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Topic: Announcement and Call: ICMI Study 22
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Jerry P. Becker

Posts: 13,471
Registered: 12/3/04
Announcement and Call: ICMI Study 22
Posted: Jul 3, 2012 9:37 AM
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Information provided by Minoru Ohtani, Kanazawa University, Japan.
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Announcement and call for papers (including Discussion Document):

ICMI STUDY 22: TASK DESIGN IN MATHEMATICS EDUCATION

In 2011 ICMI initiated Study 22 on Task Design.
The Study Conference will take place from July
22nd to July 26th 2013 inclusive at the
Department of Education, University of Oxford,
UK. A book will be prepared for publication after
the conference, based on a synthesis of the
proceedings and discussions at the conference and
subsequently.

The study aims to produce a state-of-the-art
summary of relevant research and to go beyond
that summary to develop new insights and new
areas of knowledge and study about task design.
In particular, we aim to develop more explicit
understanding of the difficulties involved in
designing and implementing tasks, and of the
interfaces between the teaching, researching, and
designing roles - recognising that these might be
undertaken by the same person, or by completely
separate teams.

Convenors: Anne WATSON, University of Oxford, UK
& Minoru OHTANI, Kanazawa University, Japan

Plenary speakers:
Marianna BOSCH, Jan DE LANGE, Toshiakira FUJII, Michal YERUSHALMY

IPC
Janet AINLEY, School of Education, University of Leicester, UK
Janete Bolite FRANT, LOVEME Lab, UNIBAN, Brazil
Michiel DOORMAN, Utrecht University, Netherlands
Carolyn KIERAN, Université du Québec à Montréal, Canada
Allen LEUNG, Hong Kong Baptist University, Hong Kong
Claire MARGOLINAS, Laboratoire ACTé, Université
Blaise Pascal, Clermont Université, France
Peter SULLIVAN, Monash University, Australia
Denisse THOMPSON, University of South Florida, USA
Yudong YANG, Shanghai Academy of Educational Sciences, China

Conference administrator: Ellie DARLINGTON

Study Conference:
The Study Conference will take place at the
Department of Education, University of Oxford,
July 22nd to 26th inclusive 2013. Places are
limited to 80 and only those whose papers are
accepted will be invited to attend. The Study
Conference will be organised so that most work
takes place in Theme Working Groups. For more
information about these read the full discussion
document at the end of this announcement.
Conference proceedings will be online.

Call for papers
Papers are invited from designers, researchers,
teacher educators, teachers and textbook authors
and we are especially interested in co-authored
papers that cross these communities. For
information about paper submission see below.

Venue and costs
The conference will take place at the Department
of Education, University of Oxford
(www.education.ox.ac.uk). The conference fee
will be £245 and this includes lunches and
refreshments for five days and a gala conference
dinner. Accommodation will be available nearby
at Lady Margaret Hall
(http://www.ox.ac.uk/colleges/colleges_and_halls_az/lmh.html)
in single rooms at £62.00 per night bed and
breakfast. These are good quality well-appointed
single rooms with ensuite. There are also a
limited number of twin rooms available at £84.00.
There are also hotels nearby at various prices.
Booking for Lady Margaret Hall will be managed by
the conference administrator when papers have
been accepted
.
Registration
Registration of interest is by submission of a
paper for review. There is no other registration
process until papers have been accepted. See
timeline below.

Call for papers
The final date for paper submission is September
1st 2012 and the final date for notification for
final acceptance will be January 1st 2013.

ICMI Study 22 is a working event, and attendance
will be by invitation to those whose papers are
accepted by a Theme Working Group. In order to
inform that work, all papers MUST contain the
following features. Papers that do not include
all these features will be rejected, however good
they are.
1. Explicit information about the role of the
author(s) in the design of the task(s)
2. Explicit information about the definition of 'task' being used in the paper
3. Explicit information about the mathematical
and epistemological perspectives adopted by the
author(s)
4. Articulation of design principles
5. Implications of the work reported in the
paper on different communities: e.g. students,
teachers, educators, designers, researchers,
publishers
6. Explicit information about the institutional,
systemic, and resourcing context of the work
being reported
7. Papers must be about research in task design,
not merely reports of particular tasks.
'Research' includes: empirical work; theoretical
work; systematic study in practice; the
development of frameworks relevant for design;
systematic development.
We particularly welcome papers that are authored
jointly by members of different design
communities, such as teachers & academics;
designers & researchers (where these are
different people).

The format of the paper should be:
1. Eight pages maximum including references,
using the ICMI Study 22 style template in Times
New Roman 12 pt single spaced in a Word document.
The template has been circulated in a separate
file with this announcement and is also available
from icmi22@live.co.uk (subject line 'request for
template'). This will allow all reviewers to make
suggestions for edits during the reviewing
process. For notes on using a template see below.
2. Supplementary material can be provided
digitally. For initial submission, digital
material should be placed on an accessible
website, password protected if necessary, and it
is the author's responsibility to do this.
Material should be such that reviewers can gain
sufficient understanding in 20 minutes to inform
their judgement about the inclusion of the paper
in the Study.
3. Papers must not have been submitted or published elsewhere.
4. The working title for the paper must contain
the author(s) names and the theme letter to which
it is submitted, for example: WatsonThemeC.

