abstract - era.library.ualberta.ca · web view, an active love, where one strives to do no harm in...

Running head: EXPERIMENTS IN AHIMSA

Mathematics Classroom Experiences and Ahimsa:

Love and Harm in the Mathematics Classroom

Master Final Project

Jayne Powell

University of Alberta

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Abstract At the core of Gandhi’s teachings is the practice of ahimsa, an active love, where one

strives to do no harm in the world, through neither word, thought, nor deed. In educational

practice, teachers would say that they strive to do what is ‘best’ for their students within their

capabilities, yet stories of harm pervade the stories of mathematical educational experiences. In

my research, I conducted a close reading of sixty grade nine student mathematical reflections on

experiences, with Gandhi’s notion of ahimsa as a framework from which to consider themes of

love and harm in relation to mathematics education. This was done in the hopes of better

understanding how I as a teacher can continue to move towards a more loving pedagogy, while

perhaps slowly helping to change the cultural narrative of mathematics education to a more

positive story. This study concluded that there is evidence of potential harm to students, such as;

being intensely focused on grades, having conflated being the ‘fastest’ and being the ‘best’,

viewing learning math as memorizing many sets of steps, believing that they have learned a topic

but merely ‘forget’ it on assessments, not feeling confident to ask questions, being acutely aware

they are not learning at the current pacing, and being singled out. It also showed students

benefiting from tying grades and speed to more than just end goals in and of themselves, setting

goals with metacognitive awareness, and by feeling that their teacher both wants them to and

works towards them succeeding.

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Table of ContentsAbstract.................................................................................................................................................... 2

Table of Contents................................................................................................................................... 3

Introduction............................................................................................................................................. 4

Literature Review.................................................................................................................................. 5On Ahimsa........................................................................................................................................................... 6On Love................................................................................................................................................................ 9On Harm........................................................................................................................................................... 11

Method................................................................................................................................................... 18

Findings and Discussion..................................................................................................................... 22Measured Performance.................................................................................................................................. 23

On Grades.......................................................................................................................................................................... 23On Speed............................................................................................................................................................................ 28

Approaches to Learning................................................................................................................................ 32On Memory....................................................................................................................................................................... 32On Awareness of Learning..........................................................................................................................................39On Not Understanding...................................................................................................................................................41On Asking Questions.....................................................................................................................................................45

Direct Teacher Interactions.......................................................................................................................... 47

Conclusion............................................................................................................................................. 49Implications...................................................................................................................................................... 55

References............................................................................................................................................. 58

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IntroductionI teach students many things through working with mathematics, but I have always been

hired to teach them mathematics. I acted as the mathematics specialist for six years in a high

school before the professional growth opportunities offered by my district were not helping me

answer my larger questions. I decided that to get the ‘answers’ to my questions I needed by

going back to graduate school to study full time. Surely in the hallowed halls of academia

someone would have the ‘answers’ I was seeking.

During my time in the program offered at the University of Alberta I came to realize

there were no ‘answers’ in the form I was expecting. However, I learned to think of and ask

questions differently, and to see the value in different forms of answers. These kinds of

‘answers’ would never be easily defined, nor would these answers ‘fix’ education; but, they

might help the educational world move towards a more positive place.

I realized that the question I really came back to school to ‘answer’ was how I could

impact students and school in a more positive way. The ‘answer’ I found in what I was reading

and thinking was to reflect on my approach to pedagogy. I should be continually (re)asking what

is important, and what does and what should have value. In the middle of considering positive

change as coming from within and not something that could be implemented from the outside, I

came across into Gandhi’s view of ahimsa. He appropriated the word from the world of religion,

which played a large part in his life, specifically Jainism, Hinduism, and Buddhism. Ahisma is

often translated as nonviolence, but can also be translated as to not harm (Parekh, 1988).

