Computer Supported Collaborative Reasoning – how to engage students´ in creative mathematical reasoning Author: Jan Olsson Abstract Research often point out traditional ways of teaching mathematics as guiding students into imitative rote-‐learning behavior, i.e. strategies of applying remembered procedures in problem solving without really understanding the intrinsic concepts. This paper is an introduction to empirical studies in upper secondary school using the characteristics of imitative and creative mathematical reasoning (Lithner, 2008) as tools to distinguish superficial rote-‐learning from conceptual learning. The aim is to produce guiding principles for using ICT-‐properties in development of tasks engaging students into creative mathematical reasoning. Initially one study has been performed. Upper secondary school students solved a task in dyads supported by GeoGebra. The analysis was focused on the way GeoGebra supported the conversation. Screen activity and conversation were recorded. Preliminary results show features of GeoGebra like multiple representations and providing of feedback guided the students´ into creative mathematical reasoning. Key-‐words: mathematical reasoning, problem solving, ICT, collaboration Introduction and background For some decades research has questioned mathematics teaching´s ability to produce problem solvers. The mainstream education has been described in similar vocabulary from different parts of the western world in words like monotonous and repetitive exercising on standard tasks. Furthermore it seems like students are supposed to learn some basic facts by heart and how to execute standard procedures and algorithms by rote (Stahl, 2011). This is not like reality where mathematics is a product of inquiry and dialog. There are several studies reporting students having difficulties solving non-‐routine mathematical problems and that they find mathematics boring (Boaler, 1998; Schoenfeld, 1985). Research often defines traditional mathematic teaching as oriented by textbooks and focusing on procedures and memorizing of facts while alternative courses means students are active in discussions, negotiating and participating in evaluating strategies while solving challenging tasks (Boaler & Greeno, 2000; Hiebert & Grouws, 2007; Scardamalia & Bereiter, 1994; Stahl, Rosé, & Goggins, 2011). Proposals of discussions and negotiating strategies lead to an interest of interaction. Boaler (1998) compared students from two schools, one defined as traditional with respect to mathematics teaching (i.e. the students were mainly working individual out textbooks) and one school were students mostly worked in small groups with project oriented open ended tasks. In a test containing task non typical for textbooks students from the un-‐
traditional school outperformed the students from the traditional school. While there was no differences in the students´ motivation Boaler suggests that traditional teaching has limitation only preparing students to use known procedures. 1 A research framework presented by Lithner (2008) defines students´ reasoning in order to understand the mechanisms behind the strategies related to mathematic task solving. The reasoning is described as the line of thoughts leading to a solution of the task. Rote learning is considered as a main obstacle for developing problem solving ability and characterized as imitative reasoning, i.e. the student tries to recall facts and procedures learned in previous lessons and apply them without arguments on tasks at hand. Research in relation to Lithner´s framework reports a regular use of imitative strategies among students in the Swedish upper secondary school. There are many examples where students try to remember an algorithm and then apply it without reflecting whether it´s usable to solve the problem or not. Students often maintain using imitative strategies when they find a task problematic, trying different algorithms randomly. Since Swedish students spend a lot of time doing exercises out of textbooks with given examples they are guided into imitative reasoning (Bergqvist, Lithner, & Sumpter, 2008). Boesen, Lithner & Palm (2010) find the design of the task important for what reasoning the students will engage in. In a study where students were solving tasks from the national tests students most often choose imitative reasoning when the task were similar to textbooks. Examples when students performed creative reasoning were when they did not recognize the given task as familiar and when they found a creative strategy less complicated compared to remembering strategies (e.g. a student doesn´t remember the procedure how to add fractions with different denominators, instead perform reasoning including arguments anchored in mathematic ideas and reach a solution). Even though there are many studies suggesting benefits for learning through problem solving or creative activities, investigations comparing different learning strategies are rare. A recent study (Jonsson, Lithner, Liljeqvist & Norgren, in progress) using the framework of imitative and creative reasoning investigated differences in outcomes of students learning. 