24. Mathematics Education Research
Paper
An Inquiry-Based Approach for Teaching Mathematical Modelling to Prospective Primary Teachers
Jesús Montejo-Gámez1, Elvira Fernández-Ahumada2, Natividad Adamuz-Povedano2, Enrique Martínez-Jiménez2
1University of Granada, Spain; 2University of Córdoba, Spain
Presenting Author: Adamuz-Povedano, Natividad
This contribution shows an inquiry-based learning approach to mathematical modelling, and provides a first insight into its effectiveness for prospective primary teachers. There is a consensus among different authors in mathematics education that the mathematical knowledge that should be expected from a primary school teacher goes beyond the knowledge of the content to be taught (Hill et al., 2008; Carrillo et al., 2013). However, research reports the difficulties that prospective teachers have with regard to skills such as reasoning (Kaasila et al., 2010), problem-solving (Verschaffel et al., 2005) or the application of mathematics to real contexts (Sáenz, 2009). Working with mathematical modelling tasks creates opportunities to alleviate these difficulties.
Over the last decades, modelling has become a crucial area in mathematics education (Barquero, 2019). In fact, curricula in different countries have gradually incorporated modelling competencies, and modelling is generating a growing interest in teacher education, specially in prospective Primary Teachers (Guerrero-Ortiz & Borromeo-Ferri, 2022). Likewise, there has been a proliferation of international projects aimed at designing resources that can support the learning of modelling, such as LEMA, MASCIL or MERIA (2016). In this European project whose main objective was to promote the teaching of mathematics applicable to real life. It combined the principles of Realistic Mathematics Education (Van den Heuvel-Panhuizen & Drijvers, 2014), the ideas of Inquiry-Based Learning (Maaß & Doorman, 2013; Dorier & Garcı́a, 2013) and the pedagogical approach of Didactic Situations Theory (Brousseau, 1997). The key idea developed in MERIA is to implement the principles of inquiry-based learning. In this context, teaching should provide students with just the right amount of help to support mathematical learning. These ideas can be harnessed to stimulate the autonomous development of modelling activity by pre-service teachers, thus it give rise to our research question: Does the inquiry-based learning influence the models developed by prospective teachers?
Modules and scenarios for an inquiry-based learning of modelling
For the sake of providing the “right” amount of direction to inquiry, teaching approach developed in in the MERIA project were based on two key tools: Modules and scenarios. A module is the union of a scenario with all the material needed to implement this scenario in the classroom. Likewise, a scenario is a full description of a lesson in terms of the Theory of Didactical Situations (TDS, Brousseau, 1997). Under the TDS approach, students are intended to construct new knowledge when they solve a task while adapting to what is called a didactical milieu. It consists of the task, students’ previous knowledge, and the artifacts to solve the task. The role of educators is to design such milieu and to help student to adapt to it. In this process, two kind of situations appear. The first one is composed of adidactical situations, which are those where the students are engaged in the task and explore the milieu without the teacher’s interference. The second kind of situations are the didactical ones, where students and educators explicitly interact. A balanced combination between didactical and adidactical situations leads to the inquiry process and the students' construction of new knowledge. Therefore, a proper scenario should contain such combination of situations along different phases. (i) Devolution: the educator introduces the task and explains the rules to solve it (didactical situation). (ii) Action: students are engaged with solving the task and actively work on it (adidactical). (iii) Formulation: students explicitly formulate outcomes of the action phase (adidactical). (iv) Validation: students test their hypotheses and strategies against the milieu (adidactical). (v) Institutionalisation: educator declares the institutional knowledge (didactical). In this phase, teacher may put ideas together, compares viewpoints and explains optimal strategies.
Methodology, Methods, Research Instruments or Sources UsedParticipants and instructional design
The sample is composed of 22 students enrolled in a course focused on mathematical competencies for primary their fourth year of the elementary teacher’s degree studies at the University of Granada. These students attended a set of sessions in which different modelling tasks were solved by applying the ideas of inquiry-based learning. In total, six sessions took place, and one task was solved per session. This resulted in a total of 132 written productions, of which 44 (corresponding to two tasks) are analysed for the present study, due to length limitations.
