12/18/2014
WOODBURN
SCHOOL
DISTRICT
MATHEMATICS INSTRUCTIONAL FRAMEWORK
By Elementary Teachers, Middle School Mathematics Teachers, High School Mathematics
Teachers, and Instructional Coaches in collaboration with WSD Department of Teaching and
Learning
T
able of Contents
Introduction…………………………………......……………………………………………………………………...
A historical overview of the Woodburn School District’s (WSD) journey into and through
mathematics.
3-4
Philosophy…………………………………………………………………………………………………………………
A brief statement that identifies the philosophical underpinnings and research of
mathematics in Woodburn.
Methodology…………………………………………………………………………………………………….
An explanation of the systems and processes that are used to put a philosophy into
practice.
Student Mathematical Practices………………………………………………………………………….
1. Mathematically Productive Thinking Routines
a. Habits of Mind
b. Habits of Interaction
Mathematically Productive Teaching Routines
1. Structures for mathematical Discourse
2. Organization and Planning for Instruction
a. Organization
b. Planning
References ………………………………………………………………………………………………………..
An annotated list of resources that support various components of the WSD mathematics
instructional framework.
Glossary…………………………………………………………………………………………………………….
A short dictionary of terminology used throughout the document.
Common District Agreements Appendix ……………………………………………………………
Woodburn School District mathematical norms on a variety of topics in Q&A format.
Appendix Index…………………………………………………………………………………………………
A compilation of articles and executive summaries that serve to explain or expand upon
various aspects of the mathematics framework.
Woodburn School District- Mathematics Framework – Revised 9.16.13
5-6
Page | 2
I
ntroduction
Though this is the first documentation of Woodburn School District’s thinking regarding
mathematics instruction, it is not the beginning of the work that has been done. Rather it is a
culmination of years of thinking and work by a diligent group of educators.
In the mid 1990’s, the mathematics adopted textbooks in the Woodburn School District were
very traditional. Around that time, a group of courageous and innovative teachers began using
a math curriculum with a constructivist approach, which originated from Portland State
University. This program was adopted district-wide as a supplemental program in 1995. The
adoption was the district’s first movement towards a constructivist approach to mathematics.
In 2002, WSD officially adopted Bridges, Investigations, Connected Mathematics Project (CMP)
and Interactive Math Program (IMP). The implementation of these materials continued to
move the district toward a conceptually-interconnected and constructivist approach to the
instruction of mathematics. This adoption sparked a relationship between the district and
Teachers Development Group (TDG), a non-profit organization dedicated to increasing
mathematical understanding and achievement through meaningful and effective professional
development.
In 2003, WSD developed a District Wide Math Cadre and in 2005 began participating in the
Oregon Mathematics Leadership Institute (OMLI) which had the dual focus of teacher content
knowledge and teacher mathematics pedagogy to improve student achievement as assessed
through student discourse. In 2009, a consortium of High School teachers convened to look at
text book options and switched from IMP to College Preparatory Math (CPM). As a result of the
work lead by the institute, the district partnered with TDG to provide training to teachers on
Math Best Practices through 2011.
With the national movement towards the Common Core In the 2011-2012, grade band levels
began working on alignment to the common core. In August of 2012, the district launched a
move towards a standards-based curriculum for all students in every content area that would
unify the districts constructivist approach and align to the Common Core. Teacher teams
deconstructed standards across the district that would later be used as a basis for unit planning.
In November of 2012, a group of mathematics teachers representing every grade band across
the district met and drafted a mathematics instructional framework to:
 Document the WSD’s philosophical beliefs about mathematics instruction
 Guide the district initiatives in mathematics through a continuous cycle of inquiry and
research,
 Outline a methodology to guide our practice, and
 Provide clear pedagogy and practices for teachers to utilize in classrooms.
Woodburn School District- Mathematics Framework – Revised 9.16.13
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Suggested Implementation Timeline
Elementary
Implementation
expectations
Who
Initial Professional
Development
By when
Follow Up
Professional
Development
Timeline
Additional
Resources needed
to support
implementation Page
New staff
All staff
Some staff
Few staff
Middle School
Who
Implementation expectations
Initial Professional
Development
