Strategies for Empowering Students in Elementary Mathematics

By Dr. Sherri Lorton

Welcome to the final installment of our series on the Standards for Mathematical Practice (MPs) in elementary classrooms! Our journey began with the Communicating Reasoning bundle (MP3 and MP6) and progressed through the Modeling and Data Analysis bundle (MP2, MP4, and MP5). This conclusion focuses on the Problem-Solving bundle—MP1, MP7, and MP8—which are the essential strategies and habits of mind that empower students to navigate challenges and experience genuine mathematical discoveries. Using appropriate tools strategically, MP5, also supports the problem-solving process and is often used alongside these other three practices.

As educators, we must intentionally foster opportunities for students to develop and apply problem-solving strategies to real-life situations. The goal is to equip students with a robust mathematical toolkit that enables them to make productive use of their knowledge when solving complex problems.

Getting to Know (Making Sense of) the Problem-Solving Standards

The Problem-Solving bundle (MP1, MP7, and MP8) describes how students engage in the struggle and discovery inherent in mathematics.

MP1 (The Perseverance): Make sense of problems and persevere in solving them.

This is the foundation for all deep mathematical engagement. This practice is fundamentally tied to the rigor of application, as it involves applying mathematics to solve complex, real-world problems. MP1 ensures students see mistakes not as failures, but as pathways and opportunities to work through and achieve breakthroughs.

• Finding Entry Points: Instead of immediately searching for "key words," young students look for entry points to begin solving the problem. They may plan and choose a solution pathway before diving into calculations. For example, in kindergarten, students might use concrete objects or pictures to show the actions of a word problem, such as counting out and joining two sets.
• Monitoring Progress and Changing Course: As they work, students continually ask themselves, "Does this make sense?". If they find their initial strategy doesn't work or makes no sense, they demonstrate perseverance by checking their approach, monitoring their progress, and seeking another pathway that does. A second-grader understands that if they are adding two (positive) whole numbers, if their answer is smaller than either addend, then they should rework the problem. 
• Using Simpler Forms: A vital strategy for gaining insight into a solution is to consider simpler forms of the original problem. For example, a third-grade student might replace multi-digit numbers in a word problem with single-digit numbers to better appreciate the quantities and their relationships.
• Checking the Answer: Once a student has a solution, MP1 requires them to check their answer using a different approach. For example, a fourth-grader understands that after finding the answer using division, they can check their work with multiplication.

MP7: Look for and make use of structure, and MP8: Look for and express regularity in repeated reasoning

These two practices work together, often leading to powerful moments of insight that some describe as “Lightbulb” Moments.

MP7 (The Insight) focuses on recognizing and using underlying mathematical patterns and forms to simplify complex procedures. Elementary students look for and use structures such as place value, the properties of operations, and the attributes of shapes. Younger students recognize that adding 1 always yields the next counting number, thereby identifying the basic structure of whole numbers. An older student might apply the place-value structure and the Distributive Property to find the product. Students also look for structure when they view expressions as single objects to observe and interpret.

MP8 (The Generalization) follows repeated action and focuses on generalizing a rule or shortcut based on noticed regularities. Students look for regularities as they solve multiple related problems, and then they identify and describe these patterns. For an input-output machine, students might notice a pattern in how the product changes when the first factor is increased by 1, and observe that the second number changes by 5. They can then express this regularity by saying, "When you change one factor by one, the product increases by the other factor". This practice often leads students to formulate conjectures about what they notice.

Similarities and Key Differences

Mathematical Practices MP1, MP7, and MP8 all support students in becoming flexible, confident problem solvers, but they emphasize different aspects of reasoning.

• MP1 focuses on perseverance: Students make sense of problems, choose strategies, and stick with them even when challenges arise.
• MP7 highlights the insight: These structures, such as patterns in numbers, properties, or representations, make problems easier to understand.
• MP8 directs students to make generalizations: Identifying what happens over and over allows students to reason and/or make shortcuts.

Together, these practices help students approach problems thoughtfully, look for underlying patterns, and develop efficient strategies.

Problem-Solving Practices across K-5

The central purpose of teaching problem-solving strategies (MP1, MP7, MP8) is to empower students to tackle challenges and apply those intellectual tools to real-life situations. This application begins in the earliest grades and becomes more sophisticated across the K–5 trajectory.

In kindergarten and 1^st grade, problem-solving focuses on concrete strategies and on establishing foundational structures to support the major work that revolves around place-value foundations, addition, and subtraction.