Papers should be submitted to the Theme Working
Group that most closely relates to its contents.
Papers that conform to the above requirements but
do not fit well with the emergent scope of the
Working Group to which they have been submitted
will be sent to another relevant group within our
reviewing process. Our first priority will be to
develop the work of the Theme Working Groups, but
the IPC will keep their scope under review to
ensure that papers of sufficient quality can be
included in the scientific work.
Papers should be submitted to: icmi22@live.co.uk
with the subject line 'Submission: Theme X'

All queries should be submitted to Ellie
Darlington at icmi22@live.co.uk with an
appropriate subject line.

The outcomes of the review will be one of the following decisions:
o Accept
o Accept with support for written English
o Instructions to revise
o Suggestions for development and resubmission
o Reject

Conference proceedings will be online. Verbal
presentations at the conference will be brief, at
most 5 minutes, with the expectation that
participants will have read the papers.
Presenters will focus on posing questions and
issues raised by their paper and its relation to
other papers.
The IPC reserve the right to vary the focus on
the themes, and to introduce other paper
presentation sessions, as appropriate when the
scope of submissions becomes clear.

OUTLINE OF THE DISCUSSION DOCUMENT
(If you wish to submit a paper for this Study it
is important for you to read the full version of
this document in which we review the literature
on which the study is based. You will find this
after the conference information)

There has been a recent increase in interest in
task design as a focus for research and
development in mathematics education. Task design
is core to effective teaching. This is
well-illustrated by the success of
theoretically-based long term design-research
projects in which design and research over time
have combined to develop materials and approaches
that have appealed to teachers.
One area of investigation is how published tasks
are appropriated by teachers for complex purposes
and hence how task design influences mathematics
teaching. Such tasks are often complex and
multi-stage, addressing complex purposes. We
encourage an interest also in tasks that have
more limited but valid intentions, such as tasks
that have a change in conceptual understanding as
an aim, or tasks that focus only on fluency and
accuracy.

Tasks generate activity which affords opportunity
to encounter mathematical concepts, ideas,
strategies, and also to use and develop
mathematical thinking and modes of enquiry.
Teaching includes the selection, modification,
design, sequencing, installation, observation and
evaluation of tasks. This work is often
undertaken by using a textbook and/or other
resources designed by outsiders. Textbooks are
not the only medium in which sequences of tasks,
designed to afford progressive understanding or
shifts to other levels of perception, can be
presented, and we expect that study conference
participants will look also at the design of
online task banks.

Tasks also arise spontaneously in educational
contexts, with teachers and/or learners raising
questions or providing prompts for action by
drawing on a repertoire of past experience. We
are interested in how these are underpinned with
implicit design principles.

It is important to address also the question of
sequences of tasks and the ways in which they
link aspects of conceptual knowledge. In some
sequences, the earlier tasks might be technical
components to be used and combined later; in
others, the earlier tasks might provide images or
experiences which enable later tasks to be
undertaken with situational understanding.

The communities involved in task design are
naturally overlapping and diverse. Design can
involve designers, professional mathematicians,
teacher educators, teachers, researchers,
learners, authors, publishers and manufacturers,
or combinations of these, and individuals acting
in several of these roles. In the study, we wish
to illuminate the diverse communities and methods
that lead to the development and use of tasks.

THEMES OF WORKING GROUPS
The work for the Study will take place mainly
within five working groups. We expect there to be
several aspects (such as use of digital
technology, teacher education, curriculum design)
which appear in several themes and the conference
will be designed to allow these to emerge and be
discussed.
Theme A: Tools and representations
Theme B: Accounting for student perspectives in task design
Theme C: Design and use of text-based resources
Theme D: Principles and frameworks for task
design within and across design communities
Theme E: Features of task design informing
teachers' decisions about goals and pedagogies

Timeline
Call for papers and dissemination of discussion document May 2012
Launch of website for submissions May 2012
Final date for paper submissions September 1 2012
Notification to authors of final acceptance of papers January 1 2013
Registration and payment Jan -April 2013
Conference
July 22nd to 26th inclusive 2013
Book chapter first draft deadline July 1 2014

Notes on using the template
The template is designed so that if you write
your text into the appropriate place it will
appear in a common style. For example, this
paragraph is written into the template where it
says 'write your text here'. It appears in the
ICMI Study 22 Body text style.

If you have already written your text in another
form, you can re-format it to our set of styles
using the full descriptions we have given you in
the template.

****************************************************************
TASK DESIGN IN MATHEMATICS EDUCATION

DISCUSSION DOCUMENT

Anne Watson, University of Oxford, UK & Minoru
Ohtani, Kanazawa University, Japan

IPC
Janet Ainley, School of Education, University of Leicester, UK
Janete Bolite Frant, LOVEME Lab, UNIBAN, Brazil
Michiel Doorman, Utrecht University, Netherlands
Carolyn Kieran, Université du Québec à Montréal, Canada
Allen Leung, Hong Kong Baptist University, Hong Kong
Claire Margolinas, Laboratoire ACTé, Université
Blaise Pascal, Clermont Université, France
Peter Sullivan, Monash University, Australia
Denisse Thompson, University of South Florida, USA
Yudong Yang, Shanghai Academy of Educational Sciences, China

ICMI STUDY 22
In 2011 ICMI initiated Study 22 on Task Design.
The Study Conference will take place from July
22nd to July 26th 2013 inclusive at the
Department of Education, University of Oxford, UK.
The study aims to produce a state-of-the-art
summary of relevant research and to go beyond
that summary to develop new insights and new
areas of knowledge and study about task design.
In particular, we aim to develop more explicit
understanding of the difficulties involved in
designing and implementing tasks, and of the
interfaces between the teaching, researching, and
designing roles - recognising that these might be
undertaken by the same person, or by completely
separate teams.