By When/
Who
Follow Up
Professional
Development
Timeline
Additional
Resources needed
to support
implementation
Initial Professional
Development
By When/
Who
Follow Up
Professional
Development
Timeline
Additional
Resources needed
to support
implementation
All Language
Arts Teachers
High School Implementation Timeline
Who
Implementation expectations
All Language
Arts Teachers
Woodburn School District- Mathematics Framework – Revised 9.16.13
|4
Page | 5
Participants:
Administrators
Teachers & Instructional Coaches
Facilitator
Charles Ransom
Jennifer Dixon
Victor Vegara
Geri Federico
Joe James
Casey Wooley
Ricardo Marquez
Greg Baish
Irene Novichihin
Jenny Crist
Sherrilynn Rawson
Todd Farris
Juan Larios
Sonia Kool
Cadence Fee
Mike Blackwell
Sean Shevlin
Rochelle Gutierrez-Taeubel
Mark Gano
Marcelo Peralta
Ami Silkey
Brea Cohen
Ken Joslen
Gergana Dezsofi
Serge Lopez
Sara Chaudhary
Luisa Rodriguez
Lena Baucum
Liliya Zaltsman
Felipe Lora
Douglas Grant
Ana Casas
Maria Cervantes
Lizzett Wilson
Hector Tejeda
Brad Agenbroad
Marco Bolaños
Alisha Lopez
Lynn Koenig
Cristy Juarez
Woodburn School District- Mathematics Framework – Revised 9.16.13
P
hilosophy
Mathematics is a coherent intellectual system that we use to make sense of the world around
us. Students learn this system best when they are part of a community that engages in sensemaking through inquiry and constructivism (Piaget, 1985; Vygotsky, 1978). Students are
encouraged to represent, think about and communicate their mathematical understanding
through discourse (Resnick et al., 1991) and metacognition (Flavell, 1976; Brown, 1980).
Students learn to carry out procedures flexibly, accurately, efficiently, and appropriately to
ensure procedural fluency in addition to conceptual understanding (Kilpatrick, 2001). Teachers
use multilingualism and multiculturalism as an asset to increase cognitive flexibility and
abstract thinking skills, which are key to mathematics (Han, 2009). Instruction and experiences
consistently encourage higher-order thinking skills (Resnick et al., 1991) and target a student’s
zone of proximal development (Vygotsky, 1978). Ongoing formative assessments are used to
differentiate teaching and match instructional tasks to student needs (Tomlinson, 2000).
Heterogeneous learning communities hold all students to rigorous academic standards and
allow teachers to build on different learning styles and needs. Within heterogeneous learning
communities students are grouped flexibly, allowing students to
fluidly move between groups for different learning experiences and
purposes (Oakes, 1995). Students employ research- based
technologies in preparation for the practical applications of
mathematics in the 21st century (Tamin et al, 2011).
NOTE: Additional information and resources on Philosophy can be found in the Appendix __.
Woodburn School District- Mathematics Framework – Revised 9.16.13
Page | 6
M
ethodology
In order to fully implement our district philosophy of mathematics, we simultaneously employ
the Concrete-Representational-Abstract (CRA) continuum, the 4-phase model of instruction and
Sheltered Instruction (which includes explicit vocabulary instruction, vocabulary transfer and
the assurance of comprehensible input).
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Page | 7
Concrete-Representational-Abstract (CRA) Continuum:
Mathematics is taught along a developmental continuum that ranges from the concrete, to
representational, to the abstract. At the concrete stage, manipulatives are used to enhance
students’ understanding of mathematical concepts and their ability to make sense of a
problem. At the representational stage, students are encouraged to further develop their own
understanding of mathematic al concepts by using more efficient strategies, such as visual
representations using numbers that represent the concrete materials (manipulatives). During
the representational stage, the use of concrete is weaned so that students are able to move to
the abstract stage (Kato, 2002). Students at the abstract stage apply knowledge to make
generalizations and employ strategies to solve problems using symbols. Teachers support
students as they move along this developmental continuum through the use of ongoing
assessment, metacognition and non-judgmental feedback.
“Students acquire higher levels of mathematical proficiency when they have
opportunities to use mathematics to solve significant problems as well as to
learn the key concepts and procedures of that [particular strand of]
mathematics. Research reveals that various kinds of physical materials
commonly used to help children learn mathematics are often no more concrete
to them than symbols on paper might be. Learning begins with the concrete
when meaningful items in the child’s immediate experience are used as
scaffolding with which to erect abstract ideas. To ensure that progress is made
toward mathematical abstraction, we recommend the following:

Links [between and] among written and oral mathematical expressions,
concrete problem settings, and students’ solution methods, should be
continually and explicitly made during school mathematics instruction.”
(Kilpatrick, 2001)
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4 PHASE MODEL:
Teachers must be deliberate in how they scaffold conceptual understanding as students move
into the abstract and further away from representational and concrete mathematics to ensure
Page | 9
development of mathematical reasoning (Cai, 2001). This scaffolding includes:




The launching of new concepts,
Opportunities for students to explore new concepts,
Opportunities to synthesize and practice new concepts,
And application of concepts to new tasks and real world problems, with assessment
of that application.
Four Phase Model
Launch
Explore
STUDENTS
TEACHERS
Teachers pose a
question or questions,
conduct discussions,
demonstrate, model,
and/or explain
Teachers differentiate for
individual/groups of
students through task
selection or level of
scaffolding provided for a
given mathematical task.
Synthesize & Practice
Teachers pose questions
that promote student
discourse, justifications,
and generalizations.
Teachers assess students’
individual abilities to
apply knowledge to new
problems. Assessments
can vary from informal to
formal, depending on the
immediate situation and
goal of the particular
lesson or lessons. The
teacher should also
review the goals of the
lesson with the students
and tie goals back to the
standard or standards.
Students use multiple
methods to share, discuss,
reflect, synthesize, justify,
and make generalizations
that tie mathematical
concepts together.
Additionally, they make
explicit connections
between concrete and
abstract ideas.
Students demonstrate
their ability to work
independently, transfer
their knowledge to new
tasks or situations and
justify their thinking.
Teachers circulate, and
provide feedback by
posing questions to
groups or individuals.
Students are actively
involved through
answering questions,
discussing topics, and/or
attending to and
thinking about the
teacher’s presentation.
Students explore
concepts within their
zone of proximal
development through
relevant and worthwhile
tasks.
Students engage in
mathematical discourse
to critique ideas and seek
efficient mathematical
solutions.
Apply & Assess
Modified from Mathematics Framework for California Public Schools, California State Board of Education, 2006
Originally, a “Three-Phase Instructional Model” was created based on a review of 110 high
quality experimental research projects in mathematics done by the National Center to Improve
Woodburn School District- Mathematics Framework – Revised 9.16.13
the Tools of Educators, University of Oregon (Dixon et al. 1998). In 2005, the National Research
Council published How Students Learn Mathematics in the Classroom which shed light on the
importance of metacognition in ensuring conceptual understanding in mathematics.
Consequently, in 2012, WSD adapted the 3 phase model to include a “synthesize and practice”
phase to ensure students’ metacognition.
SHELTERED INSTRUCTION:
Sheltered Instruction includes a variety of techniques to help content-area teachers make
material comprehensible for students whose language skills are not yet fully developed yet
already have some proficiency in the language of instruction. Sheltered programs have proven
successful in the development of academic competence in students acquiring a second
language because such programs concentrate on the simultaneous development of contentknowledge and language skills. Sheltered Instruction has two charges: to provide access to
core content through ensuring students receive comprehensible input and to scaffold language
production so that all students may participate fully in the classroom context. (Krashen, 1985)
Teachers utilize sheltered instructional strategies within the four phases of mathematics
instruction to ensure that students receive comprehensible input and
explicit language instruction. Students participate in predetermined
language routines that enable them to practice/incorporate new language
and vocabulary into their mathematical discourse. The teacher provides
time for students to routinely transfer vocabulary between languages.
NOTE: See Appendix ? for Sheltered Instruction/4 Phase Architectures.
Woodburn School District- Mathematics Framework – Revised 9.16.13
Page | 10
S
tudent Mathematical Practices
The Standards for Mathematical Practice describe varieties of
expertise that mathematics educators at all levels seek to
develop in their students. These practices rest on important “processes
and proficiencies” with longstanding importance in mathematics
education. The first of these are the NCTM process standards of problem
solving, reasoning and proof, communication, representation, and connections. The second are the
strands of mathematical proficiency specified in the National Research Council’s report Adding It Up:
adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical
concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own
efficacy).
The Standards for Mathematical Practice describe ways in which developing student
practitioners of the discipline of mathematics increasingly engage with the subject matter as
they grow in mathematical maturity and expertise throughout the elementary, middle and high
school years.
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem
and looking for entry points to its solution. They analyze givens, constraints, relationships, and
goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous
problems, and try special cases and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their progress and change course if
necessary. Older students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get the information
they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on
using concrete objects or pictures to help conceptualize and solve a problem. Mathematically
