Knowing Mathematics for Teaching

Contributed by:
Jonathan James
Every day in mathematics classrooms across the country, students get answers mystifyingly wrong, obtain right answers using unconventional approaches, and ask questions: Why does it work to “add a zero” to multiply a number by ten? Why, then, do we “move the decimal point” when we multiply decimals by ten? And is this a different procedure or different aspects of the same procedure—changing the place value by one unit of ten? Is zero even or odd? What is the smallest fraction?
Mathematical procedures that are automatic for adults are far from obvious to students; distinguishing between everyday and technical uses of terms— mean, similar, even, rational, line, volume—complicates communication. Although polished mathematical knowledge is an elegant and well-structured domain, the mathematical knowledge held and expressed by students is often incomplete and difficult to understand. Others can avoid dealing with this emergent mathematics, but teachers are in the unique position of having to professionally scrutinize, interpret, correct, and extend this knowledge.
1. Knowing Mathematics
for Teaching
Who Knows Mathematics Well Enough
To Teach Third Grade, and
How Can We Decide?
By Deborah Loewenberg Ball, Heather C. Hill, and Hyman Bass
implement standards and curriculum effectively, school sys-
ith the release of every new international mathe-
matics assessment, concern about U.S. students’ tems depend upon the work of skilled teachers who under-
mathematics achievement has grown. Each stand the subject matter. How well teachers know mathe-
mediocre showing by American students makes it plain that matics is central to their capacity to use instructional mate-
the teaching and learning of mathematics needs improve- rials wisely, to assess students’ progress, and to make sound
ment. Thus, the country, once more, has begun to turn its judgments about presentation, emphasis, and sequencing.
worried attention to mathematics education. Unfortunately, That the quality of mathematics teaching depends on
past reform movements have consisted more of effort than teachers’ knowledge of the content should not be a surprise.
effect. We are not likely to succeed this time, either, with- Equally unsurprising is that many U.S. teachers lack sound
out accounting for the disappointing outcomes of past ef- mathematical understanding and skill. This is to be ex-
forts and examining the factors that contribute to success in pected because most teachers—like most other adults in
other countries. Consider what research and experience this country—are graduates of the very system that we seek
consistently reveal: Although the typical methods of im- to improve. Their own opportunities to learn mathematics
proving U.S. instructional quality have been to develop cur- have been uneven, and often inadequate, just like those of
riculum, and—especially in the last decade—to articulate their non-teaching peers. Studies over the past 15 years
standards for what students should learn, little improve- consistently reveal that the mathematical knowledge of
ment is possible without direct attention to the practice of many teachers is dismayingly thin.1 Invisible in this re-
teaching. Strong standards and quality curriculum are im- search, however, is the fact that the mathematical knowl-
portant. But no curriculum teaches itself, and standards do edge of most adult Americans is as weak, and often weaker.
not operate independently of professionals’ use of them. To We are simply failing to reach reasonable standards of
mathematical proficiency with most of our students, and
Deborah Loewenberg Ball is interim dean of the School of Edu- those students become the next generation of adults, some
cation and the William H. Payne Collegiate Professor in Educa- of them teachers. This is a big problem, and a challenge to
tion at the University of Michigan. Her areas of specialization in- our desire to improve.
clude the study of efforts to improve teaching through policy, re-
form initiatives, teacher education, and mathematical knowledge 1
For example, Liping Ma’s 1999 book, Knowing and Teaching
for teaching. Heather C. Hill is associate research scientist at the Elementary Mathematics, broadened interest in the question of how
School of Education, University of Michigan. Her areas of spe- teachers need to know mathematics to teach (Ma, 1999). In her study,
cialization include educational policy, instruction, and teachers’ Ma compared Chinese and U.S. elementary teachers’ mathematical
knowledge. Producing a portrait of dramatic differences between the
content knowledge. Hyman Bass is the Roger Lyndon Collegiate
two groups, Ma used her data to develop a notion of “profound
Professor of Mathematics and of Mathematics Education in the understanding of fundamental mathematics,” an argument for a kind
Department of Mathematics and the School of Education, Uni- of connected, curricularly-structured, and longitudinally coherent
versity of Michigan. His areas of specialization include algebra knowledge of core mathematical ideas. (For a review of this book, see
(geometric methods in group theory), teacher education, and the Fall 1999 issue of American Educator,
mathematical knowledge for teaching. reports/american_educator/fall99/amed1.pdf.)
