Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Making Sense of the Number System
Standards via a Set of Tasks
Tennessee Department of Education
Middle School Mathematics
Grade 7
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
Tasks form the basis for students’ opportunities to learn what
mathematics is and how one does it, yet not all tasks afford
the same levels and opportunities for student thinking.
[They] are central to students’ learning, shaping not only their
opportunity to learn but also their view of the subject matter.
Adding It Up, National Research Council, 2001, p. 335
By analyzing instructional and assessment tasks that are for
the same domain of mathematics, teachers will begin to
identify the characteristics of high-level tasks, differentiate
between those that require problem-solving, and those that
assess for specific mathematical reasoning.
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Session Goals
Participants will:
• make sense of the Number System Common Core
State Standards (CCSS);
• determine the cognitive demand of tasks and make
connections to the Standards for Mathematical
Content and the Standards for Mathematical
Practice; and
• differentiate between assessment items and
instructional tasks.
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Overview of Activities
Participants will:
• analyze a set of tasks as a means of making sense of
the Number System Common Core State Standards
(CCSS);
• determine the Mathematical Content Standards and
the Mathematical Practice Standards aligned with the
tasks;
• relate the characteristics of high-level tasks to the
CCSS for Mathematical Content and Practice; and
• discuss the difference between assessment items and
instructional tasks.
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The Research About Students’
Understanding of Rational
Numbers
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Linking to Research
Virtually none of the making connections problems in
the U.S. were discussed in a way that made the
mathematical connections or relationships visible for
students. Mostly, they turned into opportunities to apply
procedures. Or, they became problems in which even
less mathematical content was visible (i.e., only the
answer was given).
TIMSS Video Mathematics Research Group, 2003
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Linking to Research
Once students have been introduced to the decimal
computation procedure for a particular arithmetic
operation, the type of errors they are most likely to
make seems to remain nearly constant. There are no
qualitative changes in the way students compute with
decimals after they receive their first instructional
lessons.
Hiebert and Wearne, 1985
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Linking to Research/Literature
Research has shown that children who have difficultly
translating a concept from one representation to another
are the same children who have difficulty solving
problems and understanding computations.
Strengthening the ability to move between and among
these representations improves the growth of children’s
concepts.
Lesh, Post, & Behr, 1987
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Analyzing Tasks as a Means of
Making Sense of the CCSS
The Number System
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
Setting Goals
Selecting Tasks
Anticipating Student Responses
Accountable Talk® is a registered trademark of the
University of Pittsburgh
Orchestrating Productive Discussion
• Monitoring students as they work
• Asking assessing and advancing questions
• Selecting solution paths
• Sequencing student responses
• Connecting student responses via Accountable
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Talk® discussions
Linking to Research/Literature:
The QUASAR Project
• Low-level tasks
• High-level tasks
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Linking to Research/Literature:
The QUASAR Project
• Low-level tasks
– Memorization
– Procedures without Connections
• High-level tasks
– Doing Mathematics
– Procedures with Connections
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The Mathematical Task Analysis Guide
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:
A casebook for professional development, p. 16. New York: Teachers College Press
.
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The Cognitive Demand of Tasks
(Small Group Discussion)
Analyze each task. Determine if the task is a high-level
task. Identify the characteristics of the task that make it
a high-level task.
After you have identified the characteristics of the task,
then use the Mathematical Task Analysis Guide to
determine the type of high-level task.
Use the recording sheet in the participant handout to
keep track of your ideas.
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The Cognitive Demand of Tasks
(Whole Group Discussion)
What did you notice about the cognitive demand of the
tasks?
According to the Mathematical Task Analysis Guide,
which tasks would be classified as:
• Doing Mathematics Tasks?
• Procedures with Connections?
• Procedures without Connections?
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Analyzing Tasks: Aligning with the CCSS
(Small Group Discussion)
Determine which Content Standards students would
have opportunities to make sense of when working on
the task.
Determine which Mathematical Practice Standards
students would need to make use of when solving the
task.
Use the recording sheet in the participant handout to
keep track of your ideas.
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Analyzing Tasks: Aligning with the CCSS
(Whole Group Discussion)
How do the tasks differ from each other with respect to
the content that students will have opportunities to
learn?
Do some tasks require that students use Mathematical
Practice Standards that other tasks don’t require
students to use?
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The CCSS for Mathematical Content: Grade 7
The Number System
7.NS
Apply and extend previous understandings of operations and fractions to
add, subtract, multiply, and divide rational numbers.
7.NS.A.1
7.NS.A.1a
7.NS.A.1b
Apply and extend previous understandings of addition and
subtraction to add and subtract rational numbers; represent
addition and subtraction on a horizontal or vertical number line
diagram.
Describe situations in which opposite quantities combine to make
0. For example, a hydrogen atom has 0 charge because its two
constituents are oppositely charged.
Understand p + q as the number located a distance |q| from p, in
the positive or negative direction depending on whether q is
positive or negative. Show that a number and its opposite have a
sum of 0 (are additive inverses). Interpret sums of rational
numbers by describing real-world contexts.
Common Core State Standards, 2010, p. 48, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 7
The Number System
7.NS
Apply and extend previous understandings of operations with fractions to
add, subtract, multiply, and divide rational numbers.
