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Proportion: A tablet app for collaborative learning


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Everyday computing technology is transitioning from PCs to more natural user interfaces. At the forefront of this trend are multi-touch tablets. Each year, tablets become more affordable, capable and widespread. Now is the time for research to shape how they will be used to support learning. In this paper, I introduce the Proportion tablet application as both a concrete vision of how tablets can be used to support co-located collaborative learning and as a research platform for investigating that possibility. I motivate the work, describe how the design has evolved and outline the questions this design-based research aims to address.
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Proportion: A Tablet App for Collaborative Learning
Jochen Rick
Department of Educational Technology
Saarland University
D-66123 Saarbrücken
Everyday computing technology is transitioning from PCs to more
natural user interfaces. At the forefront of this trend are multi-touch
tablets. Each year, tablets become more affordable, capable and
widespread. Now is the time for research to shape how they will be
used to support learning. In this paper, I introduce the Proportion
tablet application as both a concrete vision of how tablets can be
used to support co-located collaborative learning and as a research
platform for investigating that possibility. I motivate the work, de-
scribe how the design has evolved and outline the questions this
design-based research aims to address.
Categories and Subject Descriptors
H.5.2 [Information Interfaces and Presentation]: User
Interfaces—user-centered design
General Terms
Design, Human Factors
Tablets, collaborative learning, shareable interfaces
One of the most consistent ndings in education is that collabo-
ration makes learning more active, engaging and effective [2, 14].
With many students and one teacher, peer-to-peer co-located col-
laboration is well suited to the average classroom. Unfortunately,
PCs—the most prevalent classroom computing technology—are ill
equipped to support such collaboration. As the term “personal
computer” suggests, these devices were created for a single user
interacting with the machine through a single mouse and a single
keyboard. Consequently, the PC has not been able to support co-
located collaborative learning en masse.
Recently, ne w technologies, broadly grouped under the term natu-
ral user interfaces [15], have expanded how technology can sup-
port co-located users. In particular, research has demonstrated the
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Figure 1: Two children working together with Proportion.
benets of using interactive tabletops to support co-located col-
laborative learning [3, 4]. Two properties are fundamental: direct
input and multiple access points. Direct input means that an end
user can directly manipulate the software interface and applications
using touch, pen and / or by moving tangible objects. In compar-
ison to using a mouse to control a cursor, the cognitive distance
between intent and execution is shortened. Multiple access points
means that multiple concurrent interaction points are sensed by the
hardware and utilized by the software. This enables both multi-
point gestures, such as pinching with two ngers to zoom out, and
switching which hand to use. In addition, the access points can be
distributed among multiple participants, thereby enabling collab-
oration. As a result, interactive tabletops have been shown to be
particularly useful in supporting the collaboration of even young
children [7, 8].
Multi-touch tablets too support direct input and multiple access
points. Can they similarly enable co-located collaborative learn-
ing? That is the question that drives this work. Tablets differ from
tabletops in two important ways. First, tablets are much smaller
(e.g., the Apple iPad tablet has a 9.7” diagonal display, whereas
the Microsoft Surface tabletop has a 40” diagonal display). Table-
tops provide a large enough surface that users can work indepen-
dently but still stay informed about what others are doing [11]. A
smaller surface could lead to more confrontation over display real
estate and may make it more difcult to see what elements are in-
dicated by a partner. Second, tablets are more commercially suc-
cessful. Market analysts predict that the market for multi-touch
tablets will overtake PCs (desktops and laptops combined) as early
(a) No Support (b) Fixed Grid (d) Labeled Lines(c) Relative Lines
Figure 2: Four interfaces for supporting learners in solving problems.
as 2013 [13]. An iPad costs about $500; a Microsoft Surface costs
about $10,000. As a consequence, tablets are more likely to have
an impact on ev eryday learning practices. If research can show
ho w tablets can support co-located collaborative learning, then this
opens up promising new avenues for supporting collaborative learn-
ing in the classroom.
I designed the Proportion iPad application to research the potential
of one tablet to support collaborative learning for two co-located
learners. For this application, the tablet is positioned vertically
on a table in front of two learners, aged 9–10 (Figure 1). Learn-
ers work together to solve a series of ratio / proportion problems.