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
2. What is less obvious is the remedy. One often-proposed
solution is to require teachers to study more mathematics, Although many studies demonstrate
either by requiring additional coursework for teachers,2 or
even stipulating a subject-matter major.3 Others advocate a that teachers’ mathematical
more practice-grounded approach, preparing teachers in the
mathematics they will use on the job. Often, these advocates knowledge helps support increased
call for revamping mathematics methods coursework and
professional development to focus more closely on the student achievement, the actual na-
mathematics contained in classrooms, curriculum materials,
and students’ minds. Still others argue that we should draw
ture and extent of that knowledge—
new recruits from highly selective colleges, betting that over-
all intelligence and basic mathematics competence will prove
whether it is simply basic skills at the
effective in producing student learning. Advocates for this grades they teach, or complex and
proposal pointedly eschew formal education courses for
these new recruits, betting that little is learned in schools of professionally specific mathematical
education about teaching mathematics effectively.
At issue in these proposals is the scope and nature of the knowledge—is largely unknown.
mathematical knowledge needed for teaching. Do teachers
need knowledge of advanced calculus, linear algebra, ab-
stract algebra, differential equations, or complex variables in
order to successfully teach high school students? Middle
school students? Elementary students? Or do teachers only
need to know the topics they actually teach to students? Al-
ternatively, is there a professional knowledge of mathematics
for teaching, tailored to the work teachers do with curricu-
lum materials, instruction, and students?
Despite the uproar and the wide array of proposed solu-
tions, the effects of these advocated changes in teachers’
mathematical knowledge on student achievement are un-
proven or, in many cases, hotly contested. Although many
studies demonstrate that teachers’ mathematical knowledge
helps support increased student achievement, the actual na-
ture and extent of that knowledge—whether it is simply
basic skills at the grades they teach, or complex and profes-
sionally specific mathematical knowledge—is largely un-
known. The benefits to student learning of teachers’ addi-
For example, in the 2001 report, The Mathematical Education of
Teachers, the Conference Board of the Mathematical Sciences,
American Mathematical Society, and Mathematical Association of
America call for prospective elementary teachers to take at least nine
semester-hours on fundamental ideas of elementary school
mathematics; prospective middle-grades math teachers to take at least
21 semester-hours of mathematics, including at least 12 semester- tional coursework, either in mathematics itself or “mathe-
hours on fundamental ideas of school mathematics appropriate for the matics methods”—courses that advise ways to teach mathe-
middle grades; and prospective high school mathematics teachers to matics to students—are disputed by leading authorities in
complete the equivalent of an undergraduate major in mathematics,
the field. Few studies have been successful in pinpointing an
including a 6-hour capstone course connecting their college
mathematics courses with high school mathematics. The report appropriate mathematics “curriculum”—whether it be
recommends that prospective teachers take mathematics courses “that purely mathematical, grounded in practice, or both—that
develop a deep understanding of the mathematics they will teach,” and can provide teachers with the appropriate mathematics to
“a thorough mastery of the mathematics in several grades beyond that help students learn (Wilson and Berne, 1999). Similarly, we
which they expect to teach, as well as of the mathematics in earlier know too little about the effectiveness of recruits who bypass
grades.” traditional schools of education. What is needed are more
NCLB requires that all new middle- and high-school teachers programs of research that complete the cycle, linking teach-
demonstrate subject-matter competency by 1) passing a state academic ers’ mathematical preparation and knowledge to their stu-
subject test in each of the subjects in which they teach; or 2)
dents’ achievement.
completing an academic major, a graduate degree, coursework
equivalent to an undergraduate academic major, or advanced In this article, we describe one such program of research
certification or credentialing in each of the subjects in which they that we have been developing for more than a decade. In
teach (Public Law 107-110, Section 9101 [23]). 1997, building on earlier work (see Ball and Bass, 2003), we
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
3. began a close examination of the actual work of teaching ele- do teachers do in teaching mathematics, and in what ways
mentary school mathematics, noting all of the challenges in does what they do demand mathematical reasoning, insight,
this work that draw on mathematical resources, and then we understanding, and skill? Instead of starting with the cur-
analyzed the nature of such mathematical knowledge and riculum they teach, or the standards for which they are re-
skills and how they are held and used in the work of teach- sponsible, we have been studying teachers’ work. By “teach-
ing. From this we derived a practice-based portrait of what ing,” we mean everything that teachers do to support the in-
we call “mathematical knowledge for teaching”—a kind of struction of their students. Clearly we mean the interactive
professional knowledge of mathematics different from that work of teaching lessons in classrooms, and all the tasks that
demanded by other mathematically intensive occupations, arise in the course of that. But we also mean planning those
such as engineering, physics, accounting, or carpentry. We lessons, evaluating students’ work, writing and grading as-
then rigorously tested our hypothesis about this “profes- sessments, explaining class work to parents, making and
sional” knowledge of mathematics, first by generating spe- managing homework, attending to concerns for equity, deal-
cial measures of teachers’ professional mathematical knowl- ing with the building principal who has strong views about
edge and then by linking those measures to growth in stu- the math curriculum, etc. Each of these tasks involves
dents’ mathematical achievement. We found that teachers knowledge of mathematical ideas, skills of mathematical rea-
who scored higher on our measures of mathematical knowl- soning and communication, fluency with examples and
edge for teaching produced better gains in student achieve- terms, and thoughtfulness about the nature of mathematical
ment. This article traces the development of these ideas and proficiency (Kilpatrick, Swafford, and Findell, 2001).
