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Improving the mathematics preparation of elementary teachers, one lesson at a
time
Dawn Berk a; James Hiebert a
a
University of Delaware, USA
Online Publication Date: 01 June 2009
To cite this Article Berk, Dawn and Hiebert, James(2009)'Improving the mathematics preparation of elementary teachers, one lesson
at a time',Teachers and Teaching,15:3,337 — 356
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Teachers and Teaching: theory and practice
Vol. 15, No. 3, June 2009, 337–356
Improving the mathematics preparation of elementary teachers,
one lesson at a time
Dawn Berk* and James Hiebert
University of Delaware, USA
(Received 26 July 2008; final version received 1 October 2008)
Taylor and Francis Ltd
CTAT_A_405842.sgm
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Teachers
10.1080/13540600903056692
1354-0602
Original
Taylor
302009
15
Dr
berk@udel.edu
000002009
DawnBerk
&Article
and
Francis
(print)/1470-1278
Teaching: theory(online)
and practice
In this paper, we describe a model for systematically improving the mathematics
preparation of elementary teachers, one lesson at a time. We begin by identifying
a serious obstacle for teacher educators: the absence of mechanisms for
developing a shareable knowledge base for teacher preparation. We propose our
model as a way to address this challenge, elaborating the principles that define the
model to show its relevance. We then provide an example of the model in action,
detailing how the model was used to gradually but steadily improve a single
mathematics lesson for prospective elementary teachers. We conclude by
presenting data indicating that the model is effective in generating and vetting
knowledge that helps to improve our mathematics program over time. Although
our discussion is situated in the mathematical preparation of prospective
elementary teachers and draws on examples from mathematics, we argue that the
model could be applied to build knowledge and improve teacher preparation in
any discipline.
Keywords: teacher preparation; elementary teachers; teacher education;
mathematics; curriculum
Few readers would argue that teacher preparation in most countries is of sufficiently
high quality to ensure a steady stream of excellent teachers. In the USA, there is a
wide range of teacher preparation programs, most operating independently and without reliable evidence of effectiveness (Levine, 2006). What approaches to teacher
education might yield improvements? In this paper, we identify a major obstacle to
improving teacher preparation and describe one promising approach for surmounting
this obstacle and defining a path for gradual but steady improvement.
One of the most serious obstacles to improving teacher education is the absence of
a shared credible knowledge base for how to effectively prepare teachers (CochranSmith & Zeichner, 2005). Without reliable knowledge, teacher educators within
each program must make their own way, using their own judgments about what
proficiencies teachers need and how to help prospective teachers acquire them.
The biggest problem, however, is not that teacher educators currently lack a
knowledge base but that the field of teacher education has no infrastructure and no
widespread mechanism for acquiring one – for systematically capturing, elaborating,
and refining knowledge generated by its members. Most programs, perhaps even some
highly effective ones, have not captured the local knowledge they are acquiring so it
*Corresponding author. Email: berk@udel.edu
ISSN 1354-0602 print/ISSN 1470-1278 online
© 2009 Taylor & Francis
DOI: 10.1080/13540600903056692
http://www.informaworld.com
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D. Berk and J. Hiebert
can be shared with others. Some notable exceptions exist. For example, Korthagen,
Kessels, Koster, Lagerwerf, and Wubbels (2001) have documented the effects of a
Dutch program that helps prospective teachers become intensely reflective about their
practice. Lampert and Graziani (2009) describe in detail the processes used in an Italian language school for building knowledge for training language teachers.
The approach to teacher preparation we describe is consistent with these examples
in its focus both on improving a particular teacher education program and on accumulating knowledge from the improvement efforts that can be shared with others. In
simplest terms, we conceive of improving teacher education as synonymous with
improving teaching – the students are prospective teachers and the teachers are teacher
educators. Consequently, the key features or principles of the model we propose are
drawn from the everyday practices of classroom teachers. At some level, all teachers
ask themselves: What should students learn today? How can I tell whether or not they
have learned it? What could I do better next time? Unpacking these questions and
addressing them intentionally and systematically has produced a model for studying
and improving teacher preparation that we believe can contribute to a shareable
knowledge base.
The model we present shares many features with design research (Brown, 1992;
Design-Based Research Collective, 2003; Gravemeijer & van Eerde, 2009). Goals of
the work include both the actual improvement of practice and the generation of
shareable knowledge about that practice. The process plays out through continuing
cycles of planning, enactment, analysis, and revision, and hypotheses about connections between teaching and learning are used to drive each cycle of the process. But
the model we propose views the process as continuing indefinitely rather than for a set
period of time. Rather than producing (local) theories of instruction, we aim to
produce instructional materials (e.g., lesson plans), linked to specific learning goals,
that are embedded with increasingly useful knowledge for teaching particular
concepts to particular students. Our model slows down the process of teaching by
turning lessons into experiments (Hiebert, Morris, & Glass, 2003), thereby making
each phase of teaching deliberate and intentional.
