Development of Mathematics Teaching

192 Pages·2015·11.04 MB·English

by  Ola Helenius

Skrifter
 från
  ,
 Nr.
 10

SMDF
Editors:

Ola
 Helenius,
 Arne
 Engström,
 Tamsin
 Meaney,
 Per

Nilsson,
 Eva
 Norén,
 Judy
 Sayers,
 Magnus
 Österholm

Development
 of
 Mathematics

Teaching:
 Design,
 Scale,
 Effects

Proceedings
 of
  9
MADIF
The
 Ninth
 Swedish
 Mathematics

Education
 Research
 Seminar

Umeå,
 February
 4-­‐5,
 2014

SMDF Svensk
 Förening
 för
 MatematikDidaktisk
 Forskning

Swedish
 Society
 for
 Research
 in
 Mathematics
 Education
Skrifter
 från
  ,
 Nr.
 10

SMDF

Editors:

Ola
 Helenius,
 Arne
 Engström,
 Tamsin
 Meaney,
 Per

Nilsson,
 Eva
 Norén,
 Judy
 Sayers,
 Magnus
 Österholm

Development
 of
 Mathematics

teaching:
 Design,
 Scale,
 Effects

Proceedings
 of
  9
MADIF

The
 Ninth
 Swedish
 Mathematics

Education
 Research
 Seminar

Umeå,
 February
 4-­‐5,
 2014

SMDF Svensk
 Förening
 för
 MatematikDidaktisk
 Forskning

Swedish
 Society
 for
 Research
 in
 Mathematics
 Education
Skrifter från Svensk Förening för MatematikDidaktisk Forskning, Nr 10

ISBN 978-91-973934-9-2
ISSN 1651-3274

© Given to the authors 2015

SMDF
Svensk Förening för Matematikdidaktisk Forskning
c/o Nationellt Centrum för Matematikutbildning
Göteborgs universitet
Box 160
SE 40530 Göteborg

Swedish Society for Research in Mathematics Education

For information see web page www.matematikdidaktik.org

Printed in Sweden 2015
Preface
This volume contains the proceedings of MADIF 9, the Ninth Swedish Mathe-
matics Education Research Seminar, held in Umeå, February 4-2, 2014. The
MADIF seminars are organised by the Swedish Society for Research in
Mathematics Education (SMDF). MADIF aims to enhance the opportunities for
discussion of research and exchange of perspectives, amongst junior researchers
and between junior and senior researchers in the field. The first seminar took
place in January 1999 at Lärarhögskolan in Stockholm and included the
constitution of the SMDF. The second meeting was held in Göteborg in January
2000, the third in Norrköping in January 2002, the fourth and fifth in Malmö in
January 2004 and 2006, respectively, and the sixth and seventh in Stockholm in
January 2008 and 2010, respectively. Like MADIF 9, the eighth meeting was
held in Umeå. Printed proceedings of the seminars are available for all but the
very first meeting and in 2015 also online versions will be made available both of
the present and of previous volumes.
The members of the 2010 programme committee were Arne Engström
(Karlstad University), Ola Helenius (National Center for Mathematics Education,
chair), Tamsin Meaney (Malmö Högskola), Per Nilsson (Örebro University), Eva
Norén (Stockholm University), Judy Sayers (Stockholm University), and Magnus
Österholm (Umeå University. The local organiser was Tomas Bergqvist (Umeå
University).
The programme of MADIF 9 included two plenary lectures by invited
speakers Koeno Gravemeijer and Beth Herbel-Eisenmann. There were also a
plenary panel consisting of Lisa Björklund-Boistrum, Jeremy Hodgen, Darina
Jirotkova and John Mason, moderated by Ola Helenius. As before, MADIF
works with a format of full 10 page papers and as well as short presentations.
This year was the first where the short presentation (24) outnumbered the full
papers (15). It will be interesting to see if this trend continues in 2016. As the
research seminars have sustained the idea of offering formats for presentation
that enhance feedback and exchange, the paper presentations are organised as
discussion sessions based on points raised by an invited reactor. The organising
i
committee would like to express its thanks to the following colleagues for their
commitment to the task of being reactors: Annica Andersson, Paul
Andrews, Jonas Bergman Ärlebäck, Jorryt van Bommel, Gerd Brandell, Johan
Häggström, Darina Jirotkova, Cecilia Kilhamn, Ia Kling Sackerud, Johan
Lithner, Hanna Palmér, Kerstin Pettersson, Ann-Sofi Röj-Lindberg, Frode
Rönning and Hans Thunberg.
This volume comprises summaries of the two plenary addresses, 15 research
reports (papers) and abstracts for the 24 short presentations. In a rigorous two-
step review process for presentation and publication, all papers were peer-
reviewed by at two to four researchers. Short presentation contributions were
reviewed by members of the programme committee. Since 2010, the MADIF
Proceedings have been designated scientific level 1 in the Norwegian list of
authorised publication channels available at http://dbh.nsd.uib.no/kanaler/.
The editors are grateful to the following colleagues for providing reviews:
Annika Andersson, Paul Andrews, Anette Bagger, Jonas Bergman Ärlebäck,
Tomas Bergqvist, Camilla Björklund, Per Blomberg, Jorryt van Bommel,
Andreas Ebbelind, Robert Gunnarsson, Ola Helenius, Thomas Hillman, Maria
Johansson, Annasara Karlsson, Cecilia Kilhamn, Ia Klick Sackerud, Troels
Lange, Niclas Larson, Maria Larsson, Thomas Lingefjärd, Johan Lithner, Tamsin
Meaney, Lars Mouwitz, Miguel Perez, Hanna Palmér, Eva Riesbeck, Helena
Roos, Judy Sayers, Marie Sjöblom, Håkan Sollervall, Henrik van Steenbrugge,
Görel Sterner, Allan Tarp, Anna Wernberg .

