Collaborate, Communicate, Connect: High-Leverage Practices to Turn Standards into Learning Please take a handout as you enter. Diane J. Briars Past President National Council of Teachers of Mathematics [email protected] Alabama Council of Teachers of Mathematics November 3, 2016
FYI Electronic copies of slides will be posted at nctm.org/briars or are available by request [email protected] National Council of Teachers of Mathematics www.nctm.org National Council
Mathematics National Councilof of Teachers Teachers of of Mathematics www.nctm.org www.nctm.org For $144 per year, your school will get a FREE print-only subscription to one of the following award-winning journals:
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National Councilof of Teachers Teachers of of Mathematics www.nctm.org www.nctm.org New Member Discount $20 off for full membership $10 off e-membership
$5 off student membership Use Code: BDB0616 Community. Collaboration. Solutions. Bring your team and engage in a hands-on, interactive, and new learning experience for mathematics education. With a focus on Engaging the Struggling Learner, become part of a team environment and navigate your experience through three different pathways:
Response to Intervention (RtI) Supporting productive struggle Motivating the struggling learner NCTM Interactive Institute www.nctm.org Grades PK-5, 6-8, High School, and School Leaders February 34, 2017 San Diego
2017 NCTM Annual Meeting and Exposition www.nctm.org April 58, 2017 San Antonio Agenda Identify effective teaching practices that promote students proficiency in rigorous mathematics standards,
e.g., CCSS-M/AL COS Read and analyze a short case of a teacher (Mr. Donnelly) who is attempting to support his students learning Discuss selected effective teaching practices and relate them to the case Identify high-leverage actionsthose that will produce the greatest impact for your effortin enacting the effective teaching practices in your classroom, school, and district. High Quality Standards Are
Necessary, but Insufficient, for Effective Teaching and Learning Principles to Actions: Ensuring Mathematical Success for All Describes the supportive conditions, structures, and policies required to give all students the power of mathematics Focuses on teaching and learning Emphasizes engaging students
in mathematical thinking Describes how to ensure that mathematics achievement is maximized for every student Is not specific to any standards; its universal Key Features of CCSS-M Focus: Focus strongly where the standards focus. Coherence: Think across grades, and
link to major topics Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application Standards for Mathematical Practice Key Features of CCSS-M Focus: Focus strongly where the standards focus. Coherence: Think across grades, and link to major topics
Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application Standards for Mathematical Practice Curriculum Standards, Not Assessment Standards Understand and apply properties of operations and the relationship between addition and subtraction. (1.OA) 3. Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is
also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 4. Understand subtraction as an unknown-addend problem. For example, subtract 10 8 by finding the number that makes 10 when added to 8. Curriculum Standards, Not Assessment Standards Define, evaluate, and compare functions. (8.F)
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Why Focus on Understanding?? Understanding facilitates initial learning and retention. Understanding supports appropriate application and
transfer. Key Features of CCSS-M Focus: Focus strongly where the standards focus. Coherence: Think across grades, and link to major topics Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application Standards for Mathematical Practice
Key Features of CCSS-M Focus: Focus strongly where the standards focus. Coherence: Think across grades, and link to major topics Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application Standards for Mathematical Practice
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated
reasoning. Phil Daro, 2010 Phil Daro, 2010 Other Butterflies? FOIL Cross multiplication Fraction division:
Keep-change-flip, KFC, Yours is not to reason why, just invert and multiply Integer subtraction: Keep-change-change (a - b = a + - b) Long Division: Dad, Mother, Sister, Brother, Rover Does McDonalds Sell Cheese Burgers? Key words Diane J. Briars, October, 2016
Key Instructional Shift From emphasis on: How to get answers To emphasis on: Understanding mathematics Implementing CCSS-M Requires Instructional practices that promote students development of conceptual
understanding and proficiency in the Standards for Mathematical Practice. Guiding Principles for School Mathematics 1. Teaching and Learning Effective teaching is the non-negotiable core that ensures that all students learn mathematics at high levels.
