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The Standards for Mathematical Practice
2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards The 2010 Alabama Course of Study: Mathematics contains two types of standards – practice standards and content standards. The portion of the presentation will focus on the 8 Standards for Mathematical Practice which are common to every grade level. (CLICK) The Standards for Mathematical Practice
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.” (CCSS, 2010) Read this quote. (CLICK)
National Council of Teachers of Mathematics 5 PROCESS Standards Problem Solving Reasoning and Proof Communication Connections Representations Underlying frameworks for the mathematical practice standards are found in documents from two national educational organizations, the National Council of Teachers of Mathematics and the National Research Council. The National Council of Teachers of Mathematics (NCTM) has long held the belief that learning is a process that envelopes the students’ thinking processes through: (CLICK) Problem solving – building new mathematical knowledge through problem solving, solving problems, applying and adapting appropriate strategies to solve problems (CLICK) Reasoning and proof – making and investigating mathematical conjectures, developing mathematical arguments and proofs, selecting and using various types of reasoning (CLICK) Communication – organizing mathematical thinking through communication, communicating coherently to peers, teachers, and others, using the language of mathematics precisely (CLICK) Connections – recognizing and using connections among mathematical ideas, applying mathematics in contexts outside of mathematics (CLICK) Representations – creating and using representations to organize, record, and communicate mathematical ideas, modeling mathematical phenomena, selecting, applying and translating among mathematical representations to solve problems (CLICK) NCTM (2000M). Principles and Standards for School Mathematics. Reston, VA: Author.
National Research Council Strands of Mathematical Proficiency Conceptual Understanding Procedural Fluency Strategic Competence Adaptive Reasoning Productive Disposition The National Research Council’s report, Adding It Up: Helping Children Learn Mathematics, is a summary of research on mathematics learning from prekindergarten through grade 8. The committee chose the term mathematical proficiency to capture what it means for anyone to learn mathematics successfully. Mathematical proficiency is broken down into five components, or strands: (CLICK) Conceptual understanding – comprehending mathematical concepts (CLICK) Procedural fluency – skill in carrying out procedures accurately and efficiently (CLICK) Strategic competence – ability to formulate, represent and solve mathematical problems (CLICK) Adaptive reasoning – capacity for logical thought, reflection, explanation and justification (CLICK) Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile. “The most important observation the committee makes and stresses is that the five strands are interwoven and interdependent in the development of proficiency in mathematics.” The writers of the Common Core State Standards took all these process standards and developed the 8 Standards for Mathematical Practice. (CLICK) NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.
The Standards for Mathematical Practice
Mathematically proficient students: Standard 1: Make sense of problems and persevere in solving them. Standard 2: Reason abstractly and quantitatively. Standard 3: Construct viable arguments and critique the reasoning of others. Standard 4: Model with mathematics. Standard 5: Use appropriate tools strategically. Standard 6: Attend to precision. Standard 7: Look for and make use of structure. Standard 8: Look for and express regularity in repeated reasoning. Notice that each practice standard begins with the words “Mathematically proficient students:” (CLICK X 8) to bring each standard onto screen. Read each standard. Keep in mind, these standards for mathematical practice are behaviors we want to develop in our students. (CLICK)
Questions to consider…
1. What student behaviors are included in this standard? 2. What type of activities could help my students develop this behavior? Adapted from Kathy Berry, Monroe County ISD, Michigan
Questions to consider…
What does this standard look like in the classroom? What will students need in order to do this? What will teachers need in order to do this? Adapted from Kathy Berry, Monroe County ISD, Michigan Our first activity today is a group activity to get you into the thinking mode about the standards. Some of you might know what these standards are and for others, this may be the first time you have seen them. You will be divided into 8 different groups and each group will be given a practice standard to discuss. Each group will discuss these questions for their standard. You will have approximately five minutes. There aren’t any right or wrong answers at this point. When you consider the standards, you should reflect on these questions. (CLICK) #1 What does this standard look like? (CLICK) #2 What will students need in order to do this? (CLICK) #3 What will teachers need in order to do this? (Presenter distributes handouts: one standard per group) (CLICK)
Standard 1: Make sense of problems and persevere in solving them