7.NS.A.1c
Understand subtraction of rational numbers as adding the additive
inverse, p – q = p +(-q). Show that the distance between two rational
numbers on the number line is the absolute value of their difference,
and apply this principle in real-world contexts.
7.NS.A.1d
Apply properties of operations as strategies to add and subtract rational
numbers.
7.NS.A.2
Apply and extend previous understandings of multiplication and division
and of fractions to multiply and divide rational numbers.
7.NS.A.2a
Understand that multiplication is extended from fractions to rational
numbers by requiring that operations continue to satisfy the properties
of operations, particularly the distributive property, leading to products
such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
Interpret products of rational numbers by describing real-world
contexts.
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Common Core State Standards, 2010, p. 48, NGA Center/CCSSO
The CCSS for Mathematical Content: Grade 7
The Number System
7.NS
Apply and extend previous understandings of operations with fractions to
add, subtract, multiply, and divide rational numbers.
7.NS.A.2b
7.NS.A.2c
Understand that integers can be divided, provided that the divisor is
not zero, and every quotient of integers (with non-zero divisor) is a
rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Interpret quotients of rational numbers by describing real-world
contexts.
Apply properties of operations as strategies to multiply and divide
rational numbers.
7.NS.A.2d
Convert a rational number to a decimal using long division; know that
the decimal form of a rational number terminates in 0s or eventually
repeats.
7.NS.A.3
Solve real-world and mathematical problems involving the four
operations with rational numbers.
Common Core State Standards, 2010, p. 49, NGA Center/CCSSO
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The CCSS for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
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A. Walking Task
Mary Jane and her brother Paul each go on a walk
starting from the same location. Mary Jane walks north
3 miles. Paul walks south 1.5 miles.
1. Use a number line to represent their starting and
ending points.
2. Determine their distance from each other at the end
of their walks using 2 different methods.
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B. Positive, Negative, or Neither
Consider the following expressions. Use the number line
above to determine whether each of the expressions has a
value that is positive, negative, or equal to 0. Explain your
reasoning.
a.
b.
c.
d.
e.
f.
g.
a+1
b–b
a+a
a–b
b–a
–a
ab +1
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C. Same or Different?
Explain in words and equations why each of the following
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rational numbers is or is not equivalent to –( )
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a.
3
−4
b.
−3
−4
c.
−6
8
d.
1.5
−2
e.
1 1
( )/( )
3 4
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D. Some Sum!
Points A and B are the same
distance from 0 on the number line.
What is A + B? Explain how you
determined your answer.
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E. Number Line Multiplication
1. Consider the product 2 x 5.
a. Explain how the number line below models the
product 2 x 5.
b. Write a scenario that can be modeled by the
expression 2 x 5.
2. Draw a number line model and write a scenario for
each of the following products:
a. 2 x -5
b. -2 x 5
3. Is it possible to use a number line to model the
product -2 x -5? Why or why not?
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F. Party Favors
Destiny is making party favor bags for her birthday party next week. She
has everything she needs to fill the bags except glow sticks. There are two
stores near her house that sell glow sticks.
• Party Central sells packages of 6 glow sticks for $1.98.
• Party Time sells packages of 8 glow sticks for $2.56.
a. Write number sentences using division to determine which store offers
a better deal. Explain what each value in your number sentence
represents in the problem context.
b. Destiny has $10 to spend on glow sticks. Write number sentences
using division to determine how many packages can she buy at each
store. Explain what each value in your number sentence represents in
the problem context.
c. How did the operation of division help you think about this problem?
d. Which store should Destiny buy glow sticks from? Justify your decision
using mathematics.
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Reflecting and Making Connections
• Are all of the CCSS for Mathematical Content in this
cluster addressed by one or more of these tasks?
• Are all of the CCSS for Mathematical Practice
addressed by one or more of these tasks?
• What is the connection between the cognitive
demand of the written task and the alignment of the
task to the Standards for Mathematical Content and
Practice?
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Differentiating Between Instructional
Tasks and Assessment Tasks
Are some tasks more likely to be assessment tasks
than instructional tasks? If so, which and why are you
calling them assessment tasks?
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Instructional Tasks Versus Assessment Tasks
Instructional Tasks
Assessment Tasks
Assist learners to learn the CCSS for
Mathematical Content and the CCSS for
Mathematical Practice.
Assesses fairly the CCSS for Mathematical
Content and the CCSS for Mathematical
Practice of the taught curriculum.
Assist learners to accomplish, often with
others, an activity, project, or to solve a
mathematics task.
Assess individually completed work on a
mathematics task.
Assist learners to “do” the subject matter
under study, usually with others, in ways
authentic to the discipline of mathematics.
Assess individual performance of content
within the scope of studied mathematics
content.
Include different levels of scaffolding
Include tasks that assess both developing
depending on learners’ needs. The
understanding and mastery of concepts and
scaffolding does NOT take away thinking from skills.
the students. The students are still required to
problem-solve and reason mathematically.
Include high-level mathematics prompts.
(The tasks have many of the characteristics
listed on the Mathematical Task Analysis
Guide.)
Include open-ended mathematics prompts as
well as prompts that connect to procedures
with meaning.
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Reflection
• So, what is the point?
• What have you learned about assessment tasks and
instructional tasks that you will use to select tasks to
use in your classroom next school year?
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