Woodburn School District- Mathematics Framework – Revised 9.16.13
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proficient students check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?” They can understand the approaches of
others to solving complex problems and identify correspondences between different
approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it
symbolically and manipulate the representing symbols as if they have a life of their own,
without necessarily attending to their referents—and the ability to contextualize, to pause as
needed during the manipulation process in order to probe into the referents for the symbols
involved. Quantitative reasoning entails habits of creating a coherent representation of the
problem at hand; considering the units involved; attending to the meaning of quantities, not
just how to compute them; and knowing and flexibly using different properties of operations
and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and
previously established results in constructing arguments. They make conjectures and build a
logical progression of statements to explore the truth of their conjectures. They are able to
analyze situations by breaking them into cases, and can recognize and use counterexamples.
They justify their conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that take into account
the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct, even though
they are not generalized or made formal until later grades. Later, students learn to determine
domains to which an argument applies. Students at all grades can listen or read the arguments
of others, decide whether they make sense, and ask useful questions to clarify or improve the
arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace. In early grades, this might be as simple as
writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high
school, a student might use geometry to solve a design problem or use a function to describe
how one quantity of interest depends on another. Mathematically proficient students who can
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apply what they know are comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to identify
important quantities in a practical situation and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical
results in the context of the situation and reflect on whether the results make sense, possibly
improving the model if it has not served its purpose.
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their
grade or course to make sound decisions about when each of these tools might be helpful,
recognizing both the insight to be gained and their limitations. For example, mathematically
proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can
enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are
able to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to
explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use
clear definitions in discussion with others and in their own reasoning. They state the meaning of
the symbols they choose, including using the equal sign consistently and appropriately. They
are careful about specifying units of measure, and labeling axes to clarify the correspondence
with quantities in a problem. They calculate accurately and efficiently, express numerical
answers with a degree of precision appropriate for the problem context. In the elementary
grades, students give carefully formulated explanations to each other. By the time they reach
high school they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young
students, for example, might notice that three and seven more is the same amount as seven
and three more, or they may sort a collection of shapes according to how many sides the
shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression x2 + 9x + 14, older
students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
Woodburn School District- Mathematics Framework – Revised 9.16.13
Page | 13
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed
of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a
square and use that to realize that its value cannot be more than 5 for any real
Page | 14
numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for
general methods and for shortcuts. Upper elementary students might notice when dividing 25
by 11 that they are repeating the same calculations over and over again, and conclude they
have a repeating decimal. By paying attention to the calculation of slope as they repeatedly
check whether points are on the line through (1, 2) with slope 3, middle school students might
abstract the equation (y – 2)/(x– 1) = 3. Noticing the regularity in the way terms cancel when
expanding (x – 1)(x + 1), (x – 1)(x2 + x+ 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the
general formula for the sum of a geometric series. As they work to solve a problem,
mathematically proficient students maintain oversight of the process, while attending to the
details. They continually evaluate the reasonableness of their intermediate results.
Woodburn School District- Mathematics Framework – Revised 9.16.13
M
athematically Productive Thinking Routines
All students are capable mathematical thinkers, but not all students who
enter the classroom may believe this YET. The nature of the mathematical
actions and interactions in which students engage in a classroom impacts their understanding
of mathematics and their images of themselves as mathematicians. When a teacher explicitly,
purposefully, and continuously engages all students in mathematically productive routines of
thought and interaction, students develop deep understanding, a growth mindset (Dweck,
2006), and efficacy as mathematicians – and the CCSS for Mathematical Practice become
everyday norms for all students.
Mathematical Habits of Mind
When professional mathematicians “do mathematics,” they use and extend the math they know
to make sense and solve problems, justify, and generalize. Similarly, student mathematicians
learn and develop mathematically when they:

Make Sense - To make sense of math ideas and problems, students notice and use
regularly in repeated reasoning, patterns, and structure (e.g., definitions, meanings,
mathematical properties). They create and use representations, notice and use
connections, and they draw on other math they know, examples, non-examples, and
mistakes/stuck points to generate new possibilities. They use metacognition to examine
their own thinking and disequilibrium, as well as reflection about relationships between
their thinking and other mathematicians’ ideas. Continuous support from a teacher for
these productive habits fosters an expectation by students that mathematicians can and
will make sense through effort, and hence, fosters perseverance and hunger for
mathematical understanding.

Justify -Mathematical ideas, solutions, and conjectures make sense to students when
they and use mathematical reasoning (both inductive and deductive) to justify why
those ideas, solutions, and conjectures are always, sometimes, or never true. To justify,
Woodburn School District- Mathematics Framework – Revised 9.16.13
Page | 15
students identify relevant and age-appropriate mathematical definitions, properties,
processes, examples and non-examples, and/or established generalizations to build a
robust logical argument.

Generalize – The act of generalizing is central to “doing mathematics.” Students clarify, Page | 16
deepen, and expand their thinking about math ideas, processes, and problems by
making conjectures about aspects of those ideas and processes they think are always,
sometimes, or never true. They make these conjectures based on the math they know
combined with relationships they notice while they are justifying and making sense of
problems and ideas- that is, when searching for regularity, patterns, and structure, when
creating representations and making connections, and while using megtacognition and
reflection about their own and each other’s thinking, mistakes and stuck points. They
create mathematical generalizations by justifying why conjectures (their own and
other’s) are mathematically valid in the general case, special cases, and/or different
contexts. Such mathematical activity promotes the development of a growth mindset
and a student’s self-efficacy as a mathematician. (See Appendix ___ for Student
Reflection Tool.)
Mathematical Habits of Interaction
While deep, long-lasting mathematics understanding requires that students engage in thought
and actions that are reflective of ways of thinking and working used by successful
mathematicians, the character of interactions among student mathematicians also plays an
important role in the level of student engagement, depth of student learning, and the extent to
which the learning is equitable for all students. Learning increases when norms for student
interaction emphasize: expecting private reasoning time prior to discussion of a math
problem/idea, student explanation so their mathematical reasoning, listening to understand
each other’s reasoning, using genuine questions designed to elicit thinking, exploring multiple
pathways for reasoning, comparing the logic behind each other’s reasoning, critiquing and
debating the validity of reasoning, and relying on mathematical reasoning as the authority
Woodburn School District- Mathematics Framework – Revised 9.16.13
when determining the correctness or sensibility of a solution or idea. (See Appendix ___ for
Student Reflection Tool.)
Page | 17
Woodburn School District- Mathematics Framework – Revised 9.16.13
M
athematically Productive Teaching Routines
Page | 18
Woodburn School District- Mathematics Framework – Revised 9.16.13
S
tructures for Mathematical Discourse
Page | 19
Woodburn School District- Mathematics Framework – Revised 9.16.13
O
rganization and Planning for Instruction
Page | 20
Woodburn School District- Mathematics Framework – Revised 9.16.13
T
