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To Tailor PD, D.C. Looks to Groom Teacher LeadersAn initiative in the nation's capital aims to give select teachers more time to work with colleagues

By Anthony Rebora

September 28, 2015


On a sweltering morning in late July, dozens of stalwart District of Columbia teachers were gathered around tables in an air-conditioning-challenged elementary school gymnasium, laptops and water bottles spread before them.

"I want you to close your eyes … and imagine you are a teacher and one of your colleagues is going to be your coach," said presenter Matt Radigan, a leadership coach with the New Orleans-based nonprofit Leading Educators. "What would you want in an initial conversation?"

It was an apt reflection. The educators in the room were all newly minted teacher leaders, charged with providing hands-on mentoring to their peers and helping their schools reach their learning objectives in the year ahead. Over the course of this weeklong training at HD Cooke Elementary School, they would dissect some of the finer points of instructional coaching, learning how to craft conversations with skeptical colleagues, gather evidence from classroom observations, and provide nuanced feedback.

The summer coaching intensive is a key component of the District of Columbia's Teacher Leadership Innovation program, or TLI, a 2-year-old initiative designed to provide structural support for teacher-leader roles in schools. At the heart of the undertaking is the idea that, given time and resources, expert teachers can play an instrumental role in driving overall school improvement at a time of mounting instructional challenges, including integration of the Common Core State Standards.

"First and foremost, it's about expanding the reach of great teachers and ensuring we get more feedback to more teachers," said Katie Michaels Burke, the district's director of teacher leadership. "It's a way of pushing great practice."

The program in the District of Columbia is one of a small but growing number of initiatives around the country aimed at giving more concrete and practical form to "teacher leadership," a term that has often remained vague in U.S. schools despite its rising cachet.

Formal efforts to give some teachers advanced roles outside the classroom are not new, extending back at least to the 1980s. But experts say that, though they are not without inherent risks, such programs have gained standing in recent years as school leaders and policymakers have sought both to provide principals with greater bandwidth to make instructional changes and give accomplished teachers more opportunities for career development. They've also been given a distinct boost by the Teacher Incentive Fund, or TIF, a federal grant program established in 2006 to provide financing for districts to establish performance-based-pay programs for teachers.

More Than a Title

The District of Columbia's initiative is backed in large part by a five-year TIF grant and operated in partnership with Leading Educators, which specializes in creating leadership-development models for teachers. The organization supports similar programs in Kansas City, Mo.; Memphis, Tenn.; New Orleans; and New York.

The program in the nation's capital is voluntary for schools. A total of 29 of the district's 111 schools—with 105 teacher leaders—are participating, up from 21 schools last year.

One explicit goal of the TLI is to make "teacher leader" more than an adulatory moniker. Teachers chosen for the position are given an annual stipend of $2,500—not a life-changing amount, district officials concede, but on par with raises given for other promotions in the system. Perhaps no less important, the teachers have significant stretches of release time—as much as half their total hours—built into their schedules to observe and work with their colleagues.

There is also the training aspect. The coaching intensive was the second of two weeklong sessions held for new teacher leaders this past summer. In the first, the teachers worked with facilitators and their schools' leadership teams to frame out how the new arrangement would work in the year ahead, developing school and individual action plans, detailing teacher-practice and student-learning goals, and aligning schedules. (Picture classroom walls strewn with dense networks of sticky notes and marked-up easel-pad pages.)

During the school year, the teacher leaders meet weekly with coaches from Leading Educators to review their progress, analyze data, and hone their techniques. They also attend quarterly group-training sessions organized by the district on coaching and leadership.

While some prominent national teacher-leadership efforts are predicated on giving accomplished teachers a greater voice in education policymaking, the District of Columbia's TLI program focuses on harnessing their instructional know-how.


District of Columbia teacher leaders study documentation during the summer training on instructional coaching at HD Cooke Elementary School. The district’s Teacher Leadership Innovation initiative is intended leverage teacher leaders’ experience to create more customized, school-specific professional-development opportunities.

—T.J. Kirkpatrick for Education Week

In particular, the program seeks to leverage teacher-leaders' experience to create a model for more customized, school-specific professional-development opportunities in schools.

Since the teacher leaders are working alongside their mentees, the "PD [is] much more targeted to schools' needs, and much more honed to the teacher-practice needs," said Burke, the district's teacher-leadership director. "This isn't some person from central office coming in and saying, 'This is what we're doing now.' "

The educators involved in the program tend to subscribe to this notion instinctively.

Rhonda Ferguson, a new teacher leader in early childhood at Turner Elementary, a high-poverty school on the city's southeast side, believes that the teachers she's supervising will have an advantage she didn't have in her first years in the classroom.

"They'll have someone who understands the work we do and what they're going through," she said. "I'll be helping them figure out how to make these things work—able to spell out priorities while also being able to step back if needed and look at the individual needs of the teachers."

For school administrators, meanwhile, the prospect of expanded instructional-leadership capacity within a school can be difficult to pass up. Ferguson's principal, Eric Bethel, says that the primary reason he chose to adopt the program this year is that "the work of turning around a school is too much for just a principal and a leadership team and [one] coach."

Bethel is in his second year at Turner Elementary, tasked with making significant achievement gains in a school that is among the city's lowest performing. He said that, by freeing up three of his top teachers to guide their colleagues, the TLI program "essentially builds capacity to do more, faster," including deepening use of the common standards.

"Last year, we kind of attacked deficits at a smaller scale," Bethel said. "With TLI, we can extend the work. We can concentrate on using data [more effectively] to target areas of need and develop strategies to grow."

Inherent Challenges

So far, at least as judged by educators' perceptions, the program appears to be meeting expectations.

The district has not yet parsed the student-outcome data in connection with the TLI, but according to an internal survey conducted last winter, 90 percent of teachers who were supported by teacher leaders said the program has had a positive impact on their practice, while 95 percent of teacher leaders said their new role has allowed them to expand their influence and reach more students. Fully 100 percent of the principals who responded said the program has improved collaboration and instruction in their schools.

But experts caution that, despite their common-sense appeal, high-quality teacher-leadership or -advancement programs can be difficult to sustain. Historically, they have even been "a little bit fraught," said Julia E. Koppich, a San Francisco-based education consultant who has provided technical assistance to recipients of TIF grants.

Koppich said that one common problem with career-ladder programs for teachers is that they often rest on the assumption that "just because you're a good teacher ... means you're going to be a good mentor."

"But some great teachers don't necessarily know how to work well [in advising] adults," she said. "Not everyone's a good mentor—which maybe can't be learned in a week."

Another potential risk is that, particularly in high-pressure situations, school leaders can come to rely on teachers with outside-the-classroom roles to take on administrative responsibilities for which they have not been trained or paid.

"Exploitation has happened in a number of cases," said Koppich. "There's been some abuse of programs by principals, though not usually intentionally. They just saw another person who could help with the tasks that needed to be done."

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But the coordinators of the District of Columbia's initiative believe that, particularly after making some early recalibrations, they have been deliberate enough in their planning to counter such complications.

Burke pointed to the program's explicitly team-oriented developmental approach, saying that the ongoing training is designed to deepen both teachers and principals' understanding of their roles and their school and individual action plans. She also noted that the district conducts quarterly reviews of each participating school to make sure the program is on track and to make adjustments as needed.

"We know this is incredibly hard work," she said.

Coverage of policy efforts to improve the teaching profession is supported by a grant from the Joyce Foundation, at Education Week retains sole editorial control over the content of this coverage.

Vol. 35, Issue 06, Pages s19,s20

Mathematical Habits of Mind for Teaching: Using Language in Algebra Classrooms


TME, vol10, no.3, p. 735
The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, no.3, pp.735-776
2013©The Author(s) & Dept. of Mathematical Sciences-The University of Montana

Mathematical Habits of Mind for Teaching:
Using Language in Algebra Classrooms

Ryota Matsuura2
St. Olaf College
Sarah Sword
Education Development Center, Inc.
Mary Beth Piecham
Education Development Center, Inc.
Glenn Stevens
Boston University
Al Cuoco
Education Development Center, Inc.

ABSTRACT: The notion of mathematical knowledge for teaching has been studied by many researchers, especially at the elementary grades. Our understandings of this notion parallel much of what we have read in the literature, but are based on our particular experiences over the past 20 years, as mathematicians engaged in doing mathematics with secondary
teachers. As part of the work of Focus on Mathematics, Phase II MSP, we are developing, in collaboration with others in the field, a research program with the ultimate goal of understanding the connections between secondary teachers’ mathematical knowledge for teaching and secondary students’ mathematical understanding and achievement. We are in the early stages of a focused research study investigating the research question: What are
the mathematical habits of mind that high school teachers use in their professional lives and how can we measure them? The main focus of this paper is the discussion of the habit of using mathematical language, and particularly how this habit plays out in a classroom setting.

Our Philosophy and Approach
Building on two decades of prior work, the Focus on Mathematics (FoM) Math and Science
Partnership program (MSP) has, over the last decade, developed and refined a distinctive
framework for a mathematics‐centered approach to developing teacher leaders, and it has
built a mathematical community based on that framework. The FoM approach involves
teachers, mathematicians, and educators working together in professional development
activities. The common thread running through this tightly connected set of activities is an
explicit focus on mathematical habits of mind.

