Mathematical Modeling In Mathematics Education: Basic .
Educational Sciences: Theory & Practice 14(4) 1621-1627 2014 Educational Consultancy and Research Centerwww.edam.com.tr/estpDOI: 10.12738/estp.2014.4.2039Mathematical Modeling in Mathematics Education:Basic Concepts and Approaches*aAyhan Kürşat ERBAŞMahmut KERTİLMiddle East Technical UniversityMarmara UniversityBülent ÇETİNKAYAcMiddle East Technical UniversityCengiz ALACACIeİstanbul Medeniyet UniversitybdErdinç ÇAKIROĞLUMiddle East Technical UniversityfSinem BAŞİstanbul Aydın UniversityAbstractMathematical modeling and its role in mathematics education have been receiving increasing attention inTurkey, as in many other countries. The growing body of literature on this topic reveals a variety of approachesto mathematical modeling and related concepts, along with differing perspectives on the use of mathematicalmodeling in teaching and learning mathematics in terms of definitions of models and modeling, the theoreticalbackgrounds of modeling, and the nature of questions used in teaching modeling. This study focuses on twoissues. The first section attempts to develop a unified perspective about mathematical modeling. The secondsection analyzes and discusses two approaches to the use of modeling in mathematics education, namelymodeling as a means of teaching mathematics and modeling as an aim of teaching mathematics.KeywordsMathematics Education, Mathematical Model, Mathematical Modeling, Problem Solving.*Work reported here is based on a research project supported by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) under grant number 110K250. Opinions expressed are those of the authors anddo not necessarily represent those of TUBITAK. Ayhan Kursat Erbas is supported by the Turkish Academy ofSciences through the Young Scientist Award Program (A.K.E./TÜBA-GEBİP/2012-11).aAyhan Kürşat ERBAŞ, Ph.D., is currently an associate professor of mathematics education. His researchinterests include teaching and learning of algebra, mathematics teacher education, teacher competencies,technology integration in mathematics education, and problem solving and modeling. Correspondence:Middle East Technical University, Faculty of Education, Department of Secondary Science and MathematicsEducation, 06800 Ankara, Turkey. Email: erbas@metu.edu.trbMahmut KERTİL, Ph.D., is currently a research assistant of mathematics education. Contact: MarmaraUniversity, Atatürk Faculty of Education, Department of Secondary Science and Mathematics Education,34722 İstanbul, Turkey. Email: mkertil@marmara.edu.trcBülent ÇETİNKAYA, Ph.D., is currently an associate professor of mathematics education. Contact: MiddleEast Technical University, Faculty of Education, Department of Secondary Science and MathematicsEducation, 06800 Ankara, Turkey. Email: bcetinka@metu.edu.trdErdinç ÇAKIROĞLU, Ph.D., is currently an associate professor of mathematics education. Contact: MiddleEast Technical University, Faculty of Education, Department of Secondary Science and MathematicsEducation, 06800 Ankara, Turkey. Email: erdinc@metu.edu.treCengiz ALACACI, Ph.D., is currently a professor of mathematics education. Contact: İstanbul MedeniyetUniversity, Faculty of Educational Sciences, 34700 İstanbul, Turkey. Email: cengiz.alacaci@medeniyet.edu.trfSinem BAŞ, Ph.D., is currently an assistant professor of mathematics education. Contact: İstanbul AydınUniversity, Faculty of Education, Department of Elementary Education, 34295 İstanbul, Turkey. Email:sinembas@aydin.edu.tr
EDUCATIONAL SCIENCES: THEORY & PRACTICEIn the last two decades, mathematical modeling hasbeen increasingly viewed as an educational approachto mathematics education from elementary levelsto higher education. In educational settings,mathematical modeling has been considered a wayof improving students’ ability to solve problems inreal life (Gravemeijer & Stephan, 2002; Lesh & Doerr,2003a). In recent years, many studies have beenconducted on modeling at various educational levels(e.g., Delice & Kertil, 2014; Kertil, 2008), and moreemphasis has been given to mathematical modelingin school curricula (Department for Education[DFE], 1997; National Council of Teachers ofMathematics [NCTM], 1989, 2000; Talim ve TerbiyeKurulu Başkanlığı [TTKB], 2011, 2013).The term “modeling” takes a variety of meanings(Kaiser, Blomhoj, & Sriraman, 2006; Niss, Blum,& Galbraith, 