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Chapter 2
Creating Practical Theories ofTeaching
JamesHiebert andJamesW.Stigler
Abstract In this chapter we propose a way to create theories of teaching that are
useful for teachers as well as researchers. Key to our proposal is a new model of
teaching that treats sustained learning opportunities (SLOs) as a mediating construct
that lies between teaching, on the one hand, and learning, on the other. SLOs become
the proximal goal of classroom teaching. Rather than making instructional decisions
based on desired learning outcomes, teachers could focus on the kinds of SLOs stu-
dents need. Because learning research has established reliable links between specic
types of learning opportunities and specic learning outcomes, theories of teaching no
longer must connect teaching directly with learning. Instead, theories of teaching can
become theories of creating SLOs linked to the outcomes teachers want their students
to achieve. After presenting our rationale for moving from theories of teaching to theo-
ries of creating SLOs, we describe the benets of such theories for researchers and
teachers, explain the work needed to build such theories, and describe the conditions
under which this work could be conducted. We conclude by peering into the future and
acknowledging the challenges researchers would face as they develop these theories.
Keywords Teaching mediator · Teaching theory · Useful theory · Learning
opportunities · Teaching effects
1 Creating Practical Theories ofTeaching
Imagine the challenges faced by Lucy Scott, a sixth-grade teacher planning a unit
on equivalent fractions. She is deciding what tasks to use and how to discuss them
with her students. Ms. Scott has taught these lessons before and knows she needs to
make some changes. The last time she taught the lessons, the students seemed
J. Hiebert (*)
University of Delaware, Newark, DE, USA
e-mail: hiebert@udel.edu
J. W. Stigler
University of California, Los Angeles, CA, USA
e-mail: stigler@ucla.edu
© The Author(s) 2023
A.-K. Praetorius, C. Y. Charalambous (eds.), Theorizing Teaching,
https://doi.org/10.1007/978-3-031-25613-4_2
24
confused, leaving Ms. Scott unsure, at the end, whether or not her students really
understood what it meant for fractions to be equivalent. How can Ms. Scott decide
what changes to make? Are there theories that could help her predict, for example,
what might happen if she chose one task over another? Do such theories even exist?
We begin this chapter with the unusual proposition that it is possible to build
theories of teaching—practical theories—that are useful for teachers. At the heart
of our argument is the concept of learning opportunities, specically learning
opportunities that can be sustained within and across daily classroom lessons.
Rather than assuming that teaching behaviors have a predictable impact on student
learning, we argue that it is the sustained learning opportunities (SLOs) created by
these behaviors that predict student learning. In order to help students achieve par-
ticular learning goals, teachers need to create SLOs aligned with these goals. The
creation of these opportunities provides a more proximal goal for teachers than the
achievement of learning outcomes. Focusing on SLOs opens the possibility for
teachers to reason in cause-effect terms because it is easier to anticipate the effects
of teaching behaviors on SLOs than on learning outcomes.
We develop our argument by rst discussing a simple model that has often guided
research on teaching and its effects on learning. We then describe a more complex
model, created to x the simplistic assumptions of the simpler model. Although both of
these models have generated a number of useful theories and programs of research, we
claim that theories of teaching effectiveness based on these models have reached their
limits for generating research that will take the eld beyond where it is now. In addi-
tion, we do not believe these models can support theories that teachers can use to make
instructional decisions. We argue that a different model is needed to further advance
theories of teaching effectiveness and bring them closer to the work of teachers.
Our alternative model inserts a new, single, mediating construct—sustained
learning opportunities, or SLOs—between teaching and learning. SLOs can be
dened as the temporarily stable systems that emerge during classroom lessons
from the interactions of multiple mediating variables to create the contexts in which
learning occurs. A SLO is a unit of analysis that provides the pathway through
which teaching leads to signicant learning. We present our alternative model by
elaborating the construct of SLOs, clarifying how the model differs from previous
mediating variables models, and examining the essential role this new construct
plays in mediating the connections between teaching and learning. Our aim is to
present a convincing argument that a theory of teaching most useful for teachers
will be a theory that guides the creation of SLOs.
We continue by explaining how our model drives the shift from theories of teaching
to theories of creating SLOs, and we lay out the key ingredients of these theories. We
present an example of a mini-theory that could be knit together with other mini-theo-
ries to create larger theories, and we step back to imagine ways in which teachers and
researchers, as well as partnerships they might form, could use the construct of SLOs
to build usable theories. We conclude our argument by acknowledging the challenges
of developing theories of creating SLOs while still setting this goal as a worthy pursuit.
Kurt Lewin is credited with saying, “there is nothing as practical as a good
theory” (Greenwood & Levin, 1998). Although theories of teaching effectiveness
J. Hiebert and J. W. Stigler
25
have usually been treated as guides for researchers, we interpret Lewin’s phrase as
a hypothesis that “good theories” could exist for practitioners as well as researchers.
This is not to say that good theories for practitioners would not also be useful for
researchers; quite the opposite. As we describe, developing theories of teaching for
teachers opens new lines of investigations for researchers.
Throughout this chapter, we use the terms model and theory as proposed by
Praetorius and Charalambous (this volume). Following the Oxford Dictionary, they
dened models as simplied descriptions of systems for assisting researchers in mak-
ing predictions and theories as elaborations of models that describe the systems them-
selves—interrelated sets of ideas—intended to explain something of interest. Or, to
quote another idea that strikes us as useful: “good theory helps identify what factors
should be studied and how and why they are related” (Hill & Smith, 2005, p.2).
Our analysis is shaped by our interest in understanding how classroom teaching
can support students’ learning of valued content. We appreciate that the purposes of
teaching include more than acquiring knowledge (Biesta, this volume) and that the
theories of teaching can address more than its effectiveness (Herbst & Chazan, this
volume). However, we believe there is value in theorizing about teaching effective-
ness for learning content, especially in ways that are usable by teachers.
2 Moving Beyond aSimple Model ofTeaching
Research on teaching has a long and illustrious history. It is fair to say that much of
the work has treated as axiomatic the importance of investigating the effects of
teaching on student learning outcomes (Floden, 2001). In fact, the credibility of
theories of teaching is often based on whether the theory predicts learning outcomes
(Farnham-Diggory, 1994; Herbst & Chazan, 2017). The basic model on which these
theories are based looks roughly like the one pictured in Fig.2.1. Teachers engage
in teaching behaviors, and these behaviors impact what students learn. We know
from value-added research that who students have as a teacher explains a good deal
of the variance in how much they learn (Nye etal., 2004; Sanders & Rivers, 1996).
It is reasonable to conclude that different teachers act differently, and that these dif-
ferences help to explain what students learn.
The problem with this model is that it hasn’t worked very well. Despite decades
of research, and many innovations in how researchers describe the “what teachers
do” part of the equation, they have generally found very low correlations between
teacher actions, on one hand, and what students learn, on the other (Brophy & Good,
1986; Dunkin & Biddle, 1974; Hiebert & Grouws, 2007; Oser & Baeriswyl, 2001).
One of the largest and most ambitious studies conducted based on this model—the
What What
teachers do students learn
Fig. 2.1 A simple,
common model for
research on teaching
2 Creating Practical Theories ofTeaching
26
Bill & Melinda Gates Foundation Measures of Effective Teaching study—found few
correlations between anything observable in teachers’ actions and the learning out-
comes of their students (Kane etal., 2013; Kane & Staiger, 2010). This leaves Lucy
Scott and her colleagues without much guidance for planning instruction that could
predictably help students learn, say, to understand equivalent fractions.
Beginning in the 1970s, researchers recognized that the simple model’s lack of
explanatory power could be attributed, at least in part, to the students’ role in deter-
mining what they learned from instruction (Doyle, 1977; Rothkopf, 1976). Students
do not simply stand between teaching and learning as passive recipients but actively
process information and represent events that unfold during classroom lessons.
Even simple cognitive tasks require students to actively process information and
formulate a response (Shulman, 1986). The mediating role that students play could
be pictured by inserting a box between what teachers do and what students learn, as
shown in Fig.2.2. This elaborated model was proposed as a way to move beyond the
simpler process-product model (Gage, 1972; Rosenshine, 1976) to represent the
more complex relationship between what teachers do and what students learn.
