- Professor Jill Adler: Revisiting resources as a theme in mathematics (teacher) education
- Professor Jinfa Cai: Mathematical Problem Posing, Curriculum Development, and Professional Learning
- Professor Birgit Pepin: Mathematics (E-)Textbooks: Help or Hindrance for Innovation?
- Professor Susanne Prediger: Enhancing and understanding students’ processes of mathematization and active knowledge organization - Didactical Design Research for and with textbooks in the KOSIMA-Project
Professor Jill Adler
(University of the Witwatersrand, South Africa; King’s College London, UK)
Revisiting resources as a theme in mathematics (teacher) education
Almost twenty years ago, I offered a reconceptualization of resources as a theme in mathematics teacher education (Adler, 1998, 2000). The reconceptualization had three dimensions to it. First was the suggestion that we needed to shift from viewing ‘resource’ only as a noun, and consider it also as a verb. This shift brings into view teachers-working-with-resources, or simply resources-in-use; and more openness in understanding how teachers re-sourced their practice as they enacted, for example, curriculum or pedagogic change. Secondly, the conceptualization of the notion of resources extended beyond the material and physical (e.g. textbooks, texts, chalkboards, artefacts) to include socio-cultural resources like language and time. As less visible, but nevertheless means to enabling teaching and learning, these too should be considered as further resources (or obstacles?) in school mathematics practices. Third was a theoretical orientation, discussed below, to resources-in-use informed by an understanding of school mathematics as a ‘hybrid’ practice, and of the accessibility of resources being a function of their ‘transparency’.
This reconceptualization emerged from a research and development teacher professional development project in post-apartheid South Africa, with participating teachers coming from schools serving learners in low income communities. In this context, the availability of and access to resources, while being addressed by the post-apartheid state, was still severely limited. In addition, curriculum reform was underway, with advocacy for learner centered pedagogic practices. This entailed, on the one hand, devolving autonomy for learning to learners’ activity; and on the other, in mathematics, connecting mathematical activity to learners’ everyday realities, resulting in what I referred to as “hybridization” – enacted practices emerging across these continua. I argued then that texts (e.g. worksheets with activities) and artefacts (e.g. geoboards) while produced with mathematical intentions, did not have mathematics “shining through” them. In Lave & Wenger’s (1991) terms, these resources needed to be “transparent”, visible so that they could be used, and simultaneously invisible, enabling access to mathematics. School mathematics required mediation, and critically so if meaning-making was to be through learner activity with ‘new’ material resources, and contextualized in practices other than mathematics.
At the same time – and as a multilingual country – there was focused work shifting the discourse away from learner’s main languages being considered “a problem”, to their languages being “a resource” for learning and teaching (e.g. Adler, 2001; Setati, 2005). Here too were arguments for the importance of ‘transparency’, where deliberate attention to language (making language visible) was in tension with language simultaneously needing to be invisible, providing access to mathematics (Adler, 1999).
In the past two decades there has been extensive research and development on resources in mathematics education, focused, for example, on teachers’ use of curriculum materials (Remillard, Eisenmann, & Lloyd, 2009) their documentation practices and on resources as “lived” (Gueudet, Pepin, & Trouche, 2012) . Renewed interest in research textbooks in mathematics education was manifest in two special issues of ZDM (2013 and 2018), and studies of the teacher-textbook relation (e.g. Leshota & Adler, 2018). In addition, there are a number of reviews of this accumulating work on resources and textbooks in edited book volumes (Fan, Trouche, Chunxia, Rezat, & Visnovska, 2018; Styilianides, 2016; Trouche, Gueudet, & Pepin, forthcoming). And as Fan & Schubring (2018) show, research related to mathematics textbooks has a far longer history.
In this presentation I will draw from this extensive literature and reviews of the field to ask:
What have been the developments related to resources in mathematics education, empirically, methodologically and theoretically since the early 2000s? How does this wider range of research on resources relate to focused work on textbooks, particularly in the current era of proliferating electronic resources? Where and how have these developments emerged and evolved? This retrospective will provide the landscape for revisiting the conceptualisation of resources previously offered and for critical reflection on its current salience, and with textbooks in view. My initial work suggests a number of issues come to the fore. These will form the substance of my presentation at the ICMT-3 conference.
This work is based on the research supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation (Grant No. 71218). Any opinion, finding and conclusion or recommendation expressed in this material is that of the author and the NRF does not accept any liability in this regard.
- Adler, J. (1998). Resources as a verb: recontextualising resources in mathematics education In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Annual Conference of the International Group for the Psychology of Mathematics Education (pp. 1-18). Stellenbosch: University of Stellenbosch, Faculty of Education.
