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In today’s world students who are truly successful in mathematics can not only solve problems, but can explain:
 what each problem is asking,
 how they will solve the problem,
 why their solution plan makes sense, and
 how they know their answer is correct.
In other words, students must be able to think metacognitively about each and every problem and explain their thinking.
The Standards for Mathematical Practice present a helpful framework for this complex and important work. That’s why for eight weeks in April and May we are providing you with a metacognitive study of the mathematical practices—what this thinking looks like at each grade level, a stepbystep approach to teaching each practice, and a reflection guide to support students as they “think about their thinking.”
This week we are focused on…
STANDARD FOR MATHEMATICAL PRACTICE 3:
Construct viable arguments and critique the reasoning of others.
What This Thinking Looks Like at Each Grade Level
Because the 8 Standards for Mathematical Practice are the same for all grade levels, they are not specific enough to provide much practical guidance for classroom implementation. So, we’ve provided the standard and its learning progression for every grade from PreK through High School to better understand how the thinking builds in tandem with students’ cognitive development and the demands of mathematical content, as shown in the example below.
A StepByStep Approach to Teaching Practice 3
Fostering metacognition requires a balance of explicit instruction, teacher modeling, studentcentered exploration, and responsive coaching that helps students first learn the kinds of questions and thought processes they can apply, and then grow to use them on their own. These metacognitive skills come naturally to some students but not to others. Teachers must play an active role in teaching them and helping students own their mathematical learning.
Use these two processes to teach, model, and guide students in mastering Practice 3.
Process to construct viable arguments

 Explain what the problem is asking you to do with evidence.
 Explain your plan to solve the problem and your reasons.
 Explain your answer and why it is correct.
 Explain your thinking in both writing and speaking.
Process to critique the reasoning of others

 Ask clarifying questions as someone explains what the problem is asking them to do.
 Ask clarifying questions as someone explains their plan to solve the problem.
 Ask clarifying questions as someone explains their answer and how they solved the problem.
 Cite similarities and differences between strategies for how the problem could be solved.
 Locate the error in an incorrect answer and explain why with evidence.
A Reflection Guide to Support Students as They “Think About Their Thinking”
As we said before, mathematics is much more than just solving the problems and checking to see if the answer is correct. It also involves each student’s understanding of what they are learning, how they can learn it, and what they have learned about their mathematical thinking along the way.
When you’re focusing on Practice 3, teach your students this routine and then encourage them to use it for each and every math problem they solve. In doing so, you will support them in thinking about, reflecting on, and sharing their thoughts as the develop ownership of their mathematics.
Foster Ownership in Mathematics
Owning their mathematics won’t be automatic. On the contrary—most students need a great deal of explicit instruction, modeling, and coaching before they develop the metacognitive habits that allow them to take ownership of their learning.
Click here to download a resource with the processes and reflection for the Standard for Mathematical Practice 3: Critique viable arguments and critique the reasoning of others. The more students are able to understand their own learning process and articulate it, the easier it will be for them to improve their conceptual understanding, procedural skill and fluency, and applications. In other words, students will own their mathematics with metacognition.
The Learning Brief
In this article you learned…
 What the metacognitive thinking looks like at each grade level for Mathematical Practice Standard 3.
 A stepbystep approach to teaching Mathematical Practice Standard 3.
 A reflection guide to support students as they “think about their thinking” around Mathematical Practice Standard 3.
Can you imagine building an environment full of motivated, engaged, and eager students who own their learning?
We can.