Reflections on Professional Coaching: Eight Mathematics Teaching Practices

May 24th, 2018 | Category: Elementary/Middle School Education, Lutheran Education Commentary, Secondary Education
By Adam Paape

“It gave me a little more thoughtful and purposeful way of coming up with the lessons. My teaching had kind of stayed the same. Doing the actual planning, I thought about what was going to make this lesson shine.” This quote is from Paulson, a fourth grade teacher. He made this reflection after I had coached him for a semester through the lens of the National Council of Teachers of Mathematics’ (NCTM) eight effective mathematics-teaching practices. Table 1 is a list of NCTM’s effective mathematics teaching practices. In this exploratory study, I used the effective teaching and learning practices from NCTM’s book, Principles to Actions: Ensuring Mathematical Success for All as the framework for coaching five elementary school teachers throughout an entire semester. According to Dylan Wiliam (2015, p.17), “Teachers don’t lack knowledge. What they lack is support in working out how to integrate these ideas into their daily practice, and this takes time, which is why we have to allow teachers to take small steps.” My goal in this study was to create a collaborative process whereby my participants could take small steps in improving their mathematics instruction. In addition, I wanted to provide consistent professional-development opportunities for my participants. NCTM suggests that teachers need to, “continually grow in knowledge of mathematics for teaching, mathematical pedagogical knowledge, and knowledge of students as learners of mathematics” (NCTM, 2014, p.116). This article will highlight the experiences of two of my participants as they grew in their understanding of what it means to be an effective teacher of mathematics.

The Eight Effective Mathematics Teaching Practices

Practice 1

Establish mathematics goals to focus learning

Practice 2

Implement tasks that promote reasoning and problem solving

Practice 3

Use and connect mathematical representations

Practice 4

Facilitate meaningful mathematical discourse

Practice 5

Pose purposeful questions

Practice 6

Build procedural fluency from conceptual understanding

Practice 7

Support productive struggle in learning mathematics

Practice 8

Elicit and use evidence of student thinking

Coaching as professional development

For many teachers, professional development is equivalent to the one-day in-service with which we are all too familiar. I intended to extend my participants’ understanding of professional development to include this coaching experience. Garet, Porter, Desimone, Birman, and Yoon (2001), determined through a study of over one thousand mathematics and science teachers that teachers perceived professional development to be of higher quality when two features were present – sustainment over time, and substantial number of hours. My purpose in having a full semester to coach was to collaborate with my participants in a way that allowed them to best implement the effective teaching practices identified by NCTM.

Researchers have studied the effects of coaching from two different perspectives—peer coaching (e.g., Murray et al., 2009) and expert coaching (e.g., Polly, 2012). In my position as a university professor, it seems logical to perceive my role in the coaching exchange as that of an expert. However, I took steps to make the coaching experience much more collaborative than authoritarian. During our post-observation debriefing sessions, I was especially mindful to encourage my participants to think through their personal highlights and areas of improvements from the lesson they had just taught. While I would often recommend strategies to improve particular aspects of their instruction, I was intentional to give the teachers ample opportunities to reflect on their instructional growth.

The coaching cycle

My participants and I followed a specific coaching cycle for each of the lessons I observed. Due to time constraints, we were not always able to meet prior to the lesson I observed. In an effort to still have a pre-observation experience, I emailed my participants a series of reflection questions. The pre-observation reflection prompts included the following questions:

1. What do you hope to accomplish in this lesson?

2. What aspect of this lesson might be most challenging to the students?

3. What strategies might you try in case this difficulty occurs?

4. What would you encourage me to focus on? (NCSM, 2013)

My participants emailed me their reflections prior to my lesson observation. These teacher-created goals for the upcoming lesson helped to frame my focus for the observation. Through the participants’ pre-observation prompt responses, I was also able to anticipate how the teachers would establish mathematics goals to focus learning. In addition, I was able to anticipate how the teachers planned to support students in productively struggling with the mathematical concepts of the lesson.

