Principal's Workshop: Looking for Evidence of the Standards for Mathematical Practice NEKSDC Thursday, October 11, 2012 Presenter: Elaine Watson, Ed.D.

Introductions Share What feeds your soul personally? What is your professional role? What feeds your soul professionally?

Volunteers for Breaks I need volunteers to remind me when we need breaks! Every 20 minutes, we need a 2-minute “movement break” to help our blood circulate to our brains. Every hour we need a 5-minute bathroom break.

Formative Assessment How familiar are you with the CCSS Standards for Mathematical Practice?

Goals for this Workshop You will leave with a deeper understanding of: The 8 Common Core Practice Standards Recognizing the Practice Standards in action by the STUDENTS in a math class. Recognizing TEACHER MOVES that elicit the Practice Standards being used by students. The types of tasks that build students’ ability to “practice” the Practice Standards

We will accomplish these goals by: Looking closely at each of 8 Practice Standards Verbal descriptions Videos Activities Discussions

We will accomplish these goals by: Using the SMP Template to look for “teacher moves” and “evidence of students using the practice” Looking at other resources for observation tools rich mathematical tasks

CCSSM Equally Focuses on… Standards for Mathematical Practice Standards for Mathematical Content Same for All Grade Levels Specific to Grade Level

Hunt Video:

1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: Explain to self the meaning of a problem and look for entry points to a solution Analyze givens, constraints, relationships and goals Make conjectures about the form and meaning of the solution

1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: Plan a solution pathway rather than simply jump into a solution attempt Consider analogous problems Try special cases and simpler forms of original problem

1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: Monitor and evaluate their progress and change course if necessary… “Does this approach make sense?”

1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: Persevere in Solving by: Transforming algebraic expressions Changing the viewing window on a graphing calculator Moving between the multiple representations of: Equations, verbal descriptions, tables, graphs, diagrams

1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: Check their answers “Does this answer make sense?” Does it include correct labels? Are the magnitudes of the numbers in the solution in the general ballpark to make sense in the real world?

1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: Check their answers Verify solution using a different method Compare approach with others: How does their approach compare with mine? Similarities Differences

2. Reason Abstractly and Quantitatively Mathematically proficient students: Make sense of quantities and their relationships in a problem situation Bring two complementary abilities to bear on problems involving quantitative relationships: The ability to… decontextualize to abstract a given situation, represent it symbolically, manipulate the symbols as if they have a life of their own contextualize to pause as needed during the symbolic manipulation in order to look back at the referent values in the problem

2. Reason Abstractly and Quantitatively Mathematically proficient students: Reason Quantitatively, which 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

3.Construct viable arguments and critique the reasoning of others Mathematically proficient students: Understand and use… stated assumptions, definitions, and previously established results… when constructing arguments

3.Construct viable arguments and critique the reasoning of others Mathematically proficient students: Understand and use… stated assumptions, definitions, and previously established results… when constructing arguments

3.Construct viable arguments and critique the reasoning of others 10 minute video

4. Model with Mathematics Modeling is both a K - 12 Practice Standard and a 9 – 12 Content Standard.

4. Model with Mathematics Mathematically proficient students: Use powerful tools for modeling: Diagrams or graphs Spreadsheets Algebraic Equations

4. Model with Mathematics Mathematically proficient students: Models we devise depend upon a number of factors: How precise do we need to be? What aspects do we most need to undertand, control, or optimize? What resources of time and tools do we have?

4. Model with Mathematics Mathematically proficient students: Models we devise are also constrained by: Limitations of our mathematical, statistical, and technical skills Limitations of our ability to recognize significant variables and relationships among them

Modeling Cycle The word “modeling” in this context is used as a verb that describes the process of transforming a real situation into an abstract mathematical model.

Modeling Cycle Problem Formulate Compute Interpret Validate Report

Modeling Cycle Problem Identify variables in the situation Select those that represent essential features Problem Identify variables in the situation Select those that represent essential features

Modeling Cycle Formulate Select or create a geometrical, tabular, algebraic, or statistical representation that describes the relationships between the variables Formulate Select or create a geometrical, tabular, algebraic, or statistical representation that describes the relationships between the variables

Modeling Cycle Compute Analyze and perform operations on these relationships to draw conclusions Compute Analyze and perform operations on these relationships to draw conclusions

Modeling Cycle Interpret Interpret the result of the mathematics in terms of the original situation Interpret Interpret the result of the mathematics in terms of the original situation

Modeling Cycle Validate Validate the conclusions by comparing them with the situation… Validate Validate the conclusions by comparing them with the situation…

Modeling Cycle Validate Re - Formulate Report on conclusions and reasoning behind them

Modeling Cycle Problem Formulate Compute Interpret Validate Report

6. Attend to precision Mathematically proficient students: Try to communicate precisely to others: Use clear definitions State the meaning of symbols they use Use the equal sign consistently and appropriately Specify units of measure Label axes