We define mathematical habits of mind (MHoM) to be the web of specialized ways of
approaching mathematical problems and thinking about mathematical concepts that
resemble the ways employed by mathematicians (Cuoco, Goldenberg, & Mark, 1997, 2010;
Goldenberg, Mark, & Cuoco, 2010; Mark, Cuoco, Goldenberg, & Sword, 2010). These habits
are not about particular definitions, theorems, or algorithms that one might find in a
textbook; instead, they are about the thinking, mental habits, and research techniques that
mathematicians employ to develop such definitions, theorems, or algorithms. Some
examples of MHoM follow:

  • Discovering the structure that is not apparent at first by experimenting and seeking regularity and/or coherence.
  • Choosing a useful representation—or purposefully toggling among various representations—of a mathematical concept or object.
  • Purposefully transforming and/or interpreting algebraic expressions (e.g., rewriting x2  6x 10 as (x  3)2 1 to reveal its minimum value). TME, vol10, no.3, p. 737
  • Using mathematical language to express ideas, assumptions, observations, definitions, or conjectures.

Our work over the past decade has convinced us of the importance of MHoM for
students and for teachers of mathematics, particularly at the secondary level. These habits
foster the development and use of general purpose tools that make connections among
various topics and techniques of secondary school mathematics content; they can bring
parsimony, focus, and coherence to teachers’ mathematical thinking and, in turn, to their
work with students. In this sense, we envision MHoM as a critical component of
mathematical knowledge for teaching (Hill, Rowan & Ball, 2005) at the secondary level (i.e.,
the knowledge necessary to carry out the work of teaching mathematics).

We begin this paper by describing the mathematical community that we have built
and the impact that it has had on our teachers, in particular, the impact on teachers’
mathematical understanding and instructional practices. Then we discuss the research that
grew out of our desire to study scientifically how MHoM might be an indicator of teacher
effectiveness. Lastly, we shed light on one habit that emerged prominently in our
research—using mathematical language. We examine how a teacher might use this habit in
a classroom, possible implications for student learning, and how use of the habit relates to
teachers’ use of other mathematical habits in the classroom.

We end this section with a few remarks. Although we describe our research on
MHoM, the emphasis of this paper is not on our study, on its particular outcomes, or on the
measurement instruments in development. Instead, we intend to illustrate, using examples,
our motivation for why we think these mathematical habits are important. Hence, the main
focus of the paper is the discussion of the habit of using mathematical language, and
Matsuura et al. particularly how this habit plays out in a classroom setting. We include a detailed
discussion of the FoM MSP, partly to situate our work within the MSP context in this special
issue of The Mathematics Enthusiast. We also want to provide background for the research
that emerged from and is motivated by our ongoing MSP work with secondary teachers.
Indeed, our study of teachers’ MHoM and corresponding instrument development arose
from our desire to measure progress in and continue to improve our work with our own
FoM teachers.

Focus on Mathematics

Focus on Mathematics (NSF DUE 0314692) is a targeted MSP funded by the National
Science Foundation since 2003. Our partnership is devoted to improving student
achievement in mathematics through programs that provide teachers with solid contentbased
professional development sustained by mathematical learning communities in which
mathematicians, educators, administrators, and teachers work together to put mathematics
at the core of secondary mathematics education.

The original FoM district partners include the Massachusetts school systems of
Arlington, Chelsea, Lawrence, Waltham, and Watertown. These systems range from
suburban to urban, with middle and high school student populations from 1,300 to 6,000.
Over the years, FoM has offered a variety of professional opportunities for teachers,
including: (a) a public colloquium series devoted to mathematics and education; (b)
partnership‐wide mathematics seminars; (c) week‐long summer institutes for teachers;
(d) online problem‐solving courses; and (e) a new Mathematics for Teaching Masters
Program at Boston University. Two activities deserve special mention.
TME, vol10, no.3, p. 739

  • PROMYS for Teachers summer institute, a six‐week intensive immersion in mathematics, engages participants in experiencing mathematics as mathematicians do, solving problems and pursuing research projects appropriate for them. Each summer, the institute combines teachers from multiple districts, Grades 5–12.
  • Academic‐year study groups are district‐based—often building‐based—groups that meet biweekly for two to three hours over the course of a year. Though focused on doing mathematics (rather than being taught its results or how to teach it)—again, experiencing mathematics as a mathematician would—these trade the intensity and immersion of the summer institute for long‐term, ongoing study. These mathematical learning communities with core involvement of mathematicians are designed to help teachers develop the mathematical habits of mind that are central to the discipline of mathematics. Our teachers have responded enthusiastically, with comments such as: “[The study group] is the best ‘professional development’ that I have been involved in throughout my 35‐year teaching career. I guess the best testament for the success of Focus on Mathematics comes from the continued attendance of so many teachers. We continue to talk about the topics discussed at our study groups long after the weekly session is over” (Cuoco, Harvey, Kerins, Matsuura, & Stevens, 2011).
  • “The [Masters] program has expanded my knowledge of mathematics and deepened my understanding of how children learn mathematics, but—more importantly—I am now connected to people who are as passionate about children learning and doing mathematics as I am” (Cuoco, Harvey, Kerins, Matsuura, & Stevens, 2011). Matsuura et al.

To study the impact of FoM’s professional development programs on teachers’
professional lives, the Program Evaluation Research Group at Lesley University (FoM’s
evaluators) collected and analyzed teacher and student data over five years (Lee,
Baldassari, Leblang, & Osche, 2009) and conducted case studies of teachers (Baldassari,
Lee, & Torres, 2009). Below are those findings most strongly informing our current work:

  • Teacher beliefs and attitudes about the nature of mathematics: In interviews, teachers reported understanding the structure of mathematics in greater depth— how topics and ideas are connected and how they are developed through the grade levels. Teachers referred to developing a more complete picture or understanding of mathematics as a system and understanding the connections between different threads within it (Lee, Baldassari, & Leblang, 2006; Lee, Baldassari, Leblang, Osche, & Hoyer‐Winfield, 2007).
  • Teacher changes in instructional practice: Many of the instructional changes teachers reported stem from the ways in which they experienced learning through FoM (Lee et al., 2006). When teachers developed a deeper understanding of mathematics, their confidence often increased and they developed more flexibility in their teaching and the ability to adjust lessons based on student responses. Through our work in FoM, we have seen that MHoM is indeed a collection of habits teachers can acquire, rather than some static you‐have‐it‐or‐you‐don’t way of thinking. And teachers report to us that developing these habits has had a tremendous effect on their teaching. We have collected ample anecdotal evidence, but recognize the need for scientifically‐based evidence to establish that these teachers have indeed learned MHoM TME, vol10, no.3, p. 741 and that these habits have had a positive impact on their teaching practices. We also recognize the need to study student outcomes affected by teachers’ uses of MHoM. Mathematical Habits of Mind for Teaching Research Study

Focus on Mathematics, Phase II: Learning Cultures for High Student Achievement (NSF
DUE 0928735) is an MSP project that began in 2009. In FoM‐II, we continued to refine our
mathematical learning communities and began an exploratory research study focused on
teachers’ mathematical habits of mind.

As a basis for beginning the research study, we used the theoretical frameworks
developed by Clarke and Hollingsworth (2002) for their “Interconnected Model of Teacher
Professional Growth,” which is characterized by networks of “growth pathways” among
four “change domains” in teachers’ professional lives—the external domain (E), the
personal domain (K) (of knowledge, beliefs and attitudes), and the domains of practice (P)
and salient outcomes (S). Significant, from our point of view, is the Clarke‐Hollingsworth
theory of professional growth (as distinct from simple change), which they represent as “an
inevitable and continuing process of learning” (p. 947). They aptly distinguish their
framework from others: “The key shift is one of agency: from programs that change
teachers to teachers as active learners shaping their professional growth through reflective
participation in professional development programs and in practice” (Clarke &
Hollingsworth, 2002, p. 948). The agency of teachers in their own professional growth
characterizes virtually all FoM programs, so we see the Clarke‐Hollingsworth model of
professional growth as well suited for our purposes.

We illustrate our use of the Clarke‐Hollingsworth framework with an example.
Shown in Figure 1 is a change environment diagram for “Ms. Crew,” a middle school
Matsuura et al. teacher and active member of the FoM learning community. The diagram represents the
change domains as four boxes, labeled E, K, P, and S, as explained above. The solid arrows
refer to growths due to enactment, while the dashed arrows depict those due to reflection.
The loop on the box E refers to interaction between study groups and the immersion.

Figure 1. Schematic diagram of Ms. Crew’s change environment
This particular diagram depicts activity related to Ms. Crew’s research on
Pythagorean Triples and shows how this activity led to her growth, both mathematically
and as a teacher. Each arrow represents a growth in Ms. Crew that occurred as a result of a
change in her professional life. For example, arrow 6 depicts how her increased belief
about herself (a change in box K, the personal domain) leads to Ms. Crew encouraging her
students to perform more explorations (a change in box P, the domains of practice).
Moreover, arrow 6 is solid, because the change in her classroom is due an enactment, i.e., a
particular course of action that she took as a teacher. The arrows are numbered in
chronological order, so arrow 1 denotes a growth in Ms. Crew that occurred before that
depicted by arrow 2, and so on. The dashed arrow from box E to K has multiple numbers
TME, vol10, no.3, p. 743 (as does the solid arrow from K to E). Here, the dashed arrow may be interpreted as three separate arrows (arrow 1, arrow 3, and arrow 5)—we simply condensed them into one
arrow to save space in the diagram.

