Department of Mathematics & Statistics Concordia University
Title: Inquiry-based learning approaches and the development of theoretical thinking in the mathematics education of future elementary school teachers
In the years 2013-2015, together with my research assistants, Georgeana Bobos and Ildiko Pelczer, we conducted a design experiment (Cobb, diSessa, Lehrer, & Schauble, 2003) on teaching fractions with three cohorts of undergraduate students taking a “Teaching Mathematics II” course offered as part of an Elementary Education program at North-American university. Our approach to teaching fractions to these groups of in-service elementary teachers was inspired by V.V. Davydov’s “Measurement approach” to teaching fractions to children. (Davydov & Tsvetkovich, 1991). In the first year of the experiment, our pedagogy had several features of an “inquiry-based learning approach.” This appeared to be a rational choice for us, because our design of the Measurement Approach was still very sketchy at the time and we needed to be open to modifying it based on the future teachers’ response to it. To obtain the most natural and unconstrained response, we treated the future teachers more as fellow learners and teachers than as students. We tried to establish a “community of inquiry” (Goodchild, Fuglestad, & Jaworski, 2013) together with the future teachers, where we would learn from them and they would learn from us, as we all solved and posed mathematical problems to each other. Assessment in the course was not based on quizzes, tests or exams but mostly on the products of the future teachers’ creative activity as teachers: the math problems they posed and the activities they planned and simulated with their peers. At the end, the future teachers were quite satisfied with the course, some telling us that they have learned “a lot” in the course. We, too, have learned from interacting with the future teachers, and a much more mature form of the Measurement Approach was born as a result. But we were not satisfied with the future teachers’ understanding of fractions; their understanding at the end of the course hardly differed from their initial understanding of fractions as, overwhelmingly, “parts of a whole.” How did it happen? What aspects of the pedagogy (the inquiry-based learning approach), of the didactics of fractions (the Measurement Approach), and of the future teachers’ ways of thinking could be responsible for such results? In the talk, I will reflect on these questions while telling the story of how our initial approach to fractions morphed, under the influence of future teachers’ response, into the more mature Measurement Approach that we taught in the two subsequent rounds of the experiment.
|2||Jenni Back |
Honorary Visiting Fellow at the School of Education in Leicester
University of Leicester
Title: Making Numbers: supporting teachers and children to use manipulatives to understand arithmetic
Taking the objective of developing number sense as the main aim in teaching arithmetic, we explore the place of manipulatives in supporting that aim. We will present findings from the literature and a survey of elementary teachers’ use of manipulatives and offer suggestions about effective ways of using manipulatives in the classroom. We will share some resources for classroom use as well as suggest some innovative approaches to professional development for teachers that we have found provoke teachers to adopt new ways of working with manipulatives with children of all ages and attainment groups. Detailed exemplars of good practice will be analysed to illustrate some of the affordances and constraints linked with specific manipulatives in relation to a range of arithmetic concepts.
Università degli Studi di Torino
Dipartimento di Matematica
Basing on an inquiry approach to promote mathematical thinking in the classroom
In mathematics the art of proposing a questionOne of the most delicate issues in the teaching/learning of mathematics is ensuring that students acquire the mind-set for grasping the mathematical sense of the teaching situations they face: researchers like A. Schoenfeld speak of mathematical sense making1. It is the exact opposite of the image that many people have of mathematics as a set of rules and algorithms to be learned by heart to answer questions sometimes far from any real sense.
must be held of higher value than solving it.
In fact, the classes are cultural environments in which the activities and daily practices define and give a meaning to the topics that are taught: hence students develop, more or less consciously and more or less consistently, but inexorably, (a protocol of) rules to follow, for example to succeed or at least to "survive" the questions of the teachers. This is the way they develop their own sense for mathematics. The trouble is that there can be a big difference between the teachers’ intended meaning for mathematics she/he is transmitting to the students and their sense of mathematics as a result of their experiences and practices in this domain.
My lecture will illustrate how one can design suitable learning situations and pursue classroom practices, which can generate a genuine mathematical sense through the intertwining of problem solving with problem posing. The goal is to propose a figure of the teacher, who is not (any longer) a transmitter of rules, but a promoter of mathematical sense making for students. The proposed teaching-learning method is called Method of Varied Inquiry (MVI): it is based both on the Method of Variation2 and on the Logic of Scientific Inquiry, which dates back to Galileo, and can also make use of ICT.
The development of MVI in the classroom can serve both as a tool that develops an adequate vision of mathematics, recognized and appreciated as a backdrop to address significant problems, and as an antidote for the purpose of preventing / disrupting a vision of mathematics reduced to a set of rules to be stored and applied.
During the presentation I will illustrate MVI with concrete examples of classroom activities
1Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.
2Marton, F., Runesson, U., & Tsui, A. (2003). The space for learning. In: F. Marton & A. Tsui (Eds.), Classroom discourse and the space for learning, pp. 3- 40. Mahwah, NJ: Lawernece Erlbaum Associates, Inc.
Fakultät für Mathematik
Title: Adult-learning of teachers - often overlooked restrictions for continuous professional development
In the past it was self-evident that in case of a new curriculum or new contents, administration arranged some isolated in-service teaching courses for the teachers. Meanwhile, research has proven and administration has accepted that professional development is only sustainable in case the training is administered as a life-long continuous process. Thus professional development (PD) is becoming a topic of increasing importance.
But what do we know about teacher learning, which is learning of adults, based on research? So far,it has only been a marginal topic in mathematics Eucation. The presenter, who has been being active in PD for more than three decades by now, reports his rich experiences in this field and reflects on PD within the framework of adult learning.
Dos and don’ts are listed.