Researchers often inserted into the middle box one or more variables intended to
capture how students process instruction. By 1986, Wittrock (1986a) could review
numerous efforts to identify variables that mediated the relationship between teach-
ing and learning. Variables he labeled “thinking processes” included attention, com-
prehension, motivation, interpretation of feedback, self-concept, cognitive strategies,
and metacognitive strategies. Some researchers gathered multiple cognitive vari-
ables and organized them into a “cognitive mediational paradigm” (Winne, 1987).
Other researchers introduced constructs, like “student work,” to organize and inter-
relate the mediating effects of individual variables (Doyle, 1983, 1988). As Doyle
argued, the work students do during instruction determines, to a large degree, what
they will learn. Teaching that leads to one kind of work will yield a different out-
come than teaching that prompts a different kind of work.
The idea of including mediating variables between what teachers do and what
students learn has continued to inuence the eld today (Kyriakides etal., this vol-
ume; Scheerens, this volume). More complex models include different types of
mediating cognitive variables (self-regulation, motivation, and engagement) as well
as mediating social variables (teacher-student relationships, peer relationships, and
family involvement (Cappella etal., 2016). These models, and the theories based on
them, have generated numerous research programs providing important insights
into teaching and learning (see Cappella etal., 2016).
Although we strongly endorse the insights that led to the creation of this
mediation model (Fig.2.2), we see two problems that have limited its success. First,
What teachers do How students
process instruction
What students
learn
Fig. 2.2 An elaborated model for research on teaching
J. Hiebert and J. W. Stigler
27
the number of variables that lie between teaching and learning is almost limitless.
The more researchers learn, the larger the number becomes. Isolating the effects of
individual variables is of limited use because a single variable accounts for too little
variance. But, studying the effects of collections of variables quickly introduces
overwhelming complexity. Cooley and Leinhardt (1975) anticipated this problem
by noting that “the vast array of possible inuencing variables” in studies of teach-
ing results in “an unmanageable quantity of data that has produced no clear insight
as to what practices make a difference in student learning” (p.4). The problem is
exacerbated by the fact that individual variables do not independently exert their
inuence on instructional effects. Researchers must consider their interactions.
When examining the progress of research programs on aptitude-treatment
interactions (ATIs), Cronbach (1975) noted that, even with a small number of
variables, the number of interactions would take researchers into “a hall of mirrors
that extends to innity” (p.119). Theorizing about, and researching, the mediational
effects of large collections of individual variables is simply untenable.
The second problem with the mediation model is that there are few constraints on
the nature of the mediating variables, and different researchers describe mediating
variables of different grain sizes and different types. This makes it difcult for theo-
rists to piece together ndings across empirical studies to build coherent theories.
Cronbach (1975), for example, reviewed the moderating effects that macro- variables,
such as student aptitudes, have on the relationship of instruction to learning, whereas
Winne (1987) argued for the importance of micro-variables, such as “rehearse the
dening attributes of the concept” (p.343). A wide range of mediating variables
along the continuum are found in Wittrock’s (1986a) review, from motivation to
students’ perceptions of teacher expectations to reliable counting strategies for solv-
ing beginning arithmetic problems. And, the elaborated mediation model proposed
by Cappella etal. (2016) identies macro-variables and micro- variables, both cogni-
tive and social. To reiterate, we believe the concept of mediating processes has merit
but the way in which it has been operationalized does not lead toward the develop-
ment of theories that teachers could use to make daily instructional decisions.
3 An Alternative Model ofTeaching
The importance of recognizing the impact of mediating variables cannot be
overstated. The model in Fig.2.2 has resulted in a number of fruitful programs of
research. Yet, the more that is learned, the more we believe there is something miss-
ing that could simplify the sets of mediating variables without losing the insights
they have provided. The missing construct, in our view, stems from the realization
that what takes place in classrooms, as teaching unfolds, is not just the interplay of
many variables but instead is the emergence of a stable system that denes the con-
text in which learning takes place. This system, which we call sustained learning
opportunities (or SLOs), is not just a bunch of variables but is a new construct that
we insert between teaching and learning(See Fig.2.3).
2 Creating Practical Theories ofTeaching
28
Curriculam and
teaching resources
Sustained learning
opportunities Student outcomes
Teaching Learning
Fig. 2.3 An alternative model of teaching
Before we present a more detailed description of the model, it is worth making a
few general points. First, the term “sustained learning opportunities” should not be
confused with common uses of “learning opportunities,” including its meaning of
“curricular exposure” in international comparisons (McDonnell, 1995, p.306). In
addition, our use of “SLO” is very different than the acronym’s association with
“student learning objectives” (https://texasslo.org/Resources).
Second, although SLOs are created by classroom variables and their interactions,
they cannot be described in terms of these variables. Instead, a SLO is a system that
needs to be described and understood in its own right. Because SLOs represent the
sustained episodes in classroom lessons designed to help students achieve challeng-
ing learning goals, teachers recognize them more easily than individual mediating
variables.
Third, teachers don’t single-handedly create SLOs. Instead, they orchestrate
them, drawing on and coordinating all the resources they have to work with. These
resources include curricula, but also include familiar cultural routines of teaching
and learning and the beliefs that support these routines. Importantly, teachers do not
create SLOs alone; students also play a role by participating in tasks and activities
presented by the teacher (Schoenfeld, this volume; Vieluf & Klieme, this volume).
Fourth, we cannot overstate the importance of the word sustained. The learning
opportunities that dene students’ experiences and thus shape their learning are not
just occasional events that happen by chance. If learning opportunities are not delib-
erately created and sustained over time, they are unlikely to affect students’ learning
trajectories. Our interest is in understanding how students learn things that are hard
to learn, that get mastered over long periods of time.
Finally, we want to highlight one of the most important features of the alternative
model we are proposing. We have pointed out that SLOs themselves are a system, a
construct worthy of a box. But we also see three more systems in Fig.2.3 of which
SLOs are only a part. All ve components (three boxes and two arrows) comprise a
system of teaching and learning. But the rst three components (the rst two boxes
and the connecting arrow) comprise a system in its own right, as do the last three
components (the second and third box and the arrow that connects them).
Along with the construct of SLOs, it is this nested set of systems shown in
Fig.2.4 that capture the model’s most unique and signicant contribution. In par-
ticular, our claim that the rst and second box connected by the rst arrow constitute
a system of its own means that these components form a complete whole that can be
analyzed and understood separately from the other systems. This, in turn, means
that the quality of SLOs can be treated both as a dependent measure of one sys-
tem—an outcome created by the curriculum and the teaching that brings the
J. Hiebert and J. W. Stigler
29
Curriculam and
teaching resources
Sustained learning
opportunities Student outcomes
Teaching Learning
System 2: Learning
System 1: Teaching
Fig. 2.4 Two systems constitute the overall model
curriculum in touch with students—and as an independent measure of another sys-
tem when used to predict learning.
The nested characteristic of the model helps to conceptualize the relationship
between theories and research on teaching with theories and research on learning.
In our model, one theory is not embedded in the other (Openshaw & Clarke, 1970;
Snow, 1973); rather, the theories intersect around the middle box. In order to trace
relationships between what teachers do and what students learn, our model suggests
that theories of teaching must be aligned with theories of learning at this point of
intersection. This intersection is precisely what enables theories of creating SLOs to
be useful for teachers. From their point of view, the SLOs that provide the goal for
instruction are those that theories and research on learning have linked to the learn-
ing outcomes teachers want their students to achieve.
We believe the alternative model claries for theorists and researchers the task of
building practical theories that can guide teachers’ day-to-day instructional deci-
sions. Earlier, we pointed to the overwhelming number of individual variables in the
mediation model that must be coordinated as a reason for searching for an alterna-
tive. With the diagram in Fig.2.4, we can now see this problem from a new perspec-
tive. The traditional goal of connecting what teachers do with what students learn
means documenting the connections across two distinct systems. On the other hand,
building a theory of creating SLOs requires testing hypothesized connections within
only the rst system. Although it is true that traditional theories of teaching usually
focus on the rst arrow in Fig.2.3, they often are required to explain the second
arrow as well. We believe this poses a challenge that is too big for any theory that
aims to support teachers’ decision making (Vieluf & Klieme, this volume; cf.
Kyriakides etal., this volume).
