1. 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

2. 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.
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3. Underlying Frameworks
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
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NCTM (2000M). Principles and Standards for School Mathematics. Reston, VA: Author.

4. Underlying Frameworks
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.

5. 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.
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6. 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.
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7. 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.
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8. 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.
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9. 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.
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10. 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.
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11. 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.
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12. 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.
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13. 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.
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14. 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.
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15. 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)

16. The Buttons Task is intended for upper elementary or early middle school grades. As you work through this task, think about the mathematical practice standards that your students might use. (Trainer distributes one handout per person.) (CLICK)

17. Each of you has received a Buttons Task handout
Each of you has received a Buttons Task handout. You are asked to complete parts 1-3 individually. Then find a partner to compare your work with. Complete part 4 with your partner looking for as many ways as possible to solve the problem. You will have 5 minutes to solve the BUTTON TASK. Begin now.
Give 5 minutes.
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18. Draw Pattern 4 next to Pattern 3. See answer above.
How many white buttons does Gita need for Pattern 5 and Pattern 6? Explain how you figured this out. 15 buttons and 18 buttons
How many buttons in all does Gita need to make Pattern 11? Explain how you figured this out. 34 buttons
Gita thinks she needs 69 buttons in all to make Pattern 24. How do you know that she is NOT correct? How many buttons does she need to make Pattern 24? 73 buttons
Here is a slide with the answers that could be the solutions.
#1 - Pattern 4 has 12 white buttons and 1 black button. Any questions?
#2 - Pattern 5 has 15 white buttons and pattern 6 has 18 white buttons. Would someone share how they discovered this? (Could have used the pattern, pattern 2 has 6 white, pattern 3 has 9 white, etc., or they could have drawn the picture.)
#3 – Pattern 11 would need 33 white buttons plus the one black button. Some students would forget to add the 1 black button to the total. (Ask for a volunteer to share their thinking on this problem.)
Notice that is questions 1-3, we start with the pattern and are looking for the total number of buttons. Question #4 asks us to think differently. Here we are given the total number and asked to work backward to determine if the total number is correct.
#4 – Is there just one correct way to work #4? No. The important element in all these problems is the explanation that the students provide.
When you answered the questions, did you use any particular mathematical processes? Pause. CLICK

19. Here are two samples of students’ answers which can illustrate how differently students can think about finding the solution to a problem. (Give time for participants to read through).
Do you allow your students to demonstrate how they think?
These instructions for this task ask you to allow students to evaluate the responses from Learner A and Learner B. They are asked to make sense of Learner A’s and Learner B’s reasoning.
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20. Which of the Mathematical Practices were needed to complete the task?
(Discussion)
Possible answers:
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Look for and make use of structure
Look for and express regularity in repeated reasoning
You might not have used all these practice standards during this task. Do you see how this task taught ‘thinking skills’ in addition to some mathematics’ skills? (CLICK)

21. Analyzing the Button Task
The Button Task was:
Scaffolded
Foreshadows linear relationships
Requires critical thinking skills
Did not suggest specific strategy
Let’s look at the button task: (CLICK)
The task was scaffolded. Early questions were tied to diagram. The later questions required more abstract thinking. However, if the student had trouble thinking abstractly, he/she could still work the problem using either concrete examples or drawing. (CLICK)
This task used black and white buttons. This foreshadows linear relationships which include two quantities and a relationship between the quantities. (CLICK)
The tasks required two methods of thinking. We began with a pattern that asked for a total number of buttons. By the time we reached question 4, we were given a total number and asked to determine if they were correct for a certain pattern number. (CLICK)
The tasks did not suggest a specific strategy. Each student was free to develop their own strategy and then explain their reasoning.
This task was taken from the Website insidemathematics.org. This is a good resource for additional practice standards exercises.
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23. 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, Henningtsen & 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.
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Herbert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

24. 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)

25. 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 how 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)

26. 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.
Tasks can be found on several websites. Insidemathematics.org contains many tasks categorized by grade level and course. (CLICK)

27. ?? 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)
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28. Contact Information
ALSDE Office of Student Learning
Curriculum and Instruction Section
Cindy Freeman, Mathematics Specialist
Phone:
This concludes the presentation of Phase I of the 2010 college- and career-ready mathematics standards. Are there any other questions?
Would you please take a minute and complete the evaluation form? Please include any further questions you have and be sure to put your contact information if you would like a personal response.
Common questions will be posted on our Website along with answers and comments from state personnel. You may access this information at the address listed on the screen. This Website will be an excellent source of information concerning the college- and career-ready standards for mathematics and English language arts. It will include presentations, handouts, lessons and lesson ideas, links, and other valuable resources.
If you have additional questions concerning the standards, please contact either Dr. Davis or Ms. Freeman at the addresses listed on the screen.
Thank you for your attention and have a safe trip home.