Interdependency, alternative forms of mathematical agency and joy as challenges to ableist narratives about the learning and teaching of mathematics

Abstract

Catering for the mathematical needs of disabled learners equitably and productively requires the anti-ableist preparation and professional development of teachers. In CAPTeaM (Challenging Ableist Perspectives on the Teaching of Mathematics), we design tasks that emulate inclusion-related challenges from the mathematics classroom, and we engage teachers with these tasks in workshop settings. In this paper, we focus on evidence from one type of task in which participants engage in small groups with solving a mathematical problem while at least one of them is temporarily and artificially deprived of access to a sensory field or familiar channel of communication. In this paper, we focus on evidence of emerging resignification – discursive and affective shifts in the participating teachers’ sense-making about what makes the construction of mathematical meaning possible and valuably different – as they work on the tasks. By linking Vygotsky’s vision about the educational changes required to empower and include disabled learners with more contemporary ideas from embodied cognition and disability studies, our analyses show how engagement with the tasks affects participants’ realisation and appreciation of interdependencies between learners, teacher, resources, and emotions, highlights alternative forms of mathematical agency and gives opportunities to turn initial sense of impasse and despair into joy.

Full text

ORIGINAL PAPER
ZDM – Mathematics Education
https://doi.org/10.1007/s11858-024-01565-z
or remoteness, the risk of exclusion is greater still
(UNESCO, 2023).
Ensuring the catering of these learners’ needs is a sine qua
non of an inclusive education system – and merely secur-
ing access to schooling is far from a guarantee for inclusive
education. In this paper, we focus on one of the challenges
in fostering an inclusive mathematics education. We do
so through reporting from a study, the CAPTeaM project
(Challenging Ableist Perspectives on the Teaching of Math-
ematics) in which we work with practising and future teach-
ers to identify and challenge the ableist assumptions that
currently mediate our interpretations of mathematics teach-
ing and learning.
1 Introduction
Learners with disabilities, neurodiverse learners and those
who experience diculties in learning are amongst those
groups whose needs are yet to be catered for equitably, in
many countries they still face blatant educational exclusion:
Children with disabilities are over-represented in the
population of those who are not in education. […]
Children and youth with sensory, physical, or learn-
ing disabilities are two-and-a-half times more likely
than their peers to never go to school. Where disability
intersects with other barriers, such as gender, poverty,
Elena Nardi
e.nardi@uea.ac.uk
1 King’s College, London, UK
2 School of Education and Lifelong Learning, University of
East Anglia, Norwich NR4 7TJ, UK
Abstract
Catering for the mathematical needs of disabled learners equitably and productively requires the anti-ableist preparation
and professional development of teachers. In CAPTeaM (Challenging Ableist Perspectives on the Teaching of Mathemat-
ics), we design tasks that emulate inclusion-related challenges from the mathematics classroom, and we engage teachers
with these tasks in workshop settings. In this paper, we focus on evidence from one type of task in which participants
engage in small groups with solving a mathematical problem while at least one of them is temporarily and articially
deprived of access to a sensory eld or familiar channel of communication. In this paper, we focus on evidence of emerg-
ing resignication – discursive and aective shifts in the participating teachers’ sense-making about what makes the
construction of mathematical meaning possible and valuably dierent – as they work on the tasks. By linking Vygotsky’s
vision about the educational changes required to empower and include disabled learners with more contemporary ideas
from embodied cognition and disability studies, our analyses show how engagement with the tasks aects participants’
realisation and appreciation of interdependencies between learners, teacher, resources, and emotions, highlights alternative
forms of mathematical agency and gives opportunities to turn initial sense of impasse and despair into joy.
Keywords Disability · Inclusion · Ableism · Mathematical agency · Vygotsky · Embodied cognition
Accepted: 15 March 2024
© The Author(s) 2024
Interdependency, alternative forms of mathematical agency and joy
as challenges to ableist narratives about the learning and teaching of
mathematics
LuluHealy1· ElenaNardi2· IreneBiza2
1 3

L. Healy et al.
2 Mathematics teachers’ narratives about
disability and inclusion: an emerging eld of
research
Pervasive narratives about what a “normal” and ideal stu-
dent is – a by-product of the medical model of disability
(LoBianco & Sheppard-Jones, 2007) according to which
disability is a medical condition which needs to be xed so
that a person can keep up with, and t in, society – are not
conducive to introducing, and implementing, inclusive edu-
cational policies and can often legitimise marginalisation
(Healy & Powell, 2013). Such narratives can also normalise
taking disabled students’ underperformance in mathematics
as an uncritically accepted consequence of their disability
(Gervasoni & Lindenskov, 2011), fomenting decit mod-
els of disabled students1 (Tan et al., 2019). Indeed, students
labelled as “special learners” have often been oered a
mathematics education in which attention to number and
operations tends to dominate and only limited opportunities
to engage in other areas of mathematics are oered (Wood-
ward & Montague, 2002; McKenna et al., 2015).
Whilst disabled students are positioned as in need of
remediation, as lacking when compared to their peers, their
experience of mathematics education appears to be one of
exclusion from many (even most) areas of mathematical
practices. This has left cultural and institutional practices
which present barriers to an equitable mathematics learn-
ing underexamined (Lalvani, 2015). That these students
had not received much attention in mathematics education
literature related to equity and social justice until recently
(Tan & Kastberg, 2017) is perhaps surprising and certainly
concerning.
