01 Editorial - FINALRF

Mathematics Education Research Journal

2010, Vol. 22, No. 2, 1-6

Editorial Researching Applications and Mathematical Modelling in Mathematics Learning and Teaching Gloria Stillman, Jill Brown and Peter Galbraith Research into learning and teaching using mathematical applications and modelling have been part of the agenda of the international mathematics education community for more than 25 years including the four yearly International Congresses on Mathematics Education (ICME). Since 1983 the biennial conferences of the International Community for the Teaching of Mathematical Modelling and Applications (ICTMA) and the related ICTMA series of research volumes (e.g., Lesh, Galbraith, Haines, & Hurford, 2010) have been major sources of exchange of research ideas and growth that continue to push forward the research envelope. In July 2011, ICTMA 15 will be held in Melbourne. The vitality of this research field has been further evident in recent special issues in international journals (e.g., Biehler & Leiss, 2010; Kaiser, BlomhØj, & Sriraman, 2006) and the launch of new journals in the field (e.g., Journal of Mathematical Modelling and Application from the Reference Center for Mathematical Modelling in Teaching (CREMM) at the University of Blumenau, Brazil). Over the years there have been many different models of curriculum incorporating this approach to learning and teaching varying from complete courses or significant components of courses (e.g., Queensland Studies Authority, 2010), to the inclusion of some aspects of mathematical modelling within a wider curriculum (e.g., Ministry of Education, 2006). Research into teaching and learning through applications and mathematical modelling has been ongoing in the Australasian region “because of its potential to add another dimension to the mathematical experience and skill of learners” (Stillman, Brown, & Galbraith, 2008, p. 141) and its inclusion in various curriculum documents. The 14th ICMI study on applications and modelling in mathematics (Henn & Blum, 2004; Blum, Galbraith, Henn, & Niss, 2007) provided a significant documentation of the state of the art in this field of research and practice and a boost to research generally at the time which has had an ongoing ripple effect. Against this backdrop, this special issue showcases some of the latest international research on this topic including research from Australasia. It is important to note that mathematical modelling does not have a unique interpretation within educational practice. This is almost inevitable given the range of contexts and practitioners that exist internationally, but it does provide a cautionary tale for those who want to encompass its meaning and purpose within their own specific preferences. It is therefore important that readers note carefully the assumptions and intentions that characterise the work of individual authors, including those represented in this issue.

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Generally speaking two broad philosophies can be identified as drivers of modelling initiatives. One of these uses practical problem situations to motivate and introduce particular mathematical content, the learning of which remains the first priority. The other takes the view that for students to spend years learning mathematics without any sense of how to apply it in the world around them, is inappropriate. This approach actively seeks to teach a modelling process, within which learned mathematical knowledge can be productively employed. Within this approach the search for valid solutions to genuine problems cannot live entirely inside a conventional classroom (see discussion below in relation to Peled’s article). Key issues that are addressed in this special issue of MERJ include: • Epistemology of applications and mathematical modelling (Peled); • Goals of applications and modelling and their curricular embedding (Peled; Perrenet & Adan); • Implementation (Klymchuk, Zverkova, Gruenwald, & Sauerbier) and evaluation of applications and modelling curricula in practice (Perrenet & Adan); • Technology with and for applications and modelling (Geiger, Goos, & Faragher) including technology as agent provocateur; • Modelling competencies and beliefs (Houston et al.); • Classroom practice with respect to teaching/learning mathematical modelling and applications (Chinnappan; English; Geiger et al.; Klymchuk et al.; Lingefjärd & Meier; Yoon, Dreyfus, & Thomas); • Significance of prior mathematical knowledge for successful modelling (Yoon et al.); • Modelling for deepening conceptual understanding (Yoon et al.); • Conceptions held by undergraduate students concerning the place of modelling within their mathematical experience (Houston et al.). As Lingefjärd and Meier point out “working with mathematical modelling tasks gives rich opportunities to try another teaching role” which the teachers in their study interpreted as acting as a manager fostering independence preserving team work of groups and individuals. This is in stark contrast to classrooms where “teachers solve all the interesting issues for kids and present them already resolved” (Hanner, James, & Rohlfing, 2002, p. 106). “Classes applying Mathematical Modelling provide various opportunities for mathematical thinking, knowledge search, and experiences in understanding, judgement, and interpretation of problems” (Kim & Kim, 2010, p. 118). On the other hand, the teacher who was the focus of the study by Chinnappan took a more directing approach as he responded to the challenge of the age-old conundrum for lower secondary school students: “Why do we need to learn algebra?” On the spur of the moment he attempted to exploit what is a plausible authentic context, the making and sale of toy soldiers from the chewing gum stuck under the student desks in the school. The serendipitous nature of the lesson appears to be the reason for high scaffolding by this teacher of the model construction by the students through questions and prompts resulting in a high volume of teacher talk in relation to student talk. Nevertheless, the author is able to demonstrate that the Chinnappan and Thomas (2003) framework was able to identify “important elements” in the lesson “that need to be orchestrated for

