1. Supporting Rigorous Mathematics Teaching and Learning
Illuminating Student Thinking: Assessing and Advancing Questions
Tennessee Department of Education
High School Mathematics
Geometry
Overview of the Module:
Students often struggle to solve high-level tasks. Their discomfort with the struggle when they are solving the high-level task causes them to ask the teacher for assistance. In turn, the teacher provides assistance, often by doing the problem-solving for the student. The students, as a result, do not have the opportunity to learn to think, reason, and to practice engaging in problem solving and communicating mathematically. This module is designed to help teachers learn an alternative method of assisting student learning—asking assessing and advancing questions instead of telling students the steps and procedures that they can follow.
Prior Learning Necessary to Gain the Most from This Module:
This module is most effective when teachers:
have engaged in solving and discussing multiple solution paths to a high-level task. (See Module #2:_Engaging In and Analyzing Teaching and Learning
have engaged in Module #1: Deepening our Understanding of the CCSS via a Performance- Based Assessment. Participants realize that the CCSS for Mathematical Practice will measure students ability to make sense of problems, notice patterns and relationships, formulate arguments, etc., which are many of the process that teachers take over for students when students request assistance during the problem-solving process. Therefore, teachers must find an alternative way of assisting so that they do not give students the pathways for solving problems.
MATERIALS:
Facilitator’s Overview of Module.
Slides for Presentation with Notes.
Participant Handouts.
The CCSS for Mathematical Content and for CCSS for Mathematical Practice.
Posters of Student A, B, and C; or post the student work from the Participant Handout.

2. Rationale
Effective 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.
(SAY) Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them. Often, when students are given a high-level task, the potential of the task is not realized; in other words, students do not engage in the problem solving, the thinking and the reasoning required by the task. If you remember, we talked about several factors of decline, such as time, students not having the prior knowledge necessary to engage in the task, or the teacher giving too much guidance.
In this module, we will learn about an alternative method we can use that will prevent us from “doing the work for students.” We will learn about “assessing” questions—questions that will let us know what students understand, what they don’t understand, and possible misconceptions they have. Based on the insights we gain from student responses to these questions, we will consider questions that we can ask to “advance” student learning related to problem solving strategies, mathematical ideas, or the use of, and connections between, representations.
Note we never said tell students; we said ask assessing and advancing questions. Our goal is to keep students actively engaged in “figuring out” ideas and relationships all the time. In our view, teaching hasn’t occurred unless student learning is advancing; therefore, we want to strive to ask assessing and advancing questions all the time. After we have analyzed a set of questions, we will identify characteristics of assessing and advancing questions. You will be able to use these as a guide when constructing other assessing and advancing questions.

3. 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.
Directions:
Give participants a minute to read the rationale slide. or
Paraphrase the rationale, if desired.

4. 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.
Directions:
Give participants a sense of how they will spend the time working on this component.
(SAY)
We will solve and discuss solution paths for a task.
We will analyze student work in SMALL GROUPS to determine what students know and can do.
We will write questions that we might ask students if we want to assess and advance student learning. After everyone has written questions, the whole group will analyze the set of questions.
Our ultimate goal is to identify a set of characteristics that describe the assessing and advancing questions—characteristics we can then use to guide us as we plan questions for other tasks.

5. The Structures and Routines of a Lesson
MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on:
Different solution paths to the
same task
Different representations
Errors
Misconceptions
Set Up of the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/Small Group Problem Solving
Generate and Compare Solutions
Assess and Advance Student Learning
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.
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
Directions: (Optional)
(SAY) This graphic shows the phases that a high-level task goes through as it is enacted in a classroom. It assumes we have already selected a high-level task, and are aware of how the task will help students work towards or use standards for mathematical practice as a means of understanding mathematical content.
You have seen this graphic many times. This lesson structure ensures that students have individual problem solving time, small group problem solving time, and whole group discussion time. All of these opportunities are times when the teacher can assess student thinking, but they are also times when students can work out answers with each other and practice methods before they are shared with a larger group.
An important phase of the lesson is the Explore Phase of the lesson. Our work today focuses on the Explore Phase of the lesson. The question is “How can this phase of the lesson help students enter into the task and engage in problem solving?”