describes this professional knowledge of mathematics for To illustrate briefly what it means to know mathematics
teaching. for teaching, we take a specific mathematical topic—multi-
plication of whole numbers. One aspect of this knowledge is
What Does It Mean To Know to be able to use a reliable algorithm to calculate an answer.
Mathematics for Teaching? Consider the following multiplication problem:
Every day in mathematics classrooms across this country, 3 35
students get answers mystifyingly wrong, obtain right an- 3 25
swers using unconventional approaches, and ask questions:
Why does it work to “add a zero” to multiply a number by Most readers will remember how to carry out the steps of
ten? Why, then, do we “move the decimal point” when we the procedure, or algorithm, they learned, resulting in the
multiply decimals by ten? And is this a different procedure following:
or different aspects of the same procedure—changing the 1
place value by one unit of ten? Is zero even or odd? What is 2
the smallest fraction? Mathematical procedures that are au- 3 35
tomatic for adults are far from obvious to students; distin- 3 25
guishing between everyday and technical uses of terms— 3 175
mean, similar, even, rational, line, volume—complicates 70
communication. Although polished mathematical knowl- 875
edge is an elegant and well-structured domain, the mathe-
matical knowledge held and expressed by students is often Clearly, being able to multiply correctly is essential
incomplete and difficult to understand. Others can avoid knowledge for teaching multiplication to students. But this
dealing with this emergent mathematics, but teachers are in is also insufficient for teaching. Teachers do not merely do
the unique position of having to professionally scrutinize, problems while students watch. They must explain, listen,
interpret, correct, and extend this knowledge. and examine students’ work. They must choose useful mod-
Having taught and observed many mathematics lessons els or examples. Doing these things requires additional
ourselves, it seemed clear to us that these “classroom prob- mathematical insight and understanding.
lems” were also mathematical problems—but not the kind Teachers must, for example, be able to see and size up a
of mathematical problems found in the traditional disci- typical wrong answer:
plinary canons or coursework. While it seemed obvious that
teachers had to know the topics and procedures they 3 35
teach—factoring, primes, equivalent fractions, functions, 3 25
translations and rotations, and so on—our experiences and 3 175
observations kept highlighting additional dimensions of the 70
knowledge useful in classrooms. In keeping with this obser- 245
vation, we decided to focus our efforts on bringing the na-
ture of this additional knowledge to light, asking what, in Recognizing that this student’s answer as wrong is one step,
practice, teachers need to know about mathematics to be to be sure. But effective teaching also entails analyzing the
successful with students in classrooms. source of the error. In this case, a student has not “moved
To make headway on these questions, we have focused on over” the 70 on the second line.
the “work of teaching” (Ball, 1993; Lampert, 2001). What (Continued on page 20)
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
4. Knowing Mathematics Connecting Figure 1 to the full partial product version of
the algorithm is another aspect of knowing mathematics for
(Continued from page 17) teaching:
Sometimes the errors require more mathematical analysis: 3 35
1 3 25
2 3 25
3 35
3 25
3 255 1 600
The model displays each of the partial products—25, 150,
What has happened here? Teachers may have to look longer 100, and 600—and shows the factors that produce those
at the mathematical steps that produced this, but most will products—5 3 5 (lower right hand corner), 20 3 5 (lower
be able to see the source of the error.4 Of course teachers can left hand corner), for example. Examining the diagram verti-
always ask students to explain what they did, but if a teacher cally reveals the two products—700 and 175—from the
has 30 students and is at home grading students’ homework, conventional algorithm illustrated earlier:
it helps to have a good hypothesis about what might be
3 35
causing the error.