• Task: A class is sharing snacks. We have 8 apples and need some carrots to have 15 snacks total. How many carrots do we need?
• MP1 in Action: Students persevere when they look for an entry point. If counting on from 8 to 15 is not yet in their toolbox, they may choose to count all 15 objects.
• MP7 in Action: Students may apply the Make Ten strategy (structure). They recognize the relationship between 8 and 10, noticing that 8 needs 2 more to make 10.
• MP8 in Action: They may look for the structure of 7 and mentally decompose 7 into 5 and 2, then solve 8 + 2 + 5. This ability to decompose numbers is a generalization that helps them solve abstract arithmetic problems.

In Grades 2 and 3, students extend their number understanding up to 1,000 and gain an understanding of multiplication and division. Problems require manipulating larger quantities and identifying the properties of operations as structures for calculation.

• Task: A theater sells 7 tickets per row in 8 rows. How many tickets did they sell?
• MP1 in Action: The student checks their answer by trying a more straightforward problem pathway. They might choose to multiply by 4 twice: (7 x 4) + (7 x 4) or check the inverse relationship (56 ÷ 8 = 7). Perseverance involves checking solutions using a different approach.
• MP7 in Action: A student uses the known structure of multiplication (the Distributive Property) to calculate the product. They see 7 x 8 as the sum of two easier products, such as 4 x 8 and 3 x 8. They are observing how the mathematical structure allows them to break down a complex problem into familiar parts.
• MP8 in Action: Students recognize that the structures of skip-counting or repeated addition can help them solve problems of equal-sized groups.

In Grades 4 and 5, the major work shifts to fraction operations, multi-digit arithmetic, and scaling. Problem solving requires an abstract understanding, as models shift from concrete objects to abstract diagrams such as number lines or area models for fractions.

• Task: You have a rectangular kitchen floor measuring 12 feet by 8 feet. You want to tile the floor using square tiles, and you will buy one type of tile. There are three different sizes of square tiles available:
  • Small Tile: ½ foot by ½ foot
  • Medium Tile: ¾ foot by ¾ foot
  • Large Tile: 1 foot by 1 foot
    
    
• Determine the total number of tiles needed to cover the entire kitchen floor for using:
  1. all ½ foot by ½ foot tiles,
  2. all ¾ foot by ¾ foot tiles, and
  3. all 1 foot by 1 foot tiles.

MP1 in Action: Students explain the problem's meaning to themselves, look for entry points, and choose a solution pathway. If a path doesn't make sense, they look for another one.

• MP7 in Action: Students should look for a pattern or structure by recognizing that the calculation for each tile size is mathematically structured as (Total Length / Tile Length) x (Total Width / Tile Width). For the Small Tile (1/2 foot), the student might see the division 12 ÷ ½ and realize this is structurally the same as 12 x 2 (the reciprocal). When reviewing their work, they could explain how the pattern or structure was used to solve the problem. For example, they could explain that dividing by ½ is equivalent to multiplying by 2 because two half-foot pieces fit into every whole foot.
• MP8 in Action: After the repeated calculations, the student can look for both general methods and shortcuts. They might realize that the general method for finding the number of tiles is consistent across all three problems, regardless of whether the side length is a whole number or a fraction: you divide the floor dimension by the tile side length and then multiply the two results. By finding this general relationship, they are using repeated reasoning to confirm the mathematical efficiency of their solution pathway.

Building Problem-Solving Together

To achieve the goal of preparing K-5 students to apply these strategies to real-life scenarios, teachers, coaches, and administrators must commit to providing tasks that embody the necessary rigor.

Teachers

How often do you implement tasks that promote reasoning and problem-solving, allowing for multiple entry points and varied solution strategies (MP1)? When presenting a complex calculation (e.g., a multi-digit operation), do you allow students first to apply structures and properties (MP7) to invent a strategy before moving directly to a standard algorithm?

Coaches

Are students primarily seeking rules and formulas, or are they formulating their own conjectures by looking for regularity in repeated calculations (MP8)? Look for evidence that instruction supports productive struggle in learning mathematics, which is essential for developing perseverance (MP1).

Administrators

Are your instructional materials designed to provide access to rich instructional tasks that emphasize connections to the real world and require complex reasoning (Application rigor)? Are your professional learning goals designed around enabling teachers to cultivate the "ways of knowing" and "habits of mind" described by the MPs?

By intentionally cultivating the Problem-Solving strategies, we ensure that students not only find answers but also gain confidence in their mathematical authority, equipping them with the analytical thinking and problem-solving tools that are fundamental for success in middle school, high school algebra, and diverse professional endeavors.

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