The instructional design was based in the MERIA (2016) scenarios. These were reduced for the sake of simplicity, giving rise to “short scenarios” that are focused on teacher educators' actions. Prospective teachers education is based on three activities around the short scenarios: (i) Solving the tasks by taking advantage and reflecting on the scaffolds used and on the mathematical concepts needed (ii) Reflecting on the modelling skills involved in the tasks from different theoretical frameworks (iii) Developing and establishing assessment criteria for their own short scenario.
Data analysis
The data analysis is based on the characterisation of the models developed by the participants of the study in the written productions collected, and subsequent comparison of these models with those existing in the literature for the same tasks.
In order to characterise the models, the procedure set up by Montejo-Gámez et al. (2021), which is based on the description of three elements: the real system, the mathematics used by the participants and the representations employed. In this way, the analysis begins with the identification of statements involved in the elements of representation, which makes it possible to distinguish the relationships and mathematical results of the model. From these, the objects and variables used are extracted, respectively. Finally, the analysis of the results allows the abstraction of the mathematical properties and concepts involved.
In order to compare the models found with those reported in the literature, the categories obtained by Segura (2022) will be taken and common and novel elements will be identified against these categories. This will allow us to observe the influence of the scenarios implemented on the participants' productions and to draw conclusions on the relevance of these scenarios.
Conclusions, Expected Outcomes or FindingsBased on previous experiences with similar scenarios and previous literature on mathematical modelling in prospective primary school teachers, a set of ideas emerge that shape the expected outcomes of the study. In particular, it is expected that the fact that the sessions are led by educators will condition the written productions collected, a situation that may materialise in different ways. Firstly, a higher response rate to the tasks is expected than in other studies. The complexity of the problems proposed sometimes leads students to blocking, a situation that should be avoided under the didactic proposal used. The educator's action may possibly lead to a lower number of errors, which contributes to alleviating the difficulties experienced by these students when solving contextualised problems. Similarly, participants are expected to propose more accurate models, as the session promotes discussion and comparison of different ideas among peers.
As a negative effect, on the contrary, it is expected that there will be less richness of ideas than reported in the literature, since the students have all followed the same session (and, therefore, flow of ideas). In short, we expect to find indications that support the use of inquiry-based learning, as well as points for improvement of the scenarios, which should lead to simplifications of the design and implementation of the scenarios.
ReferencesBarquero, B. (2019). Una perspectiva internacional sobre la enseñanza y aprendizaje de la modelización matemática. En J. M. Marbán, M. Arce, A. Maroto, J. M. Muñoz-Escolano & A. Alsina (Eds.), Investigación en educación matemática xxiii (pp. 19-22). Universidad de Valladolid.
Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990. Kluwer Academic Publishers.
Carrillo, J., Climent, N., Contreras, L. C. & Muñoz-Catalán, M. C. (2013). Determining specialized knowledge for mathematics teaching. En B. Ubuz, C. Haser & M. A. Mariotti (Eds.), Proceedings of cerme 8, the eighth congress of the european society for research in mathematics education (pp. 2985-2994). Middle East Technical University.
Dorier, J. L. & García, F. J. (2013). Challenges and opportunities for the implementation of inquiry-based learning in day-to-day teaching. ZDM, 6(45), 837-849.
Guerrero-Ortiz, C. & Borromeo-Ferri, R. (2022). Pre-service teachers' challenges in implementing
mathematical modelling: Insights into reality. PNA, 16(4), 309-341. https://doi.org/10.30827/pna.v16i4.21329
Hill, H., Blunk, M., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L. & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: an exploratory study. Cognition and instruction, 4(26), 430-511.
Kaasila, R., Pehkonen, E. & Hellinen, A. (2010). Finnish pre-service teachers’ and upper secondary students’ understanding on division and reasoning strategies used. Educational Studies in Mathematics, 3(73), 247-261.