Over the last decade or so, this has begun to change, and
ableist narratives – which assume that deviations from a
socially constructed ideal body standard makes people unt
for activities in society (Campbell, 2001) – are starting to be
problematised and challenged within mathematics educa-
tion (Tan et al., 2022). Alongside this problematisation, and
in contrast to said medical model, models that posit disabil-
ity as a historical, social, and political phenomenon (Hall,
2019) are starting to gain space in the eld of mathematics
education (D’Souza, 2020; Lambert, 2019).
Much work so far is framed as exploring the experiences
of disabled learners and the associated pedagogies that co-
determine these experiences (Roos, 2023). Studies in this
emerging eld investigate the mathematics of disabled stu-
dents, focus on how their mathematical agency is shaped
by the dierent ways through which they interact with,
1 We use “disabled person” to refer to neurodivergent learners and
those with disabilities and learning dierences as what they have in
common are the ableist constraints of a world that does not accom-
modate them and limits their opportunities to participate and ourish.
express and experience mathematics (Healy & Fernandes,
2011; Figueiras & Arcavi, 2014; Lambert, 2015) and oer
counter-narratives to views of disability as decit along
with glimpses of creativity and brilliance (Tan & Kastberg,
2017).
This recent upturn in attention to the practices and expe-
riences of disabled learners places questions of social jus-
tice centre-eld. It signals how a commitment to inclusive
education requires that those in the teaching profession
resist any belief system that regards disabled students as
“decient and therefore beyond xing” (European Agency
for Development in Special Needs Education, 2010, p. 30).
This too is a growing a concern of those researching in the
eld of disability and mathematics teacher education, not
least because – while a recent meta-analysis of teacher atti-
tudes to inclusion suggests teachers tend to express favour-
able views (Guillemot et al., 2022) – many teachers, from
primary to tertiary education, across all subject areas and
throughout the world, do not feel adequately prepared to
teach disabled students (Sharma, 2018).
A range of strategies aimed at preparing practicing and
future mathematics teachers to create mathematical learning
scenarios based on respect and justice has been examined.
To some extent, all require some disruption of hegemonic
discourses and practices that position those who deviate
from socially constructed norms as problematic. Evidence
from research-based teacher development programmes has
revealed, for example, how engagement with culturally
responsive pedagogies can support teachers in providing
equitable and inclusive mathematics instruction (Abdulra-
him & Orosco, 2020). Most such studies have focused on
including ethnically and/or linguistically diverse students
(e.g., Grant & Sleeter, 2007; Moschkovich & Nelson-Bar-
ber, 2009), but a small number have focused on teaching
mathematics to disabled students (Shumate et al., 2012;
Healy & Santos, 2014).
More emphasis on subject-specic teacher preparation
for inclusive classrooms has also been explored, particularly
in the case of prospective teachers, given the tendency of
pre-service teacher education courses to focus only on gen-
eral pedagogical issues of inclusion (Troll et al., 2019). Tan
et al. (2022) suggest that research in this vein indicates the
role of mathematical understanding in supporting educators
to recognise how students’ mathematical ideas develop dif-
ferently, if and when they are oered learning opportunities
appropriate to their bodyminds2.
As attention to social justice in teacher education for
inclusive mathematics teaching has grown, research from
the area of disability studies is beginning to nd a place
in mathematics teacher education. The study by Tan and
2 Price (2015, p.270) denes this term as “the imbrication (not just the
combination) of the entities usually called ‘body’ and ‘mind.’”.
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Interdependency, alternative forms of mathematical agency and joy as challenges to ableist narratives about…
Padilla (2019) is one of still only a few examples. In this
case-study, participants were prospective primary (elemen-
tary) teachers who were invited to incorporate “the principle
of disability as dierence instead of decit” in their lesson
plans for mathematics. A particular aim was to motivate
the participants to “unlearn” ableist conceptions of abil-
ity in relation to knowledge construction and acquisition.
Their ndings indicate that, while the prospective teachers
did adopt approaches to planning consistent with the prin-
cipals of social models of disability, they also experienced
tensions in unlearning aspects associated with the medical
model that were contrary to these principles. The authors
end by underscoring the need for further research to docu-
ment and create concrete ways of involving teachers in chal-
lenging ableist assumptions and to support them in hearing
and enabling the voices of disabled students, whilst also
recognising the constraints imposed by current power struc-
tures within the education system.
Our own research in the CAPTeaM project can be seen
as a contribution to this call for further research. We now
briey introduce its theoretical underpinnings and then out-
line its aims and the research question this paper explores.
3 Linking the embodied, social and political:
our vygotskian inspired approach
Our interpretations of the historical-cultural perspective of
Vygotsky, and especially his work with disabled learners
(Vygotsky, 1993), sits at the heart of the theoretical framework
that underpins the CAPTeaM project. For us, a concern for
social justice permeates his perspective and, by treating disabil-
ity as a potential strength rather than an inevitable decit, he
advocated radical educational changes aimed at empowering
learners to develop capacities for a satisfying and constructive
life (Stetsenko & Selau, 2018).
