Dynamic software enhancing creative
mathematical reasoning
Jan Olsson
Licentiand thesis, Educational works
Naturvetenskapernas och matematikens didaktik
Umeå 2014
Responsible publisher under swedish law: the Dean of the Medical Faculty
This work is protected by the Swedish Copyright Legislation (Act 1960:729)
ISBN: ISBN 978-91-7601-087-7
Elektronisk version tillgänglig på http://www.nmd.umu.se/
Umeå, Sweden 2014
To to all the inspiring people who have shaped my way, both before and
during the work on this publication
Dynamic software enhancing creative
mathematical reasoning
Licentiand dessertation by Jan Olsson
Department of Science and Mathematics Education
To be publicly discussed in lecture hall N460 at Umeå University, on
Tusday june 10, 2014, at 14.30
Abstract
This thesis includes two articles and a coat. The articles present two studies
investigating students’ reasoning when they were working in pairs, solving
mathematics problems using the dynamic software, GeoGebra, The first
study shows that the students used GeoGebra as a collaborative environment
where they shared their individual reasoning to one another. Furthermore,
GeoGebra provided the students with feedback that, to some extent, became
a basis for their creative reasoning.
The second study looked more closely into the relation between students’
reasoning and their utilization of the feedback generated by GeoGebra. The
study showed that students who before entering computer commands used
creative mathematical reasoning to hypothesize what the outcome may be,
understood the feedback from software better and used it more efficiently.
The students who engaged in imitative reasoning were mainly able to use
feedback to determine if a solution attempt was correct or not, but did not
fully understand the feedback and were less able to use it to make further
progress in solving the task.
The coat explains theories and methodologies more thoroughly and
discusses the results of the two articles. In a concluding discussion the
results of the articles are linked and possible implications for teaching are
proposed. In school it is common that teachers and textbooks provide
students with algorithmic solution templates to tasks, but in the study the
didactic situation with dynamic software was found to invite students to
create their own solution methods. Furthermore the thesis suggests that it
could be beneficial for the students to be encouraged to pay more attention
to their own solving strategies, i.e. to explain and evaluate their methods and
results rather than merely looking for the correct answers.
Acknowledgement
Writing this thesis has been a challenge accompanied by joy all the way. The
co-workers and fellow phd-students at the Department of Science´s and
Mathematics Education and Umeå Mathematic Education Research Center
have all been sources both for deep scientific discussion and easygoing
everyday conversations. The main reason for all I have learned and that I
have taken the first steps into the world of research with an increasing
engagement is that I have had Carina Granberg and Johan Lithner as
supervisors. I have always been encouraged, taken seriously, and I have
always been comfortable with to be told when I ended up on the wrong track.
I can´t thank you too much! Many thanks also to the members of the project
group LICR, and members and leaders of the research school Development
of Mathematics Education, especially to my closest colleagues Lotta, Maria,
and Helena.
The work with the thesis does not only take support by expertize. All aspects
of life must go on and my wife Lena has always been on my side, as our
children Emil, Elina, Evylinn, and Joel, even though the last ones think it is a
bit suspicious that a grown up man voluntarily takes up education. Our
grandson, little Sigge who came to us a month ago does not say so much yet
but is of course a source encouragement only by joining us.
Table of Contents
ii Abstract
iv Acknowledgement
i Table of Contents
1 Part 1
1 1. Introduktion
2. A brief history of interactive computer applications in mathematics
education
4 5 3. Frameworks
6 3.1 Imitative and creative reasoning
3.1.1 Mathematical thinking processes and reasoning sequence
6 7 3.1.2 Imitative reasoning
3.1.3 Creative mathematical reasoning
3.2 Joint problem space
10 11 12 3.2.1 Turn taking
12 3.2.2 Social distributed production
12 3.2.3 Repairs
13 3.2.4 Narration
13 3.2.4 Language and action
14 3.3 Formative feedback
14 3.3.1 Feedback and digital technology
14 3.3.2 Formative feedback
15 3.3.3 Features of formative feedback
3.3.4 Dynamic software and feedback as verification and elaboration
3.4 A framework for macroscopic analysis of problem solving protocols
4. Methodology
16 16 18 18 4.1 On using multiple frameworks
19 4.2. Why these frameworks?
20 4.3 Methodological considerations
20 4.3.1 Article 1
21 4.3.2 Article 2
23 4.4 Didactical design
5. Summary of the results of the articles
5.1 Article 1
25 25 25 5.2 Article 2
26 6. Discussion
6.1. Dynamic software affecting students´ reasoning
6.2. Students´ reasoning affecting utilization of computer features
6.3. Implications for teaching
26 28 29 31 References
1 Part 2, the articles
i
ICT-supported problem solving and collaborative creative reasoning:
Exploring linear functions using dynamic mathematics software
1 The relations between reasoning, feedback from software and success in solving mathematical tasks31 ii
iii
Part 1
1. Introduktion
Research on educational technology often sees students’ learning as active in
terms of knowledge construction, reasoning, interactive, etc. Consider the
following quotes:
“Our approach … has been to design a computational microworld that
supports 11-14 year-old students by providing them intelligent feedback
during their [knowledge] construction” (Noss et al., 2011, p. 63)
“Computer application that encouraged students to reason deeply about
mathematics increase learning“ (J. M. Roschelle, Pea, Hoadley, Gordin, &
Means, 2000, p. 78)
“Composing [visual geometric elements] has been used as an interaction to
promote deductive reasoning“ (Sedig & Sumner, 2006, p. 22)
All these quotes suggest that computer applications may affect students´
actions and learning. It is also emphasized that students should be active
when they are learning. Pedagogical software that aims at initiating learning
has developed from pure programming applications, through microworlds
that are designed to provide the student with specific mathematical tasks
that the student may explore and manipulate (Noss et al. 2011), and dynamic
software, that allow students to form and manipulate the environment
themselves (Ferrara, Pratt, Robutti, Gutierrez, & Boero, 2006). Interaction
with dynamic software is a question about interplay. In addition to the way
software affect the students’ actions, yet another perspective should be
added: the way the students act on software affects the way students learn
from the contribution from software. This thesis presents two articles, one
investigating the way software supports reasoning and another investigating
the way reasoning affects the utilization of feedback from software.
Consider the following example where two pairs of students are working with
the same task in the environment of a dynamic software, GeoGebra:
The students A and B are trying to create a function whose graph is
perpendicular to the graph of y=2x-2 and student C and D are also creating a
perpendicular graph, but to y=2x-1. Both pairs decide to try y=-x-2. A and B
state that the graphs are not perpendicular and, without articulated
1
evaluation, decide to change y=2x-2 into y=x-2, which results in
perpendicular graphs. Likewise, C and D state that the graphs are not
perpendicular and then argue that y=-x-2 must have less slope and that the
slope depends on the x-coefficient. Their conclusion is that they must have a
larger x-coefficient than (-1) and decide to try (-0.5), which gives
perpendicular graphs. The pairs´ actions are practically equal, the visible
information from the software is equal, and they both manage to create
perpendicular graphs but their utilization of information from the software is
different. The explanation to these differences may be found in the way the
two pairs prepared the computer activity. The preparation for and utilization
of computer activities are in large extent conveyed by their dialogues that
contain products of their thinking that may be interpreted as reasoning. The
two articles of this thesis focus on students reasoning with the purpose to
understand causes and consequences of different use of dynamic software
while solving mathematical tasks.
Educational research associated to interactive software often highlights
computer features like multiple representations, provider of non-judging
feedback, processor of calculations, etc. as beneficial for students’ reasoning
and a large number of publications suggest a broader view of mathematical
reasoning than for instance just as formal logical (Barwise & Etchemendy,
1998; Jones, 2000) Some propose that computers enhance the development
from everyday reasoning into more formal deductive reasoning, and other
suggests that interactive software allow students to use less formal reasoning
independently from strict logical mathematical reasoning. Lithner’s
framework (2008) offers characterizations of reasoning as imitative or
creative. The framework is not in particular directed to ICT but the two
studies of this thesis show that Lithner’s suggestion of separating the
observable reasoning sequence from the thinking processes that created it, is
useful for investigating reasoning during software-aided task solving. The
assumption that the thinking processes that create imitative reasoning are
fundamentally different from thinking processes that create creative
reasoning processes is appropriate to investigate the way students’ reasoning
affect their use of dynamic software. A reasoning sequence separated from
thinking processes can be combined with a collaborative perspective on
reasoning, in the sense that two or more students together constructs a
reasoning sequence in their attempt to solve a task. Conversation is a
fundamental part of collaboration and may be interpreted as reasoning.
Collaborative reasoning is a main component in article 1 and discussed later
in this coat. In article 2 the information the software delivers as a result of
students´ computer actions is considered as feedback. This is, feedback was
not delivered from one person to another, it was not prepared, and had no
2
purpose. The effect of the feedback depended on the way students utilized it.
This is discussed in article 2 and later in this coat.
The aim of article 1 is to develop insights into how GeoGebra can be used as a
means of supporting collaboration and creative reasoning during a problemsolving process. The research questions posed were; “To what extent do
students use GeoGebra to collaborate during problem solving?” and “What
characteristics of GeoGebra might contribute to or obstruct their creative
reasoning?” Article 2 aims at developing understanding of the relations
between reasoning, feedback, and success in task solving. The posed
questions were; “What is the relation between the students’ way of using the
feedback that GeoGebra generates and the students´ reasoning?” and “How
do students´ ways of reasoning and utilization of feedback from GeoGebra
relate to their success in problem solving?”
Article 1 builds on research showing that creative mathematical reasoning is
beneficial for learning. A study investigating learning effects of imitative and
creative reasoning shows that students practicing on tasks encouraging
creative mathematical reasoning had less correct answerers while practicing
than students practicing on tasks leading to imitative reasoning, but
performed better on a test a week later (Jonsson, Liljeqvist, Norqvist,
Lithner, in preparation). This indicates importance of designing tasks
leading to creative reasoning. Therefore the focus in article 1 was not
whether students succeeded or not, but on the characteristics of their
reasoning. Anyhow, the results of article 1 indicated that the task (provided
the students did not know how to solve it in advance) was only possible to
solve through creative reasoning. For example, finding one example of
perpendicular lines in a Cartesian system might be possible through trial and
error. But to answer the question about the relation between the xcoefficients when two graphs are perpendicular (that their product is -1) the
student must either know that relationship in advance or create and justify
well-founded strategies. In the latter case, the students must carry out
creative mathematical reasoning to find the relation. Out of this the interest
in article 2 turned to relations between reasoning, feedback, and success in
solving the task. In purpose to say something that shape the path through
task solving the interest must turn to the activities that produce the answer.
A difficulty is that the path towards a solution may be relatively complex and
it is a risk of focusing on objects outside the scope for the study.
As indicated so far, more than one perspective has been taken in the two
articles and different theories have been combined and used as framework in
these studies. This needs to be taken into consideration when methods of
analysis and drawing of conclusions are designed. It is crucial that there is a
3
clear connection between research questions, method of analysis, and
theories (Gellert, 2010; Niss, 2007; Radford, 2008). The investigations are
driven by the interest in students’ reasoning associated to the use of
interactive software. This will be further discussed in chapter 4.
2. A brief history of interactive computer
applications in mathematics education
Since computers were introduced into mathematics education there have
been high expectations for improved learning. On of the most ambitious
proposals is that the computers ability to support reasoning and
computation invites other than a few privileged experts into the world of
complex mathematics, a democratization of the access to knowledge (Ferrara
et al., 2006). However, even though many studies have shown promising
results, the expectations on a broad impact on students’ learning have never
been fulfilled (Bottino, 2004).
Computers have been used in mathematics education for different purposes,
for example as drilling practice when calculating and memorizing of facts, as
providers of mathematical problems, as tutorial support, etc. Many of these
applications have been digital variants of already existing teaching methods
and textbooks (Bottino, 2004). Features of interactivity has been considered
as an extension or replacement of existing teaching, for example
programming, manipulating of provided mathematic features in artificial
microworlds, and interactions with dynamic software (Ferrara et al. 2006).
In the seventies and early eighties computers were used in teaching projects.
Programming languages as Logo, Pascal, Basic, and others were used with
the purpose to help students to make use of and learn about mathematical
concepts (Ferrara et al., 2006). The interest for programming waned when it
was questioned whether the complexity of learning the programming
language was seen as counterproductive to learning mathematics. It was
found difficult to separate features connected to programming from specific
features of mathematics (Samurcay, 1985).
In the nineties, a wide range of software, constructed to bring students to
practice a particular mathematical content, was introduced. For example, in
line with studies suggesting that students´ errors associated with algebra
often could be related to inattention to expression structure, special
education software, as Expression, were developed. Expression allowed the
students to submit incorrect expressions but the computer would not carry
4
out the associated calculations unless the expressions were correctly
submitted. I.e. the students could continue to submit expressions until a
correct expression was found. In a study, students who had practiced using
Expression were found to internalize the mathematical structure and make
less errors (Ferrara et al., 2006).
A further development of these kinds of software is the introduction of
microworlds, i.e. applications providing virtual contexts with integrated
problems and tasks. But in contrast to software like Expression these
applications could provide students with hints, associated to the students’
action, which guided them how to proceed. Among these we find for example
SimCalc (Ferrara, 2006).
Another approach of using technology in mathematics education is the
development of dynamic software like GeoGebra, Cabri and Sketchpard. This
kind of Software do not provide problems or tasks to solve, but could be used
as tools to explore specific mathematical concepts and relations, directed by
tasks designed outside the software environment. When dynamic software is
used in mathematics education, the teacher needs to define the learning
target and create appropriate tasks to the students to solve (Olive et al.,
2010). Dynamic software, like GeoGebra, provide an environment where
students, for example, may explore the relationships between functions
representations; algebraic, graphical and tabular. Given that the teacher
assigned them to do so. These dynamic representations afford a more
interactive experience that emphasizes meaning-construction rather than
symbolic manipulation (Ferrara et al., 2006).
The two articles in this thesis investigate in what way GeoGebra may
enhance creative mathematical reasoning.
3. Frameworks
The basis for the thesis are the two articles, both using combined
frameworks. The framework for creative and imitative reasoning (Lithner,
2008) has been the starting point but the ICT-perspective has resulted in
consideration of further frameworks. In the article 1 students working in
pairs are examined and therefore parts of Roschelle and Teasly´s (1994)
framework of collaboration were used. Article 2 focuses on feedback from
software and success in task solving, which resulted in merging parts of
Shute’s (2008) framework of formative feedback and parts of Schoenfelds
(1985) protocol analysis into the framework of the study. Since the articles
present only parts of these frameworks a more thorough presentation of each
of these frameworks follows.
5
3.1 Imitative and creative reasoning
The framework addresses the problem of rote learning in the perspective of
key-aspects of imitative and creative mathematical reasoning. Relating
reasoning to thinking processes, students´ competencies, and the learning
milieu offers possible explanations for consequences of different types of
reasoning. The background of the framework is several empirical studies (J
Lithner, 2000; J. Lithner, 2003) investigating what made students fail or
succeed in solving different practice and test tasks. The focus came to be on
the disadvantages of being restricted to imitative reasoning. The framework
is trying to separate the reasoning sequence from the thinking process that
created it. The reasoning sequence is a product of the thinking processes,
which are assumed to be fundamentally different in creative mathematical
reasoning and in imitative reasoning.
3.1.1 Mathematical thinking processes and reasoning sequence
The notion of creativity is not restricted to genius or experts. Instead
ordinary students´ task solving can be based on creative thinking processes
that are flexible, fluent, admitting different approaches, and not hindered by
fixation. Superficial and imitative thinking allow students to follow solving
schemes suitable to specific tasks, which means that analytical and
conceptual thinking processes may not be necessary to solve the task. The
lack of analytical support makes the choice of solution method haphazard. In
order to separate reasoning from thinking processes, the choice is to see
reasoning as a product that appears in a reasoning sequence, starting with a
task and ending with an answer. The reasoning sequence may consist of
written solutions, interviews, think-aloud protocols, etc.
In order to exclude creative thinking like brainstorming and creative
behaviors like trial and error from creative mathematical reasoning
conditions of argumentation and anchoring were added. Argumentation is
the part of reasoning that has the purpose of convincing you or someone else
that the reasoning is appropriate. Argumentation is predictive (why will the
chosen strategies contribute to the solving of a task) and verificative (why did
the strategies solve the task). The notion of anchoring is about using relevant
mathematical properties of the components of the reasoning. The
components are: objects (numbers, variables, functions, etc.),
transformations (an object is transformed through a process, e.g. finding
max and min of a polynomial), and concepts (a central idea build on a set
objects, transformations, and their properties). Anchoring may be in
6
intrinsic or surface properties. Anchoring is associated to the purpose of a
transformation, e.g. finding out which is the largest number of 4/5 or 12/18
must be anchored in the intrinsic property of the ratio, not in the surface
properties of the size of the numbers.
3.1.2 Imitative reasoning
Empirical studies behind the framework have identified two main types of
imitative reasoning, memorized and algorithmic.
Memorized reasoning
Lithner gives an example where 150 students had an examination task, “state
and prove the Fundamental Theorem of Calculus”. Half of them got full
credit and their answers were copies of the two-page example in the
textbook. Most of them who failed had answers that in large parts were as in
the textbook but some pieces were missing or different. In a post-test the
students were asked to explain a minor part of the proof. Even though they
had managed to memorize the whole proof they were only able to explain
some sequences. A task asking for a proof may be suitable for memorizing.
Two other examples are tasks asking for facts (e.g. “How many dm2 is a
square-meter?”) and definitions (e.g. “What is a tetrahedron?”). Memorized
reasoning fulfills the following conditions (Lithner, 2008. p. 258):
1.
2.
The strategy choice is founded in recalling a complete answer The strategy implementation consists only of writing it down Another type of memorized reasoning builds on established experiences
from learning environment including apprehensions of facts, concept
images, and beliefs. Examples are arguments like “The slope of the line
should be a smaller number because it usually is less than 5”.
Algorithmic reasoning
An algorithm can be determined in advance and is not necessarily only an
explicit chain of executable instructions (e.g. the division algorithm). A wider
notion is that all pre specified procedures are algorithms (e.g. zooming in the
intersections with x-axis using a graphing calculator to find the zeros of a
quadratic function). The main point is that conceptually difficulties are taken
care of by the algorithm and only the easy parts are left to the student.
Algorithmic reasoning fulfills the following conditions (Lithner, 2008, p.
262):
7
1.
The strategy is to recall a solution algorithm. The predictive
argumentation may be of different kinds, but there is no need to
create s new solution.
2. The remaining reasoning parts of the strategy implementation are
trivial for the reasoner, only careless mistakes can prevent an answer
from being reached.
Algorithms make it possible for students to carry out advanced mathematics
with limited understanding, for example it is quite reasonable to learn 7-year
old children to differentiate simple polynomials but it is not likely that they
understand the mathematics behind the procedure. In algorithmic reasoning
the main difficulty is to identify a suitable algorithm. Three common ways
will now be described.
Familiar algorithmic reasoning
The strategy of finding a suitable algorithm is often based on established
experiences that certain textual, graphical, and/or symbolical features are
related to corresponding algorithms. For example a student solving the task
“how many m3 are 287 dm3” knows that when you convert cubic measures
from dm3 into m3 you divide with 1000. As soon the algorithm is identified
287/1000 is implemented. Algorithmic reasoning in a task solution fulfills
the following criteria (Lithner 2008, p.262):
1.
The reason for the strategy is that the task is seen as being of
familiar type and can be solved by a corresponding known
algorithm.
2. The algorithm is implemented.
The arguments that convince the reasoner of the strategy are often based on
surface properties, like similarity to practice tasks or the established
experience that certain tasks are associated to certain textual and graphical
features. As long as the right algorithm is chosen only careless mistakes may
hinder reaching an answer. In the example above the student chooses the
right algorithm and there is no anchoring in the properties of cubic measure
leading to the division with 1000. The student may have remembered
wrongly and choose to divide for example with 100 without noticing that this
led to the wrong answer. Since familiar algorithmic reasoning not necessarily
is based on anchoring in intrinsic mathematic properties it is not reliable in
problematic situations.
8
Delimiting algorithmic reasoning
Ronald, a 4th grade student, solves the task “how many apples will 12 persons
have each if they share 8 apples?” He uses the calculator to divide 8 with 12
and gets the answer 0.66…. He considers this shortly and then divides 12
with 8 and sees 1.5 apples per person as an answer. His argument for the
change is that the larger number is usually the one you start with. Delimiting
algorithmic reasoning is used when familiar algorithmic reasoning does not
work and guidance is not available (e.g. in a test session). It fulfills the
following conditions (Lithner, 2008, p.263).
1.
An algorithm is chosen from a set that is delimited by the reasoner
through the algorithms surface relation to the task. The outcome is
not predicted.
2. The verificative argumentation is based on surface considerations
that are related to the reasoner’s expectation of the requested answer
on solution. If the implementation does not lead to a (to the
reasoner) reasonable conclusion it is simply terminated without
evaluation and another algorithm may be chosen from the delimited
set.
Guided algorithmic reasoning
There are two common variants of guided reasoning, text-guided and person
guided. They are used when familiar or delimited algorithmic reasoning does
not work. In text guided reasoning the following conditions hold (Lithner
2008, p.263).
1.
The strategy choice concerns identifying surface similarities between
the task and an example, definition, theorem, rule, or some other
situation in a text source.
2. The algorithm is implemented without verificative argumentation.
A structure of a solved example followed by a set of similar tasks is common
in Swedish textbooks (Lithner 2008). Text guided algorithmic reasoning is
also found as common in small-groups learning situations (Lithner 2003).
Person guided algorithmic reasoning fulfills the following conditions
(Lithner 2008, p.264).
1.
All strategy choices that are problematic for the reasoned are made
by a guide, who provides no predictive argumentation.
9
2. The strategy implementation follows the guidance and executes the
remaining
routine
transformations
without
verificative
argumentation.
Lithner offers an example where a students want assistance to solve the task
“How much is 15% of 90?”. The teacher writes 0.15 multiplied with 90 as the
standard algorithm in the student’s notebook. Then the teacher tells the
students how to calculate every step in the algorithm. There is no
argumentation why this calculation and no anchoring in the intrinsic
properties of percentage. After the calculation is processed the teacher leaves
without further comments.
3.1.3 Creative mathematical reasoning
Two students, Olga and Leila are working on a task where they are supposed
to find the rule for two linear functions creating perpendicular graphs. They
have managed to create y=2x-1 and y=-0.5x-1. They hypothesized that one xcoefficient must be the negative fourth of the other. They tried this on several
examples with different x-coefficients, all failures. A little later the following
conversation took place.
1.
2.
3.
4.
5.
6.
7.
8.
Olga: what is common for our two examples (y=x-1 and y=-x-1,
y=2x-4 and y=-0.5x-4)….
Leila: they are like opposites….
Olga: one divided in two is zero dot five….
Olga: yes… and one divided in one is one…. but minus…
Leila: that’s it… one divided in one but minus…
Olga: then something times something must be one… but minus….
say a number…
Leila: six….
Olga: then the other one must be…. one divided to six…. but minus
[submit y=6x and y=-1/6x]…
They create a strategy, choose an x-coefficient and divide one with the
chosen x-coefficient and turn it into negative to have the other x-coefficient.
The argumentation (line 6-8) is predictive. They implement the strategy (line
8, submitting y=6x-1 and y=-1/6x-1) and after the conversation above they
state they were right. They verified that by testing on several examples.
Finally they draw the conclusion that the product x-coefficients must be (-1)
to create perpendicular graphs. On the question from the researcher of why
they were right they answered, “one x-coefficient must be minus and if one of
them is for example (5) the other one is (-1/5) because they must go in
different directions and if one slope is steep the other must be less steep”.
10
This is an example of Creative mathematical reasoning, which is defined as
fulfilling the following criteria (Lithner 2008, p.266)
1.
Novelty. A new (to reasoned) reasoning sequence is created or a
forgotten one is re-created.
2. Plausibility. There are argumentation supporting the strategy choice
and/or strategy implantation motivating why the conclusion are
plausible.
3. Mathematical foundation. The arguments are anchored in intrinsic
mathematical properties of the components involved in the
reasoning.
Creative mathematical reasoning does not have to be a challenge and the
definition also includes elementary reasoning.
3.2 Joint problem space
Focusing on socially negotiated knowledge elements, Roschelle and Teasly
(1994) investigated computer aided collaborative problem solving. Their
framework focuses the process of joint work rather than the outcome. The
definition of collaboration is “a coordinated, synchronous, activity that is the
result of a continued attempt to construct and maintain a shared conception
of a problem” (Roschelle & Teasly 1994, p.70). Roschelle and Teasly make a
distinction between collaboration and co-operation, the former is a mutual
engagement in solving a problem together and the latter when the problem
solving is divided among co-workers. Furthermore, Roschelle & Teasly
discuss the distinction between synchronous and asynchronous
collaboration. The authors point out that collaboration in face to facesituations is always synchronous. Roschelle and Teasly (1994) formed a
framework on the proposal that collaborative problem solving takes place in
a socially negotiated and shared conceptual space, constructed through
external language, situation and activity, i.e. a joint problem space (JPS).
Since the framework aims at investigating collaboration supported by
software the computer´s potential to assist students´ explanations and
production of shared knowledge is emphasized.
To examine joint problem solving it´s necessary to examine not only the
content of students’ dialogues, but also how their conversations result in
shared knowledge. Roschelle and Teasly (1994) propose a method to
structure the conversations and activities during problem solving in a joint
problem space (JPS). The JPS is a pragmatic structure into which students
introduce and accept knowledge, monitor ongoing activities for evidence of
divergences, and repair divergences that impede the progress of
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collaboration. Categories of events during the conversation that are useful
for analysis are; turn taking in general (flew, content, structure), socially
distributed productions, repairs, narration, and language and actions. These
categories will be presented in the following paragraphs.
3.2.1 Turn taking
The structure of the turn taking sequences in a conversation may indicate the
degree of students sharing of a common problem. The flow and content are
indicators of whether the participants understand each other. During
successful collaborative activities, the participants’ turn taking is smooth and
builds upon each other’s utterances. However, even successful collaborative
problem solving usually includes periods of disagreements and
misunderstandings. Furthermore, participants may occasionally withdraw
from the interaction when their thoughts are too ill formed or too
complicated to articulate. In successful collaborative problem solving these
periods of misunderstandings or silence are followed by intense interaction
to transform individual insights into shared knowledge.
3.2.2 Social distributed production
Roschelle and Teasly (1994) define social distributed production as one way
of engaging in turn taking, that is when one participant initiates an idea or a
sentence and the partner completes it. An example is sentences havening the
form of IF-THEN. One collaborator expresses the IF-part and the other the
THEN-part. This means the partners have opportunity to accept or repair
contributed knowledge. When both are satisfied with the sentence shared
knowledge is produced.
3.2.3 Repairs
Repairs mean the collaborators negotiate and solve divergent meanings or
misunderstandings. It is about dealing with troubles in speaking, hearing,
and comprehension of a dialogue. A collaborative process also includes
individual activities, which may lead to conflicts when individual ideas are
negotiated with respect to the shared work. In a successful collaboration the
intention of the conversation is to reduce such conflicts and maintain shared
understanding. That is what Roschelle and Teasly (1995) describe as repairs.
Without successful repairs breakdowns in the mutual understanding will
continue longer. Unsuccessful repairs may cause students to abandon the
current problem, try a new approach, or continue around the impasse
without repair.
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3.2.4 Narration
When two students use interactive software sharing one computer they need
to take turn in using the keyboard or the mouse. The actions made by one
student using the keyboard may be difficult for the partner to interpret.
There is a need to describe (narrate) the intention of the activity to one
another. These dialogues enhance the partner’s possibility to recognize
differences in shared understanding. Continued attention and engagement
in one another’s to narration may indicate shared understanding.
Interrupting narrations is a possibility to rectify misunderstanding.
Narration is useful when one partner signals that an action is not directly
intended to contribute to the shared goal; an utterance like “I´m just trying
this” signals that the student is exploring something that is not directly
related with the task at hand (e.g. find out how a quadratic function works in
GeoGebra even though the task at hand concerns linear functions).
3.2.4 Language and action
When working with computer support students are not dependent only on
verbal communication to maintain shared understanding. The computer
environment provides visualizations of actions and gestures. An action (e.g. a
submitting an expression into software, manipulating of an existing
expression in software, etc.) or gesture can serve as acceptance when one
partner interprets the other´s utterances. The production of an appropriate
action confirms a shared understanding. Actions and gestures can also serve
as presentations of new ideas. On partner demonstrates an idea through
computer action or gesture and if the other partner successfully interpret the
idea through utterances this indicates mutual understanding of the idea.
Effective collaboration to reach shared understanding could be achieved
when students’ activities and language complement each other. While one
partner concentrates on carrying out activities the other produces utterances
that make intentions behind the activity available for commentary and
repair.
Roschelle and Teasly (1994) used the framework to structure and analyze
data gathered in a study investigating students´ process of collaborative
problem solving. The results suggest that shared conception is the basis of a
successful problem solving. It is furthermore assumed that the process of
solving the task and maintaining the joint problem space co-exist. Finally,
Roschelle & Teasly discuss conversation, as the key activity to successfully
solve the problem and simultaneously construct and maintain the joint
problem space.
13
3.3 Formative feedback
When students solve problems using dynamic software their planning of and
their evaluating of the computer activities will be affected by the feedback
that the software generates. Students will use feedback in different ways,
which may affect the way they are solving the task. Thus it is important to
describe different attributes of feedback.
3.3.1 Feedback and digital technology
In terms of feedback, interactive software offers several benefits. The
outcome is always accurate and in line with the executed commands, it is
delivered instantaneously, and responds to students’ actions without
judgment (Sacristán et al., 2010). The capability of digital media to invite
students to immediate test and reflect on existing knowledge, enhance
exploration of ideas and promotes reasoning and learning (Chance, Garfield,
& delMas, 2000; Sacristián et al., 2010). Weir (1987) discusses the use of
digital technology in educational situations as trying out something,
watching for effects, and responding to feedback. Kieran and Drijvers (2006)
stress the tension awakening from differences in output (the feedback), and
students´ expectations as most valuable for learning. The features of
feedback from dynamic software mentioned above are mostly associated to
effects on learning, which is synonymous to formative feedback (Shute,
2008).
3.3.2 Formative feedback
A definition for formative feedback is “information communicated to the
learner that is intended to modify her thinking or behavior to improve
learning (Shute, 2008. p.258). Furthermore, formative feedback is
information to a learner in response to some action from the learner´s part.
Effective formative feedback should be non evaluative, supporting, timely,
and specific (Shute, 2008). With the overarching goal to identify the features
of formative feedback that, according to research, are most effective and
efficient in promoting learning, Shute presents a comprehensive literature
review, that presents guidelines for generating formative feedback. Article 2
adopted parts from Shute´s review that deals with feedback as verification
and elaboration associated to feedback on task level. This will be presented
in the following.
The review (Shute, 2008) is written on the premise that good feedback can
significantly improve learning processes and outcomes, if delivered correctly.
Emphasis is on the last three words. The vast literature reveals a wide range
14
of formative feedback, e.g. accuracy of solution, topic contingent, partial
solutions, etc., and different variables have been identified and scrutinized.
The results of investigating the same variables are often divergent and
different variables are often shown to interact with other variables, such as
students´ achievement-level, task-level, and prior knowledge. Shute (2008)
focuses on task-level feedback, as opposed to general summary feedback.
General summary feedback may consist of for example information of
current understanding, while feedback on task-level is specific and timely
information about for example a particular response to a problem or a task.
In research, feedback is quite often found to have negative effects. Features
of such feedback are often described as; criticizing, controlling, providing
grades and indicating students standing negative to peers. Such feedback is
however not described as formative (Shute, 2008).
3.3.3 Features of formative feedback
Black and William (Black & Wiliam, 1998) propose that formative feedback
has two main functions; to direct and to facilitate. Directive feedback tells
the student what to correct or revise. Facilitative feedback provides guidance
to the students on their revision and conceptualization. In a cognitive
perspective formative feedback has the purpose of reducing uncertainty and
cognitive load. To receive information about how well a student performs on
a task may reduce his or her uncertainty, which may lead to higher
motivation and more efficient task strategies (Shute, 2008). A novice or a
struggling student can be cognitively overwhelmed due to high performance
demands. Supporting feedback, e.g. explanations, can reduce cognitive load.
Feedback that may be useful for correcting inappropriate strategies,
procedural errors, or misconceptions seems to be effective when the
provided information is more specific. Shute (2008) describes this as
feedback specificity, which will be discussed in next paragraph.
Feedback specificity is the level of information presented in feedback
messages. That is, specific feedback provides information about particular
responses or behavior in addition to being accurate and tends to be more
directive than facilitating (Shute, 2008). Research has reported that
feedback is more effective when it provides details about how to improve the
answer than just indicating whether students are correct or incorrect.
(Bangert-Drowns, Kulik, Kulik, & Morgan, 1991; Mason & Bruning, 2001)
Formative feedback may be divided into verificative and elaborative.
Verificative feedback provides information whether an answer is correct or
not and can be delivered in different ways, e.g. explicitly as a check-mark, or
implicitly as an unexpected result of a simulation. Elaborative feedback has
several variations, e.g. to address the topic, to provide worked examples, etc.
15
Research has reported that response specific elaborative feedback is effective
since this kind of feedback addresses the question why the answer is correct
or not.
Even though research have isolated and proven specific characteristics of
formative feedback as effective for learning, there are no straight and simple
advices how to perform feedback. The success of feedback is dependent on
other components in education such as students´ achieving-level, complexity
of tasks, character of issue, etc. For example there are research describing
so-called delayed feedback as beneficial for learning, however only to high
achieving students Low achieving students on the other hand, benefit the
most from feedback with low complexity. Shute (2008) suggests that a
reasonable advice in the case of feedback in mathematics education, when it
comes to conceptual tasks, is to keep the feedback specific and clear.
3.3.4 Dynamic software and feedback as verification and
elaboration
The process of using software to solve mathematical tasks could be described
as: trying out something, watching for effects, and responding to feedback
(Weir, 1987. That kind of feedback could be labeled as verificative. However,
the software does not provide explicit information whether something is
correct or not. The students have to interpret the feedback to retrieve useful
information. The students’ planning for computer activities is crucial to
understand the feedback generated from software. The more prepared the
students are the more likely will the feedback be used for
elaboration.,(Sacristán et al., 2010)
3.4 A framework for macroscopic analysis of problem
solving protocols
Schoenfeld (1985) proposed a framework for macroscopic analysis of
problem solving protocols focusing on decisions on the executive or decision
level. Decisions at the control level are those that affect allocation of problem
solving resources. The method provides a way to identify important points of
decisions during a problem solving process and to examine the ways
individuals´ behavior shape the way the process evolve. Three types of
decision points are described: when there are major shifts in the resource
allocation, when new information through problem solving activities are
coming up, and when difficulties are indicating that something is wrong with
the approach. Due to the protocol analysis the conversations are partitioned
into chunks of consistent behavior called episodes. Episodes are periods
16
when students mainly are engaged in the same type of activity such as
reading, analysis, exploring, planning, implementing, and verifying.
Changing from one episode to another is considered as a major shift in
allocation and as a possible decision point. The junctures between episodes
are decisions points when the students may change direction of the problem
solving activity significantly. When new information or possibilities of taking
a different approach comes to attention, the problem solver has the
possibility to make decisions shaping the problem solving process. New
information may arise in the middle of an episode and may not, at least not
immediately, be considered. The third possible decision point is when the
process of problem solving has been accompanied by minor difficulties for
some time, indicating that something is wrong with the approach.
All episodes are labeled into reading, analysis, exploration, planning,
implementation, and verifying. For each label and the transitions between
episodes the framework provides relevant questions. The questions are of
various types; some can be answered objectively (e.g. “are the action driven
by the goals of the problem?”), others call for judgment of problem solving
behavior (e.g. “does the problem solver assess the current stage of her
knowledge?”), and some ask for reasonableness of certain behavior (e.g.
regarding the last question; “is it appropriate to do so?”). Parsing a protocol
into episodes and providing answers to the associated questions obtain a full
characterization of a protocol. Schoenfeld (1985) admit that some of these
questions can only be answered subjectively, but that such a systematic
model will increase objectivity.
In order to provide insights into differences between experts’ and novices’
problem solving processes Schoenfeld (1985) analyzed protocols of their
mathematical task solving. The study shows that an expert on the control
level is more efficient in using what she knows even if the expert doesn´t
have recent experiences of the mathematical content of the task. Compared
to students considered as novices, the experts more frequently assess and
monitor the current state of the solution and more frequently analyze and
verify parts of the solving procedure. Schoenfeld (1985) noted that one type
of expertise could be defined as someone who knows in advance what kind of
information and procedures that are needed to solve familiar tasks. Another
type of expertise, on novel problem solving, could be defined as someone
who can solve problems of an unfamiliar domain using general problem
solving techniques and strategies. Traditionally, the view had been that the
reason that novices were less proficient problem solvers was that they lacked
content knowledge. Schoenfeld’s (1985) study on the contrary showed that
students who recently had studied the mathematical content of the problem
17
were less successful than experts who had no recent experiences of the
mathematical content but instead had better general problem solving skills.
4. Methodology
This section will discuss the combining of different theories into a research
framework, methodological considerations for the two articles, and a
background for the design of the didactical situation that was used in both
studies.
4.1 On using multiple frameworks
The aim of this thesis is to extend our knowledge about students´ reasoning
associated to task solving supported by dynamic software. The research
questions are central and formulated on basis of what is considered as
problematic in relation to this aim. However, the questions are not restricted
to any specific theory or method. Therefore it´s important that the questions
are associated to relevant theories, methods, and analyses. Lester (2005)
distinguishes between theoretical and conceptual frameworks. A theoretical
framework is built on established coherent ways of explaining certain kinds
of phenomena and relations (e.g. Piaget´s theories of intellectual
development and Vygotsky´s sociocultural theory are two prominent ones).
Using established theories as fundaments in a framework means that the
research questions and the method should be phrased in terms associated
with the theory and so would the argumentation, expressions and use of
conventions. This has obvious advantages like facilitating communication,
sharing and presenting the process among others working with similar
questions. Lester points out some problems about using theoretical
frameworks. The data may be explained by theoretical decree rather than
evidence, the data may be stripped of context and local meaning to fit the
theory, the researcher may set a standard for scholarly discourse that is not
functional outside the academic discipline and using a single theory may
exclude the possibility of triangulation. Like theoretical frameworks,
conceptual frameworks are built on previous research but on an array of
several sources. It may be based on different theories dependent on what the
researcher considers as relevant for the current research problem. A
conceptual framework is more for justification than explaining. It is arguing
for the appropriateness of concepts chosen for the investigation and whether
they are useful given the investigation problem (Lester, 2005).
The purpose of using more than one theory and/or framework should be that
you might discover further aspects, not to ensure the possibilities of
18
justifying your discoveries (Radford, 2008). Schoenfeld (1992) points out
that it is on the researchers responsibility to document and justify when
developing methods. Investigating what one finds interesting may mean
there are no standard methods. If new methods are not explained and
described where they come from and how to be used, the associated
investigation can lead to “ad-hoc empiricism” that is theoretically shallow
(Schoenfeld, 1992). Niss (2007) states that the increasing complexity of the
mathematics education research field has led to more complex research
frameworks. Niss also points out that it is crucial to have clear connections
between theories, methods, and research questions. Radford (2008),
referring to Niss, suggests that a theory can be seen as a way of producing
understanding based on: basic principles that delineate the perspective of
the research, a methodology including techniques for collecting and
interpreting data, and a set of schematic questions, which will generate
specific research questions. This gives a context of designing theories for a
study working simultaneously with all three aspects. When research
questions appear more clearly it will have consequences for methodology
and the way basic principles for the study are justified through theories
(Gellert, 2010).
4.2. Why these frameworks?
Lester (2005) emphasizes the importance of explaining and justifying in
what ways different theories will be used for the analysis. Lester argues that
a “Grand theory of everything” will never be developed in the field of
mathematics education. Instead the focus should be on using smaller, more
focused theories and models of teaching. A conceptual framework can be
viewed as a source of ideas that can be appropriated and modified for
purposes of mathematical education (Lester, 2005). The basic idea in both
studies that are included in this thesis is that causes for students´ behavior
associated to task solving can be examined through investigating of their
reasoning. The tasks are designed with the purpose to represent non-routine
tasks and that solving them will engage students´ in creative reasoning. The
framework of imitative and creative reasoning (Lithner, 2008) offers
structures for capturing articulated results of thinking processes in a
reasoning sequence, a path through task solving, and definitions to classify
creative and imitative reasoning (see chapter 3 for further details).
In the context of collaborative task solving aided by software students need
to structure and co-ordinate conversations and computer activities in order
to understand each other. There are parts that are mutually silently
understood, parts that are mutually understood through visualization, and
the outspoken language is sometimes cryptic. The framework of joint
19
problem space (Roschelle & Teasly, (1994) offers a construct that combine
conversations and computer activities into analyzable sequences (see chapter
3 for further details).
Task solving in collaboration and aided by dynamic software implies that the
students prepare a computer action, resulting in information that may be
utilized as feedback. Computer feedback itself in the studies of this thesis is
neutral, i.e. it is either directly verifying or gives guidelines how to proceed.
It is up to the students to determine whether an answer is correct or not, and
to evaluate the feedback looking into questions like; why did (not) our idea
work. Again, it is the student who chooses how to utilize the feedback. To
explain different ways of using feedback it is appropriate to label different
forms of feedback that can be associated to different ways of using it.
Definitions of Shute (2008) describing feedback on task level as verificative
and elaborative has been used in article 2 (see chapter 3 for further details).
In article 2 one of the objects of analysis is students’ success in problem
solving. With respect to the design of the tasks, the trivial part is to
determine whether the answers are correct or not. What is more critical is to
have a view of the way the path through the problem solving is shaped.
Schoenfeld (1985) suggests a focus on possible decision points is an
appropriate way to find moments when problem solving takes new directions
(see chapter 3 for further details). Schoenfeld’s (1985) protocol analysis is in
article 2 used as a method to find potential and and actual decision points
which in turn helps to find the parts of the task solving sequence that are
crucial with respect to reasoning and utilization of feedback.
Central during the work with both studies has been to maintain a clear
connection between research questions, analysis, and justification of results.
This will be described in next paragraph.
4.3 Methodological considerations
In this chapter consequences of the chosen theories and methods are
discussed. With respect to article 1 the choice to study social activities is
argued for. In association to article 2 the view of feedback and the
contribution of detecting decision points in the task solving sequence are
discussed.
4.3.1 Article 1
The purpose of article 1 is to develop insights into how GeoGebra can be used
as a means of supporting collaboration and creative reasoning during a
20
problem-solving process. Collaboration associated to working with dynamic
software (Hoffkamp, 2009; Rakes, Valentine, McGatha, & Ronau, 2010), and
creative mathematical reasoning (Jonsson et al, in preparation) has each
independently been suggested as beneficial for learning. Therefore it´s
important to investigate the way dynamic software affects reasoning and
collaboration. The research questions posed were; “To what extent do
students use GeoGebra to collaborate during problem solving?” and “What
characteristics of GeoGebra might contribute to or obstruct their creative
reasoning?” Due to the two objects of study, reasoning and collaboration, it
was considered to combine Lithner´s (2008) framework with the virtual
construct “joint problem space” (Roschelle & Teasly, 1994). Pilot studies
were carried out with students working both individually and in pairs.
Students working in pairs seemed to create rich but also more complex data.
The collaborative setting and the addition of Roschelle and Teasly´s (1994)
framework brought more complexity since the approach is to focus on the
social interaction. Collaboration is considered as constructing shared
knowledge through interaction within a social context. Stahl (2005) suggests
there is no reason to deny individual thinking and learning in a group
activity, but it is more informative to study the processes on group level.
Furthermore shared knowledge arising in a group activity cannot be
attributed to individuals, although the contributions are individual
utterances and shared knowledge are internalized by individuals (Stahl,
2005). Roschelle and Teasly (1994) have a similar view, they don´t deny
individuals contributions and internalization but emphasize the social
interaction. Reasoning starts in a task and ends in an answer and can be
observed in a reasoning sequence as written solutions, interviews, think
aloud protocols, etc. The study suggests that collaboration could be added to
of the reasoning sequence, including social actions and interactions with
software. The approach in the study is to see the components of the
collaborative reasoning sequence as results of individual thinking and social
collaboration. To gather appropriate data a non-routine task suitable to solve
through GeoGebra was designed (see article 1). Students were instructed to
work in pairs using one computer, and the conversations, computer
activities, and gestures were recorded. In order to answer the question of
collaboration the framework of joint problem space (Roschelle & Teasly,
1994) was used and the reasoning was analyzed through the reasoning
framework (Lithner 2008).
4.3.2 Article 2
Article 2 aims at understanding the relations between reasoning, feedback,
and success in computer aided task solving. Research proposed that dynamic
software enhance reasoning, deliver feedback, and as a tool for problem
21
solving (källor). Therefore it is important to investigate the relation of these
aspects, not only from the perspective that it is the computer that affects
reasoning and provides feedback, but that the students´ actions affect the
utilization of feedback. The posed questions were; “What is the relation
between the students’ way of using the feedback that GeoGebra generates
and the students´ reasoning?” and “How do students´ ways of reasoning and
utilization of feedback from GeoGebra relate to their success in problem
solving?” Methodologically experiences from article 1 were adapted, in the
sense that students worked in pairs, solving a non-routine task, and their
conversations, computer activities, and gestures were recorded. Data were
structured by the reasoning framework, and since collaboration was not an
object of analysis the joint problem space framework (Roschelle & Teasly,
1994) was excluded. Even though collaboration was not an object of analysis
students were instructed to work together and data forming the reasoning
sequence were results of collaborative activities. The computer was used for
task solving and the information that computer activities generated was
considered as feedback. In order to distinguish different forms of utilization
of feedback definitions of verificative feedback (whether an answer is correct
or not) and elaborative feedback (why an answer is correct or not) from
Shute (2008) were used. Feedback in the study was considered as a result of
students’ planned computer activities and the way they used the result in the
subsequent task solving process. This view of feedback reflects Brousseau´s
(1997) theories of didactical situations suggesting that feedback not
necessarily is information from one person to another. Feedback may as well
be result of students´ acting on the learning environment, resulting in
changed conditions of the learning environment, which mean the student
have reason to change learning behavior. Shute (2008) defines feedback that
in research has been considered as effective for learning. Feedback on task
level is found effective when it is directed to the answer that students’
(instead of for example to the way students perform or to the topic of the
task) produce and is divided in two variants; verifying and elaborating.
Verifying means that the feedback informs whether an answer is correct or
not and elaborating means that the feedback in addition informs why an
answer is correct or not. In the study the choice is to consider feedback from
computer as on task level (i.e. it is directed to the task at hand, not to for
example how to solve tasks of the topic in general), and the use of feedback
as verificative, elaborative or both. Thus the components for analysis are
verificative and elaborative use of feedback. During the work through task
solving, students face situations where they have to decide what in what
direction they will continue. To identify points of decisions that shape the
way through solving the task, Schoenfelds (1985) framework for protocol
analysis was used. The transcribed data was parsed into episodes, i.e. periods
where the problem solvers are engaged in activities of the same type or
22
character. The junctions of such episodes, when new information comes up,
and when solving is accompanied with difficulties are considered as possible
decision points. Identifying decisions gives an opportunity to understand
how decisions shape the way of solving the task, and can help the researcher
to find reasons behind the students’ task solving success or failure.
Schoenfeld points out that interpreting problem solving through systematic
protocols of verbal data is about dealing with subjectivity. Using such a
framework for protocol analysis helps the researcher to understand the task
solving processes and offers the reader opportunities to form a view of the
approach.
Only selected parts of the frameworks and theories mentioned in this thesis
have been used in the analyses. The choice of what´s regarded as important
has been guided by the research questions. For example students’ planning
and evaluation of computer activities could have been explained by those
parts of Schoenfeld’s protocol analysis that deals with components like
monitoring and assessing, and the amount of components associated to use
of feedback could have been largely extended. In summary the research
questions have been answered by classifying reasoning as imitative or
creative (Lithner, 2008), classifying use of feedback as verificative and/or
elaborative (Shute, 2008) and identifying protocol analysis decision points
that affect task solving success and failure (Schoenfeld, 1985). To gather
suitable data a didactical situation was designed, which will be presented in
next paragraph.
4.4 Didactical design
The didactical situations in both studies, were designed to bring the students
to collaborate, working in pairs and to use GeoGebra to solve the given
problem. The propositions for the didactical design of these studies are built
on theories of Schoenfeld (1985), Brousseau (1997), and on research
investigating collaborative problem solving (Lou, Abrami, & d’Apollonia,
2001; Mullins, Rummel, & Spada, 2011).
Schoenfeld (1985) argues that students need to work with mathematical
problems that to some extent are new to them to develop problem-solving
skills. In addition the task must constitute an intellectual challenge.
Schoenfeld formed a framework based on four key components: resources
(basic knowledge), heuristics (rules of thumbs for non-standard problems),
control (metacognition, monitoring, and decision making), and beliefs
(mathematical world view). In empirical studies Schoenfeld found that
novices often have sufficient resources but are lacking the other three
components. In order to provide students with an intellectual challenge,
23
tasks should not include examples or instructions providing solution
methods and they should contain requests to explain why an answer is
correct.
Brousseau (1997) points out the importance of the devolution of
responsibility to the students for solving a mathematical task. It is the
students who shall create the solution. The problem is formulated so that the
learning target for the problem (e.g. the area of the circle, rules for
arithmetic, properties of quadratic functions, etc.) must be considered by the
students in order to reach a solution. The students must be informed of the
circumstances of the problem, like rules of a game, but if they will be
instructed how to solve the problem they will not reach the learning target.
Research examining students working in small groups aided by computers
has shown divergent results. Identified components that affect outcomes of
computer aided group activities concern for example; whether the tasks are
focused on procedures or conceptual understanding, if the tasks are asking
for numerical results or explanations, if the studies are measuring group
results or individual results, if studies are measuring individual learning
outcome and features of software (Mullins et al, Lou et al.). Considered as
beneficial for both group result and individual learning are small groups (2-3
participants) working on a single computer, tasks aimed at conceptual
understanding requiring explanations and interactive software (Mullins et al,
2011). Roschelle and Teasly (1995) argue that the collaborative constructing
of shared knowledge makes students deal with more advanced learning
objects compared to what they do on their own.
Several tasks were initially designed and pilot tests were performed to
determine whether or not they constituted a challenge to the students. The
pilots included settings with individual students and pairs. The data was
shown to be richer when pairs were solving the task in collaboration. The two
tasks chosen for the two studies provided students in the ages of 16-17 a
suitable challenge and were reasonable to solve using GeoGebra. The
learning target for both were the components of the formula y=mx+c for
linear functions and the rule for the choice of x-coefficients to create
perpendicular graphs. The task required only minor instructions in order to
reach a sufficient devolution of the problem.
24
5. Summary of the results of the articles
5.1 Article 1
This article investigated the way dynamic software, GeoGebra, may support
students´ collaboration and creative reasoning during mathematical
problem solving. The research questions posed were: “To what extent do
students use GeoGebra to collaborate during problem solving?” and “What
characteristics of GeoGebra might contribute to or obstruct their creative
reasoning?” Students´ were found to use GeoGebra as a shared working
space within all their actions (pointing, sketching, submitting, etc.) were
situated and shared. Their individual reasoning was shared and
synchronized through collaborative activities where they were mixing
language and actions, always using the information from GeoGebra as
reference. For example, John and Mike attempted to create a vertical line by
using a large x-coefficient, y=1000x-2. John was dubious but Mike insisted
to try. Using the information from software they together stated the graph
associated to y=1000x-2 still must intersect the y-axis at (0, -2).
Furthermore the students´ used GeoGebra as a visualizer to share individual
reasoning and to monitor the task solving process. The software offered an
environment that is controlled by the student and they may construct and
change formulas in line with their reasoning.
The interactive features of GeoGebra were both guiding reasoning and
provided feedback. The task was designed to invite the students to submit
the algebraic expression of the function and the graph associated to the
function was displayed. This guided the students into hypothesis before and
discussions after the computer activity. This also meant they received
feedback on their actions. The characteristic of the feedback was that it was
neutral, meaning it did not tell wright or wrong, but it was up to the student
to utilize it. However, there were examples when students´ used the
information to verify rather shallow ideas. There are some examples when
the authors invited the students´ to reflection, which in turn engaged the
students into deeper reflections.
5.2 Article 2
The study is focusing on the way students use feedback from dynamic
software. The research questions posed were: “What is the relation between
the students´ way of using the feedback that GeoGebra generates and the
students´ reasoning?” and “How do students´ ways of reasoning and
25
utilization of feedback from GeoGebra relate to their success in problem
solving?” This was investigated by observing students who in pair solved a
task, which main question was “Find a role how to choose x-coefficient and
constant term for two linear functions in a way that their graphical
representations are perpendicular”. Feedback was considered as verificative
or elaborative. Common for all pairs who reached an answer to the main
question was that they reasoned creatively and they used feedback
elaborately. The results indicated that predictive argumentation was
particularly significant for using feedback elaborately.
The article suggests that students´ reasoning affect the way they use
dynamic software. Students´ who reason creatively seems to in larger extent
utilize feedback more than just verifying compared to students who´s
reasoning is characterized as imitative. In research the matter is often
discussed the other way around, features of dynamic software affect
students´ reasoning. Considering characterize of reasoning as affecting
learning means it is of interest investigating characteristics of reasoning in
relation to use of dynamic software.
6. Discussion
The articles in this thesis build on the idea that the character of students´
reasoning is related to students´ learning behavior in a learning
environment. The use of dynamic software has been considered out of two
slightly different perspectives; the way dynamic software affect students
reasoning and collaboration (article 1) and the way students’ reasoning affect
their utilization of feedback generated by the dynamic software (article 2).
These perspectives will be discussed in the light of the results of the studies.
The chapter concludes with some implications for teaching.
6.1. Dynamic software affecting students´ reasoning
The didactic situation used in both studies was designed to encourage
creative reasoning among the students. The given task was designed to
constitute a challenge to the students and the provided instructions gave no
examples or useful algorithms how to solve the task. The students were
supposed to create their own solving methods, merely supported by the
dynamic software, GeoGebra. During the task solving process, GeoGebra
provided the students with opportunities to examine their ideas using the
software’s multiple representations and feedback. Finally, the students
worked in pairs, which is considered to invite students to engage in
reasoning since they need to communicate their ideas with one another
26
(Sacristán 2010; Roschelle & Teasly, 1995). The suggestions of computer
application supporting reasoning are often combined with a broader view on
reasoning than for example strict formal and logical. Reasoning is described
as “exploration of a space of possibilities” (Barwise & Etchemendy, 1998,
p.18) or “the process of organizing, comparing, or analyzing concepts and
relationships” (Moore-Russo, Viglietti, Chiu, & Bateman, 2013, p. 98).
Reasoning, in this sense, is contributing to individual understanding and not
only to procedural manipulations. Reasoning in the following paragraphs is
understood as the line of thought guided and limited by the students´
competencies (Lithner, 2008)
The results of article 1, show that the students used GeoGebra as a
collaborative environment within which they shared their reasoning with one
another. Furthermore, the study shows that the context of the task solving
(two students, one computer, and instructions directing them to use
GeoGebra for solving the task) engaged the students to create solving
methods and shared goals for their activities. That is, the students proposed
ideas, they negotiated what to submit into the software, and they predicted
the result of the activity. Since GeoGebra took care of the calculations and
drawings, the students didn´t have to carry out these procedures. Instead, as
soon as they submitted the algebraic formula into the software they could
focus on the feedback generated by GeoGebra. The results of article 1 show
that the students used GeoGebra to monitor and evaluate their problem
solving process. The easiness to submit expressions into GeoGebra and the
quick and informative response from GeoGebra invited the students to
create and test their ideas. Furthermore, GeoGebra was used when students
did not agree with one another, became uncertain, or when their idea did not
work. For example two students, Will and Sam, were uncertain whether the
graph of an algebraic formula having a negative x-coefficient would have a
positive or negative slope. This was easy tested by simply submitting y=2x+3, and examining the slope of the graph. Another example is Emma and
Zoe. Emma disagreed on Zoe´s assumption that the constant term
represented the intersections of the x-axis. This was sorted out by submitting
y=2x+4, and interpreting the corresponding graph. These exemplified
results make it reasonable to conclude that GeoGebra has affected the
students reasoning by providing multiple representations (showing algebraic
and the associated graphic expressions simultaneously) and offering shared
space for visualizing and monitoring thoughts and ideas (e.g. through
testing).
The results of article 1 show that students often discussed and predicted the
outcome when they prepared a computer activity. After submitting their
formulas they compared the outcome with their predictions. This was
27
discussed as “creative feedback”, in that sense that the feedback, unlike
textbooks, does not provide correct answers and unlike teachers does not
offer hints or guidelines. Instead the students had to evaluate the outcome of
computer activities and determine whether they were right and decide if and
how the results contributed to the solution. In article 1 the feedback
generated by the software was found to contribute to students´ creative
reasoning. However there were examples when GeoGebra was used to
perform merely trial and error activities. In some of these situations the
students did not argue predictively or evaluate the results of the activities,
instead they just tried something else. These relations between reasoning
and feedback became the focus for article 2. That is, the focus of the study
shifted from the way software affect reasoning, to the way reasoning affect
utilization of software.
6.2. Students´ reasoning affecting utilization of computer
features
Article 2 investigated the relationships between students’ reasoning,
feedback and success in their task solving. The students used GeoGebra to
test their created solution methods and the main purpose of their computer
activities observed in the study was to receive information about their ideas,
i.e. feedback on their activities.
The generated feedback was labeled verificative (whether an answer is
correct or not) or elaborative (why an answer is correct or not). The results of
article 2 show that creative reasoning leads to extended use of feedback
elaborately and that the only examples of students who solve the task are
those who engaged in creative reasoning and who used feedback elaborately.
It seems like those who predict and argue for their computer activities before
entering the commands prepare themselves to continue their argumentation
verifiably when they receive the feedback from the computer. This indicates
that it is important to encourage students to argue predictively for their
actions in order to make them evaluate their ideas and finally solve their
tasks. There are two examples in article 2 of students who changed behavior,
from not predicting the outcome of the computer activities and not
elaborating on feedback, into arguing predictively and using feedback
elaborately. For example, Olga and Leila spend 40 minutes of fruitless trials,
without evaluating their mistakes. When they changed their strategy and
argued for their planned computer activities and used feedback elaborately
they managed to solve the task. In article 1 there is an example where Luke
and Dan used GeoGebra to create a vertical line using a trial and error
strategy. It was not until the teacher encouraged them to explain what they
28
had done that their reasoning turned into creative reasoning. They started to
argue predictively and evaluate the feedback.
The exemplified results are indicating that students who are struggling may
turn their task solving into success if they start to evaluate and discuss their
mistakes. In the two examples the evaluation brought the students to reason
creatively. In the case of Olga and Leila, the shift into creative reasoning was
initiated by themselves and in the case of Luke and Dan it was initiated by
the teacher.
These results indicate that a didactic design would benefit from not
providing templates of solving methods such as algorithms and solved
examples. As provider of feedback, dynamic software like GeoGebra further
enhances opportunities for students to create their own strategies and argue
for them, i.e. engage in creative mathematical reasoning.
6.3. Implications for teaching
This paragraph will combine the two perspectives discussed above into
possible implications for teaching, i.e. the way dynamic software affects
reasoning and the way reasoning affects use of dynamic software. Teaching
and learning is much more complex in reality than in the context for the two
studies presented in this thesis. However, some of the results may be useful
for planning classroom activities that enhance creative reasoning.
Both studies indicate that students, working in pairs manage to work with
challenging tasks with minor guidance and instructions. The software
generally provided their need of feedback. In the two studies GeoGebra
generated feedback as a result of their computer activities. This kind of
feedback could be described as timely (Shute, 2008) since it is directly
related to the task solving. Compared to work with pen and paper without
guidance the students could always rely of the correctness in GeoGebra’s
calculations and graph drawing. That does not mean the teacher is not
important. GeoGebra gives opportunities for the teacher to encourage
students to predict and explain the result of their activities. Working with the
same task, using pen and paper, may make the teacher the one who decides
what´s correct or not. Furthermore, that kind of feedback is not necessarily
backing up the students´ argumentation for the strategy that created the
answer. But when GeoGebra generates its non-judging feedback, the
students may together with the teacher evaluate the result based on the
students’ predictive argumentation. In article 1 there are examples of
students who have come to a solution but they do not discuss the result until
the teacher encourages them to do so.
29
The results indicates that the students are not just working with superficial
algorithmic procedures but with the intended intrinsic mathematics, i.e. the
properties of the linear function y=mx+c and the conditions of perpendicular
graphs. In terms of Brousseau (1997) the didactic situation has been shown
to offer a reasonable learning environment for devolution of a mathematical
problem. This is, the students will have enough instructions to work with the
problem. Furthermore, Brousseau (ibid.) argues that if the task has an
appropriate design the students will reach the target for learning if they solve
the problem. It is important to stress that a successful design for learning
involves giving the students the responsibility to create their own solution
methods to solve the problem. The students in this study were given that
responsibility and those who solved the task worked with the intended
intrinsic mathematics, and engaged in creative mathematical reasoning.
The literature review and the results of this thesis points out the importance
of (at least sometimes) leaving the responsibility of creating solving methods
to the students. It is also stressed that students who are arguing predictively
for their strategies are more likely to use feedback from software elaborately.
Both studies indicate that GeoGebra is guiding creative reasoning and
provides feedback both for verification and elaboration. To obtain a situation
where students engage in creative reasoning and utilize feedback both
verifiably and elaborately this thesis suggests that:
•
Tasks must not contain solved examples and/or other algorithmic
templates guiding the students task solving
•
The teacher´s role is not to provide the students with solution
methods, but to encourage them to create, explain and verify their
strategies and solutions
•
Introduce students to appropriate dynamic technological tools,
which provide students with feedback and offer multiple
representations,
possibilities
to
create
mathematical
representations, and opportunities to manipulate mathematical
representations
30
References
Bangert-­‐Drowns, R. L., Kulik, C.-­‐L. C., Kulik, J. A., & Morgan, M. (1991). The instructional effect of feedback in test-­‐like events. Review of educational research, 61(2), 213-­‐238. Barwise, J., & Etchemendy, J. (1998). Computers, visualization, and the nature of reasoning. The digital phoenix: How computers are changing philosophy, 93-­‐116. Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in education, 5(1), 7-­‐74. Bottino, R. M. (2004). The evolution of ICT‐based learning environments: which perspectives for the school of the future? British Journal of Educational Technology, 35(5), 553-­‐567. Brousseau, G. (1997). Theory of didactical situations in mathematics: didactique des mathematiques 1970 to 1990 (mathematics education library/19): Kluwer Academic Publishers, Dordrecht. Chance, B., Garfield, J., & delMas, R. (2000). Developing Simulation Activities To Improve Students' Statistical Reasoning. Ferrara, F., Pratt, D., Robutti, O., Gutierrez, A., & Boero, P. (2006). The role and uses of technologies for the teaching of algebra and calculus. Handbook of Research on the Psychology of Mathematics Education Past, Present and Future, 237-­‐274. Gellert, U. (2010). Modalities of a Local Integration of Theories in Mathematics Education. Theories of Mathematics Education, 537-­‐550. Hoffkamp, A. (2009). Enhancing functional thinking using the computer for representational transfer. Proceedings of CERME 6, Lyon. Jones, K. (2000). Providing a foundation for deductive reasoning: students' interpretations when using Dynamic Geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1), 55-­‐85. Jonsson, B., Liljeqvist, Y., Norqvist, M., Lithner, J. (2014) Learning mathematics through imitative and creative reasoning. (in preparation) Kieran, C., & Drijvers, P. (2006). The co-­‐emergence of machine techniques, paper-­‐and-­‐pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11(2), 205-­‐263. Lithner, J. (2000). Mathematical reasoning in school tasks. Educational studies in mathematics, 41(2), 165-­‐190. Lithner, J. (2003). Students' mathematical reasoning in university textbook exercises. Educational studies in mathematics, 52(1), 29-­‐55. Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255-­‐276. 31
Lou, Y., Abrami, P. C., & d’Apollonia, S. (2001). Small group and individual learning with technology: A meta-­‐analysis. Review of educational research, 71(3), 449-­‐521. Mason, B. J., & Bruning, R. (2001). Providing feedback in computer-­‐based instruction: What the research tells us. Retrieved February, 15, 2007. Moore-­‐Russo, D., Viglietti, J. M., Chiu, M. M., & Bateman, S. M. (2013). Teachers' spatial literacy as visualization, reasoning, and communication. Teaching and Teacher Education, 29, 97-­‐109. Mullins, D., Rummel, N., & Spada, H. (2011). Are two heads always better than one? Differential effects of collaboration on students’ computer-­‐
supported learning in mathematics. International Journal of Computer-­‐
Supported Collaborative Learning, 6(3), 421-­‐443. Niss, M. (2007). Reflections on the state and trends in research on mathematics teaching and learning: From here to Utopia. Second handbook of research on mathematics teaching and learning, 2, 1293-­‐1312. Noss, R., Poulovassilis, A., Geraniou, E., Gutierrez-­‐Santos, S., Hoyles, C., Kahn, K., . . . Mavrikis, M. (2011). The design of a system to support exploratory learning of algebraic generalisation. Computers & Education. Olive, J., Makar, K., Hoyos, V., Kor, L. K., Kosheleva, O., & Sträßer, R. (2010). Mathematical knowledge and practices resulting from access to digital technologies Mathematics education and technology-­‐rethinking the terrain (pp. 133-­‐177): Springer. Radford, L. (2008). Connecting theories in mathematics education: Challenges and possibilities. ZDM, 40(2), 317-­‐327. Rakes, C. R., Valentine, J. C., McGatha, M. B., & Ronau, R. N. (2010). Methods of Instructional Improvement in Algebra A Systematic Review and Meta-­‐
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Part 2, the articles
Article 1 is presented as it is submitted for review and article 2 is a
manuscript. Article 1 is the second version submitted to Journal of
Mathematical Behaviour, after addressing comments from the journal’s
reviewers on the first version. This article was written in collaboration
between Carina Granberg and myself, and we estimate that each of us did
approximately half of the work. The design of the study, data collection and
analysis was done in collaboration. With respect to writing, I have done more
work in the sections on literature review, theory and method and Carina has
written more about the analysis and conclusions. But all writing was done in
several rounds of reviewing each other’s work and discussions of final
formulations.
ICT-supported problem solving and
collaborative creative reasoning:
Exploring linear functions using
dynamic mathematics software
Authors: Carina Granberg & Jan Olsson
Abstract: The present study investigates how a dynamic software
program, GeoGebra, may support students´ collaboration and creative
reasoning during mathematical problem solving. Thirty-six students
between the ages of 16 and 17 worked in dyads to solve a linear
function using GeoGebra. Data in the form of recorded conversations
and computer activities were analyzed using Lithner’s (2008)
framework of imitative and creative reasoning in conjunction with the
collaborative model of joint problem space (Roschelle & Teasly, 1994).
The results showed that GeoGebra supported collaboration and
creative reasoning by providing students with a shared working space
and feedback that became the subject for students’ creative reasoning.
Furthermore, the students’ collaborative activities aimed towards
sharing their reasoning with one another enhanced their creative
reasoning. There were also examples of students using GeoGebra for
trial-and-error strategies and students who engaged in superficial
argumentation.
Keywords: Creative reasoning, Problem
Dynamic software, Linear functions
1
solving, Collaboration,
1. Introduction
One of the challenges in mathematics education is helping students become
skilled problem solvers rather than rote learners. A research framework
presented by Lithner (2008) describes how rote learning relates to students’
line of thinking or reasoning. Reasoning based on rote learning is
categorized as imitative; during lectures, students memorize facts and
algorithms and subsequently attempt to recall them when solving tasks.
Conversely, creative reasoning engages students in instructive problemsolving processes, during which they develop well-founded and
mathematically anchored arguments for their choice of methods. Studies
have shown that students who engaged in creative mathematical reasoning
to solve non-routine problems during a training session performed
significantly better on post-tests than students who used imitative reasoning
when working with repetitive tasks. (Lithner, et al, 2013). Other similar
studies showed, based on post-test results, that students who work with
complex problems outperform students who are given traditional lectures
and practice well-structured tasks (Boaler, 1998; Kapur, 2011).
Problem solving related to functions (e.g., linear or polynomial) is no
exception to students’ tendency to use imitative reasoning and superficial
argumentation when they find this type of mathematics difficult (Even, 1998;
Hoffkamp, 2011). Similar findings as above have been reported for this type
of problem solving. Non-procedural assignments provide students with
opportunities to challenge their understanding of relations instead of
performing procedures (Ferrara et.al. 2006, Mevarech & Stern, 1997).
Moreover, several studies emphasize the value of collaborative work.
Students’ verbalization of mathematical concepts to engage in dialogue has
shown to be beneficial to enriching their conceptualizations (Hoffkamp,
2011) or establishing mathematical meaning (White, 2012). However, there
are studies that problematize these findings, pointing out obstacles to
working with non-routine problems without supporting activities (Ploetzner,
2009) and issues with students working in groups. The latter refers to
students’ tendency to cooperate, dividing the work amongst themselves,
rather than collaborating, sharing understanding and solving the problem
together (Roschelle & Teasley, 1994).
Research provides various methods of supporting students to develop
conceptual understandings as well as collaborative work. One of the
suggested methods is the use of dynamic software that allows students to
visualize functions and their representations (Rakes et al., 2010), as well as
distribute their collaborative problem solving process (Stahl, Koschmann, &
Suthers, 2006).
2
The idea of considering the appropriate support for student engagement in
collaborative problem solving and creative reasoning combined with the
proposition that technology may support these activities brings us to the
following question: How can dynamic software (in this case, GeoGebra)
support or obstruct students’ creative reasoning and collaborative work
during the problem-solving process?
1.1 Aim and research questions
The aim of this study is to develop insight into how GeoGebra could be used as a means of supporting collaboration and creative reasoning during a problem-­‐solving process. The following research questions will be addressed in this study: -­‐ To what extent do students use GeoGebra to collaborate during problem
solving?
-­‐ What characteristics of GeoGebra might contribute to or obstruct their
creative reasoning?
To examine how GeoGebra may support students’ collaboration and creative
reasoning, the didactical situation in this study was designed to allow
students to work in pairs to solve non-routine tasks while supported by
GeoGebra. The didactical situation was designed to be in line with Brousseau
and Schoenfeld’s suggestions, which will be presented in the following
section along with the theoretical frameworks used to analyze data.
2. Research framework
The following section begins by introducing the theoretical concepts used to
design the didactic situation in this study, followed by a presentation of the
theoretical frameworks for creative reasoning (Lithner, 2008) and
collaboration (Roschelle & Teasley, 1994). The latter will be used for
structuring and analyzing the data.
2.1 Designing a didactic situation, creative reasoning and collaboration
Students spend much of their time completing textbook exercises in which
examples are followed by similar tasks. Thus, students are guided into
imitative reasoning that does not give them an opportunity to argue for their
strategies (Boesen, Lithner & Palm, 2010). Solving tasks using imitative
reasoning may result in correct answers, however, to develop conceptual
understanding students need to process mathematical concepts—to struggle,
in a productive sense (Hiebert & Grouws, 2007). Schoenfeld (1985) argues
3
that learners need to work with mathematics problems that are somewhat
new to them. When students engage in challenging problem solving, they
will need, and therefore develop, their mathematical knowledge and
understanding as well as their ability to create strategies for working on
unfamiliar problems. That is, to engage in creative reasoning students need
to work with non-routine tasks for which they have no memorized procedure
to imitate to solve the task (Lithner, 2008).
Furthermore, to be engaged in creative reasoning, students need to struggle
with the problem without guidance towards a correct solution. In line with
the idea that imitating procedures is inefficient for learning, Brousseau
(1997) suggests a didactical design that leaves some of the responsibility of
the problem-solving process to the students. During this part of the
didactical situation, defined as an adidactical situation, teachers should not
interfere or guide students toward the desired answer. However, the
adidactical situation should involve feedback related to the students’ actions.
Brousseau (1997) refers to feedback as “an influence of the situation on the
pupil,” suggesting that the situation will provide each student with influence
“as positive or negative sanctions relative to her action, which allows her to
adjust this action, to accept or reject a hypothesis” (Ibid, p. 7).
In conjunction with challenging problems and adidactical situations,
collaboration is often suggested as an alternative to traditional methods
(Boaler & Greeno, 2000; Stahl, Rose, & Goggins, 2011). Students may
improve their conceptual understanding in collaboration by engaging in
discussions, mutual explanations, and elaboration of underlying
mathematical concepts (Mullins, Rummel, & Spada, 2011, Scardamalia &
Bereiter, 1994). However, having students work in small groups does not
automatically initiate collaboration, i.e., mutual engagement to solve a
problem together. Roschelle and Teasley (1995) make the distinction
between 'collaborative' and 'cooperative' problem solving. Collaboration is
understood as “a coordinated, synchronous activity that is the result of a
continued attempt to construct and maintain a shared conception of a
problem” (Ibid., p. 70). Cooperative work, on the other hand, is
accomplished by dividing the work as well as responsibility among the group
members, during which interaction as a learning activity will not take place.
The didactical design in this study builds on the following principle: creative
reasoning as well as collaboration are more efficient for student learning. To
accomplish collaborative creative reasoning, students need to work on a
challenging problem within an adidactical situation in collaboration with
other students. However, as described earlier, students need support to
address such a didactic situation, i.e., an adidactical situation should include
4
feedback, and collaboration is not automatically initiated within groups. The
focus of this study is not students’ learning, but rather the support given by a
dynamic software, or more specifically, how GeoGebra may support creative
reasoning as well as collaboration.
In this study, the students worked in pairs on a challenging problem, had a
non-guiding teacher present and used GeoGebra as the provider of feedback.
To examine how GeoGebra supported their creative reasoning and
collaboration, the following theoretical frameworks were used.
2.2 Imitative and creative reasoning – a framework
The research framework described by Lithner (2008) defines a learner’s
reasoning as his or her line of thought guided and limited by the student’s
competencies, which are created in a sociocultural milieu (figure 1). A
student’s thinking process, as an imperceptible act of the mind, may become
articulated and by that traced in the form of an observable reasoning
sequence. The reasoning sequence begins with a task; consists of all actions
taken, including a task-solving process; and ends with a correct or incorrect
conclusion or a decision to give up. The reasoning sequence can be observed
through, for example, the students’ written solutions, think-aloud protocols,
or interviews.
Figure 1. The origin of reasoning (Lithner, 2008)
The reasoning sequence can be described by a path through a task-solving
process (Figure 2). The solving process starts with a task, T, for which the
student comes up with a solving strategy, S, that is implemented. If the
process progresses to the next stage of the (sub) task, V, depends on whether
the strategy is successful. Another solving strategy will then be implemented
to solve the (sub) task and finally, if the student has not given up, the task
solving will end up with a possible conclusion Vn. The reasoning sequence
may, for example, be articulated as uttered arguments that are predictive,
answering the question of why this strategy will work, or verificative,
evaluating why the implemented strategy did or did not work.
5
Figure 2 The path of a reasoning sequence
When students solve mathematics tasks, they engage in imitative and/or
creative reasoning. Lithner (2008) defines two types of imitative reasoning:
memorized reasoning and algorithmic reasoning. In the first type, students
recall facts or a complete answer and then write it down. In the second type,
students recall or apply solution algorithms that are either memorized or
provided procedures (e.g., by the teacher or textbook) that can be followed to
reach an answer. During a task-solving process (Figure 2), reasoning is
considered imitative if it, in its articulated form, constitutes remembering a
solving strategy (S) or a procedure for how to solve the problem.
Creative reasoning, in contrast, is described by its novelty, plausible
argumentation, and mathematical foundation. Instead of recalling a
procedure for how to solve a task, students create or re-create a reasoning
sequence that, to some extent, is new to them. Given the task, T, the student
creates or recreates and then implements a solving strategy, S. The student’s
reasoning, articulated as arguments, is regarded as creative if it is supported
by plausible arguments, i.e., suggestions for how to solve the task and a
mathematically anchored justification for why the strategy will work, that is,
a predictive argument. The arguments may also be verificative, in that they
evaluate the implemented strategy and present arguments for why the
solution worked or did not work.
A problem-solving process guided by creative reasoning will, in general,
include imitative reasoning, as well. There are mathematical competences
that students benefit from having automatized: geometric facts,
multiplication tables, unit conversion etc. These types of automatized
procedures can be applied by students without burdening their working
memory and thus decrease their cognitive load (Sweller, 1994). Therefore,
these procedures enable students to engage in more cognitively demanding
mathematical problem solving through creative reasoning.
Finally, the didactical design of this study includes students working in pairs.
Because the presented framework by Lithner (2008) addresses individual
reasoning, a collaborative perspective was added.
6
2.3 Collaborative reasoning – a framework
When working in pairs, the competences of each of the two students
contributes to their combined thinking process: one student’s uttered
reasoning may influence the other student’s thinking process. Furthermore,
the feedback from the GeoGebra program may also contribute to their
individual thinking. In this way, students’ reasoning becomes an even more
complex process. The collaborative reasoning sequence consists of studentstudent interactions and student-GeoGebra interactions (figure 3), which are
observable by capturing students’ dialogue, gestures, and screen activities.
Figure 3. The origin of reasoning when learners solve problems in pairs using
GeoGebra
The suggested framework considers individual competences, resources, and
shared knowledge observed within group activities. This knowledge is shared
in the sense that the students have negotiated and agreed on knowledge they
regard as true (or at least plausible). The collaborative reasoning sequence is
situated within a joint problem space, or JPS (figure 2), that is constructed
and maintained by the students during their problem solving. The concept of
JPS will be elaborated on in the following.
2.3.1 Reasoning within a joint problem space (JPS)
As described earlier, Roschelle and Teasley (1994) defined collaboration as
“a coordinated, synchronous activity that is the result of a continued
attempt to construct and maintain a shared conception of a problem”
(Ibid., p. 70). Based on this conception, Roschelle and Teasley introduced
the concept of a joint problem space as a shared and socially negotiated
knowledge structure that consists of goals, a description of the current
problem state, and awareness of available problem-solving actions. The JPS
includes a negotiated and shared understanding of the following questions:
Where are we heading, where are we right now, and how do we get there?
7
The reasoning sequence and the JPS co-exist, beginning with a task and a
goal and, if successful, ending in a solved task and the achievement of the
goal. To construct a JPS, students must introduce individual resources and
ideas. Based on these contributions, they negotiate and create a shared goal,
shared knowledge, and solving strategies. Furthermore, to maintain their
JPS, students must monitor and control their solving process by observing
and repairing misconceptions and disagreements and deciding on suitable
strategies.
Students’ dialogue, including taking turns sharing suggestions, questions,
and explanations, is an essential part of their problem solving as they share
reasoning with one another. However, Roschelle and Teasley (1994) suggest
that students are not merely dependent on dialogue (language) to create and
maintain shared understanding; shared activities (actions), such as gestures,
drawing, or interaction with technical devices, also contribute to their
communication.
Examining how GeoGebra may support students’ collaboration could be
achieved by investigating if students are able to uphold their JPS and how
they interact with GeoGebra in situations where their JPS is created,
maintained and repaired. To study to what extent GeoGebra may contribute
to their creative reasoning, it is necessary to identify situations in which
students articulate their reasoning, establish if their reasoning is creative or
imitative, and determine what type of contributions their interaction with
GeoGebra has within these situations.
2.3.2 ICT and teaching functions
There are studies that investigate the idea that digital tools, such as graphical
calculators and dynamic software, may support students’ learning and
understandings of functions. Positive effects demonstrated by experimental
groups who used the technology performing better than control groups on
post-tests have been reported regarding graphical calculators (Tajuddin,
et.al., 2009) as well as dynamics software (Zulnaidi & Zakaria., 2012).
To explain these type of positive effects, studies have been conducted to
examine students’ problem-solving processes and identify software qualities
and learning activities that are beneficial to students’ learning. For instance,
software that performs calculations and draws geometric figures and graphs
enables students to focus on conceptual understanding rather than executing
procedures during problem solving (Nussbaum et al., 2009; Sinclair, 2005).
Furthermore, software that offers students the ability to control, design, and
manipulate mathematical content is described as beneficial because this
8
software transfers students’ problem solving from the manipulation of
symbols to the investigation of mathematical relations (Dicovic, 2009; Moss
& Beatty, 2006; Rakes, 2010). Encouraging students to explore and observe
relationships between algebraic and graphical representations has also been
shown to be beneficial to students’ understanding of functions (Brenner et
al., 1997; Ferrara et al., 2006; Mevarech & Stern, 1997), for instance, when
students interpret algebraic formulas and graphs as dynamic relationships
rather than pictures (Glazer, 2011). However, assigning students to explore
representations through interactive animations and the like does not
automatically initiate learning; on the contrary, the didactical design is
crucial. Students need to process their findings and ideas to be able to
develop conceptual understanding (Ploetzner, et.al., 2009) and to
discuss and formulate mathematical concepts (Hoffkamp, 2011).
Moreover, there is a growing field of research addressing questions about
computer-supported collaboration within mathematics education. Research
investigating computer-supported problem solving often reports that tasks
and software that support collaborative work is more efficient than tasks and
software aimed at individual work (Lou, Abrami, & d’Apollonia, 2001).
However, ICT-supported collaboration does not automatically improve
individual learning and ICT does not by itself support collaboration. The
didactical design is important. In a review synthesizing 122 reports involving
11,317 students, Lou et al. (Ibid.) investigated the outcomes of collaborative
work. The authors found that the most effective didactical design for
individual achievement was small groups in which students engaged in
reflection and interaction using explorative software to investigate intrinsic
mathematical properties. The design of tasks is therefore important and
should be aimed at producing conceptual understanding rather than
procedural skills. In the former, collaboration could support students´
learning as it encourages mutual elaboration. In the latter, students tend to
split the work amongst themselves and their collaboration becomes
cooperation (Mullins, Rummel, & Spada, 2011).
Software features that are beneficial for exploring mathematical concepts
and relations may also enhance collaboration. Limiting the amount of time
students have to devote to time-consuming drawing and calculations has
been shown to increase peer participation and mediate group discussions
among participants (Manoucheri, 2004). Furthermore, the software’s ability
to distribute the problem-solving process to all participants simultaneously
is beneficial for maintaining collaboration (Stahl, Koschmann, & Suthers,
2006). Additional gains are related to the software’s ability to record
students’ collaborative activities, which can be replayed and manipulated
and serve as a reference during students’ interactions (Stahl, 2011). The
9
presented research points at the value of encouraging students to interact
with a dynamic software as well as with a fellow student.
4. Methodology
The present study adopts a sociocultural perspective that considers
competence and knowledge to be skills that students develop through
interaction within a social setting. This study emerges from the ideas that
both creative reasoning and collaboration are beneficial to students’
learning. The framework of this study merges these perspectives and
explores the benefits of collaborative problem-solving activities as well as
students’ individual reasoning. Because the aim of the study is to examine
how GeoGebra might support collaboration as well as creative reasoning, the
didactic design, as earlier described, will include non-routine problem
solving while working in dyads and supported by GeoGebra. Furthermore,
data collection needs to capture students’ language and actions to examine
their interactions with GeoGebra, the construction and maintenance of their
JPS and finally, their reasoning as articulated through uttered arguments.
The analysis was conducted using theoretical concepts involving individual
reasoning (Lithner, 2008) and collaborative problem solving (Roschelle &
Teasley, 1994).
4.1 Designing the didactic situation
The design of the didactical situation was based on the previously described
suggestions of Schoenfeld and Brousseau. The intention was to design an
adidactic situation that included a task that students were unlikely to solve
using mere imitative reasoning and that instead offered students the
opportunity to collaborate and take responsibility for their problem-solving
process.
4.1.1. The problem
The task was designed to constitute an intellectual challenge for the
students, i.e., a problem (Schoenfeld, 1985), and include the construction
and interpretation of the algebraic and graphical representations of linear
functions. Therefore, a set of tasks that differed from those in mathematics
textbooks were constructed, and pilot tests were performed to examine
whether the tasks met the design aims, that is, the tasks could not be solved
using memorized procedures, but rather they constituted an intellectual
challenge. The problem presented below was chosen as appropriate for 16and 17-year-old students in upper secondary school (figure 4). There were no
further instructions given regarding the problem. One possible way of
executing the first part of the task is exemplified in figure 5.
10
Figure 4. The given mathematical problem.
4.1.2 Working in dyads
Lou et. al. ( 2001) argue that small groups are more likely to collaborate than
large groups. Therefore, the students in this study worked in pairs. The
didactical design provided students the opportunity to collaborate, however
because their joint work was an object of study, they were not given any
instructions on how to organize their work.
4.1.3 The role of the teachers
The researchers acted as teachers during this study, as they both designed
and delivered the ‘lesson,’ or the didactic situation. The design, as described
earlier, consisted of a challenging problem that had to be solved with a
classmate, and the students had a considerable amount of responsibility in
the problem-solving process. This lesson design, in general, differs from
students’ regular lessons, and they may have found themselves in a situation
of uncertainty in which they did not know how to proceed. Therefore, there
was a risk that they might give up easily. Thus, the design included the
option for the researchers to offer the students support, but without
eliminating the challenge.
During the problem-solving process, students were given support by
GeoGebra and the teachers. Because the support from GeoGebra is the object
of this study, the teachers’ way of interacting with the students was guided as
to not interfere with the adidactical design of the situation. The teachers
began each session with a brief introduction on how to submit functions to
GeoGebra, how to adjust submitted functions and how to use the tool to
measure angles. These were the only instructions given, and then the
problem (Figure 4) was presented to the students on paper. The students
were permitted to ask questions, however, to maintain the adidactical
design, the authors used prepared responses such as “What would you like to
do?”; “Can you explain what you have done?”; “Why do you think that the
idea did not work?”; and “Do both of you agree on this?”
11
4.2 Method
The 36 students who volunteered to participate (18 female and 18 male) were
16 to 17 years old and were enrolled in two Swedish upper secondary schools.
Half of the students were studying in the social science program, and half
were studying in the technology program. The study was performed outside
of the classroom during students’ free periods. Written informed consent
was obtained from each student, and all ethical requirements outlined by the
Swedish Research Council (2001) were followed.
The students worked in pairs, sharing one computer and using GeoGebra to
solve the given problem. None of the students had used GeoGebra before,
however, after the teachers’ short introduction they mastered how to submit
and adjust algebraic formulas, the angle tool, etc. GeoGebra allows users to
construct graphs by typing algebraic formulas into an input field. GeoGebra
then displays the algebraic and graphic representations (figure 5). Anything
added or changed in the algebraic representation is automatically visualized
in the graphical display and vice versa. The tool in GeoGebra for constructing
linear functions graphically was disabled for this study.
Figure 5: GeoGebra showing
representations of four functions
12
the
algebraic
and
graphical
Students’ dialogue and screen activities were recorded using the software BB
Flashback. Nine of the groups were observed during their work, and gestures
such as pointing to the screen were noted. The time needed to solve the task
varied from 15 to 45 minutes.
Dialogue between the students was transcribed word for word and were
related to the students’ activities at the time of the dialogue. The interactivity
with GeoGebra was described using square brackets (e.g., [submitting y =
2x - 3]). Documented gestures were added to the transcripts using
parentheses (e.g., (pointing at the origin)).
4.3 Analysis method
To answer the research questions, data were structured and analyzed using
the earlier presented frameworks of Roschelle and Teasley (1994) and
Lithner (2008). The analysis focused on students’ language (suggestions,
questions, answers, arguments etc.) and actions (gestures and interaction
with GeoGebra). The frameworks used in this analysis describe problem
solving as a stepwise process. Furthermore, according to Roschelle and
Teasley (ibid.), an important activity during their collaborative problem
solving is managing misconception and divergences. Thus, interactions of
this type were given special focus. Therefore, the first step was to structure
the data according to the frameworks’ activity sequences corresponding to
the following: 1. Receiving a task/creating a goal, i.e., initiating the reasoning
sequence and their JPS, 2. Creating, implementing and evaluating solving
strategies, 3. Observing and handling misconceptions and divergences, and
finally, 4. Solving the task/reaching the goal.
To examine if students were able to engage in collaborative work, that is,
uphold their joint problem space (JPS), and how they used GeoGebra to do
so, the first step was to identify situations where students created,
maintained, lost or repaired their JPS. The next step was to notice activities
within these situations, such as introducing individual knowledge and ideas,
negotiating a shared goal, and sharing knowledge and solving strategies.
Furthermore, activities in which students observed and repaired
misconceptions or disagreements were identified. Finally, the way students
used GeoGebra during the identified situations/activities was observed.
To look into students’ reasoning, our starting point was to identify situations
where students implemented solving strategies, uttered predictive
arguments before implementing or uttered verificative arguments evaluating
a solving strategy. To establish whether their reasoning was imitative or
creative, Lithner’s (2008) definition was used. Situations in which students
13
recalled a whole procedure were described as driven by imitative reasoning.
Conversely, students’ reasoning was regarded as creative if they created/recreated a solving strategy (not recalling a whole procedure), and (1) they
presented arguments why the strategy would work, did work or did not work,
and (2) their arguments were anchored in intrinsic mathematic properties.
As a final step, their interactions with GeoGebra during these situations were
examined.
5. Analysis
The objects of this study, students’ collaboration and reasoning during
problem solving, were analyzed as described above and the results are
presented below. Students’ episodes of silence ending in suggestions,
questions, answers etc. were interpreted as periods of reasoning. The
frameworks activity sequences during the problem-solving process has been
used as headlines that structure the presentation. The students’ construction
and maintenance of their JPS will be discussed at the end of each example.
The students’ names are fictitious and the dialogue has been translated from
Swedish.
5.1 Creating a shared goal
All 18 student groups began by engaging in activities to initiate their
collaborative work, i.e., their JPS. The given problem had an open-ended
design, and all groups initiated their JPS by creating a shared goal. They
negotiated and agreed on the appearance of the graphical representations of
the four functions needed to compose a square. This process is exemplified
by Sara and Anne, who agreed on a shared goal through a flow of turn taking,
mixing language, dialogue, and action (in brackets). They started with two
divergent suggestions: a straight square (1) and a tilted square (2).
1.
Anne: Let’s start with … hmm… a straight square, like this
(uses her hands to form a square parallel with GeoGebra’s xand y-axis).
2. Sara: …. Or ... we could make one with a slope ...
(she draws a tilted square with the mouse and becomes silent).
… or… is it possible to create an ordinary ... I mean … one with
no slope?
3. Both: Silence
To visually demonstrate their individual reasoning to one another, they
used GeoGebra as a reference tool by pointing and sketching in relation
to the coordinate axis (1-2). To respond to one another, they needed to
14
interpret and evaluate the visualized ideas, which made Sara question
whether Anne’s idea was possible to accomplish (2). Anne responded
and used GeoGebra for referencing, clarifying and anchoring her
proposition using concepts such as parallel and vertical (4). Sara
accepted by adding her own idea as the next step, and their JPS was
initiated (5).
4. Anne: Yes, you create two parallel lines like this … and two
vertical
(draws with her finger horizontal and vertical lines parallel
with the x- and y-axis).
5. Sara: Then, we can angle it … so that we will have lines with a
slope (draws a tilted square with the mouse).
Similar dialogue and activities were observed within all student groups
as they initiated their JPS and thus their collaborative work. At the
same time, by initiating their JPS, they created the first step of their
reasoning sequence. All students used GeoGebra for referencing to
visualize their reasoning, articulated as arguments, that is, proposals,
motivations, questions and answers. Through negotiation, each student
group agreed on an “imagined square” situated in GeoGebra’s graphical
field. This way of picturing a hypothetical conclusion, together with
GeoGebra’s request for algebraic input, guided the students into a
problem-solving process alongside a reasoning process for investigating
the properties of the algebraic formulas (y = mx + c) required to agree
with the hypothetical conclusion (Figure 6). During this first activity,
the students were interpreted as engaging in creative reasoning to
initiate their JPS.
Figure 6. The path of a reasoning sequence, aiming for a shared goal,
the hypothetical conclusion Vn.
In the following, the students’ reasoning when creating and implementing a
strategy (articulated as predictive arguments), the strategy itself (S), as well
15
as their reasoning when evaluating their strategies (articulated as verificative
arguments) will be given focus.
5.2 Creating, testing, and evaluating solution strategies
Like Anne and Sara, the majority of the groups (16 groups) agreed on
creating a straight square. This choice added difficulty to their solution
process because vertical lines are not functions of the form of y = f(x), and
none of the students had previously worked with this type of algebraic
formula.
Out of the 16 groups that created straight squares, 15 groups started with
horizontal lines. Ten of the groups presented propositions for how to
construct horizontal graphs anchored in the properties of m and c through
uttered and anchored arguments, such as m should equal 0 because a
horizontal line has no inclination. Three groups used arguments such as y
should equal a constant because horizontal lines imply that y should equal
the same value all of the time. The remaining three groups agreed on and
created correct formulas without presenting arguments for their solution
strategy. All 18 groups mixed language, actions (in parentheses), and
GeoGebra submissions [in brackets] to establish shared knowledge and
test ideas during their work.
The following is an example of one of the student groups who argued that
horizontal lines have no inclination. These groups began by establishing
shared knowledge by uttering their reasoning through arguments such as a
horizontal line (1) has zero slope (2); therefore, the m-value should equal
zero (3). This idea was typed and thereby visualized (4), and it was
subsequently interpreted and confirmed (5).
1. Mary: … Shall we start with the horizontal … two parallel.
2. Ella: Ok, a flat one … one with zero … slope ... (shows a horizontal line
with her hand).
3. Mary: Yes, that is … m equals zero.
4. Ella: … if we create a horizontal, I mean … yes, m equals zero, but shall
we write 0 x? [writes y = 0x ]
5. Mary: Right!
Their next step was to agree on the c-value, which they did by sharing their
reasoning and negotiating knowledge. The mathematical term for cross is
intersect (6-7), and the intersection with the y-axis (9) correlates with the cvalue (9-12). Ella showed some hesitation by asking a question (10), and
16
Mary visualized her reasoning through uttered arguments, gestures, and
referring to GeoGebra (12).
6. Mary: … However, we need to … where it starts ... (points at the y-axis)
… where it crosses.
7. Ella: Yes we need … where it intersects … how do we … the
intersection?
8. Both: Silence
9. Mary: Yes … but if this one is … 5 ... (points at 5 on the y-axis) that is
where it intersects.
10. Ella: Are we talking about m or c?
11. Both: Silence
12. Mary: … c, because we want c to be something … we want it to intersect
(points at the y-axis)… and m is the slope (points at the submission
field).
In addition to referencing and visualizing, these students used GeoGebra to
monitor their problem-solving process, gradually adding to their algebraic
formula in the input field as they negotiated their strategy (4, 13). In the end,
GeoGebra was used for verification (15).
13. Ella: Ok, it will intersect at 5 [completes y = 0x + 5] or?
14. Mary: Yes, let’s try.
15. Ella: [Submits the formula, GeoGebra draws a horizontal
line]
16. Both: Yea! High five!
The idea was interpreted as successful (16) without any further evaluation.
The following students exemplify the groups who concluded that y should
equal one value. They did not know how to execute their idea of removing
the ‘slope value’ (1-2). Kevin suggested a strategy (2), and Owen presented
predictive anchored arguments that it would fail (3). However, they agreed
to use this as a visual starting point for their problem solving (4-5)
1. Owen: We need a line with no slope-value, how do we remove that?
(silence)
2. Kevin: I do not know, but just submit a formula .. like y=2x+1.
3. Owen: That will not do, it will have a slope … there is a slope-value.
4. Kevin: I know, but then we have something to work with ... figure out
how to remove the slope.
5. Owen: Okay [submits y=2x+1].
17
The graph produced in GeoGebra, corresponding to their formula,
constituted the basis for their reasoning (6-11). Owen suggested how to
interpret that their function had a slope, anchored with verificative
arguments (7, 9) and using the graph to visualize his reasoning. Kevin
eventually agreed (10, 12) and they implemented their strategy (13).
6. Kevin: Why does this bugger have a slope?... (points at the graph and
becomes
silent).
7. Owen: Hmmm … because y has different values all along (point at the
graph at different points).
8. Kevin: How do you reason .. I do not ...
9. Owen: …. here y is 3 and … here y is 5 … (points at the graph and the yaxis)… and it becomes a slope.
10. Kevin: Okay, but could we think like that … that y should not change at
all then?
11. Owen: Yes. That is right … y is the same value … mmm … like 4 all of the
time.
12. Kevin: Lets try that … we write y=4 .
13. Owen: [submits y=4].
Both: Yes!
The strategy is considered successful, and they move on without any further
evaluation. This is a representative example of students engaging in
verificative arguments when GeoGebra presents functions that do not
correspond with their hypothetical conclusion (6-9). They then interpret and
retrieve information from the result to move on in the problem-solving
process (9-12). However, when feedback from GeoGebra confirmed their
strategies were correct, the students rarely presented any uttered arguments
verifying why the strategy worked (See Mary and Ella as well p.x). Kevin and
Owen maintained their JPS by continually sharing their knowledge and
strategies through a flow of turn taking, referring to GeoGebra, and
visualizing their reasoning for one another. Owen’s verificative arguments
(9) were interpreted and processed by Kevin, who expressed their evolved
reasoning (10).
Constructing vertical lines was a more complex process (see John and Mike
below as one example). As a result of their reasoning, the students created,
tested, adjusted, or abandoned solution strategies such as choose m-values
high enough to make it vertical (3 groups), put y or c equal to zero (3
groups), “remove” y or c (10 groups) because the line should not intersect
the y-axis, the slope should equal zero (4 groups) or the “opposite of zero” (1
group). Fifteen groups were able to construct vertical lines. Seven of the 15
groups supported their final solution with the following anchored
18
arguments: x should equal a constant because a vertical line implies that x
should equal the same value all of the time. Three groups’ reasoning,
articulated as arguments, were that because a horizontal line starts out from
the value of the intersection of the y-axis, a vertical line should do the same
from the x-axis. Five groups had the idea to swap the formula without any
uttered argumentation. Finally, the last group changed its strategy and
created a tilted square.
John and Mike agreed to create a straight square starting with vertical lines.
Their solving strategy was to choose an m-value large enough to make the
line become vertical. Of course, this strategy would fail; however, they
presented arguments that they found plausible for why their strategy might
work: a vertical line must have a large m-value because the slope is huge.
They engaged in a problem-solving process including creating, testing, and
evaluating a line of algebraic formulas corresponding to their evolving
strategies. In the following exchange, they went through three cycles, testing
y = 2x + 1, y = 4x + 1, and y = 10x + 1. All three graphs were still visible in
GeoGebra (figure 7) when they entered their fourth attempt by aiming for an
even larger m-value.
1.
2.
3.
4.
Both: Silence
John: … We need a really large one…. let’s put m = 100.
Mike: Ok [submits y = 100x + 1]. Oh no!
Both: Silence
Because no cognitive resources were devoted to procedural calculation or
drawing graphs, the students’ reasoning could focus solely on interpreting
and evaluating the feedback (1, 4). GeoGebra visualized their mistake
repeatedly, and the graphs on the screen became a way of monitoring their
work (figure 4). John’s interpretation of these graphs led him to eventually
realize their mistake. John’s reasoning is articulated through verificative
anchored argumentation of why their strategy failed (5,7). Mike eventually
interpreted and processed Johns uttered reasoning (8) and realized why
they failed (9), followed by presenting evolved reasoning through a
predictive argument (10).
19
Figure 7. The three functions
visualizing their mistake
displayed
in
GeoGebra,
5.
John: No wait … if they are vertical, they … I mean… they should not
intersect the y-axis at all... (points at the intersections).
6. Mike: … If we just angle it enough … [submits y = 1000x + 1].
7. John: [zooming in] No! ... You see, it will still intersect at 1 (points at
the intersections again) ... this will not work...
8. Both: Silence
9. Mike: Oh no, all our functions will intersect … We need to… think
again…
After a period of silence, Mike created and presented a new idea (10), and
John agreed by suggesting how to act. Their reasoning is articulated
through predictive argumentation (10-11). Their problem-solving process
continued, and eventually they reached their goal.
10. Mike: If there is no intersection with the y-axis, then y would not be
involved at all…
11. John: Let’s remove y!
John and Mike’s reasoning, just as the other students’ reasoning, can be
interpreted as recurrently creative because they created/re-created their
reasoning sequence. In other words, they needed to process knowledge,
20
negotiate strategies, test solutions and evaluate ideas. Furthermore, their
reasoning, articulated as arguments, included suggestions that were in
general anchored in mathematical properties. During this process, GeoGebra
was used to visualize their reasoning, test strategies and, in particular,
provide feedback to use as a foundation for evaluating unsuccessful
strategies. Their evaluative reasoning was articulated as verificative
argumentation used to adjust their solving strategy.
However, GeoGebra was also used for trial and error strategies. Some pairs
created strategies without any uttered arguments as to why they might work,
and because successful responses from GeoGebra were seldom evaluated,
some students ended up with a fruitful strategy without understanding why.
The following students created horizontal function that was well negotiated,
however, the vertical functions were constructed by swapping the formula
without presenting reasons for doing so. It was not until the teacher (1, 6)
interfered that they engaged in more creative reasoning (7).
1.
2.
3.
4.
Teacher: Great, and why did it work?
Leah: The horizontal ... was easy …
Pat: ... Yes, we wanted lines with no slope ... we chose m as 0 (points).
Leah: … Yes ... like this (points): y = 0x + 5, and that is y = 5 … you see
… (points at the formulas).
5. Pat: ... Mmm, and then we swapped the formula … x = 0y + 5 ... and
that is x = 5 (points).
6. Teacher: Yes, but why did the swapping work?
7. Both: Silence
After a period of reasoning (7), Pat thought of an idea that she then tested
(8). She then presented her arguments (8). Leah transferred the arguments
to the vertical lines (9).
8. Pat: Yes – no wait: [submits y = 1, y = 2, and y = 8] ... you see! ...
(points at the three lines) … yes! … (points at the line y=5) it is
horizontal at 5 because y equals 5 all of the time!
9. Leah: Yes, yes, and the vertical is straight up because x equals 5 all of
the time!
10. Pat: Super nice!
In this example, the teachers’ interactions became important to make the
students, or at least Pat, engage in more creative reasoning. Pat and Leah are
an example of students who did not engage in collaboration for a part of the
time. Pat engaged in creative reasoning by testing ideas without sharing
them, aside from her action in GeoGebra. Thanks to the graphs produced by
21
GeoGebra, however, Leah was able to follow her reasoning. Their JPS was
therefore maintained and Pat’s uttered reasoning was interpreted, processed
and developed by Leah. Leah articulated her evolved reasoning through
verificative arguments (9).
5.3 Observing and repairing divergences and misconceptions
At one time or another, all of the student groups found themselves in a
situation marked by uncertainty, divergence, or misconceptions. Such
situations might cause their JPS to cease. As in one of the previous examples,
Mike and John discovered a misconception that made them change their
solution strategy.
Furthermore, the students used GeoGebra to verify knowledge or settle
disagreements by performing tests. The boys described below, who tilted
their square, were not sure about negative m-values. One of them created a
test (2) that the other interpreted (3), whereupon that piece of knowledge
was accepted as shared (3-4):
1.
Will: However, … if m is negative … what way … I mean … is the slope
... up or down?
2. Sam: Mm … I think ... down … let’s try … mm... [submits y = -2x +
3].
3. Will: Great … ok … downhill slope...
4. Sam: Yeah!
In the following example, the students in one group disagreed with one
another. Emma suggested that m and c represent the intersections on the yaxis and the x-axis, respectively. Zoe disagreed (1), and they performed a test
(2). Emma accepted the result, and the discussion continued.
1. Zoe: .. you cannot think like that! The formula doesn’t have a number
that decides the x-axis crossing!
2. Emma: Yes it does, look! [writes y=2x+4] It will intersect on 2 and 4.
(Points) … [submits] ..No!
3. Zoe: You see, the 4 is when it crosses the y-axis but there is no number
that crosses the x-axis
4. Emma: Hmmm … (silence) .. Okay, but what is the number 2 then?
These types of short sequences, creating tests or applying known tests, were
common and important features in the students’ work. Creating tests is one
way of verifying recalled knowledge or a created hypothesis that students
need to take as the next step in their problem solving. Furthermore, these
22
actions became important for maintaining shared knowledge and ideas to
uphold their JPS.
There are also examples of students who get stuck in the process their
problem solving. The students described below had successfully created two
horizontal lines, but had difficulty constructing the vertical lines. They had
been engaged in what seemed to be a process of trial and error, attempting
different m-values with no deeper evaluation. Their unsuccessful graphs
were still visible on the screen when they gave up, their JPS ceased, and they
asked the teacher for help.
1.
2.
3.
4.
5.
Teacher: Tell me what you want to do.
Luke: We want two lines.
Dan: Horizontal and vertical.
Teacher: And what you have done?
Luke: ... Here… (points at the horizontal lines) … we did not take...
any value that made them tilt ... they are horizontal (points at the
function y=2)… but if you choose a value ... with a steep slope … it
will not ... (points at the y-axis) no! ... (silence) yes! But… let’s start
out from the x-axis instead (points at the x-axis)!
6. Dan: Yes, can we find something to write...
7. Luke: Well … let’s do like this then ... [submits x = 2]
8. Both: Yes!! ... (laughs)
These students were not able to verbalize and evaluate their work. It was not
until the teacher initiated this type of evaluation (1) that Luke was able to
verbalize his reasoning and to interpret and evaluate GeoGebra’s feedback to
create another solving strategy (5). No further evaluations were made,
however, their JPS was restored and they continued with problem solving.
All actions made during problem solving, within all groups, were situated in
GeoGebra, which became a way of monitoring the problem-solving process.
To maintain their JPS, students needed to negotiate shared knowledge and
strategies, which is closely related to their ability to articulate their own
reasoning and to interpret one another’s reasoning. GeoGebra was used to
visualize their reasoning for one another through language and actions, i.e.,
gestures, referencing, performing tests etc.
5.4 Reaching the goal
The length of the students’ reasoning sequence and the number of created,
tested, and evaluated strategies varied. Four out of the 18 groups became
stuck in their problem-solving process and needed support from the teacher.
23
However, in the end, all groups managed to construct and angle their
squares. In other words, they were able to create and maintain their JPS, the
problem-solving process was successful, and their reasoning sequence
culminated in a solved task. There were no differences found regarding
solving strategies or time needed between the students in the social science
program and the students in the technology program.
6 Conclusions
The conclusions are presented in line with the research questions and
address the properties of GeoGebra, such as a shared working place,
visualizer, interactive environment, provider of feedback etc.
6.1 Collaboration
All 18 groups were able to initiate and maintain their JPS; in other words, all
student groups engaged in collaboration during their problem-solving
process. None of the groups split the work between them; therefore, their
collaboration did not turn into cooperation. On the contrary, GeoGebra
became a shared working space for their JPS within which their actions were
situated and to which their language referred. GeoGebra showed to be the
context in which the students created and maintained their JPS by
visualizing, negotiating and establishing a shared goal, shared knowledge,
and shared solving strategies. Furthermore, they interacted with GeoGebra
to uphold their JPS, keeping a shared concept of the problem, by, for
example, testing ideas (e.g., Mag and Sus, 5.3), repairing divergences (e.g.,
Zoe and Emma, 5.3) and retrieving forgotten knowledge (e.g., Will and Sam,
5.3). The students used GeoGebra to coordinate and synchronize their
collaborative activities and their individual reasoning (e.g., Mike and John,
5.2). This method of organizing group work, so that it upholds a shared
conception of the problem, is described as crucial to creating and
maintaining a JPS and therefore to succeeding in collaborative work
(Roschelle & Teasley, 1994). Four groups found themselves in situations
where their JPS temporarily ceased and they needed support from the
teacher (e.g., Luke and Dan, 5.3).
6.2 Creative reasoning
During our study, there were shorter periods of dialogue that are fragmental
and difficult to interpret, and a stimulated recall interview would probably
have given additional valuable data. However, based on the main part of the
students’ dialogue, that is, their arguments and to what extent their
propositions were anchored in mathematical concepts, the study shows that
creative reasoning was traceable within all student groups. During their
collaborative work, the students were found to be engaged in creative
24
reasoning, articulated as predictive argumentation, and constructing solving
strategies, articulated as verificative argumentation evaluating the solving
strategies. All of the students used GeoGebra to test and develop their
solving strategies, i.e., to construct and change formulas in line with their
reasoning. GeoGebra worked as an interactive partner, visualizing their
solving strategies. However, unlike a textbook, GeoGebra did not provide the
correct answers, and unlike guidance from a teacher, GeoGebra did not offer
clues on how to proceed. Therefore, GeoGebra’s feedback could be described
as contributing to their creative reasoning, given that the students needed to
interpret and evaluate the feedback from GeoGebra and by that present
verificative arguments as to why their idea did or did not work. Their
evaluation was then used as basis for their creative reasoning, uttered as
predictive arguments, to develop their solving strategies. Even shallowly
anchored solving strategies, like trial and error attempts, were evaluated
based on GeoGebra feedback. However, verificative arguments mainly
occurred when their strategies failed, and successful strategies were in
general not evaluated (e.g., Ella and Mary, 5.2). There were cases in this
study where students implemented truly shallowly anchored, though
successful, strategies that consequently were not discussed at all. It was not
until the teacher interacted with the students by asking for this type of
evaluation that the students engaged in more creative reasoning (e.g., Pat
and Leah, 5,2).
7. Discussion
As described earlier, this study shows that students used GeoGebra as a
shared context for their joint problem space (JPS), within which they
constructed and maintained a shared conception of the given problem
(Roschell & Teasley, 1994). Thus, besides distributing the problem solving
among participants (Stahl, Koschmann & Suters, 2006), performing graphdrawing and tedious calculations, and preventing students from dividing
their work as is common in cooperation (Manoucheri, 2004), this study
suggests that GeoGebra’s main contribution, enhancing collaboration, could
be described as facilitating sharing. The students used GeoGebra for
visualizing, referencing, testing, and monitoring to negotiate shared
knowledge, ideas, solving strategies, and the current problem state.
Successful collaboration, therefore, is not only an issue of not dividing work,
it is about sharing the understanding of the questions: where are we heading,
where are we right now, and how do we get there?
The overall characteristics of GeoGebra that may enhance creative reasoning
can be described as creative feedback, or “positive or negative sanctions
relative to her actions, which allows her to adjust the action” (Brousseau,
25
1997, p.7) That is, creative feedback is a visualization of their created and
implemented solving strategies without presenting correct answers nor
further guidance. This type of feedback is presented as ‘creative’ because it
becomes the object of students’ creative reasoning, i.e., their interpretation
and evaluation of the feedback in relation to their implemented solving
strategy. This problem-solving process differs from working with interactive
animations to explore relationships when the animations are chosen and
created by the teacher (Hoffkamp, Ploetzner and others). In this case,
GeoGebra could be thought of as an empty canvas, on which students needed
to create their own formulas to receive feedback on their own actions. In
addition to creating their solving strategies, they needed to decide what
relationships were important to investigate to solve the problem.
Merging these findings leads us to the conclusion that to create and maintain
their JPS, students need to share their creative reasoning with one another.
Sharing a JPS, including the language and actions within that space, is also
sharing a reasoning sequence. The student-student interactions combined
with the student-GeoGebra interactivity enabled the students to share their
creative reasoning through uttered argumentation and actions in GeoGebra.
One students’ uttered reasoning and actions were then interpreted,
evaluated, processed, and added to the other students’ reasoning, and the
evolved reasoning was thereafter shared again (e.g., Owen & Kevin, John &
Mike, Pat & Leah, 5.2). Furthermore, this way of sharing creative reasoning
through articulating arguments is not only making one’s reasoning
comprehensible to a fellow student, but it also helps to clarify the reasoning
to one’s self. These insights were described by Vygotsky (1986) as
transforming inner speech to outer speech; as we engage in dialogue and
construct verbal utterances, we simultaneously clarify our reduced inner
speech, our un-verbalized understanding, to ourselves. That is, one student’s
uttered reasoning will impact the line of thought within both students.
Finally, this study shows that students may occasionally find themselves in
critical situations marked by a JPS ceasing or shallow argumentation, which
brings us to the role of the teacher. In addition to designing the adidactic
situation (Brousseau, 1997) and including a challenging problem to solve
(Shcoenfeld, 1985), the teacher needs to offer students feedback. Timely
feedback closely connected to their activities (Brousseau, 1997) was mainly
provided by GeoGebra; therefore, the teacher may focus his/her support on
critical situations concerning lack of ideas, stagnated dialogue, or insufficient
evaluation. The challenge is to successfully interact with the students,
encouraging them to continue their dialogue and/or creative reasoning
without transforming the design from creative reasoning to imitative, by, for
example, presenting a solving strategy. The study showed that asking the
26
students to narrate their ideas, especially by implementing similar strategies
and evaluations, could be one way to increase interaction during critical
situations. This may help students to clarify their reduced inner speech to
themselves as well as to their classmate. To make their reasoning observable
and shared and to understand ideas or insights, students need to move
forward in their collaborative problem-solving process.
27
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30
The relations between reasoning, feedback from software and success in solving mathematical tasks Author: Jan Olsson
Abstract
This study investigates the way students’ reasoning and utilization of
feedback relate to success and failure in task solving. Sixteen 16-year old
students solved a linear function task designed to present a challenge to the
students. They were instructed to use GeoGebra as mediator and they had
the responsibility to choose solution strategies. The results were analyzed
using Lithner´s (2008) framework of imitative and creative reasoning
together with Shute´s (2008) definitions of formative feedback.
Schoenfeld´s (1985) protocol analysis was used to structure the path
through solving the task. The results showed that students who were
successful in solving the task reasoned creatively and used feedback
elaborately.
Keywords: Mathematical reasoning, formative feedback, dynamic software
1. Introduction
Two students tried to create a vertical line in the graphic-field of a dynamic
software, GeoGebra, by submitting algebraic expressions (y=mx+c). Their
strategy had been to increase the x-coefficient until the slope became
vertical. “We need a really large one, let´s put in y=100x+1”…. they
performed the activity, interpreted the feedback from GeoGebra, and found
that their strategy didn´t work… “Wait, if they are vertical they should not
intersect the y-axis at all….”.
Apparently they predicted incorrectly the result of the activity, they received
feedback from the software, and they elaborated on the result of the activity,
and drew a correct conclusion. It seems like a dynamic software as GeoGebra
offers guiding to the students’ task solving in the sense that they are invited
to set up target images for their actions, and the computer’s precise feedback
of the action offers possibilities to interpret and elaborate ideas for
subsequent actions. Research has shown, that students´ discussions often
31
are mathematically shallow when they are solving tasks. One reason may be
that in regular teaching students are not encouraged to create original
solution-methods; instead they are guided into rote-learning strategies as
they are provided with examples and formulas by instructions (Hiebert &
Grouws, 2007; Lithner, 2008).
Considering rote-learning, it´s important to investigate the causes,
consequences, and alternatives. In the perspective of reasoning Lithner
(2008) defines imitative reasoning (IR), which is related to rote thinking and
its opposite, creative mathematical reasoning (CMR), which is characterized
as creating original solving methods supported by argumentation anchored
in mathematics. A study (Jonsson, Liljeqvist, Norqvist, & Lithner,
submitted) showed students practicing CMR learned better than those who
practiced IR. On the assumption that CMR is better for learning Granberg &
Olsson (submitted) performed a study investigating the way interactive
software (GeoGebra) supported CMR. It was found that GeoGebra guided
students into creating goals, planning activities, receiving feedback, and
evaluating the result of the activity. In the present study, the way of using
feedback and the associated evaluation is further investigated through
questions about the relation between students´ reasoning and using of
feedback generated by GeoGebra.
Therefor the aim of this study is to investigate the relations between
students’ reasoning and the way students use feedback from GeoGebra.
Furthermore the relationships between students’ reasoning, their utilization
of feedback and their success in solving mathematical tasks will be
examined.
2. Aim and research-questions
The aim of this study is to develop understanding about students´ utilization
of feedback from software, associated to their reasoning during joint
problem solving aided by GeoGebra.
The research questions guiding this study are:
-­‐
What is the relation between the students’ way of using the feedback
that GeoGebra generates and the students´ reasoning?
-­‐
How do students´ ways of reasoning and utilization of feedback
from GeoGebra relate to their success in problem solving?
32
To examine the students’ reasoning and the utilization of feedback generated
by GeoGebra, a didactical design (which will be presented in detail later)
used in a previous study (Granberg & Olsson, submitted) was adopted. It was
designed in line with didactical propositions of Brousseau (1997) and
Schoenfeld (1985) and was found to entail trial and error attempts, creative
reasoning, and a source for feedback. Students’ dialogues, gestures and
screen activates were recorded and used as data.
3. Research framework and background
The main components of the research questions are reasoning, feedback, and
success of problem solving. The research questions concern the relations
between those components.
To structure data Schoenfeld’s (1985)
framework for protocol analysis was used. To answer the research questions
concepts of Lithner’s (2008) framework was used to analyze reasoning and
concepts of Shute (2008) were used to analyze feedback. The paragraphs
path of reasoning and ICT and ICT and reasoning are intended as
background to discuss the results. Each part of framework and background
will be further presented in the following paragraphs.
3.1 Problem solving
Schoenfeld (1985) elaborated and extended Pólya’s (1945) four problemsolving phases to the following six: Reading the task, analysis (why
properties of a task has certain consequences), exploration (why some
outcome will be useful), and planning (why a certain approach would lead to
solution), implementing (why the problem solving is proceeding in a proper
way) and verification (why a solution is actually reached). Focusing on the
decision-making at the executive or control level, Schoenfeld (1985)
proposed a method of protocol analysis to examine the way decisions shape
the path through problem solving. The protocol is based on the six phases of
problem solving and the transitions between these phases. Protocols are
parsed into episodes, which are periods of time during which the problem
solvers are engaged in a single set of action of the same type or character
such as planning, exploration, implementing, etc. Three classes of potential
decision points are described; the junction between episodes, when new
information or possibilities to take a new approach comes to attention, and
when difficulties indicate that there is a need of considering a change of
approach. In present study Schoenfeld’s framework will be used in order to
structure the students´ solving of tasks into episodes, and to consider
whether certain decisions may be related to success of solving the task or
33
not. The conversations and computer activities associated to task solving will
be further analyzed through Lithner’s (2008) framework of reasoning.
3.2 Reasoning
Students, solving mathematical tasks, will engage in reasoning. Lithner
(2008) defines the learner´s reasoning as her line of thought, that is, the
thinking process during which learner successfully or unsuccessfully
attempts to solve a mathematical task. Reasoning is guided as well as limited
by the student’s competences and is created in a sociocultural milieu. Lithner
characterizes reasoning as imitative or creative.
During task solving students´ strategy may be to recall known facts,
algorithms, or procedures that can be followed to reach an answer. Lithner
(2008) associates these strategies to imitative reasoning, IR. One variant of
IR is memorized reasoning, to recall memorized facts or complete answers,
e.g. a proof, a definition, or that 1 liter = 1000 cm3, but mathematical tasks
that are solvable in this way are relatively uncommon in school. Most school
mathematics tasks ask for some kind of calculation or other process and such
tasks can often be solved by algorithmic reasoning (AR), to apply provided or
memorized procedures and algorithms. This is often efficient if the algorithm
is remembered correctly and then only a careless mistake may prevent the
student from reaching a correct answer. Imitative strategies are described as
memorizing and recalling, and often lead to rote learning.
Creative mathematical founded reasoning (CMR), is characterized by
novelty, plausible argumentation and mathematical foundation. That is,
instead of recalling a procedure that will solve the task, the students´ create
solution methods that, at least to some extent, are new to them. The solution
strategies may be supported by plausible argumentation anchored in
intrinsic mathematical properties of the involved mathematical components.
Lithner (2008) suggests a wide conception of mathematical reasoning. In
contrast to strict mathematical reasoning, which means distinguishing a
guess from a proof, plausible reasoning includes also distinguishing a guess
from a more reasonable guess. Plausible thus reasoning is not necessarily
strictly logical but constructive through support of plausible arguments. The
more plausible they are, the stronger the logical value.
In order to address the question of what an argument is, Lithner (2008)
introduced the notion “anchoring”, which refers to its fastening in relevant
mathematical properties of the components one is reasoning about; objects,
transformations, and concepts. The object is the mathematical component,
the transformation is what you are doing with the object (a sequence of
34
transformations is a procedure), and the outcome is another object. A
concept is a mathematical idea that builds on objects, their transformations,
and their properties. Depending on what is the purpose of a transformation,
a mathematical property may be superficial or intrinsic. Lithner (2008)
illustrates that in this example (p.261): In deciding 9/15 or 2/3 is largest, the
size of the numbers (9, 15, 2, 3) is a surface property that is insufficient to
consider while the quotient captures the intrinsic property. If the student,
instead of applying a memorized procedure creates an original solution
method (provided it´s not done by pure guesswork) it´s necessary to
construct arguments for why the method will solve the task. Argumentation
may be considered as predictive or verificative. Relating to Schoenfeld’s
problem solving phases presented above, in the phases of analysis,
exploration, and planning, the arguments are primarily predictive. The
phases’ implementation and verification include primarily verificative
argumentation (Lithner, 2008).
3.3 Feedback
The students´ activities in GeoGebra may have a more or less articulated
purpose of finding out something particular. The actual computer activity,
when the student’s input is entered and the result of GeoGebra’s processing
appears on the screen, generates feedback associated to the action. In this
study it is assumed that the computer activity has the purpose to contribute
to the solving of the task and the information from GeoGebra is feedback. It
is also assumed the student will use the feedback in different ways, e.g. to
find out if they are right or wrong, to find clues how to proceed, etc.
According to Shute (2008), information meant as feedback to a learner in
response to some action on the learner´s part can be delivered in different
ways, e.g. verification of response accuracy, explanation of a correct answer,
hints, worked examples, and can be administered at various occasions
during or after the learning process. Feedback directed to the student´s
activity is considered as having effects on student’s learning. This is known
as Formative Feedback and has the purpose of promoting learning (Shute,
2008). Shute’s definition of formative feedback is information
communicated to the learner that is intended to modify her thinking or
behavior for the purpose of improving learning (p.154). In this study the
feedback from GeoGebra is a result of an activity planned by the students,
not prepared and delivered from one person to another. But the student may
have an idea of what feedback she needs, she will shape the computer
activity in relation to that purpose and may have the opportunity to use
feedback from software to modify and improve her learning.
35
In a review Shute (2008) found that a specific form of formative feedback,
Feedback on task-level, is particularly effective for supporting learning.
Compared to general summary feedback, feedback on task level is more
specific and often provides real-time information about a particular response
to a problem or task to the student. In this study the feedback is considered
on task-level.
Formative feedback consists of two parts affecting each other. In learning
situations a teacher may give response dependent on a student’s behavior,
which in its turn may affect the student’s behavior. Brousseau argues that
feedback does not necessarily comes from a teacher or a peer; it may be a
result of the student acting on the learning situation, which in turn will
change as a result of the action. If the learning situation change the student
has to reconsider her behavior (Brousseau, 1997). Brousseau calls everything
that acts on the student or that she acts on the milieu. In the current study
one of the main parts of the milieu is the interface of GeoGebra. The dynamic
software will respond according to the student´s activity and in turn affect
the students´ actions. This will be considered as using feedback from the
interactive software.
Formative Feedback provides students with two types of information:
Verification and Elaboration. Verification is about confirming whether an
answer is correct or incorrect and can be accomplished in different ways;
explicit, e.g. a prepared peace of information from a teacher or implicit, e.g.
expected or unexpected results in a simulation. Elaboration has several
variations, e.g. to address the response, discuss particular errors, provide
worked examples or give gentle guidance. One type of elaboration, response
specific feedback is considered as particularly learning-efficient. Response
specific feedback focuses on the learners answer and may describe why or
why not an answer is correct (Shute, 2008). In this study the feedback from
software is considered as implicit and both verificative and elaborative. If the
students have articulated a prediction of the outcome of an activity and just
note whether the prediction is fulfilled or not it is defined as verification. If
the students discuss the outcome in terms why or why not the result was as
predicted or if the outcome is elaborated in some other way (above just
noting if a prediction is fulfilled or not) it is considered as elaboration on the
feedback.
36
3.4 Path of reasoning and ICT
Developing of knowledge is often described as following trajectories or paths.
Theoretical insights of the ways learning occurs may be used for planning
activities on basis of hypothetical learning trajectories as well as
understanding actual task solving activities (Sacristán et al., 2010).
Systematic use of digital technology in mathematics education may
contribute to learning trajectories with multiple representations, possibilities
for inquiry-based task, open investigate practice, etc. which may enhance
transitions between cognitive levels, such as from intuitive to formal, from
synthetic to analytic, from concrete to abstract, etc. (ibid). Use of interactive
software is based on the users existing knowledge, which will influence the
medium, and the medium will influence the user. Therefore it is important to
emphasize students’ possibilities to express, present, test, refine and adjust
their thinking during task solving (Hoyles, Noss, & Kent, 2004; Lesh & Yoon,
2004). The frameworks of Schoenfeld (1985) and Lithner (2008) used for
analysis in the present study provide structures to examine students’ path
through task solving. Next paragraphs will give a brief overview.
Schoenfeld (1985) found that competent problem solvers constantly monitor
and evaluate their solutions as they work, which novices do not. Protocols (se
the section “problem solving” above) were indicating that novices don´t read
or analyze thoroughly or not analyze at all, that they work too long with
fruitless ideas, that they don´t verify their solutions, etc. Experts are moving
between all phases, often return to a previous one, work parallel with two or
more phases, and always verify their solutions.
Lithner (2008) suggests that the thinking process is not visible but the
reasoning can be observed in form of a reasoning sequence through written
solutions, think-aloud protocols, or interviews. The reasoning sequence
begins with a task and ends up with a correct or incorrect solution or a
decision to give up. Lithner suggests that solving a task can be seen as
carrying out the following four steps (Lithner, 2008, p.257);
1.
A (sub) task is met, which is denoted problematic situation if it is not
obvious how to proceed.
2. A strategy choice is made, where strategy ranges from local
procedures to global approaches and choice is seen in a wide sense
(choose, recall, construct, discover, guess, etc.). It can be can be
supported by predictive argumentation: Why will the strategy solve
the task?
3. The strategy is implemented, which can be supported by verificative
argumentation: Why did the strategy solve the task?
37
4. A conclusion is obtained. The reasoning sequence can be understood as a path through these four
steps, containing momentary stage knowledge, from where the student takes
decisions of strategies, which are implemented between stages of momentary
knowledge. In their descriptions of path through task solving, Schoenfeld
and Lithner are not explicitly considering use of ICT. However, the presence
of GeoGebra is supposed to affect the path through task solving and
conditions for reasoning. The role of ICT in association to reasoning and
problem solving will be discussed in chapter 6 in the light of the results
according to the research questions.
3.5 ICT and reasoning
In research literature reasoning is often related to interactive use of
technology. Notions of reasoning have different definitions; they may refer to
for example deductive reasoning, visualized reasoning, symbolic reasoning,
or reasoning in more general manner. The process of reasoning is considered
as contributing to individual understanding and/or communicating of
concepts, a development from everyday expressions towards formal
mathematical reasoning or consisting of an array of visualized reasoning,
symbolic reasoning, and reasoning in general manner, where all parts have
the same importance.
Roschelle et al. (J. M. Roschelle, Pea, Hoadley, Gordin, & Means, 2000)
claims that a benefit of using interactive software in mathematic education is
that students are encouraged to reason. There are different views in what
way the features of software may contribute to students reasoning, for
example Barwise and Etchemendy (1998) states that computers make
representations much more sophisticated and allow students to reason in a
natural way without developing into formal mathematic reasoning. Jones
(2000) suggests that interactive software including multi-representations
help students to focus on relevant mathematical relationships. Heid and
Edwards (2001) highlights that the computer feature of taking care of
routine work like drawing and calculations allow students to focus on
conceptual ideas and in addition allows students to reason with confidence.
Studies on the issue of interactive software seem to advocate a broader
conception of mathematical reasoning than strict reasoning associated to
proof. Expressions like “exploration of a space of possibilities” (Barwise &
Etchemendy, 1998, p 18) and “the process of organizing, comparing, or
analyzing spatial concepts and relationships” (Moore-Russo, Viglietti, Chiu,
& Bateman, 2013, p 98) are related to features of multiple representations,
38
which through thinking and reasoning support the solving of a task (Sedig &
Sumner, 2006).
4. Method
The method was designed to answer the research questions about reasoning,
feedback, success of solving the task, and the relation between those
components. To collect data, a similar design of a didactic situation as in a
previous study was used (Granberg & Olsson, submitted). In that study the
didactic situation was found to engage students in reasoning and to use
feedback generated by GeoGebra.
4.1 The didactic situation
The didactic situation was built on three propositions: challenge,
responsibility, and collaboration. Schoenfeld (1985) argues that students
need to work with mathematical problem that to some extend are new to
them in order to develop problem solving skills and that the task must
constitute an intellectual challenge to the students. Brousseau (1997)
propose that if a task shall remain a challenge the students must have the
responsibility to create solution methods of their own. Furthermore
Brousseau suggests that the teacher should instruct students until they can
continue on their own, and then devolving the responsibility for solving the
task to the students. During student-active sessions the teacher should not
interfere by guiding the students to right answers. If a task has an
appropriate design, the students will reach the target knowledge for the task
if they solve it. If the teacher offers information how to solve the task the
students will not reach the knowledge target.
Working in small groups has been reported beneficial for learning under the
circumstances that the task is focused on relations and concepts rather than
procedures. The former invites to collaboration and the latter to cooperation (Lou, Abrami, & d’Apollonia, 2001; Mullins, Rummel, & Spada,
2011). Collaboration is understood as a coordinated activity that is the result
of a continued attempt to construct and maintain a shared conception of a
problem (J. Roschelle & Teasley, 1994). In contrast, co-operation means that
the co-operators split the task into parts and each one works with different
parts. In this study, guiding students into collaboration has the purpose of
engaging students in conversations possible to interpret as reasoning
The students worked in pairs sharing one computer using the software
GeoGebra. The task consisted of creating three pairs of linear functions
39
whose graphical representations where perpendicular and to formulate a
rule for the circumstances when the graphs of two linear functions are
perpendicular (see the appendix). The author was present and answered
questions of technical matter about how to handle GeoGebra and
encouraged the students to explain their thinking if they got stuck or
considered they had solved the task.
4.2 Sample and procedure
Sixteen students from the science program at a Swedish upper secondary
school volunteered. They were 16 – 17 years old, 8 girls and 8 boys. The task
used for the study was pilot-tested and found suitable for 16-17 years old
students. They were informed about the ethical directives from the Swedish
Research Council (2001). They had some earlier experiences of linear
functions but they had no recent teaching of the issue.
The students solved the task outside the classroom in pairs. They used a
prepared GeoGebra-file, which contained a textbox with the instructions for
the task and all tools were disabled except for the pointer, the “layer-mover”,
and the angle-tool. They had a short introduction to GeoGebra, how to
submit formulas into the input-field, how to change an algebraic expression
and how to use the visible tools. Furthermore they were informed that they
could ask for technical matters (how to handle GeoGebra). In situations
where students got stuck the author encouraged them to explain what they
had done and why they thought that their strategies worked or not. When
students considered they had solved the task (or gave up) they were asked
why they were convinced they had come to a solution and whether their
strategies had been appropriate. Data was captured through screen
recording, with integrated voice and video recording.
4.3 Analysis method
Research question 1 concerns the relation between the students’ reasoning
and the feedback generated by GeoGebra. Students' reasoning was
categorized using Lithner’s framework of creative and imitative reasoning
(2008). The way that students used GeoGebra’s feedback was examined
using the concepts verificative and elaborative feedback (Shute, 2008). The
relation between students’ reasoning and GeoGebra’s feedback was analyzed
by considering whether the students’ way of reasoning before and after a
computer activity could be related to their way of using the feedback from
GeoGebra. Research Question 2 concerns how the results from RQ1 relate to
students' problem solving success. This will be analyzed by considering
whether important decisions are consequences of certain reasoning and use
40
of feedback from GeoGebra The analysis methods indicated here will be
elaborated in the following text.
The data consisting of conversations, computer interactions, and gestures
was transcribed into written text. In order to discuss students’ reasoning and
their way of using feedback from GeoGebra in relation to their success in
problem solving the eight pairs were divided into two groups; those who
reached a reasonable solution and those who did not. The main question of
the task, as earlier described, was to: Find a rule how to choose m and cvalue in the formula y=mx+c in such a way that the graphic representations
of two linear functions are perpendicular.
Schoenfeld’s protocol-analysis provides a way to examine the way students´
decisions shaped the way that solutions evolved (Schoenfeld, 1985, p.292).
In order to structure data the transcripts were partitioned into episodes
according to Schoenfeld’s six phases of problem solving, i.e. reading,
analyzing, planning implementing, exploring, and verifying. Thereafter
possible decision points were identified, i.e. junctions between episodes,
occasions where new information arose from computer activities or
students´ discussions, and sequences accompanied by difficulties. Actual
decisions, when students’ utterances or activities indicate how to proceed
were noted. These parts were used to consider in what way the decisions
contributed to solving parts of the task and if information gained from
solving parts of the task were used to answer the main question of the task.
In order to relate students´ success in solving the task with the
characteristics of reasoning, data were analyzed through Lithner’s (2008)
framework of reasoning.
Lithner’s (2008) framework was used to classify students´ reasoning into IR
or CMR. Students’ conversation, interaction with GeoGebra and gestures
were examined and units of argumentation were identified. The
characteristics of the argumentation, i.e. the implicit or explicit justifications
of the strategy choices and the strategy implementations, were used to
determine if the reasoning fulfilled the characterizations of imitative or
creative reasoning (Lithner, 2008). The students’ reasoning was regarded as
CMR if there were signs of creating a (for the students) new solving method
and if their argumentation was anchored in intrinsic mathematical concepts.
The reasoning was categorized as imitative reasoning if the (sub) task
solutions were based on familiar facts and/or procedures.
Finally, the way students used GeoGebra’s feedback was examined using the
concepts verificative and elaborative feedback (Shute, 2008). Dialogues and
gestures before and after each computer activity were noted. A computer
41
activity in this study includes the student input and the outcome displayed
by GeoGebra. Before this moment the students will plan (planning phase)
what to submit to the software and afterwards the students may interpret the
outcome and discuss how to proceed (verificative and analytic phase). An
utterance in a planning phase when the students predicted the outcome of a
computer activity was interpreted as a preparation for using the information
from GeoGebra as verifying feedback. After a computer activity, in the
verificative phase, students could use the feedback from GeoGebra verifiably,
identified as utterances of success or failure. If they after the verification
used the information to explain, extend pre-knowledge, plan for how to
proceed with the task solving, etc. they were considered as using the
information from the software elaborately, entering the analytic phase.
Finally, the situations of preparing activities and using feedback from
GeoGebra were put into relation to whether the reasoning was considered as
CMR or IR.
To answer RQ1, the use of feedback, verifiably and/or elaborately was
associated to the characteristics of reasoning, IR or CMR during the
planning of the activity, and to the reasoning when using feedback.
To answer RQ2 the reason for students´ success or failure in solving the task
were related to decision that the students made and could have made. It was
then considered whether the success or failure was related to the
characteristics of reasoning and use of feedback.
5. Results
All eight pairs were engaged in the problem solving process, however not all
of them solved the task. Four pairs came to a reasonable solution of the main
task. They used possible decision points for solving sub problems, and used
gained information to solve the following sub problems and the main task.
Two pairs did not reach a reasonable solution of the main task. They solved
some sub problems but did in less extent use their experiences from solving
these sub tasks. The remaining two pairs started out as the less successful
pairs but changed strategy and completed the task as the more successful
pairs. In the following, sequences from one pair from each category will be
analyzed with respect to their reasoning and utilization of feedback. Since
none of the chosen pairs had a clear understanding of the formula y=mx+c
they all needed to clarify the properties of the formula. The following
examples are from such sequences.
42
5.1 Alma and Ester
Alma and Ester had an exploratory approach to the task and they solved the
main task.
5.1.1 Episodes and decision points
During their task solving Alma and Ester went through episodes of reading,
exploring, planning/implementation, analyzing, and verifying. They had
possible decision points at the junctions of episodes and when the computer
activity generated new information. Two of those decision points particularly
supported their problem solving. The first of these decision points emerged
when they realized that they did not fully understand the formula y=mx+c,
and they decided to analyze the properties of the formula. The second
decision point came up when they had difficulties to find a perpendicular
function to y=7x-1, and they decided to change the function to y=2x-1 since
(2) is easier to divide than (7). It was also clear that they used information
from these episodes of analysis later in the task solving process. In the next
paragraph their first episodes of exploring will be analyzed.
5.1.2 Reasoning
After reading the instructions they initiated an exploring episode as follows:
1. Ester: well let´s just submit something…
2. Alma: y is equal to seven….
3. Ester: That means it´s going to be very much like this (almost
vertically, in front of the screen)
Their suggestion to choose seven as the x-coefficient, is followed by a
prediction of the graphical appearance on the screen. They created the
strategy themselves and Ester’s utterance and gesture is interpreted as
predictive argumentation. This strategy of suggesting something followed by
a prediction of the result supported by argumentation, reappeared several
times during their work. Some predictions were followed up by verificative
argumentation, e.g. “m=7 means the line must increase by 7 every step to the
right”, or “this one must have m less than 1 because you go more steps
horizontal than vertical”. Their reasoning is classified as CMR.
43
5.1.3 Feedback
The following excerpt, considered as an episode of analysis, will exemplify
the way Alma and Ester used the information after submitting the function
y=-3x-1, which they predicted to have “negative but less slope than y=7x-1”:
1. Alma: This is not 90°….
2. Ester: No it´s not… but let´s measure it to see how far off we are
[uses GeoGebra’s angle tool to measure the angel]…
After a discussion ending up in a conclusion that the constant term does not
affect the slope of the function and that the slope depends only on m, the xcoefficient:
3. Alma: we must concentrate on m….
After an analysis of different examples of submitted functions Alma
summarized using y=2x-1 and y=7x-1 as references:
4. Alma: Well, if we start at minus one…. This one has m=2…. Then you
go one step to the right and then two upwards [counting squares
with the mouse]…. And this has m=7… if you go one step to the right
you go seven upwards [counting squares with the mouse]….
First they used the GeoGebra’s feedback for verification, concluding that
they did not have a perpendicular line, and then they initiated an attempt to
elaborate on the result. This led them to an episode of analysis where they
elaborated on the feedback and investigated the way m and c affect the
graphical representation. During their work these students frequently
discussed and elaborated on the received feedback according to which they
adjusted their strategies. This indicates that they were using feedback from
software both as verificative and elaborative feedback.
5.1.4 Relations between reasoning, feedback, and
success in problem solving
Alma and Ester frequently used CMR to predict the outcome of the computer
activities, and they used the feedback from GeoGebra both for verification
and elaboration. Furthermore, these students always related their
elaborations to their predictions. This indicates a relationships between
CMR and elaboration on feedback from GeoGebra. It seems that predictions
of computer activities that are founded in CMR gives ground for using the
received feedback elaborately.
44
Alma’s and Ester’s decisions to examine the formula y=mx+c and to replace
the x-coefficient of (7) with (2) are considered as important for solving the
task. Both decisions were taken in episodes of analysis and preceded by
elaboration on feedback based on CMR. Information from analysis was then
used to answer the main question of the task. These students’ engagement in
CMR, and their elaborative use of feedback in the episodes of analysis seems
important for their success in solving the task.
5.2 Bertil and Isak
Bertil and Isak had an exploratory approach, they solved some sub task but
they did not solve the main task.
5.2.1 Episodes and decisions
During their task solving, Bertil and Isak went through episodes of reading,
exploring, and planning/implementing. Possible decision points were
junctions of episodes and when the computer activity generated new
information. Their first decision was to submit y=6x-3, followed by an
utterance that the function ought to have less slope. After some
manipulation they agreed on and submitted y=x-3. Then they submitted y=x-3 which they stated was perpendicular to y=x-3. The decision to change
y=6x-3 to y=x-3 made the sub task easier. This decision allowed them to
create y=-x-3 rather easily, just changing the m-value from positive to
negative. The decision made them find a solution to the sub problem of
creating two perpendicular lines. However, no trace was found, that they
used gained knowledge to solve other sub problems or answer the mainquestion.
5.2.2 Reasoning
After reading the instruction of the task Bertil and Isak went on to
implement an example of a linear function. The following excerpt is their
first turns of conversation of the first implementing episode:
1. Bertil: if we have…. sort of…. y equal to…. six….
2. Isak: [types y=6]…. x…. isn´t it…. plus….
3. Bertil: minus…. because we want to have it down here [points with
the mouse cursor at (0, -3)]….
4. Isak: ok…. [completes y=6x-3 and pushes the enter button]…. like
this…. sort of…
5. Bertil: then we must have one going this direction [pointing with the
mouse cursor negative diagonally on the screen]…
45
The strategy of submitting a function to have a reference from where to
proceed is created by them. Line 3 predicts the intersection to the y-axis, but
there is no articulated argumentation of in what way the submitted function
will contribute to the solution. As soon the enter button is pushed they start
to seek for a perpendicular line without discussing the result of the computer
activity (y=6x-3). This is characteristic for their reasoning through the whole
procedure of solving the task. Even though they sometimes create solution
strategies and sometimes predicts outcomes of computer activities, the lack
of argumentation and shallow or absence of anchoring in mathematics
means that their reasoning cannot be classified as CMR. It is not clear what
the purpose of choosing the function y=6x-3 was. Pointing at (0,-3) seems to
predict an intersection with the y-axis (line 3), which may build on
remembering that c determines the intersection point to the y-axis and the
reason behind the choice of (6) as x-coefficient is not clear from the data.
Strategies of recalling memorized facts and procedures means there is less
necessity for argumentation, which is characteristic for IR.
5.2.3 Feedback
In the example above on line 3, there is a prediction of the intersection with
the y-axis, which is consistent with the result of the activity. However, they
do not comment on this result that the graph actually intersected at (0,-3).
This is considered as using feedback verifiably. The following excerpt is an
example from the same episode. The intersection with the x-axis(0.5, 0) for
the function y=6x-3 is not what they expected:
1.
Bertil: wait… there it is minus three [points at (0, -3)]… why is this
situated here then [points at (0.5, 0)]…
2. Isak: should we… should we have ten instead…
3. Bertil: yes… type that….
4. Isak: yes [submits y=10x-3]… this is even steeper…. but let´s have….
one…. [submits y=1x-3]…
It seems like the intersection with the y-axis is what they expected but they
question the intersection with the x-axis at (0.5, 0). Instead of trying to
understand why the intersection is at (0.5, 0) they repeatedly change the xcoefficient (line 4) until they have the 45° graph associated to y=x-1. There
are no attempts to explain why an x-coefficient gives a certain slope. This is
considered as using feedback only verifiably, not elaborately. The way of
using feedback only verifiably and replacing functions without discussion is
characteristic for this whole task solving session.
46
5.2.4 Relations of reasoning, feedback, and success in
problem solving
The relation between reasoning and feedback is that Bertil and Isak have no
argumentation in their preparations of computer activities and they are
solely using feedback verifiably. The lack of argumentation is disqualifying
the reasoning as CMR. A consequence of the lack of predictive
argumentation is that they have no clear perception of what feedback they
can expect which makes it difficult for them to elaborate on the feedback
when it appears on the screen, and this in turn is a reason behind their
failure to solve the task.
5.3 Olga and Leila
Olga and Leila’s initial strategy could be described as imitative, trying to
remember facts and procedures. During this first half of the task solving
process they solved some sub problems but they did not reach an answer to
the main question. After 40 minutes they changed strategy. They started to
create solution methods, to analyze the received feedback, and eventually
they reached an answer to the main question. The following analysis is
separated into two parts, before and after the strategy change. The second
half will be described as a summary, focusing on the main causes for their
success in solving the task.
5.3.1 Episodes and decisions first half
During the first half of the task solving session Olga and Leila went through
episodes of reading, exploring, and planning/implementing. Possible
decision points were junctions between episodes, when the computer activity
generated new information, and sequences with difficulties. Their first
decision was to implement y=2x-2, whose graph was supposed to intersect
the y-axis at (-2) and the x-axis at (2). They did not try to analyze why it did
not appear like they expected. Instead they attempted to create a
perpendicular line by submitting y=-x-1, which led to a decision to change
y=2x-2 into y=x-1. The decisions made them solve the part of the task of
creating two perpendicular lines. However, no trace was found, that they
used the gained knowledge to solve other sub problems or answer the mainquestion.
47
5.3.2 Reasoning in the first half
The extract is from their first conversation after reading the instructions. It is
considered as an exploration of the conditions for the task:
1.
2.
3.
4.
5.
6.
7.
Olga: c was where it intersected the y-axis….
Leila: yes….
Olga: yes it was…. But what is m….
Leila: m was that value in between….
Olga: yes… the difference when you go….
Leila: yes…
Olga: eh…. What should I write then….
The utterances on line 1 and 3 and the attempts to explain on line 4 and 5
indicate that these students are trying to remember the way c affects the
intersection with the y-axis and the way to calculate the x-coefficient. The
articulated facts are not coherent and there is no argumentation for why
these facts may help to solve the task. This is characteristic for imitative
reasoning. Only the utterance that “c was where it intersected the y-axis” is
used in their first implementation, exemplified in the next excerpt:
8. Leila: should we make it easy and take y=-2 and x=2 [pointing with
the mouse at (0, 2) and (2, 0)]….
9. Olga: yes… go ahead…
10. Leila: [writes y=2x+2]…. No… minus [change and submit y=2x-2]…
hm…
11. Olga: yes… and a graph perpendicular to this must go …
Line 8 indicates a prediction that the graph would intersect the y-axis at (-2)
and the x-axis at (2). Their argumentation is not anchored in mathematics.
Their idea is merely to make the implementation easier. There is no
argumentation for why the graph did not appear like expected. A few lines
down a similar behavior is observed:
12. Olga: no…. that is not perpendicular…. It is too large…. But write y=x-1….
13. Leila: [submits y=-x-1] this is not 90°….
14. Olga: no…. but we can change y=2x-2 into y=x-1….
Instead of analyzing why the graphs did not intersect perpendicularly they
changed their first function y=2x-2 into y=x-1. This seems like a decision on
intuition while there is no argumentation for why it solved the sub task. The
approach of trying to remember the way the constant term and x-coefficients
48
affect the graph and the lack of predictive and verificative argumentation
classify the reasoning as IR.
5.3.3 Feedback in the first half
The first computer activity on line 10, y=2x-2, did not result in the
intersection at the x-axis that they predicted. Feedback was not explicitly
used verifiably or elaborately. It may have been used implicitly as a reference
to plan for a perpendicular line. Feedback from next activity (line 12), y=-x-1,
was used verificative, stating that the graph was not perpendicular to y=2x2. They changed y=2x-2 into y=x-1 (line 14) without presenting any
arguments why. It seems like the visual feedback made them guess y=x-1
should be perpendicular to y=-x-1. The use of feedback, to suggest y=x-1 is
perpendicular to y=-x-1, is not considered as elaborative while there is no
articulated attempt to understand why the lines initially were not
perpendicular. This example of using feedback merely verifiably is
characteristic for the first half of Olga and Leila´s task solving.
5.3.4 Relations of reasoning, feedback, and success in
problem solving in the first half
The few predictions they articulate (e.g. that the c-value indicate intersection
with the y-axis and, wrongly, that the the x-coefficient indicate the
intersection with the x-axis) are not supported by predictive argumentation
and the feedback from GeoGebra (e.g. the graph associated to y=2x-2) is not
elaborated on. It seems like the lack of articulated predictive argumentation
may cause difficulties for the students to elaborate on feedback and to use
verificative argumentation.
The reason behind Olga and Leila´s failure in solving the task during the first
half of the session is that they did not try to understand why the feedback
from GeoGebra did not verify their predictions. There is some argumentation
but it is shallow and not anchored in mathematics (e.g. the choice of y=2x-2
because it would “make it easy”). The lack of predictive argumentation also
means they don´t have a basis for analysis of unexpected results of computer
activities.
5.3.5 Second half, changing of approach
During the first half Olga and Leila did not manage to create perpendicular
lines with other x-coefficients than (1) and (-1). The episodes were either
labeled as implementing or exploring. They increasingly used their own
49
solution methods but there was none or only shallowly anchored
argumentation and no elaborative use of feedback. The turning point,
initiating the second half, happened after some 40 minutes. They managed
by trial and error to create perpendicular lines submitting the functions
y=2x-4 and y=-0.5x-4. They hypothesized that one x-coefficient must be a
fourth of the other “but negative” to create perpendicular lines. They tried
this on several examples with other x-coefficients without success for about
10 minutes. This is what Schoenfeld describes as a possible decision point
based on information indicating that something is wrong. For the first time
Olga and Leila carried out what can be seen as an analysis:
1.
Olga: I think we started out the wrong way round…. We are looking
for a pattern that does not exist…. this one affects the slope [pointing
at the x-coefficient]… and this one the intersection with the y-axis
[pointing at constant term]
2. Leila: and m affects the angle….
3. Olga: but why are these angles equal [pointing at the examples on
the screen]…
4. Leila: but we said that c doesn´t matter, we can move them here…
and there… (pointing with her finger at different areas on the screen)
5. Olga: so it is the slope that matters… and the relation between two
different slopes…
The sequence above is crucial for the solving the task since they initiated an
analysis of the way m and c affect the graphical representation of the
function (line 1), and they decided to focus on the relationship between the
two x-coefficients of two perpendicular functions (line 5). Next excerpt
exemplifies their changed way of reasoning:
9. Olga: what is common for our two examples (y=x-1 and y=-x-1,
y=2x-4 and y=-0.5x-4)….
10. Leila: they are like opposites….
11. Olga: one divided in two is zero dot five….
12. Olga: yes… and one divided in one is one…. but minus…
13. Leila: that’s it… one divided in one but minus…
14. Olga: then something times something must be one… but minus….
say a number…
15. Leila: six….
16. Olga: then the other one must be…. one divided to six…. but minus
[submit y=6x and y=-1/6x]…
17. Both: yea….
Their strategy to find the relationship between the x-coefficient of two
perpendicular functions generated a hypothesis (line 6). To examine their
50
idea they created a computer activity (line 8) using predictive argumentation
anchored in mathematics. The reasoning in this sequence is classified as
CMR. The next excerpt exemplifies that their creative and predictive
reasoning before the computer activity prepared them to elaborate on
feedback:
18.
19.
20.
21.
Olga: all right… what do we have… six and a sixth….
Leila: and one of them is minus….
Olga: then the m:s times each other must be minus one…
Leila: let´s try m equal to five….
On line 10-11 they used feedback verifiably, stating that their prediction was
correct. On line 12 they elaborated on feedback based on their predicative
argumentation (line 4-6) and suggested an answer to the main question of
the task. On line 13 they initiated an activity to verify their idea and by that
the answer to the main question. After this excerpt they verified their idea
using several examples and concluded that the task was solved.
5.3.6 Relations of reasoning, feedback, and success in
problem solving after changing of approach
The relation between reasoning and feedback in the second half of the
session is that planning of computer activities includes creation of strategies
supported by predictive argumentation anchored in mathematics, i.e. CMR.
Then these strategies are implemented and the feedback generated by the
computer activities is elaborated in the sense that students use CMR to
explain why the feedback verifies predictions or not. The example above
shows that Olga and Leila´s predictive argumentation is the basis for the
elaboration on feedback. This indicates that the argumentation behind the
prediction prepared them for using the feedback, e.g. that the activities
(y=6x and y=-1/6x) verify the prediction of creating perpendicular lines.
The reason behind Olga and Leila´s success in solving the task is that they
after a long period of fruitless trials carried out an analysis of their examples,
y=x-1 and y=-x-1, y=2x-4 and y=-0.5x-4. This analysis initiated a change of
reasoning into CMR, i.e. they started to argue for their strategies and
predictions. When the analysis turned into implementation they started to
elaborate on feedback, for example they discussed how to choose xcoefficients to provide perpendicular lines. There is a clear distinction
between their reasoning before and after the sequence where they were
trying several examples with different x-coefficients on the assumption that
one x-coefficient should be the negative fourth of the other. As long as they
did not support this prediction by argumentation they did not come closer to
51
a solution. After analysis of the examples, they argued for the relationship
that one x-coefficient must be minus one divided to the other. Through
elaboration on feedback they continued towards a solution of the task; the
product of the x-coefficients must be (-1).
5.4 Final remarks and possible conclusions
A conclusion associated to research question 1 from of the analysis is that
only the students who´s reasoning is characterized as CMR use feedback
from software elaborately. The examples of elaborating on feedback are
mostly related to argumentation associated to predictions. For example,
Alma and Ester predicted that a perpendicular line to the graph of y=7x-1
must have a negative x-coefficient larger than (-7), which was elaborated into
the conclusion that it is the x-coefficient that affects the slope. Another
example is that Olga and Leila´s prediction that (1) divided by the first xcoefficient will give the other “but negative” was elaborated into the
conclusion that the product of two x-coefficients giving perpendicular lines is
always (-1). There are a few examples of computer activities where students
just submit a function to see what happens. Alma and Ester’s first submitted
function, y=7x-1, is one example and Bertil and Isak´s first function, y=6x-3
is another. The difference is that Alma and Ester argued predictively for the
outcome, which Bertil and Isak did not. Bertil and Isak changed into another
function without trying to understand the properties of the graph while Alma
and Ester did not replace the function until they had elaborated on the
feedback. The reasoning of Bertil and Isak and Olga and Leila (in the first
half of the session) was classified as imitative reasoning. However, both had
some parts characteristic for CMR, they both created solution methods and
they tended to predict the outcome of the computer activities. But as long as
they did not provide argumentation for their predictions or why their
methods would solve the task they did not elaborate on feedback.
A conclusion associated to research question 2 is that only the students
who´s reasoning is CMR and who´s use of feedback is elaborative are
successful in solving the task. Common for pairs who succeeded in solving
the task was that they in some sequences analyzed the outcomes of computer
activities. Alma and Ester´s reasoning was mainly CMR and sequences of
analysis were initiated by elaboration on feedback, for example after
submitting y=-3x-1 they decided to sort out the properties of the components
in the function y=mx+c. The gained knowledge of the formula was then used
to answer the main question of the task. Olga and Leila´s change from
imitative reasoning into CMR was preceded by a sequence when they were
trying several variations of the idea that one x-coefficient should be the
negative fourth of the other. This was fruitless and they realized they had to
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try a new approach, that is, they initiated an analysis of x-coefficients and
changed manners of reasoning into CMR. That included argumentation for
predictions, which was elaborated into an answer to the main question of the
task. Bertil and Isak also tried several variations but never out of the same
idea. This may be the reason why they never came to analysis of the results of
computer activities and maintained imitative reasoning and why they failed
in solving the task.
6. Discussion
This study shows that students used GeoGebra as the main environment for
their task solving. The students’ activities were focused on performing
computer activities to receive feedback that could be used to solve sub
problems and finally to answer the main question of the task: That is, to find
a rule for how to choose constants of the formula y=mx+c in such a way that
the graphical representations of two linear functions are perpendicular. The
analysis shows that the students’ process of task solving follows the following
pattern; preparing a computer activity, receiving feedback from computer
activity and finally using feedback. These steps will be discussed using the
earlier presented notions of reasoning and the path of task solving.
6.1 Reasoning and paths of learning
The didactical design, where students solve a challenging task aided by a
dynamic software puts the computer in the center of the task solving process.
The computer activity, i.e. the moment when the students submit their
function, coincide with junctions of episodes. The episodes of planning and
implementation before the computer activity will, afterwards, be replaced by
episodes of verification and elaboration. This study shows that the more the
students engage in thorough planning before the activity, the better they are
prepared to utilize GeoGebra’s feedback.
The feedback itself is just a response, a visualization of the submitted
commands. It is up to the students to choose how to use the feedback, merely
as verification or additionally for elaboration. Planning and using of
feedback is articulated through their conversations. These may be put in
relation to students’ reasoning, provided a broader view is taken of
mathematical reasoning than merely associated with formal logic.
Lithner’s (2008) definition of mathematical reasoning (see earlier chapter
“Framework and background”) is not restricted to formal logic and not
associated to genius. That is, reasoning is associated with both high stake
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and elementary tasks and not just with high achieving students. The
framework by Lithner (2008) distinguishes between different ways of
reasoning, and could be used to examine students’ motives of using feedback
from GeoGebra for verification or elaboration. Lithner (ibid.) suggests that a
reasoning sequence, when solving a task, can be understood as a path
through the following steps, 1) a problematic task is met, 2) a strategy choice
is made, 3) the strategy is implemented, 4) a conclusion is obtained (see
chapter “Framework and background). The pattern of task solving emerging
from the present study (preparing a computer activity, implementing the
computer activity, utilizing of feedback) seems to let students focus on
planning and utilizing of feedback. Entering the commands and push the
enter-key is just a short sequence and, compared to pen and paper, the
computer process calculation and drawing quickly with great accuracy and
the students may focus on the result instead of on the process. The computer
activity initiate phase 3 and is a clear distinction between 2 and 3. In a
teaching situation this may help both teachers and students to predict and
understand the path through task solving, which is beneficial for both
planning and assessing activities (Sacristán et al, 2010). In line with the
results of this study teachers should encourage students to argue predictively
and verifiably for their strategies. The recognizable visible path through task
solving enhances teachers to encourage students´ argumentation in the right
moments of the solving process, instead of providing them with solving
methods.
The analysis using Schoenfeld’s six phases of problem solving shows a
significant difference between the students who solve the task and those who
don´t. The former carry out what is interpreted as analysis in some of the
episodes. Schoenfeld (1985) found that novices, in contrast to experts, don’t
analyze the problem thoroughly enough, they tend to work too long with
fruitless ideas and they don´t verify their solutions. These shortcomings
were all observed in the present study, therefore none of the students could
be labeled as experts regarding problem solving. The student groups who
succeeded had, as far as it could be observed, similar pre-knowledge about
linear functions as the ones who failed. The differences in success seem
rather to be related to the way they made use of their knowledge. This study
did not gather more specific information about students’ pre-knowledge and
the sample is too small to draw any general conclusions. However it is still
interesting that differences in success of solving the task seems to relate to
students’ engagement in analysis, which in turn, as this study indicates, is
related to CMR.
Theoretical insights of learning as developing through paths or learning
trajectories can be used both to plan activities and to understand actual task
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solving activities. This study shows that the students who´s reasoning could
be characterized as CMR, used feedback from software elaborately and they
were more successful in solving the task. Students who´s reasoning were
characteristic as IR, didn´t elaborate on the received feedback and didn’t
solve the task. Furthermore, the results indicate that it is crucial that CMR is
present during the planning of the computer activity. Engagement in CMR
provides the conditions for using feedback elaborately, and to present
verificative argumentation. In a teaching situation these insights can be used
to encourage students’ to make predictive argumentation in the planning
phase and to refer to their predictive argumentation when they use the
received feedback.
This study suggests that the characteristics of reasoning affect the way
students use feedback from a dynamic software. Research in general
discusses this matter the other way around, that is, the use of a computer
affects the students’ reasoning (Barwise & Etchemendy, 1998; Jones, 2000).
Features like multiple representations, taking care of procedures, and
neutral feedback are supposed to contribute to reasoning (Sedig & Sumner,
2006). Common for many studies is the suggestion that interactive software
open up for a broader view of reasoning, for example not just related to
logical proofs. The combination of software features supporting reasoning
and the idea that students’ way of reasoning is important for learning, is
challenging. This study shows examples of CMR related to successful task
solving and examples of imitative reasoning related to unsuccessful task
solving. Although they are all using GeoGebra, which offers multiple
representations, taking care of procedures, and neutral feedback. Then it´s
important to investigate the characteristics of reasoning with respect to their
use of a dynamic software.
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Appendix
This is an example of the view of GeoGebra. Two functions are submitted
and the angle measure tool is used. The yellow box contains the instructions.
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