3 25
But error analysis is not all that teachers do. Students not
only make mistakes, they ask questions, use models, and 3 175
think up their own non-standard methods to solve prob- 70
lems. Teaching also involves explaining why the 70 should 875
be slid over so that the 0 is under the 7 in 175—that the
second step actually represents 35 3 20, not 35 3 2 as it Representation involves substantial skill in making these
appears. connections. It also entails subtle mathematical considera-
Teaching entails using representations. What is an effective tions. For example, what would be strategic numbers to use
way to represent the meaning of the algorithm for multiply- in an example? The numbers 35 and 25 may not be ideal
ing whole numbers? One possible way to do it is to use an choices to show the essential conceptual underpinnings of
area model, portraying a rectangle with side lengths of 35 the algorithm. Would 42 and 70 be better? What are the
and 25, and show that the area produced is 875 square units: considerations in choosing a good example for instructional
purposes? Should the numerical examples require regroup-
ing, or should examples be sequenced from ones requiring
no regrouping to ones that do? And what about the role of
zeros at different points in the procedure? Careful advance
thought about such choices is5 a further form of mathemati-
cal insight crucial to teaching.
Note that nothing we have said up to this point involves
knowing about students. Nothing implies a particular way
to teach multiplication or to remedy student errors. We do
not suggest that such knowledge is unimportant. But we do
argue that, in teaching, there is more to “knowing the sub-
ject” than meets the eye. We seek to uncover what that
“more” is. Each step in the multiplication example has in-
volved a deeper and more explicit knowledge of multiplica-
Here the student has likely multiplied 5 3 5 to get 25, but then when
the student “carried” the 2, he or she added the 2 to the 3 before
multiplying it by the 5—hence, 5 3 5 again, yielding 25, rather than
(3 3 5) 1 2 5 17. Similarly, on the second row, he or she added the 1
to the 3 before multiplying, yielding 4 3 2 instead of (3 3 2) 1 1 5 7.
Two-digit factors, with “carries,” present all general phenomena in the
multiplication algorithm in computationally simple cases. The presence
of zero digits in either factor demands special care. The general rules
still apply, but because subtleties arise, these problems are not
Figure 1.
recommended for students’ first work. For example, in 42 3 70,
students must consider how to handle the 0. In general, it is preferable
Doing this carefully requires explicit attention to units, and for students to master the basic algorithm (i.e., multiplication problems
to the difference between linear (i.e., side lengths) and area with no regrouping) before moving on to problems that present
measures (Ball, Lubienski, and Mewborn, 2001). additional complexities.
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
5. tion than that entailed by simply performing a correct calcu- A B
lation. Each step points to some element of knowing the C
topic in ways central to teaching it.
Our example helps to make plain that knowing mathe-
matics for teaching demands a kind of depth and detail that D E
goes well beyond what is needed to carry out the algorithm
reliably. Further, it indicates that there are predictable and re-
current tasks that teachers face that are deeply entwined with
mathematics and mathematical reasoning—figuring out
where a student has gone wrong (error analysis), explaining F
the basis for an algorithm in words that children can under-
stand and showing why it works (principled knowledge of al-
gorithms and mathematical reasoning), and using mathe-
matical representations. Important to note is that each of
these common tasks of teaching involves mathematical rea-
soning as much as it does pedagogical thinking. H I
We deliberately chose an example involving concepts of
number and operations. Similar examples can be developed
about most mathematical topics, including the definition of
a polygon (Ball and Bass, 2003), calculating and explaining K
an average, or proving the completeness of a solution set to
an elementary mathematics problem. Being able to carry out
and understand multi-step problems is another site for ex-
plicit mathematical insight in teaching. Each of these re-
quires more than being able to answer the question oneself. Figure 2. Candidate shapes: Which are rectangles?
The teacher has to think from the learner’s perspective and
to consider what it takes to understand a mathematical idea computer graphics, they translated “rectangle” to “box”
for someone seeing it for the first time. Dewey (1902) cap- without a blink. Teachers need skill with mathematical
tured this idea with the notion of “psychologizing” the sub- terms and discourse that enable careful mathematical work
ject matter, seeing the structures of the subject matter as it is by students, and that do not spawn misconceptions or er-
learned, not only in its finished logical form. rors. Students need definitions that are usable, relying on
It should come as no surprise then that an emergent terms and ideas they already understand. This requires
theme in our research is the centrality of mathematical lan- teachers to know more than the definitions they might en-
guage and the need for a special kind of fluency with mathe- counter in university courses. Consider, for example, how
matical terms. In both our records from a variety of class- “even numbers” might be specified for learners in ways that
rooms and our experiments in teaching elementary students, do not lead students to accept 1¹⁄₂ as even (i.e., it can be
we see that teachers must constantly make judgments about split into two equal parts) and, still, to identify zero as even.
