Final Commentary: The Challenge Continues

Final Commentary: The Challenge Continues

This edited volume started off as a practical project of the first editor to fill a self-­
identified void of not having a singular Canadian text that could be used as a source
of relevant literature in teaching future secondary school mathematics teachers.
Enter the Canadian Mathematics Education Study Group (CMESG), specifically
and perhaps appropriately, a conversation during a meal at the 40th anniversary
meeting of CMESG, in Kingston, Ontario, and a much bigger vision was born. By
the end of a scenic boat ride, we began collaborating as editors, and the project was
greatly expanded, eventually bringing together a collection of Canadian authors
who work and research within Canadian secondary classrooms in some capacity.
Broader vision and context would be provided by having a contributor who was
foundational within the Canadian mathematics education community, begin each
section with a preface. The ‘landscape’ reference was purposeful; we wanted to
represent our varying culture, geography, and context as broadly as possible. As
well as including Indigenous and Francophone perspectives, each section of chapters
also included the voice of a current classroom teacher (“A teacher’s view”), in order
to provide a practical, grassroots examination of some aspect of the section theme
from a practicing teacher’s perspective. In the end, the addition of the international
authors who graciously offered to read the pieces and use their own contexts to
comment on the works greatly added to the depth of what could be accomplished
within this single volume and allowed for us to see the possibilities for connecting
our Canadian contexts to those beyond our own borders. Below we detail some final
thoughts on how we put the sections together and take a look at how the commentaries
in each of the six parts have challenged or supported the writings within each
   The collected volume began with a highly meaningful preface by Edward
Doolittle that encompassed a fundamental vision for putting together this collection
in the Canadian context. As he noted, “Indigenous culture and issues are founda-
tional to Canada; and …continue to be a necessary part of anything Canadian”
(Preface, this volume). We were deeply honored by the connections made between

© Springer International Publishing AG, part of Springer Nature 2018                 641
A. Kajander et al. (eds.), Teaching and Learning Secondary School
Mathematics, Advances in Mathematics Education,
642                                          Final Commentary: The Challenge Continues

the work of the Canadian authors that we had compiled for this volume to this fun-
damental part of being Canadian and working in a Canadian school context.
Doolittle’s moving preface highlighted how all of the authors did in fact support this
context to some degree. In this vein, he also brought the volume to an international
audience in his final comments on how these are issues that all peoples face and
added, “the question is what we [as Indigenous peoples] have to offer to Canada,
and the world” (Doolittle, this volume). It is with this initial overview of connec-
tions to Indigeneity that situates all of the sections and chapters under one broader
theme and extends the link to secondary school mathematics.
   We chose to begin the collection of authors’ work with a look at “The changing
landscape of teaching and learning mathematics” because we have all been partici-
pants and observers in how teaching in secondary schools has changed and contin-
ues to change. We were honoured that two Canada Research Chairs (Rina Zazkis
and Nathalie Sinclair), as well as mathematician and long-standing mathematics
education advocate Walter Whiteley, contributed prefaces to this section. In this first
section, we chose to situate the volume in the historical as well as cultural landscape
of our context. Beginning the contained chapters with Peter Taylor was an obvious
choice because he has been pivotal in how Canadian mathematics education has
been shaped over the last four decades, which he describes in his chapter. As a
‘founding father’ of CMESG, an organisation with which all of the various section
preface authors have been involved, Peter has forever left a mark on teaching and
learning mathematics in Canada. We also wanted to include a strong focus on the
Indigenous knowledges that both inform and shape our Canadian classrooms. As a
more recent aspect of our context, Canadian schools are beginning to focus on emo-
tional well-being and mental health so this was also deemed a necessary aspect of
this first section. Kaino’s commentary on this section looked at how the chapters
were illustrative of the changing portions of school from his own context and per-
spective. As he noted, this section provides a chance to “re-think innovative and
better ways of teaching and learning mathematics” (Kaino, Part I commentary, this
volume). This need for innovative changes answers the call of many mainstream
media headlines that question the effectiveness and status of mathematics education
in Canada and many other areas of the world. In the end, Kaino supports the driving
vision of today’s classrooms (and this volume): to “provide ways for long-term
retention of mathematical knowledge” (Part I commentary, this volume). Part I pro-
vided the broad strokes of the challenges facing today’s secondary classrooms in
preparing the learners to be mathematics users (and not just learners), and the
remaining sections each tackle specific areas relating to classroom teaching.
   “Shifting to a culture of inclusion,” as introduced by David Pimm, a well-known
mathematics educator in Canada, focussed attention on helping all learners to
succeed in mathematics classrooms, not just those who would go on to become
mathematics educators and mathematicians later in life. Our choices for the chapters
examined those who are most at-risk in our classrooms including students who are
coming from other parts of the world. Although most of the chapters around at-risk
classrooms were Ontario based, we felt the stories were not unique to this part of the
country or even the world, and the stories could highlight learners who need the
Final Commentary: The Challenge Continues                                          643

