Transition to CCSS Mathematics in a PLC: The Impact on ...soltreemrls3.s3-website-us-west-2.amazonaws.com/.../pdf/...stlouis.p… · Transition to CCSS Mathematics in a PLC The Impact

Transition to CCSS Mathematics in a PLC: The Impact on Your K–8

Mathematics Program

Diane J. Briars

Transition to CCSS Mathematics in a PLC The Impact on Your K–8 Mathematics Program

Diane J. Briars, PhD, Mathematics Education Consultant

Past President, National Council of Supervisors of Mathematics [email protected]

“These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.”

─National Governors Association Center for Best Practices and Council of Chief State School Officers

Excerpts—Common Core Mathematics in a PLC: Grades 6–8 (Briars, Asturias, Foster, & Gale, Solution Tree Press, forthcoming Oct. 2012) One of the greatest equity issues for mathematics instruction, and instruction in general in most school districts, is that it is too inconsistent from classroom to classroom, school to school, and district to district (Morris & Hiebert, 2011). How much mathematics a student in the United States learns, and how deeply he or she learns it, is largely determined by the school the student attends, and even more significantly the teacher to whom the student is randomly (usually) assigned within that school. The inconsistencies teachers develop in their professional development practice—often random and in isolation from other teachers—create great inequities in students’ mathematics instructional and assessment learning experiences that ultimately and significantly contribute to the year-by-year achievement gap (Ferrini-Mundy, Graham, Johnson, & Mills, 1998) in your school. This issue is especially true in a vertically connected curriculum like mathematics. Five Fundamental Paradigm Shifts for Implementing CCSS in Mathematics The CCSS expectations for teaching and learning and the new consortia assessments usher in an opportunity for unprecedented change of the second-order variety. In contrast to first-order change, second-order change requires working outside the existing system by embracing new paradigms for how you think and practice (Waters, Marzano, & McNulty, 2003). There are five fundamental second-order paradigm shifts (outside of existing paradigms) required to prepare every student and teacher for the successful implementation of the CCSS in mathematics and for the general improvement of mathematics learning for K–12 students in the United States. They are:

1. Collaboration: Successful implementation of the CCSS requires a shift to a grain size of change beyond the individual isolated teacher. It is grade-level or course-based collaborative learning teams (collaborative teams), within a professional learning community culture that will develop the expanded teacher knowledge capacity necessary to bring coherence to the implementation of the CCSS.

2. Instruction: Successful implementation of the CCSS requires a shift to daily and unit lesson plans that incorporate opportunities for students to develop proficiency in the Mathematical Practices

(Transitions to the CCSS, page 1 of 2)

© Briars 2012. solution-tree.comDo not duplicate. 1

and develop deep understanding of the content standards. This change involves teaching for procedural fluency and student understanding of the grade-level CCSS standards. One should not exist at the expense of the other. This will require collaborative team commitment to the use of student-engaged learning around common high-cognitive demand mathematical tasks in every classroom.

3. Content: Successful implementation of the CCSS requires a content shift to a less (fewer standards) is more (deeper rigor with understanding) at each grade level. This shift will require new levels of knowledge and skill development for every teacher of mathematics to understand what the CCSS expect students to learn at each grade level or within each course, blended with how students are expected to learn it. What are the mathematical knowledge, skills, understandings, and habits of mind that should be the result of each unit of mathematics instruction? Schools and mathematics programs committed to helping all students learn ensure greater clarity and low teacher-to-teacher variance to the questions: What should students learn? How should they learn it?

4. Assessment: Successful implementation of the CCSS requires a shift to assessments that are a means within the teaching–assessing–learning cycle and not used as an end to that cycle. These assessments must reflect the rigor of the standards and model the expectations for and benefits of formative assessment practices around all forms of assessment including typical instruments such as tests and quizzes. How will you know if students are prepared for the rigorous assessments from the two assessment consortia: Partnership for Assessment of Readiness for College and Careers (PARCC) and the Smarter Balanced Assessment Consortium (SBAC)?

