OKALOOSA COUNTY SCHOOL DISTRICT
Math Model for Best Practices Initiative
Comprehensive Balanced Mathematics
Model
Product of Mathematics Committee – May 2006 / Updated August 2012
Chaired by: Lynda Penry
Directed by: Guyla Hendricks
Updated by: Debbie Davis
Goal: Research and adopt Mathematics Best Practices from
National Council of Teachers of Mathematics (NCTM) and the Common Core Standards For Mathematical Practice as a District Model
Math Model Example: Applying All 5 Mathematical Components to a Single Skill (Integration)
Descriptors of the Principles of Mathematics w/Classroom Best Practices
Self-reflection Survey
Math Model Resources
Comprehensive Balanced Mathematics Model
Mathematics educators and researchers suggest an activity-oriented classroom. In an activity-oriented classroom the teacher promotes reasoning
and encourages children to exchange viewpoints which foster confidence in a student’s ability to think (Kamii & Joseph, 1989). As students discuss
and exchange view points, a shift occurs in the focus of authority (Schifter & Fosnot, 1993); the classroom becomes a “community of inquiry” (p.
11). In a community of inquiry, teachers are able to promote reasoning and sense-making (NCTM, 1989). Instruction in an activity-oriented
classroom should incorporate a variety of activities which include the manipulation of materials and the opportunity for discussion and interaction.
The teacher should not only provide opportunities for individual work, but should also include opportunities for cooperative learning. If students are
to learn, they need to reflect on their own thoughts, as well as the thoughts and ideas of others (Owen & Lamb, 1996).
Component
Problem
Solving
Description
Students engage in a task for which the method for
determining the solution is not known in advance.
Problem solving enables all students to build new
mathematical knowledge; solve problems that arise in
mathematics and in other contexts; apply and adapt a
variety of appropriate strategies to solve problems, and
monitor and reflect on the process of mathematical
problem solving (Principles and Standards p. 52).
Without the ability to solve problems, the usefulness and
power of mathematical ideas, knowledge, and skills are
severely limited.
Unless students can solve problems, the facts, concepts
and procedures they know are of little use.
Problem solving is central to inquiry and application and
serves as a vehicle for learning new mathematical ideas
and skills (Schroeder and Lester 1989).
Problem solving reveals mathematics as a sense-making
discipline rather than one in which rules are given by the
teacher to be memorized and used by students.
Standard for
Mathematical
Practice 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
Teacher’s Role
Student’s Role
Establish a supportive environment in
which students are encouraged to explore,
take risks, share failures and successes,
and question one another.
Provide worthwhile problems and
mathematical tasks (p. 53).
The challenge is to build on a student’s
innate problem-solving inclinations and
establish a classroom setting that values
problem-solving.
Students need to be taught the knowledge
of strategies in order to reconsider a
problem when the initial approach fails
(e.g., break it down and look at it from
different perspectives). This process
enables students to understand a problem
better and make progress toward its
solution.
Mathematically proficient students:
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Explain to themselves the meaning
of a problem and look for entry
points to its solution.
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Analyze givens, constraints,
relationships, and goals.
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Make conjectures about the form
and meaning of the solution attempt.
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Consider analogous problems, and
try special cases and simpler forms
of the original problem.
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Monitor and evaluate their progress
and change course if necessary.
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Transform algebraic expressions or
change the viewing window on their
graphing calculator to get
information.
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Explain correspondences between
equations, verbal descriptions,
tables, and graphs.
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Draw diagrams of important features
and relationships, graph data, and
search for regularity or trends.
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Use concrete objects or pictures to
help conceptualize and solve a
problem.
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Check their answers to problems
using a different method.
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Ask themselves, “Does this make
sense?”
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Understand the approaches of
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 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.
Standard for
Mathematical
Practice 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.
Reasoning
and Proof
Students make, investigate, and evaluate mathematical
conjectures.
Reasoning is essential to understanding mathematics.
Students see and expect that mathematics makes sense.
Reasoning mathematically is a habit of mind and must be
developed through consistent use in many contexts (p. 56).