The interface has two columns (Figure 2). For each problem, users
must size the left and right columns in proportion to their respec-
tiv e numerical labels. A touch or drag on the left side of the screen
mov es the orange column to that height; likewise, touches on the
right side resize the blue column. When all touches are released,
the ratio of the column heights is evaluated; if it is accurate enough,
children are informed of their success and proceed to the next prob-
lem. Through using Proportion, learners gain competence in pro-
portional reasoning.
Ratios and proportions play a critical role in a student’s mathemat-
ical dev elopment [5]. It is a broad topic, ranging from elementary
concepts of dividing a whole into halves to being able to manipu-
late fractions to solve algebraic equalities. Because of its impor-
tance and depth, the topic is covered repeatedly and in increas-
ing sophistication in several grade levels. The cognitive develop-
ment around ratios and proportions is well documented and moves
through relatively distinct stages [9]. Proportional reasoning is re-
alized through multiple strategies, where one strategy might work
well for one set of problem but be inappropriate for another set. For
instance, in cases where the denominators are the same, the ratio of
two fractions is the same as the ratio of the respective numerators
=3:5); if the denominators are different, this strategy does
not work (
=3:5). Gaining competence in proportional rea-
soning requires acquiring strategies and understanding when and
ho w to apply them [12]. Even students who show clear compe-
tence in applying a strategy successfully to one problem might fail
to realize that the same strategy applies to another problem.
Proportional reasoning is a challenging mathematical domain. One
difculty is that the topic is usually taught and tested with mathe-
matical notation through word problems [5]. While a teacher can
give feedback about whether a student correctly solved such a prob-
lem, that feedback is temporally removed from when the student at-
tempts the problem. The student might employ the wrong strategy
(one that worked previously, but does not apply to that problem)
ov er an entire sequence of problems without realizing their miscon-
ception. Real-time feedback on task progress can allow students to
more quickly realize which of their current strategies to employ or
when to generate new ones. Consequently, physical manipulati ves
that give some level of real-time feedback (e.g., two
blocks can
be stacked together to form one
block) have been shown to be
a particularly useful technique for learning proportional reasoning
[12]. Digitally enhanced manipulatives can further enhance the ex-
perience by providing more sophisticated feedback and bridge the
gap between an embodied experience (e.g., a quarter wedge of a
circle) and its corresponding symbolic representation (
) [1, 6].
Another useful technique for supporting learning is to provide tools
that highlight specic elements of a problem. Such tools have a
number of benets. First, the tool can provide feedback on task
progress. For instance, a balance beam will only balance if the ra-
tios are correct. Second, students can gain competence in using the
tool to solve problems. Tool competence can be important in apply-
ing concepts in the real world. For instance, using a tablespoon to
keep adding increments of our and sugar to a recipe while keeping
their ratio intact is a practical cooking skill. Third, learners can ap-
ply strategies learned with the tool even when the tool is gone. For
instance, students might learn to use a measuring stick to precisely
solve a problem and later be able to use step lengths to estimate the
solution t o a similar problem.
Proportion provides several such tools implemented in four inter-
faces (Figure 2). Without any support (a), learners must estimate
the ratios. Embodied proportional reasoning, relying on rules-of-
thumb (larger denominator means smaller amount) and estimation
(9 is about twice as much as 4), are particularly important for learn-
ers to relate their everyday experiences to mathematical concepts
[1]. With a xed 10-position grid (b), learners have precise places
that they can target, thereby using their mathematical understand-
ing of the task to quickly solve problems. One strategy is for users
to select the grid line that corresponds to their respective numbers.
This works well for simple ratios, such as 4:9. For the common-
factor problem shown in Figure 2b, that strategy does not work.
As illustrated, the children tried a novel strategy of positioning the
columns based on the last digit of the number. Of course, this did
(a) Nearly Correct (b) Correct
Figure 3: Stars to pro vide performance feedback.
not work and they were able to realize that this was not a viable
strategy. With relative lines (c) that expand based on the position
of the columns, learners can use counting to help them solve the
problem. They can also learn more embodied strategies, such as
maximizing the size of the larger column to make it easier to cor-
rectly position the smaller column. When the lines are labeled (d),
other strategies can be supported. For instance, in the fraction-
based problem shown, a useful strategy is to arrange columns so
that whole numbers (e.g., 1) are at the same le vel.
