyzing Student Work


Analyzing Student Work

Use the student work to further your understanding of the task.

Consider:

• What do the students know?

• How did the students solve the task?

• How do their solution paths differ from each other?


Group A


Group B


Group C


Group D


Group E


Selecting and Sequencing Student Work


Monitoring SheetStrategy Who and What Order


Selecting and Sequencing Student Work(Small Group Discussion)

• Examine the students’ solution paths.

• Determine which solution paths you want to share during the class discussion; keep track of your rationale for selecting the pieces of student work.

• Determine the order in which work will be shared; keep track of your rationale for choosing a particular order for the sharing the work.

Record the group’s decision on the chart in

your participant handouts.


Standards and Essential Understandings

• Using coordinates of a midsegment of a triangle justifies that the midsegment is parallel to the side that it does not intersect because the slope of the segment containing the midpoints is the same as the slope of the segment connecting the endpoints of the third side of the triangle.

• Using coordinates of a midsegment of a triangle and the distance formula or Pythagorean Theorem justifies that the midsegment is half the length of the segment that it does not intersect.

• Parallel lines have the same slope because they increase or decrease at the same rate per horizontal unit increment (they have the same rise per 1 run). Coordinate geometry can be used to support this understanding.

• Coordinate Geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates.


Selecting and Sequencing Student Work(Small Group Discussion)• Each team should record their group’s sequence of

solution paths on the chart.

• Identify the student’s solution path that would be shared and discussed first, second, third, and so on.

• Be prepared to justify your response.


Selecting and Sequencing Student Work(Group Discussion)

• Listen to each group’s rationale for selecting and sequencing student work.

• As you listen to the rationale, come up with a general “rule of thumb” that can be used to guide you when selecting and sequencing work for the Share, Discuss, and Analyze Phase of the lesson.


Reflecting On Essential Understandings

Which of the sequences of student work were driven by the standards and essential understandings?


Reflecting on the Standards and the Essential Understandings

• Using coordinates of a midsegment of a triangle justifies that the midsegment is parallel to the side that it does not intersect because the slope of the segment containing the midpoints is the same as the slope of the segment connecting the endpoints of the third side of the triangle.

• Using coordinates of a midsegment of a triangle and the distance formula or Pythagorean Theorem justifies that the midsegment is half the length of the segment that it does not intersect.

• Parallel lines have the same slope because they increase or decrease at the same rate per horizontal unit increment (they have the same rise per 1 run). Coordinate geometry can be used to support this understanding.

• Coordinate Geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates.

Common Core Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.



“Rules of Thumb” for Selecting and Sequencing Student Work

What are the benefits of using the “rules of thumb” as a guide when selecting and sequencing student work for the Share, Discuss, and Analyze Phase of the lesson?


Pressing for Mathematical Understanding



Let’s focus on one piece of student work for the Share, Discuss, and Analyze Phase of the lesson.

Assume that a student has explained the work and others in the class have repeated the ideas and asked questions. Now it is time to “FOCUS” the discussion on an important mathematical idea.

What questions might you ask the class as a whole to focus the discussion? Write your questions on chart paper to be posted for a gallery walk.



EU: Coordinate Geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates.



Do a gallery walk. Review other groups’ questions.

• What are some similarities among the questions?

• What are some differences between the questions?


Reflecting on Our Learning

What have you learned today that you will think about and make use of next school year? Take a few minutes and jot your thoughts down.

Documents

© 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Students’ Solution Paths to Maximize Student Learning Supporting Rigorous Mathematics Teaching