=Paper=
{{Paper
|id=Vol-1446/GEDM_2015_Submission_8
|storemode=property
|title=Graph-based Modelling of Students' Interaction Data from Exploratory Learning Environments
|pdfUrl=https://ceur-ws.org/Vol-1446/GEDM_2015_Submission_8.pdf
|volume=Vol-1446
|dblpUrl=https://dblp.org/rec/conf/edm/PoulovassilisSM15
}}
==Graph-based Modelling of Students' Interaction Data from Exploratory Learning Environments==
https://ceur-ws.org/Vol-1446/GEDM_2015_Submission_8.pdf
Graph-based Modelling of Students’ Interaction Data from
Exploratory Learning Environments
Alexandra Poulovassilis Sergio Gutierrez-Santos Manolis Mavrikis
London Knowledge Lab London Knowledge Lab London Knowledge Lab
Birkbeck, Univ. of London Birkbeck, Univ. of London UCL Institute of Education
ap@dcs.bbk.ac.uk sergut@dcs.bbk.ac.uk m.mavrikis@lkl.ac.uk
ABSTRACT learning of algebraic generalisation [15]; and the iTalk2Learn
Students’ interaction data from learning environments has system that aims to support 8-10 year old students’ learning
an inherent temporal dimension, with successive events be- of fractions [7]. Both systems provide students with math-
ing related through the “next event” relationship. Exploratory ematical microworlds in which they undertake construction
learning environments (ELEs), in particular, can generate tasks: in MiGen creating 2-dimensional tiled models using
very large volumes of such data, making their interpretation a tool called eXpresser and in iTalk2learn creating fractions
a challenging task. Using two mathematical microworlds using the FractionsLab tool. In eXpresser, tasks typically re-
as exemplars, we illustrate how modelling students’ event- quire the construction of several models, moving from spe-
based interaction data as a graph can open up new querying cific models involving specific numeric values to a general
and analysis opportunities. We demonstrate the possibilities model involving the use of one or more variables; in parallel,
that graph-based modelling can provide for querying and students are asked to formulate algebraic rules specifying
analysing the data, enabling investigation of student-system the number of tiles of each colour that are needed to fully
interactions and leading to the improvement of future ver- colour their models. In FractionsLab, tasks require the con-
sions of the ELEs under investigation. struction, comparison and manipulation of fractions, and
students are encouraged to talk aloud about aspects of their
constructions, such as whether two fractions are equivalent.
Keywords
Exploratory Learning Environments, Interaction Data, Graph Both systems include intelligent components that provide
Modelling different levels of feedback to students, ranging from un-
solicited prompts and nudges, to low-interruption feedback
1. INTRODUCTION that students can choose to view if they wish. The aim
Much recent research has focussed on Exploratory Learn- of this feedback is to balance students’ freedom to explore
ing Environments (ELEs) which encourage students’ open- while at the same time providing sufficient support to en-
ended interaction within a knowledge domain, coupled with sure that learning is being achieved [9]. The intelligent sup-
intelligent techniques that aim to provide pedagogical sup- port is designed through detailed cognitive task analysis and
port to ensure students’ productive interaction [9]. The data Wizard-of-Oz studies [13], and it relies on meaningful indica-
gathered from students’ interactions with such ELEs pro- tors being detected as students are undertaking construction
vides a rich source of information for both pedagogical and tasks. Examples of such indicators in MiGen are ‘student
technical research, to help understand how students are us- has made a building block’ (part of a model), ‘student has
ing the ELE and how the intelligent support that it provides unlocked a number’ (i.e. has created a variable), ‘student
may be enhanced to better support students’ learning. has unlocked too many numbers for this task’; while ex-
amples of such indicators in FractionsLab are ‘student has
In this paper, we consider how modelling students’ event- created a fraction’, ‘student has changed a fraction’ (numer-
based interaction data as a graph makes possible graph- ator or denominator), ‘student has released a fraction’ (i.e.
based queries and analyses that can provide insights into has finished changing it).