Standard 1: Make sense of problems and persevere in solving them. What do mathematically proficient students do? Analyze givens, constraints, relationships Make conjectures Plan solution pathways Make meaning of the solution Monitor and evaluate their progress Change course if necessary Ask themselves if what they are doing makes sense Here’s what Standard 1 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) The Mathematical Practice Standards are standards that you can already implement this fall. Now let’s look at the other 7 practice standards. (ADDITIONAL TRAINER NOTES ON STANDARD 1. THESE ARE TO BE USED AS A TRAINER RESOURCE.) Mathematically proficient students: Explain to themselves the meaning of a problem. Look for entry points to its solution. Analyze the givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Monitor and evaluate their progress and change course if necessary. Check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” Understand the approaches of others to solving complex problems and identify correspondences between different approaches. (CLICK)
Standard 2: Reason abstractly and quantitatively
Standard 2: Reason abstractly and quantitatively. What do mathematically proficient students do? Make sense of quantities and relationships Able to decontextualize Abstract a given situation Represent it symbolically Manipulate the representing symbols Able to contextualize Pause during manipulation process Probe the referents for symbols involved Here’s what Standard 2 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 2. THESE ARE TO BE USED AS A TRAINER RESOURCE.) Mathematically proficient students: Make sense of quantities and their relationships in problem situations. Bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize – 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 Knowing and flexibly using different properties of operations and objects. (CLICK)
Standard 3: Construct viable arguments and critique the reasoning of others. What do mathematically proficient students do? Construct arguments Analyze situations Justify conclusions Communicate conclusions Reason inductively Distinguish correct logic from flawed logic Listen to/Read/Respond to other’s arguments and ask useful questions to clarify/improve arguments Here’s what Standard 3 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 3. THESE ARE TO BE USED AS A TRAINER RESOURCE.) Mathematically proficient students: Understand and use stated assumptions, definitions, and previously established results in constructing arguments Make conjectures and build a logical progression of statements to explore the truth of their conjectures Analyze situations by breaking them into cases, and can recognize and use counterexamples Justify their conclusions, communicate them to others, and respond to the arguments of others Reason inductively about data, making plausible arguments that take into account the context from which the data arose. 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 there 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. (CLICK)
Standard 4: Model with mathematics
Standard 4: Model with mathematics. What do mathematically proficient students do? Apply mathematics to solve problems from everyday life situations Apply what they know Simplify a complicated situation Identify important quantities Map math relationships using tools Analyze mathematical relationships to draw conclusions Reflect on improving the model Here’s what Standard 4 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 4. THESE ARE TO BE USED AS A TRAINER RESOURCE.) Mathematically proficient students: 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 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. Apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. 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. 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. (CLICK)
Standard 5: Use appropriate tools strategically
Standard 5: Use appropriate tools strategically. What do mathematically proficient students do? Consider and use available tools Make sound decisions about when different tools might be helpful Identify relevant external mathematical resources Use technological tools to explore and deepen conceptual understandings Here’s what Standard 5 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 5. THESE ARE TO BE USED AS A TRAINER RESOURCE.) 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. 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. 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. Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. Use technological tools to explore and deepen their understanding of concepts. (CLICK)
Standard 6: Attend to precision
Standard 6: Attend to precision. What do mathematically proficient students do? Communicate precisely to others Use clear definitions in discussions State meaning of symbols consistently and appropriately Specify units of measurements Calculate accurately & efficiently Here’s what Standard 6 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 6. THESE ARE TO BE USED AS A TRAINER RESOURCE.) Mathematically proficient students: Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning. State the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently Express numerical answers with a degree of precision appropriate for the 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. (CLICK)
Standard 7: Look for and make use of structure