Ms. Crew first encountered the concept of Pythagorean Triples while studying
Gaussian integers during her summer immersion experience. The topic left such an
impression on her (reflective arrow 1) that she pursued it (enactive arrow 2) as a research
project under the guidance of an FoM mathematician. Through months of hard work—
familiarizing herself with Pythagorean Triples through dozens of examples, making careful
data recording and analysis, discovering beautiful patterns, coming up with interesting
conjectures (some were true, some were false), and finally writing down clear and concise
propositions and proving them—she came to understand (reflective arrow 3) features of
Pythagorean triples that would have been beyond her conception before this experience.
Ms. Crew produced an independent research paper and a one‐hour mathematics talk for
her peers (enactive arrow 4).

Neither the summer immersion experience nor the independent research project
was easy for Ms. Crew, who came into our program with a rather weak mathematics
background. But completing this project had a significant effect on her mathematical selfconfidence
(reflective arrow 5). The loops of this upward spiral between domains K and E
repeated many times. Amongst her peers, Ms. Crew became one of the leaders in her study
group (4). In her curriculum planning, she now has more belief in her decision‐making
abilities (5). And in her classroom, she engages her students in performing mathematical
exploration (6). This new classroom atmosphere, as well as her new attitude towards
mathematics, led to more curiosity and questions from her students (7, 8). And while she
Matsuura et al. may not be able to answer all of them on the spot, she now welcomes mathematical dialogs
and uncertainty in her classroom (9, 10). All of this represents significant professional
growth and Ms. Crew’s change diagram enables us to see the elements of that growth at a

Looking at Ms. Crew’s change diagram, one cannot fail to notice the intense activity
taking place around the node K, which includes growth in Ms. Crew’s knowledge of
mathematics. But it seems to us that more is involved than simply knowing mathematics as
a body of knowledge. Ms. Crew is learning mathematics in a certain way. Her beliefs about
the nature of mathematics are changing. She is acquiring certain mathematical habits of
mind and she is finding these habits useful for her work in the classroom and also for
leadership roles in the school.

Applying this framework of teacher change, we began to build for ourselves a
theoretical understanding of how MHoM plays a role in the work of teaching. Recognizing
the need for a scientific approach to test the theory, and indeed investigate the ways in
which MHoM is an indicator of teacher effectiveness, we conducted an exploratory study
titled Mathematical Habits of Mind for Teaching that centers on the following question:
What are the mathematical habits of mind that secondary teachers use in their
profession and how can we measure them?

To investigate this question, we developed a detailed definition of MHoM and have been
building the following two instruments:
ï‚· A paper and pencil (P&P) assessment that measures how teachers engage MHoM
when doing mathematics for themselves.
TME, vol10, no.3, p. 745
ï‚· An observation protocol measuring the nature and degree of teachers’ uses of MHoM
in their teaching practice.

We emphasize that both instruments are needed, because in our work with teachers, we
have seen those who have very strong MHoM for themselves but do not necessarily employ
the same mathematical habits in their teaching practices.

Our current work fits into a larger research agenda that we are developing in
collaboration with leaders in the field, with the ultimate goal of understanding the
connections between secondary teachers’ mathematical knowledge for teaching and
secondary students’ mathematical understanding and achievement.

Operationalizing MHoM
To operationalize the MHoM concept, we relied on our own experiences as
mathematicians doing mathematics with secondary teachers (Stevens, 2001). We also
studied existing literature—in particular, Dewey’s (1916) and Dewey and Small’s (1897)
earlier treatments of habits and habits of mind, the Study of Instructional Improvement
(SII) and the Learning Mathematics for Teaching (LMT) projects to develop measures of
mathematical knowledge for teaching (MKT) for elementary teachers (Ball & Bass, 2000;
Ball, Hill, & Bass, 2005; Hill, Schilling, & Ball, 2004; Hill, Ball, & Schilling, 2008), and the
description by Cuoco et al. of mathematical habits of mind (1997, 2010). And we consulted
the national standards, i.e., the NCTM Principles and Standards for School Mathematics
(National Council of Teachers of Mathematics [NCTM], 2000) and the Common Core
Standards for Mathematical Practice (National Governors Association Center for Best
Practices and the Council of Chief State School Officers [NGA Center & CCSSO], 2010). But
above all, we went into the classrooms of FoM teachers, where we observed a broad
Matsuura et al. sampling of MHoM strengths. Some teachers exhibited precise use of language and careful
reasoning skills; others had strong exploration skills, were good at designing mathematical
experiments, or showed special strength at generalizing from concrete examples.

From these various sources, we began to compile a list of habits that constitute
MHoM. As the list grew, we identified four broad and overlapping categories into which our
mathematical habits naturally fell:
● Seeking, using, and describing mathematical structure
● Using mathematical language
● Performing purposeful experiments
● Applying mathematical reasoning

Indeed, these are categories of mathematical practices that are ubiquitous in the discipline.
And in order to conduct a fine‐grained study of these categories, we teased apart multiple
habits within each category that we wanted to measure, some of which were identified
earlier. That being said, we primarily envision MHoM as being comprised of the four
categories, with the list of habits within each category providing more detail and texture to
these four. By no means is our list final. In fact, we consider it an evolving document that
we will continue to revise as we obtain more data using our instruments. From our data,
we will learn which habits are more prominently used by secondary teachers, both when
doing and teaching mathematics.

Paper and Pencil (P&P) Assessment
We developed a pilot P&P assessment that measures how secondary teachers use
MHoM while doing mathematics. This assessment contains seven open‐ended problems
and is designed to be completed in one hour. In particular, we developed problems that
TME, vol10, no.3, p. 747 most teachers have the requisite knowledge to solve, or at least begin to solve. And what we are assessing is how they go about solving it. It is the choice of their approach that we
are interested in, as opposed to whether or not they have the necessary knowledge/skills
to solve it. Each item is designed to reveal what habits and tools teachers choose to use in
familiar contexts. To date, we have gone through several rounds of design, pilot‐test, data
analysis, and revision of this instrument. For our latest pilot‐test in the summer of 2011, we
administered the P&P assessment to 43 secondary mathematics teachers participating in
the NSF‐funded study Changing Curriculum, Changing Practice (NSF DRL 1019945). We will
carry out another field test with approximately 50 teachers in the summer of 2012.
To gather initial data on the role that teachers’ approach to solving mathematics
problems plays in their approach to mathematics instruction, we asked a follow‐up
question to some of our P&P assessment problems: What strategies would you want your
students to develop for a problem like this? Our 43 respondents almost unanimously
reported that they want their students to approach the problems exactly as they did
themselves. (Note: A few teachers wanted their students to appreciate a variety of
approaches.) This finding provides initial evidence that teachers’ own mathematical work
may be indicative of how they choose to explain/formulate the subject matter for their
students. Recognizing the need for further study of this hypothesis, we began to create an
observation protocol.

Observation Protocol
We are in the process of designing an observation protocol and coding scheme that
measure the nature and degree of teachers’ uses of MHoM in their classroom instruction.
To develop the instrument, we conducted live and videotaped observations of two to three
Matsuura et al. consecutive mathematics lessons collected from a total of 30 secondary teachers to identify
teacher behaviors that reflect the uses of a particular mathematical habit. In addition, we
developed a simple protocol for pre‐ and post‐ interviews with teachers we videotape. We
also collected classroom artifacts (lesson plans, in‐class worksheets, homework, and
assignments) from each classroom we observed.

An important feature of our observation protocol is that it measures how teachers
use MHoM in their instruction. Thus teachers are coded not for possessing certain
mathematical habits in the abstract, but for choosing to bring them to bear in a classroom
setting. To develop such an instrument, we are currently studying our videos and slicing
these lessons into small episodes—i.e., short instructional segments lasting 30 seconds to 4
minutes. In each episode, we determine whether there were behavioral indicators that
reflected teachers’ uses of MHoM, and we create codes that generalize and characterize
these teacher classroom behaviors. We emphasize that our current focus is on teacher
behaviors and uses of MHoM in the classroom. We are still a step away from connecting
teaching practices centered on MHoM to students’ development of MHoM and to student
achievement—partly because we do not yet have the instruments to assess these habits in
students—but impacting students, of course, is our ultimate goal.

Later, we describe three teachers from whom we gathered video data for our
observation protocol development. Specifically, we will discuss how they apply the habit of
using mathematical language in their classroom instruction. We will also consider how
teacher use of this particular habit may affect student understanding.
TME, vol10, no.3, p. 749

Relevant Literature and Related Work
The theory of mathematical habits of mind is philosophically grounded in Dewey’s
(1916) and Dewey and Small’s (1897) earlier treatments of habits and habits of mind.
Their seminal work has since encouraged educators (Duckworth, 1996; Meier, 1995) and
education researchers (Kuhn, 2005; Resnick, 1987; Tishman, Perkins, & Jay, 1995) to
further operationalize the concept of habits of mind—that is, to respond to the general
question: What do habits of mind look like in the context of learning? Not as evident in the
literature are the habits of mind that promote successful learning in specific disciplines. In
the case of mathematics, the question that has gained research attention within the last
decade is: What do habits of mind look like in the context of learning and doing mathematics?
While addressing this question is not an unfamiliar task (Hardy, 1940; Polya, 1954a, 1954b,
1962), what is less familiar is the task of gathering evidence of mathematical habits of mind
from teachers of mathematics. We began this work in our FoM‐II study; we are in the longterm
process of developing valid and reliable instruments that will allow us to more
rigorously investigate the relationship between teachers’ own MHoM, their uses of MHoM
in their teaching practice, and student achievement.

