=Paper=
{{Paper
|id=Vol-381/paper-3
|storemode=property
|title=Using Pattern Construction and Analysis in an Exploratory Learning Environment for Understanding Mathematical Generalisation: The Potential for Intelligent Support
|pdfUrl=https://ceur-ws.org/Vol-381/paper02.pdf
|volume=Vol-381
}}
==Using Pattern Construction and Analysis in an Exploratory Learning Environment for Understanding Mathematical Generalisation: The Potential for Intelligent Support==
https://ceur-ws.org/Vol-381/paper02.pdf
Using Pattern Construction and Analysis in an
Exploratory Learning Environment for
Understanding Mathematical Generalisation:
The Potential for Intelligent Support
Darren Pearce1, Eirini Geraniou2 , Manolis Mavrikis2,
Sergio Gutierrez-Santos1 , and Ken Kahn2
1
Birkbeck College {darrenp,sergut}@dcs.bbk.ac.uk
2
Institute of Education {e.geraniou,m.mavrikis,k.kahn}@ioe.ac.uk
Abstract. Few systems exist that support learners explicitly in the pro-
cess of learning mathematical generalisation. This paper presents the eX-
presser, part of a new system that seeks to address this issue by providing
the user with a microworld for the construction and analysis of general
patterns. The design includes the provision of sophisticated intelligent
support that assists learners and teachers throughout their various in-
teractions with the system. Given the open and exploratory nature of
the environment and the resultant freedom it affords, integrating such
intelligent support poses a significant research challenge. This paper de-
scribes the system in detail and discusses a variety of ways in which we
intend to meet our research goals for providing intelligent support.
Key words: microworld, generalisation, exploratory learning environ-
ments
1 Introduction
This paper presents the eXpresser, a mathematical microworld in which users
can build and analyse general patterns. Within this microworld, the user can
explore, experiment, and, through developing their own models, actively con-
struct their own understanding of the idea of mathematical generalisation. The
eXpresser is part of a larger system that is being designed and developed by
the MiGen Project.3 The aim of the project — to improve the mathematical
generalisation abilities of 11–14-year-old children — addresses a real need in sec-
ondary mathematics education since many students in this age-range perceive
generalisation and indeed algebra as difficult and unnecessary with no relevance
to the way in which they approach problem-solving (as discussed in [1]).
Despite this significant problem, a paucity of systems exist to help children
embrace algebra and understand its power. The MiGen Project is seeking to
3
See http://www.migen.org/ for more details. Funded by the TLRP, e-Learning
Phase-II; grant number: RES-139-25-0381
2 Pearce, Geraniou, Mavrikis, Gutierrez-Santos and Kahn
address this issue and, in the light of research that suggests that significant
pedagogic support is needed for the student during their interaction with such
environments (see [2]), the design of the system also includes two intelligent
components: the eGeneraliser which will model the user’s understanding of gen-
eralisation and scaffold their further exploration; and the eCollaborator which
will intelligently assist users in discussing their ideas and (mis)conceptions with
other users.
As important (and perhaps more importantly), the philosophy behind the de-
velopment of the system is not to replace teachers but to support them. As a re-
sult, all components of the system will provide appropriate potentially-intelligent
interfaces for the teacher to assist them in their goals for the lesson and to inte-
grate the system into the classroom context.
After a brief discussion of related work (Section 2), Section 3 presents the
current version of the system in detail as a basis for discussion in Section 4 of
the potential for intelligent support, both for the student and the teacher.
2 Related Work
Several learning environments have been developed and integrated in classroom
contexts over the last few years that attempt to help students in algebra and
problem-solving. However, most of them expect students to have a clear notion
of the concept of variables and an understanding of algebraic notation. There-
fore, they do not deal explicitly with the difficulties students face when they
are involved in generalisation (for a review of these difficulties see [1]) particu-
larly because they usually provide ready-made abstractions and representations
instead of allowing students to construct them themselves.
Closest to our work is research on mathematical microworlds [3, 4]. However,
the majority of these revolve around geometric concepts or are usually targeted
to students who already have a level of understanding of algebra. These tend
not to focus on algebra directly but on developing an understanding of various
representations such as tables and graphs (e.g. [5]).
One successful approach, in relation to generalisation from patterns enables
students to engage with models that they are constructing themselves. For ex-
ample, in Mathsticks [6], students work on patterns and regularities constructed
out of matchsticks using a subset of Logo commands in order to explore how
variables relate to each other and make relationships explicit using different rep-
resentational forms. Despite some successes, difficulties remain, and these tend
to coalesce around the need for significant pedagogic support from the teacher
to provide a bridge to algebraic symbolism and generalisation.