5. Intervention: Successful implementation of the CCSS requires a shift in collaborative team and school response to intervention (RTI). Much like the CCSS vision for teaching and learning, RTI can no longer be invitational to students. That is, the response to intervention becomes R²TI—a required response to intervention. Stakeholder implementation of RTI programs includes a process that requires students to participate and attend. How will you respond and act on evidence (or lack of evidence) of student learning?

Second-order change—moving beyond familiar practices—is never easy. It requires willingness to break away from the past practice of teaching one standard a day in mathematics lessons with low-cognitive demand. This change requires departure from a past practice that provided few student opportunities for exploring, understanding, and actively engaging, and one that used assessment instruments that may or may not have honored a fidelity to accurate and timely formative feedback. Now, every teacher will be required to embrace these new paradigms to meet the expectations of the CCSS.

(Transitions to the CCSS, page 2 of 2)

© Briars 2012. solution-tree.comDo not duplicate.2

Common Core State Standards for Mathematics Standards for Mathematical Practice

(Source: This material is available at www.corestandards.org/the-standards/mathematics /introduction/standards-for-mathematical-practice) The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should 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). 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 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 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

(CCSS for Mathematics, page 1 of 3)

3© National Governors Association Center for Best Practices

and Council of Chief State School Officers 2010. All rights reserved.

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 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 estimations 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 the 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.


4

© National Governors Association Center for Best Practicesand Council of Chief State School Officers 2010. All rights reserved.

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 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for learning about 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. 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. 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 + 1), and (x - 1)(x3 + x 2 + 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. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.


5

© National Governors Association Center for Best Practicesand Council of Chief State School Officers 2010. All rights reserved.

McDonald’s Claim

A recent Wikipedia article reports that 8% of all Americans eat at McDonald’s every day. Current data indicates approximately 310 million Americans and 12,800 McDonald’s restaurants in the United States.

Do you believe the Wikipedia report to be true? Create a mathematical argument to justify your position.


REPRODUCIBLE

7Modeling: Having Kittens S-1. © 2012 MARS, Shell Center, University of Nottingham.

Available for download at http://map.mathshell.org. All other rights are reserved.Please send any inquiries about derived works to [email protected].

REPRODUCIBLE

8Buttons Test 5: Form A. © CTB/McGraw Hill 2003.

Available for download at www.insidemathematics.org. All other rights are reserved.

REPRODUCIBLE



REPRODUCIBLE



REPRODUCIBLE



Levels of Demand

(Source: This page provides a summary of M. K. Stein & M. S. Smith’s 1998 article, “Mathematical Tasks as a Framework for Reflection: From Research to Practice” in Mathematics Teaching in the Middle School, 3, 268–275.)

Lower-Level Demands Memorization

a. Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts, rules, formulas or definitions to memory.

b. Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure.

c. Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to be reproduced is clearly and directly stated.

d. Have no connection to the concepts or meaning that underlies the facts, rules, formulas, or definitions being learned or reproduced.

Procedures Without Connections

a. Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction, experience, or placement of the task.

b. Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it.

c. Have no connection to the concepts or meaning that underlies the procedures being used. d. Are focused on producing correct answers versus developing mathematical understanding. e. Require no explanations or explanations that focus solely on describing the procedure that was used.

Higher-Level Demands Procedures With connections

a. Focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas.

b. Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

c. Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols and problem situations. Making connections among multiple representations helps develop meaning.

d. Requires some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.

Doing Mathematics a. Require complex and nonalgorithmic thinking—a predictable, well-rehearsed approach or pathway is not

explicitly suggested by the task, task instructions, or a worked-out example. b. Require students to explore and understand the nature of mathematical concepts, processes, or

relationships. c. Demand self-monitoring or self-regulation of one’s own cognitive processes. d. Require students to access relevant knowledge and experiences and make appropriate use of them in

working through the task. e. Require students to analyze the task and actively examine task constraints that may limit possible solution

strategies and solutions. f. Require considerable cognitive effort and may involve some level of anxiety for the student because of the

unpredictable nature of the solution process required.