3.1 Unpacking theModel
We turn now to unpack the model presented in Figs.2.3 and 2.4. The model consists
of three boxes connected by two arrows. It is worth pointing out that the arrows in
our diagram do more work than the arrows in most diagrams. Instead of represent-
ing only a ow from one box to the next, the arrows represent the processes that
create the complex relationships between the three boxes. The arrows represent
2 Creating Practical Theories ofTeaching
30
verbs, the boxes represent nouns. The boxes are things that change only when teach-
ing or learning change them. The arrows represent the processes of teaching and
learning that produce the changes.
The First Box The rst box in our model consists of all the things teachers use to
implement their lessons. These include written and supplementary materials (e.g.,
textbooks, pacing guides, concrete materials, lesson plans, etc.) designed by cur-
riculum developers to create learning opportunities for students (Remillard etal.,
2009), materials teachers create themselves, and materials they share locally and on
the Internet. This is the raw material from which teachers draw as they select, adapt,
coordinate, and implement sustained opportunities for student learning. In the pre-
vious models (Figs. 2.1 and 2.2) these things are left unspecied, though they
clearly have a major impact on the kinds of SLOs teachers are able to create.
Also included in the rst box are all of the teaching routines that are familiar to
teachers, as well as all of the content, pedagogical, and cultural knowledge of teach-
ing that teachers acquire while sitting in classrooms as students, engaged in teacher
preparation, and working as teachers. Examples include the pedagogical content
knowledge that assists teachers as they customize instructional activities for their
students, and cultural knowledge that teachers use to create and sustain the daily
classroom routines common within each culture.
The Second Box The second box in our model represents the learning opportunities
that students actually participate in and experience over sustained and repeatable
segments of time. Sustained learning opportunities are the relatively stable times
within classroom lessons during which students engage with an instructional activity
designed to help them achieve a learning goal. The fact that they are relatively
stable, even if only for a few minutes, means they can be identied and studied.
They are visible within the fast-moving and eeting interplay of variables within
classrooms.
A SLO emerges from the interaction of classroom variables as a signature
characteristic of the lesson that matters most for students’ learning. It derives its
impact (and predictive power) from the way in which the variables interact to create
its effect, not from its size or intensity. More is not necessarily better. The nal
quality of the SLO is determined by the interactions among the primary players—
teacher, students, and content (Cohen etal., 2003; Lampert, 2001).
Indeed, a SLO could be thought of as a dramatic play. Putting on a play requires
the coordination of many elements—sets, scripts, and actors, to name a few. The
quality of the play cannot be judged by each of its elements evaluated individually
but rather by the emergent qualities of the event that results from the interplay of
these elements. Teachers and students are actors in a kind of play. They each must
work to enact the play, to create a briey-sustained temporary world in the class-
room. In the case of a SLO, each actor learns from their experiences as they partici-
pate in the world they have created together.
We can clarify further the SLO construct by comparing it to related constructs.
As noted earlier, we can distinguish SLOs from “opportunity to learn” (OTL). OTL
J. Hiebert and J. W. Stigler
31
represents content covered and/or the tasks presented to students (see McDonnell,
1995, for a history of OTL and Travers, 1993, for its use in SIMS). And, opportuni-
ties presented are different than opportunities experienced (Biesta, this volume;
Praetorius etal., 2020; Vieluf & Klieme, this volume). A related distinction can be
made between SLOs and the “enacted curriculum” (Remillard & Heck, 2014; Stein
et al., 2007; Thompson & Usiskin, 2014). The emphasis in discussions of the
enacted curriculum is often on the teaching moves and behaviors that transform the
written curriculum into the learning opportunities that reach the students. SLOs
emphasize the opportunities that emerge and are experienced by students as they
participate in the enactment.
The construct we see as closest to SLOs is the construct of classroom tasks
described by Tekkumru-Kisa etal. (2020). In their formulation, a classroom task
“creates the context within which students think about the subject matter” (p.607).
Tasks move through four phases during a lesson (the life of a task). We connect the
SLO construct to the third of their four phases: “the task as perceived by each stu-
dent and as enacted by the teacher and the students is the actual intellectual work in
which students engage (i.e., the level and kind of student thinking happening during
the lesson)” (p.607).
The Third Box The third box consists of student outcomes, the most prominent of
which is student learning. In this chapter, we focus on learning outcomes aligned
with academic or content goals. Changing the focus to other goals that societies, and
teachers, often value might change what would t into the components of our model
but would not change the model itself (see Biesta, 2016, this volume, and Lampert,
2001, for examples of other important goals, such as students’ forming identities as
autonomous learners). It is also important to note that we include both immediate
and long-term goals in this box.
The First Arrow The rst arrow in our model includes much of what researchers
and educators ordinarily think of as teaching. However, it includes more than the
visible, public actions of teachers as they implement a lesson. It also includes plan-
ning for a lesson and reecting on a lesson after it is taught. It represents all of the
processes teachers use to turn the intended curriculum into the enacted curricu-
lum—the curriculum that is presented to students. “The teacher is an active designer
of curriculum rather than merely a transmitter or implementer” (Remillard, 2005,
p.214).
We focus on planning, implementing, and reecting because we see them as the
minimum processes needed to represent what the teacher does to create SLOs. We
recognize that what is involved in these activities can be unpacked in different ways
and at various levels of detail (Cai etal., this volume; Scheerens, this volume;
Schoenfeld, this volume). In fact, any single chapter cannot do justice to all the
ingredients that t into this arrow (Ball & Forzani, 2009; Grossman, 2020; Lampert,
2001). Also, although the arrows in our model ow from left to right, we can imag-
ine processes that ow in the opposite direction as well. What teachers and research-
ers learn from implementing curriculum and observing the sustained learning
2 Creating Practical Theories ofTeaching
32
opportunities, for example, could have a “backward design” effect on the way in
which the curriculum is revised and improved (Wiggins & McTighe, 2005).
The Second Arrow The second arrow represents the processes and cognitive
mechanisms that transform learning opportunities as experienced by students into
learning outcomes. Like the rst arrow, it links two boxes to form a separable sub-
system, this time consisting of interrelated elements that turn sustained learning
opportunities into learning outcomes as assessed by a wide range of measures. The
arrow establishes the types of SLOs that will become the targets teachers use to plan
and implement instruction.
Establishing connections between particular types of SLOs and particular
learning outcomes is usually the work of researchers. Researchers, however, are not
the only ones who can contribute to educators’ understanding of the second arrow.
Teachers learn about processes that produce student outcomes, for example, when
they use formative assessment tools to get a sense of what their students are thinking
and learning from the opportunities they experience (Silver & Mills, 2018; Wiliam,
2018), or when they review students’ work to get a more detailed look at students’
conceptions and misconceptions (Kazemi & Franke, 2004), or when they adminis-
ter and grade quizzes and exams to nd out what their students learned during the
lesson(s). Unfortunately, the culture and practices of education research in the
U.S. do not yet include a mechanism for routinely capturing this information.
Even though teachers do not usually contribute to more generalized knowledge
connecting SLOs with learning outcomes, they frequently use what they learn from
assessing outcomes to reect on the effectiveness of their teaching. Although teach-
ers’ reections are part of the work of teaching, and so belong squarely inside the
rst arrow, they also could contribute to our understanding of the second arrow. This
highlights the fact that the boundaries separating the two systems are not imperme-
able. There are places where work on teaching and work on learning can and should
overlap (Romberg & Carpenter, 1986).
Connecting the Two Systems The ability of researchers to document links
between SLOs and learning outcomes is crucial for the model shown in Figs.2.3
and 2.4 to function as we propose. With these links established, the work of teaching
represented by the rst arrow could set a goal of creating SLOs that have been
shown to align with desired learning outcomes. Teachers could focus on creating
SLOs with specic features if they could assume that opportunities with these fea-
tures led to the learning outcomes they intended.
It turns out that researchers have reported compelling evidence that links types of
SLOs and particular learning outcomes (Bjork & Bjork, 2011; Cai et al., 2020;
Hiebert & Grouws, 2007; Richland etal., 2012). Consider the case of mathematics.
If we specify understanding of key concepts as an important mathematical pro-
ciency and a desired learning outcome, we can identify two features of a SLO that
enable this outcome. A rst feature is often referred to as productive struggle
(Hiebert & Grouws, 2007). Mounting evidence from the learning sciences indicates
that deep and lasting learning results more often from periods of struggle and
J. Hiebert and J. W. Stigler
33
confusion than from smooth error-free learning or from the kind of Eureka! moments
educators strive to create (Harackiewicz etal., 2008; Kapur & Bielaczyc, 2012).