The goal of this paper is to outline our model for teacher preparation and to
illustrate how the model addresses the knowledge base problem. We begin by briefly
describing the context in which the model has been developed. We then elaborate the
principles that define our model for improving the mathematics preparation of
elementary teachers. Next, we provide an example of the model in action, detailing
how it was employed to gradually but steadily improve a single mathematics lesson
for prospective elementary teachers over three semesters. Finally, we present data to
show that employing the model can improve our mathematics curriculum over time
and contribute to a knowledge base for helping other prospective teachers achieve the
same learning goals. Although our discussion is situated in the mathematics preparation of prospective elementary teachers, we argue that the model could be applied to
the improvement of teacher preparation in any discipline.
The context
The model we describe has been employed by teacher educators and doctoral students at
the University of Delaware for the past seven years to design, study, and improve the
mathematics preparation of prospective elementary teachers. The mathematics component of the elementary teacher preparation program consists of three mathematics
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339
content courses and one elementary mathematics methods course required of all students
pursuing elementary (grades K-6) certification.1 Multiple sections of each course are
offered each semester and are taught by mathematics education faculty and doctoral
students in the education department.
Improvements to the preparation program are embedded in the curricula for the
four courses. To date, a set of approximately 20 elaborated lesson plans has been
developed for each course (one plan for each class session). These lesson plans,
constituting the curriculum for each course, are implemented by all course instructors
of a given course each semester. Each lesson plan specifies the learning goals, the
instructional activities, information about prospective teachers’ typical responses to
the instructional activities (e.g., common questions, responses, and misconceptions),
and suggested instructor responses. In addition, hypotheses about how the instructional activities support prospective teachers in achieving the learning goals as well as
rationales for the inclusion of particular activities over others are recorded in the
lesson plans when possible. In each course, several of these lesson plans have already
undergone multiple cycles of careful study and revision. The results of these efforts,
in the form of more elaborated and precise learning goals, revised instructional
activities, and more refined hypotheses, are also recorded in the plans.
How did such a curriculum come to be collectively developed, studied, and
improved over the last seven years, and how do we know that the current curriculum
is an improvement over the one we began with seven years ago? The answer to these
questions lies in the model we have developed for improving the mathematics
preparation of elementary teachers. In the next section, we elaborate the model by
describing the three principles that define it.
Key principles of the model
We identify three key principles of our model: (1) specify the critical learning goals
for students (i.e., prospective teachers); (2) collect and use evidence of students’
learning to drive revisions; and (3) gather and store knowledge in a shared product.
The first two principles are rooted in the ordinary practice of teachers but take on a
research stance when deployed systematically (Dewey, 1929). This inquiry orientation is compatible with the notion of teacher as a researcher (Burnaford, Fischer, &
Hobson, 1996; Cochran-Smith & Lytle, 1993, 1999) and with the literature on
reflective practice (Cruickshank & Applegate, 1980; Osterman & Kottkamp, 2004).
These two principles focus the reflective practice squarely on the relationships
between instructional activities and students’ achievement of clearly specified
learning goals.
The third principle, gathering and storing knowledge in a shared product, moves
the process from simply improving an individual program to one of generating
knowledge that can be shared with others. This principle commits teacher educators
to see their work as contributing to the improvement of the profession rather than
simply as improving their own practice.
Specify and commit to a set of targeted learning goals for prospective teachers
A key principle that defines our model is that a teacher preparation program should be
defined by a set of specific and targeted learning goals. Specifying the learning goals
up front is crucial because the learning goals should drive the curriculum design
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D. Berk and J. Hiebert
process (Wiggins & McTighe, 2005). The learning goals drive the selection and
development of instructional activities (i.e., the curriculum of the teacher preparation
program). Each learning goal should map onto one or more instructional activities that
are designed to support prospective teachers in achieving that goal; conversely, each
instructional activity should map onto one or more of the identified learning goals. A
consequence of this is that instructional activities that do not support prospective
teachers in achieving the specified learning goals, even those activities that are an
instructor’s favorite or are especially popular with prospective teachers, have no place
in the curriculum.
The learning goals also define what should constitute evidence of program
effectiveness and thus provide the measure against which changes are assessed. Only
by testing whether changes in the instructional activities help prospective teachers
achieve the learning goals more effectively can one answer the critical question: Are
the changes improvements to the program, or just changes?
Given the key role that the learning goals play in the design and improvement
process, it becomes clear that they must share two important features. First, the
learning goals must be well-specified. The more targeted or elaborated the learning
goal, the more useful it is in guiding the design of instructional activities and in
determining whether the prospective teachers achieved the goal (Jansen, Bartell, &
Berk, 2009). For example, contrast the learning goal, ‘Prospective elementary teachers
will understand subtraction’ with the learning goal, ‘Prospective elementary teachers
will be able to interpret subtraction as both a removal of a smaller quantity from a
larger quantity, and as a comparison of a smaller quantity to a larger quantity’. The
former learning goal is quite broad and underspecified. It is not clear what would
constitute ‘understanding of subtraction’ and so it is unclear how to gauge whether
prospective teachers had achieved such understanding. As a result, the goal provides
little guidance as to the types of instructional activities that would support prospective
teachers in achieving it or the types of evidence that would need to be collected to
determine achievement of it. In contrast, the latter learning goal is more elaborated,
unpacking the ways of understanding subtraction that prospective teachers are to
develop.
