Standards for Mathematical Practice: A Curriculum Specialists Guide to Building Mathematical Thinking

As an elementary math specialist with over 10 years of experience in mathematics education, both as a classroom teacher and a math coach, I have seen firsthand how the longstanding importance of effectively implementing the standards for mathematical practice builds deeper mathematical thinking. But embedding the standards while meeting the needs of all learners requires thoughtful adaptations.

Here, math expert Melanie guides you through each of the eight math standards providing insights, classroom examples and solutions to common challenges.

Whether you are new to the math practice standards or seeking to refine your implementation strategies, this guide offers valuable perspectives to help your learners become confident mathematicians.

Summary

Since 2010, the Standards for Mathematical Practice have shaped how students develop essential problem-solving and reasoning skills in their mathematics education alongside the Common Core State Standards (CCSS). Each standard for mathematical practice builds upon the others to create comprehensive mathematical thinking. For instance, MP1 connects directly to MP5 and MP6, showing how these practices work together to develop strong problem-solving skills.

What are the Standards for Mathematical Practice?

The Standards for Mathematical Practice are a set of eight practices that define the habits and approaches learners should develop to think and work like mathematicians.

Based on the NCTM Process Standards and research on mathematical proficiency across grade levels, the standards are designed to cultivate essential math skills such as:

• Conceptual understanding

• Adaptive reasoning

• Strategic competence

• Procedural fluency

• A productive disposition toward mathematics

The National Research Council’s report highlights these skills, alongside choosing the appropriate tools for math problems as the key components of mathematical proficiency.

While the Common Core establishes these standards nationwide, some states have adopted variations. Regardless of specific state standards, the principles of these eight math practices remain essential for developing strong mathematical thinkers. Below, we will explore each in depth.

The eight Standards for Mathematical Practice are:

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

These eight standards aim to close the achievement gap by developing mathematical thinkers. As students progress and learn more complex math concepts, their engagement with the mathematical practice standards evolves.

When mathematics educators integrate these standards into instruction, they facilitate problem solving skills, abstract reasoning, and the ability to apply math concepts flexibly in daily life.

Implementing the Standards for Mathematical Practice effectively

1. Make sense of problems and persevere in solving them – MP1

MP1 emphasizes understanding problems, planning a solution pathway, and persisting through challenges. Developing this skill builds strategic competence and a productive disposition toward problem solving.

Growth mindset is important for MP1. It allows students to see challenges as an opportunity to grow.

MP 1 in action: 

Math word problems are often challenging for learners. They require math and reading skills; learners have to discern what the question is asking them to do.

To demonstrate MP1 effectively, students must show that they can plan a solution pathway for a math problem, and then demonstrate perseverance as they determine the solution.

A great example of MP1 is when students first learn multi-step word problems. When I was teaching 3rd grade math curriculum, my students often had to solve multi-step word problems involving the four operations. I encouraged them to:

• Highlight key information

• Discuss possible solution strategies in pairs

• Use visual models such as bar models/tape diagrams

• Try different approaches before seeking help

Common challenges & solutions 

• Challenge: Giving up too quickly when faced with challenging math problems.

• Solution: Incorporate growth mindset strategies into daily math activities to help students see challenges as opportunities to grow and persevere.
  
  Having students retell a teacher or friend the steps and strategies they’ve tried so far is a helpful way to clarify their own thinking. Rephrase their strategies by using language such as ‘so are you saying…’ which helps learners grow their understanding.

• Appropriate Tools: Visual models such as bar models/tape diagrams, problem-solving templates or story problem graphic organizers help children understand what problems are asking them to do. Hands-on math manipulatives help children find the solution. 

  Where possible, model perseverance in the real world. For example, sticking to a budget at the grocery store, building a new bookshelf or planning the school garden.

2. Reason abstractly and quantitatively – MP2

MP2 requires understanding the relationship between numbers and real-world contexts, transitioning between concrete representations and abstract symbols.

The Concrete-Representational-Abstract (CRA) progression aims for learners to use concrete, hands-on models, progressing to visual representations and finally abstract symbolic strategies. 

MP2 in action

When teaching any new concept, follow the CRA progression to build conceptual understanding prior to procedural fluency.

When I was teaching 5th grade fractions, my students needed to start with fraction bars to help them grasp fraction division as a form of repeated reasoning.