The organising committee and the editors would like to express their
gratitude to the organisers of Matematikbiennalen 2014 for financially supporting
the seminar. Finally we would like to thank all participants of MADIF 9 for
sustaining their engagement in an intense scholarly activity during the seminar
with its tight timetable, and for contributing to an open, positive and friendly
atmosphere.

ii
Contents
Preface i
Contents iii
Plenary addresses
Design Research on Local Instruction Theories in Mathematics Education 1
Koeno Gravemeijer
Discourse and Transparency: Avoiding Agnosticism in
Our Work with Teachers 5
Beth Herbel-Eisenmann
Papers
Foundational Number Sense: A Framework for Analysing
Early Number-Related Teaching 17
Paul Andrews & Judy Sayers

Who is the Student in Need of Special Education in Mathematics? 27
Anette Bagger and Helena Roos
An Instructional Design Perspective on Data-Modelling for
Learning Statistics and Modelling 37
Jonas Bergman Ärlebäck, Per Blomberg, Per Nilsson
Brackets and the Structure Sense 47
Robert Gunnarsson and Annasara Karlsson
Analysing Instrumental and Pedagogic Situations in
Preschools using the Didaktic Space 57
Ola Helenius, Maria L. Johansson, Troels Lange, Tamsin Meaney,
Eva Riesbeck, Anna Wernberg
Preschool Teachers’ Awareness of Mathematics 67
Ola Helenius, Maria L. Johansson, Troels Lange, Tamsin Meaney,
Eva Riesbeck, Anna Wernberg
Theorising the Design of Professional Development Web Modules 77
Ola Helenius, Maria L. Johansson, Troels Lange, Tamsin Meaney,
Eva Riesbeck, Anna Wernberg

iii
Structural and Pedagogical Diversity in Swedish
Grade Six Algebra Classrooms 87
Cecilia Kilhamn, Thomas Hillman
Incorporating the Practice of Arguing in Stein et al.’s Model for Helping
Teachers Plan and Conduct Productive Whole-Class Discussions 97
Maria Larsson
Inconsistency, Regression or Development? The Professional Identity
of a Novice Primary School Mathematics Teacher 107
Hanna Palmér
When the Mathematics gets lost in Didactics 117
Judy Sayers
Designing Tasks and Finding Strategies for Promoting
Student-to-Student Interaction 127
Marie Sjöblom
Spaces of Values: What is Available to be Adopted by Students 137
Karen Skilling, John Mason
Learning Subtraction Strategies From Principle-Based Teaching
Activities 149
Håkan Sollervall
Number By Reasoning and Representations – The Design and Theory
of an Intervention Program for Preschool Class in Sweden 159
Görel Sterner, Ola Helenius
Short presentation abstracts
The Impact of a Professional Development Program in Formative
Assessment on Teachers’ Practice and Students’ Achievement 169
Catarina Andersson
The Importance Of Grammatical Style In Mathematics Tests
For Second Language Learners And Low Performing Students 169
Ida Bergvall
A Modelling Approach for Teaching Statistics and Probability 170
Per Blomberg, Per Nilsson and Jonas Ärlebäck
The Impact of a Professional Development Program in Formative
Assessment on Mathematics Teachers’ Classroom Practice 170
Erika Boström