The aim of our project is to develop tools that provide assistance to learn-
ers and advice to teachers based on analyses of students’ interactions with the
system. The rationale behind this aim is that the freedom of exploration in
microwords entails the risk that the student may not ‘accept or notice the edu-
cator’s agenda’ [4] or that they do not always manage to take advantage of the
expressive environment that the microworld provides [7]. This is a serious issue
A Pattern-Based ELE for Mathematical Generalisation: The Potential for IS 3
in a classroom situation where it is difficult to attend to the interactions and the
understandings that students develop over the course of these interactions.
Despite the existence of intelligent tutoring systems that attempt to help
students with algebra (e.g., [8]), they are again targeted to problem-solving and
are suitable for students who are following a particular procedural method of
problem solving. The closest related work is the Motions microworld [7] devel-
oped in the early 1990s. Although it does not deal with algebra explicitly, there
was an attempt to incorporate aspects of intelligence in order to address ‘mis-
conceptions of basic concepts and of generalisation’ [7]. The design proposed for
the ‘intelligent microworld’ gives the student control to decide when, and what
for, to invoke the tutor asking questions such as ‘why’, ‘predict’, ‘generalise’ and
‘challenge-me’. However, the system was not developed further. Recently devel-
oped intelligent exploratory learning environments (e.g. [9, 10]) are still within
the realm of investigating specific concepts by providing a well-defined open
learning environment that usually allows exploration of a model or a simulation.
Such environments usually allow students to work with a pre-determined list of
variables, restricting their reasoning [11] rather than engaging learners in the
construction of their own models.
3 System Description
Earlier iterative development towards the eXpresser developed ShapeBuilder, a
prototype microworld for exploring generalisation. Although many of its features
were pivotal to our thinking, ultimately, the underlying engine was not applicable
to many typical generalisation tasks within the National Curriculum. This is
due to the fact that many such tasks make use of the notion of repetition within
patterns. Figure 1 shows two typical problems, the ‘footpath’ which is a repeated
tiling and the ‘pyramid’ which not only makes use of repetition but repetition
with changes since each layer increases in size relative to the previous layer.
(a) (b) (c) (d)
Fig. 1. Example patterns. (a) the footpath pattern with 2 (dark) blue squares; (b)
with 3 blue squares; (c) the pyramid pattern with 2 levels; and (d) with 3 levels.
When investigating such tasks, students would typically be asked to work out
its general structure so that they could describe other instances of the pattern.
In the case of the footpath, for example, they would be able to describe what
it would look like if it had 10 (dark) blue squares surrounded by (light) green
squares and, similarly, in the case of the pyramid, what it would look like if it
had 6 layers. In addition to this, students are also usually asked to investigate
4 Pearce, Geraniou, Mavrikis, Gutierrez-Santos and Kahn
questions such as the general rule for the number of green squares needed to
surround any given number of blue squares in the footpath pattern.
Following from these observations, the eXpresser provides an exploratory
environment that allows the user to construct their patterns freely and analyse
these constructions so as to obtain interesting general rules. Figure 2 shows the
current interface and briefly describes each of the available tools.
Cut/copy/paste to/from the clipboard. Delete.
Zoom in/out. Set the visibility and size of the grid.
Group/ungroup. Change/Select colour. Unpattern.
Create a row/column pattern. Demonstrate a pattern.
Fig. 2. The interface of the current version of the eXpresser annotated with descriptions
of the available tools.
3.1 Basic Patterns
The user is able to create a ‘unit pattern’ which looks like a single block. However,
inspecting its properties reveals that it is in fact something more complex as
illustrated in Figure 3a. The property box in this figure shows three numeric
attributes, each of which has an icon. The cogs icon specifies the element count
of this pattern. The cogs icon with the right arrow specifies how far to the right
each shape should be from its predecessor and, similarly, the cogs icon with the
arrow downwards specifies how far down a shape should be from its predecessor.
Initially, the element count of a pattern is 1 and the other attributes 0 but by
A Pattern-Based ELE for Mathematical Generalisation: The Potential for IS 5
setting these values appropriately, various basic patterns can be obtained. Some
of these are shown in Figure 3b–e.
3 3 3 3
1 0 1 2
0 1 1 1
(a) (b) (c) (d) (e)
Fig. 3. Basic patterns. (a) A ‘single block’. Editing its attributes can result in patterns
such as (b) a row; (c) a column; (d) a diagonal; or (e) an ‘expanded’ diagonal.