- Adler, J. (1999). The dilemma of transparency: Seeing and seeing through talk in the mathematics classroom. Journal for Research in Mathematics Education, 30(1), 47-64.
- Adler, J. (2000). Conceptualising resources as a theme for mathematics teacher education. The Journal of Mathematics Teacher Education, 3(3), 205-224.
- Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht: Kluwer Academic Publishers.
- Fan, L., & Schubring, G. (2018). Recent advances in mathematics textbook research and development: An overview. ZDM Mathematics Education, 50, 765-771. doi:10.1007/s11858-018-0979-4
- Fan, L., Trouche, L., Chunxia, Q., Rezat, S., & Visnovska, J. (Eds.). (2018). Research on mathematics textbooks and teachers’ resources. Cham, Switzerland: Springer.
- Gueudet, G., Pepin, B., & Trouche, L. (Eds.). (2012). From text to 'lived' resources: Mathematics curriculum materials and teacher development. Dordrecht: Springer.
- Lave, J., & Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge: Cambridge University Press.
- Leshota, M., & Adler, J. (2018). Disaggregating a mathematics teachers’ pedagogic design capacity. In L. Fan, L. Trouche, Q. Chunxia, S. Rezat, & J. Visnovska (Eds.), Research on mathematics textbooks and teachers’ resources (pp. 89-118). Cham: Springer.
- Remillard, J. T., Eisenmann, B. A. H., & Lloyd, G. M. (Eds.). (2009). Mathematics teachers at work connecting curriculum materials and classroom instruction. UK: Routledge.
- Setati, M. (2005). Teaching mathematics in a primary multilingual classroom. Journal for Research in Mathematics Education, 36(5), 447-466.
- Styilianides, G. (Ed.) (2016). Curricular resources and classroom use: The case of mathematics. Oxford: Oxford University Press.
- Trouche, L., Gueudet, G., & Pepin, B. (Eds.). (forthcoming). The ‘resource’ approach to mathematics education.
Professor Jinfa Cai
(Changjiang Scholar, China Ministry of Education, Southwest University; University of Delaware, US)
Mathematical Problem Posing, Curriculum Development, and Professional Learning
Problem posing, the process of formulating and expressing problems based on a given situation, is an essential practice in mathematics and other disciplines (Cai & Hwang, 2019; Cai et al., 2015; Silver, 1994; Singer, Ellerton, & Cai, 2015). This is acknowledged in policy documents for school mathematics. For example, thirty years ago, the National Council of Teachers of Mathematics (NCTM) published what was to become a widely influential standards document, the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). This document contained NCTM’s first formal recommendation that students should “have some experience recognizing and formulating their own problems, an activity that is at the heart of doing mathematics” (p. 138).
Similarly, in China, one of the objectives of the curriculum reforms for 9-year compulsory education is for students to learn how to pose problems from mathematical perspectives, learn how to understand problems, and learn how to apply their knowledge and skills to solve problems so as to increase their awareness of mathematical applications (Chinese Ministry of Education, 2011). The secondary school mathematics curriculum is intended to enhance students’ abilities to pose, analyze, and solve problems from mathematical perspectives.
Yet, curriculum materials widely used in China and the United States fail to incorporate problem posing in a substantial and consistent way. For example, Cai and Jiang (2017) compared the inclusion of mathematical problem posing in popular Chinese and U.S. elementary mathematics textbooks (specifically, the textbook series published by People’s Education Press in China and Investigations in Number, Data, and Space and Everyday Mathematics in the United States).
Table 1 shows both the total number of tasks Cai and Jiang identified in each of the Chinese and U.S. elementary mathematics textbooks as well as the percentage of those tasks that could be classified as problem-posing tasks. They found only a small proportion of problem-posing activities in any of the curricula, and the proportion fluctuated across grade levels. Cai and Jiang (2017) also examined the mathematical content areas in which the problem-posing tasks were found. For all three curricula, the vast majority of the problem-posing activities (over 90% for the U.S. textbooks and nearly 80% for the Chinese textbooks) were related to number and operations. Only a few problem-posing activities were situated in the content areas of algebra, geometry, measurement, and data analysis. Even though the number and operations content area tends to occupy the most space in elementary mathematics textbooks, this distribution of problem-posing tasks remains disproportional, and it reflects a haphazard approach to including problem posing in the mathematics curriculum.
