10. Teacher Education Research
Paper
Mathematical Modelling in Upper Primary School: Finding Relevance and Value for Others Outside School
Frode Olav Haara
Western Norway University of Applied Sciences, Norway
Presenting Author: Haara, Frode Olav
For some time, mathematics education research has focused on relating mathematical literacy to students’ everyday lives (e.g., Bolstad, 2020; Freudenthal, 1973). Haara (2018) suggests that one way of combining the development of mathematical literacy and relevance for students is through pedagogical entrepreneurship. Pedagogical entrepreneurship is action-oriented teaching and learning in a social context, where the student is active in the learning process and where personal features, abilities, knowledge, and skills provide the foundation and direction for the learning processes. Such an approach entails the use of teaching methods that give students authority and activate learning awareness. It requires working methods that improve students’ creative abilities and beliefs about their own skills, provide a basis for seeing opportunities around them, and motivate them to become development stakeholders in the community (Haara & Jenssen, 2019). A focus on entrepreneurial learning, thus, requires priorities regarding both processes and products in school subjects, which in turn means that a learning environment that emphasizes authenticity and student activity is considered fundamental.
Haara et al. (2017) noted that researchers have concluded that specific attempts to work directly with mathematical literacy through mathematics alone does not work, and that it seems that teaching for mathematical literacy calls for something else than traditional mathematics teaching. Haara (2018) showed that problem-solving features, problem relevance, and student activity are recognized as valuable for the development of mathematical literacy, and that these could be emphasized through a pedagogical entrepreneurship approach. Smith and Stein (2018) provide further support for the influence of creativity and tolerance for ambiguity in school mathematics, through emphasis on problem solving and teaching based on problem-solving approaches.
However, despite thorough work within the mathematics education research community to put emphasis on mathematical modelling and unravel how this may be done with young students (e.g. Gravemeijer, 1999), this remains an issue in need of attention. According to Erbas et al. (2014), a mathematical model “is used to understand and interpret complex systems in nature” (p. 1622). When applying a modelling process in the teaching of mathematics, the underlying assumption is that students can learn fundamental mathematical concepts meaningfully during the modelling process in which they need the concepts while addressing a real-life problem-solving situation (Lesh & Doerr, 2003). Based on previous research it seems clear that emphasis on both mathematical literacy and mathematical modelling is better with a touch of relevance and real-world problem relation than with traditional word problems or quasi-real problems (Vos, 2018).
The purpose of this study is to respond to the calls regarding mathematical modelling (Blum (2002) and Erbas et al. (2014)) and mathematical literacy (Sfard (2014) and Bolstad (2020)), and to present and discuss how pedagogical entrepreneurship and mathematical modelling may be combined to pave the way for the further development of mathematical literacy in upper primary school. The area of statistics is used in this study as an example to illustrate the possibilities for such a combined effort, and the research question asked is therefore: How can pedagogical entrepreneurship and mathematical modelling combine to pave the way for learning statistics in upper primary school? Addressing this question provides the opportunity to discuss the possibilities for students’ development of mathematical literacy, with emphasis on pedagogical entrepreneurship and mathematical modelling.
Methodology, Methods, Research Instruments or Sources UsedThe fact that I was the lecturer involved in this study, places it within an action research perspective, influenced by self-study methodology (Cochran-Smith & Lytle, 2009). This method asks me as a lecturer to reflect on my practice for the purpose of improving it and the practice of others. The study involved teacher education students who carried out an assignment meant for upper primary school students, and then reflected on their own practice as teachers with an aim of improving their own and others’ practice. Together, we tried to understand the roles both as a student and as a teacher from the inside and out, rather than from the outside and in. Hence, to answer the research question in a trustworthy manner and offer mathematics teaching arguments for a “reframed thinking and transformed practice of the teacher” (LaBoskey, 2004, p. 844), I made it a priority to be as close as possible to the actual activity, and thereby sacrificed some observational distance on the altar of relevance. This means that I chose the assignment to use in the teaching, tutored student groups, organized the presentations, and was responsible for the analysis of data.
The self-study perspective offered data from two sources: teacher education students’ reports from completing the assignment, and the lecturer’s observations. The study provides close contact with actual teaching and learning experiences, as well as research perspectives on these experiences through the discussion of possibilities and necessities regarding a relationship between pedagogical entrepreneurship in mathematics, mathematical modelling, and mathematical literacy. The theme and design of this study required that the lecturer/researcher be part of the collected data and, thereby, in an unconscious manner, choose the experiences and impressions that would be subject to analysis. This may seem to be a rather unpredictable way to work in classroom research, but this is not a study about mathematics teaching and learning seen from the outside. It is an article about mathematics teaching opportunities experienced from the inside.