most support in our classrooms. Through these chapters, we reiterate the fundamental
idea that the information included is supportive of the quote by the Expert Panel on
Student Success in Ontario (2004) as well as other learning initiatives (e.g., Ontario
Ministry of Education 2013): “good for all, necessary for some” (p. 42). Beswick,
in her commentary, responds to the chapters by beginning with an overview of some
of the difficulties in enacting the ideas in the chapters in this section. The first
challenge she brings forward is the complex issue of how a teacher’s beliefs interact
(or interfere) with how pedagogy is taken up and implemented in the classroom.
This idea is a theme throughout the rest of the book as many of the chapters challenge
traditional ideas of what teaching mathematics at the secondary school level is and
should be.
    Part III in the volume, introduced by Elaine Simmt, another well-known Canadian
mathematics education researcher, discusses how relationships can be fostered in
classroom environments. As stated by Boland and Tranter in Part I, the basis of all
teaching experiences is the relationship that is formed with students and the school
community. The chapters in this section critically look at different ways relationships
can help foster greater student success in academics by focussing on behaviours and
other characteristics. Mosvold notes, “The fostering of relationships, then, goes
beyond attending to students’ mathematical thinking, and it involves getting to
know their histories, the experiences they have made in and outside of school, their
cultural background, and everything else that constitutes their identity” (Part III
commentary, this volume). His commentary on the section notes that the connections
among chapters attend to the different aspects that constitute the identities of
secondary students in the classroom. As he notes, two of the chapters specifically
focus on relationships within the classroom and student thinking; whereas, the other
three focus on the development of the whole person through the relationships.
Mosvold ends his commentary with a concern of the tendency to take theories of
learning from other fields and then use them to apply to teaching by assuming them
as theories of teaching. He turns his focus to how the chapter with the strongest link
to teaching is also the one without an explicit theoretical foundation (Newell). The
noted chapter is an interesting treatise into looking at teaching as all that a teacher
does in an effort to support the learning of students. Mosvold concludes that “more
conceptual work needs to be done in studies of mathematics teaching, and
conceptualizations of mathematics teaching should strive towards capturing the
dynamic interactions between mathematical and pedagogical aspects of the work of
teaching” (Part III commentary, this volume). The commentary ends with a
discussion about the complex nature of teaching and how this section has shown the
need to conceptualize teaching as more than just certifications. Following the more
theoretical nature of this section, the edited volume moves into specific pedagogy
for teaching mathematics.
    The part entitled “Enhancing problem-based learning” was meant to serve as a
collection of chapters that focus on specific examples of using this type of learning
environment in the classroom. All of the work in the section was meant to answer
the call of mandates by the Canadian Manifesto (Whiteley and Davis 2003/2016),
the National Council of Teachers of Mathematics (2000), and others, to include
644                                           Final Commentary: The Challenge Continues