how to define terms and whether to permit informal lan- For example, defining even numbers as “numbers that can
guage or introduce and use technical vocabulary, grammar, be divided in half equally” allows ¹⁄₄, 1¹⁄₂, ¹⁄₅, and all other
and syntax. When might imprecise or ambiguous language fractions to be considered even. Being more careful would
be pedagogically preferable and when might it threaten the lead to definitions such as, “A number is considered even if
development of correct understanding? For example, is it and only if it is the sum of an integer with itself ” or, for stu-
fair to say to second-graders that they “cannot take a larger dents who do not work with integers yet: “Whole numbers
number away from a smaller one” or does concern for math- that can be divided into pairs (or twos) with nothing left
ematical integrity demand an accurate statement (for exam- over are called even numbers.” Although expressed in sim-
ple, “with the numbers we know now, we do not have an an- pler terms, these definitions are similar to a typical defini-
swer when we subtract a large number from a smaller one”)? tion taught in number theory: “Even numbers are of the
How should a rectangle be defined so that fourth-graders form 2k, where k is an integer.” They are accessible to ele-
can sort out which of the shapes in Figure 2 are and are not mentary students without sacrificing mathematical precision
called “rectangles,” and why? or integrity.
The typical concept held by fourth-graders would lead In our data, we see repeatedly the need for teachers to
them to be unsure about several of these shapes, and the have a specialized fluency with mathematical language, with
commonly-held “definition”—“a shape with two long sides what counts as a mathematical explanation, and with how to
and two short sides, and right angles”—does not help them use symbols with care. In addition to continuing to probe
to reconcile their uncertainty. Students who learn shapes the ways in which teachers need to understand the topics of
only by illustration and example often construct images that the school curriculum, and the mathematical ideas to which
are entirely wrong. For example, in a fourth-grade class they lead, we will explore in more detail how mathematical
taught by Ball, several students believed that “A” in Figure 2 language—its construction, use, and cultivation—is used in
was a rectangle because it was a “box,” and, in an age of the work of teaching.
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
6. potheses empirically. This required us to pose many items to
Knowing mathematics a large number of teachers; to control for the many factors
that are also likely to contribute to students’ learning and
for teaching demands detect an effect of what we hypothesized as “mathematical
knowledge for teaching,” large data sets were essential. An-
a kind of depth and detail ticipating that samples of a thousand or more teachers might
be required to answer our questions, however, we quickly
that goes well beyond saw that interviews, written responses, and other forms of
measuring teachers’ mathematical knowledge would not do,
what is needed to and we set out to try to develop multiple-choice measures,
carry out the feasibility of which others doubted and we ourselves
were unsure.
the algorithm Our collaborators experienced in educational mea-
surement informed us that the first step in construct-
reliably. ing any assessment is to set out a “domain map,” or
a description of the topics and knowledge to be
measured. We chose to focus our initial work
within the mathematical domains that are es-
pecially important for elementary teaching:
number and operations. These are important
both because they dominate the school curricu-
lum and because they are vital to students’ learn-
ing. In addition, we chose the domain of patterns, func-
tions, and algebra because it represents a newer strand of the
K-6 curriculum, thus allowing for investigation of what
teachers know about this topic now, and perhaps how
knowledge increases over time, as better curriculum and
professional development become available and teachers
gain experience in teaching this domain. We have since
added geometry items and expanded our measures upward
through middle school content.
Once the domains were specified, we invited a range of ex-
perts to write assessment items—mathematics educators,
Measuring Mathematical mathematicians, professional developers, project staff, and
Knowledge for Teaching classroom teachers. We asked for items that posed questions
Using the methods described above, we could have contin- related to the situations that teachers face in their daily work,
ued simply to explore and map the terrain of mathematical written in multiple-choice format to facilitate the scoring and
knowledge for teaching. Because such work is slow and re- scaling of large numbers of teacher responses. We strove to
quires great care, we examined only a fraction of the possible produce items that were ideologically neutral; for example,
topics, grade levels, and mathematical practices teachers rejecting any items where a “right” answer might indicate an
might know. However, we believe that only developing orientation to “traditional” or “reform” teaching. Finally, we
grounded theory about the elements and definition of math- defined mathematical content knowledge for teaching as
ematical knowledge for teaching is not enough. If we argue being composed of two key elements: “common” knowledge
for professional knowledge for teaching mathematics, the of mathematics that any well-educated adult should have
burden is on us to demonstrate that improving this knowl- and mathematical knowledge that is “specialized” to the work
edge also enhances student achievement. And, as the current of teaching and that only teachers need know. We tried to
debates over teacher preparation demonstrate, there are le- capture both of these elements in our assessment.
