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 6
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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 and
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 Mathematical Content Standards
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 Standards for Mathematical Content and
the Standards for Mathematical Practice 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 Data About Students’
Understanding of Rational
Numbers
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Linking to Research
Students often think that .18 is greater than .9 because
on the whole-number scale, 18 is larger than 9.
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4
1
3
Similarly, students may read as larger than because
4 is a “bigger” number than 3 (Kilpatrick, Swafford, &
Findell, 2001).
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Linking to Research
One of the problems in learning rational numbers is that
whole-number counting strategies fail to work (Behr &
Post, 1988). With whole numbers, the idea of counting
the “next number” makes sense. The next number after
4 is 5. This idea does not make sense in the rational
numbers, which makes rational numbers difficult to
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count. For example, there is no next fraction after .
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The El Paso Collaborative for Academic Excellence, n.d., p. 6.
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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 Standards for Mathematical Practice
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 Standards for
Mathematical Practice that other tasks don’t require
students to use?
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The CCSS for Mathematical Content: Grade 6
The Number System
6.NS
Apply and extend previous understandings of numbers to the system of
rational numbers.
6.NS.C.5
Understand that positive and negative numbers are used together
to describe quantities having opposite directions or values (e.g.,
temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts,
explaining the meaning of 0 in each situation.
6.NS.C.6
Understand a rational number as a point on a number line. Extend
number line diagrams and coordinate axes familiar from previous
grades to represent points on the line and in the plane with
negative number coordinates.
6.NS.C.6a Recognize opposite signs of numbers as indicating locations on
opposite sides of 0 on the number line; recognize that the opposite
of the opposite of a number is the number itself, e.g., -(-3) = 3, and
that 0 is its own opposite.
Common Core State Standards, 2010, p. 43, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 6
The Number System
6.NS
Apply and extend previous understandings of numbers to the system of rational
numbers.
6.NS.C.6b Understand signs of numbers in ordered pairs as indicating locations in
quadrants of the coordinate plane; recognize that when two ordered pairs
differ only by signs, the locations of the points are related by reflections
across one or both axes.
6.NS.C.6c
Find and position integers and other rational numbers on a horizontal or
vertical number line diagram; find and position pairs of integers and other
rational numbers on a coordinate plane.
6.NSC..7
Understand ordering and absolute value of rational numbers.
6.NS.C.7a
Interpret statements of inequality as statements about the relative position
of two numbers on a number line diagram. For example, interpret -3 > -7 as
a statement that -3 is located to the right of -7 on a number line oriented
from left to right.
6.NS.C.7b
Write, interpret and explain statements of order for rational numbers in realworld contexts. For example, write -3o C > -7o C to express the fact that -3o
C is warmer than -7o C.
Common Core State Standards, 2010, p. 43, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 6
The Number System
6.NS
Apply and extend previous understandings of numbers to the system of
rational numbers.
6.NS.C.7c
Understand the absolute value of a rational number as its distance
from 0 on the number line; interpret absolute value as magnitude for a
positive or negative quantity in a real-world situation. For example, for
an account balance of -30 dollars, write |-30| = 30 to describe the size
of the debt in dollars.
6.NS.C.7d Distinguish comparisons of absolute value from statements about
order. For example, recognize that an account balance of less than 30 dollars represents a debt greater than 30 dollars.
6.NS.C.8
Solve real-world and mathematical problems by graphing points in all
four quadrants of the coordinate plane. Include use of coordinates
and absolute value to find distances between points with the same
first coordinate or the same second coordinate.
Common Core State Standards, 2010, p. 43, 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. Saving Money Task
5 friends are keeping track of their finances. Some of them have saved money.
Others have borrowed money from their parents.
1. Plot points on the number line below
representing the amount each friend
has saved or owes. Explain why you
Name
Money Saved
located the points where you did.
or Owed
2. Explain how you can use the points
Abbey
Saved $12.10
you placed on the number line to
determine which friend has saved the
Brandon
Saved $5.50
most money and which has the
Claire
Owes $12.10
greatest debt.
Dante
Saved $14
Elizabeth
Owes $8.25
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3. Brandon claims that the points
representing 2 people have the same
absolute value. Is he correct? Why or
why not?
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C. Location, Location, Location
Joe made a map on the coordinate plane showing the location of several places in
relationship to his home. Each tick mark represents one block.
1. Describe the location of
the school in relationship
to Joe’s home.
2. How are the positions of
the museum and park
related? Explain your
reasoning.
3. Joe’s grandmother’s
house is located at the
coordinates (-7, 5). Plot a
point to represent his
grandmother’s house.
Explain how you
determined this location.
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D. Absolute Value
Points A and B are the same
distance from 0 on the number line.
What can you determine about the
absolute values of A and B? Explain
your reasoning.
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E. Weight at Birth
The average weight of a newborn infant is 7.5 pounds. At Riverside
Hospital, every infant is assigned a number indicating their weight relative
to the average newborn weight.
1. What does it mean if an infant is assigned a positive value? A negative
value? 0?
2. On Monday, 4 babies were born at Riverside Hospital.
• Clarence weighed 6.9 pounds.
• Travis weighed 7.8 pounds.
• Brea weighed 8.9 pounds.
• Ginger weighed 6.1 pounds.
Do any of the babies have weights equally distant from the average
birth weight? Explain your reasoning.
3. On Tuesday twin boys, Larry and Harry, were born. Larry was assigned
the value -0.7 and Harry was assigned the value -0.4. Write an
inequality comparing the values assigned to each twin. Which baby’s
weight is farther from the average birth weight? Explain your reasoning.
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F. Isosceles
Xavier drew a triangle with vertices at (3, 2), (3, -5), and (-5, 2).
Teacher: How would you classify this triangle by its
angle measures and side lengths?
Xavier: This is a right triangle because the horizontal
and vertical sides meet at a right angle. It is isosceles
because the hypotenuse is the longest side and the
other 2 sides have the same length.
1. Explain how you know that Xavier’s triangle is not isosceles.
2. How could Xavier change the location of one or more
vertices to make the triangle isosceles?
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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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