Second, the learning goals must be shared. In other words, all instructors, current
and future, must commit to the learning goals. One implication of committing to a
common set of shared goals is that teacher educators must surrender the prerogative
to discard and replace particular learning goals as they move in and out of different
courses each semester. Shifting goals from one semester to the next makes the
curriculum a moving target and thus impossible to systematically improve over time.
Accumulating knowledge about prospective teachers’ achievement of particular
learning goals, and linking their learning to particular instructional activities, is a longterm process. Only by developing and maintaining a consensus around a common set
of learning goals over multiple semesters can this work take place and result in the
gradual, systematic improvement of the preparation for prospective teachers. (See
Jansen, Bartell, & Berk, 2009, for a more detailed discussion of the crucial role of
learning goals in building knowledge for teacher education).
In the context of the mathematical preparation of prospective elementary teachers,
we argue that the learning goals should target the mathematics knowledge that
teachers draw on when teaching mathematics to elementary school students (Ball,
1999; Ball & Bass, 2000; Hill & Ball, 2004; Ma, 1999). Given the limited time to
study mathematics in an elementary preparation program, only those learning goals
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that can be justified in terms of essential mathematics-knowledge-for-teaching should
be included. Because the field of mathematics education, similar to other fields, does
not have full knowledge of what elementary teachers need to know, the selection of
these goals requires substantial professional judgment. This issue provides a potential
obstacle for addressing the challenges posed in the opening paragraphs and will be
revisited later.
To reiterate, a first step in addressing the challenge of preparing prospective teachers and developing shareable knowledge for improving teacher preparation is to select
an explicit, specific, and targeted set of learning goals that are shared among the
teacher educators and are committed to for the long run. Doing so allows the teacher
educators to treat the course as an object of study, thereby generating cumulating and
shareable knowledge. As we noted earlier, a thorny question to which we will return
later is how to select the learning goals and commit to them in the absence of complete
knowledge of what subject matter knowledge is needed to the teach the subject well.
Use evidence of prospective teachers’ learning to drive revisions
A second principle of our model is that evidence of prospective teachers’ learning, and
nothing else, should drive revisions to the preparation program. This is another way
of saying that improvements should be research-based.
The principle of basing changes on prospective teachers’ learning, and only on
their learning, focuses the attention of the course (and the program as a whole), on the
developing competencies of the prospective teachers. If the learning goals are wellchosen, this unrelenting focus on prospective teachers’ learning should result in real
improvements to the program over time. In addition, the principle ensures that the
information about changes to the instructional activities saved as ‘knowledge’ is
vetted through empirical tests. This provides the kind of quality control needed for any
useful knowledge base.
As with the first principle underlying our model, this principle requires a shared
commitment by all instructors. Committing to the principle that changes to the
curriculum must be driven by evidence of prospective teachers’ achievement of the
specified learning goals prevents modifications based on instructors’ ‘hunches’ (e.g.,
I just have a gut feeling that this activity is not working) or on affective feedback from
prospective teachers (e.g., The prospective teachers really enjoyed themselves during
this activity, so we should keep it). While particular theories of learning or teaching
can inform decisions about how to revise the instruction to better support prospective
teachers’ learning, it is data on their achievement of the learning goals that should
serve as the impetus for changes to the curriculum.
This principle also requires a conscious commitment to a long-term, gradual
process of studying and improving the curriculum rather than to a series of ‘quick
fixes’ or huge overhauls. Collecting information that will provide evidence as to
whether and how certain instructional activities supported prospective teachers’
achievement of particular learning goals, and testing variations of the activities that
might more effectively support prospective teachers’ learning, take a great deal of
time and effort. The result is a process of gradual, incremental improvements to the
curriculum that slowly accumulate over time, requiring that instructors be patient,
dogged, and diligent and that they agree to give up the (often fleeting and misplaced)
satisfaction that can come from making quick, grand-scale, but insufficientlyinformed changes to the curriculum each semester.
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D. Berk and J. Hiebert
Store knowledge for preparing teachers in a shared, tangible product
The third and final principle that defines our model is that improvements to the
preparation program must be embedded in the joint construction and revision of a
shared, tangible product. This enables all instructors to participate in improving the
preparation program while preserving the evidence-based improvements accumulated
over time.
In our case, the curriculum – the elaborated lesson plans for each of the four
courses – serves as the product. Storing knowledge for preparing teachers within the
lesson plans themselves makes the knowledge readily and easily accessible to instructors and situates the knowledge in a context where it is to be used. This maximizes the
chances that new instructors will use the knowledge accumulated about the lesson in
their teaching of the lesson and in their future evidence-gathering efforts. In this way,
the lesson plan serves simultaneously as the plan for instruction, the object of study,
and the repository for knowledge acquired from such study.