Proportion provides two levels of real-time feedback (Figure 3).
If the ratio of the two columns is close to the correct answer, a
small star (a) is shown. If the ratio is within a very small zone,
then it is pronounced as correct, a large star (b) is shown and the
application moves on to the next problem. When designing this
feedback, it was important that learners not just solve the problem
based on the feedback without strategically engaging the problem.
Hence, the close feedback was designed to give no information
about which direction the correct answer lies. Concurrently, learn-
ers need enough feedback to make progress when they are testing
out or discovering a new strategy. To better support this, the sen-
sitivity of the zones is adjusted for the problems. The rst time
a new strategy is needed, the zones are relatively large, allowing
learners to more easily stumble upon the solution. As the sequence
progresses, the zones become smaller, making it uncomfortable for
learners to simply employ a stumble-upon strategy. The zones are
larger for estimation tasks (e.g., Figure 2a) where precision is dif-
cult even when learners employ a correct strategy. Conversely,
the zones are smaller when the interface should support precision,
thereby coaxing learners to take advantage of those tools.
Proportion has been through two rounds of user testing (observ-
ing two children from the intended audience using the application
for an hour) to improve the interface and ne-tune the sequence
of problems. In the initial version, a few usability problems were
found. First, the children would touch the column with their nger
but simultaneously touch t he interactive surface with their palms.
When they lifted their nger, the palm would still be touching and
the column would unintentendedly shift to that touch point. Sec-
ond, children had trouble precisely positioning a column. Fingers
produce relatively fat touch points. As a nger lifts, the center of
that touch point can unintentionally shift. Often, children would
precisely place a column, lift their nger and have the column shift
just enough to prevent it from solving the problem. While they were
able to overcome that problem, the task became one of precisely
controlling the interface rather than the intended task of solving the
mathematical problem. Third, in rare cases, a problem would be
Figure 4: Arrows to provide directional feedback.
too difcult and the star-based feedback was not enough to allow
children to make progress. These problems were addressed in the
second revision.
To avoid inadvertent palm touches, secondary touch points on a col-
umn are ignored, even when the original touch point is removed. To
address the fat nger problem, small movements at the very end of a
touch sequence are ignored if the touch point had lingered at a value
for some time. To address challenging problems, directional feed-
back was added after 15 seconds to provide more feedback (Figure
4). In the second trial, these approaches succeeded in addressing
the user problems. While learners did not usually need the direc-
tional feedback, they did occasionally over-rely on it, responding
simply to the arrows instead of engaging the problem. As such, the
time to directional feedback was changed to one minute. This is
short enough that learners can use it when they are stuck but long
enough that it becomes uncomfortable for them to rely on it for an
entire problem sequence.
One major revision of the second version was prompting for ver-
balization. In addition to the problems with star feedback, learners
were asked to complete a few problems without feedback. For these
(Figure 5), an owl would appear to ask them how to solve a problem
(a). During the task, the o wl would pique its ears when the learn-
ers were talking and would follow the column movements with its
eyes (b). Unlike the normal problems, learners received no feed-
back about their progress on the task. As such, the learners would
be more likely to need to verbalize in order to come up with an an-
swer that satised both. When done, they pressed the owl (c) and
pressed the owl again to ensure that they were done (d). These type
of problems were given for two different situations. First, it served
as a reection exercise: After learners had successfully navigated
a sequence on a specic topic (e.g., ratios of large numbers with
common factors), they would recei ve a problem to see if they could
explain their approach verbally. Second, it served as a prediction
exercise: Before learners were given a sequence, they were asked to
solve a particularly difcult one without support. During the trial,
children often forgot to approach these problems differently. In the
future, the verbal prompting will be made more noticeable (e.g., by
adding audio to the visual directions) and more demanding (e.g.,
asking afterwords why their solution was correct).