We Must Focus on Instruction Student learning of mathematics depends fundamentally on what happens inside the classroom as teachers and learners interact over the curriculum. (Ball & Forzani, 2011, p. 17) Teaching has 6 to 10 times as much impact on achievement as all other factors combined ... Just three years of effective teaching accounts on average for an
improvement of 35 to 50 percentile points. Guiding Principles for School Mathematics 1. Teaching and Learning 2. Access and Equity 3. Curriculum 4. Tools and Technology 5. Assessment 6. Professionalism
Essential Elements of Effective Math Programs Candy Jar Problem A candy jar contains 5 Jolly Ranchers (squares) and 13 Jawbreakers (circles). Suppose you had a new candy jar with the same ratio of Jolly Ranchers to Jawbreakers, but it
contained 100 Jolly Ranchers. How many Jawbreakers would you have? Explain how you know. Please work this problem as if you were a seventh grader. When done, share your work with a neighbor. Discuss: What mathematics learning could this task support? Candy Jar Task Common Core State Standards 7.RP: 2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.
5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. The Case of Mr. Donnelly Read the Case of Mr. Donnelly and study the strategies used by his students. Make note of what Mr. Donnelly did before or during instruction to support his students
learning and understanding of proportional relationships. Talk with a shoulder-partner about the actions and interactions that you identified as supporting student learning. Principle on Teaching and Learning An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their
ability to make sense of mathematical ideas and reason mathematically. Effective Mathematics Teaching Practices 1. 2. 3. 4. 5. 6.
7. 8. Establish mathematics goals to focus learning. Implement tasks that promote reasoning and problem solving. Use and connect mathematical representations. Facilitate meaningful mathematical discourse. Pose purposeful questions. Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics. Elicit and use evidence of student thinking. Effective Mathematics Teaching Practices 1. 2. 3. 4. 5.
6. 7. 8. Establish mathematics goals to focus learning. Implement tasks that promote reasoning and problem solving. Use and connect mathematical representations. Facilitate meaningful mathematical discourse. Pose purposeful questions. Build procedural fluency from conceptual
understanding. Support productive struggle in learning mathematics. Elicit and use evidence of student thinking. Establish Mathematics Goals To Focus Learning Learning Goals should: Clearly state what it is students are to learn and understand about mathematics as the result of instruction;
Be situated within learning progressions; and Frame the decisions that teachers make during a lesson. Formulating clear, explicit learning goals sets the stage for everything else. (Hiebert, Morris, Berk, & Janssen, 2007, p.57) Candy Jar Task Common Core State Standards 7.RP: 2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Goals to Focus Learning Mr. Donnellys goal for students learning: Students will recognize that quantities that are in a proportional (multiplicative) relationship grow at a constant rate and that there are three key strategies that could be used to solve problems of this type scaling
up, scale factor, and unit rate. How does his goal align with this next teaching practice? Implement Tasks that Promote Reasoning and Problem Solving Mathematical tasks should: Provide opportunities for students to engage in exploration or encourage students to use procedures in ways that are connected to concepts and
understanding; Build on students current understanding; and Have multiple entry points. Why Tasks Matter Tasks form the basis for students opportunities to learn what mathematics is and how one does it; Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information;
The level and kind of thinking required by mathematical instructional tasks influences what students learn; and Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students opportunities to learn mathematics. Why Tasks Matter Tasks form the basis for students opportunities to learn what mathematics is and how one does it;
Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information; The level and kind of thinking required by mathematical instructional tasks influences what students learn; and Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students opportunities to learn mathematics.
Comparing Tasks Finding the Missing Value Find the value of the unknown in each of the proportions shown below. How is Mr. Donnellys task (Candy Jar) similar to or different from the Missing Value
problem? Which one is more likely to promote problem solving? Core Instructional Issue Do all students have the opportunity to engage in mathematical tasks that promote students attainment of the mathematical practices on a regular basis?