BACKGROUND
In her plenary address to the International Group
for Psychology of Mathematics Education (PME)
Sierpinska (2003) identified task design and use
as a core issue in research reports and in
mathematics education research more generally.
She commented that research reports rarely give
sufficient detail about tasks for them to be used
by someone else in the same way. Few studies
justify task choice or identify what features of
a task are essential and what features are
irrelevant to the study. In some studies using
intervention/treatment comparisons to investigate
cognitive development, the intervention tasks are
often vague, as if the reader can infer what the
learning environment was like from a few brief
indications. A similar view had been expressed by
Schoenfeld (1980). Yet we learn from applications
of variation theory to learning study (e.g.,
Runesson, 2005), from studies of learning from
worked examples (e.g., Renkl, 2005), and from the
Adaptive Control of Thought model (ACT-R) (e.g.,
Anderson & Schunn, 2000) that seemingly minor
differences in tasks can have significant effects
on learning.

At the same time Burkhardt has drawn attention to
the importance of design, with the founding of an
international society and a journal, Educational
Designer (www.educationaldesigner.org).
(Schoenfeld, 2009) makes a plea for more
communication between designers and researchers,
making the point, among others, that many
designers are not articulate about their design
principles, and may not be informed by research.
In 2008, the International Congress on
Mathematics Education (ICME) hosted a topic study
group (TSG), Research and development in task
design and analysis, which provided a forum for
that kind of interaction
(http://tsg.icme11.org/tsg/show/35). Designers
had to be explicit about their principles and
demonstrate how they used them. Participants were
given the opportunity to experience various
tasks, and compare and critique design
principles. Drawing from a wide international
field, an overview of the papers makes it
apparent that:

(1) it is necessary to have theories about
learners' intellectual engagement to have
successful design; and
(2) most design principles included the use of
several representations, several kinds of sensory
engagement, and several question types.

The TSG increased its membership during the
conference, indicating that a serious, organised
look at task design was of growing interest. A
further TSG is due to take place at ICME 12 in
July 2012 in Seoul, Korea. Working groups on task
design using digital technologies, and design of
digital learning environments, proliferate, but
we are not aware of a similar level of activity
in other environments.
Mathematics educators have focused to a great
extent on the social cultures of classrooms and
designed learning environments, on patterns of
argumentation, on emotional aspects of
engagement, and on measures of learning. A
distinct mathematical contribution can be made in
understanding whether and how doing tasks, of
whatever kind, enables conceptual learning. For
example, Lagrange (2002) suggests that applying
routine techniques can achieve results, and also
provide the basis for conceptual understanding
and new theorising; (Watson & Mason, 2006) have
shown how a set of procedural exercises, seen as
one object, can provide raw material for
conceptualisation; Realistic Mathematics
Education (RME) from the Netherlands and
Mathematics in Context materials (from the United
States) show how carefully designed situational
sequences can turn a learners' attention to
abstract similarities.

Our statement that task design is core to
effective teaching is well-illustrated by the
success of theoretically-based long term
design-research projects resulting in
publications such as those from Shell Centre
(Swan, 1985), Realistic Mathematics Education (de
Lange, 1996) and Connected Mathematics (Lappan &
Phillips, 2009). In these, design and research
over time have combined to develop materials and
approaches that have appealed to teachers. In
addition, research related to the QUASAR project
(Quantitative Understanding: Amplifying Student
Achievement and Reasoning) found that the
cognitive demand of designed tasks was often
reduced during implementation (Henningsen &
Stein, 1997). A research forum at PME in Mexico
(Tzur, Sullivan, & Zaslavsky, 2008) offered
cogent explanations for the inevitability and
even desirability of teachers' alteration of the
cognitive demand of tasks. Further, Choppin
(2011) suggests how adaptation differs among
teachers. Thus, a possible area of investigation
is how published tasks are appropriated by
teachers for complex purposes. In variation
theory, a distinction is made between the
intended, enacted, and lived objects of learning.
The Documentational Approach of Didactics
(Gueudet & Trouche, 2009, 2011) also refers to
the practitioner perspective in terms of the
resources on which teachers draw.

Didactic engineering was the topic of the 15th
summer school in mathematics didactics in 2009
(Margolinas, Abboud-Blanchard, Bueno-Ravel,
Douek, Fluckiger, Gibel, Vandebrouck, & Wozniak,
2011). The discussion focused not only on various
principles of task design (see the contributions
of Bessot, Chevallard, Boero, and Schneider) but
also on the problem of the influence of task
design on the development of actual mathematics
teaching (see contributions of Perrin-Glorian,
René de Cotret and Robert). The tasks in these
references are all complex, multi-stage tasks
which address complex purposes, such as those
usefully summarised in Kilpatrick, Swafford, &
Findell (2011), namely the development of
conceptual understanding; procedural fluency;
strategic competence; adaptive reasoning; and
productive disposition.