6. 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.
Directions:
Remind participants that they have themselves engaged in this task; refer to any work in the room that was done with the task. Tell participants that we will be examining student work from this task.

7. Building a New Playground
PART A 
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.
Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture.
PART B
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.
Describe a strategy for determining a point equidistant from any three points.

8. 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?
Facilitator Notes:
If this session is separated in time from the session where participants engaged in the Building a New Playground Task, follow the directions below. If not, then move to the next slide and remind participants that they have already discussed the mathematics in the task and the CCSS for Mathematical Content addressed by the task.
(SAY) What mathematical ideas did we discuss while solving and discussing the Building a New Playground Task?

9. The CCSS for Mathematical Content CCSS 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.
Facilitator Notes:
The standards slides are here for your convenience, should you need them. All the Focus Cluster content standards appear on pages 5-6 in the handout.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO

10. The CCSS for Mathematical Content CCSS Conceptual Category – Geometry
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.
Facilitator Notes:
The standards slides are here for your convenience, should you need them. All the Focus Cluster content standards appear on pages 5-6 in the handout.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO

11. The CCSS for Mathematical Content CCSS Conceptual Category – Geometry
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.
Facilitator Notes:
The standards slides are here for your convenience, should you need them. All the Focus Cluster content standards appear on pages 5-6 in the handout.
Common Core State Standards, 2010, p. 77, NGA Center/CCSSO

12. The CCSS for Mathematical Content CCSS Conceptual Category – Geometry
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.
Facilitator Notes:
The standards slides are here for your convenience, should you need them. All the Focus Cluster content standards appear on pages 5-6 in the handout.
Common Core State Standards, 2010, p. 78, NGA Center/CCSSO

13. 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.
Facilitator Notes:
Display student work A, B, and C in the room. Hang one piece of poster paper under each piece of student work with the label “The student knows…”
If participants make inferences about what the student knows, then press participants to give you evidence from the student work to support their claims.
Directions:
Small-Group Work: Take 10 minutes to identify what students know and can do.
Group Discussion: What does each student know and what can s/he do?
Let us know how you decided that the student knows x or y or z.

14. Response A 14 Facilitator Notes:
The goal of this part of the session is to help participants become aware of the difference between evidence-based statements and inferences.
Directions:
Lead a group discussion of evidence-based observations related to each piece of student work. Record participants’ responses to the question, “What does the student know and what can he do?”.
Probing Facilitator Questions and Possible Responses Related to Response A’s Work:
What does the student know and what can he do?
The student can find the midpoint of a line segment.
The student can find two points equidistant from A and B by forming a square.
The student can label points on his/her diagram.
The student knows the equidistant points are on the perpendicular bisector.
How do you know the student is aware that other points on that line drawn are equidistant from A and B? 14

15. Response B
Probing Facilitator Questions and Possible Responses Related to Response B’s Work:
What does the student know and what can he do?
The student can find three points equidistant from A and B.
The student is forming right triangles with legs equal in measurement to find equidistant points.
The student recognizes that the lengths of the segments can be found using the Pythagorean Theorem.
How do you know if the student is aware of those facts? How can we be sure?
The student knows segments QM and AB are perpendicular and proves they are using slope. 15

16. Response C
Probing Facilitator Questions and Possible Responses Related to Response C’s Work:
What does the student know and what can he do?
The student can use a compass with radius AB to find several points equidistant from A and B.
The student connects the points s/he found.
The student can write about his/her thinking, but some of the writing lacks clarity. 16

17. 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?
Directions:
Lead a group discussion of the question on the slide.
Probing Facilitator Question and Possible Responses:
Why is it important to make evidence-based comments and not to make inferences when identifying what students know and can do?
We do not really know if students understand the strategy or the concept until we talk with the students or see evidence in their work.