In addition to recording modifications to the lesson’s learning goal(s) and the
instructional activities, we have designed two structures within the lesson plan that
enable us to store information gathered from our continual study of the curriculum and
its effects on prospective teachers’ learning. First, there are placeholders within the
lesson plan to record a brief history of the revisions a particular component of the
lesson has undergone and the rationale for those revisions. Preserving this information
allows current instructors to avoid returning to past versions of the learning goal or the
instructional activities that have already been investigated. Second, there are placeholders within the lesson plan to record hypotheses that instructors have generated for
future testing. For example, based on their study of the curriculum in a given semester,
instructors might develop a conjecture about a misconception that prospective
teachers are continuing to exhibit or about a possible instructional activity that may
better support prospective teachers’ learning. These hypotheses can then be taken up
and pursued by course instructors in future semesters. Designing the lesson plans so
that they can be used to preserve information gathered over time allows us to build on
the efforts of past instructors and gradually, systematically accumulate knowledge of
how to more effectively prepare prospective elementary teachers.
An example of the model in action
In this section, we illustrate the model in action by describing an example of several
iterations of a single mathematics lesson over several semesters. One goal in presenting this example is to highlight how the key principles described above are enacted in
the model. A second goal is to exemplify how the model can be used to both gradually
improve a particular mathematics preparation program as well as build knowledge
from these efforts that can be shared with other teacher educators.
Overview of the lesson
The lesson we describe comes from the second course in a sequence of three mathematics content courses for prospective elementary teachers. The second course
focuses on rational number concepts and operations, a topic that is well-recognized as
critical for elementary mathematics and poorly understood by many prospective and
practicing teachers (e.g., Ball, 1990; Harel & Behr, 1995; Ma, 1999; National Council
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343
of Teachers of Mathematics, 2000; National Mathematics Advisory Panel, 2008; Post,
Harel, Behr, & Lesh, 1988; Simon, 1993; Sowder et al., 1998). The key learning goal
for the lesson was to understand how to represent subtraction of fractions with a story
problem. We identified this as an important learning goal because being able to create
subtraction story problems is an authentic task for elementary teachers – this is knowledge that they would draw on when teaching mathematics to children. Moreover, the
process of writing story problems can help prospective teachers deepen their understanding of the underlying mathematics – the meaning of subtraction and ways to
represent subtraction in real-world contexts (e.g., interpreting subtraction as take away
or as comparison), ways of using fractions to represent quantities, and the role of the
referent in interpreting a fraction (e.g., a fraction of what).
From our knowledge of the literature on the teaching and learning of rational
number concepts and from our own experiences as elementary mathematics teacher
educators, we anticipated that prospective elementary teachers would struggle to
represent subtraction of fractions with appropriate story problems. In particular, we
predicted that many prospective teachers would exhibit a classic error of using different referents for the two fractions. Table 1 presents an example of a story problem task
involving subtraction of fractions, an example of a correct response to this task, and a
typical incorrect response exhibiting this classic error.
In the example of a correct response, the referent for both fractions is a pound
of coffee (5/8 pound of coffee and 1/4 pound of coffee). In the example of a typical
incorrect response, the referent for the first fraction is a pound of coffee, but the referent for the second fraction is 5/8 pound of coffee (1/4 of the coffee, meaning 1/4 of the
5/8 pound). Without employing the same referent for both fractions, the problem
requires first determining what portion of a pound of coffee has been used before
subtracting. In terms of whole numbers, this error would be similar to a story problem
asking one to subtract three apples from six oranges. In this case, three cannot be
subtracted from six because the referents are different. Although the need for like units
(referents) is obvious in the case of whole-number subtraction, both school children
and teachers often neglect to maintain like units when writing story problems for
subtraction of fractions.
Version 1 of the lesson
Research indicates that mathematics instruction that attends explicitly to key conceptual issues and that engages students with wrestling or struggling with those concepts
Table 1. Initial story problem task, sample correct response, and typical incorrect response
and associated number sentence.
Task: Write a realistic story problem for the number
sentence below.
5/8 – 1/4 = ?
Example of a correct response
Typical incorrect response
Number sentence associated
with the incorrect response
Kathy has 5/8 pound of coffee. She uses 1/4 pound of the
coffee. How much coffee is left?
Kathy has 5/8 pound of coffee. She uses 1/4 of the coffee.
How much coffee is left?
5/8 – (1/4 of 5/8) = ?
344
supports students’ development of conceptual understanding (Hiebert & Grouws,
2007). Thus, we designed the original lesson so that the initial task would elicit
prospective teachers’ misconceptions about the referent in fraction subtraction
problems and would encourage them to actively confront and resolve those misconceptions. Rather than begin instruction by modeling how to write appropriate story
problems and alerting prospective teachers in advance to the error we anticipated, we
opened the lesson simply by soliciting subtraction story problems from the prospective teachers. Prospective teachers would begin by responding to the task presented in
Table 1, and then share their story problems in small groups. An excerpt from the
initial lesson plan is shown in Figure 1.
Although we expected many prospective teachers to exhibit the referent error
described, we also predicted that at least a few would write an appropriate story
problem. Instructors would monitor the small group discussions to identify a few
prospective teachers who had written correct stories and a few who had written incorrect stories exhibiting the referent error. These prospective teachers would be invited
to share their story problems with the entire class. We conjectured that by studying
and discussing examples of correct stories from their peers, and contrasting them with
the incorrect stories, prospective teachers who had written incorrect stories would
recognize that the key issue was employing a common referent and would be
convinced that their original stories were inappropriate.
