Change in students’ conceptual
understanding of vectors using an
interactive media tool for Model-based
Reasoning
A Doctoral Thesis Proposal Document
November 2016
by
Karnam DurgaPrasad
advised by the
Thesis Advisory Committee
Dr. Aniket Sule (Guide)
Dr. Sanjay Chandrasekharan
Dr. K Subramaniam
HBCSE, TIFR Mumbai
[This is a intial draft proposal document submitted for review. Revisions based on
suggestions from the members of HBCSE shall be suitably incorporated in the final plan of
study]
TABLE OF CONTENTS
ABSTRACT
3
INTRODUCTION
Model Based Reasoning (MBR) in Science and Science Education.
Model-based reasoning in Trigonometry
4
4
4
CURRENT STATE OF AFFAIRS
6
Textbook Analysis
Teacher Interviews
Student Pretest and Interviews
6
8
8
NATURE OF THE PROBLEM
9
Integration of Mathematical Concept
Vector modelling applications in Physics
Role of Algorithmic Problem Solving
INTERACTIVE TOOL FOR MBR
Designing to create an interactive (manipulative) space
Designing to create a modelling space
Tool Description (first version)
PILOT STUDY
9
10
13
13
14
15
16
18
Data Collection
Data Analysis
Key findings
Case-studies
Case-1 (S2)
Case-2 (S5)
Conclusions from the pilot study
18
18
19
21
21
24
29
ACADEMIC PLAN
Research Questions
The Study with Timeline
Expected Outcome of the Research
29
29
30
32
REFERENCES
32
2
ABSTRACT
Reasoning about the structure and behaviour of physical phenomena using abstract
and concrete models (model-based reasoning, MBR) is a key thinking skill in science and
engineering practice. One of the key areas - MBR is introduced in the high school
curriculum, particularly the use of abstract models - is applications of trigonometry, such as
calculating heights and distances. In India, high school and pre-college (9-12 Grades)
trigonometry curricula include three broad MBR cases: i) heights and distances (ratios in
right triangles), ii) resolution and addition of vector quantities (projections in a unit circle to
give the rectangular components), and iii) periodic systems (represented as sinusoidal
functions).
Students find trigonometry and its MBR applications difficult to understand, possibly
because reasoning in this domain requires handling cognitive (internal/abstract) operations
and symbolic (external/concrete) operations simultaneously, in different and complex ways,
across these three MBR cases. A particular source of difficulty is the relationships between
these trigonometric operations, which are not clear across the three cases. Further, in the
existing curricula, trigonometric and other concepts related to vectors are scattered across 4
textbooks, and students find it hard to integrate these scattered concepts without very little
scope to perform model based reasoning.
This research proposal seeks to explore ways of supporting model-based reasoning in
pre-college students, particularly using trigonometry. As a first step in this direction, a
textbook analysis was done to characterise the use of vectors and related concepts in physics,
in relation to trigonometric applications. Teachers were then interviewed to understand
difficulties in teaching and learning of vectors. Based on the analysis and interviews, an
interactive system was developed to support students' understanding of vector operations and
the role vectors play in model-based reasoning. The design of the interactive system focused
on vector resolution and addition, a key application supporting MBR. The learning
possibilities provided by this system was then tested using a qualitative study, where grade 11
students' conceptual understanding of vectors was examined before, during, and after
working with the interactive system. The results indicate that the interactive system provides
useful support to understand model-based reasoning using vectors. Building on these results,
this proposal seeks to develop an interactive media-based research program to support
model-based reasoning using vectors.
Keywords​: model-based reasoning, vectors, trigonometry, embodied cognition, new media
3
1.
INTRODUCTION
1.1.
Model Based Reasoning (MBR) in Science and Science Education.
Science goes through many changes, in concepts, models, theories and methods.
(Kuhn, 1996) describes this historical process as a cycle of normal science followed by a
crisis, and a revolutionary paradigm change. This description provides a way of
understanding the structure of the change, but does not provide any insight into the discovery
processes, through which the changes in science emerge. ​(Magnani, Nersessian, & Thagard,
2012)​, taking a cognitive-historical perspective, develops an account of the discovery
process, suggesting that model-based reasoning (MBR) is the central cognitive process
underlying scientific discovery. ​(N. Nersessian, 1984) provides details of the MBR process,
by analysing historical cases such the Maxwell’s vortex-idle wheel model. Taking this view
further, ​(Hestenes, 2010a) argues for a modeling theory for science and math education, as
models and modeling are at the core of scientific knowledge and practice, and models are a
basic unit to construct coherent and structured knowledge in science, mathematics and
engineering. Reasoning using models is thus a central thinking skill in these fields.
According to ​(Gilbert, 2004)​, models “function as a bridge between scientific theory
and the world-as-experienced (‘reality’). They can be simplified depictions of a
reality-as-observed, produced for specific purposes, to which the abstractions of theory are
then applied. They can also be idealisations of a possible reality, based on the abstractions of
theory, produced so that comparisons with reality-as-observed can be made". ​(Hestenes,
2010a) defines a scientific model as a representation of the structure (set of relations between
objects) in a physical system or a process. This coherent structure has a central role in his
modeling theory.
Theories of MBR ​(Nancy J. Nersessian, 2002; N. J. Nersessian, 2010) draw on
cognitive science ideas such as mental modeling ​(Johnson-Laird, 1983) and metaphors ​(G.
Lakoff, 1997; G. Lakoff & Núñez, 2003)​. Model-based reasoning is thus closely connected to
key aspects of cognition, and this makes it a very effective and crucial framework while
examining the problem of science and math education ​(Vosniadou, 1996)​. ​(Ileana Maria
Greca & Moreira, 2000; Hestenes, 2010a) have systematically argued that theories of
modeling and model-based reasoning ​(Nancy J. Nersessian, 2002; N. J. Nersessian, 2010)
could provide a strong theoretical basis to address the problem of science learning, as such
theories build on an understanding of human cognition.
The above short review suggests that model-based reasoning is a critical thinking skill
in science, and science education intervention that support the development of this skill
would contribute significantly to helping students understand and practice science.
1.2.
Model-based reasoning in Trigonometry
Trigonometry is one of the key mathematical topics where students learn model-based
reasoning (MBR) in the later high school curricula. It is also a topic with wide applications in
4
advanced mathematics, physics and engineering. However, research studies ​(Gur, 2009;
Jackson, 1910; Orhun, 2004; Yusha’u, 2013) report that teachers and students find
trigonometry a hard concept to teach and learn. ​(Byers, 2010) reports a detailed study of
trigonometric representations in the Canadian curricula, listing potential sources of
difficulties for students, particularly in the transition from the secondary school to college
mathematics. ​(Gur, 2009) also identifies these problem areas, and suggests that the
difficulties come from the complex nature of trigonometric symbols. In India, trigonometry is
introduced to students during the later parts of high school, as part of Mathematics, along
with many applications across Physics. There are broadly three kinds of model-based
reasoning applications of trigonometry in the higher secondary curricula in Indian Schools.
These cases which follow the levels of understanding in geometry proposed by ​(Rebello, Cui,
Bennett, & Zollman, 2007; Van Hiele, 1986)​ are:
I. Heights and distances: Here a static real world scenario is modelled by a trigonometric
ratios in a right triangle. Trigonometric ratios are used to link the angles (of
observation) and the lengths of the triangle (heights and distances).