Facilitate Meaningful Mathematical Discourse Mathematical Discourse should: Build on and honor students thinking; Provide students with the opportunity to share ideas, clarify understandings, and develop convincing arguments; and Advance the mathematical learning of the whole class.
Facilitate Meaningful Mathematical Discourse Discussions that focus on cognitively challenging mathematical tasks, namely those that promote thinking, reasoning, and problem solving, are a primary mechanism for promoting conceptual understanding of mathematics (Hatano & Inagaki, 1991; Michaels, OConnor, & Resnick, 2008). Smith, Hughes, Engle & Stein, 2009, p. 549
Meaningful Discourse What did Mr. Donnelly do (before or during the discussion) that may have positioned him to engage his students in a productive discussion? Five Practices for Orchestrating Productive Mathematics Discussions
Anticipating likely student responses Monitoring students actual responses Selecting particular students to present their work during the whole class discussion Sequencing the students presentations Connecting different students strategies and ideas in a way that helps students understand the mathematics or science in the lesson. Smith & Stein, 2011; How did Mr. Donnelly use the 5 practices to
structure the summary discussion? Planning with the Student in Mind Anticipate solutions, thoughts, and responses that students might develop as they struggle with the problem/task. Generate questions that could be asked to promote student thinking during the lesson, and consider the kinds of guidance that could be given to students who showed one or another types of misconception in their thinking
Determine how to end the lesson so as to advance students understanding Stigler & Hiebert, 1997 Planning with the Student in Mind Strategy/ Response Unit Rate: Picture Unit Rate: Table Scale Factor: Scaling Up: Table
Scaling Up: Picture Additive Questions Students/ Group Order Elicit and Use Evidence
of Student Thinking Evidence should: Provide a window into students thinking; Help the teacher determine the extent to which students are reaching the math learning goals; and Be used to make instructional decisions during the lesson and to prepare for subsequent lessons.
Harold Asturias, 1996 Formative assessment is an essentially interactive process, in which the teacher can find out whether what has been taught has been learned, and if not, to do something about it. Day-today formative assessment is one of the most powerful ways of improving learning in the mathematics classroom. Wiliam, 2007, pp. 1054; 1091
Effective Mathematics Teaching Practices 1. 2. 3. 4. 5. 6. 7. 8.
Establish mathematics goals to focus learning. Implement tasks that promote reasoning and problem solving. Use and connect mathematical representations. Facilitate meaningful mathematical discourse. Pose purposeful questions. Build procedural fluency from conceptual understanding. Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking. Your Feelings Looking Ahead? Principles to Actions: Ensuring Mathematical Success for All 1. Teaching and Learning 2. Access and Equity 3. Curriculum 4. Tools and Technology
5. Assessment 6. Professionalism Essential Elements of Effective Math Programs Guiding Principles for School Mathematics
Professionalism In an excellent mathematics program, educators hold themselves and their colleagues accountable for the mathematical success of every student and for their personal and collective professional growth toward effective teaching and learning of mathematics. Guiding Principles for School Mathematics
Professionalism In an excellent mathematics program, educators hold themselves and their colleagues accountable for the mathematical success of every student and for their personal and collective professional growth toward effective teaching and learning of mathematics. Professionalism Obstacle In too many schools, professional isolation
severely undermines attempts to significantly increase professional collaboration some teachers actually embrace the norms of isolation and autonomy. A danger in isolation is that it can lead to teachers developing inconsistencies in their practice that in turn can create inequities in student learning. Principles to Actions, p. 100 Incremental Change The social organization for
improvement is a profession learning community organized around a specific instructional system. A. S. Bryk (2009) The unit of change is the teacher team. Collaborative Team Work An examination and prioritization of the mathematics content and mathematics practices students are to learn. The development and use of common assessments to
determine if students have learned the agreed-on content and related mathematical practices. The use of data to drive continuous reflection and instructional decisions. The setting of both long-term and short-term instructional goals. Development of action plans to implement when students demonstrate they have or have not attained the standards. Discussion, selection, and implementation of common research-informed instructional strategies and plans.