Mathematics should make sense to students.
Seeking and finding explanations for patterns helps
students develop deeper understanding of mathematics.
Students generalize from examples so teachers should
guide them to use examples and non-examples to test
whether their generalizations are appropriate (Carpenter
and Levi, 1999).
There is clear evidence that in classrooms where
reasoning is emphasized, students do engage in reasoning
and, in the process, learn what constitutes an acceptable
mathematical explanation (Lamper 1990; Yackel and Cobb
1994, 1996).
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Help students learn to make conjectures by
asking questions, to reason from what they
know.
Help students learn that several examples
are not sufficient to establish the truth of a
conjecture and that non-examples can be
used to disprove a conjecture (e.g., Can a
scalene triangle be a right triangle? Some
examples indicate no, but until a student
finds a right, scalene triangle, it disproves
the conjecture.)
Establish a risk-free climate where students
are encouraged to put forth their ideas for
examination.
Teachers and students should be open to
questions, reactions, and elaborations from
others in the classroom.
Students need to be given the opportunity
others to solving complex problems.
Notice if calculations are repeated
Look both for general methods and
for shortcuts.
Maintain oversight of the process,
while attending to the details.
Continually evaluate the
reasonableness of intermediate
results.
Mathematically proficient students:
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Work with other students to formulate
and explore conjectures and to listen
to and understand conjectures offered
by classmates (p. 57).
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Consider a range of examples and
non-examples to reason about the
general properties and relationships
they find.
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Develop descriptions and
mathematical statements about
relationships to begin to understand
the role of definition in mathematics.
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Explain and justify their thinking and
learn to detect fallacies and critique
others’ thinking.
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Understand that mathematics involves
examining patterns and noting
Standard for
Mathematical
Practice 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.
to apply their reasoning skills and justify
their thinking in mathematics discussions.
Students need to be given time,
experience, and guidance to develop the
ability to construct valid arguments and to
evaluate the arguments of others.
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Communication
Communication is a way of sharing ideas and clarifying
understanding.
Through communication, teachers and students use the
language of mathematics to express ideas precisely.
An important step in communicating mathematical thinking
to others is organizing and clarifying one’s ideas.
Communication is an essential feature as students express
the results of their thinking orally and in writing.
Speaking and writing should be more detailed and
coherent and include increasing mathematical vocabulary
to explain concepts as students progress through school.
The value of mathematical discussions is determined by
whether the students are learning as they participate in
them (Lampert and Cobb,).
Explanations should include mathematical arguments and
rationales, not just procedural descriptions or summaries
(Yackel and Cobb, 1996).
When ideas are exchanged and subjected to thoughtful
critiques, they are often refined and improved (Borasi,
1992; Moschkovich, 1998).
As students develop clearer and more coherent
Give students daily opportunities to discuss
and clarify their thinking with partners and
in small cooperative group settings.
Incorporate as an instructional goal, what is
acceptable as evidence in mathematics.
Provide adequate time and interesting
mathematical problems and materials,
encourage conversation and learning
among students.
Build a sense of community so students
feel free to express their ideas honestly and
openly.
Schedule daily opportunities to students to
talk and write about mathematics.
Provide challenging and meaningful
problems to encourage students to think
about how familiar concepts and
procedures can be applied in new
situations.
regularities, making conjectures about
possible generalizations, and
evaluating the conjectures.
Progress to the level of formulating
mathematical arguments by using
inductive and deductive reasoning.
Develop compelling arguments with
enough evidence to convince
someone who is not part of their own
learning community.
Make sense of quantities and their
relationships in problem situations.
De-contextualize (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
Contextualize (pause as needed
during the manipulation process in
order to probe into the referents for
the symbols involved).
Use quantitative reasoning that
entails creating a coherent
representation of quantities, not just
how to compute them
Know and flexibly use different
properties of operations and objects.
Mathematically proficient students:
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Work in pairs or small groups enabling
students to hear different ways of
thinking and refine the ways in which
they explain their own ideas.