II. Resolution and Addition of Vectors: In 11​th and 12​th grade physics, vector operations
involving trigonometry are used in modelling dynamic physical phenomenon with
complex network of concepts relating various physical quantities (displacement,
velocity, acceleration, force, momentum, torque, angular velocity etc).
III. Sinusoidal Systems: Trigonometric ratios (as functions) are used to model phenomena
with periodically changing physical quantities in certain (dynamic) systems. In physical
systems (like a spring mass system or a simple pendulum), chemical and electrical
systems, periodically varying physical quantities can be modelled using a sinusoidal
curve.
We examined the way trigonometry is discussed in class 9-12 textbooks. In this
investigation, we plan to focus on model-based reasoning in the case of vector resolution and
addition, which is a very key mathematical modelling tool. Difficulty in handling vectors
leads to problems in handling Newtonian Dynamics ​(White, 1983)​. ​(Byers, 2010) attributes
this difficulty to unfamiliarity of vector operations. ​(Doerr, 1996, 1997) reports an empirical
study showing how model building (in contrast to model exploring) helps in learning the
concept of force, integrating vectors and trigonometry.
The key assumption of our study was that students have trouble following the wide
range of applications of vector concepts, particularly modeling real world cases, due to a lack
of understanding of the process of model based reasoning. Students at high school and
pre-college levels in India have very little scope to do model based reasoning (see discussion
in Section 2). Resolution and addition of vectors using trigonometry, with its wide
applications in physics, provide a rich context to explore how model-based reasoning could
be supported in students, and how this could change students’ conceptual understanding of
trigonometry, and its applications in physics (Section 3).
We discuss how an interactive-media based intervention (discussed in Section 4)
5
could be used to scaffold students' understanding of MBR, in the existing curricular setting.
We present an exploratory study, tracking the changes in students’ understanding while
during and after working with this intervention. Based on the results of this study (section 5),
the plan for further studies are proposed (Section 6).
2.
CURRENT STATE OF AFFAIRS
2.1.
Textbook Analysis
We analysed the textbooks in Maharashtra state board schools, to understand the manner in
which concepts of vectors are covered. Since the topics related to vectors are spread across
mathematics and physics textbooks (grades 9-11), we were interested in documenting the
missing conceptual links, both within a text book and between text books. Figure 1 shows a
concept map of how topics are covered and applied in the physics curriculum. These include
identification of a vector quantity, its definition and representation as a geometrical entity
with magnitude and direction (predominantly in two- dimensional space). But in practice,
algebraic representation using rectangular (x-y-z) components is also prominent. Further,
addition of vectors is introduced geometrically using the Triangle Law and Parallelogram
Law of vector addition. The Triangle law is just stated, and no connection is made to
properties of vectors. Equivalent textbooks published by NCERT are careful defining vectors
as mathematical entities, following triangle law of addition. In the case of Maharashtra board,
the conceptual gap thus gets carried over to the Parallelogram and Polygon Laws, which are
proven based on the Triangle Law. Addition of vectors is thus not properly scaffolded in
these text books.
Different representations of the same vector as a geometrical entity (an arrow mark ↗
with magnitude and direction) and as an algebraic entity (the rectangular components form)
are interrelated using the operation called Resolution of vectors. The textbook does not
clarify how both these representations denote the same vector. This could lead to students
treating a vector under geometrical and algebraic descriptions differently, without a unified
perspective. Similarly adding two vectors geometrically (using triangle and parallelogram
laws) and algebraically (adding rectangular components) may be perceived as two entirely
different operations, and hence need scaffolding to understand how they lead to the same
resultant vector. The text book does not provide an integrated understanding of geometric and
algebraic representation of a vector.
Further, textbooks don’t emphasize the notion of Resolution as an inverse operation
of addition (adding the component vectors back will give the initial resolved vector). This
leads to a weak understanding of the nature of these operations, and difficulty in
understanding the nature of vectors and components in situations such as a changed frame of
reference (like a rotated frame in an inclined plane) and also the possibility of non rectangular
components of vectors. This could have potential problems when students resolve physical
quantities like force or torque. The conceptual issues here will be carried over to all the
6
connected chapters (right block in Figure 1, various chapters in ​mechanics as well as
electricity and magnetism).
A central finding from this analysis was that a key transition in learning vectors -understanding the translation between the geometric mode and the algebraic mode -- is not
well scaffolded. The role of trigonometric ratios, which are employed in this transition, is
also not discussed. Given the way the chapters are sequenced in the physics and the math
curriculum, students have little support to understand and master the application of
trigonometry in the context of vectors.
Figure 1. Vector concepts covered in Physics and Mathematics Text books (Maharashtra).
An analysis of the Mathematics text books in the previous grades (blocks to the left in
Figure 1) showed that trigonometry is introduced first in Grade 9. Till Grade 10, the text
covers the basic definitions, and applications to the problem of calculating heights and
distances. A brief mention of trigonometric ratios with varying angles is made in Grade 10.
However, these connections are not emphasized enough, for the student to apply
trigonometry in the context of resolution of vectors when they move to Grade 11. The
mathematics textbook for Grade 11 discusses trigonometric functions, but with no direct
applications in the context of vectors.
Given the current state of the textbooks and curricula, the students would be left
without enough scaffolding to understand the vector concepts. These improperly understood
concepts, when carried ahead to further abstract topics, potentially lead to conceptual
confusions, and an inability to make sense of symbolic manipulations. The only way out for
them is to rote learn or devise tricks like Fatima rules ​(Aikenhead & Jegede, 1999; Larson,
1995)​.
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2.2.
Teacher Interviews
To check if issues identified by the textbook analysis resonated with teachers, we did a round
of interviews with 3 grade 11 physics teachers and 1 mathematics teacher. The focus was the
set of problems they found when teaching, and gaps in students’ understanding, in the context
of vectors and its applications. The interviews were semi-structured, and the teachers used
paper to explain when needed. All the discussions were recorded for further analysis.
The inputs from teachers varied with teaching experience. For example, all teachers
noted students' difficulty in resolution and addition of vectors, and finding dot and cross
products. But the reasons cited by the teachers varied from procedural aspects (inability to
solve a determinant to find cross product; confusions about trigonometric ratios in vector
resolution; using the formulae for dot and cross products) to conceptual aspects (not
understanding the notion of direction of a vector, and the geometrical addition in a triangle
giving the resultant). This along with the evidence from the text book analysis gives little
hope of any scope for model based reasoning in the classroom. Interestingly, when the
teachers were shown the student responses to a follow-up questionnaire (discussed in the next
section), they noted, that questions of that exact kind are not dealt in their classrooms and that
they will include problems of that kind in their future lectures. This is another evidence to
show that there is little scope for the students to do model-based reasoning. Teachers also
discussed limitations of the curriculum and the schedules, particularly the way they limit
opportunities to engage with the content conceptually, and with rigour.
2.3.
Student Pretest and Interviews
The textbook analysis and the discussions with the teachers helped identify the potential links
that are problematic for students' understanding of vectors. We also validated these identified
cognitive gaps by directly discussing with students. A questionnaire was created with a set of
problems, specifically targeting each of the potential cognitive jumps (gaps) identified in the
earlier steps. The answers to these questions was expected to provide a picture of the actual
problems students face in understanding these topics.
The same set of issues was also probed using a set of questions in a written pretest.