Observations and recollections from instructors shared during weekly course
instructor meetings revealed that the lesson was not particularly effective at supporting the prospective elementary teachers in achieving the learning goal. Although the
instructors’ prediction that some prospective teachers would write correct story
problems while many others would create stories exhibiting the referent error was
confirmed, the instructors had difficulty helping the prospective teachers resolve the
debate that ensued during the whole-class discussion. At the conclusion of the lesson,
many prospective teachers were still unable to distinguish the two story problem types
(correct story problems and story problems exhibiting the referent error). Others could
distinguish the two story problem types, but struggled to understand why the incorrect
story problem was wrong. Even instructors who eventually made explicit to prospective teachers that the source of the error was in using different referents for the two
fractions reported that many prospective teachers were unable to understand why this
was an issue. Additional evidence collected from responses to items on the course
exams also indicated that the majority of prospective teachers had failed to achieve the
learning goal. When asked to write an appropriate story problem for a given fraction
subtraction sentence on these assessments, many prospective teachers continued to
exhibit the classic error of using different referents.
Figure 1.
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D. Berk and J. Hiebert
An excerpt from Version 1 of the lesson.
Version 2 of the lesson
The following semester, the course instructors used the evidence collected about
prospective teachers’ learning to make targeted revisions to both the learning goal and
the lesson (see Figure 2 for excerpts of the revised lesson). The instructors hypothesized that the prospective teachers’ difficulties in achieving the learning goal stemmed
primarily from a failure to understand and appreciate the role of the referent and used
this hypothesis to elaborate the learning goal. Consequently, the learning goal,
initially described in the lesson as, ‘prospective teachers will understand how to
represent subtraction of fractions with a story problem’ was revised to ‘prospective
Figure 1.
An excerpt from Version 1 of the lesson.
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345
An excerpt from Version 2 of the lesson.
Note: Text in bold italics denotes proposed additions to the lesson. Text that has been crossed out denotes proposed deletions.
Figure 2.
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D. Berk and J. Hiebert
Teachers and Teaching: theory and practice
teachers will understand how to represent subtraction of fractions with a story
problem. This involves understanding the need to employ the same referent for each
fraction as well as being able to distinguish story problems in which the referent is the
same from those in which the referent is different’. The revised learning goal was
recorded in the written lesson plan.
Instructors viewed the elaborated learning goal as a significant improvement, in
that it made explicit a key idea involved in understanding how to write story problems
for subtraction of fractions, and in doing so provided guidance as to how the lesson
might be revised to more effectively support prospective teachers’ learning. In particular, the elaborated learning goal suggested the need for additional instructional tasks
that would more effectively engage prospective teachers in identifying and understanding the role of the referent and in explicitly comparing and contrasting story
problems that employed the same referent with those that did not. Instructors conjectured that perhaps comparing pairs of subtraction story problems that involved ‘friendlier’ (i.e., more familiar) fractions would help prospective teachers distinguish correct
story problems from incorrect story problems because the different answers resulting
from the two stories would be readily apparent to them. Tasks involving more familiar
fractions might also help prospective teachers discriminate between stories in which
the referent for both fractions was the same and stories in which the referent for both
fractions differed.
Working from the elaborated learning goal and from their conjectures about
instructional tasks that would better support prospective teachers in achieving that
goal, the course instructors revised the initial lesson plan. Given its success in eliciting
story problems that exhibited the key misconception, the original opening task was
retained. Instructors would again have prospective teachers discuss their story problems in small groups, then solicit both correct story problems and story problems
exhibiting the referent error for prospective teachers to consider in a whole class
discussion. However, instead of immediately trying to resolve the ensuing debate,
instructors would then present prospective teachers with a new task in which prospective teachers were given two pairs of subtraction story problems and asked to solve
and write the appropriate number sentence for each story. Within each pair, both story
problems would involve the same contextual situation and the same two fractions.
However, one story problem would employ the same referent for each fraction, while
the other story problem would exhibit the referent error.
Figure 2 displays the new instructional task. By employing friendlier fractions, the
new story problems enable the solver to readily identify the correct answer to each
story. For example, for the first pair of story problems, prospective teachers should be
able to determine upon quick inspection that the correct answer to the first story
problem should be 0 pounds of chocolate and the correct answer to the second story
problem should be 1/4 pound of chocolate. Since the two resulting answers are
different, the two story problems must be represented by different number sentences.
In this way, the new task might help convince prospective teachers that the two types
of story problems are indeed different, and that the difference stems from using
different referents for the fraction being subtracted. After solving and discussing the
new task, prospective teachers would be asked to revisit and reconsider their responses
to the initial task. The course instructors conjectured that the prospective teachers
who had solved the initial task incorrectly would be able to identify that the source of
their error was in using different referents and would subsequently revise their story
problem.
FigureText
Note:
2. in
Anbold
excerpt
italics
from
denotes
Version
proposed
2 of theadditions
lesson. to the lesson. Text that has been crossed out denotes proposed deletions.
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D. Berk and J. Hiebert
To investigate the effects of the revisions on prospective teachers’ achievement of
the elaborated learning goal, the course instructors observed each other teach the
revised lesson. Each observer joined a different small group of prospective teachers as
they solved the initial and new tasks. Certain instructors were also videotaped as they
taught the lesson, so that the prospective teachers’ responses to the lesson could be
studied more carefully outside of class by the instructors. Copies of prospective
teachers’ solutions to the instructional tasks, as well as their responses to items on the
course exams, were also collected.
