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 PenryDirected by: Guyla Hendricks

Updated by: Debbie DavisGoal: 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 Description Teacher’s Role Student’s Role

Problem Solving

Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them.

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.

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

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: Explain to themselves the meaning

of a problem and look for entry points to its solution.

Analyze givens, constraints, relationships, and goals.

Make conjectures about the form and meaning of the solution attempt.

Consider analogous problems, and try special cases and simpler forms of the original problem.

Monitor and evaluate their progress and change course if necessary.

Transform algebraic expressions or change the viewing window on their graphing calculator to get information.

Explain correspondences between equations, verbal descriptions, tables, and graphs.

Draw diagrams of important features and relationships, graph data, and search for regularity or trends.

Use concrete objects or pictures to help conceptualize and solve a problem.

Check their answers to problems using a different method.

Ask themselves, “Does this make sense?”

Understand the approaches of others to solving complex problems.

Standard for Mathematical Practice 8: Look for and express regularity in repeated reasoning.

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.

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.

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.

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

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

Mathematically proficient students: Work with other students to formulate

and explore conjectures and to listen to and understand conjectures offered by classmates (p. 57).

Consider a range of examples and non-examples to reason about the general properties and relationships they find.

Develop descriptions and mathematical statements about relationships to begin to understand the role of definition in mathematics.

Explain and justify their thinking and learn to detect fallacies and critique others’ thinking.

Understand that mathematics involves examining patterns and noting

Standard for Mathematical Practice 2: Reason abstractly and quantitatively

1994, 1996).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.

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.

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.

Mathematically proficient students: 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.

Become the audience for one another’s comments, listening to a number of peers and joining group discussions to clarify, question, and extend conjectures.

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.

Understand and use stated assumptions, definitions, and previously established results in constructing arguments.

Make conjectures and build a logical progression of statements to explore

Standard for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

Standard for Mathematical Practice 6: Attend to precision.

communication (using verbal explanations and appropriate mathematical notation and representations), they will become better mathematical thinkers.

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.

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.

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.

Connections 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

Emphasize the interrelatedness of mathematical ideas so students may learn the usefulness of mathematics. Plan lessons so that skills and concepts are

Mathematically proficient students: Understand how mathematical ideas

interconnect. Reflect on and compare solutions as a

Mathematical Practice7: Look for and make use of structure.

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.

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.

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.

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.

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.

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

Mathematically proficient students: Learn to record or represent thinking

in an organized way. Learn to use equations, charts, and

graphs to model and solve problems. Use informal representations such as

drawings highlight various features of problems.

Use physical models to represent and understand ideas (external models as well as mental images).

Choose specific representations in order to gain particular insights or achieve particular ends.

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.

others in a particular situation. 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.

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

Mathematically proficient students Consider available tools when solving

a mathematical problem. Are familiar with tools appropriate for

their grade or course to make sound decisions about when each of these tools

Detect possible errors by using estimations and other mathematical knowledge.

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.

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. 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 PrincipleExcellence 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 instructionDifferentiated instruction (single concept presentation using variation in vocabulary, models, tasks/assignments, materials, and time)Hands-on activitiesManipulatives in usePeer interaction/supportTeacher-student conferencing with feedbackQuestioning techniques, strategies, assignments, and verbal interaction which require higher levels of Bloom’s/Webb’s taxonomies

The Curriculum PrincipleA 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 curriculumTeacher-directed lessons which include important elements such as terminology, definitions, notation, concepts, and skills covering the big ideas of mathematicsRoom 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 collaborationHigher order questioning techniques to promote mathematical understanding Display of student workModeling and sharing of thought processes

The Learning PrincipleStudents 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 tasksStudents actively engaged in discourseUse of graphic organizers by which students organize their thoughtsJustification of answers by students Writing journals which include reflection and examples of short/extended responsesRespect shown among students (e.g., positive, supportive comments)Hands-on activitiesOpen-ended tasksCollaborative groups

The Assessment PrincipleAssessment 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 progressVaried 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 PrincipleTechnology 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 intuitionComputers being used to enhance student learning opportunitiesProvide daily practice at the individual levelAdvance mathematical knowledge through real-world connections and application of mathematical skills and conceptsAid 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 CengageYou 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.