Strategies for Infusing Instruction
with Mathematical Practices
Samuel Otten
University of Missouri
ottensa@missouri.edu
National Council of Teachers of Mathematics
Regional Conference in Louisville, KY
November 8th, 2013
Introduction

Math education is more about what we
have students doing than it is about what
content they are learning.
◦ For example, if we’re teaching content
standard N-RN.1, are we going to have
students sit quietly and receive information or
are we going to activate students as thinkers
and problem solvers?
2
Introduction

Research over several decades has shown
that the way students engage in math
class impacts their attitudes and their
learning of content. (Boaler & Staples, 2008;
Hiebert & Grouws, 2007; Stein, Grover, & Henningsen, 1996)

The mathematical practices are the
official encapsulation of what students
should be doing.
3
Introduction
KEY QUESTION: What can we be doing
as teachers to infuse these practices into
our teaching?
 Or… What more can we be doing to
better infuse these practices into our
teaching?

4
Session Overview
Quick look at the practices
 Levels of Implementation

◦
◦
◦
◦

Classroom Culture
Discourse Patterns
Teacher Questions & Discourse Moves
Task Design and Selection
Conclusion
5
Standards for Mathematical Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere
in solving them
Reason abstractly and quantitatively
Construct viable arguments and
critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in
repeated reasoning
Common Core State Standards for Mathematics (2010)
6
MP6. Attend to Precision

Two types of attention to precision
◦ Numerical or measurement
 Clever estimation (and awareness that it’s an
estimate)
 Awareness of exact answers
 Significant digits and measurement error
◦ Language
 Precise definitions
 Say what you mean and mean what you say (both in
words and in symbols)
 Communication Process Standard
7
MP7. Look for and Use Structure
Students must first realize that there is
structure to be found or else they won’t
know to look for it. Math makes sense.
 Looking for structure is a habit of mind
that can be very helpful for learning.

◦ The structures themselves are often the key
mathematical ideas that we want students to
see.
◦ Structures also often unlock problems or can
be the basis of reasoning.
8
MP7. Look for and Use Structure

Examples of structures
◦ Components of algebraic expressions
◦ Factors of polynomial coefficients
◦ Symmetries in graphs or geometric objects

Process-Object distinction (Sfard, 1991)
◦ What is initially learned as a process (e.g., taking a
square root, using a function rule) eventually
becomes a mathematical object in its own right
(e.g., a radical term, a function that can be added,
multiplied, or composed with other functions).

This practice also involves students shifting
perspective and seeing the bigger picture.
9
M8. Look for and Express Regularity
in Repeated Reasoning

Noticing repetitions or regularity
◦ Most patterns/repetitions in mathematics are not
coincidental
◦ Expressing a repetition or pattern (in multiple ways?)
can be a vehicle for moving mathematical ideas
forward.

Most common form of this practice is
having students generalize and make
conjectures
◦ Leads nicely to Practice 3: Constructing viable
arguments.
10
M8. Look for and Express Regularity
in Repeated Reasoning

Reasoning itself can have regularity (e.g.,
problem types, inverse operations, proof
approaches)
◦ Great topic for Review or Going Over Homework

Metacognition
◦ stepping out and looking at the outcomes and
process of reasoning
11
Are MP7 and MP8 distinct?
I say “yes,” others say “no.” The answer
may not matter because the important
thing is for practices to be happening, not
identifying which specific practice it might
be.
 But here’s my take…

◦ MP7 (Structure) focuses on mathematical
objects, whereas MP8 (Regularity) focuses on
repetitions in process or thinking.
◦ Although distinct, they do often co-occur.
12
CLASSROOM CULTURE
13
General Characteristics of
Classroom Culture
•
Safe environment to share ideas
•
Errors or confusions are met with
excitement as learning/thinking
opportunities
14
Math-Specific Characteristics of
Classroom Culture
•
Careful thinking is pervasive
•
Students have openings and time to
communicate their mathematical ideas (and
to consider or respond to other’s ideas)
15
Culture Should Not Be…
•
Answer-focused
•
Correctness-focused
•
Rushed
Grouws et al. (2013) regarding “coverage”
16
Learning from One Another
•
What strategies have you found successful
in promoting a practices-oriented
classroom culture?
17
DISCOURSE PATTERNS
18
Initiate-Respond-Evaluate
(I) Teacher asks a question
• (R) Students gives an answer
• (E) Teacher evaluates the answer
•
•
The predominance of this interaction pattern
tends to emphasize answers (R) and
correctness (E). Can be efficient but also
makes things feel “on the clock.”
19
Herbel-Eisenmann & Breyfogle (2005)
Funneling
•
Interaction wherein a person (teacher)
asks a series of questions but the questions
themselves contain the important
mathematical ideas and the student’s
answers are low-level or unrelated to the
important ideas.
Example: Solving 6x + 18 = 36 – 12x
•
The asker is coopting the practices and
lowering the cognitive demand on the other(s).
20
Herbel-Eisenmann & Breyfogle (2005)
Focusing
•
Rather than funneling, a person (teacher)
asks questions designed to focus the
student’s attention on the important
mathematical ideas or on something that is
likely to help the student move forward.
•
The asker is offering help but still leaving the
mathematical practices for the student.
21
Going Over Homework

The typical discourse of homework review in
middle school and high school math classrooms
involves attention on one problem at a time.
An alternative discourse pattern is to look for
patterns across problems, compare/contrast
problems, or attend to the mathematical ideas
of the assignment as a whole.
 This alternative leads to learning gains and can
promote practices such as MP1, MP7, and MP8.