We would like to encourage an interest in tasks
that have more limited but valid intentions, such
as tasks that have a change in conceptual
understanding as an aim, or tasks that focus only
on fluency and accuracy. Research can investigate
how students perceive and conceptualise from the
examples they are given, or on which they work.
Most mathematics learners world-wide learn
procedures and possibly concepts through
'practice', regardless of the de-emphasis on
procedures held by reform enthusiasts. Thus, the
design of sequences of near-similar tasks
deserves attention. For reasons of global reality
and equity, the study conference shall also focus
on textbook design partly because textbooks are
often informed by tradition or by an examination
syllabus rather than through research and
development (Valverde, Bianchi, Wolfe, Schmidt, &
Houang, 2002), but also because in some countries
textbooks are the major force for change.
Textbooks are not the only medium in which
sequences of tasks, designed to afford
progressive understanding or shifts to other
levels of perception, can be presented, and we
expect that study conference participants will
look also at the design of online task banks.

Work from Sullivan indicates the need to educate
new teachers in the use of complex tasks
(Sullivan, 1999) and it is inevitable that
teacher education will cross several of our
suggested areas. A volume of the Handbook of
Mathematics Teacher Education was devoted to the
tasks and processes of teacher education (Tirosh
& Wood, 2009). A particular relationship between
teacher education and task design is the design
of tasks for teacher education purposes.
Mathematics teacher education, as a subfield of
mathematics education, has paid significant
recent attention to the nature, role and use of
tasks with a triple special issue of the Journal
of Mathematics Teacher Education (volume 10, 4-6)
edited by Mason, Watson and Zaslavsky, and a book
edited by Zaslavsky & Sullivan (2011)

THE MEANING OF 'TASK'
The word 'task' is used in different ways. In
activity theory (Leont'ev, 1975) task means an
operation undertaken within certain constraints
and conditions (that is in a determinate
situation, see Brousseau (1997)). Some writers
(Christiansen & Walter, 1986; Mason &
Johnston-Wilder, 2006) express 'task' as being
what students are asked to do. Then 'activity'
means the subsequent mathematical (and other)
motives that emerge from interaction between
student, teacher, resources, environment, and so
on around the task. By contrast, in some
professional traditions, 'activity' means a
situation set up by the teacher in which a
student has to engage in a certain way. Other
traditions (e.g.Chevallard, 1999) distinguish
between tasks, techniques, technology and
theories, as a way to acknowledge the various
aspects of a praxeology. We are also aware that
'task' sometimes denotes designed materials or
environments which are intended to promote
complex mathematical activity (e.g. Becker &
Shimada, 1997), sometimes called 'rich tasks'. In
this study, we use 'task' to mean a wider range
of 'things to do' than this, and include
repetitive exercises, constructing objects,
exemplifying definitions, solving single-stage
and multi-stage problems, deciding between two
possibilities, or carrying out an experiment or
investigation. Indeed, a task is anything that a
teacher uses to demonstrate mathematics, to
pursue interactively with students, or to ask
students to do something. Task can also be
anything that students decide to do for
themselves in a particular situation. Tasks,
therefore, are the mediating tools for teaching
and learning mathematics and the central issues
are how tasks relate to learning, and how tasks
are used pedagogically.

TASK DESIGN
The design and use of tasks for pedagogic
purposes is at the core of mathematics education
(Artigue & Perrin-Glorian, 1991). Tasks generate
activity which affords opportunity to encounter
mathematical concepts, ideas, strategies, and
also to use and develop mathematical thinking and
modes of enquiry. Teaching includes the
selection, modification, design, sequencing,
installation, observation and evaluation of
tasks. This work is often undertaken by using a
textbook and/or other resources designed by
outsiders.

The extent and detail of design varies widely
among those who work on task design. For some
(e.g., Shell Centre) design includes full
necessary materials, task sequences and advice
about effective choices, and detailed pedagogic
advice about ways of working, verbal
interventions, likely misconceptions and possibly
extensions. For others (Ainley, Bills, & Wilson,
2004, 2005) there may be provision of a
question, or a microworld, or some physical
material, with no written object to describe 'the
complete task', but rather a series of things
that the teacher might say, perhaps supported by
some written prompts. During the resulting
activity, learners may ask questions or make
comments to which the teacher needs to respond,
and part of the design is trying to anticipate
these and have a general picture of the shape of
responses which would complement the task design.
Another form of design is to refine a question or
problem-situation until it is most likely to
promote intriguing mathematical reactions (e.g.,
(ATM, various dates)). Sullivan, Zevenbergen, &
Mousley (2006) have identified a need to design
whole lesson sequences around certain types of
tasks. All of these approaches have implications
for implementation, with some relying on
teachers' existing skills, some providing advice
to extend teachers' skills, and others dependent
on teachers maintaining or adapting the original
task intentions (see, e.g., Kieran, Tanguay, &
Solares, 2011).

Tasks also arise spontaneously in educational
contexts, with teachers and/or learners raising
questions or providing prompts for action by
drawing on a repertoire of past experience. We
are interested in how these are underpinned with
implicit design principles.

TASK SEQUENCES
This discussion of tasks may lead readers to
assume that we are focused only on tasks as
single events, but it is important to address
also the question of sequences of tasks. There
are different aspects embedded in the design of
sequences and, while this is an obvious
consideration when designing textbooks, it also
stretches across the whole field of task design.

