Title: Principles
to Actions: Ensuring Mathematical Success for All
Author: National Council
of Teachers of Mathematics
Published: April
2014
NCTM Stock#: 14861
ISBN #: 9780873537742
Effective Teaching and Learning
Mathematics
Teaching Practice: Establish mathematics goals to focus learning.
 Tasks
and Questions for Reflection:
 Work
together with your team to provide an example of a goal statement of the type
described (p. 12–14, Discussion).
 How
can the development of specific math goals support other practices for
effective teaching and learning? (p. 12–14, Discussion)
 In
figure 2 (pp. 14–15), Mrs. Burke says that she wants students to “better
understand these different types of word problems and be able to solve them.”
Find solutions for each of the three problems in the figure. What equations
could be written to solve each problem? Which equations match the story
situation? Discuss how these three problems offer different ways to think about
subtraction.
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles).
What productive beliefs are demonstrated in the conversation in figure 2? How
do those beliefs move the planning forward?
 Connections
to Other Mathematics Teaching Practices:
Implement
tasks that promote reasoning and problem solving. Describe
teacher actions that would support or undercut reasoning and problem solving in
the case of the three math problems in figure 2.
Use
and connect mathematical representations. Solve
the three problems in figure 2 in more than one way. Compare your strategies
with those of others. Discuss how they are related.
Pose
purposeful questions. Review
the teachers’ mathematical goal for the lesson in figure 2. What questions can
the teachers plan to ask students during the lesson to advance them toward this
goal?
 Application
to practice:
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
16) to evaluate how the lesson applies the Mathematical Teaching Practice Establish mathematics goals to focus
learning. What evidence do you see of the teacher and student actions identified
in the chart? Where do you see missed opportunities for these teacher and
student actions? Give specific examples of evidence of this Mathematics
Teaching Practice and ways to enhance the practice in future lessons.
Mathematics
Teaching Practice: Implement tasks that promote reasoning and problem solving.
 Tasks
and Questions for Reflection:
 Work
individually or with a partner to solve task A: Smartphone Plans in figure 5
(p. 20). After explaining your decision about which phone plan is better, solve
the problem in a different way. Share solutions with your team and discuss the
different strategies used, the multiple entry points that the problem offers,
and the different ways that the task promotes reasoning. Compare task A with
task B in these respects.
 What
prior knowledge or experience would students need to solve the tasks in Figures
5, 6, and 7?
 What
are the characteristics of a task that places a highlevel cognitive demand on
students?
 How
could you take a lowlevel task and increase its cognitive demand? (Consider
rewriting a sample task or textbook problem.)
 What
types of questions could you ask, or what types of moves could you make, to
support students who struggle to get started on a problemsolving task, without
diminishing the cognitive demand of that task?
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles)
and examine figure 8 (p. 23), which presents two algebra classrooms using the
Smartphone Plans task. What beliefs are evident in Ms. Carson’s and Ms.
McDonald’s classrooms? What impact do those beliefs have on students’
opportunities for reasoning and problem solving in the lesson?
 Connections
to Other Mathematics Teaching Practices:
Establish
mathematical goals to focus learning. Consider
the problem in figure 7. Identify a mathematical goal that this problem might
support.
Use
and connect mathematical representations. Review
the Smartphone Plans task (task A in figure 4) or the task on number pairs that
make 10 (figure 7). What representations could students use to solve the
problem? Show how students might use different representations in solving.
Discuss the relationships among all the representations generated for the
problem.
Pose
purposeful questions. Review
the synopsis of Ms. McDonald’s classroom in figure 8. What questions does Ms.
McDonald ask her students and why? How do these questions engage and challenge
her students?
 Application
to practice:
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
24) to evaluate how the lesson applies the Mathematical Teaching Practice Implement tasks that promote reasoning and
problem solving. What evidence do you see of the teacher and student
actions listed in the chart? Where do you see missed opportunities for these
teacher and student actions? Give specific examples of evidence of this
Mathematics Teaching Practice and ways to enhance the practice in future
lessons.