Through two cycles of user testing, the interface and the curriculum
has been polished to where children will be able to use Proportion
without external support. That curriculum contains 215 problems
split into 21 sequences. Each sequence targets a different propor-
tional reasoning strategy, from comparing simple whole numbers
(1:5) to complex fractions (
). This broad range was cho-
sen to better support the research. At an average of 25 seconds
per problem, learners would be able to nish the entire problem
(a) Start of Problem (b) During Problem (d) After Two Taps(c) After One Tap
Figure 5: Prompting learners to verbally reect on their approach.
sequence in about 90 minutes; however, that is not how Propor-
tion will be used. As a research application, it is intended to be
used to compare multiple conditions, such as one without verbal
prompting versus one with verbal prompting. As time on task is
a dominant factor in learning success, this work aims to control
for that variable. All groups will work for an hour. Even high
performing groups are unlikely to nish as the problems go well
beyond the targeted grade level. For instance, participants had not
yet learned how to verbalize more sophisticated fractions, saying
“one seven” instead of “one seventh. Remarkably, they still made
good progress on such problems.
The research with Proportion aims to shed light on two broad re-
search topics. First, it will investigate how children communicate
to collaborate. Previous work on interacti ve tabletops has demon-
strated that children readily use their interactions with the interac-
tive surface to communicate with their partners [11]. This work
aims to tease apart the role of verbal and gestural communica-
tion. Second, it will investigate issues of equity of collaboration
for tablet-based collaboration. On tabletops, it becomes difcult
for users to access all parts of the surface; therefore, users tend to
concentrate their interactions in areas closer to their position at the
tabletop [10]. Such separation is not possible for a tablet: Every
user has good access to all parts of the interactive surface. Pro-
portion was designed to have an interface split across the users.
Children quickly grasp that they should control the column on their
side. Do children tend to adhere to this convention? What happens
when the convention breaks down? How does this affect the equity
and effectiveness of the collaboration?
I would like to thank Michael Gros of the Saarland LPM (Lan-
desinstitut für Pädagogik und Medien) for facilitating the access
to schools and those schools for supporting our development and
research efforts.
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Taking computer technology away from the desktop and into a more physical, manipulative space, is known that provide many benefits and is generally considered to result in a system that is easier to learn and more natural to use. This paper describes a design solution that allows kindergarten children to take the benefits of the new pedagogical possibilities that tangible interaction and tabletop technologies offer for manipulative learning. After analysis of children's cognitive and psychomotor skills, we have designed and tuned a prototype game that is suitable for children aged 3 to 4 years old. Our prototype uniquely combines low cost tangible interaction and tabletop technology with tutored learning. The design has been based on the observation of children using the technology, letting them freely play with the application during three play sessions. These observational sessions informed the design decisions for the game whilst also confirming the children's enjoyment of the prototype.
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We explore the possibility of creating an interactive system which can foster fantasy play in preschool children in a tabletop environment. This paper reports our experiences designing and testing two prototypes with young children aged 3-4 years old. In the first study, we focused on understanding the similarities and differences between the type of play afforded by real objects and virtual objects. In the second study, we focused on testing solutions for the interaction difficulties evinced in the first study to see how to provide an engaging experience for children. Data were collected by observing children while they played with the study materials. Both quantitative and qualitative methods were used for data collection and analysis.
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Previous research has demonstrated the capacity of interactive table-tops to support co-located collaborative learning; however, these analyses have been at a coarse scale---focusing on general trends across conditions. In this paper, we offer a complimentary perspective by focusing on specific group dynamics. We detail three cases of dyads using the DigiTile application to work on fraction challenges. While all pairs perform well, their group dynamics are distinctive; as a consequence, the benefits of working together and the benefits of using an interactive tabletop are different for each pair. Thus, we demonstrate that one size does not fit all when characterizing how interactive tabletops support collaborative learning.
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Recent, empirically supported theories of cognition indicate that human reasoning, including mathematical problem solving, is based in tacit spatial-temporal simulated action. Implications of these findings for the philosophy and design of instruction may be momentous. Here, we build on design-based research efforts centered on exploring the potential of embodied interaction (EI) for mathematics learning. We sketch two emerging, reciprocal contributions: (1) a sociocognitive view on the role of automated feedback in building the perceptuomotor schemes that undergird conceptual development; and (2) a heuristic EI design framework. We ground these ideas in vignettes of children engaging an EI design for proportion. Increasing ubiquity and access to mobile devices geared to avail of EI principles suggests the feasibility of mass-disseminating materials evolving from this line of research.