Standard 7: Look for and make use of structure. What do mathematically proficient students do? Discern patterns and structures Use strategies to solve problems Step back for an overview and can shift perspective Here’s what Standard 7 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 7. THESE ARE TO BE USED AS A TRAINER RESOURCE.) 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 X 3, in preparation for learning about the distributive property. In the expression X^2 + 9X +14, older students can see the 14 as 2 X 7 and the 9 as Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. 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. (CLICK)
Standard 8: Look for and express regularity in repeated reasoning
Standard 8: Look for and express regularity in repeated reasoning. What do mathematically proficient students do? Notice if calculations are repeated Look for general methods and shortcuts Maintain oversight of the processes Attend to details Continually evaluates the reasonableness of their results Here’s what Standard 8 looks like. Do you agree with this list of student behaviors? What will students need in order to do this? What will teachers need to do this? (Allow time for discussion) (ADDITIONAL TRAINER NOTES ON STANDARD 8. THESE ARE TO BE USED AS A TRAINER RESOURCE.) 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, when attending to the details. They continually evaluate the reasonableness of their intermediate results. (CLICK)
The Standards for [Student] Mathematical Practice
SMP1: Explain and make conjectures… SMP2: Make sense of… SMP3: Understand and use… SMP4: Apply and interpret… SMP5: Consider and detect… SMP6: Communicate precisely to others… SMP7: Discern and recognize… SMP8: Note and pay attention to… Remember, the Mathematical Practice Standards describe behaviors we want to develop in our students. It is up to the teacher to provide tasks that give students an opportunity to develop and display these behaviors. This slide provides a quick overview of the behaviors we want to see in our students. Now, that we’ve had a chance to discuss the mathematical practice standards, what do they remind you of? Aren’t the Practice Standards just Best Practices that we’ve all been using for years? It’s just good teaching. These practice standards are not just ‘something else’ to be taught. They are Best Practices that we pass on to our students through meaningful mathematical tasks. (CLICK)
CONNECTION and BALANCE
Mathematical Practice Mathematical Content Our ultimate goal is to make sure the students make a connection between the Standards for Mathematical Practice and the Standards for Mathematical Content. Let’s look at a Task to see how the Standards can be implemented. (CLICK)
Algebra Task 3 Sorting Functions
This task gives students the chance to: Find relationships between graphs, equations, tables, and rules. Explain reasoning for answers. We will view an Algebra task from the website insidemathematics.org about functions. (Participants will work through this task in a group of 3-4 people.) As you work through this task, think about the mathematical practice standards that your students might use. The information provided in the following slides is for professional development only. (CLICK)
Algebra Task 3 Sorting Functions
The task in this assignment is to match each graph on the next page with an equation, a table, and a rule. You will use this page to record your group’s findings. Notice that an explanation is required in addition to writing answers in the table provided. (CLICK)
Algebra Task 3 Sorting Functions
Use this form to answer the questions found on the previous page. (Groups will work for five minutes.) (After 5 minutes have passed, ask the small groups to share out with the entire group. In the discussion of the task, you might ask the following questions: How could you differentiate this task for students who are working at a lower level in your classroom? How could this task be enhanced for students who have a good grasp of the concept? Students will probably ask you if they can use their graphing calculator or the Internet to find the answers. What would be your response?) (CLICK)
Algebra Task 3 Sorting Functions
The Student Task is described just as in the handout. Notice how the Core Ideas 1 and 3 bring into focus the objectives you are trying to accomplish. (CLICK) Algebra – 2008 Copyright © 2008 by Noyce Foundation. All rights reserved. The information provided in the following slides is for professional development only.
The mathematics of this task:
Making connections between different algebraic representations: graphs, equations, verbal rules, and tables Understanding how the equation determines the shape of the graph Developing a convincing argument using a variety of algebraic concepts Being able to move from specific solutions to thinking about generalizations Would you agree that these statements adequately describe the mathematics of this task? (Allow time for participants to read these statements and then ask for comments.) Which of the Standards for Mathematical Practice will students demonstrate in this task? (The next slide has the 8 Mathematical Practice Standards listed on it. It might be easier for participants to determine which practice standards are used in this task while looking at the list) (CLICK) Algebra – 2008 Copyright © 2008 by Noyce Foundation. All rights reserved.
