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T
he Many Colors of Algebra
Engaging Disaffected Students Through Collaboration and Agency.
Jo Boaler, Stanford University, Tesha Sengupta
-
Irving, UCLA, Jack Dieckmann, Stanford University
& Nicholas Fiori, Yale University.
Draft Paper
.
INTRODUCTION
The numbers of
students in the United States
who dislike or fear mathematics and leave school mathematically
underprepared has prompted widespread concern (Glenn, 2000). Many
students
even those who are successful
develop negative
ideas about math and see the subject as something that is ultimately uninteresting and quite separate
from their lives (Boaler & Greeno, 2000; Madison & Hart, 1990; Seymour & Hewitt, 1997).
Additionally
,
mathematics has a wider “gap” across socio
-
economic
and racial lines than any other academic subject (Secada,
Fennema & Adajian, 1995; Tate, 1997; RAND, 2002). Persistent failure and disinterest in mathematics is of
particular concern given the growing importance of mathematical reasoning an
d ‘quantitative
literacy’ (Stee
n,
1997
) to people’s lives and work, and increasing knowledge of the ways that unsuccessful mathematics experiences
can impact students well beyond the classroom (Moses & Cobb, 2001; Thompson, 1995
; Boaler, 2005
).
Despite the developmen
t of mathematics education as a field of research in the last few decades, and the
identification of
features of learning environments that bring about mathematical interest and high achievement,
traditional teaching of mathematics endures
(
Rosen,
2001; Hiebert &
St
igler
, 2000
).
Positive characteristics of
mathematics classrooms
include reasoning about complex problems, discussing mathematical ideas, and
active
ly
engaging in
mathematical learning
(e.g.,
Malloy, 2009;
Boaler,
2008a, 2002a
; Kieran, 1994)
.
As Gutstein
(2003)
demonstrates
, for example, engaging students in
mathematically
complex ideas with real
-
world implications
fundamentally change
s
students’ orientations
to
wards
math
ematics
.
In North America, such features of
mathematical learning environments are r
are
and c
lassroom
environments
more typically
involve a teacher
presenting examples while students sit quietly, watch, and listen, before practicing similar problems (
Hibert &
Stigler, 2
000; Boaler, 2008a).
A critical feature of acti
ve mathematical engagement that has
gain
ed
recognition in
recent years is the opportunity for student agency (Gutstein, 2003; Boaler, 2002b). Pickering (1995) studied the work
of mathematicians and scientists in order to understand the interplay of knowled
ge and agency in the production of
new conceptual systems. He proposed that mathematical advances require an inter
-
change of human agency and
what he calls the ‘agency of the discipline’ (1995, p116).
In
consider
ing
some of the world’s important mathematic
al
advances
, he
identifies the times
when
math
ematicians use their own agency,
in creating initial ideas or
in extending
established ideas
for example
. He also describes when
they
surrender to
the ‘agency of the discipline’,
follow
ing
standard procedur
es of mathematical proof
or
subjecting their ideas to widely agreed methods of verification.
Pickering
refers to the interplay between human and disciplinary agency as
‘the dance of agency’ (1995, p116)
, a
process central to conceptual advances and mathematical
work
.
Classrooms
where
students
engage in
a
dance of
agency
when working on
complex mathematical problems have been shown to encour
age student interest (Boaler &
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Greeno, 2000
;
Engle &
Conant, 200
2
;
Martin, 2009
) as well as high achievement and persistence in the discipline
(Boaler & Staples, 2008). Such classrooms also offer students opportunities to engage with authentic mathematical
work, rather than simply rehearse procedures that they might need at some point in their mathematical career.
While some classrooms in the United States offer students opportunities to solve complex problems and to act with
agency as they choose, use and
adapt mathematical methods (eg Ball, 1993; Lampert, 2001; Maher & Martino,
1996), such classrooms are rare and there is a tendency in the teaching world to restrict such opportunities to high
-
level classes.
When teachers inherit a class of students
who ha
ve been identified as low achievers, or those who
struggle with mathematics
, they often assume that procedural, low
-
level remediation is most appropriate
(
Anyon,
1980, 1981;
Haberman, 1991
)
. In this article, we
present a case of teaching that took the opp
osite approach
a class
in which a diverse, heterogeneous group of students, many of whom had persistently failed mathematics classes,
were invited to solve complex mathematical problems,
act with agency
,
and
reason about mathematics
. O
ur goal
in
th
is paper
is to
increase
understanding
s of how productive engagement in mathematics can be fostered among
students
even
when they have been low achieving and
mathematically disaffected.
Engle
&
Conant
(2002, p.406)
describe four features of ‘disciplinary engagement’
instances when
students
pay
sustained attention to
the
important aspects of
subject matter
. They highlight the need to:
(1)
Problema
tize content
giving students the opportunity to question, be curious, and conduct their own
inquiries;
(2)
Enable student authority
giving students an active role in defining, addressing and resolving problems;
(3)
Hold students accountable to others and to d
isciplinary norms
encouraging students to listen to each other
and to seek reasons for explanations
that are
accountable to the norms of the discipline
(which, in
mathematics includes such norms as reasoning, and logic)
(4)
Provide r
elevant resources
giving students access to
resources
such as time and materials, which enable
the first three
features
(2002, p406).
The four features Engle and Conant describe were all central to our teaching interven
tion and the associated research
that we presenting this paper, that considered the ways that mathematical responsibility, agency and authority offered
new opportunities for mathematical engagement.
Like Engle
& Conant (2002)
our goal
in this paper
is not
to
promote
this particular
teaching
intervention
but
rather
to contribute to
increased
understanding
s of the different
ways in which
productive disciplinary engagement may be fostered’ (
2002
p401).
Amidst concerns about the
number of students failing alg
ebra and the
move
to introduce algebra to younger
and perhaps mathematically
underprepared
students,
we present
a
case of
transformative
student
engagement
that
might
otherwise be
difficult to
locat
e in the current
educational
climate
.