Effective Mathematics Teaching Practices

NCTM's first effective teaching practice is to establish goals to focus learning. The most poignant aspect of this teaching practice is that mathematics teachers must be more focused on meaningful shifts in student understanding rather than just what students are expected to be doing. This is the difference between a learning goal and a performance goal. Compare the following goal statements:

Goal A: Students will recognize that the area of the square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on the legs and will conjecture that c²=a²+b².
Goal B: Students will be able to (SWBAT) use the Pythagorean Theorem (c²=a²+b²) to solve a series of missing value problems.

Goal A is a learning goal, as it describes the reasoning and understanding we want students to learn as the result of instruction. Goal B is a performance goal, as it falls short of describing what learning is being done and instead essentially says, “Students will solve problems using a theorem.” While part of the lesson activity may include solving a series of problems, our goals should focus on the learning to be developed, not the tasks to be performed. Doing so sets up everything for the rest of the lesson: the tasks that are chosen, the representations students use, questions to be asked, the focus of classroom discussion, and assessment strategies.

The Practice in Action

One of the ways NCTM (2014, p. 16) summarizes the practice of establish mathematics goals to focus learning is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Establishing clear goals that articulate the mathematics that students are learning as a result of instruction in a lesson, over a series of lessons, or throughout a unit.
• Identifying how the goals fit within a mathematics learning progression.
• Discussing and referring to the mathematical purpose and goal of a lesson during instruction to ensure that students understand how the current work contributes to their learning.
• Using the mathematics goals to guide lesson planning and reflection and to make in-the-moment decisions during instruction.

What are students doing?

• Engaging in discussions of the mathematical purpose and goals related to their current work in the mathematics classroom (e.g., What are we learning? Why are we learning it?)
• Using the learning goals to stay focused on their progress in improving their understanding of mathematics content and proficiency in using mathematical practices.
• Connecting their current work with the mathematics that they studied previously and seeing where the mathematics is going.
• Assessing and monitoring their own understanding and progress toward the mathematics learning goals.

Most of NCTM’s attention to task selection and implementation focuses on the cognitive demand of mathematical tasks (Smith & Stein, 1998). Tasks with low cognitive demand are those that simply require memorization or ask students to perform a procedure without connecting that procedure to something that reveals deeper conceptual understanding. Tasks with high cognitive demand are those that don’t give a procedure to follow, thus requiring students to develop their own, or a given procedure includes requirements to connect the procedure to conceptual understanding.

Research has shown us that students in the United States either (a) lack access to high cognitive demand tasks (cite Teaching Gap, QUASAR) and (b) when given high cognitive demand tasks, teachers often lower the demand of tasks through the use of hints and unnecessary support (cite QUASAR, etc.). While students aren’t expected to only engage with high-demand tasks, regular opportunities to engage in high-demand tasks is associated with stronger achievement in mathematics (Stein & Lane, 1996). This requires teachers to have a well-trained eye for selecting materials. Online resources such as lesson and task repositories give teachers many sources to choose from, but a poor choice of task may not align well with standards, interrupt the learning progression and design of curricular units, and lead to less learning overall (Larson, 2016).

The Practice in Action

One of the ways NCTM (2014, p. 24) summarizes the practice of implement tasks that promote reasoning and problem solving is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Motivating students’ learning of mathematics through opportunities for exploring and solving problems that build on and extend their current mathematical understanding.
• Selecting tasks that provide multiple entry points through the use of varied tools and representations.
• Posing tasks on a regular basis that require a high level of cognitive demand.
• Supporting students in exploring tasks without taking over student thinking.
• Encouraging students to use varied approaches and strategies to make sense of and solve tasks.

What are students doing?

• Persevering in exploring and reasoning through tasks.
• Taking responsibility for making sense of tasks by drawing on and making connections with their prior understanding and ideas.
• Using tools and representations as needed to support their thinking and problem solving.
• Accepting and expecting that their classmates will use a variety of solution approaches and that they will discuss and justify their strategies to one another.

Most of NCTM’s attention to the practice use and connect mathematical representations focuses on helping students make connections between contextual, visual, verbal, physical, and symbolic representational forms. By seeing and representing mathematics in different forms, students are more able to connect and make sense of mathematical ideas, and offer opportunities for understanding and communicating for all students, particularly those who may struggle due to learning difficulties or because the language used in the classroom is not their first language. The ability to flexibly move between different representations is also important. A quality curriculum should provide students purposeful and regular opportunities to move between representations. Some of these moves could be considered a part of mathematical modeling, such as taking contextual forms and producing a graph, table, or equation, but moving in the other direction—such as starting with a table or graph and conjecturing about possible details of a context—are equally important.