Because current curriculum materials do not incorporate significant and consistent experiences with problem posing for students, it is unreasonable to expect that problem posing will spontaneously be given much attention in classrooms (Lloyd et al., 2017). Teachers already face multiple demands on their time and attention. On any given day, they cannot devote large amounts of time preparing for significant changes in their upcoming lessons. Thus, to gain buy-in from teachers (and students), any strategy to integrate problem posing more effectively in mathematics classrooms should avoid being burdensome or perceived as a radical change in practice that would require a lot of time to adapt to. Instead, integrating problem posing should build on existing, common practices.
In that vein, I propose three recommendations to better integrate problem posing into the school mathematics curriculum: 1) empowering teachers as curriculum redesigners to reshape existing curriculum materials in simple ways that create learning opportunities for mathematical problem posing; 2) enhancing existing curricula with additional problem-posing tasks that include support in the form of sample posed problems; and 3) encouraging students to pose problems at different levels of complexity. These three recommendations are practical and feasible based on evidence from teacher professional development focused on problem posing to teach mathematics (Cai et al., 2019). In fact, we are conducting a longitudinal study to investigate how teachers learn to teach mathematics using problem posing, and then their teaching impact on students’ learning (Cai et al., 2019).
Through teacher learning, teachers increase their knowledge and change their beliefs, and then change their classroom instruction aiming to improve students’ learning. In the research project, the problem-posing workshops have been designed and focused on changing teachers’ beliefs and increasing their knowledge of problem posing and teaching mathematics through problem posing. The problem-posing workshop was designed and framed in the context of effective teacher professional development. Although the field of mathematics education knows little about how to support teachers to become better problem posers and teach mathematics through problem posing, substantial evidence has emerged concerning the features of PD that have a positive impact on teachers’ instructional practice and students’ learning (e.g., Vescio et al., 2008). These features include: (a) a focus on content, (b) building on student learning and thinking, (c) close alignment with practice, (d) building a learning community, and (e) PD that is ongoing.
We found that although many teachers may have little experience with using problem posing activities in the mathematics classroom, problem posing offers enticing benefits in the potential for deepening students’ engagement with mathematics and gaining a better understanding of students’ mathematical thinking. The relatively minimal investment in professional development that would be needed to help teachers gain confidence in using problem posing in mathematics instruction would be well spent and can see effect (Cai et al., 2019).
- Cai, J., & Hwang, S. (2019). Learning to teach mathematics through problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research. Online First. https://doi.org/10.1016/j.ijer.2019.01.001.
- Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and US elementary mathematics textbooks. International Journal of Science and Mathematics Education, 15(8), 1521–1540.
- Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 3–34). New York, NY: Springer.
- Cai, J., Chen, T., Li, X., Xu, R., Zhang S., Hu, Y., Zhang, L., & Song, N. (2019). Exploring the impact of a problem-posing workshop on elementary school mathematics teachers’ problem posing and lesson design. International Journal of Educational Research. Online first. https://doi.org/10.1016/j.ijer.2019.02.004.
- Chinese Ministry of Education. (2011). Mathematics curriculum standard of compulsory education (2011 version). Beijing, China: Beijing Normal University Press.
- Lloyd, G. M., Cai, J., & Tarr, J. E. (2017). Issues in curriculum studies: Evidence-based insights and future directions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 824–852). Reston, VA: National Council of Teachers of Mathematics.
- National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
- Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
- Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. New York, NY: Springer.
- Vescio, V., Ross, D., & Adams, A. (2008). A review of research on the impact of professional learning communities on teaching practice and student learning. Teacher and Teaching Evaluation, 24(1), 80–91.
Professor Birgit Pepin
(Eindhoven University of Technology, The Netherlands)
Mathematics (E-)Textbooks: Help or Hindrance for Innovation?
Textbooks may help education innovation as they can support teachers to enact renewed curriculum intentions in classroom processes. At the same time textbooks may also hinder real innovation as they limit teachers’ opportunities to (re)design the curriculum and develop curriculum design capacity (e.g., Pepin, Gueudet, & Trouche, 2017), by an overdose of detailed scripts that reduce teachers to ‘technical slaves’. E-textbooks are heralded to be interactive and to support teachers in their everyday classroom practices, as well as in their curriculum (re-)design, through innovative and collaborative work with colleagues. However, access to useful subject-didactical resources does not always lead to curriculum renewal and innovative practices.
In this presentation I use the theoretical notion of connectivity (e.g., Pepin et al., 2016; Gueudet et al., 2018) in association with Documentational Approach to Didactics (e.g., Trouche, Gueudet, & Pepin, 2019), and selected concepts from the field of curriculum theory, to propose in which ways e-textbooks can be used, and what needs to be considered for their use, to further mathematics teacher professional development, to lead to curriculum renewal by teachers and innovative practices.