Blum’s (2015) four reasons for emphasis on mathematical modelling and three key factors for a pedagogical entrepreneurship approach (Haara & Jenssen, 2019) produced the analytical framework. However, I regard the phenomenological condensation of impressions produced through the work done by the teacher education students, and my observations related to their work, to be inspired by the constant comparative analysis method (Glaser, 1965).
Conclusions, Expected Outcomes or FindingsIn this study I focus on how a pedagogical entrepreneurship approach combined with fundamental elements of mathematical modelling may be used to strengthen students’ development of mathematical literacy in upper primary school. This has called for a review of the relationship between pedagogical entrepreneurship in mathematics, mathematical modelling, and mathematical literacy. It involved identification of elements from pedagogical entrepreneurship and their relation to mathematical modelling, and presentation of a best practice example in which the pedagogical entrepreneurship approach and mathematical modelling were used. The conclusion is that through emphasis on mathematical modelling and a scientific approach based on pedagogical entrepreneurship, we may have expectations towards increase of upper primary school students’ development of mathematical literacy. Problem solving and scientific rigor are key in both mathematical modelling and pedagogical entrepreneurship, and the idea behind both is to interpret one’s results and apply them in real-world practice. Therefore, key elements in pedagogical entrepreneurship like authenticity, relevance, and value for others enrich the mathematical modelling process, and provide valuable stepping-stones for the upper primary school students’ development of mathematical literacy. The reported study shows that it is possible to plan for learning of scientific approaches, data collection, mathematical modelling, and value for others, while learning statistics, in upper primary school. This planning needs to be based on the acknowledgement of compulsory school students as a resource when they are in school. They do not have to wait until they have finished school but can help move society forward while they learn mathematics and how to work scientifically. Development of mathematical literacy occurs through emphasis on relevance, which is identified as the application of mathematical modelling and real-life viability checks of mathematical work, and through providing value for others, identified as the application of pedagogical entrepreneurship in mathematics for local sustainability and development.
ReferencesBlum, W. (2002). ICME Study 14: Applications and modelling in mathematics education – Discussion document. Educational Studies in Mathematics, 51(1–2), 149–171.
Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In The Proceedings of the 12th International Congress on Mathematical Education (pp. 73–96). Springer.
Bolstad, O. H. (2020). Teaching and Learning for Mathematical Literacy (Ph.D. thesis). University of Agder.
Cochran-Smith, M., & Lytle, S. L. (2009). Inquiry as stance: Practitioner research for the next generation. Teachers College Press.
Erbas, A. K., Kertil, M., Çetinkaya, B., Çakiroglu, E., Alacaci, C., & Bas, S. (2014). Mathematical modeling in mathematics education: Basic concepts and approaches, Educational Sciences: Theory and Practice, 14(4), 1621–1627.
Freudenthal, H. (1973). Mathematics as an educational task. D. Reidel.
Glaser, B. G. (1965). The constant comparative method of qualitative analysis. Social Problems, 12(4), 436–445.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal
mathematics. Mathematical thinking and learning, 1(2), 155–177.
Haara, F. O. (2018). Pedagogical entrepreneurship in school mathematics: An approach for students’
development of mathematical literacy. International Journal for Mathematics Teaching and Learning, 19(2), 253-268.
Haara, F. O., & Jenssen, E. S. (2019). The influence of pedagogical entrepreneurship in teacher
education. In J. Lampert (Ed.). The Oxford encyclopedia of global perspectives on teacher education. Oxford University Press.
Haara, F. O., Bolstad, O. H., & Jenssen, E. S. (2017). Research on mathematical literacy in schools –
Aim, approach and attention. European Journal of Science and Mathematics Education, 5(3), 285-313.
LaBoskey, V. K. (2004). The methodology of self-study and its theoretical underpinnings. In J. J. Loughran, M. L. Hamilton, V. K. LaBoskey & T. Russell (Eds.), International handbook of self-study of teacher education practices (pp. 817–869). Kluwer.
Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Lawrence Erlbaum.
Sfard, A. (2014). Why mathematics? What mathematics? In M. Pirici (Ed.), The best writing of mathematics 2013 (pp. 130–142). Princeton.
Smith, M. S., & Stein, M. K. (2018). 5 practices for orchestrating productive mathematics discussions (2nd ed.). Corwin.
Vos, P. (2018). “How real people really need mathematics in the real world” – Authenticity in mathematics education. Education Sciences, 8(4), 1–14.