130 students were divided into two groups after matching tests establishing cognitive index for each individual. One of the groups went through a training session on tasks designed for guiding imitative reasoning IR, and the other trained on tasks designed for engaging in creative mathematical reasoning, CMR. One week after the training session the students had the same follow up test where they who had trained on creative tasks outperformed the students from the other group. The results preliminary indicate benefits for CMR-‐students with low cognitive index as well as for them with high index. Therefore this paper is focused on how to design tasks that engage students into creative mathematical reasoning. From mainly being used in individual drill and practice application the use of ICT in education has developed into more frequently supporting interactions (Bottino, 2004). In a review drawing on 122 studies Lou, Abrami, & d’Apollonia (2001) examined ICT-‐supported group works. They found students working in groups in general had better results than students working individually but there were also several reports of the opposite and there were also less difference in the individual learning. The most effective setting for individual achievement was found as small groups, exploratory/tool programs, and instructions requiring students to take turns, agree on answers, and to summarize and explain their group work. Under those circumstances students learned better individually compared to individual exercising. The indications of creative mathematic reasoning as beneficial for learning combined with foundlings of ICT as supporting creative interactions lead to an interest of in what way computer software may facilitate students´ engagement in creative mathematical reasoning. This paper 2 will present initial work forming research aiming at creating guiding principles for using dynamic software as a component in tasks for mathematic education. Overall aim and issues guiding the research The aim of this research is to elaborate design principles for using dynamic software as a means for supporting collaborative creative reasoning during students’ problem solving processes. Issues that will be considered: -‐ Students choice to use dynamic software in order to collaborate, i.e. communicate, negotiate solving strategies, and monitor their process etc. in order to develop shared knowledge. -‐ Characteristics of the dynamic software that could possibly be observed as facilitating as well as obstructing the students’ creative reasoning, throughout a problem solving process, when they are creating shared knowledge. Research Framework The following will present the research framework, which is guiding the design of the studies. The main source is Lithner´s framework for imitative and creative reasoning (Lithner, 2008) which will be developed by adding a collaborative perspective using the Joint Problem Space (JPS) model (Roschelle & Teasley, 1994). The framework is meant to underpin design of studies as well as structuring and analysis of data. Reasoning The research framework described by Lithner (2008) has the purpose to produce data. It defines imitative and creative reasoning and explains the origins and consequences of each type of reasoning. Reasoning is defined as the line of thoughts adopted to produce assertions and reach conclusions in task solving. It´s not necessarily correct or formal logic but includes some (to the reasoner) sense of justifying it. While we can´t observe the thinking process, the observable result of reasoning is described as a reasoning sequence starting in a task and ends in an answer, appearing as e.g. written or oral solutions (fig. 1). This may be complemented by think aloud protocols and interviews. Imitative and creative reasoning Students´ solving non-‐routine tasks involving mathematic problem (i.e. an intellectual challenge (Schoenfeld, 1985)) will be engaged in reasoning. If the strategy is to recall a complete answer or an algorithm (e.g. instructions learned by heart) the reasoning is imitative. Solving mathematical tasks becomes a matter of remembering and recalling facts and procedures. Its opposite is creative reasoning where the student produce a new reasoning sequence, argues for if the answer is right or not, and the arguments is anchored in intrinsic mathematic ideas. Creativity is often synonymous to genius. The framework of Lithner presents creative 3 mathematical reasoning as possible not only for special talented students and not only usable for solving advanced tasks. Collaborative computer supported reasoning Collaboration will be understood as: “… 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). The reasoning is furthermore accompanied by the use of technology, introduced as a mediator and as a provider of feedback. The reasoning will therefor become a more complex process compared to individual reasoning. The resources, heuristics and beliefs will contribute to the reasoning process. Resources and problem solving strategies will be negotiated and shared, and the students´ options to monitor and control will focus the collaboration on the problem solving process. Then the reasoning sequence will consists of interactivity between the students as well as interactivity with GeoGebra. The collaborative reasoning sequence could be observed by capturing dialogue, gestures and screen activities. The framework will by then consider individual competences and resources situated in the students´ minds as well as shared knowledge observed within group activities, shared in the sense that the students´ have negotiated and agreed on knowledge as true, or at least plausible. The shared knowledge is established through interactions student – student as well as students – GeoGebra. A way of making collaborative reasoning visible is to structure dialogs, gestures and screen-‐activities in a Joint Problem Space, (fig. 2) which will be discussed in next part. Reasoning within a Joint Problem Space (JPS) The concept of Joint Problem Space (JPS) was introduced by Roschelle & Teasly (1994) as a shared and negotiated knowledge structure consisting of goals, descriptions of current problem state and awareness of available problem solving actions, i.e. negotiated and shared understanding of the questions: Where are we heading? Where are we right now? and How do we get where we want? The students´ collaborative reasoning while solving the task will create the JPS, which in turn will constitute the context that, enables meaningful reasoning about how to solve the problem. The construction of the JPS will start with the students’ individual contributions, and will continue by negotiating and agreeing on shared knowledge suitable to solve the problem. To maintain the problem solving process as well as the JPS the students need to monitor and control the problem solving process, i.e. observe and repair divergences, misconceptions, and to