We re-vision his work from the 1920s and 30s through
the lenses of recent commentators on his ideas (e.g. Roth
& Jornet, 2016), along with contemporary views from both
embodied cognition (Barsalou, 2008) and disability studies
(Valle & Connor, 2011).
This revisioning has foregrounded Vygotsky’s monist
tendency, in which there is no separation of mind from
body, or of intellect from aect. It also motivated us to
include tools of the body alongside the material and semi-
otic tools he argued to shape activity. This underscores how
the mathematics we do and know depends on the tools we
use to practise it. In relation both to such shaping processes
and to operationalising the unity he ascribes to cognition
and emotion, we nd his distinction between “meaning” –
how concepts are congured and conveyed in a particular
sociocultural context – and sense – “the aggregate of all the
psychological facts that arise in our consciousness” in rela-
tion to a concept (Vygotsky, 1987, p. 275) – useful. Meaning
can be thought of as a subset of sense, with the appropria-
tion of mathematical meaning involving sensing and mak-
ing-sense of both a culturally endorsed body of knowledge
and oneself in relation to it.
From the eld of embodied cognition, we borrow the
notion of simulation as a way to explain how we become
able to act in the present by drawing on our senses of past
activities. Simulation involves the re-enactment of actions,
emotions and sensations “acquired during experience with
the world, body, and mind” Barsalou (2008, p. 618). In
the context of mathematical activity, this would suggest
that as we reuse a previously experienced concept, it is not
just some decontextualised version of its meaning that is
recalled. We also re-enact other aspects associated with our
sense of the concept, the processes by which we came to
know it and how we felt during these processes.
In short, in learning mathematics, cognition and emo-
tion are inevitably entwined. This raises questions about
sense-making that occurs when learners’ experiences of
mathematics are accompanied by negative emotions, by
feelings of inadequacy or exclusion. Here, we see a meet-
ing of critical disability theory with both embodied and
sociocultural approaches. We can expect such feelings to
become part of how mathematics is experienced and of
a learner’s sense of themselves as a mathematics learner:
the disabling discourses and practices that accompany and
frame mathematics learning experiences also become part
of the sense refracted from them. The process of learning
mathematics can hence be seen as an ongoing process of
refraction (Vygotsky, 1994), through which past and current
experiences are enacted, re-enacted and interpreted, either
enabling or constraining our visions of the possibilities for
future mathematical experiences. Since learning occurs in
social settings, our own sense-making also aects not only
the sense-making of others, but also of what de Freitas and
Sinclair (2014) describe as the body of mathematics and the
ways in which it is produced.
We argue that teachers’ senses of teaching mathematics
develop in a similar manner that involves them in sensing
mathematics and mathematical pedagogies, making sense
of their objects and relations, whilst also making sense of
themselves as teachers. For educating mathematics teachers,
and for practising mathematics teaching, a framework com-
bining historical-cultural, embodied, and critical perspec-
tives has signicant implications. If our perceptual-motor,
social and cultural experiences are tied inextricably with
knowing as tool-mediated, then curriculum, pedagogy and
assessment priorities in mathematics education must surely
follow suit in the ways in which they attend to appropriate-
ness of tools for all, including those whose tool-dependence
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L. Healy et al.
Specically, to explore how pre- and in-service teachers
can be encouraged to recognise and challenge ableism, and
develop pedagogies that empower rather than disable learn-
ers, we design and deploy two types of research-informed
tasks. Type I tasks provide opportunities for teachers to
reect on episodes in which disabled students engage suc-
cessfully with mathematics. Teacher engagement with Type
I tasks has been shown to encourage participants to discern
the potency of these students’ mathematical productions,
recognise that they are not mathematically decient (Nardi
et al., 2018) and even to transform their own thinking about
the mathematical objects at play (Batista et al., 2019).
Engagement with Type I tasks has also illustrated the
reservations, of at least some – and mostly the practising
teachers – about the viability of incorporating the less con-
ventional (and perhaps most creative) productions given
institutional constraints such as curriculum and assess-
ment demands. These ndings resonate with those of Tan
and Padilla (2019) in that, while the notion of disability as
decit was challenged, the (exclusionary) nature of curricu-
lar and assessment structures create tensions. Rather than
a process of unlearning, we suggest that experiencing the
contradictions these tensions cause may be integral to prob-
lematising them.
In this paper, we concentrate on teachers’ work with Type
II tasks – in which, to motivate the experiencing of said con-
tradictions, we invite participants to engage in mathematical
activities without using some of the sensory or communica-
tion means that might usually be available to them. We do
so to locate evidence of resignication regarding teachers’
sense-making of dis/ability in mathematics education that
surfaces when they engage with tasks that ask them to pro-
duce solutions to mathematical problems in collaborative
settings in which one or more participants are temporarily
deprived of access to at least one sensory or communication
means. We thus explore the following research question:
How does the experience of doing mathematics
without access to familiar bodily tools (such as eyes
for seeing or mouths for speaking) aect the teach-
ers’ sense of the roles that tools and teaching play in
enabling/disabling mathematical practices?
5 CAPTeaM workshops: Data collection and
analysis
Type II tasks aim to elicit reections on how access to medi-
ational tools dierently shapes mathematical activity. In this
sense, they are what Sannino et al. (2016) call “formative
interventions” which oer participants the opportunity to
diverges from what is seen as the norm. Building an inclu-
sive mathematics education thus necessitates: problema-
tising this norm; understanding how each student may
construct mathematical meaning through engaging with
dierent tools in diverse ways; and, designing/enacting/pro-
moting optimal practices that take on board, and respect,
student diversity.
