Anyone who reads or experiences Eureka Math2™ can see the importance we place on math discourse. Our lessons revolve around rich classroom discussions that invite all learners to express and understand complex mathematical concepts. Leading these kinds of discussions isn’t easy, but Eureka Math2 helps teachers create rich mathematical discourse by giving them tools like open-ended lessons, guidance for instructional support and differentiation, and student work samples with correct and incorrect strategies.
Our approach to supporting teachers in creating these discussions is informed by the book Five Practices for Orchestrating Productive Mathematics Discussions (2011) by Margaret S. Smith and Mary Kay Stein. Published by the National Council of Teachers of Mathematics (NCTM), Five Practices outlines a method that teachers can use to achieve their intended instructional goals through rich classroom discussions guided by students’ ideas.
The five practices of anticipating, monitoring, selecting, sequencing, and connecting represent a unified teaching sequence of careful planning, thoughtful observation, and logical presentation of student work. The book aligns with the NCTM’s recommendations for effective teaching in its 2014 Practices to Actions, and its approach is similar to Cognitively Guided Instruction (Carpenter et al. 2014) and the bansho (Kuehnert et al. 2018) method.
Five Practices follows the research-based educational theory of social constructivism, which states that “complex knowledge and skills are learned through social interactions” (Smith and Stein 2011, 1). According to this theory, each of us constructs our own knowledge by interacting with others, either through collaboration or with guidance from someone who knows a little more than we do. It’s the theory that gave us the idea of zone of proximal development (ZPD), which places the ideal level of instruction in between what a child knows independently and what they could learn with the help of a teacher or more advanced peers (Vadeboncoeur and Rahal 2013). Social constructivism also supports instructional strategies like small-group discussions and turn and talk.
We’ll look at each of the five practices that will show you where you can find each practice in Eureka Math2. And you’ll read about the evidence-based principles underlying the five practices, which we hope will give you an even better appreciation for our innovative, accessible, and research-based curriculum.
The first practice for orchestrating productive mathematics discussions is
“anticipating students’ solutions to a mathematics task” that is complex enough
to generate high-level student thinking and multiple solution paths (Smith and
Stein 2011, 7). When students explain their solutions in a well-orchestrated
class discussion, the teacher must balance eliciting student discourse with
achieving the goals of the lesson (Smith and Stein 2011, 2). Allowing too much
unstructured student discourse risks having a discussion with no clear resolution,
but overemphasizing lesson goals focuses the discussion on teacher explanation
instead of student exploration. So preparing for a well-orchestrated discussion
begins with anticipating what ideas students might present and considering how
those ideas align to the lesson’s goals.
Anticipating happens during lesson preparation, which Smith and Stein say is an
essential part of implementing the five practices (Smith and Stein 2011, 7). The
Eureka Math² Teach book helps teachers to think about students’ solution paths
by using overviews and the Why sections to explain the mathematics covered in
the upcoming topic, module, and lessons. The overviews and the Why section
describe the mathematical coherence and key understandings. They also provide
insights about different ways to approach problems as well as the efficiencies of
various strategies and models. The information within the overviews and the Why
section allows teachers to understand how the lessons relate to each other and to
anticipate the thinking and work that students are likely to produce as they move
through each one.
Once they understand the overall purpose of the math and their students’ potential
strategies, teachers work through the problems that their students will solve and
discuss in class. Beyond just finding the answer, teachers need to think about
the multiple solution paths that their students might take. They should consider
the ways that they can guide students who choose those paths, including how to
address incorrect or misapplied strategies (Smith and Stein 2011, 8, 36). Smith
and Stein (2011) recommend that teachers make a chart of these different paths
(9). Later on, we’ll learn more about this chart and how helpful it is to a well-
orchestrated discussion.
The authors of Five Practices also point out that the complex instructional task required for a rich mathematical discussion should engage all students (Smith and Stein 2011, 15–19). To help teachers achieve this goal, each Eureka Math² lesson highlights at least one low-floor, high-ceiling task, eliminating the challenging step of task selection or creation for teachers, so that rich and equitable mathematical discussion can blossom. For students who are English learners or not yet fluent readers, the concepts in Eureka Math² lessons are supported by digital interactives, and wordless context videos.
Once teachers look at the first problem, they can see that students can approach
it in diverse ways. That’s because the prompt guides students to draw a model but
doesn’t dictate the method:
Novice teachers will appreciate these possible solution paths, while their more seasoned colleagues may find their own examples. Teachers can use the strategies to make a chart of potential solution methods, completing this first practice and preparing them for the second practice, monitoring, which we’ll look at next.
In the first practice, anticipating, teachers read about the math in the lesson they
are about to teach and consider the different solution paths students might take,
creating a chart to summarize those paths. This work happens before the lesson
begins, and the rest of the practices occur during a lesson that’s structured around
a complex, accessible instructional task. The structure of this lesson, the authors
point out, typically contains phases to launch, explore, and then come together to
summarize the learning (Smith and Stein 2011, 1). If that sounds familiar, it should—
the Launch, Learn, Land instructional design of Eureka Math² lessons lends itself
to this mixture of collaborative work and whole-class discussion around complex
mathematical tasks.
