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5 Practices for Orchestrating Productive Mathematics Discussions [NCTM]
Learn the 5 practices for facilitating effective inquiry-oriented classrooms
Anticipating what students will do--what strategies they will use--in solving a problem
Monitoring their work as they approach the problem in class
Selecting students whose strategies are worth discussing in class
Sequencing those students' presentations to maximize their potential to increase students' learning
Connecting the strategies and ideas in a way that helps students understand the mathematics learned
This book presents and discusses an framework for orchestrating mathematically productive discussions that are rooted in student thinking.
The 5 Practices framework identifies a set of instructional practices that will help teachers achieve high-demand learning objectives by using student work as the launching point for discussions in which important mathematical ideas are brought to the surface, contradictions are exposed, and understandings are developed or consolidated. By giving teachers a road map of things that they can do in advance and during whole-class discussions, these practices have the potential for helping teachers to more effectively orchestrate discussions that are responsive to both students and the discipline.
Includes a Professional Development Guide.
Anticipating what students will do--what strategies they will use--in solving a problem
Monitoring their work as they approach the problem in class
Selecting students whose strategies are worth discussing in class
Sequencing those students' presentations to maximize their potential to increase students' learning
Connecting the strategies and ideas in a way that helps students understand the mathematics learned
This book presents and discusses an framework for orchestrating mathematically productive discussions that are rooted in student thinking.
The 5 Practices framework identifies a set of instructional practices that will help teachers achieve high-demand learning objectives by using student work as the launching point for discussions in which important mathematical ideas are brought to the surface, contradictions are exposed, and understandings are developed or consolidated. By giving teachers a road map of things that they can do in advance and during whole-class discussions, these practices have the potential for helping teachers to more effectively orchestrate discussions that are responsive to both students and the discipline.
Includes a Professional Development Guide.
104 pages, Paperback
First published January 1, 2011
34 people are currently reading
248 people want to read
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Displaying 1 - 28 of 28 reviews
August 2, 2017
It took me forever to realize that, while I really enjoy theory, I’m not very good at learning theory in the absence of practical examples. My “aha!” moments tend to come grudgingly when I have to put the jigsaw together without the picture on the front of the fox, as it were. Sorry, weak analogy.
This is, I think, why I burned out on mathematics back in my school years. Math up through trigonometry was sufficiently intuitively practical that I just loved it (it is even telling that I seem to have slept through what was taught on hyperbolic functions), but my calculus teacher, as brilliant as she was, taught the subject with a spiritual orientation. I distinctly remember her encouraging us to close our eyes and imagine the loneliness of zero during the lesson on limits. She had her eyes closed, blissful and enraptured, while all her students looked at each other in wide-eyed bewilderment. I tried again as a senior, but tragically my second calculus instructor decided over Christmas that he had to go help his relatives in Argentina protest the coup, and never returned.
I compounded the mistake by trying to be a math major at university, but it just kept getting more abstract and boring. I probably would have failed my final quarter of differential equations if my roommate hadn’t been in the same class, and warned me when tests were coming. Switching to computer science was a life-saver.
Oh, anyway — this book. Designing a lesson plan is a lot less abstract than university-level math, but I still struggled with seeing how to apply the theory to the actual classroom.
This book is, frankly, brilliant at the fairly small thing it sets out to do: design lessons in such a way as to encourage students to participate in their own learning, and that of their peers. It combines the step-by-step theory with walk-throughs of examples using that framework.
The key is to, during lesson planning, anticipate where students will stumble, make mistakes, and otherwise show signs of struggle, and design the lessons to use what might otherwise be problems as a scaffold. Anticipating is only the first of the titular Five Practices, but it is the critical lead-in.
Short and sweet: buy this book! Or, if you’re in science land, I’m sure the parallel text by the same author is just as good: 5 Practices for Orchestrating Task-Based Discussions in Science.
This is, I think, why I burned out on mathematics back in my school years. Math up through trigonometry was sufficiently intuitively practical that I just loved it (it is even telling that I seem to have slept through what was taught on hyperbolic functions), but my calculus teacher, as brilliant as she was, taught the subject with a spiritual orientation. I distinctly remember her encouraging us to close our eyes and imagine the loneliness of zero during the lesson on limits. She had her eyes closed, blissful and enraptured, while all her students looked at each other in wide-eyed bewilderment. I tried again as a senior, but tragically my second calculus instructor decided over Christmas that he had to go help his relatives in Argentina protest the coup, and never returned.
