In this study, contextaware ubiquitous learning was used to support 4th grade students as they
learn angle concepts. Contextaware ubiquitous learning was provided to students primarily through
the use of iPads to access realworld connections and a Dynamic Geometry Environment. Gravemeijer
and van Eerde’s (
Data collection included semistructured clinical interviews, observations, student artefacts, video recordings and lesson reflections. This study of technology is grounded in a subject content area (mathematics) so the researchers could clearly state the advantages of using this approach in an educational context. A review of the findings indicates that contextaware ubiquitous learning proved useful in avoiding many common errors and misconceptions that students have in learning these concepts, and students demonstrated growth in their understanding of angle and angle measure beyond what is typically expected. From this study, the researchers present four design guidelines and a full set of contextaware ubiquitous activities.
The use of technology is becoming ubiquitous throughout today’s society. As philosophies
and practice move toward learnercentred pedagogies, technology, in a parallel move, is now able to
provide new affordances to the learner, such as mobile learning (mlearning), that can be used to
provide learning that is personalized, contextualized, and unrestricted by temporal and spatial
constraints (
Geometry, the mathematical concept chosen for this study, is a complex subject incorporating many
challenging mathematical concepts. Angle concepts are particularly difficult for students to grasp
(
There is a paucity of research to explore how mobile devices can be used in this way to support students’ understanding of angle. The purpose of this study is to ameliorate this gap in scholarly understanding and to develop an empiricallybased instruction theory of how contextaware ulearning can be used to support the teaching and learning of angle and design guidelines of developing contextaware ulearning activities.
Mobile Learning (mlearning) offers many new opportunities in the evolution of technology
enhanced learning (
Contextaware ulearning is an emerging sub category of mobile learning. Hwang et al. (
The learner’s situation or the situation of the realworld environment in which the learner in location can be sensed, implying that the system is able to conduct the learning activities in the real world . . . contextaware ulearning can actively provide supports and hints to the learners in the right way, in the right place, and at the right time, based on the environmental contexts in the real world. (p. 84)
This is the way contextaware ulearning is being identified in this study. To further explicate
contextaware ulearning, Hwang et al. provided a
Models and examples of contextaware ulearning activities. Adapted from
Model  Context Aware Ubiquitous Learning Examples 



Learning in the real world with online guidance  The students are learning in the real world and are guided by the
system, based on the realworld data collected by the sensors. 
Learning in the realworld with online support  The students learn in the real world, and support is automatically
provided by the system based on the realworld data collected by the sensors. 
Collect data in the real world via observations  The students are asked to collect data by observing objects in the
real world and to transfer the data to the server via wireless communications. 
Identification of a realworld object  Students are asked to answer the questions concerning the
identification of the realworld objects. 
Observations of the learning environment  Students are asked to answer the questions concerning the observation
of the learning environment around them. 
Cooperative data collecting  A group of students are asked to cooperatively collect data in the
real world and discuss their findings with others via mobile devices. 
Cooperative problem solving  The students are asked to cooperatively solve problems in the real
world by discussion using mobile devices. 
In the examples provided, the students are interacting with the device and the environment to
learn particular concepts. The environments described in these examples are atypical classroom
environments, although they could also take place in the classroom. The premise of contextaware
ulearning is that students use portable devices to learn by physically exploring and interacting
with the real world (
Geometry forms the foundations of learning in mathematics and other academic subjects (
As students move on to angle measurement, many students believe that the size of the angle is
determined by measuring the length of the line segments that are the rays of the angle (
For centuries, scholars have advocated the importance of connecting mathematics to the real world
(e.g.
Technology has also been used to support students’ understanding of concepts. Dynamic
Geometry Environments provide the students with figures and basic tools to create composite figures.