Maaß, K. & Doorman, L. M. (2013). A model for a widespread implementation of inquiry-based learning. ZDM, 6(45), 887-889.
MERIA (2016). MERIA project: guide, guidelines for teachers and teaching scenarios. https://meria-project.eu/
Montejo-Gámez, J., Fernández-Ahumada, E., Adamuz-Povedano, N. (2021). A Tool for the Analysis and Characterization of School Mathematical Models, Mathematics, 9(13). https://doi.org/10.3390/math9131569
Sáenz, C. (2009). The role of contextual, conceptual and procedural knowledge in activating mathematical competencies (pisa). Educational Studies in Mathematics, 71(2), 123-143.
Segura, C. (2022). Flexibilidad y rendimiento en la resolución de problemas de estimación en contexto real. Un estudio con futuros maestros (Doctor of Philosophy thesis). Valencia, University of Valencia, Spain.
Van den Heuvel-Panhuizen, M. & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521-525). Springer.
Verschaffel, L., Janssens, S. & Janssen, R. (2005). The development of mathematical competence in flemish preservice elementary school teachers. Teaching and Teacher Education, 1(21), 49-63.
24. Mathematics Education Research
Paper
A Study on the Effective Use of Variation in Chinese Mathematics Lessons
Wei Xin
The University of Hong Kong, Hong Kong S.A.R. (China)
Presenting Author: Xin, Wei
Over the past few years, Asian students, especially students from Shanghai, China, always obtain extremely excellent performance in mathematics competitions (e.g., PISA, TIMSS, etc.) in comparison with their peers from other countries. A growing number of scholars hope to discover what can be learned from these high-scoring Asian education systems. In particular, the Department for Education (DfE) of the UK is adopting the Shanghai Mastery Pedagogy to improve the mathematics achievement of their students (Boylan et al., 2019).
Distinguished from the English and other mathematic education practices, Shanghai whole-class interactive teaching aims to develop conceptual understanding and procedural fluency of students. Some big ideas, such as coherence, variation, representation, and structure, are promoted by mastery specialists (NCETM, 2017). Teaching is famous for its mathematically meaningful and coherent activities with well-designed models and examples systematically using variations. Actually, the characteristics mentioned above, especially the effective use of “variation”, are also noticeably emphasized in the exploration of Chinese mathematics teaching (Gu et al., 2004). Teaching with variation has almost become a common teaching routine for many Chinese mathematics teachers (Marton et al., 2004) and has been applied either consciously or intuitively for a long time in China (Li et al., 2011).
The main research question of the study is: How do Chinese mathematics teachers make use of variation to foster student learning in their teaching? While there have been extensive studies on the effective use of variation in mathematics teaching, some gaps still exist in the following aspects. Firstly, most of the studies utilized one of the variation theories as the lens to analyze mathematics teaching in China (e.g., Qi et al., 2017; Mok, 2017; Häggström, 2008). However, insufficient attempts have been made to employ an integrated variation perspective based on several variation theories. Secondly, most of the existing studies adopted the approach of quasi-experimental or lesson study with intervention (e.g., Pang et al., 2017; Al-Murani, 2007; Kullberg, 2010). Nevertheless, very few studies adopted the naturalistic perspective to explore what actually happens in more authentic and diverse situations. Thirdly, due to the limited size of the research, some studies chose one or very few excellent public lessons or experimental lessons, even if a series of lessons were collected (e.g., Mok, 2017; Pang et al., 2016; Pang et al., 2017). The mathematics structures, relationships, and coherence within and between the sub-topics are not the major factor and draw little research attention, but they are actually the key ideas of Chinese pedagogy and the very essential platform for unfolding variation. Lastly, including the movement of the UK, most practices and studies were unfolded in a relatively primary or junior stage, while the senior-level mathematics knowledge and topics were less involved.