Coaching with Mrs. Anderson

Mrs. Anderson is a first grade teacher with almost thirty years of teaching experience. As a young learner of mathematics, Anderson liked knowing the answers to math questions, but she could never explain how she knew the answers. Over the course of our coaching episodes, She showed an aptitude for drawing out descriptive explanations from her learners. She noted that she appreciated having a different perspective in her classroom during the coaching experience, especially as I encouraged her to implement more student-to-student discourse. She and I worked through the coaching cycle in five separate instances throughout the semester.

At the end of the semester, Anderson said that she had experienced a personal transformation in her teaching practice that took her from simply following the pre-defined goals of her curriculum to an emphasis on the specific learning needs of her students. During my second observation, she used dot patterns with her learners to see how they would describe eleven dots as a sum of the individual dots.

In her efforts to engage her learners in a task that promoted reasoning and problem solving, Anderson showed her class the image in figure 1. She asked her students to explain how they counted the dots. This kind of explanation is an example of a higher-level demand as defined by Smith and Stein (1998), since She required her learners to focus on the procedure to develop a deeper understanding of the concept of addition. Her activity created an environment where the cognitive effort exhibited by her students required them to make connections within the representation. The activity also required students to use the specific vocabulary needed to communicate the pattern of the dots. For example, one student said, “I see an X in the shape, and then I added two more.” Another student said, “I see 3 + 2 +1 + 2 + 3 = 11.” Anderson used these student explanations to draw out additional ideas from her other learners.

While Anderson’s prompting elicited student thinking, she immediately intervened to express her own method for seeing the dots. Anderson described how she would cover the center dot to get two fives to assist in making an addition of 5 + 5 + 1 = 11. Her “I did it this way” assertion caused the students to stop thinking. While her assertion was mathematically correct, it lowered the cognitive demand for the students and showed her learners that the teacher had a better way to determine the eleven dots. In our debriefing session, I encouraged her to allow the students to spend more time in diagnosing the pattern. This additional time for thinking would create greater opportunities for her students to determine that covering the center dot was a helpful method. Stephen Reinhart (2000) recommends that, whenever possible, a teacher should work to include student voice in mathematics instruction. This sends the students the message that the teacher expects the students to be active participants throughout the entirety of the educational exchange. Humphreys and Parker (2015) suggest that teachers work to develop both social and mathematical agency in their learners such that the learners develop a disposition towards being active in the educational exchange. In addition, Humphreys and Parker note that teachers need to “be very, very careful when we suggest things in class” (p.23). For many students, they are used to their teachers telling them how to think about mathematics. With a patient perspective that gives students time to think for themselves, teachers can provide students greater opportunities to reach the teacher-desired conclusions about the concept.

In our final lesson cycle, Anderson worked on procedural fluency in her students through first establishing the concepts of subtraction. She had her students working with partners on the floor. She gave each pair of students a collection of fourteen Unifix cubes of different colors. She also provided each pair of students with an individual whiteboard. From the start of the coaching semester, I had encouraged her to get more student-to-student discourse. This lesson is evidence of her increased focus on discourse. Anderson posed a variety of subtraction problems for the students to act out with their cubes. She directed one student within each group to select the appropriate amount of cubes of one color to represent the subtrahend. Anderson directed that same student to connect additional cubes of a different color to represent the entire minuend. She then instructed the other student in the group to disconnect the subtrahend in order to show that the leftover pieces were the difference. Once both students agreed on the correct answer, they wrote a subtraction sentence on their whiteboard to communicate the mathematics appropriately in written form.

Throughout this entire process, Anderson’s learners were seeing how the physical manipulation of the cube pieces created patterns in the subtraction process. For instance, She sequenced her questions to get the students to identify and make use of the structure behind the subtraction ideas she was emphasizing (CCSSI, 2010). An example of these structure-oriented questions was when She asked her students to represent 12 – 4 = 8. After that, she asked her learners to consider 12 – 3 = 9. One of her learners described his process as an extension of 12 – 4 = 8, by adding back on to his stack of cubes with one additional cube. He had noticed the equivalency of 12 – 4 + 1 to that of 12 – 3. Having the students write out their subtraction sentences on their individual whiteboards allowed her to quickly see their work, diagnose their understanding of the subtraction concepts, and address any specific needs of her learners at that moment. I saw her provide interventions for a number of her learners due to her eliciting of student work through the whiteboards. This translation from the physical cubes to the written subtraction sentences supported the students in their development of representational competence (Huinker, 2015). Ultimately, this translation ability between physical and written representations forms long-term mathematical understanding in the students (Marshall, Superfine, & Canty, 2010).