A review of the literature revealed that realworld contexts and Dynamic Geometry Environments are
two pedagogical approaches to supporting students learning of geometry. There are those who have
used contextaware ulearning to make the realworld connection to mathematics. For example, Elisson
and Ramberg (
Bray and Tangney (
The purpose of this study is to use a contextaware ulearning approach to support students as they learn about angle and angle measure. The research questions guiding this research are:
How can contextaware ulearning be used to extend and enhance students’ understanding of angle?
What design guidelines will inform the development of contextaware ulearning activities?
To this end, the researchers employed Gravemeijer & van Eerde’s (
Two fourth grade teachers chose to participate in the study. This determined the classes from
which students participated. There were 30 fourth grade (9–10 years of age) students in each
class, for a total of 60 student participants in the study. The study took place in the
southeastern United States. Following Gravemeijer & van Eerde’s (
The researcher acted as the teacher in both of the teaching experiments. In the DBR process it is
not uncommon for one researcher to serve as the teacher implementing the instructional intervention
(e.g.,
There are various types of DBR including those developed by BannanRitland (
For the contextaware ulearning components of this study, each student was given an iPad2 with
Sketchpad Explorer (dynamic Geometry Environment) loaded onto the device with the addon sketch
titled
Measure a Picture with Two Angles Measured Using the Dynamic Protractor.
The two macro cycles for this study are illustrated in
A Diagrammatic Representation of the Study with Points of Data Collection.
This second cycle was a repeat of the first with a new set of students. The second teaching
experiment took place two weeks after the conclusion of the first teaching experiment. There were
two retrospective analyses conducted, one at the conclusion of each macro cycle. The local
instruction theory came from the final retrospective analysis. At the bottom of
One of the distinct characteristics of DBR methodology is that the researchers develop a deeper
understanding of the phenomenon while the research is in progress. Therefore, it is crucial that the
research team generated a comprehensive record of the entire process (
Pre and postinstruction clinical interviews
coresearcher and witness classroom observations
whole class video recording
daily mini cycle reflection audiorecording with research team
artefact collection of student classwork
researcher’s daily reflection journal
retrospective analysis at the end of a macro cycle
Scally’s (
The credibility of Scally’s clinical interview has been determined with 83% reliability and
the content validity of the instrument is established. Furthermore, Scally’s (
During the teaching experiment, observation notes were collected from the research team which included the classroom teachers, mathematics and technology specialists, and one other researcher. The video recordings were also transcribed and additional observation notes were developed from the recordings.
Immediately after each instructional episode, the research team met together to discuss their observations of the lesson and changes that need to be made to the instruction for the following day. These meetings were audio recorded and used in the retrospective analysis.
Student work artefacts from the teaching experiment were collected for analysis. This included screen shots of the students angle findings and measurements as well as worksheets and any rough notes or jottings the students created.
The primary researcher completed a personal reflection journal for each of the teaching episodes
during each mini cycle. The journal was an instrument that allowed the researcher to step back from
the action to record impressions, feelings, and thoughts (
All of these data sources were used during both the daily mini cycle analysis and the retrospective analysis phases at the end of each macro cycle. Data gathered from the final retrospective analysis was used to create a more robust local instructional theory.
To develop a set of design guidelines, data from all of the sources, other than the clinical
interviews, were coded. The interviews were not included as they underwent a separate analysis
following Scally’s (
Using DBR, the researchers developed a local instruction theory involving two components; design guidelines for informing the development of contextaware ulearning activities and a set of exemplary contextaware ulearning activities for extending and enhancing students’ understanding of angle concepts.
Using Scally’s (
Macro Cycle One: Pre and PostInstruction Interview Summary.
Pre Instruction  Post Instruction  

V  VA  A  AI  I  V  VA  A  AI  I  
Draws Angles  4  4  
Identifies Angle  4  1  3  
Sorts Angle  4  4  
Angle Measure  4  1  3  
Angle Relations  4  4 
Macro Cycle Two: Pre and PostInstruction Interview Summary.