Analyses of these data indicated that the revised lesson was more effective in
supporting prospective teachers in achieving the learning goal. Despite writing an
incorrect story problem for the initial task, most prospective teachers were immediately able to correctly solve and write number sentences for the pairs of stories in the
new task. After solving and discussing the pairs of story problems, most prospective
teachers were able to recognize that the stories in each pair were different and were
able to explain that the distinction between the two stories lies in the referents used in
each story. Moreover, many were also able to repair the story problems they had
written in response to the initial task. Prospective teachers also performed better on
items on the final exam in which they were asked to write story problems to represent
subtraction of fractions.
Based on this evidence, the course instructors decided to replace the original
lesson plan with the revised lesson plan which included the elaborated learning goal
and the new task. In addition, the rationale for the new task was recorded in the lesson
plan. Recording what was learned about if and how the instructional tasks supported
prospective teachers’ achievement of the learning goal in the lesson plan itself enables
us to generate and accumulate knowledge about the teaching and learning of prospective elementary teachers that can be passed on to future course instructors. In this way,
future instructors are able to build on what past instructors have learned, rather than
start anew and replicate past efforts.
Version 3 of the lesson
Although analyses indicated that the revised lesson better supported prospective
teachers’ achievement of the learning goal, the evidence also revealed that a subset of
prospective elementary teachers still failed to develop an understanding of how to
represent subtraction of fractions with appropriate story problems. For example, careful examination of the videotapes of the lesson revealed some prospective teachers
were still unable to repair their story problem for the initial task at the conclusion of
the lesson. Others continued to struggle on tasks requiring them to write story problems of their own. Consequently, the next group of course instructors hypothesized
that prospective teachers might benefit from additional opportunities to write story
problems and that such opportunities should engage prospective teachers explicitly in
considering the role of the referent. Based on this hypothesis, the learning goal was
further elaborated to include an expectation that prospective teachers develop the
ability to distinguish fraction subtraction number sentences that employed the same
referent from those that did not, and to be able to write appropriate story problems for
both types of number sentences. Figure 3 shows the further revised lesson plan with
the refined learning goal.
Once again, instructors used the newly elaborated learning goal to inform revisions
to the lesson plan. They decided to retain the initial two tasks from the lesson, but to
FigureText
Note:
3. in
Anbold
excerpt
italics
from
denotes
Version
additions
3 of thetolesson.
Version 2 of the lesson. Text that has been crossed out denotes deletions to Version 2 of the lesson. Text that has been shaded denotes proposed additions to Version 3 of the lesson.
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An excerpt from Version 3 of the lesson.
Note: Text in bold italics denotes additions to Version 2 of the lesson. Text that has been crossed out denotes deletions to Version 2 of the lesson. Text that has been shaded
denotes proposed additions to Version 3 of the lesson.
Figure 3.
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D. Berk and J. Hiebert
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develop a new task to be posed at the conclusion of the lesson for homework. In the
new homework task, prospective teachers were presented with two pairs of number
sentences involving subtraction of fractions and were asked to write an appropriate
story problem for each. They were also asked to draw diagrams to represent and solve
each number sentence. Within a pair, each number sentence involved the same two
fractions. However, while one number sentence employed the same referent for both
fractions, the other number sentence employed different referents. In this way, the
instructors conjectured that the new task would require prospective teachers to
explicitly take up the issue of the referent. Figure 3 displays the new homework task.
Anecdotal reports from course instructors suggest that the current version of the
lesson is even more effective at supporting prospective elementary teachers in
achieving the learning goal. In future semesters, course instructors will continue the
cycle of collecting more systematic evidence of prospective teachers’ learning and
using that evidence to inform additional refinements to the lesson. As additional
information is collected and new revisions are made, instructors will document these
changes in the lesson plan itself. In this way, the lesson plan serves not only as the
plan for instruction, but also as the object of study and the repository for knowledge
acquired from such study.
Summary
By presenting this example of three iterations of a single mathematics lesson for
prospective elementary teachers, we aim to illustrate how each of the key principles
that define our model can be enacted. We began by identifying and committing to a
specific learning goal for prospective elementary teachers that we believe is essential
for teaching K-6 students: understanding how to represent subtraction of fractions
with appropriate story problems. We then designed instructional tasks in line with the
research literature on learning and teaching this skill. As we implemented and studied
the lesson over the course of several semesters, we collected information about
prospective teachers’ responses to the instructional tasks and their achievement (or
not) of the learning goal. These data drove revisions to the lesson’s learning goal. The
learning goal was continually unpacked and elaborated as we became more explicit
about the key sub-concepts and skills involved in learning to write appropriate story
problems for subtraction of fractions. Evidence of prospective teachers’ learning,
coupled with more elaborated versions of the learning goal, drove revisions to the
instructional tasks. Finally, knowledge generated about the teaching and learning of
prospective elementary teachers, linked closely to the lesson’s learning goal, was
stored in the lesson plan itself so that it would be preserved and so that future course
instructors could access and build on it.