Students (N=49) were tested on questions (validated by a trial run on junior undergraduate
students) that probed prerequisites, addition, resolution and components of vectors, and
applications of resolution and components in physics. The students were also encouraged to
write down if they were confused by any particular question, and the reasons for the
confusion, being as specific as possible.
Based on the responses to this test, 14 students were short-listed for interviews. The
selection was based on their performance on pre-requisites and the explicitness of their
responses. Of these, only 8 students came for the interview, and they were probed further to
understand their reasoning and conceptual frameworks, based on their responses to the
questionnaire. The interviews were conducted individually, and were semi-structured. They
8
were allowed to correct their responses and write and explain their reasoning. The entire
discussion was video recorded for analysis.
Students had problems with understanding the relation between resolution and
components in connection to addition. All the students understood components as only
rectangular components (x and y components). Also, students were not able to understand
addition of vectors holistically. That is, some of them assume each law of vector addition
(triangle and parallelogram laws) giving different answers. Moreover, the relation between
performing addition geometrically and algebraically (using rectangular components) is not
appreciated. This thus reinforces the insights from the textbook analysis and the teacher
interviews pointing at the inability to connect the geometrical and algebraic denotations of
the same vector. This gap is carried over to dot products also. The responses in the answer
script and the interviews also point at a predominant usage of formulae which were
memorised. This again is an evidence for a lack of model based reasoning.
This data is only indicative, and a more systematic survey of other curricula and
students' understanding based on these textbooks is needed to get a better perspectve on the
learning issues related to vectors. This analysis will be part of the proposed main study
(Section 6).
3.
NATURE OF THE PROBLEM
The above analysis suggests that there are two aspects to the learning of the vector problem.
One pertains to the ‘mathematical conception of vectors using the varied systems of
representation’ and the other is related to the realisation of vectors as ‘a modelling system to
solve problems of physics’. Both are not well supported by the existing state of practices in
the education system.
3.1.
Integration of Mathematical Concept
With any mathematical topic, the abstract concepts are accessed through the symbolic
systems. Hence, an understanding of external representations and how these correspond to
internal representations and concepts has been a central question of interest as can been seen
in the studies of ​(Janvier, Girardon, & Morand, 1993; Pande & Chandrasekharan, 2016; Pape
& Tchoshanov, 2001)​. Along similar lines, for vectors, the relation between the
manipulations made at the concrete level (externally and procedurally), and at an abstract
level (internally and conceptually) needs to be clarified.
(Tall et al., 2001) analyses student learning paths for the bifurcation between the
operations performed mechanically at the procedural level and the conceptual level and the
role symbols (representations) play in bridging the process and concept, characterising a
spectrum of performance in carrying out mathematical processes from procedural to
proceptual. ​(Subramaniam, 2008) explore elaborately the distinctions in meaning making of
symbols in mathematics (numbers and algebra). ​(Sfard, 1991) unifies mathematical
9
conception, characterised to be of a dual nature (dynamic process of operations and a static
concept), leading to the famous idea called ‘procept’, saying that the process and the concept
are different sides of the same coin.
(Goldin & Kaput, 1996) in their carefully made analysis, distinguish between these
two domains internal and external in the context of mathematical knowledge and bring out
the horizontal and vertical connections between different systems of representations
(symbols). For example, a graphically representation of a curve and its equation (algebraic
representation) are horizontally connected systems. And internal or abstract mental entities
embody this understanding an interpretation of any or both of these formal systems of
representation. In the context of vectors, we need to characterise the way in which the
geometric (a ray with magnitude and direction in space) systems of representations and
algebraic (rectangular components using trigonometry) are connected (horizontally) and the
way these are connected (vertically) with the internal abstract representations (rather the
meaning of the representations). A further analysis of the nature of problem is warranted and
is planned as discussed in the Sections 3.4 and 6.
In modelling terms, one can make a parallel to the series of symbolic manipulations
(in external modelling system offered by vectors) and corresponding mental models (internal
representations) as described by ​(Ileana María Greca & Moreira, 2002)​. The following
section discusses in more detail the modelling aspects of applications of mathematical tools
in Physics.
3.2.
Vector modelling applications in Physics
Looking at the potential of vectors to be a modelling system to solve problems of physics,
there have been many analyses ​(Aguirre & Erickson, 1984; Doerr, 1996; Nguyen & Meltzer,
2003) on the links between vectors and topics in physics. ​(Doerr, 1997) in his study on the
conception of force, shows using an empirical study how building models effect the
integration of understanding of vectors, trigonometry and force concept. Vectors were shown
by ​(White, 1983) to be the critical link in better understanding of Newtonian Dynamics. He
uses arguments from ​(DiSessa, 1982/1)​, on connections to students’ other knowledge and
gives three instances where students could find it difficult to use vectors in understanding
force and motion. Firstly, the students hold ideas (about force and motion) which are
conflicting to the inherent assumption of vector applications similar to common sense
concepts as described by ​(Halloun & Hestenes, 1985)​. Secondly, scalar arithmetic does not
easily transfer to vector arithmetic and students struggle. Finally, vector representation in
itself brings a notion if direction attached to it, and the length signifying magnitude, which
students are unfamiliar with.
(Ileana María Greca & Moreira, 2002) as shown in the Figure 2(a), tries to organise
the relations between the physical phenomena, physical model, mathematical model and a
scientific theory and understanding of it. A physical model, based on a theory, has a
mathematical model embedded in it and a mental model of the scientific theory unites the
physical model and the mathematical model. And this study also argues that students finding
10
difficult it to make appropriate mental models being the reason for their inability to handle
physics theories.
Figure 2 (a). A concept map outlining what is meant by comprehension of a physical phenomenon
(Ileana María Greca & Moreira, 2002)
In contrast, a framework that we developed as depicted in Figure 2(b) could help us in
understanding the nature of problem in a different way, involving emergence of an
intermediate model from the problem of explaining a physical phenomenon and the available
mathematical models. In physics, typically, there are various physical phenomena that we try
to analyse, explain and understand. One such example is the motion of a body. For example,
students deal with questions like the system of bodies on an inclined plane or a pulley system.
Now as can be seen in the Figure 2(b) below, these physical phenomena can’t be analysed
and understood as such. They need to be converted into an intermediate model like a
free-body diagram. Studies like ​(Rosengrant, Van Heuvelen, & Etkina, 2009) show how
freebody diagrams are critical for successful solving of the problems in physics and how low
achieving students used it just as problem solving aid, where as high achieving ones used to
evaluate their work which can be closely associated with the learning criteria ‘​having
acceptable understanding of what a model is’ as put forth by ​(Gilbert, 2004)​.
(Doerr, 1996) quotes ​(Counts, 1989) a MSEB, National Research Council’s report
(p.36) - “mathematics provides abstract models for natural phenomenon”. In this perspective,
the free body diagram, here is something which is amenable to be manipulated in the
modelling space of vectors and thus taking advantage to model the physical phenomenon like
a body on an inclined plane (a problem of physics). From a synergy between the possible
configurations afforded by this mathematical modelling space (provided by the vectors) and
11
the possibility of the physical phenomena to be modelled (translated) to this modelling space
through an intermediate model (like a freebody diagram), a solution to the problem emerges.
Here the solution is an explanation or an analysis of the given physical phenomenon.