more tasks and explorations in the classroom. The focus in the section is on using
problems in a way that allows students to explore the mathematics, and not just use
routine problems to practice previously known formulas. Tom Kieren, long standing,
influential mathematics education scholar, provided the preface. The chapters in this
section are illustrative of how the mathematics comes from the “doing,” rather than
for students to repetitively use something that a teacher has shown them to do.
Leikin and Mason provide two very different commentaries on the chapters from
different international perspectives. Leikin takes the viewpoint of looking at the
chapters as unified through the concepts presented in a model of characteristics that
determine mathematical challenge (Part IV commentary, this volume). Her presented
model focusses on the conceptual characteristics of the problem, socio-mathematical
norms, instructional setting, and individual characteristics of participants as the four
foci, and uses the model as a framework to assess the mathematical challenge of
tasks. For her, the chapters in this section present ideas that can be summarized
through this model and notes that each of the chapters shows the authors sharing
their “authentic experiences” in using problem solving methods in the classroom
(Leikin, Part IV commentary, this volume). Mason frames the chapters in this
section through the lens of “doing” mathematics in a way that is more than just an
execution of mathematical procedures. As he notes, the unifying theme in the
section is that the authors “are trying to make a difference, trying to get students
engaged with mathematics and involved in mathematical thinking” (Mason, Part IV
commentary, this volume). Through his discussion of the history of mathematical
problem solving, Mason notes that these chapters differ from the idea that teaching
mathematics “is assumed to be about training student behaviour so that learners can
carry out operations on numbers” (Part IV commentary, this volume). Both Mason
and Leikin key into the role of context within many of the chapters in this section of
the volume, albeit from different lenses. Leikin focusses on the social justice and
citizenship aspects in the chapters, and Mason approaches the ideas from a context
standpoint with a goal of reaching for social justice. Mason explicitly links this idea
to the concerns raised by Russell (Part IV, this volume) in making sure that the
social justice themes do not eclipse the mathematics. Mason concludes by reiterating
that the chapters in this section are not meant to “prove” their approaches but rather
to serve as a description of how the authors in the chapters have used the techniques
in their own experiences.
    Part V, prefaced by Carolyn Kieran, another very prominent mathematics educa-
tion researcher, particularly well known for her work in algebra, provided concrete
examples of planning and assessment to bring directly into secondary classrooms.
These chapters were intentionally chosen to challenge the ideas that teaching is all
about direct transmission of knowledge from teacher to student and that assessment
is all about exams and quizzes that “test” how well students have retained what has
been shown to them. Reddy and Sriraman both provide commentaries on the chap-
Final Commentary: The Challenge Continues                                          645