By the end of our coaching experience, Anderson made a shift to including more student-to-student interactions. In our exit interview, She said that, based upon the coaching experience using the eight effective mathematics teaching practices, she had seen more students experiencing “light bulbs going off” in her classroom. She also reflected on her shift from being the primary explainer of the mathematics. She said, “Another big change is not saying this is the way we do it [the math], to just letting the students do it. It is still a big change in having them figure it out first or [having them] tell me how they do it, so the other kids can see the different ways of doing it. I’m doing a lot more of ‘You could do it this way or this way or this way.’” This shift in instruction is an example of a teacher growing in the fourth trajectory of Hufferd-Ackles, Fuson, and Sherin’s (2004) framework for creating a classroom community centered on discourse—the shared responsibility of students for their own learning and for the learning of others around them.

Coaching with Mr. Paulson

Mr. Paulson, a fourth grade teacher with over thirty years of teaching experience, remembered being a bright, but unmotivated student. Math was easy for him prior to his experience with Algebra. During his Algebra experience, he struggled and began to doubt his abilities. As a teacher, Paulson had great command of his mathematics classroom. He was willing to implement new, student-focused practices. While Paulson was the most experienced teacher in this study, he was eager to put into practice the strategies that I suggested to strengthen his teaching.

In my first observation of him, it was apparent that he had developed social norms within his classroom. He held students accountable for explaining their understanding and for asking questions when they did not understand. He was also very good at praising his students when they would participate in class. This created a safe place for his students to explore mathematical ideas (Stephan, 2014). From the beginning of the semester, it was evident that he had put forth significant time and effort to create a positive learning environment for his students.

When Paulson would ask a question, the majority of his students would raise their hands to give an answer. In our debriefing session after his first lesson, I encouraged him to think through strategies to engage the students who did not raise their hands. One strategy I suggested to him was to have his learners perform a turn-and-talk with a partner after he had posed a question to the entire class. During his next lesson, which covered decimal comparisons, he had his students perform a turn-and-talk during the first minute of class when he asked them to think of a number between 75.1 and 75.2. This simple move was a direct application of facilitating meaningful mathematical discourse through creating, “the purposeful exchange of ideas through classroom discussion” (NCTM, 2014, p. 29). He successfully encouraged all of his learners, not just those eager to raise their hands, to be a part of the active learning process. In addition, he was now intentionally engaging his learners in the mathematical practice of constructing viable arguments and critiquing the reasoning of others through this direct student-to-student discourse (CCSSI, 2010).

I recommended to Paulson that he should consider revising his pronoun choice in many of his questions. He frequently asked questions like, “Can you tell me how you solved that?” and “Tell me how I should finish this addition.” One of our goals in the coaching experience was to create a collaborative culture in the classroom where the learners saw themselves as active agents in the learning experience. I recommended that he change his questions to be more inclusive, “Can you tell us how you solved that?” and “Tell us how we should finish this addition.” These small pronoun changes altered the purpose of his questions. The pronouns “me” and “I” when used by a teacher can send the unintended message to the learners that the teacher is the center of the learning exchange—that the teacher is the keeper of all knowledge. The shift to more inclusive pronouns in teacher questioning encourages students to begin to see themselves and their classmates as vital to the learning experience. This message is essential in a discourse-oriented classroom.

For my fifth and final observation of Paulson, I encouraged him to use mixed-ability groupings. I had him do this as an application of Jo Boaler’s Complex Instruction model to increase the opportunities for all learners to contribute to their group’s work. The four tenets of Boaler’s Complex Instruction are:

1. Multidimensionality: students work to express their mathematical understandings in a variety of ways (e.g., calculations, asking questions, proposing ideas, connecting strategies, making use of different representations).