Figure 2 (b). Nature of the Problem (left panel - vectors; right panel - general case). The dotted
arrows denote loose scaffolding
In this scenario, firstly, the conversion of physical phenomenon to intermediate model
is something which is not scaffolded enough in the textbooks and hence could be a potential
cognitive jump for the students. Added to this, the handling of the free body diagram with
physical quantities as vectors by various operations, as can be seen from the textbook
analysis, is again not scaffolded enough. Also, integration of various mathematical ideas like
the resolution, addition and components of vectors, towards creation of a holistic modelling
space of vectors, is something key to handle vectors in free body diagrams. This is one of the
sources of difficulties that ​(White, 1983) identifies as the reason for students' poor handling
of Newtonian dynamics.
This analysis can be generalised to various topics in physics and mathematics, where
physical phenomena are explained and understood using mathematical modelling spaces via
an intermediate model. Poor connections between the physical phenomenon and the
intermediate model, and poor integration of mathematical concepts towards a coherent
mathematical modelling space, result in poor inter-relations between the symbolic
manipulations performed and the conceptual processes that they refer to. For now, we shall
focus on the effects of poorer formation of mathematical modelling space (vectors) leading to
poorer handling in physics. ​(Hestenes, 2010b) also in connection to these learning difficulties
(grounded in the non-alignment of cognitive and scientific models) proposes a revolutionary
modelling tool, geometric algebra to unify the high school geometry, algebra and
trigonometry.
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3.3.
Role of Algorithmic Problem Solving
From the teachers comments that students lack enough practice solving problems leading to
conceptual gaps, it can be realised that teachers and students rely more than required, on the
problem solving approach for not only solving a given question or situation, but also to
integrate and bridge various concepts. Problem solving approach of teaching has been found
to be ineffective in conceptual understanding by studies like that of ​(Gerace & Beatty, 2005)​.
Further, ​(Edwards & Hamson, 1989) characterising model based reasoning using
mathematics, point out that:
“​the mathematical representation is generally limited to algebraic equations with a single
right answer. This highly linear problem-solving heuristic is often unduly limited to the symbolic
representations of algebra and the subsequent manipulation of those symbols. The modelling process
is often described as iterations of this linear problem-solving approach: understand the particular
phenomenon to be modeled; define the context and constraints; identify the key variables; explicitly
define the relationships among the variables; translate those relationships to an appropriate
computer implementation; analyze and interpret the results; and then refine the model and one’s
understanding through an iterative process by repeating the above steps.”
Similarly in algebra, ​(Fischbein & Barash, 1993) reports that the algorithmic approach
leads to poor understanding, due to it creating conflicting mental models.
As hinted in the teacher discussion (add transcripts as appendix), the analysed
curriculum (Mahrashtra state board), with many conceptual links not made and the gaps left
unscaffolded, directs the students to rely heavily on practising and solving more problems.
This leads to an algorithmic way of understanding things, leading to a syntactic way of
meaning making, as described in ​(Subramaniam, 2008)​. This is evident from students’ pretest
responses, where they cite inability to recollect formulae or procedures as reasons for finding
the questions difficult.
The cognitive jumps (gaps or loosely made conceptual links) in the MBR process is
required because these links are not scaffolded by textbooks or conventional teaching
methods. These jumps lead to poor understanding in students. Moreover, it results in a larger
problem – approaching the formal system of science as a mechanical and procedural exercise,
and not as a modelling system. This is in contrast to the characterisation of science as a
practice based on modelling ​(Hestenes, 2010a; Magnani et al., 2012; N. Nersessian, 1984)​,
and arguments for developing science education practices that parallel the cognitive changes
based on modelling reasoning in scientific discoveries.
4.
INTERACTIVE TOOL FOR MBR
Now moving towards our objective of change in the way these concepts are handled in the
conventional system, which leads to the lack of holistic understanding or the perceived
hardness among the students, we propose a new media intervention. ​(Hoyles & Noss, 2003;
13
James J. Kaput & Roschelle, 2013; J. J. Kaput, 1992; J. Kaput, Noss, & Hoyles, 2002)
advocate using technology towards new way of handling the mathematical content. Also new
media has been gaining some support from the embodied cognition studies like that of
(Abrahamson & Sánchez-García, 2016; Sinclair & Heyd-Metzuyanim, 2014) talking about
the dynamic and interactive nature of human learning. Studies like that of ​(Ottmar,
Weitnauer, Landy, & Goldstone, 2015) and papers like that of ​(L. W. Barsalou, 2003; Joyce,
Richards, Cangelosi, & Coventry, 2003; Landy, Allen, & Zednik, 2014; Newell, 1980) press
for a new approach for the symbolic systems as perceptual physical systems on the line of
embodied cognition. Studies like ​(F.H. Lotfi & Mafi E, 2012; Zengin, Furkan, & Kutluca,
2012) have already attempted using new media to handle trigonometry, dealing with very
preliminary concepts and has limited theoretical basis from learning sciences.
Bridging these two key aspects of representing mathematical content and dynamizing
the content based on interaction can be incorporated into the design of the new media tool.
This new media tool is expected to scaffold the cognitive jumps (integrate the gaps) for the
student. Here, new media tool is expected to play the ​role of an intervention to move
students’ reasoning from algorithmic way to a model-based way. Here, control on the design
of the tool is important so that, it is based on the theoretical principles of embodied
interaction and model based reasoning (discussed in detail in the later parts of this section).
Besides this, we also propose to trace the trajectory of the students understanding
during the interaction. Here, to meet this objective of the research, the new media tool
expected to play the ​role of a probe. ​Being a probe, the tool needs to suit the specific
requirements and situations that arise during the research. This would be feasible if the
control on the design is with us. Further, tracing the trajectory would require us to record by
keeping a log of all actions that students do. And control on the design of the tool is again
useful, especially for data-analysis purposes.
4.1.
Designing to create an interactive (manipulative) space
Manipulation with physical objects has been shown in various studies, in the context of even
abstract mathematical concepts, to play a key role in improved learning. ​(Martin & Schwartz,
2005) examine about how interaction in the physical world helped in development of fraction
concepts, reporting systematically done experiments on fractions learning using
manipulatives and transfer. ​(Rahaman, Subramaniam, & Chandrasekharan, 2013) explored on
similar lines the effect of physical manipulations in learning the area concept. ​(Sowell, 1989)
in an overarching review of 60 studies on various mathematical topics from elementary math
till higher math, shows an overall improvement in students’ achievements and attitude
towards Mathematics by using concrete instructional materials.
Another interesting finding is that using pictures and diagrams did not show any
difference from symbols, which leads to our next assumption of symbolic reasoning similar
to actions on physical object. ​(Uttal, Scudder, & DeLoache, 1997) also report evidence for
manipulatives and symbols handled in a similar way. ​(L. W. Barsalou, 2003) ​(Landy et al.,
2014)​, argued Symbolic reasoning is accomplished by sensory motor systems. Though, we
14
don’t directly yet subscribe to this, the notion of understanding external representations
(symbols) as concrete physical entities in the environment is found useful. Arguments of
(George Lakoff & Núñez, 2000) have as mentioned earlier, brought much needed connection
of doing mathematics (conceptual system) with perceptual (also actions) system.
(Domahs, Moeller, Huber, Willmes, & Nuerk, 2010) show embodied (coupled with
the finger counting) understanding of numbers. ​(L. Barsalou, 2003) talks how conceptual
systems are non-modular (not modular and distanced from the real situations / concreteness)
and de-contextualised (but situated). Other identical claims are made by ​(Shiller et al., 2013)
(Glenberg & Kaschak, 2002) experimenting and arguing for language (seen as another
symbolic system) and action interlinks in the context of verbs. Besides these, there are
numerous neuropsychological studies triangulating the same conclusion.