Principles to Actions, pp. 103-104 Collaborative Team Work An examination and prioritization of the mathematics content and mathematics practices students are to learn. The development and use of common assessments to determine if students have learned the agreed-on content and related mathematical practices. The use of data to drive continuous reflection and instructional decisions. The setting of both long-term and short-term instructional
goals. Development of action plans to implement when students demonstrate they have or have not attained the standards. Discussion, selection, and implementation of common research-informed instructional strategies and plans. Principles to Actions, pp. 103-104 Effective Mathematics Teaching Practices 1.
2. 3. 4. 5. 6. 7. 8. Establish mathematics goals to focus learning. Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations. Facilitate meaningful mathematical discourse. Pose purposeful questions. Build procedural fluency from conceptual understanding. Support productive struggle in learning mathematics. Elicit and use evidence of student thinking. Planning with the Student in Mind Anticipate solutions, thoughts, and responses that
students might develop as they struggle with the problem/task. Generate questions that could be asked to promote student thinking during the lesson, and consider the kinds of guidance that could be given to students who showed one or another types of misconception in their thinking Determine how to end the lesson so as to advance students understanding Stigler & Hiebert, 1997
Planning the Candy Jar Lesson Collaborative Team Work An examination and prioritization of the mathematics content and mathematics practices students are to learn. The development and use of common assessments to determine if students have learned the agreed-on content and related mathematical practices. The use of data to drive continuous reflection and instructional decisions. The setting of both long-term and short-term instructional
goals. Development of action plans to implement when students demonstrate they have or have not attained the standards. Discussion, selection, and implementation of common research-informed instructional strategies and plans. Principles to Actions, pp. 103-104 Collaborative Team Work An examination and prioritization of the mathematics content and mathematics practices students are to learn.
The development and use of common assessments to determine if students have learned the agreed-on content and related mathematical practices. The use of data to drive continuous reflection and instructional decisions. The setting of both long-term and short-term instructional goals. Development of action plans to implement when students demonstrate they have or have not attained the standards. Discussion, selection, and implementation of common research-informed instructional strategies and plans.
Principles to Actions, pp. 103-104 Guiding Principles for School Mathematics Assessment An excellent mathematics program ensures that assessment is an integral part of instruction, provides evidence of proficiency with important mathematics content and practices, includes a variety of strategies and
data sources, and informs feedback to students, instructional decisions and program improvement. Guiding Principles for School Mathematics Assessment An excellent mathematics program ensures that assessment is an integral part of instruction, provides evidence of proficiency with important mathematics content and
practices, includes a variety of strategies and data sources, and informs feedback to students, instructional decisions and program improvement. How Would You Assess This Standard? 6.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context
of solving real-world and mathematical problems. What assessment tasks would you use to assess students proficiency with this standard? How Would You Assess This Standard? 6.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context
of solving real-world and mathematical problems. Compute area of different figures? Explain the relationship between the areas of different figures? Find a missing side of a rectangle or base/height of a triangle, given the area and another side? How Would You Assess This Standard? 6.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing
into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. What applications? A rectangular carpet is 12 feet long and 9 feet wide. What is the area of the carpet in square feet? County Concerns 1. The Jackson County Executive Board is considering a proposal to conduct aerial spraying of insecticides to control the mosquito population. An agricultural
organization supports the plan because mosquitoes cause crop damage. An environmental group opposes the plan because of possible food contamination and other medical risks. Here are some facts about the case: A map of Jackson County is shown here. All county boundaries are on a S north south line or an eastwest line. The estimated annual cost of aerial spraying is $29 per acre. There are 640 acres in 1 square mile. Plan supporters cite a study stating that for every $1 spent on insecticides, farmers
would gain $4 through increased agricultural production. a. What is the area of Jackson County in square miles? In acres? b. What would be the annual cost to spray the whole county? c. According to plan supporters, how much money would the farmers gain from the spraying program? County Concerns 2. The sheriff of Adams County and the sheriff of Monroe County are having an argument. They each believe that their own county is larger than the other county. Who is right? Write an explanation that would settle the
argument. Tasks Clarify Expectations Range of content Depth of knowledge Type of reasoning and evidence of it
Types of applications Assessments that Support Valid Inferences Learning goal: Understanding the definition of a triangle. Performance task: Draw a triangle. Analyzing Assessment Tasks
To what extent does the task: Provide valid information about students knowledge? Provide information about students conceptual understanding? Provide information about students proficiency in mathematical processes: problem solving, reasoning and proof, communication, connections, and representation?