Gandhi had a radical take on ahimsa; he considered ahimsa in the positive as love, to not

do harm in word thought or deed (Parekh, 1988). This idea of love as not doing harm on any

level resonated within me; this is what it means to love. A loving pedagogy is first and foremost

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an attempt to do no harm in word, thought, or deed. Because my previous teaching experience

mostly occurred in the context of mathematics, I began to consider love and harm in the

mathematics classroom. There seems to be a cultural narrative that having to learn mathematics

is harmful, at least to those that don’t feel they are ‘good’ at mathematics. When talking to

people about what it is that I do, they feel an incredibly strong need to tell me either how good

they were at mathematics or the stories of their mathematical scars. There is slow to the flow of

conversation until they are able to explain to me their mathematics education story. It seems that,

to continue our conversation, their stories of mathematical experience must be told first; to them

it is the only way I, as a math teacher, could come to know them. Many people seem to feel

harmed by their mathematics education. These stories of their ‘war wounds’ have led me to

wonder about harm, mathematics, and love as seen in people’s memories. My question(s) sprang

from these experiences of mathematics education and the notions of love and harm in Gandhi’s

ahimsa.

During my research my thinking will be guided by the question: When grade nine

student’s memories of previous mathematical experiences are considered through Gandhi’s

notion of ahimsa, what indications of love and harm emerge?

This question is complex, and through considering these questions sub-questions emerge,

such as: What is harm? What does it mean to be harmed by mathematics education? How can

someone be harmed by mathematics education? What is love? What does love look like in

mathematics education? What does practicing ahimsa look like in mathematics pedagogy? These

questions will focus my unpacking of love and harm in mathematical education experiences, so I

can consider what Gandhi’s ahimsa looks like in the mathematics classroom.

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Literature ReviewThe stories of mathematical experiences in school elicit strong emotional responses in

individuals. As a way to start interpreting these experiences, I am drawn to Gandhi’s notion of

ahimsa, to not do harm in word, thought nor deed. By considering ahimsa as a framework has led

me to ask the question: When grade nine student’s memories of previous mathematical

experiences are considered through Gandhi’s notion of ahimsa, what indications of love and

harm emerge?

In researching and writing this literature review, I am interested in starting to unpack how

to approach Gandhi’s work, and the ideas and writings surrounding ahimsa. In entering into

discussion with the ideas of ahimsa I will also have to consider both harm and love. Much is

written about violence or harm in education; in mathematics education, this is often named as

mathematical anxiety, but there is less written about love, it is often written about as empathy, or

caring, but the word love is rarely used. This can be seen in Noddings’ (2013) work on the ethic

of care. When using Gandhi’s notion of ahimsa as a framework it is as important attempt to look

for indications of love in education, as well as harm and violence.

On Ahimsa.Parekh (1988) outlines Gandhi’s notion of ahimsa within Indian tradition, through

discussion of his writings and examples throughout his life. The author concludes that Gandhian

philosophy is too broad to know exactly what Gandhi’s response would be to today’s problems;

however, there are reoccurring themes such as decreasing violence, taking action, and loving as

much as is possible within very complex situations and systems. This article effectively places

Gandhi in the traditions in which he based his ideas on, and provides an outline Gandhi’s version

of ahimsa, within multiple contexts, including some of his writings and actions. Indian traditions

prior to Gandhi “did not assign active love an important place in their moral theory; for Gandhi it

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was the highest moral value. Not surprisingly he took over the concept of ahimsa and defined it

in a radically novel manner” (Parekh, 1988, p. 198). Parekh (1988) discusses Gandhi’s ahimsa as

revolutionary, because he does not hold with Indian traditions that attempt to not cause violence

through inaction or removing oneself from the world. Instead, Gandhi believed that active love,

in the positive, was necessary to promote the well being of all. It is compassion and active love

that decrease the violence and harm in our world.

Although the author ultimately feels Gandhi should have set up a more rigid morality for

it to be of any practical use, I see the intentional flexibility of Gandhi’s philosophy as what

makes it powerful. It is not about clearly defining right and wrong, and censoring those who fall

outside the lines. Gandhi’s thinking on the notion of ahisma is broad and adaptive to the

situation, which necessitates critical thinking and love to be placed as important over clearly

defined reactions for every situation. This includes situations within education, for “ahimsa is the

attribute of the soul, and, therefore, to be practiced by everybody in all the affairs of life. If it

cannot be practiced in all departments it has no practical value” (Gandhi & Attenborough, 2000,

p. 36).