Otten, Herbel-Eisenmann, & Cirillo (in press)
Jitendra et al. (2009)
22
Learning from One Another
•
•
What experiences have you had with
focusing interactions or other discourse
patterns that promote the practices?
In what ways do you structure your
homework review to promote the
practices?
23
TEACHER QUESTIONS
& DISCOURSE MOVES
24
Types of Teacher Questions
Inauthentic Questions
•
•
•
The asker already knows the answer
Example: “What is the y-intercept of that graph?”
Function: Mini-quiz of responder’s knowledge
Authentic Questions
•
•
•
•
The asker does not already know the answer
Example: “How did you think about that graph?”
Function: Invite the responder into dialogue
More aligned with the infusion of the math practices
25
A Simple Fact
“Why” questions from teachers…
…lead to “Because” responses from students.
26
Teacher Discourse Moves (TDMs)
Inviting student participation
 Waiting
 Revoicing
 Asking students to revoice
 Probing a student’s thinking
 Creating opportunities to engage with
another’s reasoning

www.mdisc.org
27
Teacher Discourse Moves (TDMs)
This set of discourse moves can be used
to increase the quantity of talk in the
classroom and also channel that talk in
mathematically productive directions.
 The original “talk moves” have been tied
to significant learning gains in urban
districts in math and in English!

www.mdisc.org
Chapin, O’Connor, & Anderson (2009)
28
Learning from One Another
What questions or discourse moves have you
used to promote the mathematical practices?
29
TASK DESIGN &
SELECTION
30
High Cognitive Demand Tasks

Can provide opportunities to engage in the
mathematical practices.
Smith & Stein (1998)
31
Doing Mathematics Tasks…






Require complex and nonalgorithmic thinking—not predictable
or well-rehearsed approaches.
Require students to explore and understand mathematical
concepts, processes, or relationships.
Demand self-monitoring or self-regulation of one’s own thinking.
Require students to access relevant knowledge and experiences
and make appropriate use of them in working through the task.
Require students to analyze the task and actively examine task
constraints that may limit possible solutions.
Require considerable cognitive effort and may involve some level
of anxiety because of the unpredictable nature of the solution
process.
Smith & Stein (1998)
32
Low Cognitive Demand Tasks

…can also be great opportunities to engage
in the mathematical practices, especially
MP8 – Look for and express regularity in
repeated reasoning
◦ As students complete exercises or execute
procedures, they can be thinking about…
 Short-cuts
 Patterns
 Generalizations

Attending to these things can also raise the
cognitive demand as things play out.
33
Reversing Problems

Most problem-types have a canonical
“direction.” For example…
◦ Start with an equation and find x.
◦ Start with a Given & To Prove and write a proof.
◦ Start with some information and find the missing
information
◦ Start with a series and express the pattern

Reversing that direction can be a great
way to infuse the mathematical practices
into a lesson
34
Background on the Task



Grades 6–7
Standards for Mathematical Practice
◦ MP1: Problem Solving
◦ MP6: Attend to Precision
◦ MP7: Look For and Make Use of Structure
Content
◦ 6.SP.3 – Recognize that a measure of center for a numerical data
set summarizes all of its values with a single number, while a
measure of variation describes how its values vary with a single
number.
◦ 6.SP.5c – Summarize numeral data sets, such as by giving
quantitative measures of center and variability, as well as describing
any overall pattern and any striking deviations from the overall
pattern with reference to the context of the data.
◦ Note: “Mode” is not explicitly in the Common Core Standards.
Reversed Data Set Task

Make up a set of eight numbers that
simultaneously satisfy these constraints:
◦
◦
◦
◦
Mean: 10
Median: 9
Mode: 7
Range: 15
http://mathpractices.edc.org
Reflecting on our work

What are differences between this
problem and one that gives a data set and
asks for the statistical measures?

How do the differences impact students’
engagement in the practices?
A Few Other Ideas About Tasks
Engage students in the process of “welldefining” a problem
 Build in time to look back at students’
work on a task to make explicit to them
that they were engaging in mathematical
practices
 Look across problems to promote
practices and deepen learning

38
CONCLUSION
39
Conclusion



The CCSSM practices are in danger of falling into
the background, with the content standards
dominating the foreground
But the practices are arguably the most important
aspect of CCSSM in terms of promoting student
learning and attitudes toward mathematics
Although we are already implementing the
practices in certain ways, we can all continue to
improve in this area by focusing on
◦ Classroom Culture;
◦ Discourse Patterns;
◦ Teacher Questions and Discourse Moves; or
◦ Task Design and Selection
40
Thank you!







ottensa@missouri.edu
Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable
teaching approach: The case of Railside School. Teachers College Record, 110, 608-645.
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on
students' learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics
teaching and learning (pp. 371-404). Charlotte, NC: Information Age Publishing.
Koestler, C., Felton, M. D., Bieda, K. N., & Otten, S. (in press). Connecting the NCTM
Process Standards and the Common Core State Standards for Mathematical Practice to
Improve Instruction. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
National Governors Association & Council of Chief State School Officers. (2010).
Common Core State Standards for Mathematics. Washington, DC.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on
processes and objects as different sides of the same coin. Educational Studies in
Mathematics, 22, 1-36.
Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for
mathematical thinking and reasoning: An analysis of mathematical tasks used in reform
classrooms. American Educational Research Journal, 33, 455-488.
41