12© National Council of Teachers of Mathematics 1998. Used with permission.

Do not duplicate.

Examples of Higher and Lower Cognitive Demand Tasks Lower-Level Demands Higher-Level Demands

Memorization

What is the rule for multiplying fractions?

Expected student response:

You multiply the numerator times the numerator and the denominator times the denominator.

or

You multiply the two top numbers and then the two bottom numbers.

Procedures with Connections

Find 16

of 12

. Use pattern blocks. Draw your answer

and explain your solution.


Procedures without Connections

Multiply:

2 343

5 76 8

4 359

x

x

x


2 3 2 3 643 3 4 12

5 7 5 7 356 8 6 8 48

4 3 4 3 125 459 9 5

xxx

xxx

xxx

Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? Expected student response:

The formula for area is l x w. 15 x 10 = 150. She will need 150 square feet of carpet.

First you take half of the whole, which would be one hexagon. Then you take one-sixth of that half. So I divided the hexagon into six pieces, which would be six triangles. I only needed one-sixth, so that would be one triangle. Then I needed to figure out what part of the two hexagons one triangle was, and it was 1 out of 12. So 1/6 of 1/2 is 1/12.

Doing Mathematics

Create a real-world situation for the following

problem; 2 343

x .

Solve the problem you have created without using the rule, and explain your solution.

One possible student response:

For lunch Mom gave me three-fourths of a pizza that we ordered. I could only finish two-thirds of what she gave me. How much of the whole pizza did I eat?

I drew a rectangle to show the whole pizza. Then I cut it into fourths and shaded three of them to show the part Mom gave me. Since I only ate two-thirds of what she gave me, that would be only two of the shaded sections.

Mom gave me the part I shaded.

This is what I ate for lunch. So 2/3 of ¾ is the same thing as half of the pizza

PIZZA

(Source: This page is adapted with permission from M. K. Stein & M. S. Smith’s 1998 article, “Mathematical Tasks as a Framework for Reflection: From Research to Practice” in Mathematics Teaching in the Middle School, 3, 268–275.)

13© National Council of Teachers of Mathematics 1998. Used with permission.

Do not duplicate.

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CCSS Resources

Mathematics Common Core Coalition—MC3 (nctm.org/standards/mathcommoncore/) Site contains or links to CCSS information and resources from the eight coalition organizations: Association of Mathematics Teacher Educators (AMTE), Association of State Supervisors of Mathematics (ASSM) the National Council of Supervisors of Mathematics (NCSM) and the National Council of Teachers of Mathematics (NCTM), the Council of Chief State School Officers (CCSSO), the National Governors Association (NGA), and the two CCSS assessment consortia, Partnership for Assessment of Readiness for College and Careers (PARCC) and the Smarter Balanced Assessment Consortium (SBAC), as well as to other resources such as some of the websites listed below. The Hunt Institute Mathematics Videos (youtube.com/user/TheHuntInstitute#p/u/14/BNP5MdDDFPY) This site features short video segments by the standards authors about different aspects of the standards.

Illustrative Mathematics Project (illustrativemathematics.org) Site provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating “the range and types of mathematical work that students will experience in a faithful implementation” of CCSS-M. Inside Mathematics (insidemathematics.org) Supported by the Noyce Foundation, this site provides resources for classroom teachers, school-based administrators and district mathematics leaders developed by the Silicon Valley Mathematics Initiative. The Mathematics Assessment Project (MAP) (map.mathshell.org.uk/materials/) Developed by the Shell Center/MARS, University of Nottingham and University of California, Berkeley through funding from the Gates Foundation, this site contains formative and summative assessment tasks and lessons for Grades 7–12 specifically designed to support CCSS-M implementation.