Parekh (1988) also makes it clear that Gandhi could ‘understand’, ‘excuse’ and ‘pardon’

minimal violence, but for him any use of violence or harm could never be justified. For Gandhi,

the ideals of ahimsa must remain absolute: there can be no justification for violence, it is

important to try to live up to your ideals. Therefore ahimsa is a practice, that could never be done

or complete, it is a constant way of being in the world. If a person should stumble, not living up

to the ideal, it is not cause to become demoralized but, according to Gandhi, should illicit feeling

of possible improvement. By this thinking, a teacher can not live up to an ideal all the time;

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however, by striving to be better and to be someone who cares, they would be moving closer to

love and practicing ahimsa.

When reading work that takes up Gandhi’s ideas, authors use his work in vastly different

ways. Some, such as Parekh (1988), try to use his work as a tight moral guide, while others, such

as Allen (2007) use what they consider to the overarching intention of his work. Allen (2007)

rejects both essentialist and non-essentialist readings of Gandhi’s work and believes both

approaches reduce the value of Gandhi’s work by either making it too rigid and structured or too

open and not useful. He believes there is a middle reading that does not look for a static truth but

is concerned with finding truth that can offer insights to help us interpret violence and peace

education. All of Gandhi’s ideas surrounding ahisma, are tied closely with ideas of Truth.

“Without Ahimsa it is not possible to seek and find Truth. Ahimsa and Truth are so intertwined

that it is practically impossible to disentangle and separate them. They are like two sides of the

same coin, or rather of a smooth unstamped metallic disc” (Gandhi, 1954, p. 8-9). Truth for

Gandhi was an absolute, and by seeking Truth through Ahimsa, he was seeking God.

Considering Allen’s (2007) perspective to take up Gandhi’s ideas, I should neither dismiss these

discussions surrounding Truth and God, nor should I feel I have to use these ideas as a rigid

frame. There is a middle reading where Gandhi’s notion of Truth, as absolute, can offer insights

into my work.

Allen (2007) takes a hermeneutical orientation in reading Gandhi where there is no ‘true’

or static view of Gandhi’s philosophy, how writers or researchers interpret Gandhi is wound

together with their current world and situation, and this is unavoidable. This state does not have

to be overcome because every reading of Gandhi should be recognized as a (re)reading. Allen

(2007) finds his research focus on violence and peace education not in Gandhi’s specific writings

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about education, but in Gandhi’s larger philosophical orientation and framework. Using Gandhi

as a framework or as a lens does not mean I need to fit experiences into predefined philosophy

boxes: interpretation is possible, allowable, and unavoidable.

To think of improvements in education as setting up frameworks to continually ‘fix

problems’ until eventually arriving at a utopic scene is setting oneself up for disappointment.

According to Indian philosophical tradition, to live is to cause violence; there is no escape from

this violence as even killing oneself to be removed from the world would be an act of violence.

According to Gandhi, our duty is to do as little harm as possible, in the active sense, by loving

one another and the world around us. Gandhi does not promote a system where all the answers

are given, he leaves it to each individual to consider and discuss how to decrease violence or

harm in their own sphere. For teachers, becoming a more loving teacher means reflecting on

one’s own practice and making changes based on who one is, who one’s students are, and what

one’s context promotes. These actions, in turn, will decrease harm and violence. “Gandhi often

remarked that it was not his job to tell his followers how to behave, and that each of them must

sincerely work out for himself how best to practice ahimsa” (Parekh, 1988, p. 216). This is not a

one-size-fits-all ‘fix’; in fact, it is not a ‘fix’ at all. To practice ahimsa is to slowly progress

towards greater love and thus, hopefully, decreased harm.

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On Love.Ahimsa is the greatest love of all, and thus all writing surrounding ahimsa should be

considered writings on love. However, I also want to look at how love is taken up in educational

literature. It is important to call out violence and the harm that it creates, but it is also important

to focus on love and all the positive things that happen in mathematical educational experience.

These include stories of accomplishment, stories of positive change, and stories victories, large

and small. Thus I ask, “What is love?” and “What does love look like in mathematics

education?”

Noddings (2013) researches ideas surrounding the notion of caring, which are deeply

entwined with my questions surrounding love. She expertly unpacks the notion of caring: caring

for, caring about, and caring’s negative and positive connotations through hypothetical case

studies. As Noddings (2013) writes, “It may be that much of what is most valuable in the

teaching-learning relationship cannot be specified and certainly not pre-specified. The attitude

characteristic of caring comes through in acquaintance” (p. 20). In the mathematics classroom,

love and caring come through the relationships of the people involved. So much time and

discussion surrounding mathematics education deals with the content, the progression, the

assessment, and the resources, and although these things are important and valuable

conversations, they sometimes overshadow the reason all these things are important, the student.