18. Supporting Students’ Exploration of a Task through Questioning
Imagine 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).
FACILITATOR NOTE:
Hang two pieces of poster paper, one with the label “assessing questions” and another with the label “advancing questions” under each piece of student work.
Give participants sentence strips and magic markers.
Directions:
Read the directions on the slide.
Small group work: (SAY) Work with a partner or in groups of three to write one assessing and one advancing questions for two pieces of student work. (Assign groups either A & B, B & C, or C & Cannot get Started, etc. so that each piece of work is assigned to at least two groups for review). Post your assessing and advancing question under the appropriate student work.
See responses related to Student C on the next notes page.
Possible Responses for Student A’s Work
Assessing Questions
Advancing Questions
Tell me your thinking about points E and F.
Are any of the other points on line EF equidistant from A and B? How can you prove if they are or are not?
Tell me your thinking about point C. Why did you find half of five? What were you trying to find?
You added 2.5 to the x-coordinate of B and subtracted from the y-coordinate of A. Will that always work? Why/why not?
Tell me about how you determined the length of your line segments.
What other tools or strategies for measuring lengths do you know?
Possible Responses for Student B’s Work
Tell me your thinking about points P, Q, and R.
Is it possible to use a similar process to find points on the other side of AB? How about on the extension of MP past point P? How many such points can you find?
You have shown QM is perpendicular to AB. Tell me the process you used to find the numbers you used to determine slope
What can you tell me about point M? What about line PM?

19. 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?
Possible Responses for Response C’s Work
Assessing Questions
Advancing Questions
Tell me about points C and D. How do you know they are equidistant from A and B?
Can you use a radius smaller than AB? Larger? Why or why not?
You say you drew other circles. Where are the center of those circles and what are their radii? How did you then find E and F?
How can you be certain that E and F are equidistant from A and B?
What observations can you make about line CD?
Can you be sure all the points on CD are equidistant from A and B? What relationship does CD have to AB? How can you be sure?
Possible Responses for “Cannot Get Started” Work
Can you tell me what you know about the problem and what you are asked to find?
Suppose you were walking from one school to the other. When would you be equidistant from each school? What does ‘equidistant’ mean?
Suppose there is no cross-street between School A and School B, and that all the streets are represented by the lines on the grid. Is it possible to walk from one school to the other and not walk in a straight line, but still find a point equidistant from each school? How can you start to test that idea out?

20. 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?
Directions:
Label two posters, one as “Characteristics of Assessing Questions” and the other as “Characteristics of Advancing Questions”. Gather participants’ responses.
The responses for assessing questions should resemble those listed below.
Probing Facilitator Questions and Possible Responses:
How are the assessing questions alike? What makes and assessing question an assessing question?
Make a list of the similarities identified by participants.)
All of the questions point to something on the student work.
The questions are asked to gain clarification.
The teacher gets to learn about the student thinking.
What might be the benefits of referring specifically to something on the student’s paper when asking an assessing question? What might be the benefit of making sure the student knows you are referring to his/her work by saying, “I notice you wrote..”?
Students feel honored that you are referring to their work. You will find out if students understand their response.
Encourage participants to identify questions that they feel are advancing questions but they have been put in the assessing category. Misplaced questions are usually those that advance student learning instead of finding out what students know.

21. 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?
Probing Facilitator Questions and Possible Responses:
How are the advancing questions alike? What makes an advancing question an advancing question?
The questions challenge students to go further in their work.
The questions move student work further.
The students usually do not have the work that is being requested on their paper.
The questions may cause students some anxiety because they present a challenge.

22. 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?
Probing Facilitator Questions and Possible Responses:
How are some students’ assessing questions other students’ advancing questions?
The students are at different places in their learning. One group may choose a simple translation and may need to be prompted about how to find the lengths of segments that are not parallel to an axis, while another group may have already done that. One group may understand concepts or strategies, whereas another group might need to be asked an advancing question to prompt them to consider a concept or a strategy.
Why do all students deserve to have an assessing question and an advancing question?
If you ask some students only assessing questions but not advancing questions, you are telling the students that they can’t handle advancing questions.
What if a student has completed all parts of a task? Why do you still need to ask this student an assessing question and an advancing question?
Students can always be challenged further. You may have to step outside of the particular problem or make an extension, but the student still deserves to be challenged.

