The Standards for Mathematical Practice

Standards for Mathematical Practice such as those described in the CCSSM are deeply embedded in the fabric of Investigations. Every session of the curriculum calls on students to make sense of mathematics; to reason and use what they know; and to communicate their thinking. (Learn More about Investigations and the Math Practices.)

What follows are videos from Investigations classrooms that illustrate what it looks like when elementary teachers and students work on tasks that focus on important mathematics and require the use of the Math Practices, particularly Math Practices 1 and 6. Given that these are the “overarching habits of mind of a productive mathematical thinker,” these should be happening all of the time. (See the diagram on Structuring the Mathematical Practices, written by one of the CCSSM authors.)

Thoughts and questions specific to each video are included below. Questions that apply to all of the videos include:

  • What math ideas are students working on?
  • What evidence is there of Math Practice [1, 6]?
  • What’s the role of the task? Of the teacher? Of the student?

Math Practice 1: Make sense of problems and persevere in solving them.

This Practice talks about the importance of “explaining…the meaning of a problem and looking for entry points;” planning a way to solve a problem; monitoring the solution process; and double checking the solution to ensure that it makes sense. It describes how “younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem” and explains that students should understand and compare different approaches to the same problem.

Read the full text of MP1.

“Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.” (CCSSM, p. 7.)

K.OA.1-4, K.CC.4-5 All K videos.

What evidence do you see of Math Practice 1 in this clip? What is the teacher’s role? What evidence do you see that he is focused on Math Practice 1?

3.OA.2, 3.OA.5-6 All Gr. 3 videos.

What evidence do you see of Math Practice 1 in this clip? What is Ms. T’s role? How is she helping Nashon understand how his array shows 9 groups of 3 in a different way? How does Nashon use “concrete objects…to help conceptualize and solve” the problem?

Math Practice 6: Attend to precision.

This Practice talks about the importance of “communicating precisely to others” and of calculating “accurately and efficiently” with an appropriate “degree of precision.” It says that K-5 students should “give carefully formulated explanations to each other.”

Read the full text of MP6.

“Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.” (CCSSM, p. 7.)

 

1.OA.1, 1.OA.5-6, 1.NBT.1 All Gr. 1 videos.

What evidence do you see of Math Practice 6 in this clip? What is Ms. H’s role?

2.NBT.5 All Gr. 2 videos.

What evidence do you see of Math Practice 6 in this clip? In what ways is Sam calculating “accurately and efficiently?”

5.NBT.5, 5.OA.2, 5.NBT.1 All Gr. 5 videos.

What evidence do you see of Math Practice 6 in this clip? How does Olga’s “carefully formulated explanation” help Katie and Ruby compare their strategies to Olga’s?

As is evident in the clips above, the teacher, the student, and the curriculum each have a role to play in making a classroom that embodies the Standards for Mathematical Practice come to life.

The Curriculum presents a sequence of tasks designed to focus on particular content standards and to foster the use of the Math Practices. When looking at a particular task, questions to think about include:

    1. What kind of task is it? (e.g. a decontextualized computation problem, a problem set in a context, a problem with more than one answer)
    2. What math does the task focus on?
    3. Does the task suggest the use of one or more MPs? In what ways do the MPs support students’ mathematical thinking and understanding of the math content?

Teachers use their knowledge of the math, their students, and the curriculum to implement the curriculum. They facilitate the creation of an environment in which the Practices can flourish. Sometimes they purposefully focus on one particular Practice; other times, they capitalize on an opportunity that arises in the classroom. Teachers must be able to see and recognize the various Math Practices, and they must be “mathematically proficient.”

Students are the critical piece of the puzzle. Where they are with an idea, and how they approach a problem, affects the Math Practices that will be used. For example, a student who counts all to solve an addition problem is using different Practices than a student who is using a strategy that relies on numerical reasoning (e.g. 8+5=8+2+3).

The teacher, the student, and the task each have a role to play. Successful implementation of the Standards for Mathematical Practice relies on the interaction of these three factors. A task might be rich and complex, but if students aren’t ready for the ideas or it’s taught very procedurally, it won’t result in deep and varied use of the Practices. On the other hand, what can seem like a simple computation problem can blossom into a rich and interesting task, depending on how the teacher presents the problem and what students do with it.

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Learn more about Investigations and the Math Practices.