Robert and Elizabeth Bjork coined the term “desirable difculties” to refer to a
body of research showing that introducing difculties into the learning process can
produce deeper and longer-lasting learning, despite the fact that students often
describe the experience as less enjoyable and believe that they have learned less
(Bjork, 1994; Bjork & Bjork, 2011). If mathematics educators want students to
understand a concept, they must nd ways to engage students in struggling to make
sense of the concept.
Of course, struggling by itself won’t produce deep learning of signicant
mathematics. The struggle must be focused on the right things. This leads to a
second feature of SLOs that predict conceptual learning outcomes: explicit
connections (Hiebert & Grouws, 2007). To develop understanding, students must
focus their efforts on making the connections that lend coherence to a domain and
that result in knowledge that is both exible and transferable. In particular, students
must work to forge connections among core concepts, representations, and the
world to which the concept applies (Fries etal., 2020; Hiebert & Carpenter, 1992;
National Research Council, 1999, 2001; Roth & Garnier, 2006). These connections
don’t usually spring forth spontaneously. They must be made explicit, by students
or the teacher, and they must be made at the right time, when students are prepared
to recognize and construct these connections for themselves (Dewey, 1910).
Explanations, comparisons, analogies, and visual representations are all tools that
teachers can incorporate into SLOs that help students create connections and
develop deeper understanding (Richland etal., 2004, 2012).
As we noted earlier, learning things that are hard to learn requires SLOs to be
sustained over observable periods of time. To develop conceptual understanding,
students must practice struggling productively with making important connections
in the domain. Because this is not the usual form of practice, often called repetitive
practice, it has been labeled deliberate practice, a term that comes from research on
expertise (Ericsson, 2006). Deliberate practice involves a planned sequence of
opportunities that stretch over more than one lesson, sometimes over many lessons.
In fact, researchers have reported the kinds of sequential variation in mathematical
problems that create these SLOs (Carpenter etal., 2017; Clements & Sarama, 2014;
Fries etal., 2020; Huang & Li, 2017; Kullberg etal., 2017; Pang & Marton, 2009).
Treating SLOs as the proximal goal for teaching depends on documenting more
connections between features of SLOs and desired learning outcomes.
3.2 An Advantage oftheModel forTeachers (and Researchers)
A driving motivation for writing this chapter was to explore whether reimagined
theories of teaching could be more useful for teachers. By dividing the larger system
of teaching and learning into two smaller systems, we have imagined a way for
theorists and researchers to narrow their focus to one system or the other. This, we
2 Creating Practical Theories ofTeaching
34
argue, can suggest new lines of research that previously might have been hidden by
the expectation that researchers trace transformations from the written curriculum
through the enacted curriculum, through the learning opportunities experienced by
students and, nally, into learning outcomes. Research questions change if one
works within a system (Fig.2.4). As just noted, for example, research in System 2
can focus on connections between SLOs with particular features and desired learn-
ing outcomes. Lines of research in System 1 could focus on describing the class-
room conditions that yield SLOs with particular features.
Theories of creating SLOs offer teachers a clear theory of what they need to
create and what their creations should look like as they implement instruction.
While teachers are implementing instruction, it is almost impossible to keep their
eyes both on the instruction they are enacting and the evidence needed to judge
whether students are achieving the learning goals of the lesson. Although still
challenging, it is conceivable that teachers could monitor students’ immediate
responses to the learning opportunities being enacted because these are proximal to
implementing the planned opportunities. Based on these responses and on theories
of creating SLOs, teachers could adjust instruction to improve the quality of SLOs
(see Biesta, this volume).
Walter Doyle previewed this idea more than four decades ago when assessing the
usefulness of the process-product framework (similar to the simple model—in
Fig.2.1):
In the event that the presentation did not accomplish its objective, the process-product
formulations would offer no further guidance. Just knowing the relation of a technique to
terminal performance fails to supply sufcient information about immediate contingencies
in the classroom (Doyle, 1977, p.126).
He then said that if teachers could focus on activating an intermediate response from
students, they could “experiment” with instructional strategies to see which worked
best. This, Doyle (1977) said, “enables a teacher to practice what Cronbach (1975)
has called ‘short-run empiricism,’ in which one monitors responses to the treatment
and adjusts it” (p.126).
To reiterate, a theory of SLOs and how to create them could open new lines of
research and could help teachers know what to look for when observing students’
responses to instruction and what changes they might consider as they seek to
improve the quality of the SLO they are creating. A theory of creating SLOs is our
answer to the question posed in the rst paragraph of this chapter: “Are there theo-
ries that could help Ms. Scott predict, for example, what might happen if she chooses
one task vs. another?” We believe our alternative model of teaching could spawn
theories of creating SLOs that are usable by Ms. Scott and her professional
colleagues.
J. Hiebert and J. W. Stigler
35
3.3 Limitations oftheModel
Although we believe our model of teaching and learning captures meaningful
aspects of teaching and provides teachers with a manageable system to which they
can apply their efforts to improve, even this alternative model includes only a frac-
tion of the full system of teaching. One only has to look through the extensive
Handbooks of Research on Teaching (3rd edition, Wittrock, 1986b; 4th edition,
Richardson, 2001; 5th edition, Gitomer & Bell, 2016) and the chapters in this vol-
ume to see the vast and rich legacy of relevant research and theory that address vari-
ous aspects of the immensely complex system of teaching. Our model does not
touch many factors, both outside and inside the classroom, that contribute to stu-
dents’ experiences in school (Cobb et al., 2018; Cohen, 2011; Creemers &
Kyriakides, 2008; Kyriakides etal., this volume; Lampert, 1985, 2001).
We acknowledge that our model is located within a much larger multi-level
system of schools, districts, and so on (Cobb etal., 2018; Scheerens, this volume;
Scheerens etal., 2003; Strom etal., 2018), all of which impact teaching in some
way. We attend to only a small portion of these factors and to only one level of the
system of schooling. Because of the mind-bending complexity of teaching, every-
one who wades into this domain must nd ways to simplify the problem and put
boundaries on their search for solutions, leaving large portions of the domain
untouched. We are no exception.
One of the challenges facing those who presume to investigate teaching is how
to simplify teaching to make it more tractable without, at the same time, losing its
essential character. Grossman and McDonald (2008) express the challenge this way:
A framework for teaching would require a careful parsing of the domain …. This effort to
parse teaching would need to respect the difculty of breaking apart such a complex system
of activity and the dangers of doing irreparable harm to the integrity of the whole by making
incisions at the wrong places (p.186).
We sought to strike this balance between simplication and ecological validity in
two ways. To simplify, we chose to focus on the major components of teaching that
commonly fall under the control of educators whose work is purposefully directed
toward improving teaching and learning. Classroom teachers, instructional leaders,
curriculum developers, and education theorists and researchers, all fall into this
group. To retain ecological validity, we preserved minimal elements of a system of
teaching, as we understand it. Within this system, we focused on those factors that
are of most concern to classroom teachers and over which they have some control.
We wanted, in other words, to specify a model that could generate theories that
would be useful for Lucy Scott and her fellow teachers.
2 Creating Practical Theories ofTeaching
36
4 Building andUsing Theories ofCreating Sustained
Learning Opportunities
We turn now to discuss theories that could be built using the model shown in
Figs.2.3 and 2.4. We focus here on the rst system in the model, the system that
turns written curriculum and other teaching resources into specic types of SLOs
linked with particular learning outcomes. We reect on what this process might look
like—what the main elements of such theories would be, and how the elements
might be related. Our aim is to develop what Hill and Smith (2005, p.2) called a
“good theory”: one that “helps identify what factors should be studied and how and
why they are related.” We start by presenting a sample hypothesis that illustrates the
nature of these theories. We then imagine the kind of work researchers and teachers
might do to build and use theories of creating SLOs.
