SMP #3 Construct Viable Arguments &
Critique the Reasoning of Others
The North Carolina Unpacked Standards provide a summary statement about the SMPs for each grade level:
Kindergarten: Mathematically proficient students in Kindergarten accurately use mathematical terms to construct arguments and engage in discussions about problem solving strategies. For example, while solving the task, “There are 8 books on the shelf. If you take some books off the shelf and there are now 3 left, how many books did you take off the shelf?” students will solve the task, and then be able to construct an accurate argument about why they subtracted 3 form 8 rather than adding 8 and 3. Further, Kindergarten students are expected to examine a variety of problem solving strategies and begin to recognize the reasonableness of them, as well as similarities and differences among them.
1st Grade: Mathematically proficient students in Grade 1 accurately use definitions and previously established answers to construct viable arguments about mathematics. For example, while solving the task, “There are 15 books on the shelf. If you take some books off the shelf and there are now 7 left, how many books did you take off the shelf?” students will use a variety of strategies to solve the task. After solving the class, Grade 1 students are expected to share problem solving strategies and discuss the reasonableness of their classmates’ strategies.
2nd Grade: Mathematically proficient students in Grade 2 accurately use definitions and previously established solutions to construct viable arguments about mathematics. In Grade 2 during discussions about problem solving strategies, students constructively critique the strategies and reasoning of their classmates. For example, while solving 74 + 18 – 37, students may use a variety of strategies, and after working on the task, can discuss and critique each others’ reasoning and strategies, citing similarities and differences between strategies.
3rd Grade: Mathematically proficient students in Grade 3 may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking. For example, when comparing the fractions 1/3 and 1/5, students may generate their own representation of both fractions and then critique each others’ reasoning in class, as they connect their arguments to the representations that they created.
4th Grade: Mathematically proficient students in Grade 4 construct arguments using concrete representations, such as objects, pictures, and drawings. They explain their thinking and make connections between models and equations. Students refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking through discussions and written responses.
5th Grade: Mathematically proficient students in Grade 5 construct arguments using representations, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking through both discussions or written response.
Thinkmath.edc.org has a discussion about SMP#3.
InsideMathematics has videos from 5th grade classrooms showing SMP #3 in practice.
Wisconsin has additional resources here.
Kindergarten: Mathematically proficient students in Kindergarten accurately use mathematical terms to construct arguments and engage in discussions about problem solving strategies. For example, while solving the task, “There are 8 books on the shelf. If you take some books off the shelf and there are now 3 left, how many books did you take off the shelf?” students will solve the task, and then be able to construct an accurate argument about why they subtracted 3 form 8 rather than adding 8 and 3. Further, Kindergarten students are expected to examine a variety of problem solving strategies and begin to recognize the reasonableness of them, as well as similarities and differences among them.
1st Grade: Mathematically proficient students in Grade 1 accurately use definitions and previously established answers to construct viable arguments about mathematics. For example, while solving the task, “There are 15 books on the shelf. If you take some books off the shelf and there are now 7 left, how many books did you take off the shelf?” students will use a variety of strategies to solve the task. After solving the class, Grade 1 students are expected to share problem solving strategies and discuss the reasonableness of their classmates’ strategies.
2nd Grade: Mathematically proficient students in Grade 2 accurately use definitions and previously established solutions to construct viable arguments about mathematics. In Grade 2 during discussions about problem solving strategies, students constructively critique the strategies and reasoning of their classmates. For example, while solving 74 + 18 – 37, students may use a variety of strategies, and after working on the task, can discuss and critique each others’ reasoning and strategies, citing similarities and differences between strategies.
3rd Grade: Mathematically proficient students in Grade 3 may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking. For example, when comparing the fractions 1/3 and 1/5, students may generate their own representation of both fractions and then critique each others’ reasoning in class, as they connect their arguments to the representations that they created.
4th Grade: Mathematically proficient students in Grade 4 construct arguments using concrete representations, such as objects, pictures, and drawings. They explain their thinking and make connections between models and equations. Students refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking through discussions and written responses.
5th Grade: Mathematically proficient students in Grade 5 construct arguments using representations, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking through both discussions or written response.
Thinkmath.edc.org has a discussion about SMP#3.
InsideMathematics has videos from 5th grade classrooms showing SMP #3 in practice.
Wisconsin has additional resources here.
Noristown Unified School District in Pennsylvania has published some outstanding newsletters which focus on each of the Standards for Mathematical Practice. SMP Newsletter #3 can be found here.
