supporting rigorous mathematics teaching and learning

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education High School MathematicsAlgebra 1

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.


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


Bike and Truck Task

A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.D

ista

nce

from

sta

rt of

road

(in

feet

)

Time (in seconds)


Bike and Truck Task

1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision.

2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description.

3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words.

4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack.


The Common Core State Standards (CCSS) for Mathematical Content : The Bike and Truck Task

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

The CCSS for Mathematical ContentCCSS Conceptual Category – AlgebraCreating Equations

A–CEDCreate equations that describe numbers or relationships.A-CED.A.1 Create equations and inequalities in one variable and use them to

solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

Reasoning with Equations and Inequalities A–REISolve equations and inequalities in one variable.A-REI.B.3 Solve linear equations and inequalities in one variable,

including equations with coefficients represented by letters.

A-REI.B.4 Solve quadratic equations in one variable.

A-REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

A-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.


The CCSS for Mathematical ContentCCSS Conceptual Category – AlgebraReasoning with Equations and Inequalities A–REIRepresent and solve equations and inequalities graphically.A-REI.D.10 Understand that the graph of an equation in two variables is the set of

all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

★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.


The CCSS for Mathematical ContentCCSS Conceptual Category – FunctionsInterpreting Functions F–IFInterpret functions that arise in applications in terms of the context.F-IF.B.4 For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

★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 State Standards, 2010, p. 69, NGA Center/CCSSO


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 Bike and Truck 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 the Bike and Truck 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 use of and connection between 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 Bike and Truck 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 next school year?

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.

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