4.1 A Sample Hypothesis inaTheory ofCreating Sustained
Learning Opportunities
Theories are built from a set of related hypotheses. One hypothesis that could be
part of a theory of creating sustained learning opportunities is what we call the
struggle-rst hypothesis. Based on analyses of Japanese mathematics classrooms
and on experimental research carried out in the United States, this hypothesis sug-
gests that students will create connections among concepts more effectively if they
engage in productive struggle before they are given direct instruction than they
would if the direct instruction came rst. When students are given direct instruction
rst, says the hypothesis, it removes the motivation to struggle because students
already have the solution they need, thus short-circuiting the deeper learning that
can occur during productive struggle.
It is worth noting how the struggle-rst hypothesis differs from the mediating
variables approach alluded to earlier. It we took the mediating variables approach,
we might identify productive struggle as an important variable to measure. But sim-
ply measuring the amount of struggle in a lesson would not take into account the
importance of how the struggle ts within a SLO.The same mediating variables can
take on different meanings when they are part of different lessons (Janssen etal.,
2015), a fact that becomes even more salient when comparing lessons across coun-
tries with different pedagogical traditions (Stigler etal., 1996; Stigler & Hiebert,
1999). Many variables, in addition to productive struggle, have been found to have
different effects when embedded in different pedagogical systems (e.g. Kawanaka
& Stigler, 1999).
The struggle-rst hypothesis was initially formulated by researchers, and
researchers have generated empirical evidence in support of the hypothesis. But to
be useful for teachers, the hypothesis needs to be elaborated. Numerous secondary
hypotheses need to be generated, tested and rened before the struggle-rst
J. Hiebert and J. W. Stigler
37
hypothesis could guide teachers’ actions across a wide variety of contexts and con-
tent. Because this variation in context occurs in classrooms, teachers must be part of
the work that formulates, tests, and renes the hypotheses. As researchers and
teachers esh out the struggle-rst hypotheses, they might ask questions such as,
“What kinds of tasks work best for students who are encountering the topic for the
rst time?” or “What kinds of tasks work best for specic connections between core
concepts of a specic content domain?”
It might be that beginning students struggle most productively to make
connections when they are solving problems that have a particular level of cognitive
demand (Stein & Lane, 1996; Tekkumru-Kisa etal., 2020), or with tasks that vary
in particular ways from tasks students already have completed (Huang & Li, 2017;
Marton, 2015; Pang & Marton, 2009). Teams of researchers and teachers might
focus their classroom investigations on how the task is initially presented to stu-
dents, often called the “launch” (Wieman, 2019); or on how subsequent class dis-
cussions should be orchestrated during and after the task is completed (Smith &
Stein, 2018); or on the role of well-timed hints (Stigler & Hiebert, 1999); or on how
best to sustain students’ efforts to complete the task in the face of difculties, and
perhaps frustration (Mukhoiyaroh etal., 2017; Tulis & Fulmer, 2013).
These are just a few of the hypotheses that teams of teachers and researchers
could generate and investigate. These secondary hypotheses get lled in and rened
as teachers experiment with different strategies in their classrooms to shape the
learning opportunity so students derive maximum benet from productive struggle.
Formulating, testing, and rening hypotheses is an iterative process that engages
teachers and researchers in the kind of cause-effect reasoning that is essential for
improving teaching. Notice also that these secondary hypotheses represent only a
fraction of the work needed to elaborate and rene the main hypothesis; they focus
only on the “struggle” part of the struggle-rst hypothesis. Teacher-researcher
teams must formulate and test additional hypotheses to explore the best ways to
help students make explicit the connections that complete this sequence.
As teachers test secondary hypotheses in their own classrooms, researchers can
gather the ndings and look across classrooms for patterns in what teachers do and
how students respond. Are there ways of implementing a task, for example, which
leads to productive struggle for most students in most classrooms with specic char-
acteristics? As researchers guide the renement of secondary hypotheses by orga-
nizing the incoming results, sharing them with other researchers working on similar
problems, and suggesting other forms of these secondary hypotheses for teachers to
test, a mini-theory begins to take shape around how to create SLOs that support
making key connections through productive struggle.
Other primary hypotheses, such as “understanding requires repeated struggle,”
trigger the development of other mini-theories that guide, for example, the sequenc-
ing of tasks and activities both within and across lessons. As mini-theories begin
forming around hypotheses that t together, the mini-theories expand in scope and
incrementally move toward larger theories of creating SLOs. Although building
these mini-theories takes considerable time (years rather than weeks or months), the
2 Creating Practical Theories ofTeaching
38
work of teachers and researchers can be accumulated, coordinated, and aggregated
to gradually but steadily move toward more useful theories of creating SLOs.
As a mini-theory is forming for how to create productive struggle with making
key connections, one can imagine teachers drawing on the mini-theory as they plan
a lesson. As teachers internalize the mini-theories, they will be able to represent
them as mental models. Teachers can run these mental models during planning as a
means of predicting what the consequences will be as they weigh various options
for an upcoming lesson. As the mental models become richer and stronger, teachers
will be able to run their mental models on the y, as they teach, as a means of pre-
dicting how students with different characteristics will respond to different parts of
a lesson. As teachers practice making and testing predictions of this sort, they are
engaged in a high-quality professional learning process that some authors have
referred to as deliberate performance (Fadde & Klein, 2010).
4.2 Imagining theWork ofBuilding Theories ofCreating
Sustained Learning Opportunities
The problem of how best to build theories of teaching has received relatively little
attention (Praetorius & Charalambous, this volume). There are no clear precedents
to follow as we outline a possible path for building theories of creating SLOs.
Nevertheless, we move beyond the example just presented and propose some gen-
eral processes and guidelines that could help build these theories in order to provide
a more complete picture of the theories we have in mind.
We begin by asking, “If theorists and researchers wish to create and test theories
of SLOs, what might they encounter and how might their work lives change?”
Although this kind of work has not been attempted in any kind of serious way, it
might be useful to imagine what kind of work would be entailed in order to envision
the kinds of changes researchers could expect. We can identify several changes that
researchers, and teachers, might decide to make. But, we anticipate there are many
more and each of the ones we identify would likely have ripple effects through the
educational system.
Changing Roles for Teachers and Researchers Researchers and teachers have
long worked toward different goals and have played different roles in the educa-
tional system. Our example of building even a mini-theory around the struggle-rst
hypothesis suggests that these groups might need to adopt shared goals and change
their professional roles (and identities).
For some time, the eld has recognized that it is ineffective for researchers to
develop theories and then hand them to teachers. Researchers simply don’t know
enough about the processes and conditions that determine how teaching behaviors
and routines work to create SLOs in classrooms. In a eld such as education, where
good practices can run ahead of good theories, “the experience and intuition of
J. Hiebert and J. W. Stigler
39
practitioners” becomes especially important (Lipsey, 1993, p. 12). If researchers
want to be better positioned to engage seriously in solving problems of practice
(Burkhardt & Schoenfeld, 2003; Cai etal., 2018; Cohen & Mehta, 2017) they will
need to nd ways to blur the boundaries between themselves and teachers (Akkerman
& Bakker, 2011; Cai etal., 2018, this volume; Cohen-Vogel etal., 2015; Penuel
etal., 2011). Perhaps researchers will nd ways to work side-by-side with teachers
to ensure they are addressing instructional problems that teachers actually face as
they implement and evaluate learning opportunities.
We can envision three unique roles for researchers to play. First, they could
suggest hypotheses for how SLOs with particular features might be created.
“Struggle- rst” was formulated by looking across multiple settings, even multiple
cultures. Teachers are not usually in a position to do this work, but researchers are.
They could look across classrooms and search for patterns in the effectiveness of
various teaching behaviors for creating similar SLOs and, conversely, they could
search for patterns across classrooms in the conditions that turn similar teaching
behaviors and routines into different SLOs. Second, researchers could identify prior
research that would provide a starting point for teachers’ work on developing the
types of mini- theories outlined above (see the citations in the earlier example of
“struggle-rst”). Third, researchers could interpret data on learning outcomes that
emerge across classrooms in order to evaluate and rene the links between features
of SLOs and learning outcomes (the second arrow in Figs.2.3 and 2.4).
Teachers might want to expand their traditional roles as well. Teachers uniquely
have intimate knowledge of their students, enabling them to both formulate predic-
tions and test the predictions by observing students’ responses to instruction. This is
not new work for teachers. They constantly make predictions, often intuitively and
tacitly, about how students are likely to respond to particular instructional activities.