As mentioned earlier, we envision MHoM as an integral component of MKT at the
secondary level. The notion of MKT has been studied by many researchers (Ball, 1991; Ball,
Thames, & Phelps, 2008; Heid, 2008; Heid & Zembat, 2008; Heid, Lunt, Portnoy, & Zembat,
2006; Hill et al., 2008; Kilpatrick, Blume, & Allen, 2006; Leinhardt & Smith, 1985; Ma, 1999;
Stylianides & Ball, 2008). Our understandings of this notion parallel much of what we have
read in the literature, but are based on our particular experiences over the past 20 years, as
mathematicians engaged in doing mathematics with secondary teachers.
Matsuura et al.

As mathematicians working in schools and professional development, we have come
to understand some of the ways in which teachers know and understand mathematics.
These fit into four large and overlapping categories:
(1) Teachers know mathematics as a scholar: They have a solid grounding in classical
mathematics, including its major results, its history of ideas, and its connections to
precollege mathematics.
(2) Teachers know mathematics as an educator: They understand the thinking that
underlies major branches of mathematics and how this thinking develops in
(3) Teachers know mathematics as a mathematician: They have experienced a sustained
immersion in mathematics that includes performing experiments and grappling
with problems, building abstractions from the experiments, and developing theories
that bring coherence to the abstractions.
(4) Teachers know mathematics as a teacher: They are expert in uses of mathematics
that are specific to the profession, including the ability to “think deeply of simple
things” (Jackson, 2001, p. 696), the craft of task design, and the “mining” of student

The first two of these ways of knowing mathematics are common to most pre‐service and
in‐service professional development programs. FoM has paid particular attention to the
last two, which typically receive less emphasis. We have become convinced that (3) greatly
enriches and enhances the other ways of knowing mathematics and that many teachers
who go through such an experience develop the habits of mind used by many
mathematicians. Furthermore, we have seen that participation in a mathematical learning
TME, vol10, no.3, p. 751 community helps such teachers “bring it home” in the sense that they create strategies for helping their students develop the mathematical habits that they themselves have found so

Other researchers are developing instruments to assess secondary teachers’ content
knowledge and use of mathematics in their classrooms (Bush et al., 2005; Ferrini‐Mundy,
Senk, McCrory, & Schmidt, 2005; Horizon Research, Inc., 2000; Measures of Effective
Teaching Project, 2010; Piburn & Sawada, 2000; Reinholz et al., 2011; Shechtman,
Roschelle, Haertel, Knudsen, & Vahey, 2006; Thompson, Carlson, Teuscher, & Wilson, n.d.).
In developing our own instruments, we have drawn insight from all of these projects. But
we have most closely followed the model developed by Ball and Hill—specifically, their
MKT assessment and Mathematical Quality of Instruction (MQI) protocol for documenting
MKT in elementary teachers (Hill et al., 2005; Learning Mathematics for Teaching, 2006).
Their instruments measure “specialized” mathematical knowledge, that is, knowledge that
teachers use, as distinct from the mathematical knowledge held by the general public or
used in other professions, whose components include representation of mathematical
ideas, careful use of reasoning and explanation, and understanding unique solution
approaches. These skills resemble the kinds of mathematical habits that we are interested
in studying at the secondary level.

The collective efforts of the field will all contribute to what we know about MKT, but
there are important differences between our instruments and those of others. The
differences are listed below.
ï‚· A focus on MHoM—the methods and ways of thinking through which mathematics
is created—rather than on specific results (Cuoco et al., 1997). It is impossible, even
Matsuura et al.
in three or four years of high school mathematics aligned with the Common Core, to
equip students with all of the facts they will need for college and career readiness.
But learning to think in characteristically mathematical ways is a ticket to success in
fields ranging from business, finance, STEM‐related disciplines, and even building
ï‚· The core involvement, at every level, of mathematicians who have thought deeply
about the implications of their own habits of mind for precollege mathematics
curricula, teaching, and learning (Bass, 2011; Schmidt, Huang, & Cogan, 2002).
Our instruments are, therefore, aimed at discerning the extent to which secondary
classrooms are centered on the practice of doing mathematics rather than on the specialpurpose
methods that often plague secondary curricula (Cuoco, 2008). In our work with
teachers, we have seen how expert teachers use core mathematical habits of mind in their
profession—in class, in lesson planning, and in curricular sequencing. And, as the Common
Core becomes the nationally accepted definition of school mathematics, teachers will be
expected to make the development of mathematical habits an explicit part of their teaching
and learning agenda. Our work, therefore, makes a unique contribution to the field’s
increasing level of attention to secondary mathematics teaching.

Using Mathematical Language
In this section, we will focus on a specific mathematical habit—using mathematical
language—and examine how teachers use this core habit in their instructional practice. We
will also consider its potential implications for student learning, and how this habit may
work in conjunction with other mathematical habits in the classroom.
TME, vol10, no.3, p. 753

In particular, we will discuss examples of three teachers whose Algebra 1
classrooms we observed in our research study. We will begin with Mr. Hart, who uses
mathematical language to encapsulate the experiences, observations, and discoveries of his
students. Second, we will look at Ms. Graham, who uses precise and operationalizable
language as a way of promoting conceptual understanding and ease of problem‐solving.
And third, we will describe an example of a teacher, Mr. Braun, whose choice of language
can interfere with students’ engagement in activities designed to promote other MHoM.
All three of these teachers have shown evidence of strong MHoM in their own doing
of mathematics. Mr. Hart has held formal and informal leadership roles in a number of
FoM’s mathematical learning communities; and in those roles, he has exhibited strong
MHoM. The other two teachers performed well on our P&P assessment. The names of these
teachers have been altered to protect their identities.

Mr. Hart
We consider Mr. Hart, an Algebra 1 teacher who uses mathematical language to
encapsulate the underlying structure that students discovered through experimentation.
The mathematical topic of the day is recursive rules. The class begins with students
working on the following warm‐up problem.
A function follows [this rule] for integer valued inputs: The output for a given input is 3
2 greater than the previous output. Make a table that matches the description. Can you
make more than one table?
Note that the rule is incomplete, because it is missing the base case. Students experiment
with this rule, creating input/output tables and trying to derive closed‐form equations.
Matsuura et al.
Because of their different choices of base cases, they come up with different functions
defined by expressions of the form f (x)  3
2 x  b . Students conclude that the graphs of these functions are parallel lines with different y‐intercepts. Mr. Hart also asks, “So what’s the part where you get to be creative in making these tables?” He then explains, “So you get to pick one number, and then everything else is decided by the part that I gave you [in the
warm‐up]. But there’s still an awful lot of different numbers.” Here, he is foreshadowing the
need to fix the base case.

Then Mr. Hart formally introduces the notions of recursive rule and base case to
summarize students’ experiences and to capture the underlying structure they observed
when working on the warm‐up problem. He says, A recursive rule, that’s just the description that tells us how to get from an output—to an output from the previous ones. So basically, what we were doing. Now as you
saw, there’s another piece that’s not really enough information. It’s just me telling you how to get from one, to the next, to the next. To have a complete rule, we also need to know where to start. Because otherwise, we won’t know if we have the rule that—the first rule, the second rule, the third rule, or some other rule completely.
(Video transcript, February 14, 2011.)

Next, the class studies the function described by the following table:
n f (n)
0 3
1 8
2 13
3 18
TME, vol10, no.3, p. 755
4 23
5 28
6 33
In this table of data, students recognize the +5 pattern, i.e., “You add 5 to the output.”
Through discussion, Mr. Hart guides them to articulate the relationship more precisely:
f (5)  f (4)  5. Using this concrete example, students are able to derive a general equation:
f (n)  f (n 1)  5.

To make sense of this recursive rule, Mr. Hart points out that the equation
f (n)  f (n 1)  5 “lets us relate any output to a previous one.” In essence, it is the symbolic
representation of what he told students in the warm‐up problem. Then he describes the
need for the base case, saying, “But that wasn’t quite enough because lots of you wrote
down different rules. And [Student 1] had one, [Student 2] had a different one, [Student 3]
had a different one probably, and so on. So we need something else to sort of fix it in place.”
Here, a student interrupts and proposes a closed‐form rule: f (n)  5n  3. There are
now two ways to describe the function at hand, namely the (still incomplete) recursive rule
f (n)  f (n 1)  5 and the closed form rule f (n)  5n  3. He says, “[The recursive rule] tells
us how to work our way down the table. If I know one value, I know 23, I can find the next
one really easily. Now this one’s [points to the closed‐form rule] nice too because it lets me
work across the table. If I know the input, I can say the output really quickly.” In this short
episode, Mr. Hart uses the symbolic representation of each rule to discuss its underlying

Mr. Hart returns to the equation written on the board (i.e., f (n)  f (n 1)  5) and
says, “But still, this—this rule almost tells me the whole table, but it doesn’t quite because
Matsuura et al.

I’m missing one critical piece of information.” A student chimes in, “Well, you don’t know
what you started with.” Mr. Hart responds with, “That’s a good point. Yeah, so like
[Student]’s saying this 3 in the table, that’s where we’re starting. So we kind of need to
know that. So the way (pause) a good way that we can sort of keep track of this and write
our rule...” Almost 20 minutes into the lesson, Mr. Hart finally introduces the complete
f (n)  3 if n  0,
f (n 1) 5 if n  0.
 

He explains this new equation by saying, “So this formula captures exactly what we did. The
key part is the recursive part that we had written down already. And this just adds that last
bit, the base case, so we can summarize it into one compact rule.”
Instead of being a starting point, this notation is the culmination of the structures
that students discovered through their experimentation and the follow‐up discussion.
Students readily make sense of the new notation and the accompanying ideas that it
encapsulates, because the experience gained through their “struggles” allows them to
connect the new language to already‐established ideas.

