By Constance Hallemeier
Welcome to the final installment in our three-part series exploring the Standards for Mathematical Practice (MPs) in the secondary classroom! While the content standards specify what students must know, the MPs describe the ways of learning and the habits of mind that mathematically proficient students exhibit. The MPs emphasize process and expertise, cultivating mature mathematical thinkers.
Here, we focus on the practices central to problem-solving, structural thinking, and generalization:
- Make Sense of Problems and Persevere in Solving them (MP 1)
- Look for and Make Use of Structure (MP 7)
- Look for and Express Regularity in Repeated Reasoning (MP 8).
Standard 1: The Perseverance Tool
Make Sense of Problems and Persevere in Solving Them
This standard is the easy practice standard, assign students a lot of word problems, right? However, Mathematical Practice 1 requires students to do more than just word problems. It requires them to interpret and make meaning of a problem while looking for starting points. Proficient students plan a solution pathway instead of jumping immediately to a solution and continually ask themselves, "Does this make sense?". Whereas less proficient students just jump immediately to a solution. As they work, they monitor their progress and change their approach if necessary.
MP 1 in Action
Students who master Making Sense of Problems and Persevering in Solving Them can:
- monitor their progress and change their approach if necessary.
- relate current situations to concepts or skills previously learned.
- connect mathematical ideas to one another.
- understand various approaches to solutions.
- use multiple representations to develop meaning: visual diagrams, manipulatives, graphs, tables, three-dimensional models, equations.
- consider different solution pathways, both their own and those of other students, to identify and analyze correspondences among approaches.
Practical Classroom Strategies
Successful implementation requires tasks that require cognitive effort, where the pathway is not explicitly suggested, ensuring multiple entry points are available. Tasks must also require students to engage with the conceptual ideas underlying the procedures.
-
- Plan problem-solving opportunities: Allow students time to initiate a plan and use question prompts only as needed to assist. Productive struggle helps students test the plan, but the question prompts will or should guide them in the direction they need to go. “What do you notice/wonder?”, “Would you Rather?” and “Which One Doesn’t Belong?” questions are effective teaching strategies for engaging students and focusing their thinking on the desired math topic.
- Require students to defend their solutions: Continually ask students if their plans and solutions make sense. Teachers should consistently ask students to defend and justify their solution by comparing solution paths. Provide students with two different solutions and ask students to identify which one is most efficient. When a student asks, “Is this answer right?” do not just answer yes/no, ask the students to explain why they think it’s right or wrong.
- Encourage students to make connections to previous knowledge: Students have prior knowledge from earlier grade levels, outside experiences, and previous lessons; teachers need to capitalize on this information so students can make connections.
- Provide students with instructional routines they can rely on for problem-solving:
- The 3 Reads instructional routine is designed to develop MP 1 by deconstructing the process of reading mathematical situations, thereby creating a rule of thumb for sense-making in story problems. The 3 Reads strategy involves reading a math word problem three distinct times—first to grasp the context, second to identify the quantities, and third to clarify the question—to ensure deep comprehension before attempting to solve.
- The Stronger and Clearer Each Time instructional routine enhances student responses and gives conversation purpose. Students first think or write a response, then use structured pairing to clarify and refine it through discussion, and finally revise their original work. Subsequent drafts prompt students to incorporate specific details, new ideas/language, and improve precision, communication, expression, examples, and/or mathematical reasoning.
Standard 7: The Structure Tool
Look for and Make Use of Structure
Students just need to learn the algorithm for all the operations, and they will be fine, right? NO! Mathematical Practice 7 encourages students to look for the overall structure and patterns in mathematics, question how the numbers are related throughout a problem and use that relation to drive a solution. Students can see complicated things as a single object or as composed of several objects.
MP 7 in Action
Students who master looking for and making use of structure can:
- look closely to discern patterns.
- For instance, in the expression , they recognize the structure by seeing 14 as and 9 as .
- take a complex idea and identify, then use component parts to solve problems.
- recognize and identify structures from previous experiences and apply the knowledge to a new situation.
- step back for an overview and shift perspective.
- For example, viewing as 5 minus a positive number times a square, which allows them to immediately realize its value cannot be more than 5.
Practical Classroom Strategies
Task design must require students to look for the structure within mathematics to solve the problem (e.g., decomposing numbers by place value or working with properties). Tasks should ask students to take a complex idea and then identify and use the component parts.
- Chunk complicated math objects: Challenge students to compose or decompose geometric figures, identifying properties, and apply that information in identifying solution paths. Rework expressions into equivalent forms or alter the number's format to potentially reveal valuable insights.
- Expect students to explain the overall structure of the problem: To get to the big mathematical idea used to solve it, students can draw a picture, highlight and underline key information, create a table to organize the information, or sort the information in a different way to see connections.
