Rigor in the Math Classroom through the Implementation of the 8 Standards for Mathematical Practice

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1 MACC.K12.MP.1.1 Make sense of problems and persevere in solving them. (Includes + time for students to take responsibility for their learning) How would you describe the problem in your own words? What do you notice about? What are some other strategies you might try? What are some other problems that are similar to this one? How else might you organize represent sho w? Teacher input Provides time and facilitates discussion in problem solutions. Allows students to struggle with a task prior to modeling. Expects students to monitor and evaluate their progress through a task. Provides opportunities for students to explain themselves. Task Require students to explore and understand the nature of mathematical concepts, processes, or relationship. Requires students to explain how they arrived at their solution. Models using a think aloud making sense of a problem/task with connection to meaning. Allows students to struggle through a problem/task. Expects students to check and show their work. Task with more than one entry point or way to solve the problem Requires students to check solutions for errors using one other solution path. Teaches strictly procedures without connection to meaning. Integrates little wait time during instruction. Procedural task Reproducing previously learned facts, rules, formulas, or definitions. Write to explain your solution. Plan a solution pathway to solve the problem. Explain how visual representations relate to the problem. Output Use their prior knowledge and experience to make sense of the task while persevering through it. Actively engaged in solving problems and thinking is visible. Monitor and evaluate their understanding and change course if needed. Check and explain their answer. Plan a solution pathway. Continually ask does this make sense? Check their work. Unable to explain the reason for the specific procedure. Focus is on producing a correct answer.

2 MACC.K12.MP.2.1 Reason abstractly and quantitatively. What do the numbers or symbols used in the problem represent? What is the relationship of the quantities? What is the relationship between and? What properties could we use to find a solution? Could you have used another operation or property to solve the task? Why or why not? Write to explain the relationship between and. Justify why you needed to use. Compare and contrast two strategies used to solve the problem. Write a contextual problem that the equation/expression could represent. Teacher input (Includes + time for students to take responsibility for their learning) Provides a range of representations of math problem situations Prompts students to make connections between the procedures and a realistic context in which the procedure applies. Task The task has relevant context. The task provides a context or situation for students that allow them to abstract the situation and represent it symbolically. Output s are able to put an equation into context and take context and give it a mathematical representation. s accurately and flexibly manipulate representing symbols and operations in problems. Teacher models the connection between the procedures and a realistic context. Provides structure for students to connect procedures to meaning. Teacher models how to symbolically represent a problem. The task has realistic context. The task requires students to manipulate the symbolic representation. Task has solutions that can be represented by multiple representations. s use different properties of operations and objects to reason through a problem. Does not connect the procedure to context. Task is out of context. Does not represent the solution.

3 MACC.K12.MP.3.1 Construct viable arguments and critique the reasoning of others. What evidence will support your solution? Will it still work if? What is similar and different about? How do you know your approach worked? How does your strategy compare to? Construct a viable argument supporting your solution and solution pathway. Compare two arguments and determine correct or flawed logic. (Includes + time for students to take responsibility for their learning) Teacher input Assists students with differentiating between a conjecture and an assumption. Provides opportunities for students to evaluate peer arguments. Expects students to explain and justify their conjectures. Task The task requires students to explain and justify their conjecture. The task requires students to evaluate other solution pathways. output Explains and justifies their conjecture. Evaluates the arguments of their peers. Asks appropriate questions to clarify or improve arguments. Models making conjectures and justifying them. Expects students to explain their conjectures. The task avoids single steps or routine algorithms. Provides their answer with an explanation. Listens to the argument of others. Expects the students to solve the problem following a procedure without opportunities to follow other appropriate paths. Have students make conjectures, but not explain or justify them. The task requires only an answer. Provides an answer with no explanation.

4 MACC.K12.MP.4.1 Model with mathematics. How would you represent? What formula could apply to this situation? How would a diagram, graph, table help? How do the important quantities in your problem relate to each other? How has your model served its purpose? Write to explain your visual representation. Interpret the results of a mathematical situation. Explain how you might improve your model to better serve its purpose. (Includes + time for students to take responsibility for their learning) Teacher input Provides opportunities for students to model the connection between the math and real world context. Provides opportunities for students to evaluate the appropriateness of a model. Task Requires students to justify the reasonableness of their results and procedures within the context of the task. output s are able to model the problem in a real world context and justify the appropriateness of the model. Models the connection between the math and real world context. Asks prompting questions to facilitate the discussion of what an appropriate model is. Requires students to illustrate the relevance of the mathematics involved. Requires students to identify extra information or missing information. s are able to identify extra information. s are able to identify appropriate models. Teacher explains the appropriateness of the model. Requires students to perform necessary computations and identify variables/parts of a problem. s perform the computation without using an appropriate model.