This visual helped them:

• Reason about the size of the fractions they were using

• Evaluate if their quotient was reasonable

• Build conceptual understanding of dividing fractions

Eventually, they transitioned to solving equations without visual aids.

Common challenges & solutions 

• Challenge: Students struggle to connect symbolic expressions with real-world meanings. For example, when learning the standard algorithm for multiplication, learners often make errors because they have not connected the abstract symbolic strategy to the meaning behind each step. 

• Solution: Use real-world story problem contexts to introduce new concepts. When introducing multiplication of larger numbers, these contexts bring meaning to each number, allowing learners to evaluate their product’s reasonableness.  

Using the ‘go slow, to go fast’ mentality on building conceptual understanding is a helpful way for students to build towards abstract strategic problem solving. Sometimes this progression happens across grade levels, such as beginning multiplication in second grade and intoducing the standard algorithm in the fourth grade.

In other instances, this progression is shorter. But either way, connecting to real world examples, such as number of students per bus on a field trip, or number of gallons of water in each bucket helps deepen understanding.

3. Construct viable arguments and critique the reasoning of others – MP3

MP3 is the ability to justify reasoning and evaluate arguments based on logic and evidence. When students engage with others’ thinking, they clarify their own justifications and learn new problem solving strategies.

MP3 in action

In the 4th grade math curriculum and 6th grade math curriculum, students learn about prime numbers. A discussion about prime numbers can elicit many different responses as to why students believe a number is or isn’t prime.

Facilitate this discussion by teaching respectful language and signals to agree and disagree with the ideas of others. Pre-teaching this language is key to the success of these arguments. 

Common challenges & solutions 

• Challenge: Students hesitate to engage in mathematical discourse. They are often nervous to share their thinking and don’t want to disagree with their friends. 

• Solution: Create sentence stems for these class discussions. Allowing students to have sentence stems in front of them is helpful,l especially as they first learning to engage in mathematical debates.
  
  I always like starting with small group discussions so students can build confidence in sharing their ideas prior to moving to a large group discussion. 

The ability to construct viable arguments and engage in respectful mathematical discourse is a life skill. Students must learn to support their ideas with evidence, support their answer with a step by step solution, and be able to explain their thinking clearly to others. Engage in this type of discussion with cross-curriulalry and with other topics such as favorite foods or best sports teams.

4. Model with mathematics – MP4

MP4 applies mathematics to real-world problems using models, graphs, and equations. To model effectively with mathematics, students learn to identify what a real-world problem is asking them to do, and then choose a model appropriately.

For example, when solving a division problem, they should be able to determine if it is measurement or partitive division, and then draw their model accordingly. 

MP4 in action

In 7th grade, students use proportional reasoning to calculate ingredient adjustments in various recipes. This is a real-world mathematics requires students to conceptually understand what is happening so they can manipulate the recipe numbers whether they are whole numbers, fractions or decimals.

Students then learn different models to show this proportional relationship and choose one that is best suited for this problem. For example, students may learn to use a double number line, a table, and a graph to show this relationship. If students need to triple the recipe, then they must triple the ingredients and the relationship between the numbers must remain consistent.

Common challenges & solutions 

• Challenge: Students need support translating real-world situations into mathematical models and expressions.

• Solution: Scaffold the process of modeling real-world mathematics problems using an I do, we do, you do approach. This is a great strategy anytime a new concept is introduced, especially when students are using new visual models that require skill and practice. 

When I first taught my third graders to connect fraction tiles to a number line, I had to scaffold their understanding of using these models for representing various fraction problems. The fraction tile model was more appropriate for fractions of a whole chocolate bar, whereas the number line was more efficient for finding fractions of a mile or other distances.

Students must learn how to represent a fraction using both models, and how to use the models to solve problems (such as equivalent fractions or fraction addition).

Scaffolding the gradual release of responsibility to learners is a great way to build efficiency with modelling with mathematics. 

5. Use appropriate tools strategically – MP5

MP5 outlines that students must select and use tools effectively to problem solve. These tools are sometimes physical tools such as a ruler, or yard stick, fraction tiles or base ten blocks, and other times they may be digital tools such as a calculator.

To demonstrate MP5 profieiciency, students must be strategic in their tool use, knowing that one tool may be better suited for a particular problem than another. 