The basis of the theoretical framework is the Variation Theory (VT) of Ference Marton. With the help of variation and invariance, students could “discern” the “critical aspects” of an “object of learning” with certain “patterns of variation” (Marton, 2015). The “critical aspect” in VT is considered identical with a dimension of variation (Pang & Ki, 2016). Watson and Mason (2005) further developed this concept with the term “dimension of possible variation”, associated with the notion of “range of permissible change” on the extension of Marton’s originally general notion. This extension captures the qualities of variation arising in mathematics (Mason et al., 2009) and better fits the nature of mathematics. Meanwhile, their concepts of example and example space are also elaborated in a mathematical manner. In addition, the analysis is also inspired by the Chinese theory “Bianshi Jiaoxue” (teaching with variation), which is developed by Chinese mathematics expert Gu Lingyuan (Gu et al., 2004).
Methodology, Methods, Research Instruments or Sources Used To address the gaps mentioned previously, the current research aims to employ an integrated variation perspective based on several theories of variation to analyze the teaching of a mathematics topic over a series of around ten lessons in a naturalistic setting in Mainland China.
Specifically speaking, the topic of function in the senior high school curriculum is chosen as the research target, which contains three consecutive sub-topics, namely power function, exponential function, and logarithmic function. The rich and complex mathematics relationships and connections (similar expressions, inverse relationships, etc.) between them enable the exploration of variation in an intertwined mathematics structure.
The teachers participating in the study were six ordinary mathematics teachers in the local schools of three cities in China. The schools and teachers were chosen under the following criteria -- (1) following the national curriculum guide, (2) possessing high teaching standards, (3) being comparable between teachers (similar education background and teaching experience), (4) being comparable between classes (similar student achievements in mathematics).
During the whole process of all lessons conducted in all classes, the video recording was used to collect the complete data, together with the semi-structured, qualitative classroom observations and field notes. Then, teachers were interviewed with the use of the technique of video-stimulated recall in the semi-structured approach. They were requested to discuss the reason for specific learning activities and their reflections on the incidents that happened during the lessons. Meanwhile, the issues observed by the researcher were further validated in the interviews. Student performances were collected by pre-test and post-test, the school’s mid-term and monthly tests. Furthermore, the survey to collect student-generated examples also provided the researcher with an effective approach to examine the example space of students.
The data analysis was based on the integrated theoretical framework mentioned in the last section. Within each lesson, the analysis was carried out in detail in each teaching activity and example to explore how pedagogical actions enable students thoroughly experience the task and variation. After transcribing the video and audio recordings and calculating the test results, these data from different resources were aligned with and linked to the corresponding teaching activity. In accordance with the analysis, the comparisons were further conducted from various perspectives and layers, including the comparison within and across the teaching of different mathematics sub-topics of the same teacher and that of teachers in the same school and across schools.
Conclusions, Expected Outcomes or Findings This study provides an in-depth and extensive understanding of the effective use of variation in Chinese mathematics practices through the lens of a Chinese researcher. By employing an integrated variation perspective, the current research contributes to the development and refinement of theoretical frameworks and better fits the nature of mathematics learning and teaching. The analysis of variation can thus be unfolded comprehensively. The lessons conducted in a naturalistic setting enable the investigator to explore the authentic and rich teaching designs in the Chinese mainland without being limited by the existing theories. The thick descriptions and detailed interpretations allow readers to generalize and improve their research and teaching practices.
Furthermore, special attention has been paid to the mathematics structures and relationships within and between the sub-topics, allowing a more systematic and intertwined perspective of variation. Based on the preliminary analysis, several teachers thoroughly used the connections between sub-topics to achieve transfer and coherence. From the perspective of variation, the same dimension(s) of variation was opened up in different sub-topics. For example, teachers constructed a similar routine to teach the properties, such as domain, range, monotonicity, parity, etc., of every type of function in a coherent way. Meantime, the different types of functions can be viewed as various values of the dimension of variation of function. Furthermore, the concept of “exponential function” was linked to its easily-confused concept of “power function”. The comparison between them highlighted the critical aspect of the independent variable (varied in each function) and also showed the same requirements of the critical aspects of coefficient and constant. The teaching of logarithmic function based on its inverse relationship with the exponential function enabled students to understand the mathematical essence of associated critical aspects. Therefore, it is meaningful to analyze the use of variation in a more comprehensive manner.
