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Principles to Actions
Effective Mathematics Teaching Practices
The Case of Jennifer DiBrienza
and the Equation Strings Task
Grade 1
This module was developed by DeAnn Huinker, University of Wisconsin-Milwaukee; Victoria Bill, University of Pittsburgh
Institute for Learning; and Amy Hillen, Kennesaw State University. Video courtesy of New York City Public Schools and the
University of Pittsburgh Institute for Learning.
These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the
project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn
Huinker, Stephen Miller, Lynn Raith, and Michael Steele.
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Overview of the Session
• Watch a video clip of a first grade class engaged in
a whole class discussion of an equation string.
• Discuss what the teacher does to support the
students’ learning of mathematics.
• Relate teacher actions in the video to the effective
mathematics teaching practices.
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Teaching and Learning Principle
An excellent mathematics program requires effective
teaching that engages students in meaningful
learning through individual and collaborative
experiences that promote their ability to make sense
of mathematical ideas and reason mathematically.
National Council of Teachers of Mathematics. (2014). Principles to actions:
Ensuring mathematical success for all. Reston, VA: Author. (p. 7)
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Ms. DiBrienza’s
First Grade Classroom
“Equation Strings Task”
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Ms. DiBrienza’s
Mathematics Learning Goals
Students will understand that
• Fluent addition strategies use number relationships
and the structure of the number system;
• Numbers can be decomposed and added on in
parts, not just by ones; and
• Noticing regularity in repeated calculations leads to
shortcuts and general methods for adding numbers.
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The Equation Strings Task
Solve the set of addition problems. Each time you solve a problem,
try to use the previous problem to solve the next problem.
7 + 3 = ___
17 + 3 = ___
27 + 3 = ___
37 + 3 = ___
37 + 5 = ___
After you have solved all of the problems, describe some patterns
that you notice in the sequence of equations.
Show how you might represent student reasoning with a drawing
or on a number line so that students could visually see the
relationships among the quantities.
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Connections to the CCSSM Grade 1
Standards for Mathematical Content
Number and Operations in Base Ten (NBT)
Use place value understanding and properties of operations to add
and subtract.
1.NBT.C.4 Add within 100, including adding a two-digit number and a onedigit number, and adding a two-digit number and a multiple of
10, using concrete models or drawings and strategies based on
place value, properties of operations, and/or the relationship
between addition and subtraction; relate the strategy to a
written method and explain the reasoning used. Understand that
in adding two-digit numbers, one adds tens and tens, ones and
ones; and sometimes it is necessary to compose a ten.
1.NBT.C.5 Given a two-digit number, mentally find 10 more or 10 less than
the number, without having to count; explain the reasoning used.
National Governors Association Center for Best Practices (NGA Center) and Council of Chief
State School Officers (CCSSO). (2014). Common core state standards for mathematics.
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Retrieved from http://www.corestandards.org/Math/Content/1/NBT
Connections to the CCSSM
Standards 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.
National Governors Association Center for Best Practices (NGA Center) and Council of Chief State
School Officers (CCSSO). (2014). Mathematics. Common core state standards for mathematics.
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Retrieved from http://www.corestandards.org/Math/Practice
Standards for Mathematical Practice (SMP)
SMP 7. Look for and make use of structure.
Mathematically proficient students at the elementary grades use
structures such as place value, the properties of operations, other
generalizations about the behavior of the operations (for example,
the less you subtract, the greater the difference), and attributes of
shapes to solve problems. In many cases, they have identified and
described these structures through repeated reasoning (SMP 8).
SMP 8. Look for and express regularity in repeated reasoning.
Mathematically proficient students at the elementary grades look for
regularities as they solve multiple related problems, then identify
and describe these regularities.... proficient students formulate
conjectures about what they notice.
Illustrative Mathematics. (2014, February 12). Standards for Mathematical Practice: Commentary and Elaborations for K–5.
Tucson, AZ.. Retrieved from http://commoncoretools.me/wp-content/uploads/2014/02/Elaborations.pdf (pp.18-19)
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Classroom Context for the Video Segment
Teacher: Jennifer DiBrienza
Grade:
1
School:
PS 116
District:
New York Community School District 2
The students are seated in a large circle on the
classroom rug. The teacher poses a sequence of
equations for the students to solve. The students
share their strategies and discuss “what they notice”
about the equations and the answers.
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Lens for Watching the Video: Viewing #1
As you watch the video
• Make note of what the teacher does to support
student learning of mathematics, and
• Identify the mathematical insights that surfaced
for students.
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Student Learning of Mathematics
In small groups, discuss the following:
1. Identify mathematical insights that surfaced
for students.
2. Summarize ways the teacher engaged students in
the mathematical practices: SMP 7 and SMP 8.
3. Summarize ways the teacher advanced student
understanding toward the content standards:
1.NBT.C.4 and 1.NBT.C.5.