ters in this section through their differing experiences and contexts. Reddy (this
volume) provides an interesting commentary on both the commonalities within the
six chapters in this section, but also how the ideas relate to the struggles of second-
ary teaching in South Africa. He begins with a look at the vast differences between
Canada and South Africa in terms of the economic and political struggles of his
home country; however, strengthens his position on the importance of education to
move the country forward in an effort to alter the glaringly negative headlines that
Reddy mentions in his commentary, as well as to improve the future conditions of
the students. He points to the difficulties with the Curriculum 2005 (Reddy, Part V
commentary, this volume) that highlighted a challenge with the skill set of the
teachers who would be implementing the curriculum, a difficulty that is taken for
granted as not an issue within the chapters in this section of the book. This com-
mentary reminds us of the relative position Canada maintains in providing second-
ary teachers with rigorous schooling to prepare them to enact the content in this
section on planning and assessment. The “learning outcome” discussion of Reddy
in relation to the South African curriculum notes the curriculum revision included
“skills and the processes of learning, without sufficient specification of content and
knowledge” (Reddy, Part V commentary, this volume) serving as an interesting par-
allel with Canadian curriculum as a blend of the two: based on what students should
be able to do and know by the end of the school year. Reddy concludes with a dis-
cussion of the social inequities facing the South African school system and mounts
a challenge: “As Canada becomes more diverse, I would imagine it would be chal-
lenged to take into consideration, more significantly, outside classroom contexts to
improve the learning for students” (Part V commentary, this volume). This state-
ment echoes the underpinning themes within Part II of this volume, so although not
specifically addressed in this section, shows a need for Canadian teachers to focus
on more than just content when working with students. Sriraman approaches the
chapters in this section from a different vantage point: an overarching question of
what is actually being measured and notes that the chapters in the section leave
concerns over how these ideas can be implemented on a larger scale beyond the
single classroom that they seem to address (and in some cases beyond the single
cases discussed in the chapter to an entire classroom). His initial commentary
pushes back on the chapters as not being supportive of the use of multiple choice
questions in testing student understanding. He attests that the chapters imply there
is a link between open tasks and conceptual understanding, and multiple-choice
questions and procedural understanding. He ends with a commentary on what
assessment really is and asks for the reader to question the possible win-­loss sce-
narios inherent in testing. He calls for assessment to be aligned to learning as well
as deep memory in order to create a “win-win” situation for both the student and the
teacher. He further extends this to state that memory in mathematics is not simply
646                                          Final Commentary: The Challenge Continues

rote memorization or recitation but something deeper, flexible, and stronger for stu-
dents to retrieve, adapt, and use the information in situational experiences. His final
comments seem to suggest agreement with the chapters’ push to consider assess-
ment (and planning) as not just to prepare for or use multiple choice tests.
    The final part of this book, “Broadening mathematical understanding through
content,” focusses on specific instances of content within the secondary curriculum.
The goal in placing these chapters together was to look at example spaces within the
secondary classroom that are closely tied to the content students are expected to
learn. Gila Hanna, known internationally for her work in many areas particularly
around notions of proof in mathematics education research, provides the preface,
while Moreira provides the international context to the section through his view on
the chapters and their ability to “broaden” mathematical understanding as the
section claims. He begins with a separation between the “professional practice of
mathematicians” and the “professional practice of mathematics schoolteachers” in
order to frame the chapters (Moreira, Part VI commentary, this volume). He does
caution that although these are distinct practices and sets of knowledge, they are not
opposing, though the knowledge of the mathematicians is not “necessary nor
sufficient” for the knowledge needed by schoolteachers (Moreira, Part VI
commentary, this volume). He provides an additional challenge to each of the
chapters in his description meant to push these understandings of mathematics
towards the knowledge of mathematicians and where this adds to or separates from
the knowledge of mathematics schoolteachers. He notes that the issues raised by
Burazin and Lovric, although based in a Canadian context, parallel international
issues connected to his own work. In the end, Moreira comments on the complexities
of what teachers must know and do to teach mathematics effectively, which provides
an overall statement that reinforces the entirety of the collection of authors in this
    While not all invitees were able (or willing) to contribute, the breadth and range
of contributing voices (totalling 85 individual contributors in all), including many of
the founders of CMESG, editors of various mathematics education journals in
Canada and from around the world, authors of current curriculum documents, new
and experienced classroom teachers, and a breadth of international scholars from
five continents, left us deeply humbled.
    This volume was not meant to serve as an exhaustive collection of all the current
issues and challenges facing Canadian secondary school teaching and learning.
Rather, it was meant to showcase the variety and range of research and resources to
both expand and deepen the conversations around the issues and challenges facing
both the research community as well as today’s secondary school teachers of
mathematics. The speed with which technology is changing makes it impossible to
know precisely how to prepare students for the future, hence developing the ability
to think and reason mathematically may be the best possible preparation for students
to be ready to face the currently unknown mathematical challenges that are awaiting
us all in the coming years.
Final Commentary: The Challenge Continues                                                   647


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