However, making these predictions purposefully and explicitly would be new for
most teachers. Similarly, the observations needed to test predictions about SLOs are
different than the kinds of observations teachers make every day. We could imagine
that teachers who are involved in this work would gradually adopt an experimental
orientation toward their work (Hiebert etal., 2003). By experimental orientation, we
mean simply learning from “experience carefully planned in advance” (Fisher,
1953, p.8) and bringing the power of causal thinking into their practice (Gallimore
etal., 2009).
Imagine teachers and researchers developing teams, or partnerships, to meet the
challenge of creating theories of SLOs. The promise of researcher-practitioner part-
nerships has been realized in professional elds outside of education (Bryk etal.,
2015; Morris & Hiebert, 2009, 2011). From auto manufacturing to the repair of
Xerox machines to clinical medicine to the wind turbine industry, this multiple
expertise model has been used effectively to improve practices across a range of
professions (Douthwaite, 2002; Gawande, 2007; Kenney, 2008; Langley et al.,
2009; Rother, 2009). When teachers and researchers form partnerships around
shared problems of practice, they can realize similar successes (Bicknell & Young-
Loveridge, 2017; Coburn & Penuel, 2016; Donovan & Snow, 2018; Quartz
etal., 2017).
2 Creating Practical Theories ofTeaching
40
The challenge for teacher-researcher partnerships would be to retain the richness
and ecological validity of the information from individual teachers’ classrooms
while surmounting the contextual uniqueness of each classroom. Every teacher
faces somewhat different challenges because there are many factors that inuence
how students take up the opportunities teachers intend (Biesta, this volume; Clarke
etal., 2006; Nuthall, 2004; Stigler & Hiebert, 1999; Vieluf & Klieme, this volume).
The same teaching moves that work in one classroom might not work in another
classroom. And, somewhat different teaching moves might be needed in different
classrooms for students to experience the same learning opportunities.
Researchers will need to nd ways to aggregate what is learned by multiple
teachers across many classrooms into more generalized hypotheses that can provide
guidance to all the teachers trying to solve the same instructional problem. The
concept of “networked improvement communities” (Bryk etal., 2015) will undoubt-
edly play an important role in gradually formulating generalized hypotheses that
can guide teachers’ predictions. Researchers will play an especially important role
in looking across classrooms for patterns that link particular teaching moves with
desired learning opportunities. However, it is too early to speculate how these
approaches will play out and what additional, perhaps still unknown, approaches
might be needed.
It goes without saying that changing roles in these or other ways is not trivial for
either group (Cai etal., 2018; Yurkovsky etal., 2020). But, teachers and researchers
might decide it is worth the effort if they see the payoff in sustainable improvements
in teaching and richer learning for students.
Slowing Down the Cycle of Teaching In addition to changing the roles of teachers
and researchers, building theories of creating SLOs will require slowing down, at
least for some lessons, the common cycle of planning, implementing, and reecting
on classroom lessons. These activities are part of the work teachers do every day,
but planning and reecting are often done quickly, sometimes only as teachers enter
and leave classrooms. This is not sufcient because building, using, and rening
theories takes time—at the moment and over the long run. Teachers who invest in
this work will need time to plan and reect on specially targeted lessons each year.
Of course, teachers would not be able to make this happen on their own. Educators
at various levels of the larger system (e.g., building and district administrators)
would need to create the time and space for teachers to do this kind of work.
Additional time would be needed even though it would not be necessary to slow
down the cycle of teaching for more than a few lessons each year. The goal is not
the creation of a full curriculum of lessons, but the development of theories that can
be applied across multiple lessons. Over time teachers’ work could be accumulated
to yield gradually improving theories of creating SLOs.
Planning and Predicting Thoughtful planning of a lesson necessarily involves
anticipating how students will respond to particular instructional tasks. A natural
way for teachers to anticipate students’ responses is to run the lesson in their heads,
imagine how students will respond at key moments, and adjust their plans accord-
J. Hiebert and J. W. Stigler
41
ingly. Because teachers’ knowledge is often implicit, they will need to work with
researchers to make explicit the hypotheses that underlie their predictions.
A teacher might hypothesize, for example, that students’ will engage more with
an initially-challenging problem if they see how it relates to a similar problem they
have recently learned to solve. A researcher could help to clarify the hypothesis and
design an experiment that could lead to useful information related to the hypothesis.
The teacher could then select or design specic tasks that would work within the
research design. In this way, designing, implementing, and observing students’
responses to a task is not only an act of teaching by teachers, but one of hypothesis
testing by teachers and researchers.
Although anticipating how students will respond at key moments in a lesson is
often something teachers do subconsciously, it is not always easy. Teachers, espe-
cially those with experience and especially as they get to know their students well,
are likely to have good intuitions about how their students might respond to particu-
lar tasks. But, forming predictions about students’ thinking during lessons across a
range of topics will be difcult. Fortunately, teachers (with researchers’ help) can
draw ideas from the long and rich legacy of research on teaching and learning to
formulate predictions about students’ thinking in particular task situations.
In highly researched domains, such as mathematics, the predictions that teacher-
researcher partnerships make can be informed by research ndings that detail
students’ likely ways of thinking about problems of various types. For example,
research on young children’s arithmetic performance provides primary grade teach-
ers with information on likely solution strategies children might propose to most
arithmetic problems if teachers present them in certain sequences (Carpenter etal.,
1996; 2014; Sarama & Clements, 2009). Teachers can use this information to do
more than predict students’ thinking; they can use it to select mathematical prob-
lems and implementation strategies that are likely to engage students in the intended
SLOs (Carpenter etal., 2014; Clements etal., 2020). Promising work on learning
trajectories provides increasingly ne-grained descriptions of children’s thinking
and could be used by teachers to plan instruction on some topics (Clements &
Sarama, 2014; Clements etal., 2004; McGatha etal., 2002; Steffe, 2004).
Before moving to the second phase of the cycle, we should clarify the nature of
the hypothesis testing process we are describing. We do not want to enter the con-
tinuing debate in education about the most useful methods for improving practice
(Bulterman-Bos, 2008; Jacob & White, 2002; Moss etal., 2009) but rather want to
alert the reader that we have in mind the kind of “short-run empiricism” (Cronbach,
1975) or “piecemeal tinkering” (Popper (1944/1985) that involves repeated small
tests of small changes (Morris, 2012; Morris & Hiebert, 2011). In this approach, a
hypothesis is formulated about the relationship between teachers’ actions in the
classroom and the SLO that is created, predictions are made about how students will
respond, and just enough data are collected to assess the viability of the hypothesis.
Proposed by Popper (1944/1985) as the best scientic method for improving
socially-embedded professional practices, we see this kind ofsmall-scale hypothesis
2 Creating Practical Theories ofTeaching
42
testing, with accumulation of results over multiple replications, as an appropriate
method for teacher-researcher partnerships to employ.
Implementing and Observing The point of making predictions about the SLOs
students will experience is to set up the next phase of the cycle—observing the
impact of the implementation and assessing the accuracy of the prediction. Whether
predictions are accurate is, of course, an empirical question. Predictions must be
tested and then hypotheses rened. To build theories of creating SLOs, checking the
predictions means observing the kinds of learning opportunities experienced by
students.
Not just any observations will do. Needed are observations focused on whether
the learning opportunities that were experienced by students possessed the desirable
features identied during the planning phase, and whether changes in instructional
choices (e.g., of the task presented) improved the quality of students’ experience in
the predicted ways. Because students’ experience is mainly an internal affair, it is
not easy to draw completely accurate conclusions. Observing the individual
responses of 30 students and trying to accurately infer what they are thinking is
unrealistic. Teachers’ judgments will be estimates, without the psychometric prop-
erties of systematic and formal assessments. Over time, however, repeated judg-
ments by skilled teachers will lead toward accurate-enough inferences. It is useful
to remember that researchers have long called for teachers to make instructional
decisions based on inferences about students’ thinking (Carpenter et al., 2014;
Dewey, 1929; Lampert etal., 2010; Nuthall, 2004; Wittrock, 1986a). We are simply
suggesting that these inferences be made based on planning and thoughtfully con-
sidered purposes.