- Provide students with instructional routines they can rely on for finding structure:
- The Contemplate then Calculate routine is designed to shift attention away from mindless calculations toward necessary structural interpretations, fostering MP 7. Students are encouraged to analyze the problem, find shortcuts based on math properties, and then calculate.
- The Connecting Representations routine also helps students think structurally by articulating the underlying mathematics that links different representations. Teachers can pose questions to help students see the connections: How is the graph related to the table? How is the ratio related to the drawing?
Standard 8: The Pattern Tool
Look for and Express Regularity in Repeated Reasoning
Give students a bunch of pattern blocks, and we have this math practice standard covered, right? Not quite, mathematical practice 8 means students see repeated calculations and look for generalizations and shortcuts. They understand the broader application of patterns and see the structure in similar situations. As they work to solve a problem, they continually evaluate the reasonableness of their intermediate results.
MP 8 in Action
Students who master looking for and expressing regularity in repeated reasoning can:
- notice when calculations are repeated and look for both general methods and shortcuts. For example, by noticing the regularity in how terms sum to zero when expanding expressions like (X - 1) (X + 1) , they might be led to the general formula for the sum of a geometric series.
- maintain oversight of the entire process while attending to the details.
- understand the broader application of patterns and use the structure in similar situations.
- continually evaluate the reasonableness of their answers.
Practical Classroom Strategies
Tasks must present several opportunities to reveal patterns or repetition in thinking so students can make a generalization or rule. Tasks should require students to see relationships to develop a mathematical rule and connect to previous tasks to extend learning.
- Ask Questions: Ask students what math relationships or patterns can be used to assist in making sense of the problem. For example, when students are learning the exponent rules for the first time, have them look for patterns in the operations they use. Likewise, use this same strategy to help students see the relationship between logarithms and exponents.
- Make Predictions: Ask for predictions about solutions at midpoints throughout the solution process. Encourage students to share their predictions and thinking to gauge understanding of the mathematical rule they are developing.
- Require multiple attempts for solutions: Question students to help them create generalizations based on repetition in their thinking and procedures. Have different groups try the process with various problems and compare the similarities.
- Provide students with instructional routines they can rely on for finding repetition: The Recognizing Repetition routine supports the road to generalizing (MP8) by having students attend to repetition in their counting, calculating, and constructing processes, leveraging repeated reasoning to support abstract generalizations.
Building the Power of Process Together
For these standards (MP 1, MP 7, and MP 8) to be truly implemented, the entire educational system must align planning, delivery, and professional development around the core ideas of the practices.
These questions can guide implementation for teachers, instructional coaches, and administrators:
MP 1 (Perseverance) Questions
How are we ensuring tasks require cognitive effort and provide multiple entry points, rather than simple rote procedures? How are we consistently requiring students to defend and justify their solutions by comparing solution paths?
MP 7 (Structure) Questions
How are we ensuring tasks require students to look for and explain the underlying structure of mathematics (properties, composition of objects)? How are we supporting teachers in using routines like Contemplate then Calculate to shift student attention toward structural interpretations?
MP 8 (Regularity) Questions
How are we presenting several opportunities within a task sequence to reveal patterns or repetition in thinking, allowing students to generalize? How are we building the habit in students to continually evaluate the reasonableness of their intermediate results?
Secondary math teachers should prioritize the Mathematical Practice Standards (MPS) 1, 7, and 8 to deepen students' mathematical understanding and critical thinking. By focusing on MPS 1: Make Sense of Problems and Persevere in Solving Them, teachers empower students to fluently interpret problems, create a plan of attack, and understand both the process and the context of a problem. Emphasizing MPS 7: Look For and Make Use of Structure guides students to identify structures and underlying relationships within complex expressions or problems, allowing them to simplify and solve tasks efficiently. Finally, promoting MPS 8: Look For and Express Regularity in Repeated Reasoning, encourages students to generalize from repeated examples, enabling them to discover, articulate, and apply general mathematical rules or formulas. Together, these practices equip students with the sophisticated habits of mind necessary for success in higher-level mathematics and real-world problem-solving.
How will you adjust your lesson plans this week to prioritize structures and perseverance in your classroom?
References
- Implementing standards for mathematical practices. Achieve The Core. (2012, May). https://achievethecore.org/peersandpedagogy/wp-content/uploads/2016/06/Implementing-Standards-for-Mathematical-Practices-Updated-2016.pdf
- Massachusetts Mathematics Curriculum Framework — 2017. Massachusetts Department of Elementary and Secondary Education. (2017). https://www.doe.mass.edu/frameworks/math/2017-06.pdf
- Mathematical practice standards: Inside mathematics. The University of Texas Dana Center. (n.d.). https://www.insidemathematics.org/common-core-resources/mathematical-practice-standards
- National Governors Association Center for Best Practices and Council of Chief State School Officers. (2010). Standards for mathematical practice | common core state standards initiative. https://www.thecorestandards.org/Math/Practice/
About The Author: