supporting rigorous mathematics teaching and learning

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education High School MathematicsGeometry

Illuminating Student Thinking: Assessing and Advancing Questions

RationaleEffective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Asking questions that assess student understanding of mathematical ideas, strategies or representations provides teachers with insights into what students know and can do. The insights gained from these questions prepare teachers to then ask questions that advance student understanding of mathematical ideas, strategies or connections to representations.

By analyzing students’ written responses, teachers will have the opportunity to develop questions that assess and advance students’ current mathematical understanding and to begin to develop an understanding of the characteristics of such questions.

© 2013 UNIVERSITY OF PITTSBURGH

Session Goals

Participants will:• learn to ask assessing and advancing questions based

on what is learned about student thinking from student responses to a mathematical task; and

• develop characteristics of assessing and advancing questions and be able to distinguish the purpose of each type.


Overview of Activities

Participants will:• analyze student work to determine what the

students know and what they can do;• develop questions to be asked during the Explore

Phase of the lesson;– identify characteristics of questions that assess

and advance student learning; – consider ways the questions differ; and

• discuss the benefits of engaging in this process.


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


Building a New Playground Task

The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown.


Building a New PlaygroundPART A 1. Determine at least three possible locations for the park

that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools.

2. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture.

PART B3. The City Planning Commission is planning to build a third

elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park.

4. Describe a strategy for determining a point equidistant from any three points.


The Common Core State Standards (CCSS) for Mathematical Content : The Building a New Playground Task

Which of CCSS for Mathematical Content did we address when solving and discussing the task?

The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry

Congruence G-CO

Understand congruence in terms of rigid motions.

G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Common Core State Standards, 2010, p. 76, NGA Center/CCSSO


Congruence G-CO Prove geometric theorems.G-CO.C.9 Prove theorems about lines and angles. Theorems include:

vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.



Similarity, Right Triangles, and Trigonometry G-SRT

Define trigonometric ratios and solve problems involving right triangles.

G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

★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.



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.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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.



What Does Each Student Know?

Now we will focus on three pieces of student work.

Individually examine the three pieces of student work A, B, and C for the Building a New Playground Task in your Participant Handout.

What does each student know?

Be prepared to share and justify your conclusions.

Response A

14

Response B

15

Response C

16


What Does Each Student Know?

Why is it important to make evidence-based comments and not to make inferences when identifying what students know and what they can do?


Supporting Students’ Exploration of a Task through QuestioningImagine that you are walking around the room, observing your students as they work on Building a New Playground Task. Consider what you would say to the students who produced responses A, B, C, and D in order to assess and advance their thinking about key mathematical ideas, problem-solving strategies, or representations. Specifically, for each response, indicate what questions you would ask:

– to determine what the student knows and understands (ASSESSING QUESTIONS)

– to move the student towards the target mathematical goals (ADVANCING QUESTIONS).


Cannot Get Started

Imagine that you are walking around the room, observing your students as they work on the Building a New Playground Task. Group D has little or nothing on their papers.

Write an assessing question and an advancing question for Group D. Be prepared to share and justify your conclusions.

Reminder: You cannot TELL Group D how to start. What questions can you ask them?


Discussing Assessing Questions• Listen as several assessing questions are read

aloud.

• Consider how the assessing questions are similar to or different from each other.

• Are there any questions that you believe do not belong in this category and why?

• What are some general characteristics of the assessing questions?


Discussing Advancing Questions• Listen as several advancing questions are read

aloud.

• Consider how the advancing questions are similar to or different from each other.

• Are there any questions that you believe do not belong in this category and why?

• What are some general characteristics of the advancing questions?


Looking for Patterns

• Look across the different assessing and advancing questions written for the different students.

• Do you notice any patterns?

• Why are some students’ assessing questions other students’ advancing questions?

• Do we ask more content-focused questions or questions related to the mathematical practice standards?


Characteristics of Questions that Support Students’ Exploration

Assessing Questions• Based closely on the

work the student has produced.

• Clarify what the student has done and what the student understands about what s/he has done.

• Provide information to the teacher about what the student understands.

Advancing Questions• Use what students have

produced as a basis for making progress toward the target goal.

• Move students beyond their current thinking by pressing students to extend what they know to a new situation.

• Press students to think about something they are not currently thinking about.


Reflection

• Why is it important to ask students both assessing and advancing questions? What message do you send to students if you ask ONLY assessing questions?

• Look across the set of both assessing and advancing questions. Do we ask more questions related to content or to mathematical practice standards?


Reflection

• Not all tasks are created equal.

• Assessing and advancing questions can be asked of some tasks but not others. What are the characteristics of tasks in which it is worthwhile to ask assessing and advancing questions?


Preparing to Ask Assessing and Advancing Questions

How does a teacher prepare to ask

assessing and advancing questions?


Supporting Student Thinking and Learning

In planning a lesson, what do you think can be gained by considering how students are likely to respond to a task and by developing questions in advance that can assess and advance their learning in a way that depends on the solution path they’ve chosen?


Reflection

What have you learned about assessing and advancing questions that you can use in your classroom tomorrow?

Turn and Talk


Bridge to Practice

• Select a task that is cognitively demanding, based on the TAG. (Be prepared to explain to others why the task is a high-level task. Refer to the TAG and specific characteristics of your task when justifying why the task is a Doing Mathematics Task or a Procedures With Connections Task.)

• Plan a lesson with colleagues.• Anticipate student responses, errors, and misconceptions.• Write assessing and advancing questions related to the student

responses. Keep copies of your planning notes.• Teach the lesson. When you are in the Explore Phase of the lesson,

tape your questions and the student responses, or ask a colleague to scribe them.

• Following the lesson, reflect on the kinds of assessing and advancing questions you asked and how they supported students to learn the mathematics.

supporting rigorous mathematics teaching and learning

Documents

students ability

student thinking

student work

mathematical practice

student learningasking

posters of student

process of thinking

mathematical content