Understanding a Concept Explain it to someone else Represent it in multiple ways Apply it to solve simple and complex problems Reverse givens and unknowns Compare and contrast it to other concepts Use it as the foundation for learning other concepts
Common Assessment Planning Process Plan Develop Administer and Analyze Students Performance Critique Revise http://map.mathshell.org/
http:// www.insidemathematics.org/ performance-assessment-tasks Access and Equity Principle An excellent mathematics program requires that all students have access to a high-quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their
learning potential. 83 Access and Equity Principle An excellent mathematics program requires that all students have access to a high-quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their
learning potential. 84 Students Mathematics Identities Are how students see themselves and how they are seen by others, including teachers, parents, and peers, as doers of mathematics. Aguirre, Mayfield-Ingram & Martin, 2013
Mathematics Identity Mathematics identity includes: beliefs about ones self as a mathematics learner; ones perceptions of how others perceive him or her as a mathematics learner, beliefs about the nature of mathematics, engagement in mathematics, and perception of self as a potential participant in mathematics (Solomon, 2009).
Micromessages Small, subtle unconscious messages we send and receive when we interact with others. Negative micromessages cause people to feel devalued, slighted, discouraged or excluded. Positive micromessages cause people to feel valued, included, or encouraged. National Alliance for Partnerships in Equity, 2016 What Messages Are We Sending About
Mathematical Identity? What dont you understand? This is so simple. It is immediately obvious that . . . . Which students participate in class? How do they participate? What kind of questions are they asked? Wait time? Opportunities for
elaboration/explanations? Responses to students questions? To their answers? Feedback: Effort-based vs intelligence-based praise. Non-verbal communicationsmiles, nods, etc. Peer Observations to Uncover Micromessages and Unintentional Bias Observe the classroom experiences of males/females and/or individuals of certain races or ethnicities Number of interactions
Amount of wait/think time Nature of questionshigher vs lower level Nature of feedback Amount of eye contact Use of language Collaboration is the Key to Improving Mathematics Teaching and Learning High-Leverage Practices Teaching for understanding, instead of how to get answers.
Collaborative lesson planning to implement effective teaching practices Collaborative assessment planning Collaborative peer observations to uncover micromessages. Principles to Actions: Ensuring Mathematical Success for All 1. Teaching and Learning 2. Access and Equity
3. Curriculum 4. Tools and Technology 5. Assessment 6. Professionalism Essential Elements of Effective Math Programs
http://www.nctm.org/PtA/ Principles to Actions Resources Principles to Actions Executive Summary (in English and Spanish) Principles to Actions overview presentation Principles to Actions professional development guide (Reflection Guide) Mathematics Teaching Practices presentations Elementary case, multiplication (Mr. Harris) Middle school case, proportional reasoning (Mr.
Donnelly) (in English and Spanish) High school case, exponential functions (Ms. Culver) Principles to Actions Spanish translation http://www.nctm.org/PtAToolkit/ http://www.nctm.org/PtAToolkit/ Collaborative Team Tools Available at nctm.org
The Title Is Principles to Actions What actions will you take to collaborate, communicate and connect to improve mathematics teaching and learning? Thank You! Diane Briars
[email protected] nctm.org/briars