CAPTeaM aims to contribute to building an inclusive
mathematics education in precisely this manner. We now
outline the project’s aims and the research question this
paper explores.
4 The rationale and aims of the CAPTeaM
project
In CAPTeaM, we seek to explore how engaging teachers of
mathematics with challenges they are likely to face in class
regarding the inclusion of disabled learners might serve as
an eective professional development approach. CAPTeaM
– a collaboration that involves researchers and pre- and in-
service teachers in Brazil and the UK and combines the dif-
ferent research foci and methodological expertise of two
research teams – sets out from the assumption that, rather
than being the consequence of internal, individual factors,
disabled students’ oft-reported underperformance in math-
ematics can result from explicit or implicit exclusion from
mathematics learning. In CAPTeaM, we design situation-
specic tasks (Biza et al., 2018) which challenge ableist
assumptions about the teaching and learning of mathematics
and we engage teachers with these tasks in reective work-
shop settings.
In contrast to narratives that emphasise the mathematical
diculties experienced by disabled students, our prior stud-
ies demonstrate what students can achieve when working in
appropriately designed learning situations (e.g., Fernandes
& Healy, 2016). Innovative mathematical contributions
emerge from such work that question and often surpass
teacher expectations (Healy & Santos, 2014). These stud-
ies have suggested that “by promoting meetings between
dierences, possibilities emerge to experience dierences
as similarities and to learn to inhabit bodies with capaci-
ties that might dier from our own.” (Nardi et al., 2018, p.
150). The overarching aim of the tasks we design, then, is
to aect teachers’ senses of how dis/ability is constructed
in the processes of mathematics education. We do so by
engineering encounters with lessons we have learnt from
our own research and by stimulating shifts in the participat-
ing teachers’ senses about the enabling and constraining of
mathematical activity in the tasks they were working on and
beyond – this is a process we call resignication.
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Interdependency, alternative forms of mathematical agency and joy as challenges to ableist narratives about…
types of bodily involvement observed in their interactions
and what communicational channels they deployed dur-
ing these interactions. We identied four strategies (count-
ing ngers; tracing the sum; negotiating signs to indicate
place value; decomposing numbers into hundreds, tens, and
ones3) all involving haptic constructions of number in com-
municating and carrying out the calculation. As we coded
forms of bodily involvement, evidence of resignication
started to emerge. Such shifts were observable in two forms:
●discursive (language, visual or other mediation, engage-
ment with mathematical routines and comments about
mathematics, its teaching and dis/ability of self and
others).
●aective (movements between negative and positive
emotive states (Liljedahl & Hannula, 2016), such as
manifestations of helplessness, disempowerment, de-
spair, amusement, elation, and joy).
We then systematically searched for indications of discur-
sive and aective shifts. All three authors independently
recorded these indications in all of the UK data, before
reconvening to triangulate and discuss how the evidenced
shifts might be grouped into themes4. Three themes emerged
from these discussions:
●alternative forms of mathematical agency (including
collaborations between humans as well as between hu-
mans and tools)
●realisation and appreciation of interdependencies be-
tween teachers, learners and tools
●the entwinement of cognition and emotion (as emotional
engagement aected mathematical strategies and/or re-
ection on inclusive teaching practices– and vice versa).
To present our ndings, we zoom in on two episodes from
the data that evidence resignication. We do so with caution
not to overclaim resignication from singular episodes and
while attending to frequency of occurrence across our data-
sets. Our selection of episodes resonates with what Coles
and Sinclair (2019) call “telling” episodes, namely episodes
that aim to “sensitis[e] the reader to new possibilities […]
rather than asser[t] causal connections […] looking in detail
at particular cases in order to draw out more general prin-
ciples” (ibid., p. 182). We see these episodes as “paradig-
matic” examples (Nardi, 2008, p. 18–23): while they brim
with the contextual specicity, situational particularity and
uidity aorded by narrative approaches, they also mirror
3 For example, representing 300 with three ngers followed by two
sts.
4 The data collected in Brazil were subsequently analysed by the rst
author.
engage in activities that “can lead to generative, novel out-
comes” (p. 606)– in this case, discoveries about the capacity
of our bodies to participate in mathematical activity in many
and varied ways. Also resonant with this notion of formative
experience in designing Type II tasks is the participatory,
situated notion of “embodiment” in activities for the prepa-
ration of teachers (Ord & Nuttall 2016) as a means to bypass
the alienation often generated by the theory-practice divide.
Type II task participants work in groups of three. One
group member acts as observer. A second group member has
a student role and is asked to solve a mathematical problem
whilst, temporarily and articially, deprived of using a par-
ticular sensory eld and/or communicational mode (in the
task in this paper: seeing). The third member has a teacher
role, communicating the problem and intervening as judged
necessary, but without access to another sensory eld or
communicational mode (in the task in this paper: speaking).