It is in relation, the places between people, that love is found.

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In education the word love rarely is spoken. It eludes to a crossing of professional and

ethical boundaries, however Loreman (2011) feels that learning happens best with relationships

that are “intimate, safe, caring, and warm” (p. 1). He adamantly names this type of relationship

as love. Through psychological, religious, and philosophical frameworks he argues that learning

should not be learning to comply through conditioning. That by “using love as pedagogy is an

antidote to superficial learning” (Loreman, 2011, p. 8). Learning should not be training, but

education, and becoming a more critical thinker, problem solver, and empathetic person.

Loreman (2011) draws heavily on the work of Cho (2005) who argues that “love has the

power to inspire students to seek after knowledge, love can unite the teacher and student in the

quest for knowledge, and the love of learning can even empower students to challenge

knowledge thereby pushing its limits” (p. 79). Love involves kindness, empathy, intimacy,

bonding, acceptance, and community. Both Loreman (2011) and Cho (2005) argue that passion

infuses all aspects of love. Passion for students, passion for learning, passion for ones subject,

passion for the future, passion for social justice, passion for hope, passion for change, passion for

diminishing harm, passion for leaving the world a better place then you found it. Here I find

love. But can an institution like school be loving? Can you see love in the mathematics

classroom? I would say yes, although school is a structure, is a structure built of people and it is

through the relationships of these people you can see ahimsa, you can see love.

On Harm.By considering mathematical educational experiences through the framework of ahimsa,

two major ideas need to be unpacked: love and harm. Although much written about ahimsa

phrased as non-violence instead of non-harm, and although I at times use the word violence, I

choose to ask my questions from the perception of harm. Violence is the action that is done,

while harm considers the act and the legacy of that violence. I choose to ask, “What is harm?”

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instead of “What is violence?” because I am interested in the on-going struggles as well as the

initial act(s). Thus, I am also asking, “What does it mean to be harmed by mathematics

education?” and “How can someone be harmed by mathematics education?”

Few people would disagree that they entered the profession of teaching hoping to make a

positive impact in students’ lives; yet, students are harmed. Some of this violence is born of a

mainly rigid system looking to produce students who conform to their version of an acceptable

citizen; but additionally, classroom teachers are daily perpetuating harm. Sihra & Anderson

(2009) look at the social violence perpetrated in the classroom, mostly unwittingly, by teachers:

they focus on dsyconsciousness, arrogant perception, and normalization as forms of violence.

Dysconsciousness is presented as an unwillingness to see social violence and the justification of

inequality and exploitation by accepting them as ‘the way things are.’

For Sihra & Anderson (2009), it is of the utmost importance that we as teachers examine

our own lenses and to find out who and what our lens renders visible and invisible. Arrogant

perception is presented as the inability to identify with another person “leading one to ignore,

ostracize, render invisible, stereotype, isolate, or interpret as crazy those who are perceived as

different from the self” (Sihra & Anderson, 2009, p. 381). Finally, Sihra & Anderson (2009)

discuss the normalization of dysconsciousness and arrogant perception of inequity as violence.

They call for a noticing of the violence and the adoption of ahimsa, the largest love, and

philosophical humility so as to realize that there are things that we cannot know.

Sihra & Anderson (2009) highlight the importance of noticing violence. Although

dwelling on the undesirable can seem to cloud a world negatively, it is also important to notice

and give voice to harm that has been done. Noticing and then responding to this harm, so it might

not happen again. By noticing, contemplating, and discussing, we might push violence into the

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light, otherwise the harm will continue to be reproduced in the shadows. Although the discussion

of harm in my work will be entwined with love, it is important not to shy away from experiences

that demonstrate harm done to someone because of my profession. Although harm might not be

done purposefully, it happens and it is important to ask why and how to diminish these negative

experiences for future students.

Joshee (2006) is also interested in noticing violence. In her article on peace education she

discusses what is means to be peaceful and places peace education within Ganhdian pedagogy

and three sub-ideas of ahimsa: “sarvodaya (the uplift of all), satyagraha (the power that comes

from acting in ways consistent with the principles of ahimsa), and trusteeship” (p. 7). She

discusses ways of assisting peace education with three pedagogic ideas and discusses her

personal experiences working against structural violence.