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the model to emerge”. The modelling process that was being engaged in by both students and teacher was clearly a vehicle to mediate the consolidation of content but in the process demonstrate the utility of that mathematical content. In this context, the classical view of orchestration is evident as the teacher manipulates the lesson elements to construct the model he perceives will consolidate learning; however, let us not ignore the more serendipitous nature of the choice of direction and task for the lesson that the teacher spontaneously took onboard—teacher moves which are more improvisatory and more often typical of the jazz genre (Tanner et al., 2010). Peled raises interesting and important issues in her thought provoking (Fish) Food for Thought. The significance of these is not widely enough understood. Peled’s approach emphasises the basic modelling fundamental, that (possible alternative) assumptions need to be articulated at the start of a problem, and the mathematics employed is influenced by these. In thinking about the examples used in this article, a couple of other 'real' real-life examples come to mind. One is that young children are natural modellers when sharing lollies in the playground "2 = 1" is often an acceptable mathematical equivalence based on size. Unfortunately, a lot of real life understanding of young children is supplanted by school mathematics. The issues Peled highlights have a very practical realisation in the Australian taxation system, when considering personal income tax on the one hand and GST/VAT on the other. For example, 1. John hires a plumber for a job that costs $250 of which $25 is GST? Bill hires the plumber for a job whose total cost is $375. How much of this is GST? 2. June earns a net income of $30 000 and pays tax of $3 600. How much tax should Janette expect to pay on a net income of $60 000?

Those raised on a celebration of success with Fish and Eel Problems would likely go automatically into a proportional mode to get the first one right and the second one wrong - ignoring the social content of the latter concerning differential tax rates. Peled’s article also touches on the notion of authenticity. Some individuals have claimed that once a problem from the outside world is introduced into a classroom situation its authenticity is lost - almost by definition. What they are doing is privileging their conception of what school mathematics, and life in classrooms is about, and making modelling fit the stereotype and subject to associated restrictive practices. What modelling, properly conducted can do, is to challenge some of those norms and assumptions, mathematical, situational, and pedagogical. Our perception is that Peled is taking such an approach, in other words, challenging readers to confront some sacred cows! In application questions posed in classrooms the issue of whether or not students undertaking the task are familiar with the context is regarded as a significant one. However, as Tourniaire (1986) pointed out previously, it is not simply a matter of being familiar with a particular task context but rather familiarity with the use of the particular mathematical concepts and procedures in that context that allows some, but not others, to access these tasks. This point is particularly pertinent to the small study by Klymchuk et al. in this issue where university students had difficulty with what, to their lecturers, was a relatively straightforward calculus application.

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English takes a modelling as vehicle approach as she uses modelling situations to provide opportunities for Grade 1 students to construct and develop ideas related to making sense of data. The three classes of students in this study were required to identify attributes that could be used to classify given sets of data and use these attributes to classify the data. Furthermore, the student groups were to present their data using a representation of their choice. English notes the importance of task design in mathematical modeling. Her tasks are explicitly set to challenge and extend the mathematical reasoning of students as they engage with data. Statistics of the world around us are increasingly complex but this study demonstrates that young students are able to engage with, and make sense of, some of this complexity that is part of their world. English calls for all students from the early years onwards to have such experiences. With respect to technology use in modelling activities, Geiger et al. in their article confirm that “student-student-technology related activity takes place during all phases of the mathematical modelling cycle” rather than as they had theorised previously (see Galbraith, Renshaw, Goos, & Geiger, 2003) “only at the solve juncture”. This is in keeping with the position taken by other researchers (e.g., Galbraith, Stillman, Brown & Edwards, 2007). Geiger et al. also provide classroom examples where one teacher deliberately uses CAS-enabled technology to provoke learning with the technology being given the role of agent provocateur whilst the other teacher incidentally gave the technology this role by grasping what was perceived as a teachable moment. These examples are seen by the authors as a case of “CAS mediating productive social interaction within the context of mathematical modelling activity”. In a study drawing data from five national contexts, Houston et al. explore conceptions held by undergraduate students concerning mathematics. They identified common responses across all research settings that formed a hierarchical ordering of the student conceptions. Abstract and modelling conceptions represented two respective viewpoints at higher levels of the hierarchy, which provided stepping-stones to a view of mathematics as a way of life, for those reaching this higher plane. The authors consider implications for teaching that aim to facilitate such transformations of student appreciation of the purpose and power of mathematics. The necessity of graduates developing such appreciations is highlighted in the rationale for Eindhoven University of Technology introducing a modelling track into its undergraduate education as described by Perrenet and Adan in their article. This is imperative for graduates being able to tackle the multi-disciplinary problems they will be confronted with and have to mathematise in their future employment. Their case study shows that a mathematical modelling sub-program, even when embedded in a more abstract curriculum, should be taught and learnt in realistic contexts. Their careful profiling and evaluation of the competencies developed indicate that such a program has academic integrity. Yoon et al. use a model eliciting activity to investigate whether these “are more productive instructional activities than application problems”. Model eliciting activities are classroom activities “designed to mimic the kinds of real world problems encountered in … mathematics-heavy fields”. Previously it was claimed that the timing of implementation impacted their

