Rigor in the Math Classroom through the Implementation of ......Rigor in the Math Classroom through the Implementation of the 8 Standards for Mathematical Practice MACC.K12.MP.2.1

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

MACC.K12.MP.1.1 Make sense of problems and persevere in solving them.

High Rigor (Includes Moderate Rigor + time for students to take responsibility for their learning)

Moderate Rigor (Teacher models most of the work)

Low Rigor (Teacher does most of the work)

Questions 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…show…?

Writing Write to explain your

solution. Plan a solution pathway

to solve the problem. Explain how visual

representations relate to the problem.

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.

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.

Teaches strictly procedures without connection to meaning.

Integrates little wait time during instruction.

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.

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.

Procedural task Reproducing previously learned

facts, rules, formulas, or definitions.

Student 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.


MACC.K12.MP.2.1 Reason abstractly and quantitatively.




Questions 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?

Writing 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

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.

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.

Does not connect the procedure to context.

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.

The task has realistic context. The task requires students to

manipulate the symbolic representation.

Task is out of context.

Student Output

Students are able to put an equation into context and take context and give it a mathematical representation.

Students accurately and flexibly manipulate representing symbols and operations in problems.

Task has solutions that can be represented by multiple representations.

Students use different properties of operations and objects to reason through a problem.

Does not represent the solution.


MACC.K12.MP.3.1 Construct viable arguments and critique the reasoning of others.




Questions 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_________?

Writing Construct a viable

argument supporting your solution and solution pathway.

Compare two arguments and determine correct or flawed logic.

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.

Models making conjectures and justifying them.

Expects students to explain their conjectures.

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.

Task The task requires students to explain and justify their conjecture.

The task requires students to evaluate other solution pathways.

The task avoids single steps or routine algorithms.

The task requires only an answer.

Student output

Explains and justifies their conjecture.

Evaluates the arguments of their peers.

Asks appropriate questions to clarify or improve arguments.

Provides their answer with an explanation.

Listens to the argument of others.

Provides an answer with no explanation.


MACC.K12.MP.4.1 Model with mathematics.




Questions 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?

Writing 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.

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.

Models the connection between the math and real world context.

Asks prompting questions to facilitate the discussion of what an appropriate model is.

Teacher explains the appropriateness of the model.

Task Requires students to justify the reasonableness of their results and procedures within the context of the task.

Requires students to illustrate the relevance of the mathematics involved.

Requires students to identify extra information or missing information.

Requires students to perform necessary computations and identify variables/parts of a problem.

Student output

Students are able to model the problem in a real world context and justify the appropriateness of the model.

Students are able to identify extra information.

Students are able to identify appropriate models.

Students perform the computation without using an appropriate model.


MACC.K12.MP.5.1 Use appropriate tools strategically.




Questions 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?

Writing 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.

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.

Models using think aloud how to choose the appropriate tool.

Allows students to choose the tool that would work for the problem.

Tells students what tool to use to solve the problem.

Task Requires multiple tools to be used in solving the problem.

Multiple tools may be used and allows students to choose the tool appropriate.

Uses a specific tool, provided by the teacher.

Student 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.


MACC.K12.MP.6.1 Attend to precision.




Questions 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?

Writing 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.

Teacher input Expects students to identify when others are not being precise with their communication, and not addressing the question completely.

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.

Uses lower level vocabulary. Expectations for precise

explanations and precise work are not addressed.

Task Includes assessment criteria for communicating the solution to the problem (Rubric).

There are precise instructions.

Students are required to write precisely to answer the problem.

The instructions are overly detailed or wordy.

Does not require precision in the answer.

Student output

Students should write to justify their work precisely.

Students should ask the prompting questions to their peers to obtain a more precise answer.

Students 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.)


MACC.K12.MP.7.1 Look for and make use of structure.




Questions 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…? Writing 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.

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.

Models how to analyze a given task and find multiple approaches.

Guides students in analyzing a task through prompting questions.

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.

Task Requires student to identify the most efficient approach to the task and justify their approach.

Requires students to analyze the task before applying an algorithm.

Requires students to automatically apply an algorithm to a task.

Student output

Students justify the approach that is most appropriate for them.

Students analyze the task, and identify more than one approach to the task.

Students are able to choose the approach that is appropriate for them.

Uses the same procedure to answer multiple questions without evaluating the approach.


MACC.K12.MP.8.1 Look for and express regularity in repeated reasoning.




Questions 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?

Writing Write to explain the

connections made between prior knowledge and new content.

Write to explain the short cut and justify why it works.

Teacher input Guides students to make connections between prior knowledge and new content.

Prompts students to recognize the pattern or structure.

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.

Task The task addresses and connects to prior knowledge.

The task is overly repetitive. The task is disconnected from prior and future concepts.

There is no logical progression.

Student output

Addresses and connects prior knowledge to new content.

Continually evaluate their intermediate results.

Students generate their own questions based on the current task.

Students needed prior knowledge along with new skill to complete the task.

Students look for and explain short cuts.

Has no logical progression. Is overly repetitive without

developing a pattern.


Web links to assist with implementation and integration of the Mathematical Practice Standards

Inside Mathematics Inside Mathematics has video of classroom instruction integrating the Mathematical Practice Standards.

Mathematics Proficiency Matrix This proficiency matrix is based on the Mathematical Practice Standards and the evidence students should exhibit with each practice standard.

MARS Mathematical Assessment Project

The Mathematical Assessment Project contains tasks aligned to the Common Core Content Standards as well as tasks aligned to the Mathematical Practice Standards.

Mathematical Practice Learning Community Templates

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

http://www.insidemathematics.org/index.php/math-standards-together

http://www.mathleadership.com/sitebuildercontent/sitebuilderfiles/standardsoftudentpracticeinmathematicsproficiencymatrix.pdf

http://map.mathshell.org/materials/stds.php

http://map.mathshell.org/materials/stds.php

http://schools.utah.gov/CURR/mathsec/Common-Core/MathematicalPracticesLearningCommunityTemplates.aspx

http://schools.utah.gov/CURR/mathsec/Common-Core/MathematicalPracticesLearningCommunityTemplates.aspx


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

http://www.cpalms.org/Standards/PublicPreviewBenchmark6327.aspx




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.

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





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 7 × 5 + 7 × 3, 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 2 + 7. 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.



Documents

Rigor in the Math Classroom through the Implementation of ......Rigor in the Math Classroom through the Implementation of the 8 Standards for Mathematical Practice MACC.K12.MP.2.1