Structural violence is rampant in the mathematics classroom as it is in all classrooms; the

world is not left outside if we merely close our doors. Teachers and students should be aware of

these types of violence and how to hopefully engage and reduce them. Although social violence

is pervasive and woven into our world, to attempt to call them out and to address them is

important: it is a form of love. Although I am not directly discussing peace education, there may

be overlap that exists between this work and that of peace educators. We are both concerned with

recognizing structural violence and using our own lenses to move towards a more loving

pedagogy or ahimsa.

Labaree (2011) discusses the possibilities for a Hippocratic oath for education. First and

foremost, teachers should strive to do no harm. Although straightforward, doing no harm is

easier said then done. The notion of harm is complex; what is wonderful for one is harmful for

another. Current positive reforms can become future systemic issues. I see this complexity

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through a childhood memory. On a hot day in early fall, as I can clearly see the golden straw

fields, post harvest around my home. I was playing near the old red barn when one of our

families’ cats caught my eye. He was batting at something happily in the long grass. As I moved

closer to investigate I discovered the cat was toying with a small mouse. Without thinking, I

swooped down and picked up the tiny creature in my hands, thinking to myself what a nice thing

that was for me to do for such a small helpless creature. How grand I was. The mouse in

response to my magnanimous action bit me. Hard enough to make me cry out, “Fine, if you want

to be that way then I will just let the cat have you”. I placed the mouse back on the ground in

front of the cat, the pride I had felt in my supposed savior hurting. I then turned my back and

walked away. Here the memory of the mouse ends for me, and this sudden ending continually

haunts quiet moments. Over time I have constructed multiple possible endings to this tale. In

one, I race and get my father’s work gloves and, now safe from the mouse’s bite, carry it to

safety. In another, the guilt engulfs me and I turn around and change my plan of action to

removing the aggressor in this situation, the cat. In another, I bravely pick up the mouse again

and move her to safety, for she did not know how to ‘act any better’. Yet, in another version of

the conclusion, I simply walk away, perhaps it is not my place to get involved. This is why this

memory haunts my silent mind; what action or non-action did I take that day? What was the

conclusion to this story for the mouse, the cat, and for me? What has been constructed in my re-

living and what actually occurred? Did I let my damaged pride get the better of me? Did I do the

‘right’ thing? What was the ‘right thing’? Because mice are considered pests on the farm, was

the cat merely fulfilling its purpose? Did I interfere where I did not belong? I do not know, nor

will I ever, but I have lived it though a thousand times never coming to a conclusive ‘best’ action

or non-action.

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So I ask, what is harm and what does it look like in the mathematics classroom and

mathematical educational experience? Harm in mathematics education literature seems to be

often described in terms of the creation of mathematical anxiety, although the word harm or

violence is rarely used. Jackson & Leffingwell (1999) examine how instructors’ behavior

contributes to or exacerbates anxiety in mathematics’ learners. By reading pre-service teachers’

responses to their question, “Describe your worst or most challenging mathematics classroom

experience from kindergarten through college.” Jackson and Leffingwill (1999) found three

focused clusters of increased anxiety: (1) Elementary level, especially grades three and four; (2)

High school level, especially grades nine through eleven; and, (3) College level, especially

freshman year. They analyzed their data by breaking it into common themes and analyzed

whether these themes were overt of covert. Their analysis concluded that there were eight

specific things teachers could do to reduce anxiety: sharing their previous anxiety, project their

love of the subject, offering extra help, making a pervasive mutual respect rule, assisting in the

review process before exams, and offering alternative testing times.

There is a large overlap with Jackson & Leffingwell’s (1999) work on instructors’ roles

of contributing to feeling of anxiety and my focus on teacher’s role in creating feelings of harm

and or love in mathematical education experience. However, there is no tie to theory. Although

my work will be similar, my use of Gandhian theory to discuss and unpack the data will possibly

extend some of the ideas discussed in this article.

This creation of mathematical anxiety through negative experiences is tied up in issues of

equity. Multiple articles discussed in this literature review, Shira & Anderson (2009), Joshee

(2006) and Labree (2011), have made me realize that I could be writing more about equity in

mathematics education experience then I initially thought I would be. The aim of equity is not to

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just make everyone into economically productive workers but also into critical citizens and

thinkers, capable of enacting positive change. Studying mathematics can help learners become

more socially empowered and can assist in developing agency.