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Become the audience for one
another’s comments, listening to a
number of peers and joining group
discussions to clarify, question, and
extend conjectures.
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Present and explain the strategy used
to solve a problem and to analyze,
compare, and contrast the
meaningfulness and efficiency of a
variety of strategies.
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Understand and use stated
assumptions, definitions, and
previously established results in
constructing arguments.
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Make conjectures and build a logical
progression of statements to explore
communication (using verbal explanations and appropriate
mathematical notation and representations), they will
become better mathematical thinkers.
Standard for
Mathematical
Practice 3:
Construct viable
arguments and
critique the
reasoning of
others.
Standard for
Mathematical
Practice 6:
Attend to
precision.
Connections
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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.
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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.
Students’ ability to experience mathematics as a
meaningful endeavor that makes sense rests on
connections; connections between different mathematics
topics, between math and other subject areas, and math
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Emphasize the interrelatedness of
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mathematical ideas so students may learn
the usefulness of mathematics.
Plan lessons so that skills and concepts are
the truth of their conjectures.
Analyze situations by breaking them
into cases
Recognize and use counterexamples.
Justify their conclusions,
communicate them to others, and
respond to the arguments of others.
Reason inductively about data,
making plausible arguments that take
into account the context
Compare the effectiveness of
plausible arguments
Distinguish correct logic or reasoning
from that which is flawed
 Elementary students construct
arguments using objects,
drawings, diagrams, and actions..
 Later students learn to determine
domains to which an argument
applies.
Listen or read the arguments of
others, decide whether they make
sense, and ask useful questions
Try to communicate precisely to
others.
Use clear definitions in discussion
with others and in their own
reasoning.
State the meaning of the symbols they
choose, including using the equal sign
consistently and appropriately.
Specify units of measure and label
axes to clarify the correspondence
with quantities in a problem.
Calculate accurately and efficiently,
express numerical answers with a
degree of precision appropriate for the
context.
 In the elementary grades,
students give carefully
formulated explanations to each
other.
 In high school, students have
learned to examine claims and
make explicit use of definitions.
Mathematically proficient students:

Understand how mathematical ideas
interconnect.
and every-day life.
With connections, students build new understandings on
previous knowledge.
Students must view mathematics as a connected and
integrated whole (pg. 65).
Connections help students realize the beauty of
mathematics and its function as a means of more clearly
observing, representing, and interpreting the world around
them.
When students use the relationships in and among
mathematical content and processes, they advance their
knowledge of mathematics and extend their ability to apply
concepts and skills more effectively.
Connections help students see mathematics as a unified
body of knowledge rather than as a set of complex and
disjointed concepts, procedures, and processes.
Mathematical
Practice7:
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 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 numbers x and y.
Representation
Representations serve as tools for thinking about and
solving problems as well as communicating ways of
thinking.
If mathematics is the “science of patterns” (Steen, 1998),
representations are the means by which those patterns are
recorded and analyzed.
It is important to encourage students to represent their
ideas in ways that make sense to them, even if their first
representations are not conventional ones (pg. 67).
Students can develop and deepen their understanding of
mathematical concepts and relationships as they create,
compare, and use various representations.
taught not as isolated topics but rather as
valued, connected, and useful parts of
students’ experiences.
Set the expectation for students to reason
mathematically and communicate clearly
about significant mathematical tasks.
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Analyze students’ representations and
listen carefully to their discussions to gain
insights into the development of
mathematical thinking.
Gain insight by examining, questioning, and
interpreting their representations.
Help students represent aspects of
situations in mathematical terms, by using
more than one representation.
Model the process of representation as
students work through problems.
Discuss with students why some
representations are more effective than
Reflect on and compare solutions as a
means of making connections.
Describe mathematical connections.
Look closely to discern a pattern or
structure.
 Young students might notice that
three and seven more is the
same amount as seven and three
more.
 Later, students will see 7 x 8
equals the well-remembered 7 x
5 + 7 x 3, in preparation for the
distributive property.
 In the expression x2 + 9x + 14,
older students can see the 14 as
2 x 7 and the 9 as 2 + 7.