Imagine Lucy Scott presenting a cognitively demanding task on equivalent
fractions and observing whether her students are engaged in productive struggle to
connect the concept of equivalence with the numerical patterns in the written
fractions. What might she look for? It is rst important to recognize that, if it is
possible to make accurate-enough observations, Lucy Scott is the person who could
make them. Observing and interpreting students’ behavior with reasonable accuracy
requires extensive knowledge of students’ past performance, their tendencies to
respond to new challenges in particular ways, what their outward behaviors indicate
about their internal struggles, and so on. Ms. Scott is the only person with this kind
of intimate knowledge of her students.
Because productive struggle involves particular kinds of work, there are
guidelines that Ms. Scott could use when observing her students. She could look for
whether her students were wrestling with the task—(not immediately nding the
answer but continuing to try), whether they were asking questions that were relevant
to the key ideas of equivalent fractions, whether they were experiencing moments of
confusion but sustaining their efforts, whether they were developing partial solu-
tions that were on the right track (Brown, 1993; Ermeling etal., 2015; Hiebert etal.,
1996; SanGiovanni etal., 2020).
In addition, teachers like Ms. Scott are likely to nd that their observations of
students’ responses are enabled by the planning they did during the rst phase of the
J. Hiebert and J. W. Stigler
43
cycle. Along with planning instruction, teachers can plan what kinds of observa-
tions they need to test their predictions. In some cases, what teachers look for will
be visible (for example, in students’ written work, or in their behavior while solving
a problem); in other cases, teachers will need to elicit student thinking (for example,
by asking pointed questions and asking students to share their thinking). The more
that teacher-researcher partnerships learn, specically, about the manifestations of
productive struggle with equivalent fractions and with other topics, the more
informed will be the guidelines for observing student responses.
Reecting and Rening The third phase in the cycle of teaching is reecting on
the observations made during instruction in light of the predictions that were posed.
Teachers frequently reect on the success of a lesson but often do so quickly and
without much thought. Participating in a teacher-researcher partnership and using a
theory to guide reection encourages teachers to slow down the process and make
it explicit and systematic. As with planning and observing, theories play an impor-
tant role in the reecting phase of the teaching cycle. In the reecting phase, teach-
ers and researchers can work together to gure out how the results of a lesson can
be used to revise a particular hypothesis or to suggest the creation of new hypotheses.
Looking back to see links between teaching strategies used during the lesson and
learning opportunities experienced by students would enable researchers and teach-
ers to examine the accuracy of their predictions, to learn from “experience carefully
planned in advance” (Fisher, 1953, p.8). It would reinforce for everyone the realiza-
tion that the lessons for which they choose to slow down the teaching cycle are
carefully planned experiments that can be seen through a cause-effect lens
(Gallimore etal., 2009).
Because predictions are based on unproven hypotheses, many of the initial
versions will not be very accurate. However, over the years, as researchers and
teachers become more explicit about their predictions, gather more information, and
reect on this information to propose revisions, the soundness of the hypotheses and
the accuracy of the predictions will gradually improve. As researchers gather
information provided by individual teachers, examine emerging patterns, share
these with other partnerships and suggest additional tests of best predictions, the
robustness of hypotheses will grow and theories could be gradually built and rened.
To reap the benets of many teachers individually testing and rening hypotheses,
and many researchers assisting with gathering, organizing, and analyzing incoming
data, there must be ways to record, store, and share the ongoing ndings and the
best current practices. This brings us to our third big change that teachers and
researchers might make if they become invested in building theories of creating SLOs.
Creating Artifacts Long ago, Dewey (1929) observed that one of the saddest
things about American education is that teachers take their best ideas with them
when they retire. Educators have no good way to preserve what individual teachers
learn from their experience. Thousands of teachers like Lucy Scott drive to school
every day ready to tackle similar instructional problems (e.g., how to help students
understand equivalent fractions), but the current education system in the U.S. pro-
2 Creating Practical Theories ofTeaching
44
vides no way to record and share their hypotheses, predictions, and observations so
as to benet other teachers and their students (Rothkopf, 2009).
A promising approach to recording, preserving, and sharing information across
classrooms is to agree on an artifact into which teacher-researcher partnerships
could record what they learn from the process we have described. A variety of arti-
facts are possible, including a record of the presentation of a particular task plus the
ways in which students work on the task. Tekkumru-Kisa et al. (2020) argue that
examining “the enactment of a particular task, from beginning to end … allows
researchers to see, organize, and analyze students’ opportunities to learn in mean-
ingful ways” (p.607). For us, however, lesson plans have a special appeal (Morris &
Hiebert, 2011, 2015; Stigler & Hiebert, 1999). Cai etal.(this volume) recommend a
similar artefact using a different name, “teaching cases”). For one thing, lesson plans
are written at a grain size that is recognized across countries and cultures. Based on
our analyses of the TIMSS Video Study lessons, we believe it is the smallest unit of
instruction that preserves the system of creating SLOs (the system of teaching in
Fig. 2.4) (Stigler & Hiebert, 1999). Although a single lesson usually does not
develop a mathematical topic fully, it can be analyzed as a unit that stands on its own.
Teachers might nd that written lesson plans have several advantages as a shared
artifact (Morris & Hiebert, 2011). Because almost all teachers use lesson plans in
some form, they are a familiar instructional tool indexed to content topics. By anno-
tating lesson plans with the current and best teaching strategies for that lesson, teach-
ers have access to this knowledge just when they need it. This knowledge consists of
the most rened predictions at the time for how to create SLOs that have been found
to help students achieve the lesson learning goals(s) along with the hypotheses that
provide the rationale for these predictions. Accessible rationales increase the likeli-
hood that the strategies will be implemented as intended and decrease the likelihood
that future predictions will repeat the same mistakes as previous predictions.
Lesson plans also provide a natural receptacle for what partnerships learn as
teachers enact the plans. And, because annotated lesson plans provide a storage
place for knowledge, they can carry the profession’s memory, providing a way for
new teachers to pick up where the previous generation left off. Shared, updated les-
son plans can prevent the profession from suffering “collective amnesia” (Shulman,
1987, p.11), forcing every new teacher to start over. John Dewey would be pleased.
Finally, lesson plans provide a type of an instructional artifact around which
teachers, researchers, and others with relevant expertise can collaborate to solve
common instructional problems (Morris & Hiebert, 2011). Modiable, shareable
artifacts uniquely enable collaborative learning by becoming the public focal point
for the exchange of information and ideas (Bereiter, 2005). A consequence of this
collaborative activity is that teachers could experience a cultural shift from treating
teaching as an individual private enterprise to treating it as a collaborative, public,
and reective activity. This would be a signicant change, in part because it can
encourage teachers to recognize they are capable of sustained growth as true profes-
sionals (Franke etal., 1998).
J. Hiebert and J. W. Stigler
45
5 Conclusions
We have proposed a new conceptual model to guide research on teaching and
learning. Although we built on the groundbreaking work of others who explored the
idea of mediating variables between teaching and learning, our conceptualization is
not common. Many researchers and practitioners still imagine that researchers will
discover links between what teachers do and what students learn, with perhaps
some mediating variables in between. Given the historical challenges of applying
this traditional model to the day-to-day problems of classroom practice, we proposed
an alternative model as a way to move theories of teaching closer to the work of
Lucy Scott and her colleagues. Rather than trying to extract more from the traditional
models, we believe efforts would be better spent eshing out the alternative model
that sets SLOs as the proximal goal of teaching.
The brief descriptions we have provided of the model, of the theories that could
be built from the model, and of the processes that might be used to build the theories
are intended to provide a glimpse into the possibilities. But the descriptions do not
resolve many issues of which we are aware and even more issues that are sure to
arise. It is clear that, in addition to the massive work that will be required to build
out theories of creating SLOs, more work will be needed within the second system
in the model (Fig.2.4)—the transformation of SLOs into learning outcomes. More
complete and better specied theories of learning are needed to tie SLOs to learning
outcomes. These theories will require more sophisticated ways of dening and
assessing what SLOs could look like in classrooms. Because theories of creating
SLOs are dependent on specifying their features and linking them to well-dened
learning outcomes, work within both systems must proceed together.
If the model we have proposed is taken seriously, researchers and teachers will
need to work together to explore its ramications and to build useful theories of
creating SLOs. We have described some possibilities of the form this work might
take, but we are curious to see what conditions teachers and researchers decide are
critical for doing this work and for sustaining it over the time.