The example shows how the model can work to help prospective teachers achieve
a particular learning goal with increasing effectiveness over time and how the
cumulating knowledge provides the basis for supporting such improvements. But, this
is only one learning goal. Does the model work for other goals? And, won’t it take
forever to improve the full curriculum?
Accumulating evidence to show the model works
We have argued that a key component of our model is that revisions to the curriculum
are driven by evidence of prospective teachers’ learning, tied to particular learning
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D. Berk and J. Hiebert
goals. In the example described above, such evidence consisted of lesson-level
measures – instructors’ observations of prospective teachers’ responses and questions
during the lesson, prospective teachers’ written work collected during class, and
prospective teachers’ responses to relevant exam items assessing their achievement of
the lesson’s learning goals. In addition to collecting these data, we have undertaken
additional efforts to investigate prospective teachers’ learning as they progress
through the program.
One such effort has been the design and administration of a written assessment to
all prospective teachers at the beginning of each of the four mathematics content and
methods courses each semester, enabling us to collect longitudinal data over time and
to compare the performance of different cohorts of prospective teachers as they move
through the teacher preparation program. The assessment consists of a range of items
measuring prospective teachers’ mathematical knowledge for teaching. One set of
items measures prospective teachers’ ability to represent operations on fractions with
appropriate story problems. Prospective teachers are asked to write an appropriate
story problem for a subtraction of fractions number sentence and for a division of
fractions number sentence.
There are two ways in which we use the data from this longitudinal assessment to
investigate whether our prospective teachers are achieving the learning goals. One
way is to track a single cohort of prospective teachers as they enter and progress
through the mathematics component of their teacher preparation program. For
example, we can investigate whether prospective teachers’ ability to represent operations on fractions with story problems improves as they complete each course. Table
2 displays the percentage of prospective teachers in Cohort 1 (the first cohort of
prospective teachers we followed after beginning to employ our model) who
completed all three content courses and who wrote a correct story problem at each
administration of the assessment. These data indicate that the percentage of prospective teachers who wrote a correct story problem increased significantly at each
administration of the assessment. For example, while 43.8% wrote a correct subtraction story at the beginning of Course 2, 83.6% wrote a correct subtraction story at the
beginning of Course 3, a significant increase (χ2(1) = 43.925, p = .000). Similarly, the
percentage of prospective teachers who wrote a correct division story increased
significantly from the beginning of Course 2 to the beginning of Course 3, from 35.9
to 65.6%, respectively (χ2(1) = 22.568, p = .000).
These data suggest that lessons like the subtraction lesson described earlier are
relatively effective in supporting prospective teachers in becoming better able to
represent operations on fractions with story problems. In this way, we can use data
from the longitudinal assessment to investigate whether the courses are having some
positive, cumulative effect on prospective elementary teachers’ learning as they
progress through the program.
Table 2. Percentage of prospective teachers (n = 128) in Cohort 1 completing all three content
courses who wrote a correct story problem at the beginning of Course 1, 2, and 3.
Subtraction item: Write a story problem for 23/8 − 2/3 =?
Division item: Write a story problem for 13/4 ÷ 1/2 =?
Course 1
Course 2
Course 3
29.7
16.4
43.8*
35.9***
83.6***
65.6***
*p < .05; ***p < .001.
Note: Significant differences are between consecutive courses, for a particular story problem item.
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A second way we have used data from the longitudinal assessment is to compare
the performance of different cohorts of prospective teachers as they advance through
the program. If engaging in the model we have described is effective at generating
more effective curricula for prospective elementary teachers over time, then future
cohorts of prospective teachers should perform better on the assessment than their
peers in earlier cohorts. To enable us to produce these types of comparisons, we began
our administration of the assessment with Cohort 0, the cohort of prospective teachers
who entered the program the year before we began implementing our model. All
prospective teachers enrolled in any of the four courses during the fall semester of that
year took the assessment. We can then compare the performance of prospective teachers in Cohort 0 with the performance of later cohorts who received versions of the
curricula that had undergone the improvement process. For example, Table 3 shows
the percentage of prospective teachers in Cohort 0 and Cohort 1 who wrote a correct
story problem at each administration of the assessment. These data indicate that
prospective teachers in each cohort exhibited similar performance on both of the story
problem items upon entering the program. However, prospective teachers in Cohort 1
significantly outperformed their peers in Cohort 0 on the subtraction item at the
beginning of both Course 2 (χ2(1) = 3.989, p = .046) and Course 3 (χ2(1) = 11.212,
p = .001). Similarly, significantly more prospective teachers wrote a correct division
story problem in Cohort 1 than in Cohort 0 at the beginning of Course 3 (χ2(1) =
8.547, p = .003). These findings suggest that the revisions we have made to the lessons
by engaging in our model have been actual improvements to the program and not just
changes.
Concluding remarks
We began by identifying a major challenge for teacher preparation programs: developing mechanisms for building a knowledge base for teacher preparation. We
proposed a model for addressing this challenge and described an example of how
engaging in this model has enabled us to build knowledge for preparing elementary
mathematics teachers, linked to particular learning goals. We also described how
engaging in this model has enabled us to gradually improve our elementary teacher
preparation program. We conclude by acknowledging some challenges our model
poses and offering some conjectures about the next steps. In doing so, we return to a
Table 3. Percentage of prospective teachers in Cohort 0 and Cohort 1 who wrote a correct
story problem at the beginning of Course 1, 2, and 3.