MP5 in action

In the first grade math curriculum, students learn to use appropriate measuring tools strategically. They build on a foundation of using nonstandard measurement units (such as cubes) from Kindergarten and they begin learning to use measuring tools such as rulers.

Second graders are expected to choose the appropriate measuring tool for a task. For example, when measuring a piece of paper, learners are expected to pick a ruler rather than a yard stick or meter stick. On the contrary, when measuring the height of a door, a yard stick is more appropriate.

With this, comes understanding the difference in measurement units such as inches, feet and yards, or millimeters, centimeters and meters.

Common challenges & solutions 

• Challenge: Students are overwhelmed with tool options and don’t know which is the appropriate math tool. 

• Solution: Provide guided choices and ample opportunities to practice using each tool for student success.

  Start by explicitly teaching the purpose of each tool, using anchor charts or reference guides. Then, model tool selection by thinking aloud, demonstrating why a specific tool is best for a given problem.

  When students to choose their own tool, provide structured choices by limiting available tools based on the problem type and gradually increasing autonomy.

  A Tool Selection Checklist with guiding questions can help develop confidence and flexibility in choosing math tools independently.. Questions for your checklist may include:

  • What do I need to solve?

  • What tool will help me visualize or calculate more easily?

  • Have I used this tool before? Was it helpful?

Third Space Learning’s one-on-one math tutoring encourages students to use a range of appropriate tools in every lesson. Highly trained math specialist tutors encourage learners to choose the right tool for each math problem. If students struggle to select an appropriate tool, tutors use effective questioning to guide students in selecting the appropriate tool for each math question.

6. Attend to precision – MP6

MP6 is the ability to make accurate calculations and be precise in explanations. Attending to precision means using clear and precise numbers and strategies that are easy to read and follow, and lead to accurate answers. Additionally, this refers to use of precise math vocabulary both in written and verbal explanations, labels and answers to story problems. 

MP6 in action

Precision in mathematics goes beyond just calculating correctly. It includes clear communication of reasoning and results. Labelling answers to story problems is a great example of this at any grade level.

When students label their answers with the appropriate unit (e.g., 5 apples, 12 inches, or $20), they show a full understanding of the problem’s context and ensure their answer makes sense.

For example, if a student solves a problem about buying apples and simply writes “5”, the answer lacks clarity. By labeling it as “5 apples”, they make their response precise and meaningful.

Precision also helps prevent common errors, such as forgetting to convert measurements or misinterpreting the question.

Encouraging students to consistently label their answers fosters accuracy, strengthens their mathematical communication, and aligns with real-world problem-solving skills.

Common challenges & solutions 

• Challenge: Students often forget to label their answers to math story problems.

• Solution: Help learners remember to label their answers in math story problems by consistently modeling the practice and emphasizing its importance.
  
  Explicit instruction and thinking aloud while solving problems, reinforces the habit of including units.

  Visual reminders like anchor charts with common labels and structured practice with sentence frames (e.g., “The answer is ___ [unit].”) provide additional support.

  Encouraging students to use a simple checklist—Did I solve the problem? Did I label my answer?—or engage in peer reviews fosters accountability. 

As with every math standard, relating to the real-world helps students utilize these math practices. Relating labeling to real-world applications, such as measuring ingredients or handling money, helps students see its relevance. Integrating these strategies consistently builds students’ precision and mathematical communication skills.

7. Look for and make use of structure – MP7

MP7 refers to mathematical structures, which are patterns, relationships, and properties that help simplify complex problems.

When students develop this skill, they learn to recognize and use these structures to make problem-solving more efficient. This could include noticing number patterns, properties of operations, geometric relationships, or recurring sequences in calculations.

By understanding these structures, students become more strategic thinkers. Instead of solving problems through rote memorization or step-by-step computation alone, they can apply known patterns and relationships to find solutions more quickly and accurately.

MP7 in action

By engaging in hands-on exploration of a concept such as odd and even numbers in first and second grade, students start to internalize the structure of even and odd numbers. Matching sock pairs is a great example of this type of exploration. This lays the foundation for future concepts like addition patterns, division, and multiplication.

Through these experiences, students begin to see math not as a series of isolated problems but as an interconnected system where recognizing structure simplifies their work.

This type of activity can be adapted for higher grades with more complex structures, such as recognizing properties in number operations (e.g., distributive property), patterns in multiplication tables, or algebraic expressions.