Philosophers have used tool analogy for Symbolic systems like ​(Wittgenstein, 2003)
argues that language is a tool in the game with meaning emerging out of usage. ​(Vygotsky,
Hanfmann, & Vakar, 2012) understands gestures, language and also symbols as the
psychological tools. In this context, we too use the analogy of vector symbols with physical
tools, in a pursuit to take advantage of the improvement in mathematical reasoning with
manipulations on physical objects. Thus interaction is one of the key principles guiding the
design of the tool. New media can enhance the potential for interaction, especially with
abstract entities with the flexible options of interface (mouse, touch and other possibilities).
4.2.
Designing to create a modelling space
Figure 4. Intervention Design Framework- (left panel - vectors; right panel - general case)
15
The conventional handling of these concepts using the algorithmic problem solving approach
has been shown to pose various problems in itself. In addition the curriculum itself has poor
scaffolding structures. In connection to these problems, we will here see what is expected out
of the new media intervention tool to promote model based reasoning. In a similar way to the
previous problem analysis, we will focus firstly on the specific case of vectors.
The new media tool must be able to help students move towards a modelling approach
of handling these concepts, and can also be seen as a scaffolding tool in the existing teaching
learning processes. There has already been some promise of the computer based interactive
systems to promote model based reasoning shown like in the studies of ​(Raghavan & Glaser,
1995)​. In the above Figure 4, we can see on the left the specific case of the vectors and to the
right a generalization. Here, the cognitive jumps are two-fold. One of them is the transition
from the physical phenomena to the modelling space, wherefrom an intermediate model
could emerge. Two, the integration of various mathematical concepts related to vectors, to
create a holistic understanding of vectors as modelling space.
The current focus is on the second one to come up with a new media tool that
attempts to provide a platform, which is dynamic, moving towards understanding the
symbolic systems as modelling space (vector space). The same can be told about the general
case in the right panel. New media intervention tool developed on the basis of the above
principles, will help in integrating the concepts thus integrating the gaps also develop
modelling approach of understanding the formal symbolic systems. There is a notion of
implicit learning that this kind of intervention expects to accomplish similar to the learning of
number sense in the case of Touch Counts by ​(Sinclair & Heyd-Metzuyanim, 2014)​.
4.3.
Tool Description (​first version​)
The new media tool is an interactive platform with an objective to give the user access to the
modelling space (vector space) to perform addition and resolution of the vectors. The user
gets to work with the various symbolic representations of the vectors to arrive at an integrated
understanding of the geometric and algebraic versions of vectors and relations between
resolution, addition and components.
The tool was built on a javascript based platform by Harshit, an engineer, from IIT
Roorkee. The tool can run both on a normal computer or laptop (using keyboard and mouse)
and on a touch interface on a web-browser. The tool allows the user to create a vector, change
its magnitude and direction. User can create right triangle and then see the rectangular
components emerging from the right triangle (represented as an animation). Figure 5 below
with vectors A, B and C shows these possibilities of the vector. We can see the corresponding
changes in the symbolic representation of the vector. The algebraic notations are also
presented below each of the vector.
The side panels always show the right triangle projected on the x-axis and the circles
of all the vectors on the screen. Now changing of the vector is always done using either the
16
magnitude or direction and that corresponding effect can always be seen in the rectangular
components.
Figure 5. Creating a vector, changing its direction and magnitude and resolving it into rectangular
components in the new media intervention tool
Another feature that this tool supports is addition of vectors. Two vectors can be
added to see the resultant using the triangle law of vector addition.
Figure 6(a). Addition of two vectors (top left)
Figure 6(b). Right Triangles superimposed (top
right)
Figure 6(c). Rectangular Components getting added
(bottom right)
From the above Figure 6, we can see the
various snapshots of the addition of vectors.
Here again, the changes in the magnitude and the direction of any of the component vectors
results in a corresponding change in the resultant vector.
One of the key integrating aspects is the connection between the geometric and
algebraic denotations of the vectors due to the change in the text when resolving and
corresponding changes in the x and y components (magnitudes). This tool through the
presence of a circle (unit circle) connected to the vector can be a good modelling space,
17
where the components are bound by the unit circle strengthening the links between geometric
and algebraic representations of the vectors. Also, it allows dynamic real time interaction.
After a point there will a strong implicit embodied learning of the nature of the vectors and
the nature of the operations like resolution and addition and the connection to the
components. Thus this could be good starting point to let the students interact with the tool as
a first iteration.
5.
PILOT STUDY
5.1.
Data Collection
In the pilot study, from the group of 8 students who were interviewed, 6 students
attended the intervention sessions, which involved performing tasks on the interactive vector
system (new media tool) for about 70-90 minutes. There were about 11 tasks used for the
exploration of the intervention tool. These tasks were designed to make the students explore
various features in the tool. The initial set of tasks was simple to make the student used to the
controls and familiar with the equipment. Then there were certain tasks, where they had to
manipulate both the direction and magnitude of the vector to reach a vector with given x and
y components. A similar set of tasks were given for addition of vectors too. There were tasks,
where any set of two vectors need to be created which will add up to a given target vector.
Students were asked to create as many sets as they can. Then one of the component vectors is
also fixed and the target vector. The rigour of constraints kept increasing as the session
progressed. Their actions were recorded using video, written scripts (rough work), screen
capture, and eye tracking (Tobii X2-60). After about a week, these students were given a post
test similar to the pretest (but without the questions on prerequisites). All 6 students were
then interviewed, to probe their understanding in the context of the responses given in the
post test.
5.2.
Data Analysis
In the process of entire pilot study, Figure 7 shows the topics probed, categorised into
three Broad Concept Areas (BCAs), which are further categorised into 5 sub concept areas
(SCAs). These concept areas constitute 16 links between concepts (CLs). The pre and post
test answers of all the 6 students were each analysed by 3 raters, and ratings were given to all
relevant CL-question pairs. A 5-point rating scale for conceptual understanding (1 = no
indication, 5 = strong indication) was developed for each of these concept links. This
structure is based on studies examining the shift from conventional problem solving approach
(prescribed in the text book) to a more conceptually sound explanation and judgements
(Niemi 1996; Besterfield-Sacre et al. 2004; Gerace et al. 2001). The scale does not measure
the correctness of the response, but rates the conceptual clarity of that particular concept link,
as expressed in the answer to a given question.
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Figure 7. Concepts and the linkages
If 2 or more raters found a concept link-question pair irrelevant, that pair was deemed
irrelevant, irrespective of the third rater's rating. Inter-rater reliability was estimated using a
weighted proportion of agreement (‘2/3’ for two raters agreeing, and ‘1’ for all the three
raters agreeing). If only one rater found a pair irrelevant, the agreed rating of the other two
raters was used. The final score for each concept link-question pair is taken as the mode of
the three ratings for the cases with agreement. For cases where all the ratings varied,
discussions led to 2 raters coming to a consensus. This exercise ensured an inter rater
reliability of more than 67% across all the 6 raters and the CL-question pairs. These ratings
were then converted into percentages. These percentages denote the strength of the CLs in
Figure 7. This data provides a comprehensive picture of the change in student understanding
of various concepts and strengthening of specific concept links after interacting with the
system.