3.2 Using the Pattern Tools
Although interesting patterns can be created using such a basic mechanism, the
underlying engine is far more powerful since a pattern can be based on any shape
whether it is a pattern itself or a group. Currently the interface provides three
ways in which to access this generic facility: creation of a horizontal sequence,
creation of a vertical sequence and creation based on a ‘demonstration’ of the
first two elements. These first two elements implicitly indicate what happens
from one element of the pattern to the next. Using this information, the system
is able to create the pattern. For example, given two shapes with attributes
as shown in Figure 4a and Figure 4b, specifying 4 to the demonstrator tool
(Figure 4c) results in a triangle pattern (Figure 4d).4
3.3 Creating Dependencies
With the facilities described so far, the user is able to build complex patterns with
ease. However, there is also a need to make patterns depend on one another. For
example, consider the case where the user is constructing the footpath pattern. If
they decide to work row by row, they may first create a pattern of blue squares as
shown in Figure 5a. The number of green squares needed on that row is one more
than the number of blue squares. By naming the first pattern, this relationship
can be specified iconically as shown in Figure 5b. The other rows of the pattern
depend in a similar way on the number of blue squares as shown in Figure 5c.
4
Note that the two shapes used in this example also implicitly specify a movement
downwards from one element to the next. At present, this is not shown explicitly in
the attribute list so as to prevent user confusion.
6 Pearce, Geraniou, Mavrikis, Gutierrez-Santos and Kahn
(c)
(a) (b) (d)
Fig. 4. The demonstrator tool. (a) The first element in the pattern; (b) the second
element; (c) The number of elements to be in inferred pattern; (d) The resulting
pattern with 6 elements.
(a) (b) (c)
Fig. 5. Constructing the footpath pattern using dependencies.
3.4 Messing-up
Being able to relate attributes of shapes in this way is fundamental to the con-
struction process and through it, the power of generalisation becomes apparent to
the user.5 Figure 6a shows how the pattern (as constructed in Figure 5) changes
when the number of blue squares is changed to 5. As expected, the layout of the
green squares changes appropriately, thus demonstrating the generality of the
construction. In contrast, the construction in Figure 6b does not look correct
since the number of squares in its top and bottom rows were specified in abso-
lute terms when there were originally 3 blue squares. In the case of Figure 6c,
no aspects of the construction are general; all parts of the pattern were created
specific to when there were 3 blue squares.
For constructions that are not entirely general, changing the number of blues
in this example ‘breaks’ the pattern. This potential for ‘messing up’ [13] by
changing various parameters of the problem provides a simple yet powerful mech-
anism for the student to judge whether a pattern is general or not. It also provides
a pedagogic strategy for the teacher (or potentially the system) to challenge stu-
dents to construct a solution that is impervious to ‘messing up’ in this way. This
‘incentive to generalise’ [14] provides students with the opportunity to realise
5
In some of our earlier work [12], this was referred to as ‘building with n’.
A Pattern-Based ELE for Mathematical Generalisation: The Potential for IS 7
that there is an advantage to thinking in terms of abstract characteristics of the
task rather than specific numbers.
(a) (b) (c)
Fig. 6. Messing-up. (a) A general construction; (b) A partially-general construction;
(c) An entirely non-general construction.
3.5 Allocations
The features above provide the user with mechanisms for constructing general
patterns but no explicit way to analyse them so as to obtain general rules such
as the number of greens needed to surround the blues. This issue is addressed by
allowing the user to specify the number of squares of each colour that they need
for their construction. As a concrete illustration of this feature, Figure 7 shows
the step-by-step construction of the footpath interleaved with updating such
colour allocations. In this example, the user starts off by considering the instance
of the footpath pattern in which there are 3 blue squares. Since they know
that (currently) they need that many blue squares, they specify this explicitly
(Figure 7a). At this stage, since the number of green squares needed is unknown,
they leave this at its initial value of 0 (Figure 7b). Figure 7c shows their first step
in the construction process in which they create the pattern of blue squares.6
(d) (e)
(a)
(f ) (g)
(c)
(b) (h) (i)
Fig. 7. Using allocations for the footpath pattern. (a) Initial blue allocation; (b) Initial
green allocation; (c) Using the blue allocation for pattern construction; (d) Adding
the enclosing green squares pattern; (e) Increasing the allocation; (f ) Adding the top
row pattern; (g) Increasing the allocation to cover the top row; (h) Adding the bottom
row; (i) Increasing the allocation to cover the bottom row.