2007). It is important for readers whowant to study modeling to be cognizant of thesedifferences. Therefore, the purpose of this study istwofold: (i) Presenting basic concepts and issuesrelated to mathematical modeling in mathematicseducation and (ii) discussing the two main approachesin modeling, namely “modeling for the learningof mathematics” and “learning mathematics formodeling.” The following background informationis crucial for understanding the characterization ofmodeling, its theoretical background, and the natureof modeling problems.Mathematical Modeling and Basic ConceptsModel and Mathematical Model: According toLesh and Doerr (2003a), a model consists of bothconceptual systems in learners’ minds and theexternal notation systems of these systems (e.g.,ideas, representations, rules, and materials). Amodel is used to understand and interpret complexsystems in nature. Lehrer and Schauble (2003)describe a model as an attempt to construct ananalogy between an unfamiliar system and apreviously known or familiar system. Accordingly,people make sense of real-life situations andinterpret them by using models. Lehrer andSchauble (2007) describe this process as modelbased thinking and emphasize its developmentalnature. They also characterize the levels of modelbased thinking as hierarchical.Mathematical models focus on structural featuresand functional principles of objects or situationsin real life (Lehrer & Schauble, 2003, 2007;Lesh & Doerr, 2003a). In Lehrer and Schauble’shierarchy, mathematical models do not include1622all features of real-life situations to be modeled.Also, mathematical models comprise a range ofrepresentations, operations, and relations, ratherthan just one, to help make sense of real-lifesituations (Lehrer & Schauble, 2003).Mathematical Models and Concrete Materials:In elementary education, the terms mathematicalmodel and modeling are usually reserved forconcrete materials (Lesh, Cramer, Doerr, Post, &Zawojewski, 2003). Although the use of concretematerials is useful for helping children developabstract mathematical thinking, according toDienes (1960) (as cited in Lesh et al., 2003), in thisstudy, mathematical modeling is used to refer toa more comprehensive and dynamic process thanjust the use of concrete materials.Mathematical Modeling: Haines and Crouch(2007) characterize mathematical modeling as acyclical process in which real-life problems aretranslated into mathematical language, solvedwithin a symbolic system, and the solutionstested back within the real-life system. Accordingto Verschaffel, Greer, and De Corte (2002),mathematical modeling is a process in which reallife situations and relations in these situations areexpressed by using mathematics. Both perspectivesemphasize going beyond the physical characteristicsof a real-life situation to examine its structuralfeatures through mathematics.Lesh and Doerr (2003a) describe mathematicalmodeling as a process in which existing conceptualsystems and models are used to create and developnew models in new contexts. Accordingly, a modelis a product and modeling is a process of creating aphysical, symbolic, or abstract model of a situation(Sriraman, 2006). Similarly, Gravemeijer and Stephan(2002) state that mathematical modeling is not limitedto expressing real-life situations in mathematicallanguage by using predetermined models. Itinvolves associating phenomena in the situationwith mathematical concepts and representationsby reinterpreting them. To be able to express a reallife situation in mathematical language effectively,students must have higher-level mathematicalabilities beyond just computational and arithmeticalskills, such as spatial reasoning, interpretation, andestimation (Lehrer & Schauble, 2003).The Mathematical Modeling Process: No strictprocedure exists in mathematical modeling forreaching a solution by using the given information(Blum & Niss, 1991; Crouch & Haines, 2004; Lesh &Doerr; 2003a). Researchers agree that modeling is acyclical process that includes multiple cycles (Haines
