ArticlePDF Available

Proportion: A tablet app for collaborative learning

Authors:

Abstract and Figures

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.
Content may be subject to copyright.
Proportion: A Tablet App for Collaborative Learning
Jochen Rick
Department of Educational Technology
Saarland University
D-66123 Saarbrücken
j.rick@mx.uni-saarland.de
ABSTRACT
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
Keywords
Tablets, collaborative learning, shareable interfaces
1. LEARNING WITH TABLETS
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
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for prot or commercial advantage and that copies
bear this notice and the full citation on the rst page. To copy otherwise, or
republish, to post on servers or to redistribute to lists, requires prior specic
permission and/or a fee.
IDC 2012, June 12–15, 2012, Bremen, Germany.
Copyright 2012 A CM 978-1-4503-1007-9...$10.00.
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
5
8
2
3
16
12
2
5
2
1
3
2
3
2
1
5
2
5
1
5
3
5
4
5
6
5
7
5
8
5
1
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.
2. PROPORTIONAL 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
8
:
5
8
=3:5); if the denominators are different, this strategy does
not work (
3
7
:
5
4
=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
1
4
blocks can
be stacked together to form one
1
2
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
4
) [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.
3. THE PROPORTION APPLICATION
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).
4. FUTURE WORK
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 (
11
2
:
4
3
). 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?
5. ACKNOWLEDGEMENTS
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.
6. REFERENCES
[1] D. Abrahamson and D. Trimic. Toward an
embodied-interaction design framework for mathematical
concepts. In Proceedings of IDC ’11, pages 1–10, New York,
2011. ACM Press.
[2] E. G. Cohen. Restructuring the classroom: Conditions for
productive small groups. Review of Educational Research,
64(1):1–35, 1994.
[3] P. Dillenbourg and M. Evans. Interactive tabletops in
education. International Journal of Computer-Supported
Collaborative Learning, 6(4):491–514, 2011.
[4] S. E. Higgins, E. M. Mercier, E. Burd, and A. Hatch.
Multi-touch tables and the relationship with collaborative
classroom pedagogies: a synthetic review. International
Journal of Computer-Supported Collaborative Learning,
6(4):515–538, 2011.
[5] S. J. Lamon. Ratio and proportion: Connecting content and
children’s thinking. Journal for Research in Mathematics
Education, 24(1):41–61, 1993.
[6] Z. A. Leong and M. S. Horn. Representing equality: A
tangible balance beam for early algebra education. In
Pr oceedings of IDC ’11, pages 173–176, New York, 2011.
ACM Press.
[7] E. I. Mansor, A. De Angeli, and O. De Bruijn. The fantasy
table. In Proceedings of IDC ’09, pages 70–79, New York,
2009. ACM Press.
[8] J. Marco, E. Cerezo, S. Baldassarri, E. Mazzone, and J. C.
Read. Bringing tabletop technologies to kindergarten
children. In Proceedings of HCI ’09, pages 103–111,
Swinton, UK, 2009. British Computer Society.
[9] G. Noelting. The development of proportional reasoning and
the ratio concept: Part I differentiation of stages.
Educational Studies in Mathematics, 11:217–253, 1980.
[10] J. Rick, A. Harris, P. Marshall, R. Fleck, N. Yuill, and
Y. Rogers. Children designing together on a multi-touch
tabletop: An analysis of spatial orientation and user
interactions. In Proceedings of IDC ’09, pages 106–114,
New York, 2009. ACM Press.
[11] J. Rick, P. Marshall, and N. Yuill. Beyond one-size-ts-all:
How interactive tabletops support collaborative learning. In
Pr oceedings of IDC ’11, pages 109–117, New York, 2011.
ACM Press.
[12] F. Tourniaire and S. Pulos. Proportional reasoning: A rev iew
of the literature. Educational Studies in Mathematics,
16:181–204, 1985.
[13] UPI.com. Tablets giving pcs a run for the money. http://
outcomemag.com/business/2012/03/05/
tablets-giving-pcs-a-run-for-the-money/,
March 2012.