Mathematics
Teaching Practice: Use and connect mathematical representations.
 Tasks
and Questions for Reflection:

Revisit
the three problems for secondgrade students in figure 2 (pp. 14–15). Show how
students might solve each problem by using different representations. Discuss
the relationships among all the representations generated for each problem.
 Look
at the parenthetical example on p. 26 about describing a realworld situation
for 3 x 29 or y = 3x + 5. Use contexts or representations
to show how the expression 3 x 29 is related to the equation y = 3x
+ 5.
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles).
What productive beliefs are evident in the Mr. Harris’s classroom, shown in
figure 10 (pp. 27–28)? How do those beliefs support students in making
connections among different representations of the problem?
 Analyze
samples of student work from a lesson that you have taught this year. Find
examples in which students have used different representations to solve the
same problem. Make a plan to connect those representations explicitly in future
lessons. Find relationships between and among the representations and think
about how you could use the students’ work to develop their understanding of a
concept.
 Application
to practice:
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
16) to evaluate how the lesson applies the Mathematical Teaching Practice Use and connect mathematical representations.
What evidence do you see of the teacher and student actions identified in the
chart? Where do you see missed opportunities for these teacher and student
actions? Give specific examples of evidence of this Mathematics Teaching
Practice and ways to enhance the practice in future lessons.
Mathematics
Teaching Practice: Facilitate meaningful mathematical discourse.
 Tasks
and Questions for Reflection:
 Simply
having students talk does not necessarily advance the mathematical goals of a
lesson. How can the five practices identified on page 30, as described by Smith
and Stein (2011), support and facilitate the purposeful exchange of ideas in
the mathematics classroom?
 Review
the Candy Jar task in figure 12 (p. 31) and the conversation from Mr.
Donnelly’s implementation of the Candy Jar task, shown in figure 13 (pp.
33–34). What do the authors mean when they say, “Mr. Donnelly facilitates
rather than directs” this discussion (p. 34)? Give specific examples.
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles).
What productive beliefs are evident in the Mr. Donnelly’s classroom (see fig.
13)? What impact do those beliefs have on the classroom discourse?
 Discuss
how the use of different talk structures (wholeclass, smallgroup,
teacherled, studentled, etc.) can affect mathematical discourse in the
classroom.
 Connections
to Other Mathematics Teaching Practices:
Establish
mathematical goals to focus learning. Review
the illustration of Mr. Donnelly’s seventhgrade class, shown in figure 13 (p.
33–34). What is the mathematical goal of this task? Discuss how Mr. Donnelly’s
actions move the class toward this goal.
Elicit
and use evidence of student thinking. Discuss
how the purposeful exchange of ideas in the mathematics classroom can create
opportunities to elicit and use evidence of student thinking.
 Application
to practice:
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
35) to evaluate how the lesson applies the Mathematical Teaching Practice Facilitate meaningful mathematical discourse.
What evidence do you see of the teacher and student actions identified in the
chart? Where do you see missed opportunities for these teacher and student
actions? Give specific examples of evidence of this Mathematics Teaching Practice
and ways to enhance the practice in future lessons.
Mathematics
Teaching Practice: Pose purposeful questions.
 Tasks
and Questions for Reflection:
 Teachers
use a variety of questions in their instruction (see fig. 14, pp. 36–37),
including questions that should elicit mathematical reasoning and
justification. Unfortunately, teachers too often employ these questions in a
“funneling” manner (see fig. 16, pp. 39–40). Brainstorm with your team to
identify barriers that might prevent teachers from moving from “funneling” to
“focusing” questions.
 Learning
involves a cognitive reorganization of individual beliefs. This reorganization
demands some degree of dissonance. How do funneling questions discourage
dissonance and how do focusing questions encourage dissonance?