The data we draw on in this paper originate in datasets
collected in Brazil and the UK from workshops in four
dierent universities with a total of 91 pre- and in-ser-
vice teacher-participants (70 from Brazil and 21 from the
UK). Bar a small number of in-service mathematics teach-
ers– none with Special Educational Needs and Disability
(SEND) coordinator responsibilities– participants in the UK
were pre-service mathematics teachers, enrolled on a Sec-
ondary Mathematics Post-Graduate Certicate in Education
(PGCE) programme. Participants in Brazil included four
practising teachers with some Special Education responsi-
bilities, 10 teachers who were also undertaking a two-year
Masters in Mathematics Education course, 38 undergradu-
ate students on a four-year course in Mathematics Educa-
tion (future mathematics teachers) and 18 undergraduate
students studying on a four-year course in Education (to
become primary teachers).
In the workshops we draw on in this paper, participants
completed four tasks (three Type I; one Type II) in three-
hour sessions. In the Type II task, participants were asked to
communicate and carry out a multiplication of two numbers,
a three-digit number and a two-digit number (e.g. 347 × 36).
The data consist of audio / video recordings from four
dierent institutions, three in Brazil and one in the UK: 27
small-group Type II sessions and four plenary discussions.
Data collection was carried out once ethical approval by
the Research Ethics Committees in both the UK and Brazil
institutions had been granted. We note that the participants
whose photographic images are used in this paper consented
also, and specically, to this use.
Analysis of the data aimed to identify how working on a
mathematical activity in the absence of familiar mediational
tools aected participants’ sense-making about teaching
mathematics to disabled learners. In our search for partici-
pants’ strategies for coping with the task, we explored the
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L. Healy et al.
suggested verbally by the student, while in the remaining
thirteen cases, initiation of the sign came from the teacher.
To communicate the times symbol, Juliana oers her two
index ngers in the shape of a cross, which made sense to
Daniela. Daniela then suggested that Juliana holds out her
own ngers for the digits of the next number, as she found
the tracing on her arm dicult to decipher.
Following Daniela’s lead, Juliana easily communicated
67 and her joy when Daniela verbalised the requested calcu-
lation was palpable (Fig. 2).
Daniela immediately tried to eectuate the calculation as
if she was (mentally) completing the paper and pencil algo-
rithm– by far the most common calculation strategy emerg-
ing in some form at some point in 22 of the 27 sessions.
Daniela started her attempt to perform the algorithm, qui-
etly muttering the steps. But, as invariably happened in the
sessions in which this was attempted, she kept losing track
of the numbers (Fig. 3).
As in many other sessions, having communicated the
required calculation, Juliana’s rst reaction was to sit back,
passing the responsibility to Daniela. In three cases, the per-
son acting as a teacher made a conscious decision not to
intervene once the student knew the requested calculation,
even following requests for help. This was justied in the
one of the post-task discussions with the argument that it is
patterns in our participants’ narratives about inclusion and
disability which signal evidence of the resignication our
analysis aims to explore.
6 Teachers’ shifting sense-making about
disability and inclusion: two “telling”
resignication episodes
The two episodes we present in 6.1 and 6.2 are drawn from
the same workshop, but we signal the representativity of the
approaches involved in them across the data set as a whole,
by identifying the frequency of the coded strategies.
6.1 Desperation turns to joy
In this episode, Juliana assumed the role of teacher and Dan-
iela the role of student. Their exchanges were lmed by Wel-
lington. The calculation given to this group was 305 × 67.
Juliana was very unsure how to start. This is evidenced in
expressions of despair and insecurity in the rst three min-
utes (Fig. 1).
After these initial displays, Juliana took hold of Dan-
iela’s hand and separated out three ngers. Finger count-
ing was an extremely common strategy, used in 25 of the
27 Type II sessions, although in this particular case, Juliana
changed almost immediately, as she was unsure about how
to deal with zero. Her second strategy involved tracing out
the rst number in the calculation on Daniela’s arm (a strat-
egy observed in nine sessions overall). Daniela complained:
“wait, here…go slower, it’s too fast”. Following this instruc-
tion, Juliana slowly traced out the number 305 and was
delighted when Daniela correctly identied it, putting both
her thumbs up and waving them in the air. Although Daniela
could not see the gesture, she sensed from Juliana’s jubi-
lation that this was the right response, and then suggested
that Juliana tap her back whenever she made a correct inter-
pretation. The negotiation of shared signs such as this one
occurred in all but three of the 27 sessions that were vid-
eoed and seemed to have been a critical factor for successful
completion. In this case, as in eleven others, the sign was
Fig. 2 Juliana’s jubilation upon Daniela’s verbalising the requested
calculation correctly
Fig. 1 Daniela on the left (not allowed to see), Juliana on the right (not allowed to speak), initial sense of despair
1 3

Interdependency, alternative forms of mathematical agency and joy as challenges to ableist narratives about…
20,100. Daniela chose to write it down, to “hold onto it”.
Juliana then nger-communicated the nal step of 5 × 67,
which Daniela calculated again by writing the sum on paper
and pencil and verbalising each step to arrive at 335. She
nally added 335 to the now familiar 20,100 to obtain the
nal result of 20,435, leaving both equally pleased (Fig. 4).
6.2 You helped me more than I helped you
Alberto (in the role of teacher) had the task of communi-
cating and assisting Iara (in the role of student) to solve
347 × 36. They were observed by Zaíra. Alberto also used
the popular strategy of counting ngers to indicate the digits
in the rst number. It took some time for Iara to grasp that
the 3, 4 and 7 composed a three-digit number, initially dis-
tracted by the fact that 3 + 4 = 7. This provoked Alberto to
oer thumbs up and thumbs down as signs for right / wrong.
