Standards for Mathematical Practice | Common Core State Standards Initiative (2024)

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. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Standards in this domain:

CCSS.Math.Practice.MP4

CCSS.Math.Practice.MP5

CCSS.Math.Practice.MP6

CCSS.Math.Practice.MP7

CCSS.Math.Practice.MP8

CCSS.Math.Practice.MP1 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.

CCSS.Math.Practice.MP2 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.

CCSS.Math.Practice.MP3 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 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.

CCSS.Math.Practice.MP4 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.

CCSS.Math.Practice.MP5 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.

CCSS.Math.Practice.MP6 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.

CCSS.Math.Practice.MP7 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.

CCSS.Math.Practice.MP8 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.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

Standards for Mathematical Practice			| Common Core State Standards Initiative (2024)

FAQs

What are the 8 mathematical practices in Common Core? ›

Standards for Mathematical Practice
  • Make sense of problems and persevere in solving them. ...
  • Reason abstractly and quantitatively. ...
  • Construct viable arguments and critique the reasoning of others. ...
  • Model with mathematics. ...
  • Use appropriate tools strategically. ...
  • Attend to precision. ...
  • Look for and make use of structure.
Jul 25, 2023

What are the 5 Nctm content standards? ›

This practical guide includes three 11" x 17" sheets to display the expectations across the four grade bands for each of the five Content Standards: Number and Operations, Algebra, Geometry, Data Analysis and Probability, and Measurement.

What are the Common Core State Standards CCSS? ›

Since 2010, a number of states across the nation have adopted the same standards for English and math. These standards are called the Common Core State Standards (CCSS). Having the same standards helps all students get a good education, even if they change schools or move to a different state.

What is the difference between Nctm standards and common core standards? ›

The NCTM Standards put forth a broad vision for school mathematics, including many examples and advice for implementation, while CCSS provides a detailed set of grade-by-grade standards that can be immediately adopted as a state curriculum document.

What is the difference between Common Core math and regular math? ›

While traditional math teaching strategies focus extensively on formula memorization and topic-specific learning patterns, Common Core works to give your student a deeper level of knowledge by introducing broader, more foundational methods of thinking as well as strategies that align with a more in-depth learning ...

How to implement standards of mathematical practice? ›

Make sense of problems and persevere in solving them.

Interpret and make meaning of the problem looking for starting points. Analyze what is given to explain to themselves the meaning of the problem. Plan a solution pathway instead of jumping to a solution. Monitor the progress and change the approach if necessary.

What are the 6 principles of Nctm? ›

Schoenfeld and Douglas Clements. The resulting document sets forth a set of six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) that describe NCTM's recommended framework for mathematics programs, and ten general strands or standards that cut across the school mathematics curriculum.

What is the difference between content standards and the standards for mathematical practice? ›

Content standards describe the knowledge that a student must be able to recall and understand; process/practice standards provide an opportunity for students to demonstrate the skill using what they know. Simply put, content is what you know while process/practice is what can you do. They are both assessed differently.

What are the NCTM mathematical practices? ›

Take a deeper dive into understanding the five practices—anticipating, monitoring, selecting, sequencing, and connecting—for facilitating productive mathematical conversations in your high school classrooms... read more. Enhance your fluency in the five practices—anticipating, monitoring, selecting, ... read more.

Why is Common Core controversial? ›

Common Core was not benchmarked to international high-achieving countries despite claiming that this was so; Common Core standards were less clear than the California 1997 standards; Common Core had significant gaps in its content coverage; and, perhaps most obviously, despite its explicit promise to expect algebra and ...

What is the difference between Common Core and state standards? ›

What are some key differences between the current California Content Standards and the Common Core Standards? English Language Arts: CA Common Core State Standards expose students to more complex text, meaning what they read will have a higher level of difficulty.

What is the main focus of the Common Core State Standards? ›

The common standards define the rigorous skills and knowledge in English language arts and mathematics that need to be effectively taught and learned for all students to be ready to succeed academically in credit-bearing, college-entry courses and workforce training programs.

What are the 5 NCTM process standards? ›

They were based on five key areas 1) Representation, 2) Reasoning and Proof, 3) Communication, 4) Problem Solving, and 5) Connections. If these look familiar, it is because they are the five process standards from the National Council of Teachers of Mathematics (NCTM, 2000).

What is the purpose of the Common Core State Standards for mathematics? ›

Common Core math is a set of standards that outlines the mathematics concepts for students in K-12th grades. The standards aim to provide clarity and specificity for the concepts included in math instruction.

Why are the NCTM standards important? ›

Principles and Standards

It emphasizes the need for well-prepared, well-supported teachers and administrators, acknowledging the importance of a carefully organized system for assessing students' learning and a program's effectiveness.

What are the 8 math language routines? ›

ELL Mathematical Language Routines
  • MLR1: Stronger and Clearer Each Time. ...
  • MLR2: Collect and Display. ...
  • MLR3: Clarify, Critique, Correct. ...
  • MLR4: Information Gap. ...
  • MLR5: Co-Craft Questions. ...
  • MLR6: Three Reads. ...
  • MLR7: Compare and Connect. ...
  • MLR8: Discussion Supports.

What is Common Core math 8? ›

In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using ...

What are the mathematical principles Common Core? ›

  • 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.

What is the Common Core math method? ›

Compared to traditional math, Common Core math has a narrower focus on mathematical concepts. Additionally, there is a progression of concepts over time, where new lessons build on previous lessons. There is also an emphasis on real world problems and practical application, and less of an emphasis on rote memorization.

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