echniques for Mathematics Instruction
Page | 21
The instructional components within our Mathematics Instructional Framework are the structures that
support our WSD Methodology. These structures support teaching and learning by ensuring that
students will achieve proficiency in conceptual understanding, procedural fluency, strategic competence,
adaptive reasoning and productive disposition. These five strands are interwoven and interdependent in
the development of proficiency in mathematics.
Pre-Lesson Planning
Ensure Relevancy of a
Worthwhile Task
“Mathematical problems that are truly problematic and involve significant mathematics have the
potential to provide the intellectual contexts for students’ mathematical development. However,
only “worthwhile problems” give students the chance to solidify and extend what they know and
stimulate mathematics learning…Regardless of the context, worthwhile tasks should be intriguing
and contain a level of challenge that invites speculation and hard work. Most important,
worthwhile mathematical tasks should direct students to investigate important mathematical
ideas and ways of thinking toward the learning goals (NCTM, 1991).” (NCTM, 2010)
Assess and Launch
Technique
Description of Technique
Introduce New Concepts
Situate or contextualize
new concepts
Activate Schema and/or
Review old Concepts
New understandings are constructed on a foundation of existing understandings and
experiences. The understandings children carry with them into the classroom, even before
the start of formal schooling, will shape significantly how they make sense of what they are
taught. While prior learning is a powerful support for further learning, it can also lead to the
development of conceptions that can act as barriers to learning. (Council, 2005)
Pre-load vocabulary
Explore
Technique
Scaffold learning
through questioning
Description of Technique
The teacher is a model of critical thinking who respects students' viewpoints,
probes their understanding, and shows genuine interest in their thinking. The
teacher poses questions that probe students to question, support, justify their
results, and use critical thinking skills. Teacher frames questioning method in a
Woodburn School District- Mathematics Framework – Revised 9.16.13
way that challenges student thinking in a safe and engaging manner to
encourage discourse.
Scaffold CR
Page | 22
Synthesize and Practice
Technique
Description of Technique
Apply and Reassess
Technique
Description of Technique
Woodburn School District- Mathematics Framework – Revised 9.16.13
O
rganization
Page | 23
Quality of instruction is the single most important component of an effective mathematics
program (Beaton et al. 1996). In order to provide quality instruction for students, instruction
should ensure adequate time to implement all components of the 3 phase instructional
method. Within the school day, sufficient time should be devoted to mathematics instruction
to enable students to develop understanding of the concepts and procedures involved. Time
should be apportioned so that all strands of mathematical proficiency together receive
adequate attention (Kilpatrick, 2001).
Here are a couple of examples of how the math instruction time could be organized (see
student and teacher roles found on page ___):
Students without Fact Fluency
60 Minutes of Mathematics Instruction Daily
Time
Instruction
(Minutes)
10
25
15
10
As appropriate
Launch
Explore
Synthesis
Fact Fluency
Apply/Independent (homework)
Students with Fact Fluency
60 Minutes of Mathematics Instruction Daily
Time
Instruction
(Minutes)
10
30-35
15-20
As appropriate
Launch
Explore
Synthesis
Apply/Independent (homework)
by age
by age
Students without Fact Fluency
Students with Fact Fluency
Woodburn School District- Mathematics Framework – Revised 9.16.13
20
Launch
Students
Work independently or in
small groups around
differentiated tasks