Subtraction item: Write a story
problem for 23/8 − 2/3 =?
Course 1
Course 2
Course 3
Division item: Write a story problem
for 13/4 ÷ 1/2 =?
Cohort 0
Cohort 1
Cohort 0
Cohort 1
26.7 (n = 232)
33.7 (n = 104)
62.9 (n = 124)
25.7 (n = 307)
45.2* (n = 248)
80.5** (n = 169)
10.3 (n = 232)
21.2 (n = 104)
46.8 (n = 124)
14.3 (n = 307)
27.4 (n = 248)
63.9** (n = 169)
*p < .05; **p < .01.
Note: Significant differences are between cohorts, for a particular story problem item and a particular
administration of the assessment. The size of n for Cohort 1 in this table differs from Table 2 because Table
2 includes only those prospective teachers who completed all three content courses.
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D. Berk and J. Hiebert
‘thorny problem’ raised earlier – the identification of learning goals in a discipline that
comprise the knowledge needed to teach the discipline well.
One challenge lies in considering what demands the model places on teacher
educators. What knowledge, skills, and dispositions would teacher educators need
in order to employ the model we propose? Two of the key features of the model –
specifying learning goals and collecting evidence to determine students’ achievement
of those goals – are undertaken regularly by all teachers (including teacher educators).
For these features, the challenge lies in engaging in these activities more deliberately
and systematically. As noted earlier when citing Dewey (1929), these features infuse
the everyday tasks of teaching with a research perspective. Lessons become experiments and students’ learning becomes evidence for a lesson’s effectiveness. Although
the expertise needed to treat lessons in this way might be somewhat new for teacher
educators, it is not different, in kind, from the expertise developed by all thoughtful
teachers. Indeed, we would argue that it is exactly the kind of expertise that teacher
educators must develop if they hope to use a professional knowledge base.
The third feature of the model – gathering and storing the knowledge developed in
shareable products – might represent a new expectation and role for many teacher
educators. It is this third feature that asks teacher educators to contribute to the development of a knowledge base rather than just use knowledge generated by others. By
collaborating with colleagues who are more experienced at representing what is
learned in forms useful for others, we believe teacher educators can acquire the skills
needed to participate in this knowledge-building process. In our opinion, these skills
are exactly those that all teacher educators must develop over time. We would
argue that membership in a profession, such as teacher education, entails exactly this
activity – using and contributing to a shared knowledge base for the profession.
The experiences we shared in this paper only address the use of the model within
a single institution. A second challenge will be building bridges across institutions so
that the field of teacher education, as a whole, steadily improves its performance over
time. Based on our experience, the selection and commitment to precisely stated
learning goals is a precondition to collaboratively generate and share knowledge to
improve teacher preparation. We believe this will be as true across institutions as we
have found it true within an institution. Knowledge shared across institutions will
have no market unless the teacher educators share the same learning goals. What
someone else learns about how to help their prospective teachers learn will be of
interest only if we are trying to help our prospective teachers learn the same things.
Can teacher educators in a particular discipline, say mathematics, agree on the
same learning goals? This is an open question. A hopeful fact is that a useful
knowledge base for teacher preparation need not be all or nothing. Useful knowledge
is tied to particular learning goals. As long as teacher educators agree on some chunk
or sequence of learning goals, they can collaboratively build and share knowledge
pertaining to those goals. One can imagine communities of teacher educators forming
around particular sets of learning goals and, overtime, building a useful knowledge
base for those goals (Jansen, Bartell, & Berk, 2009).
A final challenge is how to select and commit to learning goals in the absence of
complete knowledge about the proficiencies elementary teachers need in each
discipline. There is no easy answer to this challenge because complete knowledge will
never be available. In addition, the choice of learning goals is complicated by the fact
that goals are dependent on values as well as on empirical data (Hiebert, 1999).
Instead of waiting until more information becomes available or until the debates about
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355
learning goals are settled, we believe it makes more sense to interpret the best
evidence available, to consult the most recent and informed policy statements, to
engage in serious conversations with teacher educators at other institutions, and then
to select and commit to a coherent set of learning goals. As better information
becomes available, learning goals can be updated and replaced even though this means
that the related knowledge acquired will need to be modified and perhaps regenerated.
For too long, the field of teacher preparation has drifted along, either satisfied with
current practices or waiting for a final solution to appear. We believe it makes more
sense to ask, ‘if we wanted to ensure that we will be preparing teachers more
effectively 20 years from now than we are today, what would we do tomorrow’? Our
answer lies in the descriptions of our work provided in this paper.
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Acknowledgments
The research reported in this paper was supported in part by a grant from the National Science
Foundation (Grant 0083429 to the Mid-Atlantic Center for Teaching and Learning Mathematics). The views expressed in the paper are those of the authors and not necessarily those of the
Foundation.
Note
1. Students who choose to pursue certification in middle-school mathematics (e.g., Grades 6–
8) are required to take an additional seven mathematics courses and an additional mathematics methods course.
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