The Practice in Action

One of the ways NCTM (2014, p. 29) summarizes the practice of use and connect mathematical representations is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Selecting tasks that allow students to decide which representations to use in making sense of the problems.
• Allocating substantial instructional time for students to use, discuss, and make connections among representations.
• Introducing forms of representations that can be useful to students.
• Asking students to make math drawings or use other visual supports to explain and justify their reasoning.
• Focusing students’ attention on the structure or essential features of mathematical ideas that appear, regardless of the representation.
• Designing ways to elicit and assess students’ abilities to use representations meaningfully to solve problems.

What are students doing?

• Using multiple forms of representations to make sense of and understand mathematics.
• Describing and justifying their mathematical understanding and reasoning with drawings, diagrams, and other representations.
• Making choices about which forms of representations to use as tools for solving problems.
• Sketching diagrams to make sense of problem situations.
• Contextualizing mathematical ideas by connecting them to real-world situations.
• Considering the advantages or suitability of using various representations when solving problems.

NCTM’s effective mathematics teaching practice facilitate meaningful mathematical discourse has come to be seen as somewhat of a “meta-practice” that includes several others, such as pose purposeful questions, use and connect mathematical representations, and elicit and use evidence of student thinking. All of these practices come together to shape the way thinking is supported and communicated in a mathematics classroom.

The most widely used guide for facilitating discourse in mathematics classrooms is 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein. (Yes, this gets a bit confusing, as this one NCTM effective teaching practice leans heavily on the work of a book called 5 Practices, which maybe is better thought of as an instructional model for teaching a lesson.) The five practices are (pp 9-10):

1. Anticipating likely student responses to challenging mathematical tasks and questions to ask to students to produce them;
2. Monitoring students’ actual responses to the tasks (while students work on the tasks in pairs or small groups);
3. Selecting particular students to present their mathematical work during the whole-class discussion;
4. Sequencing the student responses that will be displayed in a specific order; and
5. Connecting different students’ responses and connecting the responses to key mathematical ideas.

Whether or not a teacher is following the 5 Practices instructional model or not, the goal is to shift teaching away from the teacher being the dominant speaker and information authority and to build a shared culture where students lead conversations themselves and the teacher is there to probe into thinking more deeply, connect students' ideas, and sharpen the students' collective understanding.

The Practice in Action

One of the ways NCTM (2014, p. 35) summarizes the practice of facilitate meaningful mathematical discourse is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Engaging students in purposeful sharing of mathematical ideas, reasoning, and approaches, using varied representations.
• Selecting and sequencing student approaches and solution strategies for whole-class analysis and discussion.
• Facilitating discourse among students by positioning them as authors of ideas, who explain and defend their approaches.
• Ensuring progress toward mathematical goals by making explicit connections to student approaches and reasoning.

What are students doing?

• Presenting and explaining ideas, reasoning, and representations to one another in pair, small-group, and whole-class discourse.
• Listening carefully to and critiquing the reasoning of peers, using examples to support or counterexamples to refute arguments.
• Seeking to understand the approaches used by peers by asking clarifying questions, trying out others’ strategies, and describing the approaches used by others.
• Identifying how different approaches to solving a task are the same and how they are different.

Questioning is one of the most difficult teaching skills and one where our powers of improvisation regularly fail us. Mathematics teachers too often fall into the traditional routine of initiating a student response with a question, getting a simple response from a student, and then evaluating the student’s response. This pattern is so ubiquitous it’s simply known as the I-R-E pattern (Mehan, 1979), and in many mathematics classrooms it sounds like this:

Teacher: Devin, what did you get for Question 17?
Devin: Is it x = 27.5?
Teacher: That’s correct.

One quick way to notice such patterns is that students will rarely say a second sentence in response to a teacher. Posing purposeful questions in the mathematics classroom must go beyond simple recall questions and dead-end questioning patterns. Instead of simply funneling students towards reaching a predetermined conclusion, a teacher should ask focusing questions that press students to communicate their thinking completely and clearly and make students link their ideas to those of classmates or to previously learned material (Boston, Canela, & Dixon, 2019; Herbel-Eisenmann & Breyfogle, 2005). Shifting to become a better questioner is not easy, and will likely require careful planning of some questions in advance and reflection to see if those questions are achieving their desired effects.

The Practice in Action

One of the ways NCTM (2014, p. 41) summarizes the practice of pose purposeful questions is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Advancing student understanding by asking questions that build on, but do not take over or funnel, student thinking.
• Making certain to ask questions that go beyond gathering information to probing thinking and requiring explanation and justification.
• Asking intentional questions that make the mathematics more visible and accessible for student examination and discussion.
• Allowing sufficient wait time so that more students can formulate and offer responses.

What are students doing?

• Expecting to be asked to explain, clarify, and elaborate on their thinking.
• Thinking carefully about how to present their responses to questions clearly, without rushing to respond quickly.
• Reflecting on and justifying their reasoning, not simply providing answers.
• Listening to, commenting on, and questioning the contributions of their classmates.