5.3.
Key findings
Figure 8 shows the proportion of students whose understanding of concept links in a
Sub Concept Area improved across 5 SCAs (SCA1 - Triangle Law; SCA2 - Parallelogram
Law, SCA3 - Rectangular components, SCA4 - Non rectangular components, SCA5 Application in context of forces). The colored lines capture the conceptual understanding of
each student. The slope of each line captures the growth achieved by the student in the
understanding of the links in that particular SCA.
19
Figure 8. Conceptual understanding growth trajectories across SCAs of the 6 students
Students improved in all SCAs, but in different ways. SCA1, SCA4, SCA5 show 3
students improving in their conceptual understanding. Four students improved in SCA1
(triangle law), which is expected, as the interactive system is predominantly based on triangle
law. The parallelogram law (SCA2) is not very explicitly expressed in the system, but 3
students improved in this SCA. The two drops were about 2-3%. S6's drop in performance
could be attributed to disruption in concept areas SCA3 and SCA4, related to components
and addition. Surprisingly, only 3 students improved in the rectangular component (SCA3),
even though rectangular components were part of the system. The improvement for this SCA
was not more than 5-6%, this suggests this aspect of the tool needs to be redesigned. All the
20
students whose performance dropped had pretest percentages around or more than 60%. This
suggests that the system disrupted their existing concepts. SCA3 and SCA4 are closely
related to components of a vector, and 5 students showed conceptual growth in SCA4, which
pertains to non-rectangular components. Interestingly, all 3 students with weakened
conceptual understanding in SCA3 (S2, S5, S6) have shown growth in SCA4. SCA5
(applying vectors and vector operations in the context of forces) improved in all students.
Qualitative analysis of interviews showed comments supporting the above observations.
The above data suggests that the interaction with the system helped students improve
their understanding of vectors in two ways: 1) understanding of triangle law and the
non-rectangular components, and 2) the related disruption in their understanding of
rectangular components. The students' interaction process is currently being analyzed for
further insight to redesign the system.
5.4.
Case-studies
With the preliminary analysis that we did till now, we present two student cases with
some evidence to the influence of the tool on the conceptual understanding of students.
5.4.1.
Case-1 (S2)
S2 is a student who has finished the grade 11. He appears to get coaching for IIT-JEE,
a hard entrance examination for IITs, a sought out group of technological institutes in India,
known to assess students’ conceptual understanding.
His responses in the pretest were interesting as he shows some hints of conceptual
understanding, but commits procedural mistakes. He shows a command in his understanding
of vector and scalar, by correctly identifying physical quantities like weight, pressure, time.
Further, there is a sense of confidence and haste in the way things are written and thought out
in the answer script of the pretest.
To start with, we will take the the case of the expression for the magnitude of
resultant or the dot product (using components) as figure 9 below. In the problem on the left,
he initially has some confusion when dot products needs to be calculated using the
rectangular components. He starts of using the determinant form (used for the cross product).
Even when he rectifies, he directly writes some expression. Similarly, in the problem on the
right, he wanted to compare procedurally the magnitudes of resultant and the initial vectors,
but he uses the expression of dot product by mistake for the expression of the resultant. This
shows that he does not have a strong conceptual basis for the expressions (formulae) used in
the vectors and also hints at a possible rote learning of the procedures. However, he seems to
express the role of trigonometry in handling vectors due to their geometrical nature. This
understanding also pop up when he explains pressure being a scalar by saying that it does not
follow vector addition.
21
Figure 9. Responses of S2 in Pretest
Further from the ​interview after the pretest, he seems to have a strong sense of
confidence in his conceptual understanding. He, when interacted with after the pretest, started
off, saying that the test was boring. We discussed the question (to the right in the figure 9,
where the relation between |​A​| and |​P​3​|​+|Q​3​| ​was probed, ​when ​A=P​3​+Q​3​) which he said, he
found tricky, as he had to think of the extreme cases. We probed further for his conceptual
understanding of vectors. When explaining he said that they magnitudes of should be equal
(as they follow the triangle law of vector addition) clearly indicating that he missed it being a
magnitude. In fact even in the narrative that he used, he referred to them as vectors but not
magnitude of vectors. However, as discussed earlier, he confused the formulae for the
resultant and the dot product. However, once, he has the formula written, he tries to bring in
interestingly some principles of trigonometric function in quadrants.
He seems to have a fair amount of overall picture of the concepts involved in
trigonometry and its applications in physics. He shows how the rectangular components can
add up to give the initial vector, geometrically. He also shows an understanding of vectors as
geometrical entities helping in reasoning out certain problems in physics. He draws examples
from resolutions of forces, projectile motion and mathematics (circle and conics). Also, he
does not seem to display a finer understanding in the pretest answer script, he rectifies his
conceptual stance during the discussion. One of the key corrections is his acknowledgement
of presence of non-rectangular components, which is displayed by him being careful in using
the words in the discussion. Though he has a breadth in his narrative pertaining trigonometry,
this seems to break when poked in detail, as in the case of the question Pr.5d.
The student S2 was able to complete all the tasks on ​intervention session​. But, there
was a fundamental difference in the way, he accomplished the tasks. Different students made
use of the tool to different extents to help their reasoning. Some relied completely on the
geometrical entities on screen to guide their actions towards the goal. Some relied on the
algebraic entities on screen to guide their actions. Some tried to bring in their school
knowledge. The peculiar thing about S2 is that he despite the presence of the tool, relied on
the pen and paper based calculations to arrive at the solutions to the problems (tasks) and then
directly created them on the tool. When asked to create a target vector 50î+80ĵ, he made a
22
rough estimate of the magnitude using the formula, and then fine tuned it to arrive at target.
He follows a similar interesting approach when asked to create a set of two vectors, which
add to give a resultant of magnitude 120 at a direction 60° (anti clockwise) with x axis. He
resolved the vector into rectangular components and used them as two vectors which add up
to the resultant. This is a very interesting strategy hinting at the conceptual clarity of addition
and resolution of vectors. This along with the confusions in the responses in the pre-test, does
not sink well to suggest presence of model based reasoning. This could just be one of the
other broad pictures that he has developed, with little connection yet to the actual model.
However, the further tasks where he was asked to create other possible vector sets
which add up give the same resultant and follow-up constraints of them being perpendicular
to each other, were demanding his exploration of the geometrical affordances of the tool. Yet,
he was trying hard to find ways, in which he could use some formulae to arrive at a rough
estimate of the required vectors like in the case of a set of perpendicular components adding
to give a target, he assumes them to be of equal magnitude and then estimates their
magnitude as shown in the figure 10 below. Further in a follow-up discussion, when asked
about reasoning using calculations with the pen and paper, the student calls it as fundamental
understanding and associates it with theory, and
calls the interaction with the tool as an experiment.
Figure 10. A section of the rough work performed by S2
during intervention session
The student S2, in the context of a problem based on
relations between the magnitudes of vectors on the
magnitude of the resultant, in the pretest uses
formula. Whereas, in the post test​, S2 handled a
similar problem as shown in the figure 11(a) below,
directly by changing the angle dynamically and
considering the extreme conditions to arrive at an answer and attempting to use diagrams.
Figure 11 (a). A
response by S2 in the
post-test
23
Figure 11 (b). S2 using gestures in the post-test
interview to show a dynamic change in the direction of
vector
This clearly shows a shift in the reasoning
approach from a formula based algorithmic mode
of problem solving to a more model based
approach. The effect of the tool was evident in the
way, S2 used gestures as shown in figure 11(b) to
dynamically change the vectors. However, he still
articulates this reasoning as being another method/
algorithm to effectively solve the problems of that kind.
Figure 11(c) A response by S2 in post-test
To another question as shown in the figure 11(c), he says- “ ​in his mind, I just I drew
two axes and then I shifted the axes like this (gesturing and rotating his hands shown in
figure 11(d)) .. perpendicular, perpendicular, perpendicular.. oh!! There can be many!!”.
Later, he explicitly says that this is something
that he realised when he was performing a task,
where he was asked to come up another set of
vectors, which add up to a given target vector,
during the intervention session.
Figure 11(d). S2 using gestures in the post-test
interview to show a how component vectors change
to give the target vector.
There were many such instances like when
needed to create all possible sets of vectors to reach a target, he explicitly mentioned how the
interaction with the tool has helped him. However, again, the limitations of the students
understanding of components and rectangular components emerged. When solving certain
questions on components (both rectangular and non-rectangular), S2 was still looking for
formulae and some confused state made him not attempt the questions at all. We need to
further analyse the the intervention data (eye tracking data as well as the screen recordings) in
detail for a clear trajectory of the changes in the understanding of S2.
5.4.2.
Case-2 (S5)
The next case is that of S5, a girl who finished her 11th grade from the same college as S2.
She appears to take some remedial help from coaching classes (but not preparing for IIT-JEE)
24
after her regular college. She is an interesting case, as all here responses were neatly and
carefully written, and much care was taken to represent them as clearly as possible.
She too has shown a good understanding of the prerequisites from her responses in the
pretest​. She could differentiate all physical quantities as vectors and scalars including
pressure, time and friction, correctly, except for the physical quantity ​weight.
She committed the mistake of ignoring the order of vectors in the case of triangle law
of vector addition as shown in the figure 12(a) below. However, during the discussions in the
interview, it struck to her that the vectors need to be ordered in a particular way in triangle
law as shown in 12(b). But, immediately contradicted herself, as she found the geometrical
figues similar (both obtuse triangles). Discussing the questions of related to addition of
vectors, she could express a superficial connection (how they related as geometries) between
the triangle and parallelogram laws of vector addition.
Left: Figure 12(a). S5’s response in a pretest question (improper ordering of vectors triangle law of
addition). Right: Figure 12(b). S5’s writing in the pretest interview in the same context (proper
ordering of vectors). The student was confused on the ordering of vectors, as the above figures look
geometrically similar (obtuse triangles).
She fumbled in solving questions related to adding using rectangular components.
When asked to add vector after resolving into components, she could resolve them using
trigonometry, but could not figure out how to move ahead as shown in the figure 13(a) below.
The relation between rectangular components and the initial vector, seems not be very clear,
from the inaccurate representations in the figure (the lengths of the projections made
randomly) given the care taken by her to write.
When probed the same during the interview, she says that x component is the
projection of vector on x-axis and similarly with y-component. But, when asked to write the
relation between the vector and the rectangular components, she write the expression
C​cosθ+​C​sinθ = 1, as shown in figure 13(b). But when hinted using various prompts from the
questionnaire and the text book, she agreed on ​C​cosθî+​C​sinθĵ = ​C. This hints at some vagues
recollection of identities from mathematics relating sin and cosine ratios, but no strong
conceptual links in the context of vector components.
25
Figure 13 (a).Response of S5 to a question in the pretest
Figure 13(b). S5’s writings during the pretest interview
When probed for her general understanding of components using a set of questions on
addition of vectors and triangle law and the components (figure 14(a) below) she was not
clear with the diagram, as she expected components to be only x and y components. And her
confusion is articulated in the response she gave to the same set of questions (in figure 14(b)).
Figure 14(a). A sample question from student pretest.
Figure 14(b). SS5’s response to the set of questions shown in figure 14(a)
26
In another exercise in rotated frames of reference as shown in the figure 15 below, she
could correctly and confidently represent the x and y components in both the frames of
reference. Her understood of their relation breaks again, when she says the length of x
component ​A​cos80 is infinite.
Figure 15. S5’s response on a pretest question during interview
Further, the picture (shown in figure 16) that she draws with a right triangle and the
sinusoidal wave in the same graph shows a mixed up set of
confusions that the text book diagrams could make. She tries to
somehow connect the right triangle narrative of trigonometric
ratios, with rotating angle, with the sinusoidal wave form. This
clearly shows a lack of unit circle notion.
Figure 16. S5’s drawing in the pretest interview
So, these are the various instances in the case of S5 as described
above, where the understanding of components and the addition
of vector and also the role of trigonometric ratios being a broken.
During the ​intervention session​, S5 took a lot of time to start of getting comfortable
with the controls. Hence, she could not complete all the tasks, but she could explore all the
features of the tool by doing some extra tasks. In the initial tasks of arriving at a target vector
like 50î+100ĵ, she tries changing only the direction of the vector a long time, and hence never
ending up at the target. Even when she was hinted to try changing the magnitude, she could
not arrive at a systematic way to arrive at the target vector. She was not trying to control both
magnitude and direction, but always ends up controlling just one of them. She was able to
move ahead, only when a case was demonstrated, focussing just on changing both the
direction and magnitude, without speaking out the systematic approach, after which she
started arriving at the target vectors relatively easily. Another interesting thing about her
interaction was that, she relied completely on the screen controls and did not use any pen and
paper calculations. Even during the tasks, where she needed to create a set of vectors which
add up to a given target vector, she just kept manipulating the vectors, until she stumbled
upon the target vector. During the discussion after the intervention, she says the tasks were
simple and as she kept doing tasks, they seemed simpler. Further, she says with some
excitement- “​the tool helped in the understanding addition of vectors.. ummm.. components
(meant rectangular components), like the way it is shown here (while drawing). Firstly the
vectors, which were inside a circle… Earlier I was not used to how the components come.
27
Like it is shown here (pointing to the tool), the lines appear and then this line moves here,
forming the y component. Resolution of vectors… umm.. angle… I liked this very much.”
Towards the end, as a suggestion, she points out that the tool now supports only triangle law,
but would love to have parallelogram law too and tried to hint at some way of implementing
it. And as an overall comment, she says - “​it (interaction with tool) has made a lot of things
easier. As earlier I could not draw things out on her own for addition of vectors and all…
angles and others I feel are now easier…”
Figure 17. S5’s response in post test
However, a contrasting picture appears in the ​post-test and the follow-up discussion​.
The post-test responses showed a drop in her understanding of components. Right in the
beginning of the interview, she says that there were confusions in components. She is still has
no proper connections made between triangle and parallelogram laws, which is of course not
directly expected from the tool. Regarding addition of vectors using rectangular components,
she shows some progress from the pretest (see figure 13(a)), where she could not move ahead
after resolving the vectors geometrically. Here in the post test, she said, that she could add the
y components of both vectors, but was unclear of the x components (as they were in opposite
directions), as shown in figure 17.
When asked about certain questions on the relations of magnitudes (see figure 11(a)),
she says, she understood the problem when done on tool, but not here. this clearly hints at the
problem of transfer. She made many inconsistent statements too like in the figure 18 below.