Step back for an overview and can
shift perspective.
See complicated things, such as
some algebraic expressions, as single
objects or composed of several
objects.
Mathematically proficient students:
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Learn to record or represent thinking
in an organized way.
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Learn to use equations, charts, and
graphs to model and solve problems.
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Use informal representations such as
drawings highlight various features of
problems.
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Use physical models to represent and
understand ideas (external models as
well as mental images).
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Choose specific representations in
order to gain particular insights or
achieve particular ends.
others in a particular situation.
Standard for
Mathematical
Practice 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 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.
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Standard for
Mathematical
Practice 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
Understand that an object can be
better understood when viewed
through multiple lenses (i.e., different
representations support different ways
of thinking about and manipulating
mathematical objects).
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.
Simplify a complicated situation,
realizing that these may need revision
later.
Identify important quantities in a
practical situation
Map their relationships using such
tools as diagrams, two-way tables,
graphs, flowcharts and formulas.
Analyze those relationships
mathematically to draw conclusions.
Interpret their mathematical results in
the context of the situation.
Reflect on whether the results make
sense, possibly improving the model if
it has not served its purpose.
Mathematically proficient students
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Consider available tools when solving
a mathematical problem.
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Are familiar with tools appropriate for
their grade or course to make sound
decisions about when each of these
tools
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Detect possible errors by using
estimations and other mathematical
knowledge.
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Know that technology can enable
them to visualize the results of varying
assumptions, and explore
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.
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consequences.
Identify relevant mathematical
resources and use them to pose or
solve problems.
Use technological tools to explore and
deepen their understanding of
concepts.
Math Model Examples: Applying All Five Mathematical Components to a Single Skill
K-2 (Number sense/Base Ten)
 Problem Solving: I have two tens and 5 ones. You give me one ten and 8 ones. How many do I have now?
 Reasoning and Proof: Using the 100’s number chart, make a conjecture about counting by tens. What would the next row of numbers across the bottom
be? Justify your reasoning.
 Communication: Turn to your partner and explain with words or pictures how you solved the problem.
 Connections: Use two colors of connecting cubes to determine how many ways you can make 4. (Teachers: help students generalize and predict how
many ways there are to make 5, 6, 7). (pg. 133)
 Representation: Record a method to find 17 + 25 in two different ways.(pg. 140)
3-5 (Changing Dimensions/Patterns)
 Problem Solving: Using dot paper, draw six different squares beginning with the least area to the greatest area.
 Reasoning and Proof: What patterns do you see as the squares become larger? Explain the pattern.
 Communication: Express the patterns you see in the “growing squares” in mathematical sentences. Make predictions about what will happen if the
sequence is continued.
 Connections: Through discussion, students apply information obtained from the squares task to develop or suggest a formula for area and/or square
numbers.
 Representation: Does your pattern work for a square of any size? Create a chart to show your thinking.
6-8 (Changing Dimensions / Equations)
 Problem Solving: Create 4 towers (single cube wide) of differing heights using 1-inch cubes, beginning with a single cube. Determine the surface area of
each tower.
 Reasoning and Proof: Describe the pattern relating the surface areas to the towers as they become taller. Build one more tower to test your pattern.
 Communication: Express the pattern (relationship) in a mathematical sentence (equation).
 Connections: How would the pattern change if the towers are double wide rather than single? Draw 4 double width towers on dot paper to test your
prediction.
 Representation: Create a table of values and graph representing the relationship of the height of the single-wide towers and their corresponding surface
areas. You may use computer software or a graphing calculator.
9-12 (Exponential Decay)
 Problem Solving:
o Drop 100 pennies onto a table. Remove all the pennies that land tails up. How many heads-up pennies are left?
o Drop the remaining pennies onto the table. Once again remove all the tails-up pennies and count the remaining pennies.
o Again, drop the remaining pennies onto the table. Once again remove all the tails-up pennies and count the remaining pennies. Determine a
function that models this situation.
 Reasoning and Proof: Describe with your small group, the pattern of remaining pennies that you see happening. Repeat the process 3 more times to test
your prediction.