National Council of Supervisors of Mathematics (mathedleadership.org) Site contains resources for mathematics education leaders, including latest news and project reports about CCSS-M, such as:

• Improving Student Achievement in Mathematics position papers that summarize key research results and describe specific actions leaders can take to put these results into practice.

• CCSS Curriculum Materials Analysis Tools and Professional Development Materials, a set of tools to assist K–12 textbook selection committees, school administrators, and teachers in analyzing and selecting curriculum materials that support faithful implementation of the CCSS-M.

• Illustrating the Standards for Mathematical Practice, a set of ready-to-use professional development modules designed to help teachers understand the CCSS Standards for Mathematical Practice and implement them in their classrooms.

• Archived NCSM webinars addressing CCSS-M implementation. • Links to other websites with resources for mathematics education leaders

National Council of Teachers of Mathematics lessons (illuminations.nctm.org/) Illuminations provides standard-based resources to improve the teaching and learning of mathematics for all students. These materials illuminate the vision for school mathematics set forth in NCTM’s standards


documents, i.e., Principles and Standards for School Mathematics, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, and Focus in High School Mathematics: Reasoning and Sense Making. Tools for the Common Core State Standards (commoncoretools.me) This is Bill McCallum’s blog on tools that support the CCSS-M, as well as other CCSS-M news, including the latest information about Illustrative Mathematics and links to the most recent Standards Progression documents, which are narrative documents describing the progression of topics within a CCSS-M domain across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. Progressions documents are available directly at: ime.math.arizona.edu/progressions

Partnership for Assessment of Readiness for College and Careers (PARCC) (parcconline.org) Smarter Balanced Assessment Consortium Resources and Frameworks (k12.wa.us/SMARTER/default.aspx) Solution Tree (solution-tree.com; go.solution-tree.com/leadership for reproducible pages and other resources) Site includes reproducible resources to accompany Solution Tree’s publications, along with other resources that support principals’ leadership work.

References and Resources Briars, D. J., Asturias, H., Foster, D., Gale, M. A.

(Forthcoming: 2012, Oct.). Common core mathematics in a PLC: Grades 6–8. Bloomington, IN: Solution Tree Press.

Council of Chief State School Officers. (2010). Common core state standards. Washington, DC: Author.

Ferrini-Mundy, J., Graham, K., Johnson, L., & Mills, G., (1998). Making change in mathematics education: Learning from the field. Reston, VA: National Council of Teachers of Mathematics.

Kanold, T. D., & Larson, M. R. (2012). Common core mathematics in a PLC (K–12): School administrators and leaders. Bloomington, IN: Solution Tree Press.

Kanold, T. D, Briars, D. J., & Fennel, F. (2012). What principals need to know about teaching and learning mathematics. Bloomington, IN: Solution Tree Press.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Larson, M.R., Fennell, F, Adams, T. L., Dixon, J. K., Kobett, B. M., & Wray, J. A. (2012). Common core mathematics in a PLC: Grades K–2. Bloomington, IN: Solution Tree Press.

Larson, M.R., Fennell, F, Adams, T. L., Dixon, J. K., Kobett, B. M., & Wray, J. A. (2012). Common core mathematics in a PLC: Grades 3–5. Bloomington, IN: Solution Tree Press.

Morris, A. K., & Hiebert, J. (2011, January/February). Creating shared instructional products: An alternative approach to improving teaching. Educational Researcher, 40(1), 5–14.

Waters, T., Marzano, R. J., & McNulty, B. A. (2003). Balanced leadership: What 30 years of research tells us about the effect of leadership on student achievement. Aurora, CO: Mid-continent Research for Education and Learning.


Documents

Transition to CCSS Mathematics in a PLC: The Impact on ...soltreemrls3.s3-website-us-west-2.amazonaws.com/.../pdf/...stlouis.p… · Transition to CCSS Mathematics in a PLC The Impact