I compounded the mistake by trying to be a math major at university, but it just kept getting more abstract and boring. I probably would have failed my final quarter of differential equations if my roommate hadn’t been in the same class, and warned me when tests were coming. Switching to computer science was a life-saver.
Oh, anyway — this book. Designing a lesson plan is a lot less abstract than university-level math, but I still struggled with seeing how to apply the theory to the actual classroom.
This book is, frankly, brilliant at the fairly small thing it sets out to do: design lessons in such a way as to encourage students to participate in their own learning, and that of their peers. It combines the step-by-step theory with walk-throughs of examples using that framework.
The key is to, during lesson planning, anticipate where students will stumble, make mistakes, and otherwise show signs of struggle, and design the lessons to use what might otherwise be problems as a scaffold. Anticipating is only the first of the titular Five Practices, but it is the critical lead-in.
Short and sweet: buy this book! Or, if you’re in science land, I’m sure the parallel text by the same author is just as good: 5 Practices for Orchestrating Task-Based Discussions in Science.
October 18, 2018
An excellent guide with examples and real scenarios where teachers have implemented questioning strategies in the classroom.
Planning is essential to open-ended tasks. What are your learning goals (not doing goals) for students? What do you want your students to know after the lesson is over? Use these goals to guide your class discussion.
Are the tasks you have chosen high level tasks? Can the students use their creativity to solve the problem? Are the tasks open ended and lend themselves to student choice?
The teacher needs to spend time anticipating what kinds of answers the students will come up with.
The teacher is in control of giving all students equity in access to information and cognitive tasks—who you call on and who you allow to explain are equity issues. Take the power you have as the teacher and use it wisely to promote equal access to rigor.
Students need time to formulate ideas.
Teachers need high expectations of all students before starting a high-level task.
Determine the order in which students are selected to share their work and mix it up. It should follow some kind of logical progression.
Teachers as the facilitator of learning and discussions.
Planning is essential to open-ended tasks. What are your learning goals (not doing goals) for students? What do you want your students to know after the lesson is over? Use these goals to guide your class discussion.
Are the tasks you have chosen high level tasks? Can the students use their creativity to solve the problem? Are the tasks open ended and lend themselves to student choice?
The teacher needs to spend time anticipating what kinds of answers the students will come up with.
The teacher is in control of giving all students equity in access to information and cognitive tasks—who you call on and who you allow to explain are equity issues. Take the power you have as the teacher and use it wisely to promote equal access to rigor.
Students need time to formulate ideas.
Teachers need high expectations of all students before starting a high-level task.
Determine the order in which students are selected to share their work and mix it up. It should follow some kind of logical progression.
Teachers as the facilitator of learning and discussions.
April 30, 2025
I hated this book. Helpful for first couple chapters then got so repetitive. I got so bored one time I had to take a break and write my grocery list in the margins. Did I truly hate this book or did I hate writing a paper on every chapter every week? Probably both.
July 15, 2016
There's a movement in modern-day mathematics teaching that promotes problem-based lessons. It's a movement away from the mini lesson, in which a teacher models a skill and students practice it to mastery. Problem-based learning instead starts with an open-ended problem, to which students apply their own strategies to arrive at a solution, and it all ends in a carefully orchestrated discussion.
It's the "discussion" part of a problem-based lesson that is the most daunting to me. And since this book is currently topping NCTM's best-seller list, I know I'm not alone. Along the discussion vein, this book has a lot going on. The 5 Practices provide a solid framework for considering the elements that go into a strong discussion. And the Thinking Through a Lesson Protocol (TTLP) gives me a concrete way to implement the 5 Practices into a lesson.
Perhaps the most frustrating part of problem-based discussions is that they seem to be based more off research than reality. Where are the teachers that lead these discussions? This is where the book most excels: it provides the actual dialogue from class discussion case studies. I want more of these; Smith and Stein surely must be holding out on us. But the ones they do provide offer a much needed lens into these elusive discussions.
A solid, timely professional resource for teachers pursuing problem-based learning. For more on problem-based learning, check out Van De Walle's Elementary and Middle School Mathematics. The two resources complement each other nicely.
It's the "discussion" part of a problem-based lesson that is the most daunting to me. And since this book is currently topping NCTM's best-seller list, I know I'm not alone. Along the discussion vein, this book has a lot going on. The 5 Practices provide a solid framework for considering the elements that go into a strong discussion. And the Thinking Through a Lesson Protocol (TTLP) gives me a concrete way to implement the 5 Practices into a lesson.
