Supporting Rigorous Mathematics
Teaching and Learning
Illuminating Student Thinking: Assessing and
Advancing Questions
Tennessee Department of Education
High School Mathematics
Algebra 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.
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.
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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.
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The Structures and Routines of a Lesson
Set Up of the Task
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
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
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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.
Missing Function Task
If h(x) = f(x) · g(x), what can you determine about g(x)
from the given table and graph? Explain your reasoning.
x
-2
-1
0
1
2
f(x)
0
1
2
3
4
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The Common Core State Standards
(CCSS) for Mathematical Content : The
Missing Function Task
Which of CCSS for Mathematical Content did we
address when solving and discussing the task?
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The CCSS for Mathematical Content
CCSS Conceptual Category – Number and Quantity
The Real Number System
N-RN
Extend the properties of exponents to rational exponents.
N-RN.A.1 Explain how the definition of the meaning of rational
exponents follows from extending the properties of
integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For
example, we define 51/3 to be the cube root of 5
because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must
equal 5.
N-RN.A.2 Rewrite expressions involving radicals and rational
exponents using the properties of exponents.
Common Core State Standards, 2010, p. 60, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Seeing Structure in Expressions
A–SSE
Write expressions in equivalent forms to solve problems.
A-SSE.B.3
Choose and produce an equivalent form of an expression to
reveal and explain properties of the quantity represented by
the expression.★
A-SSE.B.3c Use the properties of exponents to transform expressions for
exponential functions. For example the expression 1.15t can
be rewritten as (1.151/12)12t ͌ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
A-SSE.B.4
Derive the formula for the sum of a finite geometric series
(when the common ratio is not 1), and use the formula to solve
problems. For example, calculate mortgage payments.★
★ 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. 64, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Arithmetic with Polynomials and Rational Expressions A–APR
Understand the relationship between zeros and factors of
polynomials.
A-APR.B.2
Know and apply the Remainder Theorem: For a polynomial
p(x) and a number a, the remainder on division by x – a is
p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A-APR.B.3
Identify zeros of polynomials when suitable factorizations
are available, and use the zeros to construct a rough graph
of the function defined by the polynomial.
Common Core State Standards, 2010, p. 64, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Building Functions
F–BF
Build a function that models a relationship between two quantities.
F-BF.A.1
Write a function that describes a relationship between two
quantities.★
F-BF.A.1a
Determine an explicit expression, a recursive process, or
steps for calculation from a context.
F-BF.A.1b
Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
F-BF.A.2
Write arithmetic and geometric sequences both recursively
and with an explicit formula, use them to model situations, and
translate between the two forms.★
★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. 70, 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 Missing Function Task in your
Participant Handout.
What does each student know?
Be prepared to share and justify your conclusions.
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Response A
13
Response B
14
Response C
15
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 can do?
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Supporting Students’ Exploration of a
Task through Questioning
Imagine that you are walking around the room, observing
your students as they work on the Missing Function 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).
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Cannot Get Started
Imagine that you are walking around the room, observing
your students as they work on the Missing Function 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?
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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?
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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?
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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?
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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.
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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?
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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?
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Preparing to Ask Assessing and
Advancing Questions
How does a teacher prepare to ask
assessing and advancing questions?
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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?
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Reflection
What have you learned about assessing and
advancing questions that you can use in your
classroom tomorrow?
Turn and Talk
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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.
•
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.
Anticipate student responses, errors, and misconceptions.
Write assessing and advancing questions related to the student
responses. Keep copies of your planning notes.
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