Because monitoring occurs before the whole-class discussion, the teacher
has a chance to shape that student discourse in advance. This happens in two
important ways: collecting information about students’ solution paths and
asking questions that help students navigate and explain their chosen strategy.
Understanding which students are using which strategies will help teachers with
selecting and sequencing, the next two practices, so the authors recommend that
teachers record this information in a simple three-column chart that they began in
the first practice:
| Strategy | Who and What | Order |
During the anticipating practice, teachers recorded as many of the solution
strategies as they could imagine in the Strategy column, adding an “Other” line for
strategies that they didn’t anticipate. In the monitoring practice, the teacher notes
which students or groups are using each strategy in the second column, along with
some quick notes about how they applied the strategy to the problem. We will see
how those notes are used later on.
The practice of monitoring also allows the teacher to affect the upcoming
discussion in a different way, by asking probing questions to learn more about
students’ thinking or to guide them onto a more productive solution path. Learning
more about how a student is thinking about solving a problem is essential
formative assessment feedback. It’s also useful information for the teacher to
bring up again during the whole-class discussion to highlight the thinking behind
certain strategies. Asking questions before the discussion gives students a
chance to revise their answers (Smith and Stein 2011, 10), allowing everyone to
participate comfortably.
Asking students the right guiding questions is a delicate art. Smith and Stein
explain that good monitoring questions can guide students away from an
“unproductive or inaccurate pathway” or encourage other students to think more
deeply about their strategies (Smith and Stein 2011, 37). But questions that require
the teacher to provide the correct answer or that merely provide hints won’t
generate student-focused deep thinking and rich classroom discussion (Smith and
Stein 2011, 62). That’s why chapter 6 of Five Practices provides great guidelines
on asking questions that strike the right balance between teacher authority and
student autonomy (Smith and Stein 2011, 61–74).
Eureka Math² provides assistance on crafting questions for the monitoring practice, which begins when students start to work on the complex mathematical task, typically during the Learn segment of a lesson. Along with guidance for the next practice, selecting, this excerpt from grade 5 module 1 lesson 7 (also included in the digital platform's grade 5 module 1 lesson 7 PowerPoint Lesson Slides' Presenter Notes) offers some suggestions for advancing questions that teachers can use to uncover their students' thinking (Smith, Steele, and Raith 2017):
solution strategy when they encounter it, including how to guide students toward
applying the strategy appropriately. To help teachers with both anticipating and
monitoring, Eureka Math² includes descriptions of alternative solution paths, along
with suggestions of how to respond to different approaches. This can occur in
sample student dialogue and teacher directions, as shown in the excerpt below
from grade 5 module 2 lesson 10 (also included in the digital platform’s grade 5
module 2 lesson 10 PowerPoint Lesson Slides’ Presenter Notes):
Eureka Math² makes the monitoring practice effortless and prepares teachers for the practice of selecting.
Although it may appear shorter than the other practices in the book, selecting
requires preparation and represents a crucial component of orchestrating
productive class discussions. As the authors point out, thoughtfully choosing
student work to share allows the teacher to keep student discussion moving toward
the main ideas of the lesson.
Teachers who call on student volunteers to present their solutions are forced
to think on their feet. Sometimes this means trying to figure out an unexpected
solution path. Other times, they risk confusion by overwhelming students with
so many solution paths that they bury the point of the lesson. By choosing work
purposefully, the teacher can move the discussion in a productive direction toward
the lesson goal. Making a conscious selection in advance also benefits equity, since
the teacher can call on a student who might not otherwise volunteer, or who might
be slower to gather their courage to raise a hand (Smith and Stein 2011, 44).
Smith and Stein offer an example of a second grade classroom where the typical
“Who wants to present next?” teaching strategy went awry. As various student
volunteers shared correct approaches of counting by twos, the progression of ideas
was suddenly derailed by a student volunteer who offered a solution that was an
odd number. As she paused discussion to determine the origin of the error, the
class became focused on one student’s answer, rather than the original objective
of counting by twos.
The class was left confused about the main idea of the lesson. Had the teacher
monitored and selected student answers in advance, she could have guided the
student privately to a better solution and not asked him to present his work, thus
moving the class discussion toward its conclusion (Smith and Stein 2011, 44).
Choosing student work thoughtfully is essential in ensuring the class discussion
progresses logically toward the main idea of the class.
Teachers can make even more thoughtful choices if they have made the chart of
student strategies that the authors recommended during the first practice. Here is
what that chart might look like after the teacher has observed a few students.
| Strategy | Who and What | Order |
| Tape diagram | Shen—annotated with incorrect units |
|
| Blake—grouped correctly | ||
| Area model | Leo—unique arrangement |
|
| Riley—calculation error | ||
| Equation | Kayla—showed all steps clearly |
During the practice of selecting, teachers can fill out the second column with the student's name along with any notes about how the students are using this strategy. This method takes some preparation, however, and Eureka Math² makes the selecting practice easier in several ways.