Perhaps the most frustrating part of problem-based discussions is that they seem to be based more off research than reality. Where are the teachers that lead these discussions? This is where the book most excels: it provides the actual dialogue from class discussion case studies. I want more of these; Smith and Stein surely must be holding out on us. But the ones they do provide offer a much needed lens into these elusive discussions.
A solid, timely professional resource for teachers pursuing problem-based learning. For more on problem-based learning, check out Van De Walle's Elementary and Middle School Mathematics. The two resources complement each other nicely.
January 21, 2013
Discourse is not an accident. This book will help you create meaningful mathematical discussions in you classroom.
July 28, 2024
Book #37 of 2024. "5 Practices for Orchestrating Productive Mathematics Discussions" by Margaret Smith and Mary Kay Stein.
4/5 rating. 94 p.
This beautifully short book highlights the best ways to promote fruitful discussion and learning in math classrooms. If you are a math teacher who would love to help your students succeed in more life-oriented problem solving and a deeper understanding of mathematical concepts, this is the book for you!!
I read Peter Liljedahl's book about thinking classrooms a few years ago and wish I had found this at the same time. It focuses on many of the same tools for promoting discussion; specifically the 5 practices of:
1) Anticipating likely student responses
2) Monitoring actual responses
3) Selecting particular students to present their ideas
4) Sequencing the student responses
5) Connecting the different responses to the key mathematical idea
This book sparked so many ideas in how to better serve my students in class. I loved the talk of the importance of understanding what you need the students to know, and thinking through tasks that are a good medium to touch on each of those goals.
One of the most useful portions of the book is that the chapters are built around lessons that were given so that you can see what the process of each step is like. How do you select the order that ideas are presented? How do you connect the student work to your specific Learning Objectives?
I highlighted a ton in here, and wrote even more notes down about how to structure my year. If you want to help your students become more deep thinkers in math, better problem solvers, and build a flexibility that can help them in the future, I would strongly recommend this book!
Quotes:
"Most schoolwork consists of assignments composed of 'problems' for which students have been taught a preferred method of solving. There is little engagement of student 'thinking' in such tasks, and only the straightforward application of previously learned skills and recall of memorized facts. It is unrealistic to expect students to learn to grapple with the unstructured, messy challenges of today's world if they are forced to sit silently in rows, complete basic skills worksheets, and engage in teacher-led 'discussions' that consist of literal, fact-based questions and answers."
"In mathematics classrooms, high-quality discussions support student learning of mathematics by helping students learn how to communicate their ideas, making students' thinking public so it can be guided in mathematically sound directions, and encouraging students to evaluate their own and each other's mathematical ideas. These are all important features of what it means to be 'mathematically literate.'"
"Students quickly get the message - often from subtle cues - that 'knowing mathematics' means using only those strategies that have been validated by the teacher or textbook; correspondingly, they learn not to use or trust their own reasoning."
"For example, the teacher might want to have the strategy used by the majority of students presented before those that only a few students used, to validate the work that the majority of students did and make the beginning of the discussion accessible to as many students as possible. Alternatively, the teacher might want to begin with a strategy that is more concrete (using drawings or concrete materials) and move to strategies that are more abstract (using algebra). This approach - moving from concrete to abstract - serves to validate less sophisticated approaches and allows for connections among approaches."
"Rather than having mathematical discussions consist of separate presentations of different ways to solve a particular problem, the goal is to have student presentations build on one another to develop powerful mathematical ideas."
"To ensure that a discussion will be productive, teachers need to have clear learning goals for what they are trying to accomplish in the lesson, and they must select a task that has the potential to help students achieve those goals."
"Without explicit learning goals, it is difficult to know what counts as evidence of students' learning, how students' learning can be linked to particular instructional activities, and how to revise instruction to facilitate students' learning more effectively. Formulating clear, explicit learning goals sets the stage for everything else."
"There are many...different ways that student responses could be selected and sequenced that could be equally productive. The point is that the method selected must support the story line that the teacher envisions for the lesson so that the mathematics to be learned emerges in a clear and explicit way."
"Hence, the questions must go beyond merely clarifying and probing what individual students did and how. Instead, they must focus on mathematical meaning and relationships and make links between mathematical ideas and representations."
"It is the questions and their close connection to the context - the actual solutions produced by students - that can advance students' understanding of mathematics."
4/5 rating. 94 p.
This beautifully short book highlights the best ways to promote fruitful discussion and learning in math classrooms. If you are a math teacher who would love to help your students succeed in more life-oriented problem solving and a deeper understanding of mathematical concepts, this is the book for you!!