The Standards for Mathematical Practice
Mathematically proficient students: Standard 1: Make sense of problems and persevere in solving them. Standard 2: Reason abstractly and quantitatively. Standard 3: Construct viable arguments and critique the reasoning of others. Standard 4: Model with mathematics. Standard 5: Use appropriate tools strategically. Standard 6: Attend to precision. Standard 7: Look for and make use of structure. Standard 8: Look for and express regularity in repeated reasoning. (Some possible answers that participants might share: Standard 2: Reason abstractly and quantitatively. Standard 3: Construct viable arguments and critique the reasoning of others. (Students are required to explain how they matched the graphs with the equations, tables, and rules) Standard 5: Use appropriate tools strategically. ) Have you implemented these standards in the past without knowing you were doing so? Do you reflect on your teaching practice regularly? (CLICK)
Student A Look at these student work samples. Student A uses some interesting language to describe differences between Graphs A and B, giving more details about parabolas. For Graph D, the student makes an argument about symmetry. Notice how asking students to explain their reasoning requires students to think about their answers and the reasons for them. For the teacher, it provides insight into the level of understanding that the student possesses. Student B has a unique explanation for matching the equation to Graph D. (Read his explanation aloud.) Once again, asking students to explain their reasoning adds additional rigor to the task. This task was taken from the Website insidemathematics.org. This is a good resource for additional practice standards exercises. (CLICK) Student B
The Standards for [Student] Mathematical Practice
“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” Stein, Smith, Henningsen, & Silver, 2000 “The level and kind of thinking in which students engage determines what they will learn.” As you can read on the slide, all tasks are NOT created equal. Engaging the students in higher order thinking determines what they will learn and will inspire them to think “outside the box”. And, it’s not just about having the student ‘do’ a task, it’s the nature of the task and the relationship between the task and the opportunities the tasks provides for the student to learn. (CLICK) Herbert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
But, WHAT TEACHERS DO with the tasks matters too!
The Mathematical Tasks Framework Tasks are enacted by teachers and students Tasks as they appear in curricular materials Tasks are set up by teachers It is important to recognize that what teachers do with the tasks matters. The tasks that appear in curricular materials, along with tasks that are set up and enacted by teachers, all impact student learning. Do you delve into higher order thinking skills? Do you allow open ended questioning? Do you encourage discovery? (CLICK) Student Learning Stein, Grover, & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen, & Silver (2000)
Standards for [Student] Mathematical Practice
The Standards for Mathematical Practice place an emphasis on student demonstrations of learning… Equity begins with an understanding of how the selection of tasks, the assessment of tasks, and the student learning environment create inequity in our schools… Equity in our classrooms is essential and we must provide opportunities for all students to demonstrate these mathematical practices. To ensure equity, select tasks that can be differentiated. Some will need to be enhanced to meet the needs of students with a greater understanding of the content. While other tasks will have to be broken down into smaller parts for other students to work. Help students reason strategically. Encourage them to ‘think outside of the box’. Choose the best assessments that provide an accurate picture of the student’s understanding and development. (CLICK)
Mathematical Practice Standards
Leading with the Mathematical Practice Standards You can begin by implementing the 8 Standards for Mathematical Practice now Think about the relationships among the practices and how you can move forward to implement BEST PRACTICES Analyze instructional tasks so students engage in these practices repeatedly You can begin the process of implementing the Standards for Mathematical Practice now. Become familiar with the practice standards and the behaviors that you want to develop in your students. Select tasks that provide opportunities for your students to demonstrate desired behaviors. Where can you find mathematical tasks? Some are embedded in your instructional materials (textbooks, additional resources)..As you begin the process of selecting new curricular materials, be aware of the tasks that are provided in the resources. (CLICK)
?? Questions ?? Are there any questions concerning the Standards for Mathematical Practice? If not, we will continue with the Literacy Standards in the 2010 Alabama Mathematics Course of Study. (Collect Practice Standards Cards) (CLICK)
Contact Information ALSDE Office of Student Learning Curriculum and Instruction Cindy Freeman, Mathematics Specialist Phone:
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