the ways that students are using the affordances of the sys-
tem and the effects of system interventions on students’ be- Teacher Assistance tools can subscribe to receive real-time
haviour. Our case studies are two intelligent ELEs: the information relating to occurrences of indicators for each
MiGen system, that aims to foster 11-14 year old students’ student, and can present aspects of this information visu-
ally to the teacher [8]. Indicators are either task independent
(TI) or task dependent (TD). The former refer to aspects of
the student’s interaction that are related to the microworld
itself and do not depend on the specific task the student is
working on, while the latter require knowledge of the task
the student is working on, may relate to combinations of
student actions, and their detection requires intelligent rea-
soning to be applied (a mixture of case-based, rule-based and
probablistic techniques). Detailed discussions of MiGen’s TI
and TD indicators and how the latter are inferred may be Event EventType
found in [8]. dateTime eventID
taskID eventStatus
occurrenceOf
In this paper we explore how graph-based representation of constrID eventCat
userID
event-based interaction data arising from ELEs such as Mi- sessionID
Gen and FractionsLab can aid in the querying and analysis
of such data, with the aim of exploring both the behaviours next
of the students in undertaking the exploratory learning tasks
set and the effectiveness of the intelligent support being
provided by the system to the students. Data relating to
Figure 1: Core Graph Data Model
learning environments has often been modelled as a graph
in previous work, for example in [10] for providing support
to moderators in e-discussion environments; in [16, 18] for There is a relationship ‘next’ linking an instance of Event
supporting learning of argumentation; in [17] for modelling to the next Event that occurs for the same user, task and
data and metadata relating to episodes of work and learning session. There is a relationship ‘occurrenceOf’ linking each
in a lifelong learning setting; in [1] for learning path discov- instance of Event to an instance of the EventType class.
ery as students “navigate” through learning objects; in [3] for
recognising students’ activity planning in ELEs; and in [23] The instances of the EventType class include: startTask,
for gaining better understanding of learners’ interactions and endTask, numberCreated, numberUnlocked, unlockedNum-
ties in professional networks. berChanged, buildingBlockMade, correctModelRuleCreated,
incorrectModelRuleCreated, interventionGenerated, interven-
Previous work that is close to ours is the work on interac- tionShown, in the case of eXpresser (see [8] for the full list);
tion networks and hint generation [6, 21, 20, 4, 5], in which and startTask, endTask, fractionCreated, fractionChanged,
the graphs used consist of nodes representing states within a fractionReleased, inverventionShown, in the case of Frac-
problem-solving space and edges representing students’ ac- tionsLab.
tions in transitioning between states. This approach targets
learning environments where students are required to select We see that instances of the EventType class have several
and apply rules, and the interaction network aims to rep- attributes, including:
resent concisely information relating to students’ problem-
solving sequences in moving from state to state. Our focus
here differs from this in that we are using graphs to model • eventID: a unique numerical identifier for each type of
fine-grained event-based interaction data arising from ELEs. indicator;
In our graphs, nodes are used to represent indicator occur-
rences (i.e. events, not problem states) and edges between • eventStatus: this may be -1, 0 or 1, respectively stat-
such nodes represent the “next event” relationship. Also, ing that an occurrence of this type of indicator shows
rather than using the information derived from querying and that the student is making negative, neutral or posi-
analysing this data to automatically generate hints, our fo- tive progress towards achieving the task goals; an addi-
cus is on investigating how students are using the system tional status 2 is used for indicators relating to system
and the effects of the system’s interventions in order to un- interventions;
derstand how students interact with the ELEs and improve
their future versions. • eventCat: the category into which this indicator type
falls; for example, startTask and endTask are task-
related indicators; interventionGenerated and inter-
2. GRAPH-BASED MODELLING ventionShown are system-related ones; numberCreated,
Figure 1 illustrates our Graph Data Model for ELE interac- numberUnlocked, unlockedNumberChanged are number-
tion data. We see two classes of nodes: Event — represent- related; and fractionCreated, fractionChanged, frac-
ing indicator occurrences; and EventType — representing tionReleased are fraction-related.