iv
The Quality of Supervised Group Discussions within the Frame
of Cooperative Learning 171
Gerd Brandell
Multiple Mathematical Practises Figuring in a Lecture About Assessment 171
Andreas Ebbelind
Student Teachers’ Reasoning About the Mathematical Content in
Pupils’ Solutions 172
Birgit Gustafsson
Compulsory School Students’ Experiences of Mathematic Teachers’
Assessment Practice with a Focus on Communication 172
Lena Heikka
Mathematics Communication within the Frame of Supplemental
Instruction SOLO & ATD Progression 173
Annalena Holm
Young Pupils’ Way of Explaining and Arguing in the Discourse
of Mathematics 173
Eva Juhlin
From Natural Numbers to Integers (N → Z) – A Learning
Study about the Importance of Identifying Critical Aspects to
Enhance Pupils ́ Learning 174
Anna Lövström
Tools for Teachers – The Issue of Developing Mapping Tests for
Primary School Mathematics 174
Guri A. Nortvedt and Andreas Pettersen
Negotiating Mathematics Teaching? A Study of a Mathematics
Teacher’s Agency in Collegial Collaboration 175
Anna Pansell
Developing Mathematics Instruction with Adaptive Conceptual
Frameworks 175
Miguel Perez and Håkan Sollervall
Algebra Tasks in a Word Problem and Non- Word Problem
Context – A Multilingual Project 176
Jöran Petersson and Eva Norén
Conventions as Obstacles for Understanding? – Pupils’ Reasoning
when Making Sense of School Mathematics Language 176
Elisabeth Rystedt

v
Integrating Writing to Support Students ́ Understanding of Reading
in Mathematics 177
Cecilia Segerby
Kompetensutveckling i matematik för pedagoger i förskola och
förskoleklass 177
Christina Svensson, Troels Lange and Anna Wernberg
Immigrant Students’ Perspective On Learning Mathematics 178
Petra Svensson
Communicating Mathematically with Images 178
Anna Teledahl and Eva Taflin
The Discursive Use Of Gestures In University Mathematics Lecturing 179
Olov Viirman
How Hard Can It Be? What Knowledge and Skills Does a
Teacher Practising Formative Assessment Use? 179
Charlotta Vingsle
The Meaning of Concept 180
Lotta Wedman
Valuing Mathematics: Translation Challenges 180
Lisa Österling and Annica Andersson
E-mail addresses to the contributors 181

vi
Design research on local instruction theories in
mathematics education
Koeno Gravemeijer
Eindhoven Technical University

Over the last decades, the view, that teachers have to transmit knowledge, has
been replaced with the view that students have to construct knowledge while
being supported by teachers and textbooks. It is, however, not immediately clear
how to guide and support students in such processes in the case of mathematics
education. In response to this problem, design research emerged as a method for
developing theories that can function as frameworks of reference for teachers.
Mark that the notion that people construct their own knowledge does not
offer a pedagogy. For it implies that students will construct their own knowledge
whatever form instruction takes. It does, however, point to the question of what it
is the students construct. Or, what we want them to construct. This brings us to
the question: What do we want mathematics to be for our students? Following
Freudenthal (1971) we argue that students should experience mathematics “as a
human activity”, as the activity of doing mathematics. According to Freudenthal
students should be supported in reinventing mathematics, which fits nicely with
the constructivist mantra of students constructing their own knowledge. But how
to help students invent or construct what you want them to invent/construct?
In answer to this problem, Simon (1995) coined the term, “hypothetical
learning trajectory” (HLT), which refers to choosing tasks with an eye on what
they might bring about, envision the mental activities of the students, and
anticipate how their thinking might help them to develop the mathematical
insights you are aiming for. Being hypothetical, the learning trajectory of course
has to be put to the test. When the HLT is enacted, one has to observe students,
analyze and reflect upon their thinking, and adjust the HLT. Following this line
of thought, we have to support teachers by helping them to design HLT’s, not by
offering them scripted textbooks. For, if we want students to reinvent
mathematics by doing mathematics, teachers have to adapt to how their students
reason and help them build on their own thinking. To do so they need a
framework of reference to base their HLT’s on. We may offer them such
frameworks in the form of “local instruction theories”—and corresponding
resources. A local instruction theory consists of theories about both the process
of learning a specific topic and the means to support that learning. The goal of