Eventually, after numerous repetitions of the sequence “3,
4, 7, x”, Iara identied the rst number as 347. The second
number (36) was then quickly communicated.
Immediately following the conrmation that her task was
to multiply 347 by 36, she says “How am I going to do this,
I am hopeless at calculating mentally, I have a really bad
memory”. She starts trying to multiply 347 by 6 but quickly
loses track and asks for help: “how are you going to teach
me to do this, Alberto?”. This request was unusual in that,
in most sessions, it seems to be initially accepted by both
participants that the calculation would be something the stu-
dent would be able to do, and, in the rst instance, attempt
on their own.
In this case, as a response to Iara’s request for help, Alberto
gave her a piece of paper and put a pen in her hand. She
rejected both, laughing while saying “What’s this Alberto?
Alberto, I can’t see.” It seems that Iara wanted to encourage
Alberto to use a strategy other than one commonly used,
one that would be appropriate for someone who might never
had had access to the visual eld. Alberto’s gestures (Fig. 5)
demonstrated that he had no idea how to help Iara and he
important for a student to do the calculation on their own. In
most sessions, however, in the face of their students’ strug-
gles, the teacher did intervene.
In this case, Juliana’s rst intervention was to place a pen
into Daniela’s hand, an action that prompted her to attempt
to complete the calculation by writing out an algorithm.
Again, this was not uncommon and appeared in ten ses-
sions. Daniela worked without help for some time, writing
out the steps that she could not see. When she did arrive
at an answer, it was incorrect. Unsure how to convey this,
Juliana began to communicate the calculation again and
Daniela quickly interpreted this to indicate that her answer
is wrong. She tried to perform the algorithm again before
lifting her head towards Juliana and noting: “I think I need
a dierent way”. After a moment’s pause, Juliana oered
her hands to suggest that Daniela calculates 7 times 300,
now using her st for the two zeros. Daniela then sponta-
neously calculated 60 × 300 and added 2100 and 18,000
resulting in 20,100. They both initially thought they had
the answer before Juliana realised they had one more step.
She tapped Daniela’s head to suggest that she remembers
Fig. 5 Iara on the left (not allowed to see), Alberto on the right (not
allowed to speak), initial sense of impasse
Fig. 4 Desperation has turned to joy
Fig. 3 Daniela struggles to remember the steps of her calculation as
Juliana sits back
1 3

L. Healy et al.
They are both pleased. Alberto’s expression especially
closely resembles pure joy. Iara opens her eyes, and they
laugh together as Alberto says “You helped me more than I
helped you” to which Iara responds “but it shows the neces-
sity of getting the student to verbalise more their point of
view, you had to tell me to do that, but you couldn’t, but we
can and should do that when we are teaching”.
resorted to repeating the calculation 347 × 36 over and over,
until Iara said “I know it’s 347 × 36, I need you teach me
how to do it”.
After a short pause, she joked “you could just show me
the numbers for the answer” and they both laughed, break-
ing the tension a little. Iara then went back to mental cal-
culations, whispering to herself before stopping when she
became unable to hold all the numbers in her memory. Per-
haps because Iara’s joke suggested to him it might help if he
did know the answer, Alberto sat back, losing tactile contact
with Iara, and worked through the calculation on paper. Iara
sensed Alberto’s distancing and asked “Alberto, what are
you doing, have you given up on me?”. His gestures now
directed at the observer suggested that he was indeed on the
point of desisting (Fig. 6).
In response to Alberto’s gestures, Zaíra (observer) tells
him not to give up. Upon hearing this, Iara reiterates “Are
you giving up on me? Don’t give up. You can’t give up”.
They sit without attempting to communicate for a short
while, after which Alberto breaks the condition of not being
allowed to speak, saying “This is really dicult, I don’t
know what to do, you need to tell me how I can help you”.
The tone of his voice indicates to Iara, who remained with
her eyes shut, his discomfort and insecurity– a contrast with
the generally light-hearted tone of the exchanges between
them before this point.
His plea turns the interaction around: Iara begins to ver-
balise more clearly the calculations as she performs them
and explicitly solicits that he both help her remember the
numbers and indicate when she was correct. She starts by
multiplying 347 by 30, letting go of the sequence usual
in performing the paper and pencil algorithm and, having
ascertained her answer of 10410 was correct, she instructs
10410, 10410, 10410. I am never going to forget this
number 10410, but write it down, my head is already
hurting.
By now, the pace of the interaction had changed as they
seemed to be working in tandem. Figure 7 shows how,
as Iara verbalised the steps in her calculation and Alberto
recorded them, they maintained physical contact.
Rather than attempting to multiply 347 by 6, Iara notices
two number facts that she could draw on: 6 is two times 3
and 30 is three times 10. This avoided using the more tra-
ditional algorithm, and she explained “30 was 10410, so 3
times would be 1041 and 6 would be double, 2082”. Alberto
vibrates with excitement as he feels the answer is in sight.
Having indicated that 2082 is correct, he places Iara’s n-
gers in form of the addition sign and she adds 10,410 to
2082. Alberto raises her hands in triumph when the correct
response is obtained (Fig. 8).