gitimate competing definitions of mathematical knowledge To measure common knowledge of mathematics, we de-
for teaching and, by extension, what “teacher quality” means veloped questions that, while set in teaching scenarios, still
for mathematics instruction. To test our emerging ideas, and require only the understanding held by most adults. Figure 3
provide evidence beyond examples and logical argument, we presents one such item:
developed (and continue to refine) large-scale survey-based
Ms. Dominguez was working with a new textbook and she no-
measures of mathematical knowledge for teaching. ticed that it gave more attention to the number 0 than her old
Our two main questions were: Is there a body of mathe- book. She came across a page that asked students to determine
matical knowledge for teaching that is specialized for the if a few statements about 0 were true or false. Intrigued, she
work that teachers do? And does it have a demonstrable ef- showed them to her sister who is also a teacher, and asked her
fect on student achievement? To answer these questions, we what she thought.
needed to build data sets that would allow us to test our hy- (Continued on page 43)
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
7. Knowing Mathematics
(Continued from page 22) The claim that we can measure
Which statement(s) should the sisters select as being true? knowledge that is related to high-
(Mark YES, NO, or I’M NOT SURE for each item below.)
Yes No I’m not sure
quality teaching requires solid
a) 0 is an even number. 1 2 3
b) 0 is not really a number. 1 2 3
It is a placeholder in
writing big numbers.
c) The number 8 can be 1 2 3
written as 008.
Figure 3. Item measuring common content knowledge
To measure the more specialized knowledge of mathemat-
ics, we designed items that ask teachers to show or represent
numbers or operations using pictures or manipulatives, and
to provide explanations for common mathematical rules
(e.g., why any number is divisible by 4 if the number
formed by the last two digits is divisible by 4).
Figure 4 shows an item that measures specialized content
knowledge. In this scenario, respondents evaluate three differ-
ent approaches to multiplying 35 3 25 and determine
whether any of these is a valid general method for multiplica-
tion. Any adult should know how to multiply 35 3 25 (see
our earlier example), but teachers are often faced with evaluat-
ing unconventional student methods that produce correct an-
swers, but whose generalizability or mathematical validity are
not immediately clear. For teachers to be effective, they must
be able to size up mathematical issues that come up in class— Although students are mentioned in this item, the question
often fluently and with little time. does not actually tap respondents’ knowledge of students, or
of how to teach multiplication to students. Instead, it asks a
Imagine that you are working with your class on multiplying
large numbers. Among your students’ papers, you notice that mathematical question about alternate solution methods,
some have displayed their work in the following ways: which represents an important skill for effective teaching.
Student A Student B Student C ased on our study of practice as well as the research
3 35 base on teaching and learning mathematics, analyses
3 35 3 35
3 25 of curriculum materials, examples of student work,
3 25 3 25
and personal experience, we have developed over 250 multi-
3 125 3 175 3 25 ple-choice items designed to measure teachers’ common and
1 75 1 700 150 specialized mathematical knowledge for teaching. Many
875 875 100 dozens more are under development. Building a good item
1 600 from start (early idea stage) to finish (reviewed, revised, cri-
875 tiqued, polished, pilot-tested, and analyzed) takes over a
year, and is expensive. However we regarded this as an essen-
Which of these students is using a method that could be used
to multiply any two whole numbers?
tial investment—a necessary trade-off for the ease, reliability,
and economy of a large-scale multiple-choice assessment.
Method would work Method would NOT
for all whole work for all whole I’m not sure
Our aim is to identify the content knowledge needed for
numbers numbers effective practice and to build measures of that knowledge
that can be used by other researchers. The claim that we can
a) Method A 1 2 3 measure knowledge that is related to high-quality teaching
b) Method B 1 2 3 requires solid evidence. Most important for our purposes is
whether high performance on our items is related to effec-
c) Method C 1 2 3 tive instruction. Do teachers’ scores on our items predict
that they teach with mathematical skill, or that their stu-
Figure 4. Item measuring specialized content knowledge
dents learn more, or better?
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
8. Is There Knowledge of Mathematics for
Teaching? What Do Our Studies Show? Teachers’ performance on our
We were fortunate to be involved in a study that would
allow us to answer this question. The Study of Instructional
knowledge for teaching questions—
Improvement, or SII, is a longitudinal study of schools en-
gaged in comprehensive school reform efforts. As part of
including both common and
that study, we collected student scores on the mathematics specialized content knowledge—
portion of the Terra Nova (a reliable and valid standardized
test) and calculated a “gain score”—or how many points significantly predicted the size of
they gained over the course of a year. We also collected in-
formation on these students’ family background—in partic- student gain scores.
ular their socioeconomic status, or SES—for use in predict-
ing the size of student gain scores. And importantly, we also
included many of our survey items—including those in Fig-
ures 3 and 4—on the teacher questionnaire. Half of these
items measured “common” content knowledge and half
measured “specialized” content knowledge. Teachers who
participated in the study by answering these questions al-
lowed us to test the relationship between their knowledge
for teaching mathematics and the size of their students’ gain
on the Terra Nova.