Pre Instruction  Post Instruction  

V  VA  A  AI  I  V  VA  A  AI  I  


Draws Angles  3  1  1  3  
Identifies Angle  1  3  4  
Sorts Angle  3  1  4  
Angle Measure  4  1  3  
Angle Relations  3  1  1  3 
The students in macro cycle one began working between the visual and the analysis level for drawing, identifying, and sorting angles. For angle measure and relations the students were working within the visual level. For the post instruction interviews, the four students improved and moved from the visual to the analysis level. The majority of the students were working fully within the analysis level (level two) at the end of the macro cycle.
Students in macro cycle two predominantly scored within the visual level in the pre instruction interview with some students working partially between the visual and analysis level. For the post instruction interview, the majority of the students moved into the analysis level of geometric thinking, however, for drawing angles and angle relations three of the four students were working between the analysis level of thinking and the informal deduction level.
Following the teaching experiment the students from both macro cycles showed improvement. However, students in macro cycle two demonstrated the greatest increase from pre to post interview scores. Arguably, this improvement is due to the revision to the activities following macro cycle one.
In the review of the literature, a number of problem areas were described as to how students can
develop misconceptions and errors as they come to understand angle concepts. Contextaware
ulearning was proposed as a pedagogical approach that may ameliorate those problems (See
Ways in Which ContextAware Ubiquitous Learning Activities Supported Students Understanding of Angles.
Problem Addressed  



Recognizing angles in different contexts. Student lack this ability as
indicated by 
By using the mobile devices to take photographs of the angles, the students were able to first see the 3D angles which helped the students connect with the realworld angles. In addition, the camera view reduced the amount of external information the student was receiving to more easily find the angles. 
Determining plausible answers  The students could look back from the device to see the physical angles which helped them determine if the final measurement was plausible. 
Angles are based on a dynamic rotation. Student lack and understanding
of this concept as indicated by 
Students were able to understand that an angle is the rotation from a point as the dynamic protractor demonstrated this movement. 
The length of the angle rays (lines) are irrelevant attributes of
angles A misconception indicated by 
Students were supported in understanding that the length of the rays
does not change the size of the angle as the rays on the app were changeable in length. In 
Orientation is an irrelevant angle attribute. A misconception
indicated by Battista ( 
As students became familiar with looking for angles in the real world,
the students realized that angle orientation did not matter. For example, the typical textbook right
angle always faced one way. In the real world as the students found right angles and measured them
using the dynamic protractor they realized that orientation did not matter. For example, using the
app (Measure a Picture), the student in 
A student uses the App to demonstrate he/she understood that orientation and ray length were not relevant angle attributes.
The results of this study provide a set of activities involving contextaware ulearning. Due to
space constraints, the full set of activities developed from this study cannot be provided within
this paper but they are included as
Data collected from this study provided a vast amount of information. These data were coded and four design guidelines emerged.
Discussion is an effective way of promoting learning. “Reflective thought and, hence,
learning is enhanced when the learner is engaged with others working on the same ideas” (
During the instructional experiment it was found that students engaged their partners in very little discussion when they were asked to share the angles they had identified. Instead the students used the features of the iPad to share the angles and provided very little verbal explanation. For example, one student was asked by their partner what angles he/she had found and the student responded by pointing to the iPad screen and using the pinch feature to zoom in and out of the image, again pointing each time they did this. The student did not make any verbal connection to the other student during this time.
During the design of these activities it is important to include a specific requirement that the students verbally interact as well as use the features of the technology to get across the information to another student or educator.
Cognitive load is a detailed field of study that is too great to go into indepth review or analysis in this paper. However, data from this research show that students struggled to learn two new independent concepts at the same time. At the beginning of the teaching experiment students are first coming to explore the meaning of the term angle and to have them learn the use of a new technological device and program (Measure a Picture) at the same time was too much information for the students to process. This was changed to have students’ first focus on the educational concept of study, then on the second day the students were introduced to the mobile devices and the program.