Fig. 8 Iara and Alberto complete the task
Fig. 7 Iara and Alberto resume working together
Fig. 6 Alberto appears to be about to give up
1 3

Interdependency, alternative forms of mathematical agency and joy as challenges to ableist narratives about…
Although alternatives to the algorithm were used in only
ten of the 27 sessions, they were discussed in all post-task
discussions. In one of the workshops, a participant, Bruno,
explained how he had calculated 247 × 35 by rst adding
three to 247, so he could multiply 250 by 30, and adding
to this the result of multiplying 247 by 5 before subtracting
the extra 3 × 30. Though clearly impressed by the creativity
of his method, another participant suggested that it worked
because he had been lucky with the numbers, while students
need to know the algorithm because “it always works, it is
general, ecient”. To this, Bruno replied laughing “not with
my eyes shut”. In the post-task discussion of another work-
shop, David, like Melanie, expressed concerns about the
appropriateness of teaching this method to someone who is
blind. He was questioned by another participant who was
worried that, if the same thing was not taught to the blind
student, then this was not inclusion. David’s view was that
the content, multiplication, could be the same but not neces-
sarily the methods.
Constraining access to the visual eld hence highlighted
the algorithm (tool)-dependent nature of multiplication,
as well as potential diculties that learners might experi-
ence if expected to work with tools that were not congruent
with their bodies. These are points which resonate with our
Vygotskian interpretation of the roles of embodied, mate-
rial, and semiotic tools in the practice of mathematics and
hence that tools oered in learning situations need to be
attuned to the bodyminds of the students.
The post-task discussion in all four institutions sug-
gested that Type II interactions had provoked many to
reassess aspects related to the teaching of multiplication,
aecting their senses of both the algorithm and other ways
of performing multiplication. Their initial approaches had
been shaped by their own experiences with multiplica-
tion. The algorithm as a tool had become an integral part
of their thinking, but there was an incongruency between
this method and the resources they had available to per-
form it. The culturally shared mathematical meanings they
had appropriated in their senses for the algorithm likely
remained stable, but meanings associated with its role in
the process of teaching, learning and practising mathemat-
ics were experienced as contested rather than xed. While
some held onto a view, generally supported in curricula
sequencing – that the formal written method was in some
respects more sophisticated than the other methods – for
many students, participation in the task appeared to provoke
discursive shifts and some resignication of their senses of
multiplication methods. Some argued for the importance of
doing things dierently with dierent students, while others
seemed to have reected deeply on the alternative forms of
mathematical agency provoked by the absence of access to
7 Mathematical agency, interdependence
and joy in collective mathematical activity
The “telling” episodes we report in 6.1 and 6.2 originate
in the same workshop. Following the small group interac-
tions on the Type II task, the whole group reconvened for
the post-task discussion. There was a sense of excitement,
and everyone was eager to share their experience and their
strategies. We note that the same excitement was evident
in the workshops that took place in the other three insti-
tutions. As we report upon these discussions, we return to
our research question, and we reect on how participants’
sense of the roles of tools and teachers in enabling/disabling
mathematical practices was aected in relation to each of
the three themes that emerged.
Mathematical agency. Perhaps not surprisingly, in post-
task discussions, a major focus was the algorithm and the
diculties of carrying this out without recourse to the visual
eld. Wellington (observer in 6.1) noted: “when she started
to calculate, she wrote out the algorithm and then the prob-
lem became the algorithm itself”. All participants in the
student role in this workshop (there were four groups) had
attempted the algorithm and all had found it dicult. Mela-
nie (observer in another group) noted:
“I think in all the groups, the person with their eyes
shut tried to reproduce the algorithm in their head. So,
I am wondering, how does a person who is blind imag-
ine this calculation?”.
The researcher (rst author) explained that some schools
have access to a grid and Braille pieces (Cubaritmo) which
enables blind learners to set out the algorithm, but that some
of the blind learners had described to her how they tend to
use decomposition or rounding and adjusting strategies.
This reminded Iara of her work with youth and adults in a
type of schooling in Brazil (Educação de Jovens e Adultos,
EJA) aimed at those who did not complete basic school-
ing with their peers. She described how some of those who
worked as builders, for example, used such strategies to cal-
culate with impressive speed.
Taking part in this activity highlighted the value and
legitimacy of strategies other than the conventional algo-
rithm, but also how dependent the participant themselves
had become on its use. Zaíra summed this up as follows:
I think this activity took us to an unknown place in the
sense that we have to really think about the operations,
perhaps we don’t really know the operations, we just
know the algorithm…The algorithm becomes more
important than the operation, than multiplication.
1 3

L. Healy et al.
other and on the tools of our cultures. Dependencies of the
disabled body tend to be more visible, since the dependen-
cies of those with bodies labelled as able, as typical, have
become so normalised as to disappear.