The results were clear: In the analysis of 700 first- and
third-grade teachers (and almost 3,000 students), we found
that teachers’ performance on our knowledge for teaching
questions—including both common and specialized content
knowledge—significantly predicted the size of student
gain scores, even though we controlled for things
such as student SES, student absence rate,
teacher credentials, teacher experience, and
average length of mathematics lessons
(Hill, Rowan, and Ball, 2005). The stu-
dents of teachers who answered more items
correctly gained more over the course of a year
of instruction.
Comparing a teacher who achieved an average score on
our measure of teacher knowledge to a teacher who was in
the top quartile, the students of the above-average teacher
showed gains in their scores that were equivalent to that of an
extra two to three weeks of instruction. Moreover, the size of
the effect of teachers’ mathematical knowledge for teaching
was comparable to the size of the effect of socioeconomic sta-
tus on student gain scores. This was a promising finding be-
cause it suggests that improving teachers’ knowledge may be
one way to stall the widening of the achievement gap as poor
children move through school. The research literature on the sample of students and schools, regardless of student race
effect of SES on student achievement indicates that there and socioeconomic status? Or, are minority and higher-
tends to be a significant achievement gap when students first poverty students taught by teachers with less of this knowl-
enter school and that many disadvantaged children fall fur- edge? Our data show only a very mild relationship between
ther and further behind with each year of schooling. Our student SES and teacher knowledge, with teachers of higher-
finding indicates that, while teachers’ mathematical knowl- poverty students likely to have less mathematical knowledge.
edge would not by itself overcome the existing achievement The relationship with students’ race, however, was stronger.
gap, it could prevent that gap from growing. Thus, our re- In the third grade, for instance, student minority status and
search suggests that one important contribution we can make teacher knowledge were negatively correlated, at 2.26. That
toward social justice is to ensure that every student has a is, higher-knowledge teachers tended to teach non-minority
teacher who comes to the classroom equipped with the math- students, leaving minority students with less knowledgeable
ematical knowledge needed for teaching. teachers who are unable to contribute as much to students’
This result naturally led us to another question: Is teach- knowledge over the course of a year. We find these results
ers’ mathematical knowledge distributed evenly across our shameful. Unfortunately, they are also similar to those found
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
9. elsewhere with other samples of schools and teachers (Hill Conclusions
and Lubienksi, in press; Loeb and Reininger, 2004). They Our work has already yielded tentative answers to some of
also suggest that a portion of the achievement gap on the the questions that drive current debates about education pol-
National Assessment of Educational Progress and other stan- icy and professional practice. Mathematical knowledge for
dardized assessments might result from teachers with less teaching, as we have conceptualized and measured it, does
mathematical knowledge teaching more disadvantaged stu- positively predict gains in student achievement (Hill, Rowan,
dents. One strategy toward narrowing this gap, then, could and Ball, 2005). More work remains: Do different kinds of
be investing in the quality of mathematics content knowl-
mathematical knowledge for teaching—specialized knowl-
edge among teachers working in disadvantaged schools. This
edge or common knowledge, for example, or knowledge of
suggestion is underscored by the comparable effect sizes of
students and content together—contribute more than others
teachers’ knowledge and students’ socioeconomic status on
to student achievement? The same can be said for building a
achievement gains.
Another arena for testing our ideas is in professional de- knowledge base about effective professional development.
velopment. If there is knowledge of mathematics for teach- Historically, most content-focused professional development
ing, as our studies suggest, then it should be possible for has been evaluated locally, often with perceptual measures
programs to help teachers acquire such knowledge. To probe (e.g., do teachers believe that they learned mathematics?)
this, we investigated whether elementary teachers learned rather than true measures of teacher and student learning (see
mathematical knowledge for teaching in a relatively tradi- Wilson and Berne, 1999). Developing rigorous measures,
tional professional development setting—the summer work- and having significant numbers of professional developers use
shop component of California’s K-6 mathematics profes- them, will help to build generalizable knowledge about
sional development institutes—and, if so, how much and teachers’ learning of mathematics. We emphasize that this
what those teachers learned (Hill and Ball, 2004). We ex- must be a program of research across a wide sector of the
plored whether our measures of teachers’ content knowledge scholarly community; many studies are required in order to
for teaching could be deployed to evaluate a large public make sense of how differences in program content might af-
program rigorously. We found that teachers did learn con- fect teachers, teaching, and student achievement.