To achieve the goal of teaching a whole
conceptual field (e.g., rational numbers), we
have to describe the different aspects of this
knowledge and the way the aspects are linked (for
interesting examples see Brousseau, Brousseau, &
Warfield, 2004a, 2004b, 2007, 2008, 2009). In
Brousseau's Theory of Didactic Situations
(Brousseau, 1997), particular situations (or
single tasks) are generated from more general
situations. The earlier tasks in a sequence
should provide experiences that scaffold the
student in the solution of later tasks, allowing
them to engage in more sophisticated mathematics
than would otherwise have been the case. In some
published sequences, the earlier tasks might be
technical components to be used and combined
later; in others, the earlier tasks might provide
images or experiences which enable later tasks to
be undertaken with situational understanding.

To understand how tasks are linked in order to
support teaching, it is important to understand
the nature of the transformation of knowledge
from implicit knowledge-in-action (see Vergnaud,
1982) to knowledge which is formulated,
formalized, memorized, related to cultural
knowledge, and so on.
However, there are different ways to create
sequences of tasks, some of them are more
commonly known by teachers themselves. One of
these types of task sequences is that in which
the problem formulation remains constant but the
numbers used increase the complexity of the task,
say moving from small positive integers (for
which answers might be easy to guess) to other
ranges of numbers for which a method might be
needed. Another type of sequence is one in which
the problem is progressively made more complex by
the addition of steps or variables, such as in a
network task where additional nodes are added. A
third type of sequence may be one where the
concept itself becomes more complex, such as in a
sequence of finding areas or progressively more
complex shapes from rectangles, to composite
shapes, to irregular shapes. These different
types of sequences, and their relation to the
teaching unit as a whole, are often the focus of
lesson study cycles, such as those reported in
for example Corey, Peterson, Lewis, & Bukarau
(2010); Huang & Bao (2006) Yoshida, (1999).

The importance of sequencing is explicit in
Realistic Mathematics Education. In that
tradition, a task sequence starts with situated
problems (Gravemeijer, 1999), like dividing large
numbers of people into smaller groups (quotative
division problems) to evoke informal strategies
and representations, and continues by changing
the focus to formalizing and generalizing
solution procedures, i.e. in this case a general
algorithm that can be used for various division
problems. In this type of task sequence the idea
of 'guidance with didactical models' from
informal to formal is important as an alternative
strategy for the increasing mathematical
complexity of problems students encounter (Van
den Heuvel-Panhuizen, 2003). The situated
problems are often already rather complex and can
be solved before you know 'the' mathematical
solution procedure, and therefore can be good
starting points for problematizing a concept.

DESIGN COMMUNITIES AND METHODS
Of course, teachers also design tasks explicitly
and deliberately. Whereas some authors think it
desirable that designing and teaching are
separate acts carried out by separate groups of
people (e.g. Wittman, 1995), the experience of
the authors of this discussion document indicates
that the communities involved in task design are
naturally overlapping and diverse. Design can
involve designers, professional mathematicians,
teacher educators, teachers, researchers,
learners, authors, publishers and manufacturers,
or combinations of these, and individuals acting
in several of these roles. In the study, we wish
to illuminate the diverse communities and methods
that lead to the development and use of tasks.
In all methods, the central consideration is the
interaction between teachers and learners through
the designed artefacts and/or the design process.
A major focus in the study will therefore be on
learning how design impacts on learners and
learning, rather than research which focuses
solely on the design process. For example,
research which identifies implicit design
principles would be of interest if connections
are made between these principles and the impact
on learning; research about identities of
different players in the design process would be
of interest if it contrasted
'teacher-as-task-designer' and
'teacher-as-task-user'.

THEMES OF WORKING GROUPS
The work for the Study will take place mainly
within five working groups. The foci of these
groups will overlap and there will be
opportunities during the Study Conference to
develop our understanding of these overlaps.
There are also some strong themes that will
pervade all groups, such as the role of ICT,
implications for teacher education, the designer
perspective, communication between communities
about tasks and so on.
Theme A: Tools and representations
Theme B: Accounting for student perspectives in task design
Theme C: Design and use of text-based resources
Theme D: Principles and frameworks for task
design within and across design communities
Theme E: Features of task design informing
teachers' decisions about goals and pedagogies

THEME A: TOOLS AND REPRESENTATIONS
Allen Leung, Hong Kong Baptist University, Hong Kong
Janete Bolite Frant, LOVEME Lab, UNIBAN, Brazil

In the mathematics classroom, concrete tools (for
example, compasses and ruler, unit blocks,
interactive ICT platforms) are usually used as
resources to enhance the teaching-learning
activity (see for example, Bartolini Bussi &
Maschietto, 2008; Maschietto & Trouche, 2010;
Radford, 2011). In this context, tools are
broadly interpreted as physical or virtual
artefacts that have potential to mediate between
mathematical experience and mathematical
understanding. This theme concerns designing
teaching-learning tasks that involve the use of
tools in the mathematics classroom and
consequently how, under such design, tools can
represent mathematical knowledge. A task here is
a teacher designed purposeful 'thing to do' using
tools for students in order to activate an
interactive tool-based environment where teacher,
students and resources mutually enhance each
other in producing mathematical experiences. On a
meta-level, it is about possible tool-driven
relationships within the design, teaching and
learning triad. In this connection, this type of
task design rests heavily on the complex
relationship between artefacts and mathematical
knowledge.

