# Mathematics Teaching Practices

#### A Professional Development Approach

###### Overview: The Skill Set of Teacher Noticing & Responsive Teaching

The Mathematics Teaching Practices section is framed around the skill set of Teacher Noticing and responsive teaching (Jacobs, et al., 2010, Richards & Robertson, 2016). Teacher Noticing is comprised of three skills:

• Attending to students’ thinking (listening, observing)

• Interpreting students’ work based on progressions in that thinking, and

• Deciding how to respond both in the moment and in one’s future planning

A lesson plan in a curriculum guide directs you in what to do. But how you look and listen to the students shapes how any lesson unfolds. A curriculum guide may provide hints as to what to listen and look for, but it is up to the teacher to notice those details. Attending to the details of students’ thinking allows you to assess in the midst of instruction where the students are at both conceptually and procedurally. This requires listening to and observing (attending) students’ actions on paper, with their use of manipulatives, or their verbal explanations. Is the explanation based on a nugget of a good idea? Is the strategy a good one but may have a computational error? Is the thinking mathematically sound but at an emergent or advanced level of understanding? What properties of operations might implicitly be understood by the students that you might draw out? These are the skills of interpreting students’ thinking based on teacher knowledge of the mathematical content and the progressions of student thinking that students pass through. Deciding how to respond both in-the-moment as well as in one’s planning for the next lesson(s) reflects your adjustments necessary to guide students forward towards comprehending and executing the targeted mathematics.

This noticing skill set empowers you, the teacher, to guide students as they navigate the pathway from partial understandings to fully formed mathematical ideas. When to elicit, when to tell, which questions to ask next to have students elaborate, clarify, justify, and/or compare and contrast their thinking, which representation might best capture the mathematical idea, all these in-the-moment decisions cannot be scripted in a text book. These are the thinking-on-your-feet skills that are learned, cultivated, and supported over time.

Such teaching environments are language rich. This system of instruction (Heibert, et al., 1997; Posey, Cast, Inc.) has a clear organized set of mathematical benchmarks as its goals. But by attending to, and interpreting students’ thinking, one’s decision-making on how to respond allows the lesson and unit plans to be responsive and fluid. The goals are clear. The pathways to those goals are adaptive based on where your students are in their level of understanding (Vilson, 2017). Responsive teaching is responsive because it takes up and pursues the substance of student thinking (Richards & Robertson, 2016).

Figure One: System of Instruction - Adapted from Hiebert, et al, 1997

The Common Core Standards for Mathematics (CCSSO, 2010) identified the eight Standards for Mathematical Practices that students are expected to develop over their K-12 learning period.  NCTM’s Principles to Actions (2014) and Catalyzing Change series, (2018, 2020a, 2020b)  lays out eight instructional practices grounded in years of research on teaching and learning. The 2022 Minnesota Academic Standards for Mathematics are anchored in these two documents.

Figure 2 lists the Standards for Mathematical Practice (what students do) and the Mathematics Teaching Practices (what teachers do). Students become adept at these practices from experiencing successive classrooms where these instructional practices that research has determined are actively used. Students need to be explicitly supported in learning these practices over time. Attending, interpreting, and deciding how to respond is the teacher noticing skill set through which the eight instructional practices are enacted. The focus of instruction is centered on students’ thinking as the basis for planning, assessment, and instruction.

###### Standards for Mathematical Practice  ←→  Mathematics Teaching Practice
 Standards for Mathematical Practices (What students do) Eight Mathematics Teaching Practices (What teachers do) MP1: Make sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure. MP8: Look for and express regularity in repeated reasoning 1. Establish mathematical goals to focus learning2. Implement tasks that promote reasoning and problem solving3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse5. Pose purposeful questions6. Build fluency from conceptual understanding7. Support productive struggle in learning mathematics 8. Elicit and use student thinking

These eight practices are organized here into two clusters and cross-referenced to the Standards for Mathematical Practice. Two additional cluster sections follow; one on language development, the other on differentiation within the mathematical classroom. Click on the links to access more detailed professional development materials exploring each of the practices.