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Communication: Express the pattern you see in the “decreasing pennies” with a graph.
Connections: Connect this activity with half-life of radioactive materials – possibly research and bring back to the group/class.
Representation: Create a table of values reflecting the experiment. Using the graph and table of values, represent the experiment with an equation.
Descriptors of the Principles of Mathematics with Classroom Best Practices in Action
The Equity Principle
Excellence in mathematics education requires equity – high expectations and strong support for all students.
Equity requires high expectations and worthwhile opportunities for all.
Equity requires accommodating differences to help everyone learn mathematics.
Equity requires resources and support for all classrooms and all students.
Best Practices:
Small group instruction
Differentiated instruction (single concept presentation using variation in vocabulary, models, tasks/assignments, materials, and time)
Hands-on activities
Manipulatives in use
Peer interaction/support
Teacher-student conferencing with feedback
Questioning techniques, strategies, assignments, and verbal interaction which require higher levels of Bloom’s/Webb’s taxonomies
The Curriculum Principle
A curriculum is more than a collection of activities: it must be coherent, focused on significant mathematical concepts, and well articulated across the
grades.
A mathematics curriculum should be coherent.
A mathematics curriculum should focus on important principles of mathematics.
A mathematics curriculum should be well articulated across grade levels.
Use the Next Generation Florida Standards, Benchmarks, and the Secondary Bodies of Knowledge to drive instructional decisions, rather than a specific
text or program.
Best Practices:
A wide and rich range of materials to support the curriculum
Teacher-directed lessons which include important elements such as terminology, definitions, notation, concepts, and skills covering the big ideas of
mathematics
Room displays which support a math-rich environment (i.e., word walls, posters, bulletin boards, and other visual aids).
Evidence of instructional planning which contain grade-level or course content standards (lesson plans, unit plans).
Math stations which include activities for practicing grade-level benchmarks or course content standards.
The Teaching Principle
Effective mathematics teaching requires the understanding of what students know and need to learn, and how to provide the challenge and support for
them to learn it well.
Effective teaching requires knowing and understanding mathematics, research-based principles applied to students as learners, and rich and varied
pedagogical strategies.
Teachers need to understand the big ideas of mathematics and be able to represent mathematics as a coherent and connected enterprise (Schifter 1999;
Ma 1999).
Effective teaching requires a challenging and supportive classroom learning environment.
Students’ ideas should be valued and serve as a source of learning (pg.145).
Rich problems, a climate that supports mathematical thinking, and access to mathematical tools contribute to students’ seeing connections (pg.359).
Effective teaching requires continually seeking improvement.
Individual professional development should be sought to build competence and confidence in mathematical understanding.
Best Practices:
Reflective practice
Student discussions and collaboration
Student seating arrangements which promotes discussion and collaboration
Higher order questioning techniques to promote mathematical understanding
Display of student work
Modeling and sharing of thought processes
The Learning Principle
Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
Learning mathematics with understanding is essential.
A hands-on, risk-free environment promotes mathematical understanding.
Conceptual understanding is an important component of proficiency, along with factual knowledge (Bransford, Brown, and Cocking 1999).
Students can learn mathematics with understanding.
From a young age, children are interested in mathematical ideas. Therefore, a mathematics program should enhance their natural desire to understand
what they are asked to learn (pg.21).
Best Practices:
Students actively engaged in tasks
Students actively engaged in discourse
Use of graphic organizers by which students organize their thoughts
Justification of answers by students
Writing journals which include reflection and examples of short/extended responses
Respect shown among students (e.g., positive, supportive comments)
Hands-on activities
Open-ended tasks
Collaborative groups
The Assessment Principle
Assessment should support the learning of important mathematics principles and furnish useful information to both teachers and students regarding
instructional decisions that will improve student outcomes.
Assessment should enhance student learning.
The learning of students, including low achievers, is generally enhanced in classrooms where teachers include attention to formative assessment in
making judgments about teaching and learning (Black and Wiliam 1998).
Assessment tasks must be worthy of student time and attention.