Common challenges & solutions 

• Solution: Incorporate hands-on activities and guided discovery. Using pattern blocks, number charts, or manipulatives, students can physically explore mathematical structures. Encouraging them to explain patterns in their own words and justify their reasoning deepens their understanding. Asking guiding questions like, “Do you notice a pattern?” or “How is this similar to what we did before?” helps students actively seek out and apply mathematical structures, making problem-solving more intuitive and efficient.

8. Look for and express regularity in repeated reasoning MP8

The core of MP8 is developing procedural fluency through a foundation of conceptual understanding. This standard encourages students to notice repeated calculations, identify patterns, and develop shortcuts or generalizations to solve problems more efficiently.

It helps students move beyond isolated procedures to recognizing mathematical structures that allow them to predict results and work more fluently.

MP8 in action

Repeated reasoning is essential for mathematical proficiency; it enables students to connect new concepts with prior knowledge. When students observe patterns in arithmetic, algebra, or geometry, they begin to generalize rules and develop strategies that simplify problem-solving.

For example, noticing that adding two even numbers always results in an even sum helps students build number sense and make predictions without recalculating every time. This allows students to use procedural fluency to build strategic competence in their problem solving. 

Common challenges & solutions 

• Challenge: Students may struggle to recognize patterns in repeated reasoning, treating each problem as a new, unrelated task rather than identifying underlying structures. This can lead to inefficient problem-solving, as they rely on rote memorization rather than developing generalizable strategies.

• Solution: Use guided questioning and visual representations to encourage students to verbalize their thinking. Ask: What do you notice happening again and again?” or “How does this pattern help you predict the next step?

  Hands-on activities, such as using number charts for skip counting or exploring multiplication patterns with arrays, provide concrete ways to visualize repetition.

  By highlighting connections between problems and encouraging students to develop shortcuts (e.g., recognizing that multiplying by 10 always shifts digits left), teachers can build students’ ability to generalize and apply reasoning efficiently across different mathematical concepts.

Measuring success against the standards

Assessment strategies

Teachers can progress monitor proficiency in the math practice standards using various methods:

• Observational data allows educators to monitor student engagement in problem-solving and mathematical discussions, evaluate their precision and ability to verbalize their strategies and critique the reasoning of others.

• Reviewing work samples provides insight into the depth of explanations and solutions, revealing whether learners are demonstrating strategic competence.

• Formative assessments, such as exit tickets and quick checks, offer immediate feedback on understanding, helping teachers adjust instruction to reinforce the math practice standards and problem-solving strategies.

Professional development for teachers

Professional development focused on collaboration, targeted training, and cross-curricular applications helps deepen understanding of the Standards for Mathematical Practice.

Engaging in peer observations and shared lesson planning allows educators to see how colleagues implement the standards in their classrooms, fostering a culture of continuous learning.

Workshops and training sessions provide valuable strategies for integrating mathematical practices effectively, ensuring that instruction aligns with best practices.

Additionally, exploring cross-curricular connections in subjects like science and engineering helps teachers see how mathematical reasoning applies beyond the math classroom, reinforcing its relevance in the real-world.

These PD opportunities help educators strengthen their instructional approach and better support students in developing mathematical proficiency.

Standards for Mathematical Practice checklist

Mathematical practice standards cannot be assessed on a percentage basis, a letter grade or even pass-fail. It is beneficial to evaluate them on a continuum or progression of development towards proficiency. 

A sample rubric for evaluation of the mathematical practice standards is included below. Refer back to the description for each standard when evaluating student proficiency in the mathematical practice standards. 

Creating an evaluation checklist for each standard may also be helpful. During class discussion I often used a checklist to evaluate student progress towards proficiency in MP3 because it was not something that could be evaluated through an exit ticket or assessing work samples.

An example of a discussion checklist for MP3 is shown below. Use check marks or tally marks during discussion to keep track of responses.

Construct viable arguments and critique the reasoning of others (MP3)

Building mathematical thinkers for life

The Standards for Mathematical Practice equip learners with the necessary skills for lifelong problem solving and critical thinking.

By building deep understanding, adaptive reasoning, and procedural fluency, mathematics educators prepare students to engage confidently with complex problems both in the classroom and in everyday life.

Thoughtful implementation of these practice standards ensures that all learners develop a strong foundation in mathematics that extends beyond equations and into meaningful, real-world applications.

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