[14] N. M. Webb and A. S. Palincsar. Group processes in the
classroom. In D. C. Berliner and R. C. Calfee, editors,
Handbook of Educational Psychology, pages 841–873.
Simon & Schuster, 1996.
[15] D. Wigdor and D. Wixon. Brave NUI world: Designing
natural user interfaces for touch and gesture. Morgan
Kaufmann, San Francisco, 2011.
... subject fluent in comparing decimals, choosing > ). Previous research implies that when the degree of distance is too high, children may focus solely on technological components of the activity (Rick, 2012) or avoid tasks requiring actions perceived as difficult (Tucker, 2018). However, during activity, many attributes are modified (Tucker & Johnson, 2017), as represented by changes in the activity and the externalized representations (e.g. ...
... Although conceptually congruent gestures can support mathematical learning (Segal et al., 2014) and have great potential when part of activity involving multi-touch technology (Baccaglini-Frank & Maracci, 2015;Sinclair & de Freitas, 2014), as in other research (e.g. Rick, 2012;Tucker, 2018), participants who repeatedly struggled to effectively perform certain gestures had limited access to some relevant aspects of the mathematics. However, most participants in this study decreased technological distance enough to attend to the mathematics, including via conceptually congruent gestures where applicable. ...
This study examines the construct of distance-the degree of difficulty of interacting with something-as part of activity involving children using touchscreen digital games to learn mathematics. Ten fifth-grade children engaged in video-recorded semi-structured task-based interviews in which they used two touchscreen digital mathematics games on a touchscreen tablet and responded to semi-structured follow-up questions. Qualitative data analysis was iterative, featuring analytic memoing and eclectic coding techniques to identify themes related to distance. In advanced coding stages, magnitude coding was used to characterize the degree of distance present. Findings provide evidence of the presence of distance , changes in distance, and interactions between distance types throughout the activity. In particular, both mathematical distance and technological distance were present, changed in various ways, and often influenced each other. Implications include the relevance of distance for designing, implementing, and researching educational technology.
... One recent example of physicalizing the interface in a learning technology is the Proportion tablet application used in a study of collaborative learning in Schmitt and Weinberger (2019) and described in detail in Rick (2012), where two primary school students work together to create a visualization of a target proportion. On the blue side of the screen one student selects a point that represents a quantity of blue on a vertical axis, and another student selects a point on the orange side. ...
Article
Full-text available
This paper describes a framework for making explicit the design decisions in the development of immersive and interactive STEM learning technologies. This framework consists of three components: (1) visual viewpoint, the location from which a visual simulation depicts observable components; (2) embodied interaction, the ways in which a learner can physically engage with the simulation interface; and (3) learners’ roles, the purpose and the participation structure the technology presents to the learner. The recent literature on the design of STEM learning technologies is reviewed with the lens of how the three components have been leveraged and what, if any, rationale is provided for the design decisions that were made. The definition and review of each component is followed by a set of reflective questions intended to prompt researchers and designers to be more explicit about these decisions and the ways they are intended to impact student learning in both the design process and the reporting of their work. The paper concludes with a discussion of how the three components interact, and how their articulation can support theory building as well as the proliferation of more effective STEM learning technology designs.
... A more interesting solution would be, as identified by Falloon and Khoo (Falloon & Khoo, 2014), to have a multi-tablet setting in which the devices were placed on a table publicly available, which would increase workspace awareness and, in turn, collaborative performance (Hornecker et al., 2008). In this respect, Proportion (Rick, 2012) is a collaborative app that places two children in front of a shared tablet to help them learn about proportions and ratios. They have good awareness of what the other participant is doing and they can communicate, but collaboration is not enforced since one user can monopolize all the interactions. ...