tent knowledge for teaching mathematics as a result of at- These results represent progress on producing knowledge
tending these institutes. We also found that greater perfor- that is both credible and usable. In the face of this, the neg-
mance gains on our measures were related to the length of ative responses we have received from some other education
the institutes and to curricula that focused on proof, analy- professionals are noteworthy. Testing teachers, studying
sis, exploration, communication, and representations (Hill teaching or teacher learning, at scale, using standardized
and Ball, 2004). In addition to these specific findings, this student achievement measures—each of these draws sharp
study set the stage for future analyses of the conditions criticism from some quarters. Some disdain multiple-choice
under which teachers learn mathematical content for teach- items, claiming that nothing worth measuring can be mea-
ing most effectively. sured with such questions. Others argue that teaching, and
One of the most pressing issues currently before us is teacher learning, are such fine-grained complex endeavors
whether specialized knowledge for teaching mathematics ex- that large-scale studies cannot probe or uncover anything
ists independently from common content knowledge—the worth measuring. Still others claim that we are “deskilling”
basic skills that a mathematically literate adult would pos- or “deprofessionalizing” teachers by “testing” them. We
sess. Analyses of data from large early pilots of our surveys argue that these objections run counter to the very core of
with teachers (Hill and Ball, 2004) suggest that the answer the critical agenda we face as a professional community.
may be yes. Often we found that results for the questions Until and unless we, as educators, are willing to claim
representing “specialized” knowledge of mathematics (e.g., that there is professional knowledge that matters for the
Figure 4) were separable statistically from results on the quality of instruction and can back that claim with evidence,
“common” knowledge items (e.g., Figure 3). In other words, we will continue to be no more than one voice among many
correctly answering the kind of question in Figure 4 seemed competing to assert what teachers should know and how
to require knowledge over and above that entailed in an- they might learn that, and why. Our claims to professional
swering the other kind correctly (e.g., Figure 3). This sug- knowledge will be no more than the weak claim that we are
gests that there is a place in professional preparation for con- professionals and deserve authority because we say so, not
centrating on teachers’ specialized knowledge. It may even because we can show that what we know stands apart from
support a claim by the profession to hold a sort of applied what just anyone would know. Isolating aspects of knowing
mathematical knowledge unique to the work of teaching. If mathematics different from that which anyone who has
this finding bears out in further research, it strengthens the graduated from sixth grade would know, and demonstrating
claim that effective teaching entails a knowledge of mathe- convincingly that this knowledge matters for students’ learn-
matics above and beyond what a mathematically literate ing, is to claim skill in teaching, not to deskill it. Making
adult learns in grade school, a liberal arts program, or even a these arguments, too, is part of the challenge we face as we
career in another mathematically intensive profession such seek to meet the contemporary challenges to our jurisdic-
as accounting or engineering. Professional education of tional authority.
some sort—whether it be pre-service or on the job—would Our research group’s experience in working from and
be needed to support this knowledge. with problems of professional practice, testing and refining
Reprinted with permission from the Fall 2005 issue of American Educator,
the quarterly journal of the American Federation of Teachers, AFL-CIO.
10. them with tools that mediate the power of our own convic-
tions and common sense, is one example of the work of try- Unless we, as educators, are willing
ing to build knowledge that is both credible and useful to a
range of stakeholders. Many more examples exist and can be to claim that there is professional
developed. Doing so is imperative in the current environ-
ment in which demands for education quality are made in a knowledge that matters for the
climate of distrust and loss of credibility. Meeting this chal-
lenge is a professional responsibility. Doing so successfully is quality of instruction and can back
essential to our survival as a profession. l
that claim with evidence, we will
The research reported in this paper was supported in part by
continue to be no more than
grants from the U.S. Department of Education to the Consor-
tium for Policy Research in Education (CPRE) at the Univer-
one voice among many
sity of Pennsylvania (OERI-R308A60003) and the Center for
the Study of Teaching and Policy at the University of Wash- competing to assert
ington (OERI-R308B70003); the National Science Founda-
tion (REC-9979863 & REC-0129421, REC-0207649, what teachers should
EHR-0233456, and EHR-0335411), and the William and
Flora Hewlett Foundation, and the Atlantic Philanthropies. know.
Opinions expressed in this paper are those of the authors, and do
not reflect the views of the U.S. Department of Education, the Na-
tional Science Foundation, the William and Flora Hewlett Founda-
tion or the Atlantic Philanthropies. We gratefully acknowledge helpful
comments by Daniel Fallon, Jennifer Lewis, and Mark Thames on an
earlier draft of this paper.
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