Mr. Hart uses the structure that students found through their experiments to motivate
the language needed to describe their observed results. For instance, students’ experiments
with the warm‐up problem, in which they propose different functions that all satisfy the
given rule, make the need for the base case come alive for them. Indeed, his mathematical
habits of mind allow Mr. Hart to create a learning environment where students build new
knowledge from their experiences (NCTM, 2000).
TME, vol10, no.3, p. 757

Ms. Graham
Through Ms. Graham, we look at how an Algebra 1 teacher uses precise and
operationalizable language as a way of promoting ease of problem‐solving. More
specifically, she helps students make sense of the objective of the given problem and,
subsequently, provides insight into how to proceed.

In this episode, a student asks about the following question:
Determine if r  2 is a solution to 6r  2 12 r.
Ms. Graham asks, “Did we not understand what they were asking?” The student confirms,
“Yeah, obviously there’s an easier way to do it, but I just didn’t know how.” Then the
following dialogue occurs, in which Ms. Graham presses for the meaning of the word
Teacher (T): All right. When we use the word “solution,” all right, we’ve talked a lot about
what a solution is. What does “solution” mean?
Student (S): Like, does—it—when it works.
T: When you said “it works,” what do you mean? Because I think you’re on the right
S: Like, does it make sense?
T: Be a little more specific.
S: I don’t know how, like…
T: What does “solution” mean, anyone know? All right.
New student (SN): The answer?
T: “The answer.” We talked about this a lot. What’s a solution to an equation?
SN: Something that can go into make an equation work.
Matsuura et al.
T: Something that makes the equation true, OK?

As we will see later in Mr. Braun’s example, “works” is often used by students and
teachers to describe what it means for a number to be a solution to an equation. Ms.
Graham does not settle for this nor other oft‐used phrases such as “it makes sense” and
“the answer.” The language used by students does not help them unravel the problem to
understand what they are being asked to do. Only after the operational definition of
“solution” has been given can Ms. Graham continue with an explanation of how to proceed.
T: We’re stating that 6r  2 will be equal to 12  r. And they’re asking, “Is r  2 a
solution?” So you got to test it out, just as I asked you to test out that one that we
just did. So 6r  2 12 r. Substitute in r  2. So 6 times 2 plus 2—does that have
the same value as 12 plus 2? And we have to test. All right? We’re asking ourselves
the question of, does this equal that? [Points to each side of the equation.] OK?
Then Ms. Graham leads the class through the process of substituting r  2 into the
equation and concluding that it is not a solution, since r  2 yields unequal values of 10
and 10 for the two sides of the equation. The student who originally inquired about this
question says, “Ok. Now I get it.” The definition of “solution” provided by Ms. Graham—
namely, “something that makes the equation true” is operational (i.e., students can use this
definition to understand and accomplish the task posed by the given question). Indeed,
once the definition has been given, substituting r  2 and checking if it makes the
equation true is a natural next step.

Ms. Graham concludes this episode by foreshadowing what students will be learning
next, by providing them with another definition:
TME, vol10, no.3, p. 759

T: We’re getting to the point where we’re going to ask you, “What is the value of r that
makes the equation true?” And that’s called solving the equation.
Throughout the lesson, Ms. Graham consistently uses language carefully. She corrects a
student who writes 828  90  3  30 5  25, calling it a “run‐on sentence in math.” When
a student describes two sides of an equation by saying, “It’s equals,” Ms. Graham
immediately responds, “They’re equal to each other.” She repeatedly tells students to check
their answer after solving an equation, reminding them what “solution” means. She is also
precise in her instructions (e.g., asking the students to “write an expression for the right
side of the equation, so that you’ve got an equation that works and is true when x  3”).

Mr. Braun
One of the issues we have encountered in the development of our observation
protocol is, “What counts as evidence of non‐use of MHoM?” In the case of the habit of using
mathematical language, we do see moments in which teachers choose less careful language.
For example, a teacher might choose to use informal language. Sometimes there is evidence
that the teacher is making this choice because the informal language seems more accessible
to students. But such choices—if not made carefully—can lead to student confusion.
In the following example, Mr. Braun is setting up an investigation that aims to lay
the foundation that the graph of an equation is a representation of the solution set of the
equation (Education Development Center, Inc., 2009b). To launch the investigation, Mr.
Braun writes the equation 3x  2y 12 on the overhead projector and asks students,
“What’s the answer?” He then describes some of the solutions students offer as “that
works” or “that doesn’t work.” The following is an excerpt from the launch of the
investigation. There are two things to note. First, Mr. Braun is modeling how students
Matsuura et al. might experiment with numbers as a way of making sense of the relationship between
graphs and equations. Second, observe how frequently he uses the word “works.”
T: 3x  2y 12. What’s the answer?
SN: It’s complicated.
T: Oh, no. What do you think?
SN: 1 and 2?
T: You think I can use 1 and 2?
S: x is 1 and y is 2.
T: x is 1 and y is 2. How would I find out if [name] is right? I could put in the numbers
that he gave me, so I’m going to put in 1 for x and I’m going to put in 2 for y, and do I
get 12, like I’m supposed to? What’s 3ï‚´1?
Students (Ss): 3.
T: What’s 2ï‚´ 2 ?
Ss: 4.
T: What’s 3 + 4?
Ss: 7.
T: Did I get 12?
Ss: No.
T: Man, [name], that’s a bummer. OK, so—
SN: Oh, I know it.
T: —that was something that didn’t work. It’s not bad to find out things that don’t
work. Sometimes, you’re going to be asked in these investigations to find things that
don’t work, so remember how we did that.
TME, vol10, no.3, p. 761

At this point, the teacher continues to take student guesses for x and y. Students
make guesses and one student suggests x  2 and y  3. Mr. Braun tries that suggestion,
and sees that indeed, 3(2)  2(3)  12.
T: OK, so we found out that 1 and 2 did not work; we found out that 2 and 3 did work.
Do you think there are any more things that don’t work?
SN: Yes.
T: A lot more things that don’t work. OK, do you think there are any more things that
do work?
S: Yes.
T: Can you think of another thing that does work? [...]
SN: 3(3)…
T: OK, if I put a three there, OK.
S: And then, the 2y is 2, 2(1).
T: 2 ï‚´1. OK, this is 9, right? Plus 2, makes 11 instead of 12. So, we found another thing
that doesn’t work. So, I—[name], you must have been right, there were more things
that do not work. Can you find anything else that does work?
SN: 4 and 1.
T: You think 4 and 1 works? Where do I put my 4, for x or for y?
S: For x, yeah.
T: OK, so I put in 3(4) + 2(1), that gives me 12 + 2 = 14. We found another thing that
doesn’t work.
S: Actually, put 3 for y, plus 1.5.
T: […] 2(1.5), what are we going to get?
Matsuura et al.
Ss: It’s 3.
T: 3, and we had 9. Is 3 + 9 = 12?
Ss: Yes.
T: Hey, look at that. All right, now, that’s the kind of thing I want you to do. You’re just
going to try some things. Some of them will work; some of them won’t work.
Mr. Braun has modeled a detailed investigation of looking for points that satisfy the
equation 3x  2y 12, using the word “works” as a substitute for “satisfies the equation.”
He uses the phrases “works” and “doesn’t work” repeatedly. He then hands out a worksheet
for investigation that includes the problems:
Each point in the following table satisfies the equation x  y  5.
a) Complete the table.
x y (x, y)
1 4 (1, 4)
 11
b) Graph the (x, y) coordinates that satisfy the equation x  y  5. [Grid supplied.]
c) What shape is the graph?
Use the equation 2x  3y  12.
TME, vol10, no.3, p. 763
a) Find five points that satisfy the equation.
b) Find five points that do not satisfy the equation.
Students begin the investigation. Some do not know what it means for a point to
“satisfy an equation.” Mr. Braun had created the worksheet based on problems in an
Algebra 1 textbook—in the book, students are reminded that “If a point’s coordinates make
an equation true, the point ‘satisfies the equation’” (Education Development Center, Inc.,
2009a, p. 251). Mr. Braun had left that reminder off of his worksheet, and some of the
students get stuck. For example:
S: … Please!
T: You just told me, though. [Laughter] What are we trying to do? What’s it asking you
to do?
S: Find this point…
T: OK, what does “satisfy” mean? That’s the same equation we played with at the
beginning of class, right?
S: I don’t know.
T: It is, right? We didn’t say “satisfy” and “not satisfy”; what were the words that we
S: I don’t know. I don’t know.
T: When [name] gave us 3 and 1.5, what did we say?
S: Decimal?
T: Well, we said they were decimals, we sighed at [name], but beside that, what else
did we say? What does this side equal?
S: x? y? What?
Matsuura et al.
T: What’s 3ï‚´3?
S: 9.
T: What’s 2 ï‚´1.5?
S: 3.
T: What’s 9 + 3?
S: 12.
T: So, what did we say? “[Name]’s solution...”
S: Works?
T: Works! “Works” is another word for “satisfies.” If you want to sound smart, you say,
“It satisfies the equation.” OK? All right.
Similarly, another student asks:
S: I don’t understand what it’s asking us! [Laughter]
T: All right, fair enough. It says, “Sketch a graph of all the (x, y) coordinates that
satisfy”—work—“in this equation,” and here’s my equation.
On one hand, this is not a big deal. The teacher can travel from group to group,
reminding them what “satisfies the equation” means, but he usually simply says that “it
means ‘works.’” However, “works” as a description is not operational. When students are
solving problems, they repeatedly ask about the phrase “satisfies the equation.” Rather
than offer the operationalizable definition: “if a point’s coordinates make an equation true,
the point satisfies the equation,” Mr. Braun returns to the phrase “works.”
It is worth noting that the following day, Mr. Braun poses a warm‐up question to his
class: “What does it mean to be a solution?” Although he does not specifically address the
TME, vol10, no.3, p. 765 definition of a point satisfying an equation (and the issue continues to persist for students), he does start working on unpacking that language for students.