ERBAŞ, KERTİL, ÇETİNKAYA, ÇAKIROĞLU, ALACACI, BAŞ / Mathematical Modeling in Mathematics Education: Basic.& Crouch, 2007; Lehrer & Schauble, 2003; Zbiek &Conner, 2006). In the literature, a variety of visualreferences describe the stages of the cyclic natureof the modeling process (Borromeo Ferri, 2006;Hıdıroğlu & Bukova Güzel, 2013; Lingefjard, 2002b,NCTM, 1989). For instance, the modeling processdescribed in the earlier Standards document byNCTM (1989, p. 138) emphasizes that mathematicalmodeling is a non-linear process that includes fiveinterrelated steps: (i) Identify and simplify the realworld problem situation, (ii) build a mathematicalmodel, (iii) transform and solve the model, (iv)interpret the model, and (v) validate and use themodel. Such types of diagrams can help readers andteachers understand the probable stages that studentsmay experience during the modeling processes.Mathematical Modeling and Problem Solving:Mathematical modeling is often confused withtraditional word problems. From the view of Reusserand Stebler (1997), traditional word problems causestudents to develop some didactic assumptionsabout problem solving. Moreover, the real-lifecontexts in these problems are often not sufficientlyrealistic and thus fail to support students’ abilitiesto use mathematics in the real world (English,2003; Lesh & Doerr, 2003; Niss et al., 2007). Whileworking on such problems, students often simplyfocus on figuring out the required operations (e.g.,Greer, 1997; Nunes, Schliemann & Carraher, 1993).Some studies focus on reorganizing word problemsto enable students to gain competence in thinkingabout real-life contexts while solving them (Greer1997; Verschaffel & De Corte, 1997; Verschaffel, DeCorte, & Borghart, 1997; Verschaffel et al., 2002).Such versions of word problems can be used aswarm-up exercises in preparation for modeling(Verschaffel & De Corte, 1997).While Lingefjard (2002b) argues that it isunreasonable to compare problem solving andmodeling, the similarities and differences betweenthem can be useful (Lesh & Doerr, 2003a; Lesh& Zawojewski, 2007; Mousoulides, Sriraman,& Christou, 2007; Zawojewski & Lesh, 2003).The following table briefly describes a few of theimportant differences between the two concepts.Mathematical Modeling ApproachesDifferent approaches have been proposed withdifferent theoretical perspectives for usingmodeling in mathematics education, and nosingle view is agreed upon among educators(Kaiser, Blum, Borromeo Ferri, & Stillman, 2011;Kaiser & Sriraman, 2006). To clarify the differentperspectives on this issue and reach a consensus,these similarities and differences should beelaborated (Kaiser, 2006; Kaiser & Sriraman, 2006;Sriraman, Kaiser, & Blomhoj, 2006). Kaiser’s (2006)and Kaiser and Sriraman’s (2006) classificationsystems for presenting modeling approaches canbe considered the leading perspective. Accordingto this scheme, the perspectives are classified as(i) realistic or applied modeling, (ii) contextualmodeling, (iii) educational modeling, (iv) sociocritical modeling, (v) epistemological or theoreticalmodeling, and (vi) cognitive modeling. Generally,modeling is also classified by its purpose inmathematics education, such as (i) modeling as thepurpose of teaching mathematics or (ii) modelingas a means to teach mathematics (Galbraith, 2012;Gravemeijer, 2002; Julie & Mudaly, 2007; Niss et al.,2007).Table 1A Comparison between Problem Solving and Mathematical Modeling (Adapted from Lesh & Doerr [2003a] and Lesh & Zawojewski [2007])Traditional Problem SolvingMathematical ModelingProcess of reaching a conclusion using dataMultiple cycles, different interpretationsContext of the problem is an idealized real-life situation or arealistic life situationAuthentic real-life contextStudents are expected to use taught structures such asformulas, algorithms, strategies, and mathematical ideasStudents experience the stages of developing, reviewing, andrevising important mathematical ideas and structures duringthe modeling processIndividual work emphasizedGroup work emphasized (social interaction, exchange ofmathematical ideas, etc.)Abstracted from real lifeInterdisciplinary in natureStudents are expected to make sense of mathematical symbolsand structuresIn modeling processes, students try to make mathematicaldescriptions of meaningful real-life situationsTeaching of specific problem-solving strategies (e.g.,developing a unique approach, transferring onto a figure)transferable to similar problemsOpen-ended and numerous solution strategies, developedconsciously by students according to the specifications of theproblem.A single correct answerMore than one solution approach and solution (model)possible1623