10. Teacher Education Research
Paper
Explorative Participation of Prospective Mathematics Teachers within the context of Equation and Inequality
Elcin Emre-Akdogan1, Fatma Nur Gurbuz2
1TED University, Turkiye; 2Middle East Technical University, Turkiye
Presenting Author: Emre-Akdogan, Elcin
The concept of equations and inequalities have a significant place in mathematics (Bazzini & Tsamir, 2002; 2004). They are intertwined in various mathematical subjects such as algebra, analysis, and linear programming (Bazzini & Tsamir, 2004; Tsamir & Almog, 2000), and many concepts of geometry are based on inequalities (Kaplan & Acil, 2015), in addition, equations, and inequalities provide a complementary perspective to each other (Tsamir & Almog, 2000). Equations and inequalities are involved in problem-solving techniques (Altun et al., 2007), abstraction, and mathematical modeling that simulates real-life situations (Karataş & Güven, 2010). It is recommended that students learn to depict cases involving equations and inequalities and how to solve equivalent expressions, equations, and inequalities by inferring their meaning (NCTM, 2000). However, it is claimed that students’ conceptual representations of the algebraic expression and equation lack completeness and accurateness, students at all levels frequently struggle with the concept of inequality, find it extremely difficult to interpret inequality solutions, and students who make more mistakes in these concepts exhibit low mathematical understanding (Stewart, 2016). Students encounter a variety of challenges when they attempt to solve equations and inequalities, such as an inadequate understanding of the role of the equals and inequality signs (Almog & Ilany, 2012; Blanco & Garrote, 2007; Knuth et al. 2006), a lack of understanding of the symbolic representation of variables and coefficients in an equation (Kilpatrick & Izsak, 2008), and changing the direction of the inequality when multiplying by the negative number (Cortes & Pfaff, 2000). According to Bazzini and Tsamir (2001), students who discovered inequalities with traditional methods (by doing algorithmically memorized steps) had difficulty when presented with non-traditional tasks. Moreover, Tsamir and Bazzini (2004) found that students think that inequalities should result in inequalities/intervals, and solving the inequalities and equations requires the same process. It is suggested that to address these challenges, solution processes of equation and inequality should not be introduced directly and quickly, that the symbols used be clearly differentiated and have meaning for the students, that the differences between the concepts of equation and inequality be made clear, and that every-day, visual-geometric and algebraic language should be used interchangeably (Blanco & Garrote, 2007; Bazzini & Tsamir, 2002).
The existing research emphasizes the difficulties students have with equation and inequality concepts. Most of the studies are based on cognitive perspectives, and some of them explore classroom interaction through sociocultural perspectives. In this study, we focus on the equation and inequalities in the classroom discourse to analyze the understanding of learners and teachers. Our study uses a commognitive perspective because it highlights the interaction in a natural classroom setting and enables us to analyze the exploration of learners and teachers. we adopted a methodological lens that Nachlieli & Tabach (2019) provide for ritual-enabling and exploration-requiring opportunities to learn. We interpreted explorative-requiring opportunities to learn as explorative participation and analyzed data on how (procedure) and when (initiation and closure) explorative participation was actualized. We aim to explore the characterization of explorative participation of prospective mathematics teachers and lecturers on equation and inequality concepts in the context of classroom discourse. We address the following question: How do the characterization of explorative participation of prospective mathematics teachers and lecturers on equation and inequality concepts in the context of classroom discourse?
Methodology, Methods, Research Instruments or Sources UsedThe data for this study was conducted in the context of a “basic mathematical concepts” course in a mathematics education department in Turkey. We collected data from classroom observations conducted in the context of a “basic mathematical concepts” course for 20 seniors studying at a mathematics education department. The lecturer is a professor with a Ph.D. degree in mathematics and works in the mathematics education department. Prospective mathematics teachers (PMT) take mathematics education content courses (such as calculus, discrete mathematics, and linear algebra), mathematics education courses (such as geometry education, algebra education, material design, and technology in mathematics education), and pedagogical courses (such as developmental psychology, classroom management, approaches and theories of teaching and learning).
In the context of a “basic mathematical concepts” course, PMTs analyze and discuss basic mathematical concepts (such as propositions, equations, inequalities, polygons, vectors, functions, and transformation). In this study, we focused on the equations and inequalities concepts. In this course, PMTs work in groups of four. PMTs investigate the origins, meaning, and history of specific mathematical concepts, then analyze and categorize the definitions of the particular concept in the literature. After PMTs examine the equation and inequalities concepts, each group presents one clear mathematical concept in the classroom and comprehensively discusses the definitions. Each group justifies and supports their ideas regarding the definitions of equations and inequalities. Finally, the presenting group provided a final and concise definition of the equations and inequalities they had discussed.