5 MACC.K12.MP.5.1 Use appropriate tools strategically. What mathematical tool could be used? Why did you choose that mathematical tool? Why is the tool you chose better than other possible tools? Why is it helpful to use? What estimate did you make for the solution? Why is estimating helpful when solving the problem? (Includes + time for students to take responsibility for their learning) Teacher input Allows students to choose the appropriate tool and has them readily available during student work time. Provides problem solving opportunities that lend themselves to using multiple tools. Task Requires multiple tools to be used in solving the problem. Models using think aloud how to choose the appropriate tool. Allows students to choose the tool that would work for the problem. Multiple tools may be used and allows students to choose the tool appropriate. Tells students what tool to use to solve the problem. Uses a specific tool, provided by the teacher. Use the available tools and reflect on the strengths and limitations of each. Write to explain why you used the specific tool to solve the problem. output Uses multiple tools to solve to reach a solution. Writes to explain the use of the tools. Can solve the problem using the appropriate tool. Checks the problem using estimation. Is able to use the specified tool to solve the problem.

6 MACC.K12.MP.6.1 Attend to precision. What mathematical terms or symbols should be used in this situation? What mathematical language, definitions, and properties can you use to explain? How do you know your answer is correct/ reasonable? How are you showing the meaning of the quantities? (Includes + time for students to take responsibility for their learning) Teacher input Expects students to identify when others are not being precise with their communication, and not addressing the question completely. Task Includes assessment criteria for communicating the solution to the problem (Rubric). Models working through a problem using precise vocabulary and content. Expects students to communicate precisely and intervenes when necessary. Prompts students to be precise through questioning strategies. There are precise instructions. s are required to write precisely to answer the problem. Uses lower level vocabulary. Expectations for precise explanations and precise work are not addressed. The instructions are overly detailed or wordy. Does not require precision in the answer. Write a precise explanation of your reasoning. Write to justify your work precisely. Write prompting questions to encourage others to write a more precise answer. output s should write to justify their work precisely. s should ask the prompting questions to their peers to obtain a more precise answer. s should answer the problem accurately and efficiently. Is precise with their answer including units, vocabulary, reasoning, and labeling axes to clarify the correspondence with quantities in a problem. Imprecise in their answer (i.e. does not include units, does not use proper math terminology, unable to articulate their reasoning behind an answer.)

7 MACC.K12.MP.7.1 Look for and make use of structure. What observations can you make about? In what ways does this problem connect to other mathematical concepts? What patterns do you find in? What are some other problems similar to this one? How does this relate to? (Includes + time for students to take responsibility for their learning) Teacher input Prompts students to identify the mathematical structure of the task. Provides opportunities for students to choose the most efficient method to solve the problem. Task Requires student to identify the most efficient approach to the task and justify their approach. Models how to analyze a given task and find multiple approaches. Guides students in analyzing a task through prompting questions. Requires students to analyze the task before applying an algorithm. Gives students the algorithm to use for a task without evaluating its appropriateness. Requires students to use the same algorithm for a task when another may be just as appropriate. Requires students to automatically apply an algorithm to a task. Write to explain the overall structure and pattern in the mathematics. Write to explain the observations made. Write to explain multiple approaches to the task. Justify the approach you took when solving the problem. output s justify the approach that is most appropriate for them. s analyze the task, and identify more than one approach to the task. s are able to choose the approach that is appropriate for them. Uses the same procedure to answer multiple questions without evaluating the approach.

8 MACC.K12.MP.8.1 Look for and express regularity in repeated reasoning. How might that strategy work in another situation? What mathematical consistencies do you notice? What would happen if? Is this always true, sometimes true, or never true? Why? Write to explain the connections made between prior knowledge and new content. Write to explain the short cut and justify why it works. (Includes + time for students to take responsibility for their learning) Teacher input Guides students to make connections between prior knowledge and new content. Prompts students to recognize the pattern or structure. Task The task addresses and connects to prior knowledge. output Addresses and connects prior knowledge to new content. Continually evaluate their intermediate results. s generate their own questions based on the current task. Builds on previously taught skill and makes the connection for the students. Models the connection between the standards/tasks performed. Does not build on prior knowledge students have of a concept prior to further instruction. Presents a task or standard in isolation. The task is overly repetitive. The task is disconnected from prior and future concepts. There is no logical progression. s needed prior knowledge along with new skill to complete the task. s look for and explain short cuts. Has no logical progression. Is overly repetitive without developing a pattern.

9 Inside Mathematics Web links to assist with implementation and integration of the Mathematical Practice Standards Mathematics Proficiency Matrix MARS Mathematical Assessment Project Mathematical Practice Learning Community Templates Inside Mathematics has video of classroom instruction integrating the Mathematical Practice Standards. This proficiency matrix is based on the Mathematical Practice Standards and the evidence students should exhibit with each practice standard. The Mathematical Assessment Project contains tasks aligned to the Common Core Content Standards as well as tasks aligned to the Mathematical Practice Standards. These templates can be used for each practice standard within a professional learning community. You would want to select the template that will have the largest impact for your teachers. They include: Developing an understanding of each Mathematical Practice Standard Examining student work for evidence of the mathematical practice standard Connecting the practice standards to proficiency What the mathematical practice looks like during instruction

10 MACC.K12.MP.1.1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. MACC.K12.MP.2.1: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MACC.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine

11 domains to which an argument applies. s 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. MACC.K12.MP.4.1: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MACC.K12.MP.5.1: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using 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. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. MACC.K12.MP.6.1: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary

12 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. MACC.K12.MP.7.1: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well-remembered , in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MACC.K12.MP.8.1: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x² + 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, while attending to the details. They continually evaluate the reasonableness of their intermediate results.