different indicator types. The instances of the Event class
are occurrences of indicators that are detected or generated Figure 2 shows a fragment of MiGen interaction data con-
by the system as each student undertakes a task. We see forming to this graph data model. Specifically, it relates to
that instances of Event have several attributes: dateTime: the interactions of user 5 as he/she is working on task 2
the date and time of the indicator occurrence; userID: the during session 9. The user makes three constructions during
student it relates to; sessionID: the class session that the this task (with constrIDs 1, 2 and 3). The start and end
student was participating in at the time; taskID: the taskID of the task are delimited by an occurrence of the startTask
that the student was working on; and constrID: the con- and endTask indicator type, respectively — events 23041
struction that the student was working on1 . and 33154. We see that the two events following 23041 re-
late to an intervention being generated and being shown to
1
The model in Fig. 1 focusses on the interaction data. The the student (this is likely to be because the student was in-
full data relating to ELEs such as eXpresser and Fraction- active for over a minute after starting the task); following
sLab would also include classes relating to users, tasks, ses- which, the student creates a number — event 24115.
sions and constructions; and attributes describing instances
of these classes, such as a user’s name and year-group, a
task’s name and description, a construction’s content and There are additional attributes relating to events, not shown
description, and a session’s description and duration. here for simplicity, capturing values relating to the student’s
23041 344712
startTask startTask
dateTime: occurrenceOf dateTime: occurrenceOf
20150331091524 eventID:0 20150215091741 eventID:0
taskID:2 eventStatus:0 taskID:56 eventStatus:0
eventCat:taskEv next eventCat:taskEv
constrID:1 constrID:4
userID:5 userID:5
sessionID:9 sessionID:1
intervention- intervention-
next Generated Shown
... fractionChanged fractionReleased
23921 344758
occurrenceOf eventID:6001 eventID:6002 occurrenceOf eventID:1002 eventID:1003
dateTime: eventStatus:2 eventStatus:2 dateTime: eventStatus:1 eventStatus:1
20150331091637 eventCat:systemEv eventCat:systemEv next 20150215091828 eventCat:sfractionEv eventCat:fractionEv
taskID:2 taskID:56
constrID:1 constrID:4
userID:5 userID:5
sessionID:9 occurrenceOf numberCreated sessionID:1 occurrenceOf interventionShown
eventID:1006 eventID:6002
next eventStatus:1 next occurrenceOf eventStatus:2
23923 occurrenceOf eventCat:numberEv 344759 eventCat:systemEv
dateTime: 24115 dateTime: 344760
20150331091638 dateTime: 33154 20150215091828 dateTime: 344761
taskID:2 20150331091655 taskID:56 20150215091832
constrID:1 dateTime: occurrenceOf constrID:4 dateTime:
taskID:2 20150331094453 taskID:56 20150215091833 occurrenceOf
userID:5 constrID:1 userID:5 constrID:4
sessionID:9 next userID:5
... taskID:2 endTask sessionID:1 next userID:5
taskID:56 clickButton
constrID:3 eventID:9999 constrID:4 eventID:3002
sessionID:9 next userID:5 sessionID:1 next userID:5
eventStatus:0 eventStatus:0
sessionID:9 eventCat:taskEv sessionID:1 eventCat:taskEv
Figure 2: Fragment of Graph Data Figure 3: Fragment of Graph Data
of a node would be represented by an edge and its value by
constructions and information relating to the system’s in- a literal-valued node. So, for example, the information that
terventions. For example, for event 24115, the value of the the taskID of event 23041 is 2 would be represented by an
number created, say 5; for event 23921, the feedback strat- taskID
edge 23041 −−−−−→ 2. The query examples in the next section
egy used by the system to generate this intervention, say assume this “classical” graph representation.
strategy 8; and for event 23923, the content of the message
displayed to the user, say “How many green tiles do you need
to make your pattern?” and whether this is a high-level in-
3. GRAPH QUERIES AND ANALYSES
terruption by the system or a low-level interruption that Because the sub-graph induced by edges labelled ‘next’ con-
the student can choose to view or not. Such information sists of a set of paths, the data readily lends itself to explo-
can be captured through additional edges outgoing from an ration using conjunctive regular path (CRP) queries [2]. A
value
event instance to a literal-valued node: 24115 −−−→ 5, 23921 CRP query, Q, consisting of n conjuncts is of the form
strategy message
−−−−−−→ 8, 23932 −−−−−− → “How many green tiles do you need (Z1 , . . . , Zm ) ← (X1 , R1 , Y1 ), . . . , (Xn , Rn , Yn )
level
to make your pattern?”, 23932 −−−→ “high”. Since graph data
where each Xi and Yi is a variable or a constant, each Zi is
models are semi-structured (and graph data therefore does
a variable that appears also in the right hand side of Q, and
not need to strictly conform to a single schema), this kind
each Ri is a regular expression over the set of edge labels.
of heterogeneity in the data is readily accommodated.
In this context, a regular expression, R, has the following
syntax:
Figure 3 similarly shows a fragment of FractionsLab inter-
action data, relating to the interactions of user 5 working R := | a | | (R1 .R2 ) | (R1 |R2 ) | R∗ | R+
on task 56 during session 1. The user makes one construc-
where denotes the empty string, a denotes an edge label,
tion during this task. We see events relating to the stu-
denotes the disjunction of all edge labels, and the operators
dent changing and ‘releasing’ a fraction. Following which
have their usual meaning. The answer to a CRP query on a
the system displays a message (in this case, it was a high-
graph G is obtained by finding for each 1 ≤ i ≤ n a binary
interruption message of encouragement “Great! Well Done”).
relation ri over the scheme (Xi , Yi ), where there is a tuple
(x, y) in ri if and only if there is a path from x to y in G
We see from Figures 2 and 3 that the sub-graph induced by
such that: x = Xi if Xi is a constant; y = Yi if Yi is a
edges labelled ‘next’ consists of a set of paths, one path for
constant; and the concatenation of the edge labels in the
each task undertaken by a specific user in a specific session.
path satisfies the regular expression Ri . The answer is then
The entire graph is a DAG (directed acyclic graph): there
given by forming the natural join of the binary relations
are no cycles induced by the edges labelled ‘next’ since each
r1 , . . . , rn and finally projecting on Z1 , . . . , Zm .
links an earlier indicator occurrence to a later one; while
the instances of EventType and other literal-valued nodes
To illustrate, the following CRP query returns pairs of events
can have only incoming edges. The entire graph is also a
x, y such that x is an intervention message shown to the user
bipartite graph, with the two parts comprising (i) the in-
by the system and y indicates that the user’s next action –
stances of Event, and (ii) the instances of EventType and
in eXpresser – was to create a number (note, variables in
the literal-valued nodes.
queries are distinguished by an initial question mark):
As a final observation, we note that Figures 1 – 3 adopt
a “property graph” notation (e.g. as used in the Neo4J (?X,?Y) <- (?X,occurrenceOf,interventionShown),
graph database, neo4j.com) in which nodes may have at- (?X,next,?Y),
tributes. In a “classical” graph data model, each attribute (?Y,occurrenceOf,numberCreated)
The result would contain pairs such as (23923,24115) from (?X,?Y,?Z) <- (?X,occurrenceOf,interventionGenerated),
Figure 2, demonstrating that there are indeed situations (?X,constrID,?C), (?X,next+,?Y),
where an intervention message displayed by the MiGen sys- (?Y,constrlID,?C), (?Y,occurrenceOf,?Z)
tem leads directly to the creation of a number by the student.
The following query returns pairs of events x, y such that The result would contain triples such as
that x is an intervention message shown to the user by the (23921, 23923, interventionShown),
system and y is the user’s next action; the type of y is also (23921, 24115, numberCreated),
returned, through the variable ?Z: (23921, 24136, numberUnlocked),
(23921, 24189, unlockedNumberChanged),
relating to construction 1 made by user 5 during session 9
(?X,?Y,?Z) <- (?X,occurrenceOf,interventionShown), for task 2 (two more events — 24136 and 24189 — relat-
(?X,next,?Y), ing to construction 1 have been assumed here, in addition
(?Y,occurrenceOf,?Z) to 23923 amd 24115 shown in Figure 2, for illustrative pur-
poses). The results would not contain (23921,33154,end-
The result would contain triples such as (23923,24115,num- Task), since event 33154 relates to construction 3.
berCreated) from Figure 2 and (344760,344761,clickButton)
from Figure 3, allowing researchers to see what types of To show more clearly the answers to the previous query in
events directly follow the display of an intervention mes- the form of possible event paths, we can use extended regular
sage. This would allow the confirmation or contradiction of path (ERP) queries [11], in which a regular expression can
researchers’ expectations regarding the immediate effect of be associated with a path variable and path variables can
intervention messages on students’ behaviours. appear in the left-hand-side of queries. Thus, for example,
the following query returns the possible paths from x to y:
Focussing for the rest of this section on the data in Figure 2,
the following query returns pairs of events x, y such that x is
(?X,?P,?Y,?Z) <-
any type of event and y indicates that the user’s next action
(?X,occurrenceOf,interventionGenerated),
was to unlock a number; the type of x is also returned,
(?X,constrID,?C), (?X,next+:?P,?Y),
through the variable ?Z:
(?Y,constrID,?C), (?Y,occurrenceOf,?Z)
(?X,?Y,?Z) <- (?X,occurrenceOf,?Z),
(?X,next,?Y), The result would contain answers such as
(?Y,occurrenceOf,numberUnlocked) (23921, [next], 23923, interventionShown),
(23921, [next, 23923, next], 24115, numberCreated),
(23921, [next, 23923, next, 24115, next], 24136, numberUn-
The result would allow researchers to see what types of
locked),
events immediately precede the unlocking of a number (i.e.
(23921, [next, 23923, next, 24115, next, 24136, next], 24189,
the creation of a variable). This would allow confirmation
unlockedNumberChanged).
of researchers’ expectations about the design of the MiGen
system’s intelligent support in guiding students towards gen-
The use of the regular expressions next and next+ in the
eralising their models by changing a fixed number to an ‘un-
previous queries matches precisely one edge labelled ‘next’,
locked’ one.
or any number of such edges (greater than or equal to 1),
respectively. However, for finer control and ranking of query
The following query returns pairs of events x, y such that
answers, it is possible to use approximate answering of CRP
that x is an intervention generated by the system and y is
and ERP queries (see [11, 17]), in which edit operations such
any subsequent event linked to x through a path comprising
as insertion, deletion or substitution of an edge label can be
one or more ‘next’ edges; the type of y is also returned,
applied to regular expressions.
through the variable ?Z:
For example, using the techniques described in [11, 17], the
(?X,?Y,?Z) <- (?X,occurrenceOf,interventionGenerated), user can chose to allow the insertion of the label ‘next’ into
(?X,next+,?Y), a regular expression, at an edit cost of 1. Submitting then
(?Y,occurrenceOf,?Z) this query:
The result would contain triples such as (23921, 23923, inter- (?X,?P,?Y,?Z) <-
ventionShown), (23921, 24115, numberCreated), ... (23921, (?X,occurrenceOf,interventionGenerated),
33154, endTask), allowing researchers to see what types of (?X,constrID,?C), APPROX(?X,next:?P,?Y),
events directly or indirectly follow the display of an interven- (?Y,constrID,?C), (?Y,occurrenceOf,?Z)
tion message by the system. This would allow the confirma-
tion or contradiction of researchers’ expectations regarding
the longer-term effect of intervention messages on students’ would return first exact answers, such as
behaviours. (23921, [next], 23923, interventionShown). The regular ex-
pression next in the conjunct APPROX(?X,next:?P,?Y) would
We can modify the query to retain only pairs x, y that relate then be automatically approximated to next.next, leading
to the same construction: to answers such as
Transition Matrix(Session 3)
(23921, [next, 23923, next], 24115, numberCreated) 6003 e s
6002 1001
at an edit distance of 1 from the original query. Following 6001 1002
this, the regular expression next.next would be automati- 5004 1003
cally approximated to next.next.next, leading to answers 5003 1004
such as 5002 1005
(23921, [next, 23923, next, 24115, next], 24136, numberUn-
locked) 5001 1006
at distance 2. This incremental return of paths of increas- 3009 1007
ing length can continue for as long as the user wishes, and 3008 1008
allows researchers to examine increasingly longer-term ef- 3007 1009
fects of intervention messages on students’ behaviours. It 3006 1010
3002 1011
would also be possible for users to specify from the outset a 1015 1014
minimum and maximum edit distance to be used in approx- 0.96342
0.00024384
imating and evaluating the query, for example to request
paths encompassing between 2 and 4 edges labelled ‘next’.