Assessment is a valuable tool for making instructional decisions.
In addition to formal assessments such as tests and quizzes, teachers should be continually gathering information about their students’ progress through
informal assessment measures (pg. 23).
Best Practices:
Immediate feedback provided during classroom discussions
Student self-assessment
Use of high-interest electronic assistance (e.g., Flash Master, Hot Dots, Math Safari) to provide daily practice and immediate feedback
Charts of student progress
Varied assessment techniques such as open-ended questions, constructed-response tasks, selected-response items, performance tasks, observations,
conversations and interviews with students, or interactive journals
The Technology Principle
Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.
Technology enhances mathematics learning.
Technology supports effective mathematics teaching.
Technology influences what mathematics is taught.
Best Practices:
Calculators being used to foster understanding and intuition
Computers being used to enhance student learning opportunities
Provide daily practice at the individual level
Advance mathematical knowledge through real-world connections and application of mathematical skills and concepts
Aid in process development (i.e., spreadsheet)
Electronic support (e.g., Flash Master, Hot Dots, Math Safari) for computational fluency
Self-reflection Survey
Comprehensive Balanced Mathematics Model
Rate your confidence level on each of the following descriptors which should be related to the consistency of use in the classroom. This information can be used
to identify the focus for staff development in mathematics (individual and/or total staff) to increase instructional competence and confidence.
Very Confident
Confident
Neutral
Not There Yet
Nowhere Near
4
3
2
1
0
Equity
_____ Incorporate hands-on activities
_____ Use manipulatives effectively
_____ Allow for small group instruction
_____ Allow for peer interaction/support
_____ Offer differentiated instruction
_____ Questioning techniques require higher levels of Bloom’s/Webb’s Taxonomies
_____ Practice teacher-student conferencing with feedback
Curriculum
_____ Room displays which support a math-rich environment
_____ Use math stations which include activities for practicing grade-level benchmarks
_____ Teacher directed lessons including important elements of mathematics
_____ A wide and rich range of materials to support the curriculum
_____ Plan for instruction using Next Generation Florida Standards, grade-level benchmarks, or Secondary Bodies of Knowledge
Teaching
_____ Display students’ work
_____ Continue to build competence and confidence in mathematical understanding
_____ Allow for student discussions and collaboration
_____ Model and discussing thinking process
_____ Questioning techniques are used to promote deep understanding embodied in the mathematical task
Learning
_____ Collaborative groups
_____ Students actively engaged in tasks and discourse
_____ Use of graphic organizers by which students organize their thoughts
_____ Use writing journals which include reflection and examples of short/extended responses
_____ Present open-ended tasks
_____ Nurture respect shown among students
Assessment
_____ Offer immediate feedback during classroom discussions
_____ Allow for student self-assessment
_____ Use varied assessment techniques
_____ Know the assessment piece before teaching the skill or concept
Technology
_____ Use calculators to foster understandings and intuitions
_____ Use computers in the classroom to enhance students’ learning opportunities
_____ Offer electronic support for computational fluency
Resources
C-PALMS:
http://www.floridastandards.org Use the tabs to search for course descriptions, Florida Standards, lesson plans and activities, and other resources
matched to the Florida benchmarks. Resources have been submitted and reviewed by Florida educators.
Common Core State Standards and Standards for Mathematical Practice:
 Common Core State Standards Initiative: http://www.corestandards.org/
 Common Core Progressions: http://ime.math.arizona.edu/progressions/
National Council of Teachers of Mathematics:
 Lessons and Resources:
http://nctm.org/resources/default.aspx?id=230
 Illuminations: http://illuminations.nctm.org/
 Principles and Standards: http://www.nctm.org/standards/
FCAT Explorer and Florida Achieves:
http://www.fcatexplorer.com/ Username and password information has been sent to each teacher.
Gale Cengage
You do not need a password if you log on at school.
Go to www.okaloosaschools.com and click on Gale Cengage under Instructional Technology. This is available to all parents, teachers, and
students. Gale is an educational data base of reference content that supports innovative teaching,
At home: Username and password information was sent to each teacher.