Common Themes in the Examples
Several observations and questions emerge for us in these examples. First, what
strikes us again and again is the complexity of teachers’ uses of MHoM. These habits cannot
be deployed independently in the classroom any more than they can be when teachers (and
mathematicians) do mathematics for themselves. In fact, we saw that the habit of using
mathematical language can either complement or get in the way of student
experimentation and inquiry, depending on how the teacher uses the habit. In Mr. Hart’s
class, the precise definition of recursive function is motivated by the structure that his
students discovered through experimentation. And, in turn, Mr. Hart plans to use this
function notation as an investigative tool to explore further topics (e.g., the connection
between linear and exponential functions). Mr. Braun also brings experimentation into his
classroom. Indeed, his students conduct an investigation to explore the relationship
between an equation and its graph. However, some students have difficulty beginning the
investigation, because they do not understand the language they encounter in the task.
Here, an operational definition of the phrase “satisfies the equation” may have led them to
understand the problem statements and given them insight into how to proceed.
Throughout these examples, we also saw how the use of mathematical language can
support students’ understanding. In Ms. Graham’s class, we see how she pushes her
students to clearly state the meaning of the word “solution.” And its definition becomes a
vehicle that facilitates the problem‐solving process. In contrast, we see Mr. Braun whose
students encounter the phrase, “satisfy the equation.” Instead of providing a usable
Matsuura et al.
definition, he offers an alternative, namely “works.” We believe Mr. Braun is wellintentioned
here. Specifically, there is evidence that he is trying to make the language less
intimidating for students by offering a more informal phrase. Indeed, he says, “‘Works’ is
another word for ‘satisfies.’ If you want to sound smart, you say, ‘It satisfies the equation.’”
But as discussed earlier, “works” is a phrase that is difficult to operationalize. It leads to
confusion for his students, because they do not know how to use it. One of the mathematical
practices advocated by the Common Core is attending to precision. The Common Core
states that, “Mathematically proficient students try to communicate precisely to others.
They try to use clear definitions in discussion with others and in their own reasoning”
(NGA Center & CCSSO, 2010, p. 7). That “usability” of language is an important part of
communicating precisely, and one that seems especially important for teachers.
In particular, the careful use of mathematical language not only helps clarify ideas
for students, as it did in Ms. Graham’s class, but it helps them understand the mathematics
itself in a deeper way. We see this in Mr. Hart’s lesson, where the recursive formula for
f (n) captures the properties of the function that students found through their
investigations. Indeed, this formula is both a product and a reflection of their experiences.
In our work with FoM teachers, we have found that encapsulating various insights into
precise language—as we saw in Mr. Hart’s class—helps one better understand the ideas

Mr. Hart also recognizes the power of precise language to drive further
investigations. Later in the school year, these students will use function notation to study
transformations of functions (e.g., stretches, shrinks, and translations). He adds, “I think
TME, vol10, no.3, p. 767 that will be a place where students will really appreciate the function notation in
representing those transformations more easily.”

Mr. Hart concludes the post‐interview by describing how today’s lesson is part of a
bigger unit and how it sets the foundation for later lessons. He plans to use these recursive
rules as a vehicle for better understanding their closed‐form counterparts. In a future
lesson, students will investigate the connection between linear and exponential functions.
“I want my students to see that recursively, exponential functions are very, very similar in
their representation to linear functions. I think that will provide a nice foundation for
studying exponents,” he says. Here, Mr. Hart is using the language of recursive functions to
shed light on the connections between their corresponding closed‐form representations.
Our own goals in watching these videos have been to better understand teachers’
uses of MHoM, and to learn about how we might measure that use. Part of our desire to
measure the use stems from our desire to understand (eventually) the link between
teachers’ uses of MHoM and learning outcomes for students, particularly if we can measure
students’ uses of MHoM or students’ facility with Common Core’s Mathematical Practices,
which include significant overlap with MHoM. Within the context of the examples in this
paper, might teachers’ use of language have an impact on student achievement? Even to
begin to answer such a question, we must have some objective way of deciding whether or
not a given teacher is using clear, usable, and precise language. This, too, is complex.
Establishing what counts as “clear, usable, and precise” language depends very much on the
classroom context. Mr. Braun uses the word “works” so consistently in his classroom
discussion, that if it did not cause confusion, surely we would want to “rate” that as totally
acceptable language, taken as shared by the whole classroom.
Matsuura et al.

Impact and Next Steps
We began our research work partly because we wanted to assess the effects of our
own MSP professional development programs using tools that were consistent with the
goals of our MSP, and partly because we wanted to understand the MHoM of secondary
teachers better. We did not find instruments that measured teachers’ MHoM—either when
doing mathematics for themselves or teaching mathematics in their classrooms—in
existence in the field, so we began to create our own. Although we expected to learn from
the data gathered using our instruments, we did not anticipate the immediate implications
that our research would have on the professional development programs in our MSP. For
example, based on what we had learned from our research, we piloted the Mathematical
Habits of Mind Shadow Seminar in the summer of 2011, geared toward teacher participants
returning to PROMYS for Teachers (our summer immersion program) for a second
summer. Through discussions, readings, curriculum analyses, and lesson designs, the goal
of this seminar was to explore (a) the ways in which secondary teachers know and use
MHoM in their profession, and (b) the effects that a learning environment that stresses
MHoM might have on secondary students. We will continue to offer and refine this course
as part of our summer immersion program for teachers.

We also did not anticipate the potential for impact on the field. While development
and validation of truly reliable tools is beyond the scope of the current FoM‐II study, we
have been laying the groundwork for our MHoM instruments—the P&P assessment and the
observation protocol—over the last few years. This exploratory phase of instrument
development also coincided with the emergence of the Common Core State Standards and
its adoption by 45 states (NGA Center & CCSSO, 2010). Our MHoM construct is closely
TME, vol10, no.3, p. 769 aligned with the Common Core, especially its Standards for Mathematical Practice, and
there is considerable overlap in the two. For example, both place importance on seeking
and using mathematical structure, uses of precision, and the act of abstracting regularity
from repeated actions. As we presented our preliminary findings at national conferences
(Matsuura, Cuoco, Stevens, & Sword, 2011; Matsuura, Sword, Cuoco, Stevens, & Faux,
2011), we received several requests to use our instruments, even though they were in the
pilot phase of development. One district leader wanted to diagnose the preparedness of her
teachers to teach from a curriculum based on the Common Core. Others wanted to use the
instruments as pre‐ and post‐ measures for evaluating professional development programs
aligned to the Common Core. We have become abundantly aware of the national need for
valid and reliable instruments to measure teachers’ knowledge and use of
MHoM/Mathematical Practices, as well as guidelines for acceptable use of such
instruments. Thus, in the next phase of our research, we plan to subject our pilot
instruments to rigorous scientific testing. The examples in this paper are exemplars of
those that provide both the content basis for the P&P assessment and the behavioral
indicators for the observation protocol.

Matsuura et al.

Baldassari, C., Lee, S., & Torres, R. T. (2009). The case of a high school mathematics teacher.
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Article Image
New Commitments to Support Computer Science Education

White House Release



The White House

Office of the Press Secretary

For Immediate Release

December 08, 2014

FACT SHEET: New Commitments to Support Computer Science Education

“I’m proud to join the students, teachers, businesses, and non-profit organizations taking big new steps to support computer science in America’s schools. Learning these skills isn’t just important for your future – it’s important for our country’s future. If we want America to stay on the cutting edge, we need young Americans like you to master the tools and technology that will change the way we do just about everything.”

-- President Obama, December 2013, on Computer Science Education Week

Last year, to kick off Computer Science Education Week, President Obama issued a call to action to students, teachers, businesses, foundations, and non-profit organizations to join the growing grassroots campaign to support computer science education in K-12 schools.

The President encouraged Americans from all backgrounds to get involved in mastering the technology that is changing the way we do just about everything, and he encouraged millions of students to learn the skills that are becoming increasingly relevant to our economy.

Today, the Administration is announcing new commitments that will help give millions of additional K-12 students access to computer science education. These include:

  • Commitments by more than 60 school districts, including the seven largest school districts in the country, to offer computer science courses to their students. Together, these districts reach over 4 million students in more than 1,000 high schools and middle schools, in partnership with
  • Over $20 million in philanthropic contributions to train 10,000 teachers by fall 2015 and 25,000 teachers to teach computer science to in time for the school year beginning in fall 2016.
  • New partnerships by the National Science Foundation (NSF), including a new Advanced Placement (AP) Computer Science course by the College Board that emphasizes the creative aspects of computing and a focus on real-world applications. Leading partners, including Teach for America and the National Math and Science Initiative, will assist in implementation and scale-up of the course.
  • New steps to increase the participation of women and under-represented minorities in computer science, including a new computer-science classroom design prize and innovative outreach efforts.

