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Editorial
Clarifying the Impact of Educational
Research on Learning Opportunities
Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang,
Victoria Robison, and James Hiebert
University of Delaware
In our last editorial, we considered the impact of research on students’ learning.
In clarifying our perspective, we answered the question of “impact of research on
what” to include both cognitive and noncognitive outcomes in students as well as
long-term impact on students that goes well beyond short-term cognitive impact.
A natural next step is to examine the conditions under which students can achieve
such broad goals. We will devote the next set of editorials to exploring ways in
which researchers can design their work to increase its impact on students’
opportunities to achieve these goals.
We begin our exploration by focusing on the learning goals and learning
opportunities that guide classroom instr uction. Although research has shown that
factors outside of school have a larger effect on students’ learning than their
experiences in the classroom (e.g., Lave, 1988; Nye, Konstantopoulos, & Hedges,
2004; Resnick, 1987; Rockoff, 2004), classroom instruction is still considered a
central component for understanding the dynamic processes and organization of
students’ thinking and learning (e.g., Bruner, 1998; Gardner, 1991; Rogoff &
Chavajay, 1995; Stigler & Hiebert, 1999). In addition, learning opportunities are
something that educators can address directly and something that can be
influenced by educational research. In this editorial, we try to clarify what we
mean by the impact of research on learning opportunities. As in our first editorial,
we begin with a story.
A Fraction Activity Using a Number Line
With the guidance of a mathematics education researcher (Ms. Research), a
fourth-grade teacher (Mr. Lovemath) introduced the following fraction task to his
students: Order the fractions 7/9, 2/4, 9/10, 6/13, 1/2, 9/5, and 3/7 from smallest to
largest, and place them on the number line. This task was part of the fourth lesson
in the fractions unit. In the first three lessons, students had been introduced to the
definition of fractions, the meaning of fraction symbols, and equivalent fractions.
Ms. Research and Mr. Lovemath chose this task for three reasons. First, this is
a cognitively demanding task. Ms. Research is well aware of studies showing that
the nature of instructional tasks determines the learning opportunities they
provide. Doyle (1988) argued that instructional tasks with different cognitive
demands are likely to offer different kinds of learning opportunities. Tasks
Journal for Research in Mathematics Education
2017, Vol. 48, No. 3, 230–236
Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in other formats without written permission from NCTM.
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Cai, Morris, Hohensee, Hwang, Robison, and Hiebert
determine not only students’ attention to particular aspects of content but also
their ways of processing information. Regardless of the context, cognitively
demanding tasks can invite exploration, ref lection, and hard work (Hiebert &
Wearne, 1993; National Council of Teachers of Mathematics, 2000; Stein, Grover,
& Henningsen, 1996). Instructional tasks that are truly problematic and involve
significant mathematics, therefore, have the potential to provide intellectual
contexts for students’ mathematical development and to engage students in
productive str uggle.
The second reason for choosing this task is the potential role the number line
can play in fostering students’ learning. In contrast to students’ learning about
whole numbers, students are expected to learn that fractions are both part–whole
comparisons and numbers (Behr, Harel, Post, & Lesh, 1992; Pantziara &
Philippou, 2012). The visual number line is often thought to help students make
the abstract concept of fraction-as-number more concrete. In fact, the Common
Core State Standards for Mathematics (National Governors Association Center
for Best Practices & Council of Chief State School Officers, 2010) explicitly calls
for students to develop an understanding of fractions as numbers on the number
line. Ms. Research and Mr. Lovemath selected this task from the curriculum
precisely because it could help address this Common Core standard as well as the
fourth-grade Common Core standard regarding comparing fractions.
Finally, the third reason for choosing this task is that it allows for multiple
strategies to be used to compare the fractions. When students use different solution
strategies, they are able to draw on any piece of knowledge they have learned and
justify their ideas in ways they feel are convincing. Therefore, students can
develop their strategic competence and adaptive reasoning (Kilpatrick, Swafford,
& Findell, 2001) as they creatively problem solve in this context. This, in turn,
affords students the opportunity to make stronger connections and develop deeper
understanding of the fraction ideas involved.