ReferencesBoylan, M., Wolstenholme, C., Maxwell, B., Demack, S., Jay, T., Reaney, S., & Adams, G. (2019). Longitudinal evaluation of the Mathematics Teacher Exchange: China-England-Final Report.
Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: perspectives from insiders (pp. 309–347). Singapore: World Scientific.
Li, J., Peng, A., & Song, N. (2011). Teaching algebraic equations with variation in Chinese classroom. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 529–556). New York, NY: Springer.
Kullberg, A., Watson, A., & Mason, J. (2009). Variation within, and covariation between, representations. In Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 433-440). Thessaloniki: PME.
Marton, F. (2015). Necessary conditions of learning. London: Routledge.
Marton, F., Runesson, U., & Tsui, A. (2004). The space for learning. In F. Marton & A. Tsui (Eds.), Classroom discourse and the space for learning (pp. 3–40). Mahwah, NJ: Lawrence Erlbaum Associates Inc.
Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10-32.
Mok, I. A. C. (2017). Teaching Algebra through Variations: Contrast, Generalization, Fusion, and Separation. In Huang, R., & Li, Y. (Eds.), Teaching and Learning Mathematics through Variation: Confucian heritage meets western theories (pp. 187-205). Rotterdam, The Netherlands: Sense Publishers.
Pang, M. F., & Ki, W. W. (2016). Revisiting the Idea of “Critical Aspects”. Scandinavian Journal of Educational Research, 60(3), 323-336.
Pang, M. F., Marton, F., Bao, J. S., & Ki, W. W. (2016). Teaching to add three-digit numbers in Hong Kong and Shanghai: illustration of differences in the systematic use of variation and invariance. ZDM, 48(4), 455-470.
Qi, C., Wang, R., Mok, I. A. C., & Huang, D. (2017). Teaching the Formula of Perfect Square through Bianshi Teaching. In Huang, R., & Li, Y. (Eds.), Teaching and Learning Mathematics through Variation: Confucian heritage meets western theories (pp. 127-150). Rotterdam, The Netherlands: Sense Publishers.
Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Erlbaum.
24. Mathematics Education Research
Paper
Statistics Teaching Practices: Errors and Imprecision
Vuslat Seker, Erdinc Cakiroglu
Middle East Technical University, Ankara, Turkiye
Presenting Author: Seker, Vuslat
The errors and imprecisions in teaching mathematics are part of mathematics classrooms and might negatively influence student learning (Ball & McDiarmid, 1990; LMT, 2011). The errors and imprecision dimension is one of five in Mathematical Quality of Instruction (MQI). The other domains are the classroom work is connected to mathematics, the richness of mathematics, working with students and mathematics, and common core-aligned student practices. MQI (2014) defined the errors and imprecision dimension as "teacher errors or imprecision in language and notation, or the lack of clarity/precision in the teacher's presentation of the content" (p.19). Most studies in the literature are interested in possible errors, misconceptions, or difficulties in learning statistics (e.g., Batanero et al., 1994; Capraro et al., 2005). To illustrate, certain studies conducted on revealing possible errors or difficulties made by students during interpretations of graphs (e.g., Aydın-Güç et al., 2022 for scatterplots; Capraro et al., 2005 for bar, line, and circle graphs; Edwards et al., 2017 for boxplots). On the other hand, there are also studies showing pre-service teachers’ errors or imprecisions on graphs as well (e.g., Işık et al., 2012 for line graphs; Ulusoy & Çakıroğlu, 2013 for histogram). However, there is a lack of research on what kinds of teacher errors and imprecision are present in teaching statistics. It is essential to explore teacher errors and imprecision to learn from them for not to transfer inaccurate information teachers possess to the students (Ball & McDiarmid, 1990). By exploring teacher errors and imprecisions in teaching practices, it is possible to identify areas of improvement, leading to more effective and engaging instruction and better student outcomes. Ultimately, understanding teacher errors and imprecisions is crucial in promoting high-quality mathematics education (LMT, 2011). In light of this gap in the literature, the purpose of this case study is to explore two 7th-grade mathematics teachers' statistics teaching with regard to errors and imprecision. The central research question guiding this study is: What types of teacher errors and imprecisions are present in 7th-grade mathematics teachers' statistics teaching?