What we evidenced in many of the Type II sessions was
the emergence of productive interdependencies, with par-
ticipants in both roles dependent on each other for dierent
aspects of the interaction. Those assigned to be the teachers
relied on instructions from those acting as their students,
while the students counted on their teachers to serve, for
example, as tools for remembering. These interdependent
interactions supported autonomy, with autonomy viewed as
“emancipation from hegemonic and hierarchical ideologies”
rather than reduced to independence (Meekosha & Shuttles-
worth, 2009, p. 52–53). While we do not contend that this
meaning was explicitly appropriated by participants in this
study, there were a number of expressions of interdepen-
dence both in the Type II tasks (emerging in 18 of the 27
sessions) and in all post-task discussions, evidencing dis-
cursive shifts in the participants’ senses of teacher-student
relationships. For example, in relation to the rst episode,
Juliana commented that Daniela (as student) “showed me
the way to interact with her”, while Daniela recognised how
“Juliana thought of decomposition, even then it was di-
cult to recollect, but she helped with this. I used her as my
memory”. Alberto and Iara’s interactions also aected their
sense of interdependencies between teachers and learners as
their last comments above illustrate.
Entwinement of emotion and cognition. The uncertainty
and anxieties about how to proceed expressed by those
assigned to act as the teacher in both the above episodes
emerged at some point in all Type II interactions. As dier-
ent forms of communicating numbers and operations were
developed, this frequently turned to a shared sense of delight.
Not all the interactions ended so positively: three partici-
pants in the teacher’s role suspended the interaction before
they had communicated the task; and, in a further seven
groups, the calculation was only ever partly completed. But,
in the cases in which student autonomy had been supported
by the teachers’ interventions, the sense of satisfaction of
both participants was palpable– as the creative enactment of
mathematical agency in forms not previously experienced
was felt with enjoyment and pleasure. These aective shifts,
we argue, are as signicant to the resignication process as
the discursive shifts related to agency and interdependency
discussed in the other two themes.
We also observed how those acting as the student seemed
able to sense how their teachers were feeling, even though
they couldn’t see them. Iara’s awareness that Alberto was on
the point of desisting aected how she felt in the moment.
It also provoked her to identify with those students who
might have diculty in appropriating mathematical ideas
the visual eld, leading to shifts in their evaluations of the
mathematical validity of such methods.
Interdependence. Another issue that was raised in post-
task discussions was the extent of teacher help that was
legitimate. We saw in the Alberto/Iara episode, Iara jok-
ingly suggesting that Alberto communicated the answer at
one point. In fact, the teacher directly giving the answer
to the student was not observed in any of the Type-II ses-
sions. Much more common was an initial expectation that
the students would do the calculation on their own. The
realisation that, under the conditions posed, it proved very
dicult for students to succeed without support, provoked
dierent reactions from those assigned to act as teachers.
In three extreme cases, no help at all was oered, except
to encourage and indicate correctness of particular calcula-
tions. In none of these cases was the result of the calculation
obtained. In all other cases, upon perceiving diculties, the
teacher helped. In a small number of cases (in six of the 27
sessions), and always involving the algorithm, the teacher
essentially took over, guiding the student through each step.
Observers of these cases reported that, although the students
correctly performed the steps, they could not tell if they
were aware of where they were in the algorithm (in three
cases, those acting as students agreed later that they were
not). Indeed, in contrast to the shared joy evident in Figs. 4
and 8, expression of satisfaction on solving the task in these
cases tended to be muted, with the teacher generally more
pleased than the student (Fig. 9).
It could be argued that students’ independence had been
restricted by the communication conditions we imposed on
the task, and that the relative lack of enthusiasm of the stu-
dents when the teacher took over reected an over-reliance
on the teacher. We oer an alternative interpretation that
challenges culturally sanctioned meanings which posit inde-
pendence as desirable and dependence as a form of weak-
ness. Disabled activists and critical disability scholars have
decried the ableist myth of independence (Goodley, 2020),
pointing to how all human beings are dependent on each
Fig. 9 Raquel, in the role of teacher, seems more satised than Arthur
1 3

Interdependency, alternative forms of mathematical agency and joy as challenges to ableist narratives about…
own senses of mathematics and its teaching and learning
have been strongly aected by the intimate connections of
emotion and cognition played out in the research activities
we have been involved in. By restricting participants from
using the resources that they are accustomed to use to com-
municate and solve a mathematical task, the Type II tasks
invite teachers to temporarily inhabit a body whose capaci-
ties are dierent from their own. Our evidence suggests that
this experience can encourage them to feel for themselves
how inclusion and exclusion might be sensed by their stu-
dents. We hope too that the analyses presented in this paper
open a window onto our feelings about inclusion, and about
how– and why– rather than deciency, we view dierence
as potential, renewal, change, resistance, and inspiration.
Acknowledgements CAPTeaM is an International Partnership and
Mobility project between institutions in the UK and Brazil funded
by the British Academy (Awards: 2014-15, PM140102; 2016-21,
PM160190) and, as part of the MathTASK programme, by the UEA
Pro-Vice Chancellor’s (PVC) Impact Fund since 2015. We thank
CAPTeaM researchers (Brazil: Solange Hassan Ahmad Ali Fernandes,
Leiliane Coutinho da Silva Ramos, Gisela Maria da Fonseca Pinto,
Érika Silos de Castro and Aline Simas da Silva; UK: Gareth Joel, Lina
Kayali, Elizabeth Lake, Angeliki Stylianidou and Athina Thoma) and
participants for their commitment to the project.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format,
as long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indicate
if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless
indicated otherwise in a credit line to the material. If material is not
included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted
use, you will need to obtain permission directly from the copyright
holder. To view a copy of this licence, visit http://creativecommons.
org/licenses/by/4.0/.
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