 In
questioning small groups of students working on a problem, a teacher noticed
that when she asked a “focusing” question, the students continued to look at
their work and continued to engage in their own dialogue. When she asked a
“funneling” question, the students looked up at the teacher. Comment on these
observations.
 Identify
a math task that you might give to your students. State the learning goal, and
then use the task to create a list of related questions using the framework in
figure 14 (pp. 36–37). It will be helpful to first anticipate likely student responses and misconceptions (see Smith
& Stein’s practice 1, p. 30).
 If your district uses a specific
framework for questioning (e.g., Bloom’s Taxonomy or Webb’s Depth of
Knowledge), compare that framework with the framework of types of questions
shown in figure 14 (pp. 36–37). Discuss any connections.
 Review
the patterns of questions about the Coin Circulation task posed in each
classroom, as shown in figure 16 (pp. 38–39). Classify the question types
according to the framework in figure 14. What do you notice?
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles).
What beliefs are evident in the two questioning examples shown in figure 16?
What impact do those beliefs have on students’ opportunities to make sense of
the mathematics and advance their reasoning?
 Connections
to Other Mathematics Teaching Practices:
Establish
mathematical goals to focus learning. Why
is it essential for teachers to have a clear learning goal when facilitating
discussion that focuses on student thinking?
Elicit
and use evidence of student thinking. Review
the patterns of questioning in figure 16. Students in both classrooms notice
that range for the middle 50 percent of the pennies is from 3 to 19 years old,
observed by student 5 (S5) in the funneling example and by student 2 (S2) in
the focusing example. Compare the reactions of the two teachers to these
students’ observations. How might the teacher in the funneling example have
responded to student 5 in a different way to gain a better understanding of
what the student or other students knew and understood?
 Application
to practice:
 Observe
or record a mathematics lesson. Use the framework for types of questions used
in mathematics teaching (fig. 14, p. 36) to evaluate questions asked during the
lesson. Tally questions by type. Provide specific examples of questions for
each type. Reflect on the distribution of questioning and how the specific
questions cited advanced or limited progress toward the mathematical goals of
the lessons.
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
41) to evaluate how the lesson applies the Mathematical Teaching Practice Pose purposeful questions. What evidence
do you see of the teacher and student actions identified in the chart? Where do
you see missed opportunities for these teacher and student actions? Give
specific examples of evidence of this Mathematics Teaching Practice and ways to
enhance the practice in future lessons.
Mathematics
Teaching Practice: Build procedural fluency from conceptual understanding.
 Tasks
and Questions for Reflection:
 Consider a right triangle in the first quadrant of the coordinate plane (see the example below). Label each point. Discuss with your team how the triangle’s measurements are related to the distance formula (shown on p. 44). Specifically, what two x values are represented in the coordinates of your right triangle? What two y values are represented? What is the difference between the two x values? How does that relate to the measurements of the triangle? What is the difference between the two y values? How does that relate to the measurements of the triangle? How would you find the length of the triangle’s hypotenuse? And so on.

 Review the different methods for multidigit multiplication shown in figure 18 (p. 45). Discuss how the methods are interrelated. For example, the traditional algorithm gives a partial product of 368. Where can you find 368 in the other methods?
 Review the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles). What beliefs are evident in Mr. Donnelly’s implementation of the Candy Jar task (p. 46, Illustration, see also figs. 12, 13, and 19)? What impact do those beliefs have on students’ opportunities for reasoning and problem solving in the lesson?
 Connections to Other Mathematics Teaching Practices:
Use and connect mathematical representations. Look at David’s and Anna’s work in figure 17 (p. 43). How could Anna’s reasoning help David understand his mistake? What other representations could the teacher use to support the students’ thinking here?
Pose purposeful questions. What are some common mistakes that students make when solving multidigit multiplication problems like 68 x 46? Give examples, and discuss questions that teachers could ask students in each case. Explain the purpose of and possible responses to each question.