Cluster One: Instructional Practices – Planning & Assessment (Interpreting, Responding)

• Establish mathematical goals to focus learning
• Implement tasks that promote reasoning and problem solving
• Build fluency from conceptual understanding (MP2: Reason abstractly and quantitatively; MP6: Attend to precision. MP7: Look for and make use of structure.)

Cluster Two: Instructional Practices – Instruction (Attending, Interpreting, Responding)

• Facilitate meaningful mathematical discourse (MP3: Construct viable arguments and critique the reasoning of others)
• Pose purposeful questions
• Elicit and use student thinking
• Use and connect mathematical representations (MP4: Model with mathematics; MP5: Use appropriate tools strategically.)
• Support productive struggle in learning mathematics (MP1: Make sense of problems and persevere in solving them; MP5: Use appropriate tools strategically.)

Cluster Three: Instructional Practices – Language Development (Access & equity)

• Developing the classroom norms and conversational skill sets for students to communicate their ideas. (MP3: Construct viable arguments and critique the reasoning of others)
• Unpacking a lesson before students start working
• Academic Language development: Content vocabulary, language functions, grammar syntax
• A definition through a culturally sustainable lens
• For all students
• For Multilingual/English Language Learners

A culturally sustainable definition of Academic Language:  Academic language, at its core, is the language students need to organize and convey their thinking to the audience with whom they are interacting. Access and acceptance within a specific community vary and a lack of understanding of a community’s language patterns can be used to exclude participation within that community. Assisting students to gain the perspective and skill set of understanding the societal language patterns of different communities is to enable them to gain entry to a wider range of societal circles of power and therefore position themselves to affect change. We acknowledge the tension between preparing teachers and their students for the language demands of academic and professional work while also respecting and honoring varying linguistic identities – J. Brickwedde & P Egan

Cluster Four: Instructional Practices – Differentiation (Access & equity)

• Access points for all students (MP1: Make sense of problems and persevere in solving them)
• Supportive interventions and extensions
• Developing student autonomy
• Planning with a vertical progression of the standards and benchmarks in mind

References

Heibert, J., Carpenter, T.P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H. Olivier, A., and Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. pp. 29-41. Portsmouth, NH.: Heinemann Press.

Jacobs, V.R., Lamb, L.L.C., Phillipps, R.A. (2010) Professional Noticing of Children's Mathematical Thinking. Journal for Research in Mathematics Education. 41 (2), 169-202.

National Council of Teachers of Mathematics. (2020). Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations. Eds. Huniker, D. Yeh, C. & Marshal A.M. National Council of Teachers of Mathematics: Reston, VA.

National Council of Teachers of Mathematics. (2020). Catalyzing Change in Middle School Mathematics: Initiating Critical Conversation.  Eds. Bush, S.B., Jackson, C., Roy, G.J. & Milou, E., National Council of Teachers of Mathematics: Reston, VA.

National Council of Teachers of Mathematics. (2018). Catalyzing Change in High School Mathematics: Initiating Critical Conversation.  Eds. Graham, K., Burrill, G., & Curtis. J., National Council of Teachers of Mathematics: Reston, VA.

National Council of Teachers of Mathematics. (2014). Principles to actions. National Council of Teachers of Mathematics: Reston, VA.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.

Posey, A. Universal Design for Learning (UDL): A Teacher’s Guide. Retrieved from: https://www.understood.org/en/articles/understanding-universal-design-for-learning

Richards, J. and Richardson, A.D., (2015). A Review of the Research on Responsive Teaching. In A. D. Robertson, R. E. Scherr, and D. Hammer (Eds.) Science and Mathematics Responsive Teaching in Science and Mathematics (pp.36-55), Routledge, London.