I read Peter Liljedahl's book about thinking classrooms a few years ago and wish I had found this at the same time. It focuses on many of the same tools for promoting discussion; specifically the 5 practices of:
1) Anticipating likely student responses
2) Monitoring actual responses
3) Selecting particular students to present their ideas
4) Sequencing the student responses
5) Connecting the different responses to the key mathematical idea
This book sparked so many ideas in how to better serve my students in class. I loved the talk of the importance of understanding what you need the students to know, and thinking through tasks that are a good medium to touch on each of those goals.
One of the most useful portions of the book is that the chapters are built around lessons that were given so that you can see what the process of each step is like. How do you select the order that ideas are presented? How do you connect the student work to your specific Learning Objectives?
I highlighted a ton in here, and wrote even more notes down about how to structure my year. If you want to help your students become more deep thinkers in math, better problem solvers, and build a flexibility that can help them in the future, I would strongly recommend this book!
Quotes:
"Most schoolwork consists of assignments composed of 'problems' for which students have been taught a preferred method of solving. There is little engagement of student 'thinking' in such tasks, and only the straightforward application of previously learned skills and recall of memorized facts. It is unrealistic to expect students to learn to grapple with the unstructured, messy challenges of today's world if they are forced to sit silently in rows, complete basic skills worksheets, and engage in teacher-led 'discussions' that consist of literal, fact-based questions and answers."
"In mathematics classrooms, high-quality discussions support student learning of mathematics by helping students learn how to communicate their ideas, making students' thinking public so it can be guided in mathematically sound directions, and encouraging students to evaluate their own and each other's mathematical ideas. These are all important features of what it means to be 'mathematically literate.'"
"Students quickly get the message - often from subtle cues - that 'knowing mathematics' means using only those strategies that have been validated by the teacher or textbook; correspondingly, they learn not to use or trust their own reasoning."
"For example, the teacher might want to have the strategy used by the majority of students presented before those that only a few students used, to validate the work that the majority of students did and make the beginning of the discussion accessible to as many students as possible. Alternatively, the teacher might want to begin with a strategy that is more concrete (using drawings or concrete materials) and move to strategies that are more abstract (using algebra). This approach - moving from concrete to abstract - serves to validate less sophisticated approaches and allows for connections among approaches."
"Rather than having mathematical discussions consist of separate presentations of different ways to solve a particular problem, the goal is to have student presentations build on one another to develop powerful mathematical ideas."
"To ensure that a discussion will be productive, teachers need to have clear learning goals for what they are trying to accomplish in the lesson, and they must select a task that has the potential to help students achieve those goals."
"Without explicit learning goals, it is difficult to know what counts as evidence of students' learning, how students' learning can be linked to particular instructional activities, and how to revise instruction to facilitate students' learning more effectively. Formulating clear, explicit learning goals sets the stage for everything else."
"There are many...different ways that student responses could be selected and sequenced that could be equally productive. The point is that the method selected must support the story line that the teacher envisions for the lesson so that the mathematics to be learned emerges in a clear and explicit way."
"Hence, the questions must go beyond merely clarifying and probing what individual students did and how. Instead, they must focus on mathematical meaning and relationships and make links between mathematical ideas and representations."
"It is the questions and their close connection to the context - the actual solutions produced by students - that can advance students' understanding of mathematics."
June 23, 2019
One of the best mathematics resources I have seen in years! The book illustrates the 5 practices (anticipate, monitor, select, sequence, and connect) with case studies, detailed instructions, and forms for planning. I have used a watered-down form of the practices in my classes; after reading this and doing some practice I feel more confident I can use the strategies to much better effect. I also think that working through some of the exercises with other teachers would make for great professional development.
November 27, 2021
I really like how this book advances conceptual thinking and understanding over rote skills. Kids really struggle with being able to do this because teachers rarely engage in this type of learning. That said, the work in this book is powerful and awesome, but extremely time consuming, which is why it doesn't happen. The book cites Japanese lesson studies - great example - they have planning time! A model the US needs to follow - then we would NOT need STEM. The book would have been a lot better had it provide corresponding website with more plans and videos.
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July 8, 2025Loved this approach to mathematics discussion! This is totally feasible and something I think that can be implemented right away. The book presents five practices for guiding discussions on mathematical concepts. The practices seem easy to implement and I can tell how they would be highly effective. I do think teachers that have the ability to teach the same grade level every year would have an advantage because these practices do take a lot of pre-planning and strong understanding of how students will approach problems to implement.
June 21, 2023
Solid pedagogical advice with am emphasis on mathematical discourse as the pathway to true conceptual understanding. I read this for professional development credit and will be applying what I learned. It bothered me that there was such a massive push to incorporate high-quality math tasks (which I definitely agree with) but there were very few resources included for actually finding such grade-appropriate tasks.