Mr. Lovemath divided his students into small groups to work on the fraction
task. However, after 20 minutes, none of the groups were able to come up with a
correct solution, much less one they could explain, and the students became
frustrated. Mr. Lovemath felt compelled to provide a hint and suggested that the
students make use of equivalent fractions using a common denominator, which
they had studied in a previous lesson. This not only reduced the cognitive demand
of the task but also made the students’ solutions quite procedural.
Why did the students encounter difficulties in solving this problem from their
curriculum? Why did the intended opportunity to learn not materialize? Whose
fault was it? Did the teacher fail to understand the situation or react to it properly?
Did the curriculum developers design a task for which the students were not
sufficiently prepared? What additional roles could Ms. Research have played?
Answers to these questions are critical because this kind of story repeats itself
thousands of times every day in classrooms that aim to provide students with
broader, richer, and more ambitious learning goals.
232 Editorial
Often, unrealized learning opportunities like these have been attributed to
teacher deficiencies. Teachers are frequently blamed for reducing the cognitive
level of the task because they do not properly facilitate students’ learning
(Henningsen & Stein, 1997). Although we might criticize Mr. Lovemath’s choice
to give a hint, this does not address the primary issue of why his students had more
difficulty than expected grappling with the task in the way the curriculum
developers intended. What do researchers need to know to assist teachers like Mr.
Lovemath in making this learning opportunity more productive and thus helping
students achieve broader and richer learning goals?
Research and Learning Opportunities
Even though we believe that this task can provide rich opportunities for students’
learning, we suspect that the pace of the lessons in which it appeared was too fast
for students to develop robust understandings of key preliminary concepts, such
as the meaning of fraction symbols, unit fractions, and equivalent fractions.
Depending on the nature of the previous lessons, it could be that the only solution
strategy available to students was to use the common denominator algorithm.
Thus, there was likely a misalignment between the rich learning opportunities the
task was intended to foster and the learning opportunities actually available to the
students. Ultimately, the students did not attain learning goals in prior learning
experiences that would have adequately prepared them to take advantage of the
learning opportunities offered by this task.
This story highlights a specific difficulty in our more general quest to increase
the impact of research on practice. As we concluded in our last editorial, the
construct of learning opportunities can provide researchers with a way to think
about inf luencing practice. Research has shown that in order to foster students’
learning, they should be provided with opportunities to engage in productive
struggle with cognitively demanding tasks that are neither too easy nor too
challenging. However, in this story, the intended learning opportunities seemed
to be inaccessible to the students because of a mismatch between the demands of
the task and the learning opportunities the students had previously experienced.
Although a mathematics task can be very rich, the learning opportunities it
offers are always defined by the prior learning necessary for students to engage
with the task. The relationships among learning opportunities and sequences of
learning opportunities form a space within which researchers can work to generate
results that are useful for practice. For any mathematical task, researchers could
ask a series of questions that might lead to increasing the learning opportunities
afforded by the task: What are the learning subgoals that a student needs to attain
in order to make progress toward achieving the main learning goal of this task?
How specific do these subgoals need to be? What are the learning opportunities
needed to achieve those subgoals? How do we help students access these prior
learning opportunities? What kinds of learning opportunities are best aligned with
specific subgoals X, Y, and Z to help students achieve primary goal A, and in what
order are they best addressed?
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Cai, Morris, Hohensee, Hwang, Robison, and Hiebert
We want to draw attention to two features of these questions. First, answering
the questions requires breaking down a primary learning goal into finer grained
subgoals. In our previous editorial, we argued for broadening the learning goals
that researchers should consider. We now argue that once major learning goals
have been determined, a good deal of empirical work remains to unpack those
major goals into smaller subgoals. Broadening learning goals is important to
opening up the research space, but so is identifying the smaller goals implicit in
the broader learning goals.
A second feature of these questions is that they are legitimate questions for
researchers to address and should not just be left to curriculum developers to f igure
out. At their heart, these are empirical questions. For the fraction task, researchers
would need to empirically investigate which subgoals are truly necessary, and
which might be helpful, for achieving the overall goal of the fraction task. Because
investigating these questions is best done in the context of actual classroom
activity, the research findings would not need a complex translation into practice.
Instead, teachers, who likely would be working alongside researchers, could
directly apply (and test) these findings in their classroom practice.