2. Roles: predefined jobs for each group member (facilitator, recorder, resource manager, and team captain).

3. Assigning competence: the teacher identifies the struggling learners and looks for ways to praise those learners when they contribute to the mathematical work of their group.

4. Shared student responsibility for learning (Boaler, 2016).

His lesson focused on developing student understanding of basic fractions. In order to promote productive struggle, he was strategic in his use of fraction representations. See figure 2 and figure 3 for two examples of the part-to-whole representations that he gave to different student groups.

Paulson used a 12 inch by 18 inch sheet of construction paper to represent the whole. From a separate sheet of construction paper, he cut out smaller pieces of varying shapes into fractions of the whole. He provided each group with one of the smaller pieces. His goal was for his students to discover how the smaller pieces related to the whole. For example, one group had to determine that their small piece was one twelfth by seeing how many of their piece would fit in the whole. His plan was that his students would eventually decide to use a pencil to see how many times they could trace the smaller pieces inside the whole. Some of his groups filled the whole with tracings of their fractional piece. Other groups, especially those with rectangular fractional pieces, determined how many columns and rows they could fill with their piece. These groups used multiplication to determine how many of their piece could fill the rectangular array that represented the whole.

Paulson purposefully selected specific fractional representations to engage his learners in conversations about key fractional concepts (Marshall, Superfine, & Canty, 2010). His lesson helped his students develop their understanding of fractions by using their geometric reasoning. However, in our debriefing session, I suggested that he add more challenge to his activity by having the groups work with more than one fractional piece. See figure 4 for an example of a potential challenge problem. In this scenario, students would need to connect their work from each of the given shapes to determine how the fractions form a common relationship to the whole. This would help students to see the benefit of common denominators in fraction addition where:

As Paulson and I concluded our time together, he reflected on how he had become a more purposeful lesson planner because of our collaborative exchanges. He had become more deliberate in setting the stage for his learners to engage with the mathematical concepts in his lessons. He also reflected on his newfound desire to make his classroom into a community of interactive learners who saw value in the problem-solving process of other students.


During each coaching cycle experience, I was able to use the eight effective mathematics teaching practices to frame the pre-observation interactions, my observations, and the post-observation debriefing sessions. While observing Anderson and Paulson, I was able to make instructional recommendations based upon the effective mathematics teaching practices that fell in line with the research found within NCTM’s Principles to Actions: Ensuring Mathematical Success for All. Since my participants and I agreed that the practices, as defined by NCTM, were beneficial to both the students and the teacher, we all experienced a positive collaborate environment throughout the semester. LEJ


Boaler, J. (2016) Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.

Common Core State Standards Initiative (CCSSI). (2010) Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief States School Offices.

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Marshall, A., Castro Superfine, A., & Canty, R. (2010). Star students make connections: Discover strategies to engage young math students in competently using multiple representations. Teaching Children Mathematics 17(17), 38-47.

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National Council of Supervisors of Mathematics (NCSM). “Pre-Conference Strategies.” Last modified 2013.

Polly, D. (2012). Supporting mathematics instruction with an expert coaching model. Mathematics Teacher Education and Development 14(1), 78-93.

Reinhart, S. (2000). Never say anything a kid can say. Mathematics Teaching in the Middle School 5(8), 478-483.

Smith, M. & Stein, M. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School 3(5), 344-349.

Stephan, M. (2014). Establishing standards for mathematical practice. Mathematics Teaching in the Middle School 19(9), 532-538.

Wiliam, D. (2015) Embedding formative assessment: Practical techniques for K-12 classrooms. West Palm Beach, Fl: Learning Sciences International.

Author Information

Adam D. Paape, Ed.D. is Associate Professor of Education and Chair of the Secondary Education Department at Concordia University Wisconsin where he teaches mathematics, mathematics education, and general education courses. Prior to serving at Concordia,
Dr. Paape served at both public and Lutheran high schools as a mathematics and theology instructor. He currently serves on the statewide board of directors for the Wisconsin Mathematics Council. He may be contacted at