The user next adds the surrounding green pattern for the middle row. Since
the allocation for green is currently zero, these squares appear differently as
6
Note that the specification of the element count here is in terms of the blue allocation.
This differs from Figure 5a in which the element count was specified explicitly.
8 Pearce, Geraniou, Mavrikis, Gutierrez-Santos and Kahn
shown in Figure 7d. The user can correct this by specifying the number of green
squares that are now needed7 (Figure 7e). The remainder of the example contin-
ues similarly for the top row (Figure 7f-g) and the bottom row (Figure 7h-i). In
this way, the user iteratively refines the expression for the green allocation, in-
terleaving construction and analysis. Ultimately, the blue allocation now ‘drives’
the pattern in that changing its value results in a different instance of the foot-
path. Moreover, the green allocation is now the general rule for the number of
green squares needed to surround the blue squares.
This example shows a very rigorously interleaved process of construction
and analysis. The advantage here is that through this interleaving, the user
may gain a deeper understanding of their construction. However, this comes
at a cost in terms of requiring quite complex interaction with the task on two
different levels. In view of this, the system is being designed so that if the user,
teacher or task designer so desires, construction can proceed without a concern
for allocations. Then, once comfortable with how to create the general pattern,
the user can, in a subsequent phase, re-build their construction paying attention
to the allocations needed. So, rather than interleaving construction and analysis,
construction happens first followed by interleaved reconstruction and analysis.
In summary, just as ‘messing up’ provides the user with an incentive to
generalise, allocations provide the user with an incentive to analyse.
3.6 Multiple Solutions
One of the design criteria for the eXpresser and one closely related to the con-
structionist approach we have adopted is that the microworld should be expres-
sive enough so that any given pattern could be constructed in a variety of ways.
Such scope for alternatives allows the learner to realize how multiple valid so-
lutions to a problem lead to different — but equivalent — expressions. There is
then an incentive to develop intuitively some of the basic rules of algebra such
as commutativity and associativity as well as provocation to collaborate with
other children who have found correct but different solutions. Figure 8 shows
two further constructions of the footpath.
Fig. 8. Different constructions of the footpath pattern. Each row shows a different
construction step by step with the most recently added shape selected.
7
We are in the process of elaborating this feature so that excessive allocations are
indicated to the user appropriately.
A Pattern-Based ELE for Mathematical Generalisation: The Potential for IS 9
4 Discussion
As Section 3 explains, the system affords a variety of ways for users to interact
with tasks. These interactions between the system and the students are shown
diagrammatically in Figure 9 by the edges annotated with ➀ and ➁. Given the
inherently open and exploratory nature of these interactions, integrating intel-
ligent support within this process is a significant challenge (see [15]) but one
that holds great potential. Possibilities that we are actively exploring include
highlighting inconsistencies, automatically prompting the user to simplify their
constructions in various ways and providing them with alternative representa-
tions so as to assist their understanding of the tasks.
➄ Student A
➃ ➀
➂
➀
Teacher ➅ System
➁
➂
➃ ➁ Student B
➄
Teacher Student A Student B
Fig. 9. Information flow between the system, the teacher and the students.
Students will be able to collaborate both during and after the construction
and analysis processes (➂) and, through intelligent modelling of the collaborative
behaviour of the students, the system will be able to recommend effective pair-
ings/groupings of students. Furthermore, there is also significant scope within
the system for the effective use of dynamic variation of the collaborative affor-
dances of the interface [16], allowing users to co-construct their patterns while
reducing dominance and encouraging high quality collaboration.
The teacher will interact with the system (➃) in order to create, design and
deploy tasks and task sequences. They will also be able to affect the experiences
of the students directly (➄) by recommending next steps, intervening or allo-
cating further tasks. Most importantly, it will also be possible for the teacher to
obtain summary information across all their students (➅). At its simplest, this
can provide the teacher with an overall impression of class progress and, with
the use of appropriate representations, can facilitate attending first to those who
are most in need of help. Both these services can be greatly enhanced by the
appropriate integration of intelligent support. For example, a summary of class
progress could take into consideration all previous user experience and provide
a much more accurate view. Moreover, with such a facility in place, it is possible
to avoid situations in which students misunderstand certain concepts and leave a
10 Pearce, Geraniou, Mavrikis, Gutierrez-Santos and Kahn
lesson with a false sense of achievement. This, in combination with other aspects
of intelligent support, balances freedom and feedback, allowing students to play,
explore, make mistakes, see the immediate outcome of their actions and learn.
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