Article
Gamification has been identified as an interesting technique to foster collaboration in educational contexts. However, there are not many approaches that tackle this in primary school learning environments. The most popular technologies in the classroom are still traditional video consoles and desktop computers, which complicate the design of collaborative activities since they are essentially mono-user. The recent popularization of handheld devices such as tablets and smartphones has made it possible to build affordable, scalable, and improvised collaborative gamified activities by creating a multi-tablet environment. In this paper we present Quizbot, a collaborative gamified quiz application to practice different subjects, which can be defined by educators beforehand. Two versions of the system are implemented: a tactile for tablets laid on a table, in which all the elements are digital; and a tangible in which the tablets are scattered on the floor and the components are both digital and physical objects. Both versions of Quizbot are evaluated and compared in a study with eighty primary-schooled children in terms of user experience and quality of collaboration supported. Results indicate that both versions of Quizbot are essentially equally fun and easy to use, and can effectively support collaboration, with the tangible version outperforming the other one with respect to make the children reach consensus after a discussion, split and parallelize work, and treat each other with more respect, but also presenting a poorer time management.
... Scenario 1: Tablet-based classroom applications are growing rapidly as reported by British Educational Suppliers Association (BESA) (2013) and are seen by many as one of 21st century's key classroom innovations. Accordingly, a number of classroom-oriented collaborative learning tools have been developed for tablets that support two simultaneous users working around the tablet (e.g., (Rick and Jochen 2012;Thinking n.d.)). Such collaborative learning tools can benefit from user identification not only to support accountability but also to show level of participation and support collective agreement on different decisions in the process for example (Kharrufa et al. 2010). ...
Article
Full-text available
User identification on interactive surfaces is a desirable feature that is not inherently supported by existing technologies. We have conducted an extensive survey of existing identification techniques, which led us to formulate a unified model for user identification. We start by introducing this model that (1) classifies existing user identification approaches in five categories according to the identification technology, (2) identifies eight characteristic identification system parameters, and (3) proposes a way for visualizing the system's characteristics as points on a radar chart to allow for quick comparison and contrast between systems. This model is then used to present our survey of existing user identification approaches and visualize their characteristics, highlighting their strengths and limitations. The model also makes it possible to visually represent requirements of systems that require user identification, identify existing approaches that can meet an application's requirements, and help report on and evaluate new approaches to user identification systematically.
... This affected how easily students could work on their circuit and how they collaborated within their pairs. Consistent with research in collaborative learning, creating a larger tool allows multiple access points for students to work more seamlessly together [13]. Therefore, the number of students to be working together should be considered when determining the size for a tool in this space. ...
Conference Paper
Building physical computing projects can enable learners to integrate computing into a range of interests and disciplines. However, the electronic portion of these projects can be difficult. Students are learning new concepts as well as how to work with new tools. This influx of information can be difficult for students to retain in their working memory as they construct their circuits. In this paper, we introduce BitBlox, a set of modular, solderless Breadboards for prototyping circuits. BitBlox attempts to decrease the cognitive load on the user by reducing the complexity found in the standard Breadboard by bringing visibility to the underlying connections within its modules. We present a comparative classroom study integrating the Breadboard and BitBlox into two different high school classes. Our qualitative analysis focuses on student errors, strategies, and collaborative practices, highlighting important dynamics for designing hardware tools.
... Difficulty performing required input such as controlled mouse movements (technological distance) while interacting with virtual Pattern Blocks can lead to unintended mathematical outcomes such as unintentionally rotating shapes instead of sliding them (mathematical distance) (Highfield and Mulligan 2007). A high degree of technological distance in the form of difficulty using appropriate input can also lead to a user focusing attention on performing the gestures (i.e., decreasing technological distance) rather than attending to the mathematical content (i.e., high degree of mathematical distance) (Rick 2012). Thus, distance plays a role in user-tool interactions. ...