Classroom observations collected through a video camera were transcribed into participants’ native language and translated from Turkish into English. The transcripts of the classroom observations included participants’ utterances and their visual mediators and actions. The data were analyzed regarding participants’ and lecturers’ characterization of explorative participation (Sfard, 2008). We adopted a methodological lens that Nachlieli & Tabach (2019) provide for ritual-enabling and exploration-requiring opportunities to learn. We interpreted explorative-requiring opportunities to learn as explorative participation and analyzed data on how (procedure) and when (initiation and closure) explorative participation was actualized.
Conclusions, Expected Outcomes or FindingsIn this study, we have explored the characterization of explorative participation of prospective mathematics teachers and lecturers on equations and inequalities in the classroom discourse. The main discussion in this classroom is driven by the definitions of the available equations and inequalities found in the literature. The main goal of this study is to analyze the definitions of equations and inequalities that enable PMTs to comprehend the flexibility and logical structure of symbolic representations of algebraic expressions and equations, inequalities signs, and their mathematical meaning. PMTs provide three themes on definitions of inequalities and five themes on definitions of equations. By exploring their definition decisions, the lecturer has orchestrated the classroom discourse. When PMTs were explaining their ideas, the lecturer asked exploratory questions. The lecturer prompted exploratory questions for classroom discussion to obtain exploratory engagement, where the actions aligned with the lecturer's goal and were applied flexibly in a logical structure (Nachlieli & Tabach, 2019). Each group has provided logical justifications for their decision-making process. Prospective mathematics teachers actively participated in the classroom discourse by producing mathematical narratives focused on expected outcomes. We discovered that the lecturer had initiated words such as what, why, find, and frequently explain, allowing PMTs to engage in exploratory practices.
ReferencesAlmog, N., & Ilany, B. S. (2012). Absolute value inequalities: High school students’ solutions and misconceptions. Educational Studies in Mathematics, 81(3), 347-364.
Altun, M., Memnun, D. S., & Yazgan, Y. (2007). Primary school teacher trainees’ skills and opinions on solving non-routine mathematical problems. Elementary Education Online, 6(1), 127-143.
Bazzini, L., & Tsamir, P. (2001). Research-based instruction: Widening students’ perspective when dealing with inequalities. In Proceedings of the 12th ICMI Study (pp. 61-68).
Bazzini, L., & Tsamir, P. (2002). Teaching implications deriving from a comparative study on the instruction of algebraic inequalities. In Proceedings of CIEAEM (Vol. 54, pp. 1-8).
Bazzini, L., & Tsamir, P. (2004). Algebraic Equations and Inequalities: Issues for Research and Teaching. Research Forum. International Group for the Psychology of Mathematics Education.
Blanco, L. J., & Garrote, M. (2007). Difficulties in learning inequalities in students of the first year of pre-university education in Spain. Eurasia Journal of Mathematics, Science & Technology Education, 3(3), 221-229.
Cortes, A., & Pfaff, N. (2000). Solving equations and inequations: Operational invariants and methods constructed by students. In Proceedings of the PME CONFERENCE (pp. 2-193).
Kaplan, A., & Acil, E. (2015). The investigation of the 4 th grade secondary school students’ construction processes in “inequality”. Bayburt Eğitim Fakültesi Dergisi [Journal of Bayburt Faculty of Education], 10(1), 130-153.
Karatas, I., & Guven, B. (2010). Examining high school students’ abilities of solving realistic problems. Erzincan University Journal of Education Faculty, 12(1), 201-217.
Kilpatrick, J., & Izsak, A. (2008). A history of algebra in the school curriculum. In C. E. Greenes (Ed.), Algebra and algebraic thinking in school mathematics (pp. 3-18). Reston, VA: National Council of Teachers of Mathematics.
Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal of Research in Mathematics Education, 37, 297-312.
Nachlieli, T. & Tabach, M. (2019). Ritual-enabling opportunities-to-learn in mathematics classrooms. Educational Studies in Mathematics, 101(2), 253-271.
NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
Sfard, A. (2008). Thinking as Communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.
Stewart, S. (2016). And the rest is just algebra. Springer.
Tsamir, P., & Almog, N. (2001). Students’ strategies and difficulties: the case of algebraic inequalities. International Journal of Mathematical Education in Science and Technology, 32(4), 513–524.
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