Reconsidering Learning Opportunities
One of the most robust findings of education research is that students learn best
that which they have the opportunity to learn (Bransford, Brown, & Cocking,
2000; Kilpatrick et al., 2001). Thus, it is incumbent on researchers to investigate
how to align learning opportunities with learning goals. Moreover, research will
have a greater impact on students’ achievement of a primary learning goal if it can
inform teachers about how learning opportunities can be created that correspond
with the relevant learning subgoals for that primary goal. Researchers should not
feel satisfied by simply unpacking primary learning goals into component
subgoals. To support practice, additional research is needed to uncover the
learning opportunities likely to help students achieve each subgoal (Morris &
Hiebert, 2011). Of course, these twin findings—learning subgoals and their
accompanying learning opportunities—will quite naturally go hand in hand as
researchers investigate the most productive subgoals for a larger goal.
Fortunately, mathematics educators already have some useful conceptions of
what this kind of research could look like. Simon’s (1995) concept of “hypothetical
learning trajectories” (p. 133) captures the idea of a carefully sequenced set of
learning opportunities that help students build toward milestone learning goals.
Our earlier claim that identifying relevant learning subgoals is an empirical issue
means that researchers can contribute to practice by turning the “hypothetical”
into the empirically supported. The works of Clements and Sarama (2007);
Confrey, Maloney, and Corley (2014); Hackenberg and Lee (2015); Lobato and
Walters (in press); Norton (2008); and Steffe (2001) illustrate this process. In
addition, methodologies that could be used in this kind of research are being
actively developed and refined. For example, design research (Cobb, Jackson, &
Dunlap, in press; Gravemeijer, Bowers, & Stephan, 2003) and approaches that
234 Editorial
build from improvement science (Bryk, Gomez, Grunow, & LeMahieu, 2015)
seem tailored for investigating these finer grained questions of identifying critical
subgoals and creating productive learning opportunties aligned with these goals.
In some countries, efforts to investigate productive learning sequences have
taken different directions from what is typical in the United States. In China, for
example, there has long been a tradition of “teaching research” ( jiaoyan) based
on a teacher−researcher model (Huang & Bao, 2006; Paine, 1990; Paine, Fang, &
Jiang, 2015). In Japan, the lesson study model represents a distinct but related form
of research (Lewis & Perry, 2017; Stigler & Hiebert, 1999). These approaches
represent research-based systems that search for increasingly productive learning
sequences that identify both an important sequence of learning goals and
corresponding learning opportunities. We are not proposing these models as the
only, or even the best, ways to identify subgoals and learning opportunities.
Rather, the existence of these models in some countries serves to highlight the
kind of research findings we are describing.
To reiterate, educational researchers need to consider how to best create the
learning opportunities needed to maximize the impact on students’ learning. Prior
research has established a basis for this line of work, but systematic research is
needed to map out the appropriate grain size for learning goals along with
productive learning opportunities. For that to happen, mathematics education
researchers would need to adopt a different perspective on conducting research
than that held by many educators today. This brings us to one of those “thought-
provoking” issues we promised to raise in these editorials. It seems to us that,
consistent with the current research model, many researchers address questions
of theoretical or personal interest. This can lead to results that need extensive
translation to reach the grain size teachers need to plan lessons. Researchers and
teachers both know the obstacles inherent in this model that prevent the use of
such research. However, the kind of work we propose—research that produces
findings about the learning opportunities that are needed to achieve particular
learning goals—is much closer to the daily work of mathematics teachers. If
researchers wish to make this kind of an impact on practice, the alternative model
we describe in this editorial is worth considering.
We stated in our first editorial (Cai et al., 2017) that we believe a critical first
step in increasing the impact of research is to seek to understand the fundamental
reasons for the divide between research and practice. In this editorial, we have
suggested that one reason for the lack of impact is that researchers sometimes fail
to recognize the small grain size teachers must consider as they help students move
from one idea to the next. Tackling issues of learning goals and learning
opportunties at this level of detail is one approach that could increase the impact
of research on practice. In our next editorial, we will explore the related question
of how research might inform practice at the level of teachers’ implementation of
instructional activities.
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