6. Attend to precision Mathematically proficient students: Try to communicate precisely to others Calculate accurately and efficiently Express numerical answers with a degree of precision appropriate for the problem context Give carefully formulated explanations to each other Can examine claims and make explicit use of definitions

What SMP’s do you see? Even or Odd Video

Use the Standards for Mathematical Practice Lesson Alignment Template. What SMPs do you see? Even or Odd Video

7. Look for and make use of structure Mathematically proficient students: Look closely to discern a pattern or structure In x 2 + 9x + 14, can see the 14 as 2 x 7 and the 9 as Can see complicated algebraic expressions as being composed of several objects: 5 – 3 (x – y) 2 is seen as 5 minus a positive number times a square, so its value can’t 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 Look for both general methods and for shortcuts Maintain oversight of the process while attending to the details.

What SMPs Do You Observe Maya Practicing? See Maya Video

Let’s Practice Some Modeling Students can: start with a model and interpret what it means in real world terms OR start with a real world problem and create a mathematical model in order to solve it.

Possible or Not? Here is an example of a task where students look at mathematical models (graphs of functions) and determine whether they make sense in a real world situation.

Possible or Not?

Questions: Mr. Hedman is going to show you several graphs. For each graph, please answer the following: A. Is this graph possible or not possible? B.If it is impossible, is there a way to modify it to make it possible? C. All graphs can tell a story, create a story for each graph.

One A. Possible or not? B. How would you modify it? C. Create a story.

Two A. Possible or not? B. How would you modify it? C. Create a story.

Three A. Possible or not? B. How would you modify it? C. Create a story.

Four A. Possible or not? B. How would you modify it? C. Create a story.

Five A. Possible or not? B. How would you modify it? C. Create a story.

Six A. Possible or not? B. How would you modify it? C. Create a story.

Seven A. Possible or not? B. How would you modify it? C. Create a story.

Eight A. Possible or not? B. How would you modify it? C. Create a story.

Nine A. Possible or not? B. How would you modify it? C. Create a story.

Ten A. Possible or not? B. How would you modify it? C. Create a story.

All 10 Graphs What do all of the possible graphs have in common?

And now... For some brief notes on functions!!!! Lesson borrowed and modified from Shodor.Shodor Musical Notes borrowed from Abstract Art Pictures Collection.Abstract Art Pictures Collection.

Pyramid of Pennies Here is an example of a task where students look at a real world problem, create a question, and create a mathematical model that will solve the problem.

Dan Meyer’s 3-Act Process Act I Show an image or short video of a real world situation in which a question can be generated that can be solved by creating a mathematical model.

Dan Meyer’s 3-Act Process Act I (continued) 1. How many pennies are there? 2. Guess as close as you can. 3. Give an answer you know is too high. 4. Give an answer you know is too low.

Dan Meyer’s 3-Act Process Act 2 Students determine the information they need to solve the problem. The teacher gives only the information students ask for.

Dan Meyer’s 3-Act Process What information do you need to solve this problem?

Dan Meyer’s 3-Act Process Act 2 continued Students collaborate with each other to create a mathematical model and solve the problem. Students may need find text or online resources such as formulas.

Dan Meyer’s 3-Act Process Go to it!

Dan Meyer’s 3-Act Process Act 3 The answer is revealed.

Dan Meyer’s 3-Act Process Act 3

Standards for Mathematical Practice Describe ways in which student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity

Standards for Mathematical Practice Provide a balanced combination of Procedure and Understanding They shift the focus to ensure mathematical understanding over computation skills

Inside Mathematics Videos hematical-practice-standards

Standards for Mathematical Practice Students will be able to: 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.

Think back to the Pyramid of Pennies. At what point during the problem did you do the following? 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.

Resources for Rich Mathematical Tasks The “Go-To” site for looking at the Content Standards and finding rich tasks, called “Illustrations” that can be used to build student understanding of a particular Content Standard.

Resources for Rich Mathematical Tasks is a website with a plethora of resources to help teachers transition to teaching in a way that reflects the Standards for Mathematical Practice. It’s worth taking the 6:19 minutes to watch the Video Overview of the Video Tours to familiarize yourself to all of the resources. There are more video tours that can be accessed by clicking on a link below the overview video.Video Overview of the Video Tours

Resources for Rich Mathematical Tasks There are several names that are associated with the website: MARS, MAPS, The Shell Center…however the tasks are usually referred to as The MARS Tasks. The link above will show tasks aligned with the Practice Standards They have been developed through a partnership with UC Berkeley and the University of Nottingham

Resources for Rich Mathematical Tasks Tools for the Common Core is the website of Bill McCallum, one of the three principle writers of the CCSSM. Highlights of this site are the links (under Tools) to the Illustrative Mathematics Project, the Progressions Documents, and the Clickable Map of the CCSSM.

Resources Northeast Kingdom School Development Center Standards of Mathematical Practice Workshop and Dinner