# 19.8: Mathematically productive instructional routines

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Mathematically Productive Instructional Routines (MPIRs) are high leverage instructional routines that focus on student ideas as central to the learning, make student thinking visible, and provide opportunities for mathematical discourse thereby allowing opportunities for students to make sense of mathematics in their own way. Consistently engaging students in these routines can change student’s dispositions about mathematics, support shifts in instructional practice, and deepen mathematical content knowledge and a growth mindset for both students and teachers.

MPIRs can be implemented with students from pre-school to college and are not tied to any curriculum. MPIRs take place at the beginning or end of a math lesson and are designed to take 10–15 minutes. It is beneficial to use routines daily, or multiple times a week. Mathematical routines can cover many different mathematical ideas, and can be used across a variety of concepts and topics.

While there are several different formats for these routines, all Mathematically Productive Instructional Routines share these common attributes:

They are routine. MPIRs are brief and used frequently. Students and teachers engage in these activities often enough that the routine itself is learned and can be engaged in quickly and meaningfully. The predictable structure creates a safe time and space for students to take risks and explore and share their ideas.

They are instructional. While classrooms also rely on routines designed to manage student behavior, transitions, and supplies, MPIRs are routines that focus on student learning. MPIRs provide an opportunity for students to share their mathematical ideas and make connections and deepen their understanding of math concepts as they listen and respond to other students. Routines also provide an opportunity for the teacher to formatively assess students.

They are mathematically productive. Prompts for each MPIR are carefully chosen to opportunities for students to enact the Standards for Mathematical Practice. Student discussions highlight central mathematical ideas. Students gain important insights and develop positive dispositions about engaging in mathematics through their participation in MPIRs..

Mathematically Productive Instructional Routines create a structure where teachers listen to, build on, and respond to student thinking. Using such routines frequently can support the development of a classroom culture in which sense-making is at the heart of all learning, and mistakes are expected, respected, and inspected.

## Number Talks

Number Talks are an example of a mathematically productive instructional routine that can support the development of a classroom culture in which students feel encouraged to share their thinking, and teachers become skilled at listening to their students’ thinking. This short mental mathematics routine can be used daily with any curricular materials to promote number fluency as well as develop conceptual understanding of numbers and operations.

In a number talk, students have the opportunity to share their thinking and learn from fellow students about multiple ways of using number relationships and structures, and visual models to perform mental computations. With number talks, teachers must listen to and represent student thinking, which not only provides them with information for determining next steps, but also deepens the teacher’s own understanding of mathematics. Number talks are the best pedagogical method for developing number sense and helping students see the flexible and conceptual nature of mathematics (Boaler, 2015).

In their recent book, Making Number Talks Matter, Number Talks pioneers and researchers Cathy Humphreys and Ruth Parker claim:

Number Talks help students become confident mathematical thinkers more effectively than any single instructional practice we have ever used.… With Number Talks, students start to believe in themselves mathematically. They become more willing to persevere when solving complex problems. They become more confident when they realize that they have ideas worth listening to. And when students feel this way, the culture of a class can be transformed.

Jo Boaler, Stanford University mathematics education professor, provides educators and parents with a 15-minute video about Number Talks that gives a full description of the practice and shares examples to help schools get started with Number Talks in every classroom.

#### References

• Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. John Wiley & Sons.
• Hiebert, J., & Morris, A. K. (2012). Teaching, rather than teachers, as a path toward improving classroom instruction. Journal of Teacher Education, 63(2), 92–102.
• Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In Instructional explanations in the disciplines (pp. 129–141). Springer US.
• Humphreys, C., & Parker, R. (2015). Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4–10. Stenhouse Publishers

## Games

Mathematics games may be used for extended learning time to support instruction and to help students meet the state standards. Some research has found that game-based learning is an effective way to enhance motivation and performance.

Choosing which game to play depends on the instructional goal and learning target. Games can be used both for instruction and practice. Games may also give students the opportunity to apply new learning. Games may not be appropriate in all situations, and are more effective if they are embedded in instruction and include debriefing and feedback. Also, games should be used as adjuncts and aids, not as stand-alone instruction.

## Technology

Mathematics games may be used for extended learning time to support instruction and to help students meet the state standards. Some research has found that game-based learning is an effective way to enhance motivation and performance.

Choosing which game to play depends on the instructional goal and learning target. Games can be used both for instruction and practice. Games may also give students the opportunity to apply new learning. Games may not be appropriate in all situations, and are more effective if they are embedded in instruction and include debriefing and feedback. Also, games should be used as adjuncts and aids, not as stand-alone instruction.

19.8: Mathematically productive instructional routines is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.