determine appropriate knowledge and strategies. The dialog is essential in collaborative problem solving and could be understood as interactive turn-‐taking including suggestions, questions, answers, acceptance, disagreement, repairs etc. Roschelle & Teasly (1994) added shared activities like gestures, drawing, and interacting with 4 some technical device as complements to the communication, used in order to clarify and strengthen arguments. Depending on the didactical design students may construct their JPS by collaborative imitative reasoning, i.e. remembered facts, algorithms and procedures, or collaborative creative reasoning, i.e. negotiating goals an solving strategies and anchoring their arguments in mathematical knowledge. Thereby the construct of JPS is suitable for investigating collaborative reasoning as a result of task-‐design. Methodological considerations The study has a sociocultural perspective, considering knowledge as something students develop through interaction and collaboration. Knowledge, on its part, could be regarded as situated in students’ minds and/or as shared and situated within group activities. This vagueness is discussed as problematic among researchers, and there are arguments that both perspectives are needed (Sfard, 1998) while others consider that as a methodological matter rather than an ontological (Stahl, 2005). These notions will be considered as interactive. Knowledge, or resources, within the individual minds will be shared and negotiated through interactions during group activities. The process of sharing, negotiation and creating shared knowledge through collaborative activities will on its part involve individual interpretation, evaluation and, if needed, an adjustment of the individual knowledge. The evolving individual reasoning will contribute as well as become influenced by the problem solving process, their joint activities and construction of shared knowledge. Individual, inner reasoning is however not possible to observe besides as utterances and actions within the group activities, this mix of language and action will constitute data of the studies. A Study of ICT supporting collaborative reasoning and a pilot test in a classroom context The presented framework has been used in a study performed the autumn 2012 a task designed for engaging students in mathematical creative reasoning was tested. 36 students enrolled in the upper secondary school participated. The following will present aim and questions of the research, a summary of the design of task, the method, and preliminary results (A report is in progress, authors; Carina Granberg & Jan Olsson). Further on the task has been tested in a regular school context in collaboration with teachers. Aim and research questions The aim of this study is to elaborate design principles for using dynamic software, in this case GeoGebra, as a means for supporting collaborative creative reasoning during students’ problem solving process, creating shared knowledge. The following research questions will therefore be focused in this study: -‐ How do students choose to use GeoGebra in order to collaborate, i.e. communicate, negotiate solving strategies, and monitor their process etc. in order to develop shared knowledge? -‐ What characteristics of the dynamic software, GeoGebra, could possibly be observed as facilitating as well as obstructing the students’ creative reasoning, throughout a problem solving process? 5 Design of the task The design principles for the task was evaluated from ideas of Schoenfeld and Brousseau and inspired by ideas of interaction. Furthermore the task would be suitable to solve in the milieu of GeoGebra. The following give a brief overview of the ideas underpinning the design. For further details see references. • Schoenfeld (1985) argues that learners need to work with mathematic problem that to some extent must be new to them, an intellectual challenge. Central areas and competencies students need to successfully engage in mathematical problem solving are; Resources: individual mathematical knowledge and understanding, Heuristics; strategies for working on unfamiliar problems, Control; decision-‐making on what resources and strategies to use, monitoring the process etc. and Belief systems; individual mathematical world view. • Brousseau (1997) suggest a didactical design where the students take responsibility. During this part of the didactical situation teachers should not interfere with the creative process by guiding to the desired answer. However feedback is important to support the students´ problem solving process. The feedback could come from the teacher, a student or computer software. • While an important component in creative mathematic reasoning is plausible argumentation anchored in mathematic concepts (Lithner, 2008), there is an interest creating learning environment were it is natural to explain for and understand each other. Scardamalia & Bereiter (1994) suggest schools should be organized with focus on knowledge building where students working in small groups produce and improve conceptual understanding in collaboration. •
Features of ICT as visualization, availability of multiple representations and performing low order tasks (e.g. simple calculations, drawing of geometric figures, etc) scaffold a problem solving approach and engagement to higher order tasks (i.e. conceptual understanding) through sharing information, joint interpretation, negotiating and exploring divergences in understanding. An appropriate software offers an accessible record of the joint work which can be replayed and modified and is a reference to explanations and arguments (Nussbaum et al., 2009; Sinclair, 2005; G. Stahl, Koschmann, & Suthers, 2006) Thus the didactical design principles was: • Intellectual challenge – the students were not supposed to recognize the task as similar to earlier experience • Responsibility – the students should have possibilities to proceed with the task solving on their own • Collaboration – the task should invite to interaction 6 The