Having the students conduct contextaware ulearning activities will typically have the students interacting with the realworld environment. Although the students may easily connect with a familiar environment, e.g. school grounds, there is a lot of visual information connected with that place when students are asked to explore it for a particular concept. For example, in this study, students were required to find angles in a realworld context. In a 360 degree view of the environment next to a school building there is a large amount of information to review to identify angles. In addition, the students are new to understanding what an angle is.
This information should be reduced and a photo viewer is a good way of reducing the information
the student is receiving. This should be included in a contextaware ulearning program to allow
students to interact with the real world in a manageable way. As the students are preparing to use
the mobile technology, to reduce the load of information students can be required to use a
nondigital technology such as a cardboard tube to look through. The students can then move from the
cardboard tube to the photo viewer.
Cardboard viewing tubes to reduce the amount of realworld information being reviewed.
It is important to have students working with contextaware ulearning activities to gain an indepth understanding of concepts with connections to the realworld. However, the contextaware ulearning activities must also be mixed with decontextualized learning to ensure the students can transfer that information. In other words, the students may recognize angles on a building in the real world, but they should also be able to recognize an angle drawn onto a piece of paper and make the connection that they are both angles.
This study resulted in an empiricallybased instruction theory of how contextaware ulearning
can be used to support students’ understanding of angle and angle measurement, and a set of
design principles for developing contextaware ulearning activities. Using a cyclical iterative
process of anticipation, enactment, evaluation, and revision (
Using Scally’s interview, that matched students’ angle understanding to the van Hiele
levels of geometric thinking (1957/
Furthermore, evidence from the multiple data sources was triangulated and it would appear that
context aware ulearning was supportive for learning about angle concepts in these ways: (a) by
using the mobile devices to take photographs of the angles, the students were able to first see the
3D angles which helped the students connect with the realworld angles; (b) as students became
familiar with looking for angles in the real world, the students realized that angle orientation did
not matter. For example, the typical textbook right angle always faced one way. In the real world as
the students found right angles and measured them using the dynamic protractor they realized that
orientation did not matter; (c) the students could look back from the device to see the physical
angles which helped them determine if the final measure was plausible; (d) students were able to
understand that an angle is the rotation from a point as the dynamic protractor demonstrated this
movement; and (e) students were supported in understanding that the length of the rays does not
change the size of the angle as the rays on the app were changeable in length. These points
connected with the misconceptions and errors that students have with angle concepts that were
initially identified in the literature review. The final set of contextaware ulearning activities
can be found in full in
From this study, four design guidelines emerged for contextaware ulearning activities. These guidelines were not specific to mathematics but for educational designers and scholars across all subject areas and all ages. The four design guidelines are:
From a thorough review of the literature, this study appears to be the first of its kind to determine how a form of contextaware ulearning can be used to support students’ understanding in learning about angle concepts. In addition, it is the first study to include the use of dynamic geometry environments in contextaware ulearning activities. Another distinct advantage of this study is that the researchers focused on a technological approach and did this by fully grounding that approach within a specific subject to determine the types of affordances this pedagogical approach can bring to learning.
Nonetheless, the specific focus of the educational concept is also a limitation to this study. Therefore, the researchers cannot claim that these types of activities will work across subject content areas as they have only been tested with students learning angle concepts. Nonetheless, the activities developed for this study and the clinical interview used to determine the efficacy of the technology is not age specific, but instead focuses on a set of mathematical understandings that can be broadly spread from young students to young adults. Therefore, the activities can be applied by teachers of all ages depending on the skill set of the learners.
Finally, as educational designers are provided with a growing number of technologies and new affordances, this study provides a set of design principles for the development of contextaware ulearning activities for extending and enhancing students’ understanding of angle. In addition, educational designers and educators are provided with a set of exemplar contextaware ulearning activities that are ready for immediate use.
The author declares that they have no competing interests.