Chapter
While extensive research has examined the outcomes of interacting with virtual manipulatives, less research has focused on constructs and relationships among constructs involved in user-tool interactions. This chapter presents the Modification of Attributes, Affordances, Abilities, and Distance (MAAAD) for Learning framework, which conceptualizes the relationships among these constructs to describe user-tool interactions, including those involving virtual manipulatives. The framework is primarily grounded in theories of representation and embodied cognition, as user-tool interactions in mathematics involve internalizing and externalizing representations through physically embodied mathematical practices. In the framework, attributes, affordance-ability relationships, and distance are interrelated, and modification of one construct contributes to modification of the other constructs. Each attribute can contribute to many affordance-ability relationships and to distance . Attribute modification can change the approach or degree of affordance access and alter the degree of distance present, which can, in turn, lead to attribute modification. This chapter illustrates the constructs and relationships among constructs that form the framework in the context of user-tool interactions in mathematics. The chapter then applies the framework to examples of children’s interactions with mathematics virtual manipulative touchscreen tablet apps . The MAAAD for Learning framework has implications and applications relevant to theory, development, implementation, and research concerning technology tools, including virtual manipulatives.
Article
One challenge of child–computer interaction research is surveying variation of children’s attitudes towards novel educational technology, which often results in an opinion ceiling effect. In this article, the authors introduce BiCo – a bipolar continuous rating scale which builds on children’s ability to draw relative comparisons. They elaborate on the development of BiCo and on how they designed it to better survey children’s attitudes. Beyond addressing the opinion ceiling effect, BiCo is suitable for surveying a wide range of concepts and its invertible design enables simulation of inverted items. The authors provide data comparing BiCo with the widely used Smileyometer instrument when used by fourth-graders (around 10 years old), demonstrating that BiCo mitigates the opinion ceiling effect. They discuss BiCo as a new tool for children’s technology evaluation and provide directions for future research.
Chapter
The purpose of this study was to examine what patterns were revealed using heatmaps with hierarchical clustering to examine preschooler’s performance, speed, and developmental progressions in counting and seriation. The chapter describes a study conducted with 35 preschoolers who used six touchscreen virtual manipulative mathematics apps in two different learning sequences: counting and seriation. The analysis employed heatmaps coupled with hierarchical clustering to highlight changes in children’s performance, speed, and developmental progressions, between a pre- and post- assessment app after using two learning apps. This method allowed for analysis of individual and whole group data examining several tasks within each app and also several apps within each learning sequence. The analysis revealed different clusters of children grouped according to their developmental progressions which were related to incremental changes in performance and speed from the Pre to Post App use.
Article
Multi-touch interfaces allow for direct and simultaneous input by several co-present learners and afford hands-on learning experiences. Additional scaffolding for strategic behavior and/or verbalizations may constructively complement collaborative learning with a multi-touch device. In this study, the tablet app “Proportion” is supposed to enable two novices (about 10 years old) to collaboratively construct an understanding of proportional relations. In a 2 × 2 factorial design (n = 162), effects of enriching Proportion with strategy prompts (with/without) and verbalization prompts (with/without) on multi-modal processes as well as near and far transfer learning gains have been investigated. The process variables include task focus, positive and negative emotions, and quality of dialogue (transactivity, epistemic quality). We found a general improvement in near transfer task types over all conditions without the two prompt types further affecting learning gains. While the strategy prompts did not significantly affect processes or outcomes, the verbalization prompts had versatile effects on learning processes: On one hand, quality of talk was improved, on the other hand, task focus and emotions were negatively affected. © 2018, Association for Educational Communications and Technology.
Chapter
Our objective in this chapter is to present a framework that can be used as a guide for designers of virtual manipulatives and for researchers who study their effects on student learning in mathematics. Because a significant amount of research has been devoted to the effects of concrete manipulatives on student learning, the crux of the framework is based on the existing literature in this area. Specifically, the framework consists of three interrelated components that align with the research on students’ learning with external representations: the surface features of the representations themselves, the pedagogical contexts that support students’ meaning making, and the students’ perceptions and interpretations of the representations. Where applicable, we integrate the research on virtual manipulatives to support the validity of the framework itself and its applicability for researchers of virtual mathematics tools.
Conference Paper
Full-text available
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.
Conference Paper
Full-text available
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.
Conference Paper
Full-text available
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.
Conference Paper
Full-text available
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.
Article
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.
Article
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.
Article
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.
Article
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)
Article
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.
Conference Paper
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.