Methodology, Methods, Research Instruments or Sources UsedThis study is a part of a qualitative case study that allows for an in-depth examination of teaching practices within the real-life classroom context (Creswell & Plano Clark, 2007). Two middle school mathematics teachers were selected through purposive sampling with the following criteria: teaching 7th grade, having an undergraduate degree in a middle school mathematics education program, having at most 12 years of experience teaching, and working at the current school for at least two years. I observed teachers' instruction while teaching statistics. Fourteen hours for the Cem teacher and 13 for the Esra teacher were video and audio-recorded in order to explore the quality of instruction, specifically the errors and imprecisions dimension for this proposal. I analyzed all videos with three elements in this dimension. Mathematical Content Errors (MCE), Imprecision in Language or Notation, Lack of Clarity in Presentation of Mathematical Content, and Overall Errors and Imprecision are the codes for the dimension. This dimension only considers the errors not corrected during the segment. I assigned Not Present (1), Low (2), Mid (3), and High scores (4) for the codes to 7.5-minute segments determined by the MQI instrument. In this dimension, Not Present (1) means that the segment is free from errors and imprecisions, and high (4) showed that the segment consists of a significant amount of errors and imprecision.
Conclusions, Expected Outcomes or FindingsThe results showed that the teachers' statistics instructions did not include errors and imprecision in most segments for all dimensions (59% for Cem, and 66.7% for Esra). The instructions included brief errors and imprecision (11.5%for Cem and 8.8% for Esra). To exemplify the Mathematical Content Error code, both defined the mode as the most frequent number instead of the value. They did not focus on the data type in their lessons and mostly worked on examples with quantitative data while teaching average. The definition does not obscure statistics in those examples. However, students made errors in the examples with categorical data; they reported frequency numbers as the mode of the data set. Teacher definitional error might lead to student error. Also, some segments included high content errors due to the inconsistencies between the graph's aims and the context in constructing a graph. Both teachers used ordinal data on the x-axis in a line graph task similar to the study of Işık et al. (2012). All in all, detecting and learning from teacher errors and imprecision might prevent possible misconceptions in student learning (Ball & McDiarmid, 1990). I will provide the results related to other codes with further discussion.
ReferencesAydın-Güç, F., Özmen, Z. M., & Güven, B. (2022). Difficulties scatter plots pose for 11th-grade students. The Journal of Educational Research, 115(5), 298-314.
Ball, D. L., & McDiarmid, G. W. (1990). The subject matter preparation of teachers. In R.Houston (Ed.), Handbook of research on teacher education (pp. 437-449). New York: Macmillan.
Capraro, M. M., Kulm, G., & Capraro, R. M. (2005). Middle grades: Misconceptions in statistical thinking. School Science and Mathematics, 105(4), 165-174.
Creswell, J. W., & Plano Clark, V. L. (2007). Designing and conducting mixed methods research. Thousand Oaks, CA: Sage
Edwards, T. G., Özgün-Koca, A., & Barr, J. (2017). Interpretations of boxplots: Helping middle school students to think outside the box. Journal of Statistics Education, 25(1), 21-28.
Işık, C., Kar, T., İpek, A. S., & Işık, A. (2012). Difficulties Encountered by Pre-Service Classroom Teachers in Constructing Stories about Line Graphs. International Online Journal of Educational Sciences, 4(3), 644-658
Learning Mathematics for Teaching Project. (2011). Measuring the mathematical quality of instruction. Journal of Mathematics Teacher Education, 14(1), 25-47.
Ulusoy, F., & ÇAKIROĞLU, E. (2013). İlköğretim matematik öğretmenlerinin histogram kavramına ilişkin kavrayışları ve bu kavramın öğretim sürecinde karşılaştıkları sorunlar. İlköğretim Online, 12(4), 1141-1156.
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