The 8 mathematical practices include problem solving, reasoning and explaining, modeling and using tools, using structure and generalizations, attending to precision, looking for and making use of structure, looking for and expressing regularity in repeated reasoning, and constructing viable arguments and critiquing the reasoning of others.
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The 8 Mathematical Practices are the core pillars of the Common Core State Standards for Mathematics. These practices are designed to develop students’ abilities to think mathematically, using problemsolving and critical thinking skills.
According to the Common Core State Standards Initiative, the 8 Mathematical Practices are:

Make Sense of Problems and Persevere in Solving Them: Students must be able to analyze a problem, identify the relevant information, and develop a plan to solve it.

Reason Abstractly and Quantitatively: Students must be able to analyze complex problems and break them down into smaller, more manageable parts.

Construct Arguments and Critique the Reasoning of Others: Students must be able to present cohesive arguments and logically critique the reasoning of others.

Model with Mathematics: Students must be able to use mathematical models to represent realworld situations, interpret the results, and draw conclusions.

Use Appropriate Tools Strategically: Students must be able to choose and use appropriate mathematical tools (such as calculators, graphs, and formulas) effectively.

Attend to Precision: Students must pay attention to detail and maintain accuracy when making calculations or other mathematical operations.

Look for and Make Use of Structure: Students must recognize patterns and structures in the problems they encounter and use this information to develop solutions.

Look for and Express Regularity in Repeated Reasoning: Students must look for patterns in their problemsolving processes and use this information to generalize those processes for use in future situations.
As noted by Jo Boaler, a Professor of Mathematics Education at Stanford University and author of “Mathematical Mindsets,” “The 8 Mathematical Practices are designed to emphasize that mathematics is a subject of inquiry, not just a set of rules to be memorized.” By developing these practices, students are better equipped to navigate complex mathematical problems and understand the realworld applications of mathematics.
Here is a table summarizing the 8 Mathematical Practices:
Practice  Description 

Make Sense of Problems and Persevere in Solving Them  Analyze a problem, identify relevant information, and develop a plan to solve 
Reason Abstractly and Quantitatively  Analyze complex problems and break them down into smaller, more manageable parts 
Construct Arguments and Critique the Reasoning of Others  Present cohesive arguments and logically critique the reasoning of others 
Model with Mathematics  Use mathematical models to represent realworld situations, interpret results, and draw conclusions 
Use Appropriate Tools Strategically  Choose and use appropriate mathematical tools effectively 
Attend to Precision  Pay attention to detail and maintain accuracy when making calculations 
Look for and Make Use of Structure  Recognize patterns and structures in problems and use them to develop solutions 
Look for and Express Regularity in Repeated Reasoning  Look for patterns in problemsolving processes and generalize for future use 
In conclusion, the 8 Mathematical Practices are essential for all students to learn and understand as they develop their mathematical minds. As stated by Boaler, “mathematical practices are what mathematics is all about – thinking, reasoning and problemsolving.” By emphasizing these practices, students can develop a strong foundation in mathematics, which will serve them well throughout their academic and professional careers.
There are alternative points of view
Breaking down the Common Core’s 8 mathematical practice standards
 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.
See a related video
The video discusses the importance of mathematical practices and how it goes beyond just knowledge and skills. It emphasizes that standards for mathematical practices are crucial as they capture the processes and proficiencies that students should possess. The speaker highlights the ability to construct arguments and reason mathematically as a crucial skill for students to succeed in their careers. The video also emphasizes the importance of modeling and applying mathematics to realworld problems, being strategic with technology, developing skills in seeing mathematical meaning, and transforming objects into something purposeful.
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 CONCEPTUAL UNDERSTANDING.
 PROCEDURAL FLUENCY.
 STRATEGIC COMPETENCE.
 ADAPTIVE REASONING.
 PRODUCTIVE DISPOSITION.