There are a few theoretical grounds on which to
build and expand this discussion. Instrumental
genesis explicates how the usage of a tool can be
turned into a cognitive instrumentation process
for knowledge acquisition. A Vygotskian approach
examines how an artefact can be turned into a
psychological tool in the context of social and
cultural interaction developed through the zone
of proximal development and internalization
processes. Semiotic mediation can be used as an
integrated approach to explore the mathematics
classroom under which a tool takes on multiple
pedagogical functions (Bartolini Bussi &
Mariotti, 2008). Embodiment theory proposes that
there are strong relationships among sensory
activities and cultural artefacts in the
appropriation of mathematical practices, and in
particular, their application to inclusive
mathematics education (Healy & Fernandes, 2011).
The guided re-invention principle of RME
(Realistic Mathematics Education) practiced by
the Freudenthal school can be used to direct the
design of tool-based mathematical tasks. These
theoretical orientations, and/or others, may
serve to facilitate discussion on tool-based task
design and representation in the mathematics
classroom.

An important question to address in this theme
is: How to design tasks that can bring about
situated discourses (hence representations) for
the mathematical knowledge mediated by tools in
the mathematics classroom and how these
discourses relate to mathematics knowledge? This
in turn comprises several additional questions.

Possible questions about tools and representation:
o What mathematics epistemological
considerations are taken into account when
designing tasks using tools?
o How do we create a tool environment for the
mathematics classroom to support the design of
teaching and learning tasks for specific
mathematic topics?
o How do different types of tools afford
different mathematical activities/tasks,
different representations and/or discourses, and
different interactions between representations?
o How do different task designs using tools
impact on students' learning and understanding of
mathematics?
o How do we design mathematical tasks that can
transform an artefact into a pedagogical
instrument?
o Are there models (theoretical or pragmatic) of
tool-based task design for the teaching and
learning of mathematics?

THEME B: ACCOUNTING FOR STUDENT PERSPECTIVES IN TASK DESIGN
Janet Ainley, School of Education, University of Leicester, UK
Claire Margolinas, Laboratoire ACTé, Université
Blaise Pascal, Clermont Université, France

It is obvious that tasks or sequences of tasks
are designed to embody mathematical knowledge in
ways that are accessible to students, and to
improve students' mathematics thinking. However,
if we look beyond the intentions of those who
design and select tasks, the actual impact on
students' mathematical learning raises important
questions. One of the aims of this thematic group
is to gain insights into students' perspectives
about the meanings and purposes of mathematical
tasks, and to better understand how appropriate
task design might help to minimise the gap
between teacher intentions and student
mathematical activity.

There is a tacit assumption that the completion
of mathematical tasks chosen or designed by the
teacher will result in the student learning the
intended mathematics. This view is persistent
despite research that suggests that this is not a
direct relationship (Margolinas, 2004, 2005).
This can result in completion of the task (rather
than mathematical learning) becoming the priority
for students and even sometimes for teachers.
This can be particularly true for younger and
lower achieving students, who are 'helped' by the
teacher to complete the task in order to 'keep
up' with their peers. Teachers are encouraged to
differentiate tasks for different students in
order to facilitate learning. However, changes
that make it easier for the student to complete
the task may have the effect of undermining the
designers' intentions, and reinforcing students'
attention of completion as the priority.

Research about learners' perceptions of the use
of contexts in mathematical tasks has suggested
that these can differ considerably from
intentions of designers (Cooper & Dunne, 2000).
Whilst designers may choose contexts to offer
real world models to think with or to illustrate
the usefulness of mathematical concepts in real
life, pedagogic practice may lead students to
adopt 'tricks' to bypass the contextual elements
(e.g. Gerofsky, 1996), Verschaffel, Greer, &
Torbeyns, 2006)), or fail to appreciate the
extent to which everyday knowledge should be
utilised in the mathematical task (Cooper &
Dunne, 2000). Tasks or sequences which draw on
real world contexts, but which do not reflect the
purposes for which mathematics is used in the
real world, may be perceived by students as
evidence of the gap between school mathematics
and relevance to their everyday lives (Ainley,
Pratt, & Hansen, 2006).

Another issue is a methodological one. One
possibility for measuring the impact of tasks or
sequences on students' learning is the use of
pre- and post-tests. However, since it is highly
likely that any teaching may result in some
outcome on posttests, it is not so obvious what
should be considered as a significant posttest
outcome. For instance, if we consider only the
mean value of an entire cohort of students, we
may not understand whether the low achieving
students (as determined by the pretest) have
really benefited from the task or sequence.
Moreover, the goal of the task or sequence may
not be easily (or even possibly) assessed in a
written test. Often, it is only by observing the
evolution of students' strategies that we can
understand the effect of a task or sequence
(Brousseau, 2008). Task design is generally
initially implemented in favourable contexts: the
teachers are members of the research team or
closely linked to the designers. In this context,
the impact on students is not only linked to the
tasks but also to the impact on teacher or
students of a collaborative way of dealing with
teaching (Arsac, Balacheff, & Mante, 1992). These
methodological issues are only examples of those
that can be addressed in our group. An aim of
this thematic group is therefore to reflect on
methodological issues related to studying task
impact on students.