In our current 21st century environment we are surrounded by technology; we cannot disconnect from technology, and this is true for textbooks too (e.g., Pepin, Choppin, Ruthven, & Sinclair, 2017). Textbooks still play a vital role, albeit they are nowadays often of a different nature – they are e-textbooks. In earlier works (e.g., Pepin et al., 2016) we have defined e-textbooks as “an evolving structured set of digital resources, dedicated to teaching, initially designed by different types of authors, but open for re-design by teachers, both individually and collectively” (p.644). These changes and developments, from static books to “dynamic” e-textbook, call for different teacher competences, and these are related to the design of teaching, and of the curriculum, with digital resources. With the change of the nature of textbooks, we need new ways of supporting teachers with their work, and this is likely to include different ways of studying their work: how can we support teachers in their endeavour of developing a coherent curriculum for their students? How can we help teachers to select suitable resources for teaching and learning the mathematics at hand?
I have chosen to introduce and use the notion of connectivity as a critical feature of curriculum coherence, which can be used -beyond the case of e-textbooks analysis- for helping mathematics teachers to connect the different components of the mathematics curriculum in time of digitalization. This choice allows me to develop a frame for developing learning/teaching trajectories (with the help of e-textbooks) for innovative teaching in a digital age. Considering the mathematics curriculum in different representations: intended, enacted, attained; and at different curriculum levels: nano - student, micro - classroom, meso - school, macro - national level (Thijs & Van den Akker, 2009), mathematics teachers can be supported/guided by the e-textbook to align their goals with the affordances of the connected resources.
What I argue in this presentation is that this kind of support needs to be deliberately and systematically addressed, in order to help teachers to develop a more coherent overview of the mathematics curriculum and its didactically sensitive/suitable tools and resources, in order to realize their mathematical and pedagogical goals in the classroom. Teachers need support to make such connections across the different components of the curriculum (for internal coherence) and between the different curriculum levels (for overall coherence). Educative curriculum materials (Krajcik & Delen, 2017), in digital format (Pepin, 2018), are likely to be helpful to provide such support. Such materials, connected to e-textbooks, may focus on a limited number of essential characteristics for curriculum renewal. They may
- help teachers to orientate on and practice with new elements in their teaching repertoire;
- guide teachers in role taking experiences that exemplify new pedagogical approaches;
- create opportunities for shared reflection by teachers that may challenge also their beliefs about appropriate teaching.
Based on such collaborative, specific experiences, teams of teachers are likely to be stimulated and supported to redesign their overall mathematics education approaches.
Such educative materials are best designed and piloted by small teams of mixed composition: teachers, teacher educators, curriculum designers, and researchers. The development process is said to be iterative, with gradually shifting emphasis in quality criteria: from relevance, to consistency, practicality, and effectiveness.
The concluding argument is that beside appropriate (e-)resources teachers need support in professional development collectives, to develop an awareness of connectivity and a capacity to increase internal and external curriculum coherence.
- Gueudet, G., Pepin, B., Sabra, H., Restrepo, A., & Trouche, L. (2018). E-textbooks and connectivity: proposing an analytical framework. International Journal for Science and Mathematics Education, 16(3), 539–558.
- Krajcik, J., & Delen, I. (2017). The benefits and limitations of educative curriculum materials. Journal of Science Teacher Education, 28(1), 1–10.
- Pepin, B., Choppin, J., Ruthven, K., & Sinclair, N. (2017). Digital curriculum resources in mathematics education: foundations for change. ZDM Mathematics Education, 49(5), 645–661.
- Pepin, B., Gueudet, G. & Trouche, L. (2017). Refining teacher design capacity: Mathematics teachers’ interactions with digital curriculum resources. ZDM Mathematics Education, 49(5), 799–812. https://doi.org/10.1007/s11858-017-0870-8.
- Pepin, B., Gueudet, G., Yerushalmy, M., Trouche, L., & Chazan, D. (2016). E-textbooks in/for teaching and learning mathematics: A potentially transformative educational technology. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 636–661). New York: Taylor & Francis.
- Pepin, B. (2018). Enhancing teacher learning with curriculum resources. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.). Research on mathematics textbooks and teachers’ resources: Advances and issues, Springer (pp. 359–374). Cham, Switzerland: Springer.
- Thijs, A., & van den Akker, J. (2009). Curriculum in development. Enschede, The Netherlands: Netherlands Institute for Curriculum Development (SLO).
- Trouche, L., Gueudet, G., & Pepin, B. (Eds.). (2019). The ‘Resource’ approach to mathematics education. Cham: Springer.