These commitments and leading organizations will be highlighted at an event today at the White House. In addition, the President released a new video message on computer science education and the President and the Vice President will jointly meet with a group of students participating in an Hour of Code.


By 2020, more than 50 percent of STEM jobs are projected to be in computer science-related fields. If current trends continue, 1.4 million computer science-related jobs will be available over the next ten years, but only 400,000 computer science graduates will be added with the skills to apply for those jobs. Yet a large majority of K-12 schools do not offer any computer programming classes, and in 25 out of 50 states, computer science classes cannot count towards math and science high school graduation requirements.

That is why as part of Computer Science Education Week last year, the President praised efforts to get more computer science into K-12 schools and issued a call to action to private sector leaders, technologists, schools leaders, and others to do more to give students access to these critical skills.

Commitments Being Announced Today: Expanding Computer Science Offerings to Millions More Students

There is a growing grassroots movement in the United States to bring computer science education to K-12 schools. With leadership from, the movement has already helped introduce more than 50 million students to computer science through the “Hour of Code,” with more than 40 percent of participants being girls, and through other projects and initiatives supporting computer science in more than 60,000 classrooms across the country.

Today, responding to the President’s 2013 call to action, philanthropic organizations, cities, non-profits, and others are announcing a major expansion of this grassroots effort, including:

  • Commitments to offer computer science by more than 60 school districts, including the seven largest school districts in the country. The New York City (NY), Los Angeles Unified (CA), Chicago (IL), Miami-Dade County (FL), Clark County (NV), Broward County (FL), and Houston (TX) school districts, and many smaller districts, together reach more than 4 million students in more than 1,000 high schools and middle schools and serve nearly 15 percent of the African American and Hispanic American student population in the United States. Each of these districts is committing to offer introductory computer science courses at the high school or middle school grade levels and to market these courses to their students and their parents. will assist districts by providing teachers with curriculum, professional development, and year-round support.
  • Over $20 million in philanthropic contributions to train 25,000 teachers to teach computer science in time for the school year beginning in Fall 2016. With support from companies including Google, Microsoft, the Omidyar Network, and, as well as philanthropists including Ballmer Family Giving, John and Anne Doerr, Bill Gates, Reid Hoffman, Drew Houston, Sean Parker, Ali and Hadi Partovi, Diane Tang and Ben Smith, and Mark Zuckerberg and Priscilla Chan, will host computer science instruction workshops for 1,000 elementary school teachers each month. Workshop participants will learn how to teach modules of computer science for grades K-5. also has committed to preparing at least 500 middle school teachers and 500 high school teachers each year to teach computer science.

Furthermore, the National Science Foundation (NSF) is announcing major steps from its non-profit partners to support computer science education. These announcements build on nearly ten years of NSF investment and fall under NSF’s CS 10K Project, a nationwide effort to get engaging and rigorous academic computer science courses into 10,000 schools taught by 10,000 well-prepared teachers and a longer-term goal to include all schools across the nation.

Over the past decade, NSF has invested in research into and development of curricula, course materials, pedagogy, scalable models of teacher preparation, and approaches to sustainable, ongoing teacher support. Today, NSF is highlighting this work by launching a new web portal that showcases the agency’s investments in computer science education.

With leadership and key support from NSF, a number of leading education non-profits are announcing major expansions in their efforts to support computer science education:

  • The College Board is announcing the launch of AP Computer Science Principles, a new multidisciplinary course designed to help recruit many more students, including women and under-represented minorities into computing. The new course will be taught in secondary schools starting in the 2016-17 academic year with the first exam administered in May 2017. The course will draw more students into the discipline by focusing on foundational computing skills and the creative aspects of computing. The inherently multidisciplinary course teaches students to analyze problems, create computer programs, and collaborate to find solutions to real-world issues. AP Computer Science Principles aims to prepare a more diverse student population—including groups typically underrepresented in computing—for the demands of STEM and computing coursework and careers. The course was created with partial funding from NSF for the development of teacher support materials and assessments. A dedicated online teacher community will enable teachers to connect, discuss teaching strategies, and share resources with each other. Furthermore, AP STEM teachers will be invited to participate in a live webinar focused on computer science education during this year’s Computer Science Education Week.
  • Teach For America will begin a nationwide push to encourage partner schools to offer computer science. Building on an NSF-funded pilot project in New York City, AmeriCorps grantee Teach For America (TFA) is beginning a nationwide push to expand computer science course offerings in the schools they serve. By 2018-2019, TFA will recruit, place, and support a diverse group of at least 75 new teachers to implement the Exploring Computer Science course in high-needs schools. TFA will also advance the President’s STEM AmeriCorps initiative by promoting opportunities for their extensive, national network of educators to engage in after-school and out-of-school computer science teaching opportunities sponsored by partners.
  • The National Math and Science Initiative will expand its professional development offerings in computer science, reaching 25 states in the next two years. The National Math and Science Initiative (NMSI) is committing to broadening access to and achievement in rigorous computer science coursework through its College Readiness Program, a comprehensive approach to raising the academic bar in U.S. schools by working with teachers, students, and administrators to set and achieve aggressive performance goals. NMSI will broaden training and learning opportunities around AP Computer Science Principles, as well as Exploring Computer Science and equivalent courses, in 25 states by the end of 2016.
  • Project Lead the Way will continue to grow its computer science offerings. Project Lead the Way (PLTW) and Verizon will enable students in 12 underserved middle schools to explore the power of computational thinking and the connection of digital literacy to their lives. Verizon will supply PLTW with up to 35 tablets equipped with data plans for each school, allowing for a 1:1 student-to-tablet ratio in each of the 12 schools participating in PLTW’s Introduction to Computer Science course. Students will use MIT App Inventor to learn fundamental computer science concepts that apply to a range of disciplines, future studies, and careers. Student teams will work collaboratively and learn the impact of computing in society, and how to use the internet safely and responsibly. 
  • NSF and announce a public-private partnership. NSF and are signing a Memorandum of Understanding (MOU) to encourage and facilitate cooperation on respective efforts to support and enable widespread computer science education throughout the United States. NSF and are already collaborating on projects such as Massachusetts Exploring Computer Science, the joint result of NSF and awards to the Massachusetts Computing Attainment Network (MassCAN) and the Massachusetts Exploring Computer Science Partnerships (MECSP). The new MOU will provide a structure through which NSF and can expand their work, and co-develop additional projects and programs.
  • Massachusetts continues to grow a unique public-private partnership to introduce computer science education in its K-12 schools.  MassCAN is a multi-partner initiative in Massachusetts working cooperatively with projects funded by both NSF and to bring computer science to high schools across the state. The purpose of the partnership is to offer professional development to K-12 teachers based on a standard-based curriculum, with a goal of training 3,000 teachers over 3 years. Recently, Massachusetts enacted economic development legislation including $1.5 million to help fund the MassCAN during the program’s first year.  Today, MassCAN is announcing that Massachusetts industries, led by the Massachusetts Competitive Partnership, the Massachusetts Business Roundtable, and the Massachusetts Technology Leadership Council, have committed to raise $300,000 in matching funds and are mobilizing to match the remainder of the State grant during 2015. 
  • New York City will implement the College Board’s new AP Computer Science Principles course in 100 high schools and will expand computer science offerings overall. With support from NSF, 100 New York City (NYC) high schools will introduce University of California at Berkeley’s “Beauty and Joy of Computing” as a new AP Computer Science Principles course in 2015. This represents a significant expansion of NYC AP computer science course offerings and a dramatic increase in the number of students exposed to computer science curricula. NYC has already taken a number of steps to help advance computer science education, including:
    • The New York City Department of Education launched a Software Engineering Pilot designed to provide multi-year sequences of computer science classes at 18 middle and high schools citywide. Today, the program is in its second year and serves a diverse body of 2,600 students, 40 percent of whom are girls.
    • With support from the New York City Foundation for Computer Science, programs like TEALS, Bootstrap, ScriptEd, and Scalable Game Design are providing NYC schools with a wide range of opportunities to introduce computer science curriculum and learning activities into the regular school day for the first time.
    • With AT&T Aspire support, students from the Academy for Software Engineering, Bronx Academy of Software Engineering and the Software Engineering Pilot participate in cross-school community events such as hackathons and showcases of student work, as well as summer learning opportunities and internships with local companies. NYC has also begun introducing students to the Maker experience by offering 3D printing classes in select schools. 