EDUCATIONAL SCIENCES: THEORY & PRACTICEModeling as the Purpose of Teaching MathematicsIn this perspective, mathematical modeling is seenas a basic competency, and the aim of teachingmathematics is to equip students with this competencyto solve real-life problems in mathematics and inother disciplines (Blomhøj & Jensen, 2007; Blum,2002; Crouch & Haines, 2004; Haines & Crouch,2001; Izard, Haines, Crouch, Houston, & Neill, 2003;Lingefjard, 2002a; Lingefjard & Holmquist, 2005).In this approach, initially, mathematical conceptsand mathematical models are provided and laterthese ready-made concepts or models are applied toreal-world situations (i.e., mathematics " reality)(Lingefjard, 2002a, 2002b, 2006; Niss et al., 2007).Mathematical models and concepts are consideredas already existing objects (Gravemeijer, 2002).Researchers adopting this perspective focus on theissue of conceptualizing, developing, and measuringthe modeling competencies (e.g., Haines & Crouch,2001, 2007). In the literature, different viewpointsexist on this issue (Henning & Keune, 2007). WhileBlomhøj and Jensen (2007) adopt a holistic approach,other studies address this issue at the micro level(Crouch & Haines, 2004; Haines, Crouch, & Davis,2000; Lingefjard, 2004). Furthermore, some studiesfocus on teaching mathematical modeling (Ärlebäck& Bergsten, 2010; Lingefjard, 2002a). Fermi problems,for example, are regarded as appropriate kinds ofproblems for teaching of modeling (Ärlebäck, 2009;Ärlebäck & Bergsten, 2010). Sriraman and Lesh(2006) contend that Fermi problems can be used aswarm-up and starting exercises in preparation formodeling.Modeling as a Means for Teaching MathematicsIn this approach, modeling is considered a vehiclefor supporting students’ endeavors to create anddevelop their primitive mathematical knowledgeand models. The Models and Modeling Perspective(Lesh & Doerr, 2003a) and Realistic MathematicsEducation (Gravemeijer, 2002; Gravemeijer &Stephan, 2002) are two examples of this approach.Models and Modeling Perspective (MMP)The models and modeling perspective is a newand comprehensive theoretical approach tocharacterizing mathematical problem-solving,learning, and teaching (Lesh & Doerr, 2003a; 2003b)that takes constructivist and socio-cultural theoriesas its theoretical foundation. In this perspective,individuals organize, interpret, and make sense1624of events, experiences, or problems by using theirmental models (internal conceptual systems). Theyactively create their own models, consistent withthe basic ideas of constructivism (Lesh & Lehrer,2003). Moreover, for productive use of models foraddressing complex problem-solving situations,they should be externalized with representationalmedia (e.g., symbols, figures).Model-eliciting activities (MEAs) are speciallydesigned for use within the MMP. In MEAs, studentsare challenged to intuitively realize mathematicalideas embedded in a real-world problem and tocreate relevant models in a relatively short periodof time (Carlson, Larsen, & Lesh, 2003; Doerr &Lesh, 2011). Lesh, Hoover, Hole, Kelly, and Post(2000) offered six principles to guide the design ofMEAs: (i) the model construction principle, (ii) thereality principle, (iii) the self-assessment principle,(iv) the construct-documentation principle, (v) theconstruct shareability and reusability principle,and (vi) the effective prototype principle. In theimplementation of MEAs, students work in teamsof three to four. They are expected to work oncreating shareable and reusable models, whichencourage interaction among students. Therefore,the social aspect of learning is another componentof the MMP (Zawojewski, Lesh, & English, 2003).According to Lesh et al. (2003), MEAs should notbe used as isolated problem- solving activities.They should be used within model developmentsequences, where warm-up and follow up activitiesare also important.The Modeling Approach in Realistic MathematicsEducationSimilar to the MMP, the modeling approachassumed by RME is based on constructivistand socio-cultural theories (Freudental, 1991;Gravemeijer, 2002). In this approach, modeling goesbeyond translating real-life problem situations intomathematics. It involves revealing new relationsamong phenomena embedded in the situations byorganizing them (Gravemeijer & Stephan, 2002).In modeling, students initially work on real-lifesituations and create their primitive models, whichare called model of. The term “model” describes notonly the physical or mathematical representationsof the phenomena, but also the componentsof students’ conceptual systems, such as theirpurpose and ways of thinking about the situation(Cobb, 2002). With the help of carefully designedreal-life problems and learning environmentsthat encourage students to discover sophisticated