ICT-‐propositions for the task was: • Providing a record of the joint work – support the students control and monitoring of the joint solving of the task by affording references for arguments, negotiating, and repairs of divergences and misconceptions • Performing calculations and constructions – the students may focus on conceptual understanding • Providing multiple representations of linear functions, easy to create and manipulate – students must be in control of the performance which allow them to test and evaluate ideas Based on this the following task was evaluated: 1/ Create four linear functions so that their graphical representations form a square 2/ Adjust the functions so that the square rotates to any degree. Method of the Study The study was performed outside the classroom. Students solved the task in dyads after a brief introduction to GeoGebra and they spent 15 – 45 minutes. The authors were present and answered questions about understanding the task and how to handle GeoGebra. Questions about how to solve the tasks were answered like “Can you explain what you have done?” or “Why did you try this” (referring to what was visible on the screen), nothing that directly was a clue for proceeding. The computer activity and conversations was recorded and for half the students their gestures were noted. The conversations and computer activity were transcribed in details. Analysis The framework described earlier in this paper was used for analysis, which was performed in following steps: 1/ Structuring the data as the Joint Problem Space, i.e. the students creating of a shared goal, creating, testing and evaluate solving strategies, observing and handling divergences in understanding and misconceptions and finally reaching the goal. 2/Focusing on collaborative interactions, i.e. language (turn taking, argumentation, negotiating, explanation etc.), actions (gestures and interactions with the software), and then relate their activity to their use of GeoGebra. 3/ The language and actions were examined in relation to the characteristics of Creative Mathematic Reasoning, which was also related to the students’ use of GeoGebra. Summary of preliminary results The study shows that the students used GeoGebra as a shared working space, a visualizer and an interactive “partner”, altogether, a creative context for their reasoning sequence and joint 7 problem space. Some of GeoGebra’s interactive characteristics became particularly usable for their creative reasoning and problem solving process, namely: guiding the reasoning, constructing and providing creative feedback. These features facilitated activities like: creating, testing, interpreting, evaluating and adjusting solving strategies, a necessity to keep a flow in their creative reasoning and by that their problem solving process. A “pilot” pilot study In order to give inspiration to further research on the theme one pilot “follow up” study have been performed as a development of the (preliminary) conclusions into a classroom activity. The students were in first year of the technological program in the Swedish upper secondary school and had experiences from linear functions. How to proceed from research results into everyday classroom practice One of the aims of the research is to produce prescriptive results useful for designing teaching. While the aim of the study came out of research indicating that students using creative reasoning learn better than those using imitative reasoning the interest was oriented to interpret if the students were engaged in creative reasoning. Not what they eventually learned. In an everyday classroom context it is anxious to have opportunities to assessment with respect to what students learn. Therefor the teachers took part of the draft of the article associated to the study described I previous part and discussed what they thought the students could learn out of the task. They evaluated both summative assessment guiding anchored in the Swedish curriculum for mathematics and they prepared formative assessment out of the NCTM´s “Five key-‐strategies for effective formative assessment” (ORDER, 2007). The questions for the intervention were: In what ways could the results of students´ creative reasoning from the study be observed in a classroom context? What occasions to formative and summative assessment are given out of the students´ reports? The design of the task was used with two added questions according to the teachers´ request; the students were asked what are the conditions for two graphic representations being perpendicular and create three functions which graphic representations form an isosceles triangle. The students worked in pairs during 60 minutes assisted by GeoGebra. But instead of recording the whole activity they were instructed to individually record a summary of the GeoGebra-‐
solution, orally commented, and submit the file to the teacher. During the activity the students were concentrated and focused on the task and they had few questions to the teacher. Even though their textbooks were available nobody were observed to use them as references. In the concluding individual reports the students presented a summary of their work where they argued for e.g. slopes of lines, intersections of lines and axis, equal length of sides, etc. This was usual for the teacher to assess the students´ knowledge and provide feedback of how they could evaluate their argumentation. An example of a misconception observed was a student claiming if two functions y=mx+c has opposite m-‐values, e.g. y=x+2 and y=-‐x+2 or y=2x+2 and y=-‐2x+2, means their graphic representations are perpendicular. This is right in the first example but not in general. The student-‐recordings allowed the teacher to discuss this both with the individual student and with the whole class. 