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 Page | 3 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). Woodburn School District- Mathematics Framework – Revised 9.16.13 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) Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 8 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 Page | 11 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 Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 12 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. Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 26 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 Page | 27 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 ) Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 28 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. Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 29 C ommon Agreements Q & A 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? Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 30 A ppendix Philosophy Methodology Student Mathematical Practices Mathematically Productive Teaching Routines Organization Woodburn School District- Mathematics Framework – Revised 9.16.13 Page | 31