Mathematical understandings allow access to cultural capital otherwise not available. To

be mathematically literate, to be seen as mathematically literate, and to consider oneself

mathematically literate allows learners a type of emancipation. “Mathematics has long been a

tool of subjugation, but it can be turned into an instrument of empowerment. It can enable

individuals to see through the inequalities and exaggerations that are so often cloaked in a

mathematics rhetoric” (Davis, 2001, p. 18). Learning mathematics can deepen one’s awareness

of the world. Whether our preferred paradigm or not, we live in an age in which scientism reigns,

and scientism is based upon mathematical understanding. Those who can understand and use

mathematics have a cultural capital which gives them power over their world. Mathematical

literacy is vital in the struggle for social change in our technological society (Davis, 2001).

Everyone needs to understand and use mathematics to be able to reject false or misleading

mathematical claims Mathematics is commonly seen as a mostly finished set of procedures and

formulas which leads to a collective understanding that mathematics knowledge is un-

challengeable, constructed, and used by an elite group of thinkers. Yet, those who are

mathematically literate come to see mathematics as more complex, accessed and used by more

than those in positions of power. Davis (2001) argues that, to use mathematics for social equity

portrays a mathematics that is useable, powerful, accessible, and do-able. This mathematics is

not less rigorous but is a mathematics more focused on critical thought than on memorization of

algorithms. This mathematics is aimed at allowing another way to honestly view our world and

consider change.

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As prominent critical mathematics curriculum thinker Frankenstein (1997) wrote: “With

hope, study, and persistence, we continue, trying to connect our work to the struggle for justice,

against inequalities” (p. 21). Those of us involved in education must not allow ourselves to be

resigned to the immutably seeming way things are. Upon entry into classrooms in the role of

teacher instead of student, it is easy to fall into teaching the way we were taught. Reproducing an

educational experience similar to our own makes a kind of sense: it “worked” for us; therefore, it

can “work” again. This method merely serves to reproduce society; yet, “knowledge is not a

copy of reality, but a process of construction” (Piaget, 1978 [1967], p. 27). Although as teachers

we can rely on what we have experienced and what we learned during our education, it is

paramount for our profession to question our role in the system. Do we merely want to reproduce

the system, or enact positive change? I see this change coming through a socially aware

pedagogy of love that hopes to help build a more equitable and just society through learning and

teaching mathematics.Thus, considering mathematical educational experiences, harm is not only

having a negative experience, often born out of the structural violence discussed in Sihra &

Anderson (2009) and Joshee (2006), nor is it merely the development of mathematical anxiety as

considered by Jackson and Leffingwill (1999). Harm is also caused by diminished opportunities

to take up mathematics in meaningful ways so as to become mathematically literate, leaving the

structures of inequity that are masked in a mathematical dialect mostly unchallenged and

unchanged.

Mathematical educational experiences present me as a researcher an entry point into

considering ahimsa in the mathematics classroom. It is in relationships, and the small things that

are done everyday that students may experience ahimsa. In entering into discussion with the

literature, I have come to see that it is important to interpret Gandhi’s theory for the current

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situation, while not diminishing it by only partially taking up his ideas. To write about ahimsa, I

will have to write about truth. Also, harm is very complex and that harm in the mathematics

classroom occurs both through negative experiences, as well as not developing mathematical

literacy. Also, ideas surrounding love are important and powerful. Allen (2007) explains the

larger value of experiences of love when he writes that:

Nonviolence is a powerful bonding and unifying force that brings us together in caring,

loving, cooperative relations; this allows us to realize and act consistent with the

interconnectedness and unity of all life. Violence, in contrast, maximizes ontological

separateness and divisiveness and is based on the fundamental belief that the other…is

essentially different from me or us (p. 302).

Gandhi calls for everyone to practice ahimsa in all the affairs of life. Perhaps experiments in

ahimsa can act as seeds for other individuals to practice ahimsa, knowingly or not. Loving

actions seeding positive change.