8 The teachers were positive to the possibilities to listen to the students´ explanations one by one (compared to a general lesson where the time does not admit listening two more than 20 students individually), the discussions that were initiated and the opportunities to correct misunderstandings. They were concerned about the time the preparations, the activity and the sum up work consumed. Discussion The difficulties to affect the traditions of teaching mathematics with respect to problem solving seems to be well documented in research and there is many reports proposing alternatives to the widespread rote learning, i.e. learning by following instructions until you remember the procedure. Anyhow, even if research has shown the problem and positive results from interventions where students learn by solving non-‐routine problems it seems to be little impact in everyday teaching. One obstacle for that could be the research often is descriptive and not often presents clear comparisons between different learning methods. In an attempt to follow up the indications of beneficial learning through tasks engaging students to Creative Mathematical Reasoning (CMR) this paper argues for prescriptive research presented as guiding principles for designing of CMR-‐tasks. The experiment of using the (preliminary) result of the ICT-‐study in an everyday classroom shows some important differences of conditions for teaching and researching. The same task was used but while the researcher can spend weeks of analyzing a few parameters the teacher has just a limited amount of time considering a wide spectrum of information concerning the students learning, development, what´s appropriate feedback for the individual, etc. The pilot-‐
study showed, not surprisingly, even though the teachers agreed on the benefits of creative mathematical reasoning, it was not enough knowing the characteristics of students´ reasoning. To be an appropriate task for everyday teaching there must be guidelines for summative and formative assessment and it must be manageable taking part of the students´ reports. One may argue that teacher are professional and in position to determine what beneficial for teaching and they should be able to adopt positive elements out of research. Anyhow, a closer collaboration teachers and researcher might be fruitful while researchers could contribute with important details while teacher considering the whole. In the pilot study of this paper the research contribution initially was the founding of how to engage students into creative mathematical reasoning while the teachers gave the demands of assessment, and finally the researcher and the teachers together sorted out the guidelines for assessment anchored in research. A suggestion as the key to the fruitful collaboration is convincing argumentation for engaging students in creative mathematical reasoning and a task design allowing assessment relevant for curriculum and individual development. The particular interest of ICT has led to insights of different views of the role of computers in education. From mainly appearing as an instructive tool or drill and practice instrument for learning facts and procedures, research nowadays often emphasizes features scaffolding interactions aiming at conceptual understanding. In the study of ICT and creative mathematic reasoning the students´ problem-‐solving process was visible through the record GeoGebra provided. They decided under negotiating what they wanted to submit into the program, dependent of what the program returned they argued jointly for the plausibility of their actions, a satisfying feedback from GeoGebra meant they could continue and if it wasn´t they could step 9 backwards investigating what was wrong. There was no need trying to remember algorithms or procedures. I suggest this is a main reason for the students´ engagement in creative mathematical reasoning. The features of GeoGebra made creative reasoning more rational than imitative reasoning. Of course this is an assumption, which must be further investigated. 10 References Bergqvist, T., Lithner, J., & Sumpter, L. (2008). Upper secondary students' task reasoning. International journal of mathematical education in science and technology, 39(1), 1-‐12. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 41-‐62. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. Multiple perspectives on mathematics teaching and learning, 171-‐200. Boesen, J., Lithner, J., & Palm, T. (2010). The relation between types of assessment tasks and the mathematical reasoning students use. Educational studies in mathematics, 75(1), 89-‐105. 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. Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. Second handbook of research on mathematics teaching and learning, 1, 371-‐404. Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255-‐276. 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. Nussbaum, M., Alvarez, C., McFarlane, A., Gomez, F., Claro, S., & Radovic, D. (2009). Technology as small group face-‐to-‐face Collaborative Scaffolding. Computers & Education, 52(1), 147-‐153. ORDER, I. (2007). Five “Key Strategies” for Effective Formative Assessment. Roschelle, J., & Teasley, S. D. (1994). The construction of shared knowledge in collaborative problem solving. NATO ASI Series F Computer and Systems Sciences, 128, 69-‐69. Scardamalia, M., & Bereiter, C. (1994). Computer support for knowledge-‐building communities. The journal of the learning sciences, 3(3), 265-‐283. Schoenfeld, A. (1985). Mathematical problem solving. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational researcher, 27(2), 4-‐13. Sinclair, M. P. (2005). Peer interactions in a computer lab: Reflections on results of a case study involving web-‐based dynamic geometry sketches. The Journal of Mathematical Behavior, 24(1), 89-‐107. Stahl, G. (2005). Group cognition in computer‐assisted collaborative learning. Journal of Computer Assisted Learning, 21(2), 79-‐90. Stahl, G., Koschmann, T., & Suthers, D. (2006). Computer-‐supported collaborative learning: An historical perspective. Cambridge handbook of the learning sciences, 2006. Stahl, G., Rosé, C. P., & Goggins, S. (2011). Analyzing the discourse of geogebra collaborations. Essays in Computer-‐Supported Collaborative Learning, 186. 11