Commitments Being Announced Today: Broadening Diversity in Computer Science

Improving the participation and success of women and underrepresented minorities in computer science is critical. The number of women completing college degrees in these fields has decreased over the last two decades, and a smaller percentage of U.S. high school students take computer science courses than they did two decades ago. Today, less than 20 percent of students enrolled in AP computer science courses are women or girls, and less than 10 percent are Hispanic or African-American. Furthermore, less than 20 percent of college graduates in computer science are women. A number of leading organizations are taking new steps to address this challenge, including:

  • The USA Science and Engineering Festival will launch a prize for computer science classroom design. The USA Science and Engineering Festival is announcing a classroom-design prize competition that will launch on January 5, 2015. Research has shown that small changes in classroom design elements can dramatically affect the attractiveness of computer-science courses to girls. The competition will engage teams of high school students around the country to create cost-effective and innovative designs for K-12 computer science classrooms that encourage more young women to study computer science and pursue careers in STEM. The competition will run throughout spring 2015, and the most innovative ideas will be awarded with cash prizes. Some of the prize winners will be considered for further in-classroom testing and potential deployment in classrooms around the country. The entries for the competition will be student-driven, and the design of the competition was led by the Youth Advisory Board to the USA Science and Engineering Festival in partnership with the Dell Youth Innovation Advisors.
  • A new nationwide initiative to engage Latinas in technology careers. Latinas represent the fastest-growing female population in the U.S. Including their perspectives and talent in information technology is vital to growing our innovation economy. In collaboration with major Latino community influencers and organizations, the National Center for Women & Information Technology (NCWIT) is launching a nationwide initiative to engage Latinas in computing and technology careers. NCWIT will leverage its research capabilities and national network of partners to design and launch a national media campaign and supporting program to give Latinas the inspiration to explore technology careers, the resources to engage in computer science, and connections to computer science support networks. Central to this initiative will be strategies to engage Latino parents, families, and influencers in supporting Latinas’ pursuit of technology education and careers. The project will launch on January 20, 2015 with a working roundtable of Latino leaders who will inform messaging and support the implementation of the campaign. 
  • #YesWeCode expands efforts to more youth from under-represented communities into coding. #YesWeCode, a national initiative of Dream Corps Unlimited to support the movement to train 100,000 low opportunity youth to become high-level computer programmers, is announcing that it will host a series of 4-6 youth-focused hackathons in key cities in 2015 including in New Orleans, Detroit, and Oakland. At these hackathons, local youth will team up with professional developers, innovators, designers and mentors to create apps to benefit their communities. This will complement #YesWeCode’s efforts to incubate a national job-training pipeline in Oakland, in partnership with the public school district, major tech employers, independent grassroots coding education groups, and other major community stakeholders. The job-training three-step pipeline is designed to guide youth from introductory coding programs, to immersive job-training programs, and eventually into employment. Once fully realized in Oakland, the plan is to replicate nationally. 

What is Conceptual Understanding



Conceptual understanding is a phrase used extensively in educational literature, yet one
that may not be completely understood by many K-12 teachers. A Google search of the
term produces almost 15 million entries from a vast arena of subjects. Over the last
twenty years, mathematics educators have often contrasted conceptual understanding
with procedural knowledge. Problem solving has also been in the mix of these two.
A good starting point for us to understand conceptual understanding is to review The
Learning Principle from the NCTM Principles and Standards for School Mathematics
(2000). As one of the six principles put forward, this principle states:
Students must learn mathematics with understanding, actively building new
knowledge from experience and prior knowledge.

For decades, the major emphasis in school mathematics was on procedural knowledge,
or what is now referred to as procedural fluency. Rote learning was the norm, with little
attention paid to understanding of mathematical concepts. Rote learning is not the
answer in mathematics, especially when students do not understand the mathematics. In
recent years, major efforts have been made to focus on what is necessary for students to
learn mathematics, what it means for a student to be mathematically proficient. The
National Research Council (2001) set forth in its document Adding It Up: Helping
Children Learn Mathematics a list of five strands, which includes conceptual
understanding. The strands are intertwined and include the notions suggested by NCTM
in its Learning Principle. To be mathematically proficient, a student must have:

• Conceptual understanding: comprehension of mathematical concepts, operations,
and relations
• Procedural fluency: skill in carrying out procedures flexibly, accurately,
efficiently, and appropriately
• Strategic competence: ability to formulate, represent, and solve mathematical
• Adaptive reasoning: capacity for logical thought, reflection, explanation, and
• Productive disposition: habitual inclination to see mathematics as sensible,
useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.

As we begin to more fully develop the idea of conceptual understanding and provide
examples of its meaning, note that equilibrium must be sustained. All five strands are
crucial for students to understand and use mathematics. Conceptual understanding
allows a student to apply and possibly adapt some acquired mathematical ideas to new

The National Assessment of Educational Progress (2003) delineates specifically what
mathematical abilities are measured by the nationwide testing program in its document
What Does the NAEP Mathematics Assessment Measure? Those abilities include
conceptual understanding, procedural knowledge, and problem solving. There is a
significant overlap in the definition of conceptual understanding put forth with both the
National Research Council and the NCTM definitions.

Students demonstrate conceptual understanding in mathematics when they
provide evidence that they can recognize, label, and generate examples of
concepts; use and interrelate models, diagrams, manipulatives, and varied
representations of concepts; identify and apply principles; know and apply facts
and definitions; compare, contrast, and integrate related concepts and principles;
recognize, interpret, and apply the signs, symbols, and terms used to represent
concepts. Conceptual understanding reflects a student's ability to reason in
settings involving the careful application of concept definitions, relations, or
representations of either.

To assist our students in gaining conceptual understanding of the mathematics they are
learning requires a great deal of work, using our classroom resources (textbook,
supplementary materials, and manipulatives) in ways for which we possibly were not
trained. Here are some examples that shed light on what conceptual understanding might
involve in the classroom.

• For grades 3 through 5, the use of zeros with place value problems is simple, but
critical for understanding. "What is 20 + 70?" A student who can effectively
explain the mathematics might say, "20 is 2 tens and 70 is 7 tens. So, 2 tens and 7
tens is 9 tens. 9 tens is the same as 90."
• In grades 5 through 6, operations with decimals are common topics. "What is
6.345 x 5.28?" A student has conceptual understanding of the mathematics when
he or she can explain that 335.016 cannot possibly be the correct product since
one factor is greater than 6 and less than 7, while the second factor is greater than
5 and less than 6; therefore, the product must be between 30 and 42.
• For grades 1 through 4, basic facts for all four operations are major parts of the
mathematics curriculum. "What is 6 + 7?" Although we eventually want
computational fluency by our students, an initial explanation might be "I know
that 6 + 6 = 12; since 7 is 1 more than 6, then 6 + 7 must be 1 more than 12, or
13." Similarly, for multiplication, "What is 6 x 9?" "I know that 6 x 8 = 48.
Therefore, the product 6 x 9 must be 6 more than 48, or 54."
• In grade 6, fractions, decimals, and percents are integrated in problem situations.
"What is 25% of 88?" Rather than multiplying .25 x 88, conceptual
understanding of this problem might include "25% is the same as 1/4, and 1/4 of
88 is 22." Concepts are integrated to find the answer.
• In grades 4 through 6, measurement of circles is started and extended. Critical to
conceptual understanding of both perimeter and area is the understanding of π.
The answer to the question "What is π?" gives teachers a very good measure of
student understanding. "π is equal to 3.14, or 22/7" lacks student understanding.
"π is the ratio of the circumference of a circle to its diameter, and is
approximately 3.14" shows conceptual understanding.
• For most states, ideas about even and odd numbers are included in grades 1 and 2.
Using manipulatives or making drawings to show and explain why 5 is an odd
number and 8 is an even number provides evidence that a student has conceptual
understanding of the terms.
"5 is an odd number because I can't make pairs with all of the cubes (squares). 8
is an even number because I can make pairs with all of the cubes?
• Prime and composite numbers are topics for grades 5 and 6. Manipulatives can
be used by students to show conceptual understanding of these terms. "5 is a
prime number since there are only two ways to arrange squares to make a
rectangle; 6 is a composite number since there are more than two ways to make a
rectangle. All primes numbers have only two ways."
• Fundamental geometry concepts are introduced in grade1; however, more specific
characteristics of shapes occur in later grades. "Draw a parallelogram that is not a
square." A student needs to reason about and integrate related concepts. "A
parallelogram is a quadrilateral with opposite sides parallel and congruent. Since
a square is a parallelogram with 4 congruent sides and 4 right angles, I need to
draw a parallelogram that doesn't have those characteristics."
"A rhombus, parallelogram, or rectangle each meets the conditions."
• In grades 3 through 6, functional relationships become critical for future work in
algebra. "One 4-leaf clover has 4 leaves, two 4-leaf clovers have 8 leaves. How
many leaves do fifteen 4-leaf clovers have?" A student with conceptual
understanding of this problem might do a couple of things to explain the answer.
"First, I can make a table and then look for a pattern. I can extend the pattern to
find the answer for 15 clovers. Maybe, I can find a rule that will help me get the
Number of Clovers Leaves
1 4
2 8
3 12
. .
15 ?
"8 is 2 x 4; 12 is 3 x 4; 16 is 4 x 4. Therefore, if I have 15 clovers, there must be
15 x 4 = 60 leaves."

Getting students to use manipulatives to model concepts, and then verbalize their results,
assists them in understanding abstract ideas. Getting students to show different
representations of the same mathematical situation is important for this understanding to
take place. Getting students to use prior knowledge to generate new knowledge, and to
use that new knowledge to solve problems in unfamiliar situations is also crucial for
conceptual understanding. As noted by the National Research Council (2001), when
students have conceptual understanding of the mathematics they have learned, they
"avoid many critical errors in solving problems, particularly errors of magnitude."
Getting students to see connections between the mathematics they are learning and what
they already know also aids them in conceptual understanding.

The NCTM Principles provides an excellent conclusion to the discussion of conceptual

  • Learning with understanding is essential to enable students to solve the new kinds of problems they will inevitably face in the future.
  • Teachers of mathematics must create opportunities for students to communicate their conceptual understanding of topics. This may involve lesson structures that require a change in pedagogical techniques. Ideas for supporting students in developing conceptual understanding of their mathematics must be provided in resources for teachers.

© Balka, Hull, and Harbin Miles

National Research Council (2001). Adding It Up: Helping Children Learn Mathematics.
Washington, DC: National Academy Press
NAEP (2003). What Does the NAEP Mathematics Assessment Measure? Online at
National Council of Teachers of Mathematics (2000). Principles and Standards for
School Mathematics. Reston, VA: NCTM.