Transcript Assessing and Advancing Questions

Supporting Rigorous Mathematics
Teaching and Learning
Illuminating Student Thinking: Assessing and
Advancing Questions
Tennessee Department of Education
High School Mathematics
Geometry
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.
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.
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Building a New Playground
PART 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 B
3. 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.
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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?
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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.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
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.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
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.
Common Core State Standards, 2010, p. 77, NGA Center/CCSSO
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.
Common Core State Standards, 2010, p. 78, 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 Building a New Playground Task in
your Participant Handout.
What does each student know?
Be prepared to share and justify your conclusions.
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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?
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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 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).
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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?
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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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