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Teacher Actions
As you reflect on the teacher actions that
supported student learning in this lesson, which
Effective Mathematics Teaching Practices did
you notice the teacher using?
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Effective Mathematics Teaching Practices
1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem
solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.
National Council of Teachers of Mathematics. (2014). Principles to actions:
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Ensuring mathematical success for all. Reston, VA: Author.
Effective Mathematics Teaching Practice
Build Procedural Fluency from
Conceptual Understanding
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Teaching Practice Focus: Build Procedural
Fluency from Conceptual Understanding
Effective teaching of mathematics
• builds on a foundation of conceptual understanding;
• results in generalized methods for solving problems; and
• enables students to flexibly choose among methods to
solve contextual and mathematical problems.
National Council of Teachers of Mathematics. (2014). Principles to actions:
Ensuring mathematical success for all. Reston, VA: Author. (p. 42)
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Teaching Practice Focus: Build Procedural
Fluency from Conceptual Understanding
Jot down your responses.
• What does it mean to be fluent with
computational procedures?
• Why is it important to build procedures
from conceptual understanding?
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Fluency
Being fluent means that students are able to choose flexibly among
methods and strategies to solve contextual and mathematical
problems, they understand and are able to explain their approaches,
and they are able to produce accurate answers efficiently.
Fluency builds from initial exploration and discussion of number
concepts to using informal reasoning strategies based on meanings
and properties of the operations to the eventual use of general
methods as tools in solving problems. This sequence is beneficial
whether students are building toward fluency with single- and multidigit computation with whole numbers or fluency with, for example,
fraction operations, proportional relationships, measurement
formulas, or algebraic procedures.
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National Council of Teachers of Mathematics. (2014). Principles to actions:
Ensuring mathematical success for all. Reston, VA: Author. (p. 42)
Conceptual Understanding and
Procedural Fluency
When procedures are connected with the underlying concepts,
students have better retention of the procedures and are more able
to apply them in new situations (Fuson, Kalchman, and Bransford
2005). Martin (2009, p. 165) describes some of the reasons that
fluency depends on and extends from conceptual understanding:
To use mathematics effectively, students must be able to
do much more than carry out mathematical procedures. They
must know which procedure is appropriate and most productive
in a given situation, what a procedure accomplishes, and what
kind of results to expect. Mechanical execution of procedures
without understanding their mathematical basis often leads to
bizarre results.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring
mathematical success for all. Reston, VA: Author. (p. 42)
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Lens for Watching the Video: Viewing #2
As you watch the video again, attend to the
teacher actions that
build procedural fluency from conceptual
understanding.
Be prepared to give examples and to cite line
numbers from the transcript to support your
observations.
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Video Observations
In what ways did the teacher actions support
students in building procedural fluency from
conceptual understanding?
Use the transcript to cite line numbers to
support your observations.
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Effective Mathematics Teaching Practice
Implement Tasks that Promote
Reasoning and Problem Solving
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Teaching Practice Focus: Implement Tasks
that Promote Reasoning and Problem Solving
Effective teaching of mathematics
• provides opportunities for students to engage in
solving and discussing tasks;
• uses tasks that promote inquiry and exploration
and are meaningfully connected to concepts;
• uses tasks that allow for multiple entry points; and
• encourages use of varied solution strategies.
National Council of Teachers of Mathematics. (2014). Principles to actions:
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Ensuring mathematical success for all. Reston, VA: Author. (p. 17)
Teaching Practice Focus: Implement Tasks that
Promote Reasoning and Problem Solving
Mathematical tasks can range from a set of routine
exercises to a complex and challenging problem that
focuses students’ attention on a particular mathematical
ideas (p. 17).
It is important to note that not all tasks that promote
reasoning and problem solving have to be set in a context
or need to consume an entire class period or multiple
days. What is critical is that a task provide students with
the opportunity to engage actively in reasoning, sense
making, and problem solving... (p. 20).
National Council of Teachers of Mathematics. (2014). Principles to actions:
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Ensuring mathematical success for all. Reston, VA: Author.
How did the teacher use the equations string task to
promote reasoning and problem solving in ways
that advanced student understanding toward the
mathematics learning goals of the lesson?
7+3=
17 + 3 =
27 + 3 =
37 + 3 =
37 + 5 =
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As you reflect on the
effective mathematics teaching practices
examined in this session, summarize one or
two ideas or insights you might apply to
your own classroom instruction.
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Extension
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Equation Strings Tasks
Work with a small group to create a string of
equations appropriate for your students.
• Select an operation and the type of numbers
(e.g., whole numbers, fractions, percents).
• Identify the mathematics learning goals which will
be supported by your equation string.
• Develop a sequence of equations that will build
toward student fluency connected to conceptual
understanding and reasoning.
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