In this episode of Room to Grow, Curtis and Joanie reconsider the balance of conceptual understanding and procedural fluency in math instruction. Although this topic has been discussed before, our hosts acknowledge that there is great nuance and many considerations in considering these two ideas in the teaching and learning of mathematics.

Curtis and Joanie discuss how inquiry-based, discovery-style learning opportunities are more open ended, are student centered, and are less teacher directed. They support these types of lessons in math instruction while recognizing that there are times when an explicit approach where teachers are sharing important information also has a place. Additionally, our hosts consider that teaching procedures and algorithms also provides and opportunity to cultivate conceptual understanding. When teachers help student find the  conceptual understanding within the procedures, they engage in mathematical reasoning. This type of reasoning through concepts and procedures contributes to a broader and more robust understanding of meaningful mathematics.

Additional referenced content includes:

·       NCTM article From Rules That Expire to Deeper Mathematical Thinking. Mathematics Teacher: Learning & Teaching PK-12 Volume 118 Issue 4. April 2025. (Membership required).

·       NCTM article Teaching Is a Journey: From Rules That Expire to a Journey Aspired. Mathematics Teacher: Learning & Teaching PK-12 Volume 118 Issue 4. April 2025. (Membership required).

·       Robert Kaplinski’s website and Open Middle website

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

This episode of Room to Grow, Curtis and Joanie speak with Pam Harris. Pam is well known and loved for her website, podcast, books, and conference sessions all based on her core belief that “Math is FigureOutAble.” Today’s discussion centers on Pam’s newest publication, Developing Mathematical Reasoning: Avoiding the Traps of Algorithms.

Pam starts with three distortions about math that are common among teachers, students, and the population, and can impact how educators engage students with math in their classrooms. Next the conversation shifts to what is meant by algorithms, and how they differ from strategies and formulas. Then the discussion focuses on the potential traps to learning that can result from teaching algorithms in mathematics. All of these ideas are based on the development of mathematical reasoning, from counting strategies to additive thinking, to multiplicative reasoning and proportional reasoning, then the functional reasoning that comprises much of the math students learn in high school.

There are so many good ideas in this episode that will challenge you and get you thinking!

Additional referenced content includes:

·       Pam Harris’ website, Math is FigureOutAble.

·       Pam’s book, Developing Mathematical Reasoning: Avoiding the Traps of Algorithms.

·       Pam’s podcast, Math is Figure-Out-Able!

·       Find Pam on all your favorite social media platforms.

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

This episode of Room to Grow brings together our 2025 series on the Mathematics Teaching Practices from Principles to Actions, an NCTM publication. After discussing each of the eight practices in isolation over the last several months, this month tackles the ideas and challenges in actually making substantive change to classroom practice. 

Our hosts share ideas for engaging with others in deeper professional learning, coaching, and peer observations, as well as ideas for individual teachers to work on making change in their classrooms. They acknowledge that change is complicated, and encourage listeners to engage in small, incremental steps that add up over time. 

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       NCTM’s Principals to Actions Toolkit – Professional learning and other resources for a deeper dive into the eight Mathematics Teaching Practices

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

This episode of Room to Grow wraps up Joanie and Curtis’ season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions. The final practice in the series is “Support productive struggle in learning mathematics.” This is defined as follows:

Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.

 Our hosts begin with discussing the connection to classroom culture, ensuring that students feel safe, emotionally and academically, in order to be willing to struggle. Additionally, the expectations set by the teacher when there is a belief that productive struggle is a valuable and important aspect of learning math. Then they linger on the word “grapple” and the connotation it creates, and how different it is than the connotation of the word “struggle.” And they emphasize that productive struggle isn’t about the math being hard, but rather thinking about the mathematics in new ways, and making connections that may not have been obvious. A key idea is the importance of educators’ asking questions to better understand student thinking, which is a necessary condition for helping them to grapple with new mathematical connections.

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       Favorite No protocol (5 minute Video)

·       Previous Room to Grow episode Helping Students Struggle Productively

·       Image of America flag made of baseballs

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating it’s 10th anniversary. This month’s practice is “Build procedural fluency from conceptual understanding.”  This is defined as follows:

 Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

 Using some discussion about multiplication, our hosts try to differentiate what is meant by conceptual understanding and by procedural fluency. They tease out the confusion that can arise by associating conceptual understanding with inquiry-based instruction and procedural fluency with direct instructional strategies. Although these types of instruction often go together, they are different, and separating them can help educators focus on how to best get to student learning.

 Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       NCTM’s position paper on Procedural Fluency (January 2023 – membership not necessary)

 Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating it’s 10th anniversary. This month’s practice is “Pose Purposeful Questions.” This is defined as follows:

Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.

 In unpacking this definition, our hosts key in on the importance of planning for questions during instruction to ensure that they are purposeful. They discuss how purposeful questions can assess student thinking by giving teacher insight to how students are reasoning about the mathematics. This might include asking a follow-up question even when a student provides a correct answer! They also consider advancing questions, that take students’ current thinking and move it beyond where they are to deeper understanding of important mathematics. With intentional and careful planning of questions, teachers can provide insight to students about what aspects of the math are most important, and encourage them to use and value their own thinking on the journey of learning math.

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       MTLT article Planning and Implementing Effective Questioning (membership required)

·       (additional articles from CCTM book study Chapter 5)

 Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating its 10th anniversary. This month’s practice is “Establish Mathematics Goals to Focus Learning.” This is defined as follows:

Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

In today’s conversation, our hosts unpack the key components of this principle. First, they discuss how learning goals, focused on important mathematical understandings, differ from procedural, process goals, which may include skills and procedures that are not directly connected to the underlying mathematical concepts. Next, the discussion turns to situating goals within a learning progression, which helps teachers stay focused on what is relevant to their grade level or course, and provides a venue for students to be active in their progress toward learning. Finally, effective mathematics goals guide instructional decisions, helping educators know which tangents to explore and which are distractions from the intended learning. We hope you enjoy the conversation, and that it extends your thinking on mathematics goals for learning.

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       NCTM’s myNCTM forums (membership required).

·       How learning goals serve as a guide – NCTM Teaching Children Mathematics blog post

·       Rachel Harrington’s appearance on the Math Learning Center podcast/blog discussing mathematical goals

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating it’s 10th anniversary. This month’s practice is “Elicit and Use Evidence of Student Thinking.” 

In Principles to Actions, NCTM describes this teaching practice in this way:

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

This meaty description provides the fodder for today’s conversation. Our hosts consider what is meant by “effective teaching,” “assessing progress,” and “adjusting instruction continually,” and tie these ideas back to the important work of classroom educators.

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       Want more ideas for eliciting student thinking in your classroom? Check these out:

o   Descriptors of teacher and student behaviors for this practice

o   Thoughts and linked resources from the Colorado Department of Education

o   A classroom observation tool focused on this practice from the Minnesota Department of Education 

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating it’s 10th anniversary. This month’s practice is “Use and connect mathematical representations.” Our hosts describe the five representations outlined in Principles to Actions, which include visual, symbolic, verbal, contextual, and physical descriptions of mathematics, but emphasize that the representations are not meant to be a check list to be covered during instruction. Rather, the different representations provide a framework for explore important mathematical concepts through different lenses, allowing students to build and deepen their understanding as they consider these ways of engaging.

In addition to deep understanding, teachers’ attending to different representations will allow different students in the class to be elevated, as their unique strengths and preferences will have the opportunity to come out and be showcased. 

 Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       Making Connections Explicit​ (NCTM requires subscription)

·       Supporting Understanding Using Representations​ (NCTM requires subscription)

·       Three Ways to Enhance Tasks for Multilingual Learners​ (NCTM requires subscription)

·       Interpreting Distance-Time Graphs – lesson referred to in this episode 

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating it’s 10th anniversary. This month’s practice is “Implement Tasks that Promote Reasoning and Problem Solving.” Our hosts being by expounding on the difference between selecting a task and implementing it, and that selecting a good task does not guarantee good implementation. They bust the idea that the only way to engage students in reasoning and problem solving is with a rich task, by considering how educators can weave together procedural learning with conceptual understanding.

Next, they connect reasoning and problem solving to the Standards of Mathematical Practice, particularly practices 7: Attend to and make use of structure and 8: Look for and express regularity in repeated reasoning. Capitalizing on students’ natural noticing and shining the light on the underlying mathematics leads to stronger connections and increasing students’ ability to generalize their understanding.

By building a foundation of reasoning and sense-making, and helping students understand that this is a resource for them tap into, that allows for the learning and engagement beyond rote classroom experiences.

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       Selecting and Creating Mathematical Tasks article from Smith and Stein

·       A Teacher’s Guide to Reasoning and Sense-Making from NCTM

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing [email protected] . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy.