=Paper=
{{Paper
|id=Vol-381/paper-7
|storemode=property
|title=Challenges for Intelligent Support in Exploratory Learning: the Case of ShapeBuilder
|pdfUrl=https://ceur-ws.org/Vol-381/paper06.pdf
|volume=Vol-381
}}
==Challenges for Intelligent Support in Exploratory Learning: the Case of ShapeBuilder==
https://ceur-ws.org/Vol-381/paper06.pdf
Challenges for Intelligent Support in
Exploratory Learning: the case of ShapeBuilder?
Mihaela Cocea, Sergio Gutierrez-Santos and George D. Magoulas
London Knowledge Lab, Birkbeck College, University of London,
23-29 Emerald Street, London, WC1N 3QS, UK
{mihaela,sergut,gmagoulas}@dcs.bbk.ac.uk
Abstract. Exploratory learning environments give a lot of freedom to
learners to explore tasks on their own. However, although this can have
a positive effect on learning, the lack of structure makes it difficult to
provide intelligent support in such systems. Furthermore, the open na-
ture of these systems makes it harder to compare the support provided
by different systems. This papers describes a series of scenarios that
demonstrate these challenges in the context of an exploratory learning
environment for mathematical generalisation and proposes a formulation
that employs cases as a form of knowledge representation for modelling
this domain.
1 Introduction
Exploratory Learning Environments (ELEs) provide students with a lot of free-
dom to explore and play, rather than being constrained or very directed. In
ELEs, learners are encouraged to explore a broad set of possibilities, and con-
struct models within them1 . They arguably have a more positive impact on the
user’s learning than structured guided environments, due to their open nature
in line with constructivist principles [2]. However, their open nature makes it
difficult to design a system that supports the user, as the possible different ac-
tions are broad and mostly unstructured. On the other hand, an exploratory
environment without any support or guidance can actually hinder learning [3].
This paper describes several scenarios that can take place in an exploratory
learning environment for mathematical generalisation, called ShapeBuilder, and
explores alternatives for providing intelligent support through scenarios. It also
describes succintly a modelling strategy that is useful for the environment to
identify when these scenarios take place.
The paper is structured as follows. The next section provides some back-
ground of the specific microworld (in the sense of [4]) that has been the main
drive behind this work. Based on preliminary pilot studies in which it was de-
ployed, several possibilities for support have been identified. The scenarios for
?
Work funded by TLRP e-Learning Phase-II; RES-139-25-0381.
1
The term ELE is sometimes used for referring to systems used in simulation-based
learning [1], as they usually allow a limited degree of exploration.
2 M. Cocea, S. Gutierrez-Santos and G.D. Magoulas
support in such an environment are described in Section 3. A knowledge repre-
sentation scheme that employs case-based reasoning is presented in Section 4 as
a a first step toward developing an engine that would underpin the provision of
intelligent support in our microword. Finally, Section 5 closes the paper, drawing
the final conclusions.
2 Background
ShapeBuilder [5] is an exploratory environment that is built to support the
development of algebraic thinking, aimed at learners between 11 and 14 years old.
From the many mathematical skills that children learn in a typical curriculum,
the ability to express general concepts through algebra is arguably the most
difficult to grasp. However, it is also one of the most important because the
ability to generalise is at the core of scientific enquiry, and the use of algebra to
express general concepts is basic in most branches of mathematics. Therefore,
there is a need for systems that support students in the acquisition of the basic
concepts of generalisation thinking, scaffolding their learning towards algebra.
ShapeBuilder gives the learners the opportunity of creating shapes (usually
rectangles, although other shapes are possible) on a square grid. The shapes
are defined by several attributes (e.g. “length”) that the learners can set and/or
modify. The interesting features of ShapeBuilder allow users to select specific at-
tributes from these shapes and use them to create other shapes.2 The attributes
of the shapes are represented by means of icon-variables, that are graphical rep-
resentations of attributes of shapes. Their graphical nature makes them more
intuitive and arguably easier to grasp for young students, that have been re-
ported to have difficulties understanding the use of letters to represent general
concepts [6]. Icon-variables can be combined to form algebraic expressions (this
will be discussed in detail in the next section, see for example Scenario 5).
Several pilot studies have been conducted in classrooms to explore how stu-
dents undertaking a common activity in mathematical curricula known as pond-
tiling work with ShapeBuilder. In this activity, students are given a pond (usaully
rectangular) and are asked to surround it with square tiles (see Figure 1a). Such
a task helps learners develop the ability to predict the number of tiles that are
necessary to surround any pond, given its dimensions: this is, in essence, an
algebraic expression. There are several ways of surrounding a pond of any size
or shape, and each one leads to a different (yet equivalent) algebraic expression.
For example, the number of tiles needed to surround a rectangular pond can be
generally expressed as “twice the length, plus twice two more than the breadth”
(i.e. 2L + 2(B + 2)) or “twice two more than the length, plus twice the breadth”
(i.e. 2(L + 2) + 2B); many others can be developed. Given these different ar-
rangements, students can be prompted to have interesting discussions about the
equivalence of the expressions derived from each viewpoint.
2
The procedure, as well as a full description of system features, is described in [5].
Challenges for Intelligent Support in Exploratory Learning 3
3 Scenarios for Intelligent Support in ShapeBuilder
There are many open problems with regard to ELEs. This section looks at some
of the challenges relevant for this workshop through user scenarios from Shape-
Builder.
3.1 Balance freedom with control
Scenario 1. In this scenario a student is working on a model that the system
identifies as being far way from a typical or “desirable” construction according to
some knowledge base that it stores. This may happen as a result of the student:
(a) coming up with an “unusual/uncommon” solution that is not stored in the
knowledge base of the system, or (b) acting very diferently from the typical path
of conducting the task (see Figure 1). Two posible strategies to follow would be:
(i) letting the learner continue with the exploration of the task, or (ii) guiding
the learner to a “desirable solution”. Unfortunately, it is unclear what would be
the best way to provide support here, as it depends on the context. In certain
instances of case (a), if action (ii) is selected, the learner would be provided a
support that is not needed. On the other hand, applying action (i) in case (b)
would result in the student needing a support that is not provided. Neither
situation can be expected to end in a better learning experience.
A possible solution for intelligent support here would be that the system
informs the teacher when this situation occurs in order to let him/her choose a
course of action, e.g. turn off feedback in situation (a) (at least for a while until
the current context becomes clearer) and let the learner explore further, or let
the system guide the student in situation (b).
Scenario 2. In this scenario, a learner initially started the construction of a
model but then suddenly moved into a totally irrelevant activity; the system
has detected this “abnormal behaviour”. Should the support take the form of a
system intervention that would prompt the learner to come back to the relevant
task, and after how long should this intervention occur? Possible ways to provide
support here are the system to: (a) intervene automatically when a time thresh-
old/interval is reached; (b) inform the teacher about this particular behaviour
and suggest an intervention, or leaves the teacher to decide the next action.
Fig. 1. Scenario 1: (a) correct innovative strategy; (b) “innefficent” and non-general
and incomplete strategy.
4 M. Cocea, S. Gutierrez-Santos and G.D. Magoulas
3.2 The value of correct/incorrect actions
Making mistakes and observing the results of “wrong actions” is considered quite
useful in exploratory learning. It encourages reflection and self-explanation [7].
In ELEs, the learner should have the freedom to explore without continuous dis-
tractions by prompts at each wrong action. However, learners do need guidance
when they are stuck or their actions are not leading to a sensible outcome. One
way to do this is to make use of the consequences of their actions “in the long
run”. What appears to be a wrong action could be caused by misconceptions,
slips or migrating bugs [8] and some of them would require immediate feed-
back (e.g. slips, random errors) or delayed feedback (e.g. misconceptions). The
following scenarios attempt to provide some examples for this type of situation.
Scenario 3. Our preliminary studies have shown that learners often start with
non-general strategies like surrounding the pond using 1 by 1 tiles or in another
non-systematic manner as illustrated in Figure 1b. Although this might be a cor-
rect way for surrounding a pond, it will not lead to generality, i.e. the solution
provided cannot be generalised to other similar cases. Intervention at this point
may be helpful especially if it points out the lack of generality of the approach.
An effective strategy from the literature of Dynamic Geometry Environments is
“messing up” [9]. This involves changing something in the figure that demon-
strates that the student’s construction, although it may look correct, does not
follow the constraints that would make it general enough in this environment.
For example, in our case the system or the teacher could change the size of the
pond. “Messing-up” could also be used as a way to support the learner by check-
ing the correctness of a solution that the learner believes is general. This process
can be triggered by the system, or can be led by the user (e.g. in a collaborative
scenario, in which a learner tests the generality of solutions produced by their
peers).
Scenario 4. For this scenario, consider the tilings illustrated in Figure 2. They
are both correct but different, which shows that the way to approach a task may
vary depending on personal preferences. One may argue that the surrounding in
Figure 2b is not exactly right, but it would be difficult to argue that it is wrong.
Situations like this can be handled with constraints on the position of shapes
and thus both ways of surrounding the pond could be identified as correct.
Fig. 2. Scenario 4 - “I strategy”: (a) shapes tied together; (b) shapes at a distance.
Challenges for Intelligent Support in Exploratory Learning 5
Scenario 5. Some learners construct the general expression for surrounding the
pond in the minimal form (Figure 3a) and other do not simplify their expressions
(Figure 3b). Although the minimal form is “desirable”, the non-simplified ones
are still correct. Depending on the goals of the task, the “expanded” expressions
could be acceptable or not.
3.3 The timing of feedback
Feedback on correctness of solution is not enough for ELEs, as support is often
needed during an activity, i.e. in ShapeBuilder during model construction not
just when construction is complete.
Scenario 6. Here the learner is working on the task and has constructed a model
that is partially correct. She makes several changes but still does not get it right.
Should an intervention occur at this point or maybe earlier/later? Should the
learner be left to explore the task further? This scenario is similar to Scenario 1,
except that the focus now is on the exact timing of an intervention should it be
needed, as opposed to the decision of whether or not to intervene as discussed
in Scenario 1. Possible strategies for support in this case could be: (a) a help
button that the learner could use (it has been reported that this might lead to
other problems such as help abuse [10] or gaming the system [11]); (b) a “what
others did next” button that shows what other learners did in a similar situation
(this might require tools to allow teachers to validate learners choices in advance
so that the next steps displayed are beneficial for learning). As certain studies
show that some learners tend to avoid system’s help [12], it would be interesting
to investigate how the “what others did next button” is used. Help on request is
a typical example of this scenario as it allows learners to request support when
they feel stuck.
Scenario 7. In this scenario support during model construction is needed depend-
ing on the stage within the task. The support can take the form of examples,
which could be partial or complete solution depending on the context. They
could also be “standard examples”, i.e. pre-encoded in the system, or models
developed by peers which have been previously validated by the teacher or by
peer learners.
Fig. 3. Scenario 5, “I strategy”: (a) simplified expression; (b) “expanded” expression.
6 M. Cocea, S. Gutierrez-Santos and G.D. Magoulas
3.4 Teacher support
Assisting teachers in developing teaching strategies, informing them of students
who are in difficulty, as well as the appearance of common misconceptions or
problematic strategies is another form of support needed in ELEs like Shape-
Builder.
As it is impossible for a teacher to attend to all students in a class, general
statistics of the whole class are particularly beneficial in identifying outliers:
pupils that are doing very well or very poorly. Knowing the overall status of the
class with respect to the task/learning goals makes it possible for the teacher to
give tasks to groups of students that are appropriate to their level or abilities,
or even challenge them.
At the same time, knowing what a student has been doing helps the teacher
to support the student. To facilitate this, the following information could be
provided to the teacher: (a) fast replays of the most recent interactions of the
student with the system; (b) screenshots taken at certain times during the task
that create a “visual diary” of some important steps of the construction process
followed by a particular student or a “scrapbook” for a student that experi-
ments with alternative solution/strategies of approaching a task ; (c) summary
statistics like current stage within the task, number of help requests, etc.
3.5 Support for collaboration
While the focus so far was on individual learning, it is acknowledged that effective
learning benefits from recognising the importance of participation in communi-
ties of practice. There is however a particular challenge when designing for col-
laborative learning in mathematics. This stems from the difficulty of identifying
engaging things for students to talk about as abstract concepts like generalisa-
tion are not necessarily intrinsically motivating. The challenge, therefore, is to
support collaboration between students around models produced by their com-
munity, help them understand what other students in the community are doing
and identify others working on the same or similar models.
Group formation is particularly important in this scenario as collaboration is
more beneficial when the learners are paired up according to educational criteria
like: (a) if the goal is to help a learner that has difficulties, the other member
of the pair should be more advanced on the same task and follow the same or a
similar strategy; (b) if the goal is to point out that the same task can be solved
with different approaches, the pairing should be done with learners that have
different models/expressions. Models’ similarity can be measured by comparing
their characteristics and the strategies used for their construction (this will be
discussed in the next section).
4 Towards Personalised Feedback and Support
Research into personalised feedback and support has primarily focused on the
learner’s knowledge “level”. While this might work well in structured learning
Challenges for Intelligent Support in Exploratory Learning 7
environments that adopt the role of an expert tutor presenting concepts grad-
ually and use questions and quizzes to assess students knowledge, it will not
suffice for our purposes with its more conceptual focus. In ShapeBuilder, the
problem is to find ways to support the process of the construction of models by
exploiting information collected during individual model construction, identify
similarities among students as expressed by the characteristics of their models
and the strategies used for their construction. Furthermore, we need to recog-
nise patterns in student behaviour during the various stages of the learning
process. This process is exploratory and investigative and leads to progressive
construction of structural knowledge. This information can then be used to pro-
vide support and facilitate collaboration with peers.
To this end we present an approach to personalised feedback that is based on
recent student actions rather than simply on their knowledge levels, and provide
different types of feedback during this progressive provision; for example, advice
on the quality of learners’ constructions, possible next actions, use of appropri-
ate construction components, and possible alternative construction strategies.
The ultimate aim is to create an intelligent support system that will foster the
learner’s articulation of general patterns and relationships, suggest exploration of
special cases, critique model solutions and seek justification for students’ models.
Typically, within a task or activity there are several sub-tasks, and the activ-
ity is sequenced within the system so as to know at any time the current context.
Attributes and relations are stored in cases, which represent characteristic com-
ponents of a learner’s construction. They carry different relevance depending on
the context, which in ShapeBuilder corresponds to different stages of the con-
structivist learning process that learners go through as they explore the various
sub-tasks within a learning activity. Through comparison of strategies the sys-
tem can identify how far a learner is from desired or alternative construction
strategies (this could be used to deal with situations like the ones described in
Scenarios 1, 2, 3 and 4), whether the construction includes the right compo-
nents, or the similarity between the models of different learners (this could be
used when informing decisions about collaboration. Comparisons are based on
defining similarity measures as described below.
4.1 Representation using case-based reasoning
In this subsection we present a formulation to represent attributes, relations and
strategies based on case-based reasoning (CBR).
Definition 1. A case is defined as Ci = {Fi , RAi , RCi }, where Ci represents
the case and Fi is a set of attributes. RAi is a set of relations between attributes
and RCi is a set of relations between Ci and other cases respectively.
Definition 2. The set of attributes is defined as Fi = {αi1 , αi2 , . . . , αiN }.
The set Fi includes three types of attributes: (a) numeric, (b) variables and
(c) binary. Variables refer to different string values that an attribute can take,
8 M. Cocea, S. Gutierrez-Santos and G.D. Magoulas
Table 1. The set of attributes (Fi ) of a case.
Category Name Label Possible Values
Shape Shape type αi1 Rectangle(/L-Shape/T-Shape)
Dimensions Width type αi2 constant (c)/variable (v)/icon variable (iv)/numeric
of shape expression (n exp)/expression with iv(s) (iv exp)
.. .. ..
. . .
Thickness type αiv c /v /iv /n exp /iv exp
Width value αiv+1 numeric value
.. .. ..
. . .
Thickness value αiw c /v /iv /n exp /iv exp
Part of P artOf S1 αiw+1 1
Strategy P artOf S2 αiw+2 0
P artOf Sr αiN 0
and binary attributes indicate whether a case can be considered in formulating
a particular strategy (1) or not (0). The set of attributes of a generic case for
ShapeBuilder is presented in Table 1. The first v attributes (αij , j = 1, v) are
variables, the ones from v + 1 to w are numeric (αij , j = v + 1, w) and the rest
are binary (αij , j = w + 1, N ).
Definition 3. The set of relations between attributes of the current case and
attributes of other cases (as well as attributes of the same case) is represented
as RAi = {RAi1 , RAi2 , . . . , RAiM }, where at least one of the attributes in each
relation RAim , ∀m = 1, M , is from the set of attributes of the current case Fi .
Two types of binary relations are used: (a) a dependency relation (Dis ) is
defined as (αik , αjl ) ∈ Dis ⇔ αik = DEP (αjl ), where DEP : αik → αjl for
attributes αik and αjl that are variables of cases i and j (where i = j or i 6= j),
and means that αik depends on (is built upon) αjl (if i = j, k 6= l is a condition
as to avoid circular dependencies) (e.g. the width type of a case is built upon
the height type of the same case; the width type of a case is built upon the
width type of another case, an so on); (b) a value relation (Vis ) is defined as
(αik , αjl ) ∈ Vis ⇔ αik = f (αjl ), where αik and αjl are numeric attributes and
f is a function and could have different forms depending on context (e.g. the
height of a shape is two times its width; the width of a shape is three times the
height of another shape, etc.).
Definition 4. The set of relations between cases is represented as RCi = {RCi1 ,
RCi2 , . . . , RCiP }, where one of the cases in each relation RCij , ∀j = 1, P is the
current case (Ci ).
Two time-relations are used: (a) P rev relation indicates the previous case
with respect to the current case: (Ci , Cj ) ∈ P rev if t (Cj ) < t (Ci ) and (b) N ext
Challenges for Intelligent Support in Exploratory Learning 9
relation indicates the next case with respect to the current case: (Ci , Ck ) ∈
N ext if t (Ci ) < t (Ck ). Each case includes at most one of each of these two
relations (p ≤ 2).
Definition 5. A strategy is defined as Su = {Nu (C), Nu (RA), Nu (RC)}, u =
1, r , where Nu (Ci ) is a set of cases, Nu (RAi ) is a set of relations between
attributes of cases and Nu (RCi ) is a set of relations between cases.
Four different similarity measures are used for comparing cases: Eucledian
distance for numeric attributes, Hamming distance for the set of variable at-
tributes, and a Jaccard index for the relation between attributes and the relation
between cases. In order to identify the closest strategy to the one employed by
a learner, the four distances are combined for each of the compared strategies.
4.2 Identifying similarities in learner’s strategies and constructions
Identifying similar strategies can provide insight on the behaviour of learners
using the system and initiate different types of feedback and support depending
on the situation. This can be eminently suitable when dealing with scenarios
such as Scenario 1, 2, 3 and 4. This idea can be extended to identify how similar
the models of different learners are in order to form collaborations among peers.
When the learner’s construction is equally similar to two strategies, the fol-
lowing options could be adopted: (a) present the learner with the two options
and let him/her choose one of the two (an approach that appears more suitable
for advanced learners than for novices); (b) automatically suggest one of the two
in a systematic way, e.g. present the one that occurs more/less often with other
learners; (c) inform the teacher about the learner’s trajectory and the frequency
of strategies and let him/her decide between the two.
5 Conclusions
Given the open nature of exploratory learning environments, it is difficult to
make fair comparisons between different systems, and between the approaches
taken for providing intelligent support in them. The goal of this paper is to pro-
vide a range of possible scenarios that can take place in a particular exploratory
learning environment for mathematical generalisation, in order to provide some
common ground on which different techniques for intelligent support can be
assessed.
Although most of the examples have been focused on this system, called
ShapeBuilder, an effort has been made to extrapolate the situations, so they can
be applied to other exploratory environments, like MoPiX [13] or SketchPad [14].
The scenarios cover different aspects, such as the difficulty of asserting what is
right or wrong in an exploratory environment, the analysis of the actions of the
students and the possibilities for collaboration. For each scenario, the different
ways in which a learner can be supported are depicted.
10 M. Cocea, S. Gutierrez-Santos and G.D. Magoulas
The paper also described a case-based reasoning formulation for the repre-
sentation of learner behaviour. This approach defines metrics that allow different
types of comparisons to be made. These can be further used to inform decision-
making for personalising the feedback and the support provided.
References
1. Swaak, J., van Joolingen, W.R. and de Jong, T.: Supporting simulation-based learn-
ing: the effects of model progression and assignments on definitional and intuitive
knowledge. Learning and Instruction, 8, 235–253 (1998).
2. Piaget, Jean.: The Psychology of Intelligence. New York: Routledge (1950).
3. Kirschner, P., Sweller, J. and Clark, R.E.: Why minimal guidance during instruc-
tion does not work: An analysis of the failure of constructivist, discovery, problem-
based, experiential and inquiry-based teaching. Educational Psychologist, 41(2),
75–86 (2006).
4. Balacheff, N. and Kaput, J.: Computer-based Learning Environments in Mathemat-
ics. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick and C. Laborde (Eds.)
International Handbook on Mathematics Education, chapter 13. Kluwer (1996).
5. Gutierrez, S., Mavrikis, M., Pearce, D.: A Learning Environment for Promoting
Structured Algebraic Thinking in Children. IEEE Int. Conf. on Advanced Learning
Technologies, 74-76, (2008).
6. Kuchemann, D.: Algebra. In Hart, K. M. (Ed.) Children’s Understanding of Math-
ematics, Antony Rowe Publishing Services, 102–119 (1981).
7. Conati C., VanLehn K.: Toward Computer-Based Support of Meta-Cognitive Skills:
a Computational Framework to Coach Self-Explanation. International Journal of
Artificial Intelligence in Education, 11, 389–415 (2000).
8. VanLehn, K.: Mind Bugs: The Origins of Procedural Misconceptions. MIT Press
(1990).
9. Healy, L., Hoelzl, R., Hoyles, C., Noss, R.: Messing Up. Micromath, 10(1), 14–16
(1994).
10. Wood, H., Wood, D.: Help Seeking, Learning, and Contingent Tutoring. Computers
and Education, 33, 153-159 (1999)
11. Baker, R.S., Corbett, A.T., Koedinger, K.R., Wagner, A.Z.: Off-Task Behavior in
the Cognitive Tutor Classroom: When Students Game the System. Proceedings of
ACM CHI 2004: Computer-Human Interaction, 383–390 (2004).
12. Aleven, V., McLaren,B.M., Roll, I., Koedinger, K.R.: Toward tutoring help seek-
ing: Applying cognitive modeling to meta-cognitive skills. Proceedings of the 7th
International Conference on Intelligent Tutoring Systems, 227–239 (2004).
13. Part of the ReMath project. Available at http://www.lkl.ac.uk/mopix/ (last ac-
cessed: July 2008).
14. Jackiw, N. and Finzer, W.: The Geometer’s Sketchpad: Programming by Geometry.
In A. Cypher (Ed.), Watch What I Do: Programming by Demonstration, 293–308
(1993).