23. 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.
Directions:
Show the list of similarities and differences on this slide after participants have created their list. (Optional)
Discuss any characteristics participants did not identify in their previous discussion.
Ask participants to compare their assessing and advancing characteristics with other’s characteristics.
Lead a discussion of the similarities and differences. Add to the participants’ chart.
Refer to this as a tool for reflecting on their practice.

24. 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?
Facilitator Notes:
This slide includes additional questions related to the characteristics of the assessing and advancing questions. The facilitator can decide if these questions will further the conversation.
Probing Facilitator Questions and Possible Responses:
Why is it important to ask both assessing and advancing questions when students are engaged in a task?
Asking both an assessing question and an advancing question signals to students that you respect what they are thinking and that you think enough about them as problem solvers to challenge them to think.
Why is it important for the questions to be linked to the particular mathematical content of the lesson?
We need to ask assessing and advancing questions focused on problem-solving strategies, mathematical ideas and the use of, and connection between, representations.
Why is this important? What might we need to be mindful of over time as we work with students?
Why are some students’ assessing questions other student’s advancing questions?
One student may already demonstrate an understanding of a concept or a procedure and need a more advanced question, whereas another student might need a question focused on something simple like setting up a problem. Students are at different places in their learning, therefore need to be advanced to different places of learning the mathematical content and standards of mathematical practice.

25. 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?
Directions:
Read the statements and question on the slide. If time permits, ask participants to turn and talk with a partner. Accept responses to the question.
Probing Facilitator Questions and Possible Responses:
What are the characteristics of tasks in which is it worthwhile to ask assessing and advancing questions?
The task must be a high-level task. It must have multiple solution paths so the teacher can assess student thinking. If the task is low-level or a procedural task, then only one pathway exists and all the student work will look the same.
What might the task look like?
The task might ask students to show more than one way to solve the task.
A graph, a table, and an equation might be possible solution paths.
Justifications of reasoning might be requested by the task.
The tasks requires students to look for patterns or to write about mathematical relationships.

26. Preparing to Ask Assessing and Advancing Questions
How does a teacher prepare to ask assessing and advancing questions?
Directions:
Read the question on the slide and engage the group in a discussion.
Probing Facilitator Questions and Possible Responses:
Were you able to ask assessing and advancing questions today? What made this possible? How did we prepare to ask assessing and advancing questions today?
We found several solution paths to the high-level task and then we analyzed student work and identified what students knew and could do.
How was this helpful in preparing you to ask assessing and advancing questions?
It is not possible to think through every lesson to this level of detail, so what can a teacher gain from doing it well a few times?
Engaging in thorough planning for one lesson can help the teacher learn practices that can be generalized to other lessons.

27. 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?
Directions:
Read the question on the slide and engage the participants in a discussion of the question.
Possible Responses:
Thinking through a lesson prior to teaching it will prepare a teacher to support student learning.
Anticipating possible responses will prepare teachers for what they might hear from students, thus making it easier to understand student thinking.
Preparing questions in advance of the lesson will ensure that more precise language is used during the lesson.

28. Reflection
What have you learned about assessing and advancing questions that you can use in your classroom tomorrow? Turn and Talk
Facilitator Note:
Listen carefully to types of responses given by participants because this is your means of finding out what they learned in the session.
Note misconceptions or comments that lack detail, because this will help you determine the focus of your next session with the teachers. Teachers often need several opportunities to engage in writing and discuss assessing and advancing questions before they understand the difference between assessing and advancing questions and feel comfortable and prepared to ask them in their classroom.
Directions:
Read the question on the slide and accept responses from participants.

29. 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.
Directions:
Read the Bridge to Practice and answer the teachers’ questions. Tell teachers that we will use a protocol called the Case Story Process to share their work at the next meeting.