July 23, 2019
Great information about the 5 math practices: anticipating, monitoring, selecting, sequencing, and connecting. I plan on using these to improve my math instruction this year. I do wish that more of the examples used in the book were at an elementary level rather than high school. That would have just been more relevant for me.
December 9, 2021
As far as pedagogical texts go, this one is quite good! We used this text for a professional development group at work, and our leader, Kim Freitag, made such a difference!!! We’ve been trying out the strategies in this book, and while our students have a long way to go, we have hope! Math instruction needs to be much more discussion based, and this book has insight into how to make that happen.
March 15, 2019
The book gives excellent examples of the 5 practices. I especially appreciated the description of how to thoughtfully lesson plan in order to engage in meaningful mathematical discussions in the classroom.
July 18, 2017
Well-organized quick read to support math teachers in leading productive, focused discussions.
July 27, 2018
Solid advice but not many specific examples for different levels of math. I wish they had a library of tasks for elementary, middle, and high school to read.
June 4, 2021
Writing was a bit dull but good ideas overall. Excellent quick summer reading for math teachers.
November 30, 2024
A really useful grounding technique for running a mathematics lesson with discussion
December 1, 2024
Maybe one of the most important books I've read for my early career. The five practices are practical and grounded in case studies throughout the book.
Having prewritten questions to advance students who are struggling and questions that make clearer mathematical relationships will really elevate discussions in my classroom.
I would recommend reading this and then reading "Building Thinking Classrooms" by Peter Lijedahl.
Having prewritten questions to advance students who are struggling and questions that make clearer mathematical relationships will really elevate discussions in my classroom.
I would recommend reading this and then reading "Building Thinking Classrooms" by Peter Lijedahl.
October 10, 2016
The explicit identification of five stages in supporting productive math discussions in the classroom is probably the greatest benefit of this book (although I think that could also have been obtained from the paper that preceded it). So often, just having names for each aspect is so helpful in terms of planning and discussing due to common terminology and understanding. As someone with a decent math and education background, I didn't find the case studies particularly illuminating...they certainly fit with the practices and did a good job of illustrating them, but they didn't do a whole lot in terms of benefiting me further. I could see how they would be helpful when using this book with teachers though - although I would then wish that they also provided elementary examples. Overall, I hope that these stages become a process that is talked about more regularly and that we would allocate more planning time to teachers to make them possible.
March 5, 2017
I love this book!
I am excited to implement the practices in my classroom.
Short and easy read that is full of practical practices to enrich instruction in a Standards-based classroom.
I am excited to implement the practices in my classroom.
Short and easy read that is full of practical practices to enrich instruction in a Standards-based classroom.
July 12, 2014
Provides a framework for teaching math more closely aligned to the CCSS Math Practice Standards. It provides a way to plan lessons to better teach students how to problem solve, instead of teaching students examples that they can imitate later. It provides ways for teachers to plan for discussion of problems in order to help students gain understanding of the intended mathematical ideas. Lots of good insight in how to plan tasks and prepare for multiple methods(both correct and incorrect) and how to incorporate student's methods in a presentation to the class so that all students may benefit from seeing how mathematical ideas are connected.
July 8, 2014
The ideas in this book are true to the changing world of teaching math. I feel that this book would be excellent for secondary teachers. I teach first grade. There were not any primary examples used in this book. Thus it is hard for me to picture how these practices would look in a primary classroom.
May 21, 2016
If you want to effectively use rich mathematical tasks In your classroom, this is the book to read! Most of the examples in this book are for upper grades, so if you teach Early years, stretch your thinking by applying the important ideas to your grade level. Find a colleague or two and work through it together.
June 25, 2012
Got this book at a summer professional development class. I really like that it's for Middle School AND math. You don't find many books out there like that. The ideas inside are practical and I hope will be easy to implement this next school year.
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September 8, 2011This is a NCTM sanctioned book that really supports teachers in engaging children in meaningful mathematics. Great ideas for teachers w second language learners.
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October 21, 2012This is one of the best books I've read about teaching math. It gives practical ideas that work in the classroom.
December 31, 2014
Enlightening ideas to consider with scenarios for you to process.
March 31, 2017
The 5 practices lead teachers into diving deeper into their goal of meaningful task and discussions on mathematics in their classroom. As we develop more student-centered instruction with the teacher taking on the roll of facilitator the 5 practices break down and guide us through the thinking involved with this shift.
Displaying 1 - 28 of 28 reviews