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effectiveness in “encouraging students to develop their own [conceptual] understandings through mathematising” with implementation preinstruction in the mathematical topics being best to fulfil this role; whilst post-instruction they served only as an opportunity to apply knowledge. As this small study demonstrates, in this instance the use of a tramping model eliciting activity after the participants had completed instruction in integration did not ensure they could approach this task as an application task. Instead, all four participants solved the problem by mathematising the context. In terms of being able to use prior mathematical knowledge, this study showed that knowledge gained in one context (i.e., integration in the context of speed) is not necessarily transferable to another and participants’ lack of representational versatility limited their use of prior mathematical skills. Clearly fluency in using mathematical tools was problematic for the participants in this study; however, the authors point out “the modelling activity … gave them an opportunity to build up their conceptual understandings through the iterative process of mathematising”. At this point in time it is relevant to consider mathematical modelling in relation to the proposed National Mathematics Curriculum for Australia. The document rationale (Australian, Curriculum, Assessment and Reporting Authority, 2010, p. 1) includes “developing … problem solving skills to enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently” and “all students … be able to apply their mathematical understanding creatively and efficiently”. Among the aims of this curriculum is to “ensure that students are confident, creative users and communicators of mathematics, able to investigate, represent, and interpret situations in their personal and work lives, and as active citizens” and “recognise connections between the areas of mathematics and other disciplines” (p. 1). Additionally, recommendations for different levels of schooling include: “children … pose basic mathematical questions about their world, identify simple strategies to investigate solutions, and strengthen their reasoning to solve personally meaningful problems” (p. 4); “students studying coherent, meaningful and purposeful mathematics that is relevant to their lives” (p. 4); and “students need an understanding of the connections between the mathematics concepts and their application in their world in contexts that are directly related to topics of relevance and interest to them” (p. 5). All these purposes, and others listed under content detail and implications for teaching and learning, can be addressed directly through aptly targeted initiatives in mathematical modelling and applications – indeed some uniquely so.

References Australian Curriculum, Assessment and Reporting Authority. (2010). Mathematics: Draft consultation version 1.1.0 Australian Curriculum. Available from www.australiancurriculum.edu.au/Documents/Mathematics curriculum.pdf Biehler, R., & Leiss, D. (Eds.). (2010). Empirical research on mathematical modelling [Special Issue]. Journal für Mathematik-Didaktik (Journal for Didactics of Mathematics), 31(1). Blum, W., Galbraith, P., Henn, H-W., & Niss, M. (Eds.). (2007). Modelling and applications in mathematics education - The 14th ICMI study. New York: Springer.