It is too early to make claims about the ultimate impact of this work, but we
believe it is sufciently promising to warrant serious attention. We recognize this is
not a quick x for putting useful theories into the hands of teachers. It will take
years to see the payoffs in terms of student learning gains. As one anonymous
reviewer of this chapter phrased it, “the promise of this work will depend on how it
gets taken up, developed and elaborated.” This can be, by itself, a reason to not take
the model seriously.
Following the TIMSS-R Video Study of Mathematics and Science Teaching, the
rst author testied before a U.S. congressional committee on education. The tes-
timony did not describe theories of creating SLOs, but it did outline the work we
have described that lies behind the creation of these theories. The next day the rst
author received a call from a U.S. senator’s ofce asking for more details about such
a plan for American schools. The senator’s assistant asked how long this would take.
When the assistant was told 15years, maybe 10 at best, he laughed and asked what
2 Creating Practical Theories ofTeaching
46
a 1–2year plan would look like. When he was told there was no such plan, he hung
up the phone and was not heard from again. Educational theorists and researchers
face major challenges in convincing themselves and others of the benets of long-
term research agendas.
6 Our Answers totheEditors’ Questions fortheAuthors
The editors of this book asked all authors to address the following questions, either
in the context of their presentation or as an additional section at the end of their
chapter. We have addressed most of the questions but our answers might be some-
what hidden and implicit. So, we will address all four questions here. If the earlier
sections contained relevant responses, our answers will be brief.
6.1 What Is aTheory (of Teaching)?
We can begin with the denition of theory that we presented and that is consistent
with the denition presented in the introductory chapter by the editors of this vol-
ume (Praetorius & Charalambous, this volume): an interrelated set of ideas intended
to explain something. The statement we borrowed from Hill and Smith (2005)—“the-
ory helps identify what factors should be studied and how and why they are
related”—helps to clarify our focus. Because we are interested in theories of teach-
ing that teachers actually can use, our theories of teaching attended to “factors” of
the classroom environment that teachers normally use to make instructional
decisions.
This bias toward theories that are usable by teachers leads us to the following
answer to this rst question. In a general sense, theories of teaching must account
for how the intended curriculum, broadly dened, is transformed into learning
opportunities that are experienced by students. This means that, in our view, theo-
ries of teaching consist of connected sets of hypotheses that predict how specic
instructional activities and tasks will produce learning opportunities experienced by
students in particular ways. That is, theories of teaching are capable of guiding the
cause-effect reasoning that lies at the core of making instructional decisions about
what kinds of tasks and activities will yield what kinds of sustained learning oppor-
tunities, and they do so with an eye toward studying and improving these decisions.
J. Hiebert and J. W. Stigler
47
6.2 Can Such aTheory Accommodate Differences Across
Subject Matters andStudent Populations Taught? If So,
How? If Not, Why?
Our response is “yes,” and “no.” Theories of teaching can be developed in general
ways that allow researchers and educators to swap out subject matter, student popu-
lations, and other contextual variables without changing the theory. Returning to our
interest in theories that are useful for teachers, such theories could help teachers
make and test instructional decisions but mostly in general and vague ways. More
helpful theories would be those developed with more specicity, and more specic-
ity requires building into the theories information about contextual variables.
For example, when Lucy Scott, our sixth-grade teacher, makes decisions about
what tasks could help her students understand equivalent fractions and how to
implement these tasks, a useful theory would contain informed hypotheses about
the kinds of experiences students need to develop conceptual understanding of key
mathematical concepts and how to create them. The more specic the hypotheses
are about developing understanding of equivalent fractions, the more useful the
theory. The hypotheses would specify features of these experiences, like struggle-
rst, that would, in turn, suggest selecting equivalent fraction tasks with consider-
able cognitive demand and situating them deliberately in lessons so as to increase
her students’ chances of experiencing productive struggle with equivalent fractions.
Theories would become increasingly useful as other teachers experimented with
similar tasks and researchers accumulated information over multiple trials.
This leads to another element of theories of teaching that makes them useful for
teachers: hypotheses are specic enough to be indexed according to the learning
goals or outcomes students are asked to achieve. Because different classroom expe-
riences are related to different learning outcomes, teachers will want to access
hypotheses about the kinds of teaching actions that will lead to experiences aligned
with the learning goals they want their students to achieve. If the hypotheses are too
general, they cannot help teachers like Ms. Scott make instructional decisions for
this lesson with this learning goal.
6.3 Do WeAlready Have aTheory/Theories onTeaching? If
So, Which Are They?
This question is difcult to answer because, in our view, theories of teaching are
necessarily so complex that they are only in progress; they are never complete. In
our view, not shared fully with some authors in this volume, the status of a theory
can be measured by the number of hypotheses that have been formulated, the range
of classroom learning events they can predict, and the state of empirical conrma-
tion of these predictions. Using these criteria, we would say the eld has theories at
the very beginning stages of development. Often, the “theories” are more like small
collections of hypotheses that still need to be fully tested.
2 Creating Practical Theories ofTeaching
48
One of the huge challenges facing theorists of teaching became clear for us while
working on the TIMSS video studies. We learned that, although it is possible within
a single culture to maintain the view that teachers’ particular actions are the causes
of particular learning outcomes, this view is hard to sustain in the context of cross-
cultural comparisons. We found that many of the variables educators have believed
are important—whether teachers lecture to the whole class or divide the class into
student work groups, whether teachers use concrete or abstract representations,
whether teachers use technological tools or just write on the chalkboard—turn out
to vary among these countries. These do not appear to warrant theorists’ attention as
individual variables. It was not the teachers’ actions, or even the problems presented
to students, that higher-achieving countries shared. Rather, it was how the elements
of a lesson were congured and the way in which students engaged with the learn-
ing opportunities.
To make things even more challenging, we found that different kinds of teacher
actions could produce similar kinds of learning opportunities and similar kinds of
teacher actions could produce different learning opportunities. What mattered was
the way in which students took up the opportunities. Across the higher achieving
countries, we saw many different instructional strategies and teachers’ actions that
resulted in the richest kinds of learning opportunities—repeated opportunities for
students to engage in productive struggle to make connections among important
mathematical concepts, facts, and procedures. These ndings help to explain why
the eld is struggling to build theories of teaching that teachers can use.
6.4 In theFuture, inWhat Ways Might It BePossible, If atAll,
toCreate a(More Comprehensive) Theory ofTeaching?
Our rst response to this question is that we have described what Lipsey (1993)
calls “small theories attempting to explain treatment processes, not a large theory of
general … phenomena” (p.11). In this sense, we have shown, at least implicitly, our
bias against “comprehensive” theories of teaching. This is due partly to our belief
that “small theories,” focused on teaching processes that lead to particular learning
opportunities for students, are the kinds of theories that will be useful for teachers.
Our interest in “small theories” also is due to our skepticism that, at this point in the
history of theory development and research on teaching, developing a comprehen-
sive theory of teaching is likely, or is even the next best step.
However, we certainly endorse the goal of creating more comprehensive “small
theories.” Our answer to the question of creating gradually more comprehensive
(small) theories is contained in our descriptions of building theories of creating
sustained learning opportunities. We can pull out a few features of this work that
seem especially important: begin with documented connections between the kinds
of sustained learning opportunities that yield speciable learning outcomes; develop
hypotheses about how teachers can create sustained learning opportunities of the
J. Hiebert and J. W. Stigler
49
targeted kinds; continuously test and revise predictions suggested by the hypothe-
ses; coordinate the work of teachers and researchers to test predictions and revise
hypotheses; aggregate ndings across classrooms and search for patterns that rise
above specic contexts; and, nd ways to create sustainable partnerships between
teachers and researchers, and build networks of partnerships. As learning theorists
and researchers continue to identify the features of sustained learning opportunities
that yield particular learning outcomes, researchers and teachers can continue to
expand the scope of their theories of teaching.
We want to repeat that the processes we have identied for building more
comprehensive theories are tailored to the values we expressed at the beginning of
the chapter and to the kind of theories in which we are most interested. Stepping
back, we recognize that the processes for building theories of teaching will result, in
large part, from the kinds of theories the community wishes to build. Authors of
other chapters in this volume outline different theory-building agendas.
Acknowledgements We want to thank Ronald Gallimore and the editors of this book for their
comments on an earlier draft of this chapter.
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