Figure 4: Transitions between Event Types
Queries based on evaluating regular expressions over a graph-
based representation of interaction data, such as those above,
can aid in the exploration of students’ behaviours as they are applied across a whole dataset, or focussing on partic-
undertaking tasks using ELEs and the effectiveness of the ular students, tasks or sessions;
intelligent support being provided by the ELE. The query
• nodes that have a high probability of being visited on a
processing techniques employed are based on incremental
randomly chosen shortest path between two randomly
query evaluation algorithms which run in polynomial time
chosen nodes have high betweenness centrality; deter-
with respect to the size of the database graph and the size
mining this measure for pairs of event type nodes (ig-
of the query and which return answers in order of increasing
noring the directionality of the ‘occurrenceOf’ edges)
edit distance [11]. A recent paper [19] gives details of an
would identify event types that play key mediating
implementation, which is based on the construction of an
roles between other event types.
automaton (NFA) for each query conjunct, the incremental
construction of a weighted product automaton from each
conjunct’s automaton and the data graph, and the use of We have already undertaken some ad hoc analyses of in-
a ranked join to combine answers being incrementally pro- teraction data arising from classroom sessions using ELEs.
duced from the evaluation of each conjunct. The paper also For example, Figure 4 shows the normalised incoming tran-
presents a performance study undertaken on two data sets sitions for a 1-hour classroom session involving 22 students
— lifelong learning data and metadata [17] and YAGO [22]. using MiGen (in the diagram, s denotes the ‘startTask’ and
The first of these has rather ‘linear’ data, similar to the in- e the ‘endTask’ event types). Event types with an adja-
teraction data discussed here, while the second has ‘bushier’ cent circle show transitions where this type of event occurs
connectivity. Query performance is generally better for the repeatedly in succession. The thickness of each arrow or
former than the latter, and the paper discusses several pos- circle indicates the value of the transition probability: the
sible approaches towards query optimisation. thicker the line, the higher the probability. Red (light grey)
is used for probabilities < 0.2 and black for probabilities
In addition to evaluating queries over the interaction data, ≥ 0.2. We can observe a black arrow 3007 → 1005, indicat-
by representing the data in the form of a graph it is possible ing transitions from events of type 3007 (detection by the
to apply graph structure analyses such as the following: system that the student has made an implausible building
block for this task) to events of type 1005 (modification of a
• path finding and clustering: this would be useful for rule by the student). Such an observation raises a hypoth-
determining patterns of interest across a whole dataset, esis for more detailed analysis or further student observa-
or focussing on particular students, tasks or sessions tion, namely: “does the construction of an incorrect building
c.f. [4]; block lead students to self-correct their rules?”. Developing
a better understanding of such complex interaction can lead
• average path length: this would be useful for determin- to improvement of the system. For this particular example,
ing the amount of student activity (i.e. the number of we designed a new prompt that suggests to students to first
indicator occurrences being generated per task) across consider the building block against the given task before
a whole dataset, or focussing on particular students, proceeding unnecessarily in correcting their rules. More ex-
tasks or sessions; amples of such ad hoc analyses are given in [14]. Represent-
ing the interaction data in graph form will allow more sys-
• graph diameter: to determine the greatest distance be- tematic, flexible and scalable application of graph-structure
tween any two nodes (which, due to the nature of the algorithms such as those identified above.
data, would be event type nodes); this would be an in-
dication the most long-running and/or most intensive
task(s); 4. CONCLUSIONS AND FUTURE WORK
We have presented a graph model for representing event-
• degree centrality: determining the in-degree centrality based interaction data arising from Exploratory Learning
of event type nodes would identify key event types oc- Environments, drawing on the data generated when students
curring in students’ interactions; this analysis could be undertake exploratory learning tasks with the eXpresser and
FractionsLab microworlds. Although developed in the con- networks: Generating high level hints based on
text of these systems, the model is a very general one and network community clustering. EDM, 2012.
can easily be used or extended to model similar data from [7] B. Grawemeyer and et al. Light-bulb moment?:
other ELEs. Towards adaptive presentation of feedback based on
students’ affective state. IUI, pages 400–404, 2015.
We have explored the possibilities that evaluating regular [8] S. Gutierrez-Santos, E. Geraniou, D. Pearce-Lazard,
path queries over this graph-based representation might pro- and A. Poulovassilis. Design of Teacher Assistance
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