Professor Susanne Prediger
(TU Dortmund University, Germany)
Enhancing and understanding students’ processes of mathematization and active knowledge organization - Didactical Design Research for and with textbooks in the KOSIMA-Project
In the Kosima Project, 16 years were spent on enhancing and understanding students’ learning processes, with a specific focus on constructing meanings by mathematization and active knowledge organization. Initiating rich processes of mathematization is a central aim for all mathematics education. However, many obstacles appear for these processes to generate solid and sustainable mathematical knowledge. For overcoming these obstacles, our textbook research was conducted as Didactical Design Research, which systematically combined two aims: (a) research-based design of teaching-learning arrangements and (b) topic-specific design-based research for a deeper understanding of the initiated learning processes. Finally, a field trial could prove effectiveness of the design.
Project Architecture: Design Research
The talk will report on the long-term design research project KOSIMA (2005-2020, cf. Hußmann et al., 2011; Barzel et al., 2013). It follows the methodology of Didactical Design Research (Gravemeijer & Cobb, 2006) with its dual aim of designing teaching-learning-arrangements for a complete middle school curriculum (Grades 5 to 10) and empirically researching the teaching-learning-processes and their conditions. The developed curriculum has been published as the textbook Mathewerkstatt (Barzel et al., 2012-2017) and a comprehensive teachers’ manual.
The collaborative project involved the design team (including 22 experienced reflective practitioners and the 4 project leaders), researchers (4 project leaders and 13 PhD students), the commercial publisher (with 2-4 copy editors), and project teachers (experimenting with the designs in their mathematics classrooms).
All teaching-learning-arrangements of the curriculum were developed in iterative cycles of design, evaluation (by expert discussions and classroom experiments), and redesign. Whereas the design and evaluation steps of the project referred to the entire textbook, the deeper research was organized in several smaller design research studies that necessarily had to address more narrow research questions for topic-specific aspects. These studies used different concrete research methods and designs (e.g., quasi-experimental controlled trials or design experiments in laboratory settings with up to four cycles, e.g., Leuders & Philipp, 2012; Prediger & Schnell, 2014). Empirical evidence of summative effectiveness was provided in a quasi-experimental field trial with pre-post-test and control group.
Design Principles for the Curriculum and Materials
The design of the Kosima curriculum followed design principles which were partly borrowed from Realistic Mathematics Education (Freudenthal, 1991) and elaborated with respect to developing conceptual understanding (Hiebert & Carpenter, 1992, Prediger & Schnell, 2014) and active knowledge organization (Barzel et al., 2013):
- support students’ sense-making processes
- conceptual understanding before procedures by collaborative meaning construction
- guided reinvention and mathematizations in cognitively demanding explorations
- support active knowledge organization in scaffolded vertical mathematization
- connect multiple representations and diverse mathematical approaches
For realizing these design principles for different mathematical topics, specifying and structuring the learning contents turned out to be a crucial working area. This involves epistemological as well as empirical analysis in order to best align students’ realized learning pathways to the intended learning trajectories (Hußmann & Prediger, 2016). Initiating rich processes of mathematization is a central aim for all mathematics education. However, many obstacles appear for these processes to generate solid and sustainable mathematical knowledge. So the iterative design cycles led to elaborating task formats which can scaffold teachers’ teaching as well as students’ learning.
- Barzel, B., Hußmann, S., Leuders, T., & Prediger, S. (Eds.). (2012–17). Mathewerkstatt 5–10. Berlin: Cornelsen.
- Barzel, B., Leuders, T., Prediger, S., & Hußmann, S. (2013). Designing tasks for engaging students in active knowledge organization. In A. Watson et al. (Eds.), ICMI Study 22 on Task Design—Proceedings of the Study Conference (pp. 285–294). Oxford: ICMI.
- Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer.
- Gravemeijer, K., & Cobb, P. (2006). Design research from the learning design perspective. In J. van den Akker, K. Gravemeijer , S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 17–51). London: Routledge.
- Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.
- Hußmann, S., Leuders, T., Prediger, S., & Barzel, B. (2011). Kontexte für sinnstiftendes Mathematiklernen (KOSIMA) – ein fachdidaktisches Forschungs- und Entwicklungsprojekt. Beiträge zum Mathematikunterricht, 419–422.
- Hußmann, S., & Prediger, S. (2016). Specifying and structuring mathematical topics. Journal für Mathematik-Didaktik, 37(Suppl. 1), 33–67.
- Prediger, S., & Schnell, S. (2014). Investigating the dynamics of stochastic learning processes: A didactical research perspective, its methodological and theoretical framework, illustrated for the case of the short term–long term distinction. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking (pp. 533–558). Dordrecht: Springer.