Possible questions might be:
o How is it possible to assess the impact of
task or sequence on students' mathematical
learning?
o What is the intended and actual impact of a
task or sequence on low achieving students?
o What do students actually do and attend to when confronted with tasks?
o How do students understand the purposes of
tasks they are given in the classroom?
o How do students' reactions influence teachers' adaptation of the task?
o Might what appears to be 'only' a change in
presentation convey a different meaning to the
student, and result in different mathematical
activity?

THEME C: DESIGN AND USE OF TEXT-BASED RESOURCES
Denisse Thompson, University of South Florida, USA
Anne Watson, University of Oxford, UK

This theme focuses on the design of textbooks,
downloadable materials, and other forms of
text-based communication designed to generate
mathematical learning. We recognise that most
teachers use textbooks and/or online packages of
materials as their total or main source of tasks.
Hence the design and use of tasks presented in
textbooks is central to many school students'
experience of mathematics. The scholarly study of
task design should include consideration of
theoretically-based textbook development, and can
take place at different grain-sizes from
individual tasks, through sequences of tasks, to
a whole textbook series (Usiskin, 2003).

Some analyses of textbooks draw attention to
differences in the use of language,
illustrations, cultural and social allusions and
some focus more on the mathematical and
epistemological content (Askew, Hodgen, Hossain,
& Bretscher, 2010; Haggarty & Pepin, 2001;
Sutherland, 2002; Thompson, Senk, & Johnson, in
press). Significant differences have been found
in the conceptual coherence, mathematical
challenge, consistency of images, and ordering of
tasks between, for example, UK and Singapore
textbooks. For example, in some textbooks a new
concept is introduced through some everyday
questions which are gradually refined to focus on
a formal presentation; in others, practice of a
technique precedes application through word
problems (Ainley, 2010). The design of the order,
development, representation and presentation of
content is therefore a suitable topic for this
ICMI study.

Another way to look at textual presentation is to
analyse the content of individual questions or
sequences of questions, and variation theory has
been used as a tool both for design and analysis
at this fine-grained level (Watson & Mason,
2006). For example, control of variation among
examples can be used to direct learners towards
inductive generalisations about concepts; example
sequencing with controlled variation can lead
learners towards some cognitive conflict.
Textual presentation could be informed by
research about how features of page and screen
layout affect learners' attention (Ainsworth,
2009; Poole & Ball, 2006).

A third way to look at textbook tasks is to view
them as the shapers of the curriculum rather than
merely presenting a given curriculum (Senk &
Thompson, 2003). The underlying commitments about
the nature of mathematics, mathematical activity,
and how mathematics is learnt, vary between
textbook series and between countries. How these
are promoted in the design and content of the
tasks in the textbook is an important area of
study because a textbook series might have more
influence on learners and learning than a
national curriculum. Different designers may
interpret national standards or recommendations
in different ways so that understanding the
principles on which they instantiate these
recommendations is an important area of study
(Hirsch, 2007). Various components of mathematics
will be prioritised or marginalised differently
through different kinds of tasks and there will
be legitimate debate about how students come into
contact with mathematical absolutes (if there are
any) (e.g. Harel & Wilson, 2011).

Authors' intentions can be different from how
tasks or sequences of tasks are used in
classrooms, and in this theme we could also look
at pedagogic suggestions, particularly for
innovative or unusual tasks, and information
about conceptual intentions (Thompson & Senk,
2010). Many textbooks now refer users to online
resources and tasks, and there is a professional
development element to their use. There may be a
difference between the adventurousness of
students and the conservatism of teachers in
their use and vice versa. (See chapters in Reys,
Reys, & Rubenstein (2010) for issues related to
curriculum and tasks in terms of intentions and
enactments.)

Throughout the following set of questions, we
consider a textbook and/or online resource to be
a collection of tasks, generally sequenced in a
given way, and often surrounded by related
narrative and/or questions:
o How do curriculum expectations influence authors' design principles?
o How does an intention to promote change influence design?
o How do designers' expectations of teacher
knowledge inform the design of dual purpose
tasks: to teach students and to facilitate
teacher learning?
o How can authors and teachers learn from
alignments and misalignments of teachers'
adaptations and authors' intent, and the
implications for students' learning?
o How can or should new digital formats
influence textbook design: e.g. use of podcasts,
twitter, and other social media; implications for
design and coherence of materials (either
original digital design or transfer from print)
if teachers are able to select tasks in varied
orders?
o How do cultural considerations about
instruction and pedagogy influence design: for
example, whether teachers are seen as
'facilitators' or 'givers' of knowledge?
o How can designers take account of the language
of instruction not being students' home language?
o What research about design of textbooks and
other materials should be undertaken to inform
the next generation of designers? In particular,
how might design experiments (e.g., Clements
(2007) or teaching experiments (such as Japanese
lesson study)) influence task design in
curriculum materials?
o How can design principles from software
design, advertising, graphical art and eye-gaze
research be used to improve text-based materials?