Elicit and use evidence of student thinking. Suppose that a student multiplies 68 x 46 and gets 2448. What error has the student made? How could a teacher use one or more of the methods shown in figure 18 to help this student?
 Application to practice:
 Observe or record a mathematics lesson. Use the “Teacher and student actions” chart (pp. 47–48) to evaluate how the lesson applies the Mathematical Teaching Practice Build procedural fluency from conceptual understanding. What evidence do you see of the teacher and student actions identified in the chart? Where do you see missed opportunities for these teacher and student actions? Give specific examples of evidence of this Mathematics Teaching Practice and ways to enhance the practice in future lessons.
Mathematics
Teaching Practice: Support productive struggle in learning mathematics.
 Tasks
and Questions for Reflection:
 Review
the problemsolving strategies suggested by Ms. Ramirez’s students in figure 21
(p. 51). Solve the problem by using each of these studentsuggested strategies.
 Review
the video “My Favorite No: Learning From Mistakes” (https://www.teachingchannel.org/videos/classwarmuproutine). Choose a common student error and create a
“favorite no” for the problem presented in figure 21. Why is this common error
useful to know?
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles).
What beliefs are evident in Ms. Flahive’s and Ms. Ramirez’s classrooms (see
fig. 21)? What impact do those beliefs have on students’ opportunities to
grapple with the mathematical ideas and relationships in the problem?
 Connections
to Other Mathematics Teaching Practices:
Use
and connect mathematical representations. Read
the problem presented in figure 21. Show how students might solve the problem
by using different representations. Discuss the relationships among all the
representations generated for the problem.
Pose
purposeful questions. Consider
the students’ strategies in Ms. Ramirez’s class (fig. 21), and give examples of
questions that teachers might pose to facilitate their reasoning and
perseverance.
 Application
to practice:
 Observe
or record a mathematics lesson. Figure 20 (p. 49) presents a chart adapted from
Smith (2000), which redefines student and teacher success in relation to
teachers’ expectations and actions and indicators of success. Use the chart to
evaluate teacher and student attitudes that are evident in the lesson. Provide
specific examples. Reflect on how teacher and student attitudes advanced or
limited progress toward the mathematical goals of the lesson
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
52) to evaluate how the lesson applies the Mathematical Teaching Practice Support productive struggle in learning
mathematics. What evidence do you see of the teacher and student actions
identified in the chart? Where do you see missed opportunities for these
teacher and student actions? Give specific examples of evidence of Mathematics
Teaching Practice and ways to enhance the practice in future lessons.
Mathematics
Teaching Practice: Elicit and use evidence of student thinking.
 Tasks
and Questions for Reflection:
 Discuss
how other mathematical teaching practices, such as selecting tasks,
facilitating discourse, and posing questions, are connected to this practice.
 Review
the “Beliefs about teaching and learning mathematics” chart (p. 11, Obstacles).
What beliefs are evident in Ms. Lewis’s classroom in figure 22 (pp. 55–56)?
What impact do those beliefs have on the teacher’s ability assess her students’
understanding and make appropriate adjustments to her instruction?
 Connections
to Other Mathematics Teaching Practices:
Use
and connect mathematical representations. Look
at Maddie’s work and Gabe’s work, shown in figure 22. How could the teacher
leverage the students’ representations to develop Maddie’s understanding of the
problem?
Build
procedural fluency from conceptual understanding Look
again at Maddie’s work and Gabe’s work, shown in figure 22. What conceptual
understandings do the children demonstrate? What understanding does Maddie
appear to be missing?
 Application
to practice:
 Observe
or record a mathematics lesson. Use the “Teacher and student actions” chart (p.
56) to evaluate how the lesson applies the Mathematical Teaching Practice Elicit and use evidence of student thinking.
What evidence do you see of the teacher and student actions identified in the
chart? Where do you see missed opportunities for these teacher and student
actions? Give specific examples of evidence of this Mathematics Teaching
Practice and ways to enhance the practice in future lessons.