Teacher embeds guidance in lessons to help teachers know what kind of examples to look for - and even what to do if they're not seeing any students use a particular solution strategy. These explanations often cover more than just one practice, like the examples below from lessons in grade 5 module 2. They tell teachers what to anticipate and select, and they provide us with some froeshadowing for the last two practices of sequencing and connecting. This highlights how interconnected the five practices are, and how it's often hard to talk about one practice in isolation.
Smith and Stein explain that sequencing accomplishes two goals: improving the
accessibility of the mathematics and creating a logical flow to the presentation
of student work (Smith and Stein 2011, 11, 44). By presenting a straightforward
example first, teachers can help make connections to more complex or unexpected
solution paths. Once this entry point is chosen, sequencing of student work is
a matter of deciding how to build to the lesson’s desired mathematical goals.
As the authors phrase it, sequencing reflects how the teacher wants “to build a
mathematically coherent story line” (Smith and Stein 2011, 44).
Just as there are many ways to tell a story, there is no one way to sequence student
responses (Smith and Stein 2011, 49). In general, teachers should start with the
most accessible strategies and proceed toward more complex and abstract ones.
However, the decision depends on a teacher’s style and the makeup of their class.
For a class familiar with using tape diagrams for division, problems might begin with
that representation to ease into a new concept. Teachers can start by highlighting
a common error or an innovative approach from a student—or save that innovation
for last.
No matter what sequence they choose, the order is going to change for each class
and each group of students. This is because the secret power of sequencing is
guiding students to tell that story in their own way. When students explain their
mathematical understanding in a way that makes sense to them, they grasp the
concepts more deeply and more clearly. And when they hear their classmates explain the ideas in their own way and then respond to those explanations, they not only hear their classmates' stories but also learn about cooperation, argumentation, and reasoning (Chapin, O'Connor, and Anderson 2009, 7).
Though it might seem like the orchestration of productive mathematical discussion is now complete, there's one more essential step to bring it all home and build mathematical knowledge effectively—and that's covered next.
These connections show how the different student work examples relate to one
another, to concepts students have previously learned, and to the new concepts
they are learning. Consistent with the underlying social constructivist theory,
however, teachers can’t just tell students what these connections are. They need
to ask students the right questions so that students can discuss and understand
the ideas themselves. And that, Smith and Stein (2011) point out, makes this the
hardest practice of all (49).
The questions are so difficult to formulate because they must be grounded in the students' work to be effective, and they must start with what students already know (Smith and Stein 2011, 50). If that sounds like a lot to pack into a question, it is. And that's why every Land segment of every Eureka Math² lesson has already formulated these questions for the teacher.
The Debrief for grade 7 module 3 lesson 17 shows how the questions invite students to connect familiar concepts—such as proportional relationships, equivalent ratios, and if-then moves for solving equations—with new learning: equations in the form a/b=c/d.
Topic overviews outline the major concepts of the module and how they build on each other. Here is part of the Topic C narrative:
Within the Fluency section of each Eureka Math² lesson, there are practice opportunities on previously learned skills that set the foundation for what’s to come.
This Differentiation Support note, shown below, incorporates a helpful and relevant strategy, first introduced in a prior grade level, to support students who may need extra help.
The lessons themselves contain many connections that teachers can make explicit to their students. In this lesson, students are asked to contrast different strategies as a way of seeing the connections between the two solution paths.
These questions are samples of what could be asked based on these specific
examples, but the questions illustrate how making connections in classroom
discussions affect students. As students are asked about their own work and the
work of others, they consider the choices they made and defend the logic behind
them. They see their own thinking validated by the teacher, even if the thinking is
flawed or applies in a different context, and they gain clarity and self-confidence
as they make sense of the math with their peers. Eureka Math² demonstrates the
approaches in every lesson with these example discussions, anchoring them in the
students’ work and explaining concepts, both to the teacher and to the student.
Whatever ideas teachers want to connect can be easily highlighted in the digital interactive by the teacher’s careful choice and sequence of student work. As they select student work, teachers have to think about how the work expresses and connects to important mathematical concepts that thread through students’ mathematical journeys. And because the Eureka Math² text makes so many of these connections explicit and explains them, it’s so much easier to see how student work illustrates these concepts.
Beyond mathematical learning and achievement, productive classroom discussions
can build many other skills, like collaboration, self-confidence, and argumentation
(Chapin, O’Connor, and Anderson 2009, 8–9, 165–7). Discussions like these help
students to write their own mathematical story, to see math through many different
lenses, and to think about themselves and the world around them in a completely
different way. There are so many reasons for teachers to build these essential
classroom discussions, and the five practices will help teachers conduct them
effectively, with plenty of support throughout Eureka Math².