THEME D: PRINCIPLES AND FRAMEWORKS FOR TASK
DESIGN WITHIN AND ACROSS COMMUNITIES
Carolyn Kieran, Université du Québec à Montréal, Canada
Michiel Doorman, Utrecht University, Netherlands
Minoru Ohtani, Kanazawa University, Japan

Considerations and principles for designing and
sequencing tasks depend highly upon the context
of the design activities. Various design
communities, such as those consisting of
researchers, teachers, professional developers
and teacher trainers, or textbook writers, have
different aims and agendas for task design. Thus,
principles for task design vary across the
context in which the communal practice is
situated. In addition, principles for task design
can vary as to whether they are applied to the
initial creation of tasks or to the shaping and
modification of existing tasks (Remillard, 2005),
as well as to whether they are applied to the
design of a single task versus a sequence of
tasks. Moreover, tasks can be designed not only
by members of a singular community but also by
groups whose members cut across two or more
design communities (see, e.g., the international
examples of such efforts (Kieran, Krainer, &
Shaughnessay, in press). For example, recent
projects where teachers are regarded as key
stakeholders in research (i.e., as (co)producers
of professional and/or scientific knowledge) and
where they have a significant role to play in the
design of tasks have been shown to yield not only
rich task designs for mathematical learning, but
also make the link between research and practice
more fruitful for both sides. In this working
group, we address the diversity and the
interactions between design principles and
communities that are involved in task design and
attempt to make explicit those principles of task
design within and across design communities that
have up to now been largely tacit.

This Working Group has a twofold aim:

o To solicit papers that delineate principles
and frameworks for task design within singular
design communities so as to illuminate
differences and commonalities across the specific
contexts of the various communities.
o To solicit papers that delineate principles
and frameworks for task design by teams that cut
across the various diverse communities so as to
illuminate the nature of, and thereby aid in
encouraging the further emergence of, such
interactive, cross-community approaches to task
design.
Papers being submitted to this working group
should specify which of the two above aims is the
main focus of the paper. Papers being proposed
for this group should also address and develop
some subset of the following questions, in
addition to whatever other issues might be
considered relevant to the given theme:
o If you identify yourself as a member of a
singular design community, which one is it? Or if
you identify yourself as a member of a design
group that cuts across communities, which ones
are they? If the latter, how did this
cross-community come to be formed?
o When you or your group engages in designing
tasks, what are you trying to achieve? What are
your primary considerations?
o Do the principles applied to task design
depend on the nature of the mathematical activity
inherent in the tasks (i.e., tasks for
exploration, concept development, practicing,
generalizing and reflection)? If so, in which
ways?
o In which ways do the principles for task
design interact with the issue of the time
factor, that is, whether a task sequence is to
occur across several lessons or within one given
lesson?
o Which theoretical, mathematical, pedagogical,
technological, cultural, and/or practical aspects
are taken into account when designing a task or a
task sequence? Which aspects are considered
primary?
o Is there a particular framework or theory of
learning that is drawn upon in designing a task
or task sequence, and how is this framework
reflected in the task design?
o What is the extent to which
individual/communal value systems and beliefs
about how mathematics is to be learned enter into
the designing of tasks?
o What is the extent to which the inclusion of
digital-technology tools within a task or task
sequence is reflected in the principles employed
in designing the task or task sequence?
o Are the designed tasks subject to revision in
later cycles of the work? If so, what is it that
specifically leads to the redesign? On what
basis and according to which principles is the
redesign carried out?
o What constitutes the main differences and
commonalities between design principles for
different design communities?
o What constitutes the main differences and
commonalities between design principles for
different age groups and school levels?

THEME E: FEATURES OF TASK DESIGN INFORMING
TEACHERS' DECISIONS ABOUT GOALS AND PEDAGOGIES
Peter Sullivan, Monash University, Australia
Yudong Yang, Shanghai Academy of Educational Sciences, China

Based on their mathematical goals for their
students, teachers choose or design tasks and
sequences of tasks, select media for presenting
tasks to students and for students to communicate
results, plan pedagogies associated with
realising opportunities in tasks, determine the
level of complexity of tasks for their students
including ways of adapting for them, and
anticipate processes for assessing student
learning. Each of these decisions is influenced
by teachers' understanding of the relevant
mathematics, by earlier assessments of the
readiness of their students, by the teacher's
experience or creativity or access to resources,
by their expectations for student engagement, by
their commitment to connecting learning with
students' lives, and informed by teachers'
awareness and willingness to enact the relevant
pedagogies. This working group invites
contributions from researchers and teachers who
have considered such issues from the perspective
of task design. The intention is to synthesise
what is known about teachers' decision making
about tasks, and to offer suggestions about task
design for teachers, teacher educators, task
designers, text and resource authors, and
curriculum developers.

Among the questions that might be considered by
authors contributing to the working group and
which can be addressed by submitted papers are:

o How do features of design influence teachers'
decisions to use particular tasks/sequences, or
adapt them, or create their own?
o How do features of tasks/sequences influence
teachers' choices about their potential for their
class, including the media used for communication
about the task?
o How does the design process influence teacher
decisions about tasks within sequences?
o How do design considerations facilitate
teacher adaptation of tasks/sequences to their
students' experiences?
o How does feedback from classroom
implementation of tasks/sequences inform future
decisions on task design and use?
o How does collaboration between teachers, or
between researchers and teachers, influence
design of tasks/sequences?
o What are the implications for initial teacher education in task design?
o What is the effect of different cultural
backgrounds on teachers' knowledge or belief on
tasks and task design?

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*********************************************
--
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244
E-mail: jbecker@siu.edu



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