Figure 18. S5’s response in the post-test (blue) and during the post-test interview (black)
In the interview she says “ ​in tool, she could change both magnitude and direction…
and thinks”, again showing that she could not transfer. But, when asked questions explicitly,
28
she could give a conclusive answer. But she now seemed to have confused with the way the
equation ​R = A + B, ​capture both direction and magnitude of the vector. ​However, there is a
deeper analysis of her understanding is needed (through the intervention data).
The case with other student S1, who was performed relatively poorer in the pretest
and could not complete all the tasks in the interaction sessions, there was an improvement in
the performance in all the SCA as shown in figure 8. However, during the interviews, we
could see some broken ways in which he was integrating his experiences of interaction with
his earlier school (text book) knowledge of vectors similar to S5. His post test responses
show some places, where he hurriedly mistakes magnitudes for actual vectors, but shows
signs of questioning unlike earlier. However, there is still a lot of to be explored in his
understanding.
5.5.
Conclusions from the pilot study
S2 and S5, the two cases above show some interesting aspects of the how the tool for model
based reasoning, could change their conceptual understanding to different degrees. S2, who
uses more pen and paper and was pushed by certain tasks to rely on the tool, ends up using
more pictures and gestures during the post-test. S5, who had no clue of how things were
interrelated other than sheer memory of formulae during the post test, has started making
some conjectures on the basis of the tool, but was struggling to transfer that understanding to
formal representations.
All are many such interesting trajectories that students take which could throw some
light on how students could move to model based reasoning. The power of the tool as a probe
will turn out to be very useful in such contexts, where we need to investigate their actions
during the interaction. Logs and eye tracking and mouse click data could help us in doing
this. Hence, beyond the analysis of the written answer scripts, the qualitative data from the
interviews and the interaction sessions, presents some interesting stories, which need to be
pursued and explained.
This adds to the promise that the tool helps in developing a model based reasoning in
the students when they approach the vectors. These also hint that the intervention, which was
of only one session, does show a influence in their conceptual understanding, and a proper
integration in the conventional system with some sustained interaction could lead to better
conceptual understanding, which shall be part of the plan discussed in the next section.
6.
ACADEMIC PLAN
6.1.
Research Questions
To further understand and address the above identified problems in the learning of
vectors and their use in model-based reasoning, we propose the following research questions.
Note that these questions have already been operationalized and addressed by the pilot study
29
we report the section 5 above. In the proposed research, these questions will be addressed
more systematically, based on the results from the pilot study. The central research question
is:
Can a systematic change in the conceptual understanding of vectors and their role in
MBR be achieved using an interactive system designed to support model based reasoning?
This question requires answering the following sub-questions:
1. What are the key conceptual transitions and scope for MBR in vectors in the
high-school and pre-college curricula?
This involves characterising the conceptual area of trigonometry and vectors
and it applications in physics. It would be similar to the text book analysis
reported earlier. Also, we shall look for the scope for model based reasoning
that the text books allow students.
2. What principles and features should a tool incorporate to provide students a scope /
chance to do MBR?
Having seen the nature of the problem in the section 3 and the pilot study of
Section 5, we shall work on the design principles for an effective intervention
tool, which would allow the students to perform MBR. This will involve
identifying the design features as detailed in the Section 4.
3. What are the changes in students’ conceptual understanding using this tool?
We shall trace the trajectories of changes in students’ conceptual
understanding of vectors and its applications in physics. The new media tool
can be used as a probe for this purpose, where, the students’ interactions with
the tool can be used to get a picture of cognitive processes. Using a pretest,
post-test paradigm, we could look for the change in their conceptual
understanding.
4. How can an intervention of this nature be integrated in the conventional system?
We shall explore with an experiment on how this could be used in the existing
system to scaffold students conceptual understanding by allowing model based
reasoning. And follow it up with discussions with teachers how this could be
integrated in the conventional system.
6.2.
The Study with Timeline
The pilot study discussed in the section 5, has shown how the interactive media-based
tool had changes the students understanding. A shift in their reasoning towards MBR can also
30
be traced in the case studies presented. Now in line with the larger questions, as described in
the above section 6.1, we shall here provide a tentative plan of study along with the timeline.
Phase-1: Characterisation of concepts and redesign of the tool ​(​6 months till May 2017​)
6.2.1.
Start with characterisation of the concept area further, by analysing other curricula.
The probable ones that we consider now are NCERT, Singapore National curriculum
and the US (Indiana) high school and early UG curricula.
6.2.2.
In parallel, further analysis of the pilot studies data by taking specific cases of the
students will be done.
6.2.3.
Moving ahead, we will redesign the tool based on the inputs that we get from the
analysis of pilot data.
Phase-2: Tracing changes in conceptual understanding of the students (10 months till
March 2018)
6.2.4.
After preliminary validation of the tool, we shall test the redesigned tool with the
students (who have finished their Class 11th or equivalent grade in April 2017) and
discuss with the teachers on how to incorporate in the classroom set up. The
methodology will be fundamentally similar to that of the pilot study, with certain
refinements and validations of the pretest and post-test questionnaires as well as the
interview protocols. The data collected will include the written manuscripts of the
students, the recordings of the interviews and interaction sessions and the eye
tracking data. (​3 months​ till August 2017)
6.2.5.
We shall analyse this data using the framework developed during the pilot study (as
described in the Section 5.2). We shall present a preliminary analysis in the
EpiSTEME (​3 months ​till December 2017)
6.2.6.
A full analysis of the data with the redesigned tool shall be reported as a journal
paper. This will be tracing the trajectory of how students reach model based
reasoning and the role the tool played in this process. This shall be submitted by
around March 2018. (​7 months ​till March 2018)
Phase-3: Integration in Classroom (8 months till November 2018) ​Now, with the
redesigned tool tested with students, we shall explore how to introduce it in the real
classroom setting along with the discussions with the teachers.
6.2.7.
We shall deploy the redesigned tool in the real classroom setting. We shall compare
it against a control group. This shall be a controlled experiment looking for the
effects of using the tool in the classroom setting. The data collected shall be pretest
and post test of the student performances in the two groups (test and control). This
will happen in the beginning of the academic year when the topics pertaining to
vectors are dealt with in the academic calendar ​(3-4 months ​till around July 2018​).
31
6.2.8.
We shall analyse this data and report in a journal paper. (​4 months ​till November
2018)
6.2.9.
We shall produce a rough draft of thesis and write a ​Synopsis - March 2019.
6.3.
Expected Outcome of the Research
The research can give key insights and have implications in multiple areas. Education
literature in vectors and students handling of them is very limited. This research could find its
place in starting a discussion about this pedagogical as well as content knowledge of vectors.
Besides, this research, being placed in the context of application in physics, also starts
exploring the question of transfer of mathematical knowledge to physics applications.
Model based reasoning is something that math and science educators who are
philosophers have been advocating recently and this could be a small instance trying to trace
the conceptual change in the process of model based reasoning ​(Nancy J. Nersessian, 1999)​.
In addition, this research also explores how that could be introduced in the mainstream
education.
New media interventions have been popping up with various studies trying to prove
its relevance in education. However, there seems not to be any coherent framework guiding
the design of these interventions. This research attempts to give some insight in that direction
drawing inspiration from embodied cognition and modelling.
Finally, the nature of the knowledge - mathematical as well as scientific (physics) - in
connection with the representations and the mechanisms of how human cognitive system
could accomplish this exercise, can also derive some fruitful engagements from this research.
7.
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