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Galbraith, P., Renshaw, P., Goos, M., & Geiger, V. (2003). Technology-enriched classrooms: Some implications for teaching applications and modelling. In Q. Ye, W. Blum, S. K. Houston, & Q. Jiang (Eds.), Mathematical modelling in education and culture (pp. 111-125). Chichester, UK: Horwood. Galbraith, P., Stillman, G., Brown, J., & Edwards, I. (2007). Facilitating middle secondary modelling competencies. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics (pp. 130140). Chichester, UK: Horwood. Hanner, S., James. E., & Rohlfing, M. (2002). Classification models across grades. In R. Lehrer, & L. Schauble (Eds.), Investigating real data in the classroom (pp. 99-117). New York: Teachers College Press. Henn, H-W., & Blum, W. (Eds.). (2004). ICMI Study 14: Applications and modelling in mathematics education. Dortmund, Germany: University of Dortmund. Kaiser, G., BlomhØj, M., & Sriraman, B. (Eds.). (2006). Mathematical modelling and applications: Empirical and theoretical perspectives [Special Issue]. Zentralblatt für Didaktik der Mathematik, 38(2). Kim, S. H., & Kim, S. (2010). The effects of mathematical modeling on creative production ability and self-directed learning attitude. Asia Pacific Educational Review, 11(2), 109-120. Lesh, R., Galbraith, P. L., Haines, C. R., & Hurford, A. (Eds.). (2010). Modeling students’ mathematical modeling competencies: ICTMA 13. New York: Springer. Ministry of Education. (2006). Mathematics syllabus: Secondary. Singapore: Author. Queensland Studies Authority. (2010). Senior syllabus: Mathematics C 2008. Brisbane: Author. Stillman, G., Brown, J., & Galbraith, P. (2008). Research into the teaching and learning of applications and modelling in Australasia. In H. Forgasz, A. Barkatsas, A. Bishop, B. Clarke, S. Keast, W. T. Seah, & P. Sullivan (Eds.), Research in mathematics education in Australasia 2004-2007 (p. 141-164). Rotterdam, The Netherlands: Sense Publishers. Tanner, H., Jones, S., Beauchamp, G., & Kennewell, S. (2010). Interactive whiteboards and all that jazz: Analysing classroom activity with interactive technologies. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education (Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia, Freemantle, Vol. 2, pp. 547-554). Adelaide: MERGA. Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics, 17(4), 401-412.

Authors Gloria Stillman, School of Education (Victoria), Aquinas Campus, Australian Catholic University, PO Box 650, Ballarat, VIC 3350. Email: Jill Brown, School of Education (Victoria), St Patrick’s Campus, Australian Catholic University, Locked Bag 4115, Fitzroy, VIC 3065. Email: Peter Galbraith, School of Education, The University of Queensland, St Lucia, QLD 4072. Email:

Mathematics Education Research Journal

2010, Vol. 22, No. 2, 7

Voices from the Field Research into learning and teaching using mathematical applications and modelling has the potential to inform classroom practice and have an impact on what is considered feasible in classrooms (primary, secondary and tertiary). In this section several practitioners from different sectors in the field from the Australian Capital Territory comment on the implications of particular papers in this issue for classroom practice and policy with respect to curriculum. The comments have been collected by Clare Byrne from the Department of Education and Training, Canberra. We thank the contributors for their comments which follow. The work explored in research by English in a primary school classroom fits well with the Quality Teaching initiative introduced in all ACT public schools. In particular, it provides scope to determine students’ conceptual understanding. The research by Chinnappan examined how a teacher in a secondary classroom could use a real-life problem-solving context to engage high school students and deepen understanding of algebra concepts during the course of a lesson. This research shows that teachers need to develop their skills and capacity to do this if mathematics is going to be seen to be relevant to students in their lives. The study by Lingefjärd and Meier highlighted the complexity of the teachers’ role in presenting modelling problems to groups of students. One major implication of this work is that to improve the use of mathematical modelling in classrooms, teachers need specific training in management of students’ engagement with the problems. The opportunity to use CAS technology to fully investigate mathematical problem solving techniques (again with immediate feedback) as articulated by Geiger, Goos and Faragher will have an impact on a student's deep understanding of the mathematics used in applications that incorporate modelling. Furthermore, use of software like this, that is very similar (or sometimes identical) to modelling software used by mathematicians, engineers and others involved in mathematical problem solving, allows students to connect their experiences with those in the real world. Overall there are opportunities for teachers to incorporate quality pedagogy by using this sort of technology (mindfully). At the top end of the classroom spectrum the paper by Klymchuk, Zverkova, Gruenwald and Sauerbier concerning university students’ perceptions of their difficulties in solving application problems in calculus highlights a longstanding difficulty students have in translating discipline knowledge to practical problems. The researchers recommend teaching basic skills in solving application problems from the beginning of the course and that teachers encourage students to write down all steps in solving a problem. In another study conducted at the tertiary level with a focus on calculus, the value of modelling and real life problem solving in deepening students’ understanding of major mathematical concepts is underscored by the research by Yoon, Dreyfus and Thomas. The study particularly highlighted the fact that there is benefit to students’ learning in exposing them to such problems at any stage of teaching a unit of content – at the beginning or at the end.