ERBAŞ, KERTİL, ÇETİNKAYA, ÇAKIROĞLU, ALACACI, BAŞ / Mathematical Modeling in Mathematics Education: Basic.mathematical models, students proceed to createmore abstract and formal models, which arecalled model for (Doorman & Gravemeijer, 2009).Accordingly, modeling is characterized as a processof moving from “model of ” to “model for,” whichis called as emergent modeling (Doorman &Gravemeijer, 2009; Gravemeijer & Doorman, 1999).Besides describing students’ learning process, thisperspective also assumes principles about howa learning environment should be designed tosupport students’ emergent modeling processes.Discussion and ConclusionIn recent years, using modeling in mathematicseducation has been increasingly emphasized(NCTM, 1989, 2000; TTKB, 2011, 2013). A varietyof different perspectives have been proposed for theconceptualization and usage of modeling (Kaiser& Sriraman, 2006). These perspectives can begrouped into two main categories: (i) modeling as ameans for teaching mathematics and (ii) modelingas the aim of teaching mathematics (Blum & Niss,1991; Galbraith, 2012). In the first perspective,students are provided with predetermined modelsand are expected to apply these models to real-lifesituations. The ultimate goal is to improve students’modeling competencies (Haines & Crouch, 2001,2007; Izard et al., 2003; Lingefjard, 2002b). In thesecond perspective, the underlying assumption isthat students can learn fundamental mathematicalconcepts meaningfully through a modelingprocess in which they need and intuitively discovermathematical concepts while addressing a real-lifeproblem-solving situation (Lesh & Doerr, 2003a).In summary, the second approach (i.e., modelingas a means for teaching mathematics) seems moredeveloped for pedagogical purposes. However,whatever approach is preferred and used,integrating modeling into mathematics educationis important for improving students’ problemsolving and analytical thinking abilities. However,few studies have been conducted in Turkeyon using modeling in mathematics education.Furthermore, there are insuff
Mathematical modeling and its role in mathematics education have been receiving increasing attention in Turkey, as in many other countries. The growing body of literature on this topic reveals a variety of approaches to mathematical modeling and related concepts, along with differing perspectives on the use of mathematical .
So, I say mathematical modeling is a way of life. Keyword: Mathematical modelling, Mathematical thinking style, Applied 1. Introduction: Applied Mathematical modeling welcomes contributions on research related to the mathematical modeling of e
2.1 Mathematical modeling In mathematical modeling, students elicit a mathematical solution for a problem that is formulated in mathematical terms but is embedded within meaningful, real-world context (Damlamian et al., 2013). Mathematical model
3. Department of Mathematics, TIT group of Institutions Bhopal Abstract In the present paper we have discussed about mathematical modeling in mathematics education. 2. Introduction and Preliminary work already done Mathematics is not only a subject but it is the language with some different symbols and relations. Mathematics
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What is mathematical modeling? – Modeling (Am -English spelling) or modelling (Eng spelling) is a mathematical process in which mathematical problem solvers create a solution to a problem, in an attempt to make sense of mathematical phenomena (e.g., a data set, a graph, a diagram, a c
A good mathematical model is one that helps you better understand the situation under investigation. The process of starting with a situation or problem and gaining understanding about the situation through the use of mathematics is known as mathematical modeling. The mathematical descriptions obtained in the process are called mathematical .
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