We have not been able yet to discover the true measure of the innumerable properties of

an article of daily use like water. Some of its properties fill us with wonder. Let us not,

therefore, make light of a force of the subtlest kind of Ahimsa, and let us try to discover

its hidden power with patience and faith. Within a brief space of time we have carried to

a fairly successful conclusion a great experiment in the use of this force. As you know I

have not set much store by it. Indeed I have hesitated even to call it an experiment in

Ahimsa. But according to the legend, as Rama’s name was enough to float stones, even

so the movement carried on in the name of Ahimsa brought about a great awakening in

the country and carried us ahead. It is difficult to forecast the possibilities when men with

unflinching faith carry this experiment further forward. . (Gandhi, 1954, p. 154 and 155)

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Method

As it is strongly suggested that the data used for a master level project should not require

the completion of the ethics process, thus I went looking online for publically available data.

Online I found one hundred and twelve grade nine student responses to three questions about

their mathematical experience. These students were instructed to write a blog post to their

teacher discussing their math experience as a way to introduce themselves It was indicated it was

to be four paragraphs long and should answer three questions:

1.) Do you consider yourself to be good or bad at math?

2.) What have been your good and bad experiences with math?

3.) What do you hope to accomplish this year in your math class?

The initial post from their teacher was made in August 2012, and the student responses occurred

from the beginning of to mid September. These are rich responses where the students shared a

multitude of experiences and feelings towards mathematics. Little can be learned about the

background of the students who posted their stories to this site based on their responses. In their

writing there are multiple references to their goals for grade nine, but they do not write about the

school they are at, or what state or province in which they reside. By knowing the teacher’s name

and the domain name of her email I was able to track this data set to a school. On this school’s

current website there is also a math teacher of the same name as the blog post teaching grade

nine math, so I feel confident that I located the school that matched my chosen data set and that it

is authentic, in the sense that students did write the entries.

From their website I learned that this is a grade six to twelve school, with class sizes of

about twenty-five students and one hundred students per grade level, that prides itself on

fostering an academically challenging, student-centered environment. The ‘about’ section of the

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school also stated that these students would have been in this teacher’s class for the first time,

and would have her not only for grade 9 math, but grade 10 math as well. This school defines

itself as an academically challenging program, where close to one hundred percent of their

graduates go on to postsecondary studies. Therefore, these responses are from a rather

homogenous group of students who have sought out this specific school, have the resources to go

through a lengthy acceptance process (including; creating a portfolio, writing entrance tests, and

an interview), and live close enough to commute to Manhattan daily.

I have chosen to refer to the students through their blog post number and not their name,

as although these posts were shared publically, I still would like to protect the student’s

anonymity.

Once these blog posts were located my first act was to read through these student’s

writings, considering their reflections in relation to ahimsa. I read about twenty posts and some

obvious possible codes started to emerge, such as grades, memory, comparing themselves with

others, metacognition, defining themselves good or bad at math, and some very specific positive

and negative memories. The emerging themes observed would be influenced by my theoretical

framework of Gandhi’s concept of ahimsa, specifically love and harm. After the initial reading I

used NVivo, a digital data analysis tool, to help organize and code the data. This tool allowed me

to organize the coding process, search, and easily adjust codes. As I read through these student’s

experiences, I heard the voices of students I have had in my own classrooms. Although these

pieces of writing are not from students I have taught, they seem to have had similar experiences

and views.

The codes that were given to different sections evolved organically as I attempted to

notice love and harm in these student’s experiences. I gave each post a general code for their

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expression of their mathematical experience, if it seemed overall positive, mixed, or negative.

Some of the first codes used included grades, which were then broken down into smaller

categories of final, assignments, and tests (which included students writing about tests, exams,

and quizzes). As I coded through more data the node of assignments seemed to not be necessary

and was removed. Other initial codes included comparing self to others, speed, goals, referencing

specific math topics, memory, metacognition, specific negative memory, specific positive

memory, practice, race, and problem solving. These codes changed as I read more posts; for

example, ‘memory’ got a specific sub section for students discussing needing to memorize steps,

and the code ‘negative moment’ gained two main subsections for catching up after being absent

and not understanding a topic. How the codes ended up being grouped together is shown in

Table1. The themes that emerged from the codes are across the top row. Below each theme is the

codes that were compressed to make up that theme. As I worked with these codes to create

themes, I considered similarities and overlap in the student’s writings.

Table 1.