Essential
Elements
Access
and Equity
 Read
the full statement of the Access and Equity Principle (the three blue lines in
italics on p. 59) three times: first, aloud at your table, second, silently to
yourself, and finally, aloud to the whole group.
 With
a partner, and in your own words, write a sentence that conveys the same ideas
as the full statement of the Access and Equity Principle (the three blue lines
in italics on p. 59).
 Referring
to the full statement of the Access and Equity Principle, explain how highquality mathematics differs from advanced mathematics?
 Justify
the assertion that every student needs to know something about exponential
growth. Not all students need to learn
the same concepts and take the same pathway to understanding. Develop a
threetiered activity to allow ALL students access to the concept of
exponential growth.
 What
are
the biggest obstacles that you face in ensuring access and equity for all
students? The authors note that, in many classrooms, the Mathematical Teaching
Practices described in this document are inconsistently or ineffectively
implemented (p. 61). Discuss how
specific changes in teaching practices can help to overcome the obstacles you
identified.
 Looking
at the “Beliefs about access and equity in mathematics” chart (pp. 63–64), can
you find an unproductive belief and a productive belief that you can relate to?
Be prepared to share your thoughts with your team.
 Briefly
discuss why Principles to Actions places
Access and Equity as the first Essential Element?
Curriculum
 If
the Common Core State Standards drive the content that you teach or will teach,
what resources would you like and need to help ALL students meet those
standards? How would these resources affect your teaching practices?
 What
needs to be in a curriculum to guarantee that students develop fluency with the
processes and practices identified in the Standards for Mathematical Practice
in the Common Core State Standards?
 What
would you need to change in your personal style of teaching mathematics to
overcome some of the obstacles to achieving the Curriculum Principle?
 With
a partner, outline steps or actions that you intend to take to reshape your
curriculum for a closer match with the vision captured in the full statement of
the Curriculum Principle (three blue lines in italics on p. 70)? Pick one
action that you will DO, and share how you will know you have reached that
goal.
Tools
and Technology
 The
opening sentence on the Tools and Technology Principal states, “For meaningful
learning of mathematics, tools and technology must be indispensable features of
the classroom” (p. 78). What does, or would, such a classroom look like and
sound like?
 Imagine
that cost is not an issue, and develop a list of personal obstacles that might
still keep you from being part of the classroom that you described above?
 How
could you or others overcome these obstacles? Make a list of ALL pathways for
overcoming the obstacles. When you have your list, circle three pathways that you
could travel down to make the classroom that you described above a reality.
 A
useful question to consider is, “Are you abusing the technology or using it to
make sense of mathematical ideas?” Define the difference between abusing technology
and using it in sense making by writing a quick lesson on a highrigor topic of
your choice.
Assessment
 Read
the “Obstacles” section (pp. 89–92) for the Assessment Principle. Underline or
highlight three ideas that resonated with you, and write down one question that
you have about obstacles to effective assessment. Share your ideas and question
with four different people.
 With
a partner, make a detailed list of actions that you or others could take to
overcome some of the obstacles shared with you.
Professionalism
 Before
reading the section on the Professionalism Principle, write out your personal
definition of professionalism.
 Read
the section on the Professionalism Principle (pp. 99–108), and then share some
of the obstacles that you face in professional collaboration.
 Using
the “Beliefs about professionalism in mathematics education” chart (pp.
102–103), engage with a partner in a friendly discussion in which one of you
defends “unproductive beliefs” and the other defends “productive beliefs.”
Alternate positions and discuss every pair of beliefs.
 Coaching
is a key to overcoming obstacles. How could coaching be (or how is it) a
positive force in your school’s efforts to overcome obstacles? How could it be
(or how is it) a positive force in your personal growth as a teacher?
 Review
your definition of professionalism and
make revisions if you wish. Share your revised definition with your team.
 With
your team, come up with at least two actions that team members believe will
improve their professionalism.