Resources for Building Fluency from Understanding

• Book: Figuring Out Fluency in Mathematics (2021) by Jennifer Bay-Williams and John SanGiovanni 
• Book: Catalyzing Change in Middle School Mathematics (2020) from NCTM (p. 61)
• US ED Report: Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008) from the US Department of Education
• Online Book: Nix the Tricks: A Guide to Avoiding Shortcuts That Cut Out Math Concept Development
• Book (digital, free): Mathematical Understanding: An Introduction in How Students Learn: Mathematics in the Classroom (2005) by Karen C. Fuson, Mindy Kalchman, and John D. Bran
• Podcast: Room to Grow - Build Procedural Fluency from Conceptual Understanding [44:22]
• Podcast: Room to Grow - Balancing Conceptual Understanding and Procedural Fluency [41:16] 
• Podcast (with Transcript): Rounding Up - Productive Ways to Build Fluency With Basic Facts [25:40] 
• Video Library: Progression of K-5 Math Thinking 
• Instructional Routine: Contemplate Then Calculate 
• Tasks: Relational Thinking Prompts
• NCTM Journal (members only): 13 Rules That Expire (2014) by Karen Karp, Sarah Bush, and Barbara Dougherty

Productive struggle is as much a goal of NCTM’s Effective Mathematics Teaching Practices as it is a practice itself. Struggle in mathematics is easy to find, but what makes it productive is that feeling of making sense of things, of seeing relationships between quantities or shapes, and making progress towards a strategy that may yield a viable solution, and, more importantly, a better understanding of the underlying mathematics. Letting students find this place of productive struggle, and to stay there for a while, often takes more patience and restraint than teachers are accustomed to having. This is why the other Effective Teaching Practices are so important—teachers are unlikely to position students for engaging in productive struggle unless they’re focused on the right goals, they’ve chosen the right tasks, they facilitate meaningful discourse, they ask purposeful questions, they encourage a range of representations, and they strategically make use of evidence of student thinking. When the teacher establishes the right conditions for student learning, it becomes far more likely that students will persist rather than give up, take risks without fear, and persevere when the work becomes challenging.

The Practice in Action

One of the ways NCTM (2014, p. 52) summarizes the practice of support productive struggle in learning mathematics is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Anticipating what students might struggle with during a lesson and being prepared to support them productively through the struggle.
• Giving students time to struggle with tasks, and asking questions that scaffold students’ thinking without stepping in to do the work for them.
• Helping students realize that confusion and errors are a natural part of learning, by facilitating discussions on mistakes, misconceptions, and struggles.
• Praising students for their efforts in making sense of mathematical ideas and perseverance in reasoning through problems.

What are students doing?

• Struggling at times with mathematics tasks but knowing that breakthroughs often emerge from confusion and struggle.
• Asking questions that are related to the sources of their struggles and will help them make progress in understanding and solving tasks.
• Persevering in solving problems and realizing that is acceptable to say, “I don’t know how to proceed here,” but it is not acceptable to give up.
• Helping one another without telling their classmates what the answer is or how to solve the problem.

One of the most important traits of successful teachers is that they are insatiably curious about what their students know and understand. This isn’t the kind of curiosity that leads teachers to wonder, “Can my students answer 80 percent of these questions correctly?” Rather, it’s the kind of curiosity that leads teachers to look at how students are thinking, how that thinking develops over time, and how the student responds to and overcomes obstacles along the way.

Looking for these deeper levels of understanding should not be left to chance. It is the teacher’s responsibility to strategically check in with students during critical moments while they work on high-quality tasks. to plan key questions in advance that focus students on the underlying mathematics and press them to explain their understanding, and to make sense of students’ mathematical representations. A teacher needs to make sense of all this and use that to support the progress students show in their mathematical understanding. Feedback should be focused on students’ understanding of a task and the underlying mathematical ideas, and not be aimed at students’ egos by simply indicating what students have done right or wrong.

The Practice in Action

One of the ways NCTM (2014, p. 56) summarizes the practice of elicit and use evidence of student thinking is by describing a set of actions expected of teachers and students that indicate engagement in this teaching practice.

What are teachers doing?

• Identifying what counts as evidence of student progress toward mathematics learning goals.
• Eliciting and gathering evidence of student understanding at strategic points during instruction.
• Interpreting student thinking to assess mathematical understanding, reasoning, and methods.
• Making in-the-moment decisions on how to respond to students with questions and prompts that probe, scaffold, and extend.
• Reflecting on evidence of student learning to inform the planning of next instructional steps.

What are students doing?

• Revealing their mathematical understanding, reasoning, and methods in written work and classroom discourse.
• Reflecting on mistakes and misconceptions to improve their mathematical understanding.
• Asking questions, responding to, and giving suggestions to support the learning of their classmates.
• Assessing and monitoring their own progress toward mathematics learning goals and identifying areas in which they need to improve.