Moving beyond the general question of effectiveness of small group learning, this conceptual review proposes conditions under which the use of small groups in classrooms can be productive. Included in the review is recent research that manipulates various features of cooperative learning as well as studies of the relationship of interaction in small groups to outcomes. The analysis develops propositions concerning the kinds of discourse that are productive of different types of learning as well as propositions concerning how desirable kinds of interaction may be fostered. Whereas limited exchange of information and explanation are adequate for routine learning in collaborative seatwork, more open exchange and elaborated discussion are necessary for conceptual learning with group tasks and ill-structured problems. Moreover, task instructions, student preparation, and the nature of the teacher role that are eminently suitable for supporting interaction in more routine learning tasks may result in unduly constraining the discussion in less structured tasks where the objective is conceptual learning. The research reviewed also suggests that it is necessary to treat problems of status within small groups engaged in group tasks with ill-structured problems. With a focus on task and interaction, the analysis attempts to move away from the debates about intrinsic and extrinsic rewards and goal and resource interdependence that have characterized research in cooperative learning.
Twenty-four sixth-grade children participated in clinical interviews on ratio and proportion before they had received any instruction in the domain. A framework involving problems of four semantic types was used to develop the interview questions, and student thinking was analyzed within the semantic types in terms of mathematical components critical to proportional reasoning. Two components, relative thinking and unitizing, were consistently related to higher levels of sophistication in a student's overall problem-solving ability within a semantic type. Part-part-whole problems failed to elicit any proportional reasoning because they could be solved using less sophisticated methods. Stretcher/shrinker problems were the most difficult because students failed to recognize the multiplicative nature of the problem situations. Student thinking was most sophisticated in the case of associated sets when problems were presented in a concrete pictorial mode.
Two related problems have to be solved before we can have a clearer picture of cognitive development:(i) Is development hierarchical, leading to higher-order systems controlling lower-order subsytems? (ii) If so, what are the mechanisms involved in a process of development? These two problems will be studied here taking as example a concept which finds its achievement only in late adolesence: the concept of proportion. Part I of this article is devoted to the first problem. It will bear on the experiment which was undertaken and the analysis of results leading to a differentiation of stages of development. These stages will be illustrated by typical protocols of each stage. Part II will be devoted to problem solving strategies at each stage, and finally to a second order analysis leading to an attempt to interpret the passage from one stage to the next in terms of increasing equilibration or adaptive restructuring of the strategies put to use to solve problems.
This paper offers a synthesis of research on cooperative learning in small groups. The main challenge for teachers who utilize cooperative learning is to stimulate the type of interaction desired according to their teaching objective. A generalization regarding student interactions is that if students are not taught differently, they will tend to operate at the most concrete level. Student participation in a task group that is structured to foster resource- or goal-interdependence appears to increase student motivation and performance. The effectiveness of the group structure depends on the task's complexity and uncertainty and on the extent to which the instructions attempt to micromanage the interaction process. Information is also offered on ensuring equity in interaction, managing the interaction, and unsettled issues, such as special curricula and assessment. Successful implementation of cooperative learning also requires staff development and principals who demonstrate effective managerial skills and instructional leadership. (LMI)
This paper presents a review of the research on proportional reasoning. Methodologies used in proportional reasoning studies are presented first. The discussion is then organized around the following topics: strategies use to solve proportion problems, including erroneous strategies; factors that influence performance on proportion problems, both task-related and subject-related; training studies. The discussion is accompanied by suggestions for educational and research applications.
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In this paper we describe the design and implementation of a tangible balance beam that we created for early algebra education. We also present data from an exploratory study with seven children (ages 9--10 years) in a local elementary summer school classroom. Our results provide insight into how students solve algebra problems using our tangible interface, how they coordinate multiple representations (both digital and physical) in the problem solving process, and how they understand the concept of algebraic equality in this context. The data suggests that our interface helps students think about equations in a relational context, which has been shown to be an important skill for understanding more advanced concepts in algebra. Whether or not the combination of physical and digital representations provided by our interface helps students apply this relational understanding to equations written using standard algebraic notation is an open question that we hope to investigate in future work.