Explore

Synthesize &
practice

and/or Apply
5
10
Synthesize
15
Synthesize & Practice
10
Teacher
Works with small
groups

Launch

Explore

Synthesize &
practice

Apply

and/or Fact and
Procedural
Fluency
20
Teacher assesses throughout math class
10
60 Minutes of Mathematics Instruction Daily
Time
Instruction
5
10
15
Instruction
Launch
Students
Work independently or in
small groups around
differentiated tasks

Explore

Synthesize &
practice

and/or Apply
Synthesize
Synthesize & Practice
*A 90 min. period for extensions may be found in Appendix…?
Woodburn School District- Mathematics Framework – Revised 9.16.13
Teacher
Works with small
groups

Launch

Explore

Synthesize &
practice

and/or Apply
Teacher assesses throughout math class
60 Minutes of Mathematics Instruction Daily
Time
Page | 24
R
eferences
(To be added once the document is completed by teams of teachers)
Woodburn School District- Mathematics Framework – Revised 9.16.13
Page | 25
G
lossary of Terminology
Abstract Thinking Skills
The ability to process ideas that process complex
visual or language based ideas that are not easily
associated with concrete ideas. Abstract ideas are
often invisible, complex and subjective. (Logsdon,
2012)
Adaptive Reasoning
Capacity for logical thought, reflection, explanation,
and justification.
CMP
Concepts
Connected Mathematics Program
Conceptual Understanding
Comprehension of mathematical concepts,
operations, and relationships.
Constructivism
Constructivism holds that humans are better able to
understand the information they have constructed
by themselves. According to constructivist theories,
learning is a social advancement that involves
language, real world situations, and interaction and
collaboration among learners. The learners are
considered to be central in the learning process.
Content
CPM
Differentiation
College Preparatory Math
Differentiation is when a teacher varies their
teaching in order to create the best learning
experience possible. This can be accomplished
through the differentiation of content, process,
products, and learning environment.
Non-judgemental Feedback
Effective feedback informs students of where they
are at, what the target is, and their next steps to
getting there. It is clear, concise and helps students
to see how to move closer to proficiency.
Fact Fluency
Math fact fluency is the ability to recall the answer
of basic math facts automatically and without
hesitation.
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Formative Assessment
Formative assessments are those that we use to
make changes in our teaching in order to support
student learning (Black & William, 1998).
Heterogeneous Learning Groups
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Higher-Order Thinking Skills
Higher-order thinking skills include critical logical,
reflective, metacognitive and creative thinking.
They are activated when individuals encounter
unfamiliar problems, uncertainties, questions or
dilemmas. (King F.J. et al., ___)
IMP
Inquiry
Interactive Mathematics Program
Inquiry implies student involvement that leads to
understanding. This means possessing skills and
attitudes that permit students to seek resolutions to
questions and issues while constructing new
knowledge.
Justification
Learning Environment
Mathematical Argument
Metacognition
The ability for one to think about his or her own
thinking and analyze one’s own thinking process.
Metacognition is used to reach higher-order
thinking.
Methods
Woodburn School District- Mathematics Framework – Revised 9.16.13
Multiculturalism
A person who is at ease with and understands the
values and norms of other cultures; is able to
engage in and navigate cultural boundaries.
Multilingualism
The ability to communicate in more than one
language either by speaking, writing, reading, or
signing.
Procedural Fluency
Skill in carrying out procedures flexibly, accurately,
efficiently, and appropriately.
Process
Product
Productive Disposition
Habitual inclination to see mathematics as sensible,
useful, and worthwhile, coupled with a belief in
diligence and one's own efficacy.
Proficiency Based Learning
The practice of enabling students to reach a high
level of proficiency by use of specific targets within
a proficiency through the use of formative and
summative assessments. Formative assessments
are used to inform student reflection and teacher
practice. Summative assessments are used to show
final proficiency.
Relevant Tasks
Research-Based Technologies
Scaffolding
Technologies are any research-based apparatus that
assist in learning and/or the connecting of
mathematics to real world application.
Technologies may consist of: graphing calculators,
scientific calculators, computers, computer
programs (i.e. Geometer's Sketchpad), infocus,
ipads, excel, etc.
Scaffolding refers to the idea that specialized
instructional supports need to be in place in order
to best facilitate learning when students are first
introduced to a new subject. (Wood et al., 1976 )
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Strategic Competence
Ability to formulate, represent, and solve
mathematical problems.
Summative Assessment
Summative assessments are those that we use at
the end of learning to measure the progress of
students and the impact of our teaching (Black &
William, 1998).
Technologies
Technologies are any apparatus that assist in
learning and/or the connecting of mathematics to
real world application. Technologies may consist of:
graphing calculators, scientific calculators,
computers, computer programs (i.e. Geometer's
Sketchpad), infocus, ipads, excel, etc.
Worth-while tasks
Good tasks are ones that do not separate
mathematical thinking from mathematical concepts
or skills that capture students' curiosity and that
invite them to speculate and to pursue their
hunches (NCTM, 1998).
Zone of Proximal Development
ZPD is delineates what a learner can accomplish
with help and what they cannot accomplish without
help. Teaching should lie slightly above their
independent level which a student should be able to
attain with the help of scaffolding.
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C
ommon Agreements Q & A
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How do we use calculators at different grade levels/bands?
How are students who are gifted in mathematics challenged?
How do we group students in mathematics classes?
How do we place students in math courses?
What language do we instruct students in at different grade levels?
What mechanisms are students expected to use to hold their thinking? (Notes, learning logs,
etc)
What is the expectation around the use of manipulatives?
What vocabulary do we use for mathematics in Russian, English, and Spanish?
How should a math class be organized at different grade bands?
How many minutes should we be spending teaching math each day at each grade level?
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A
ppendix
Philosophy
Methodology
Student Mathematical Practices
Mathematically Productive Teaching Routines
Organization
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