Codes Used in Analyzing Data

Overall

Attitude

Speed Grades Memory Metacognition Negative

Moments

Positive

Moments

Specific

Math

Topic

Mixed First one

done

Assignments Failed memory When ‘stuck’ I… Being absent Teacher

Neutral Compare

Self to

others

Final Grades Memorization Practice Not

understanding

Singular

Experience

Positive Tests Developed Goal

Setting

Singular

Experience

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Negative Compare self

to others

Grade Goals

After coding about forty entries I was finding that no new codes were being produced. I

then coded twenty more. While coding this group, no more new codes emerged, so I ended up

coding sixty of the one hundred and twelve posts.

Next, I considered the codes that had emerged and grouped them into three categories:

measured performance, approaches to learning, and direct teacher interactions. I went back

through and reread the coded sections and transferred them to a word document, being careful to

catalog the number of the student who had written each quote, so if I needed to refer to the entire

entry at a later date, it would be possible. I then printed all of the chosen quotes and cut them out

individually. Having each coded quote on a separate piece of paper allowed me to manipulate

them and organize them into groups, and allowed me to construct a flow of ideas from one to the

next. I also used a pen and a few highlighters to record thoughts I had while organizing these

paper slips. Through this process, many of the quotes that I valued were discarded as they either

duplicated ideas, or did not best illustrate what I was seeing. The remaining quotes and ideas

were then transferred to a word document where I analyzed and discussed these reflections. I

used more direct quotes then paraphrasing as I believe that having the student’s own voice is

powerful when writing and reading about personal memories. I often chose to correct grammar,

punctuation or spelling when I used direct quotes from students as to not distract the reader.

However, when correcting errors in sentence structure, spelling, or grammar seemed to change

the voice of the student, I left the quotes unchanged.

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I approached this analysis as a previous student and current teacher. The findings and the

discussion are biased by my experiences. This bias is not necessarily negative as having worked

with students whose voices I hear in these reflections. I am empathetic and have an

understanding that I would not have without this bias. As I am writing this project for myself and

possibly other teachers, therefore, these experiences that color my reflections is what connects

this work to the intended audience.

Findings and DiscussionMany factors can influence a person’s school experience, from the individuals they

encounter, to their location, the institution, political climate, as well as many socioeconomic

aspects. In the findings and discussion of these student experiences, the student reflections have

caused me to direct my attention to the love and harm students are encountering that are the

result of systemic factors of the education system. Other experiences of love or harm appear to

be caused directly by a single teacher or experience and also by a practice or tone in instruction.

In considering these student memories of previous mathematical experience through a

framework based on Gandhi’s notion of ahimsa, it is important to recognize that the practice of

ahimsa has two phases. Firstly, one needs to reflect, notice and name harm and love in

educational practice. Then actions need to be taken to decrease harm and grow love. While both

phases are important to a pedagogy based in ahimsa, this study focuses on the first phase,

noticing.

Measured Performance

On GradesOne of the first themes to develop when reading through these remembrances of

mathematical experiences was how often and how emotionally students reflected on grades.

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Grades were sometimes discussed in the context of something larger, as a result of an action,

either their own or something done to them. However, many students spoke of grades as a static

fact about themself, not tied to anything beyond a number or letter. Their grade was the defining

statement of their overall mathematical experience.

None of the questions directly asked students to specifically write about grades, yet

nearly all of the student responses referenced grades at least once. Without direct prompting,

their previous grades are listed as their answer to the questions posed by their teacher about their

previous good or bad experiences with math, such as, “My best experience is getting straight A's

on all my homework assignments for the third trimester of 8th grade” (s1). This student’s grade

on homework assignments is their best experience after hours and hours or work and experiences

both in and outside of the classroom. Another student wrote that their good experience was that

they were “one of the few people in [their] school to get 4’s in the [state] math exams repeatedly,

grade after grade” (s6). These remembrances seem positive but are not tied to how or the why

they were successful on these measures.

For a number of students, grades seem to be a litmus test for how ‘good’ or ‘bad’ they are

at mathematics. “My bad experience is that most of the time I don’t understand math. I have

never received an A on math. I have always been one of the worse or worse person I know in

math” (s19). In this student’s personal narrative, she merely names an inability to score A’s to

prove she is ‘bad’ at math. She assumes the grade speaks enough to communicate her challenges

without reflecting on or speaking to how, or why this might be the case. While another student

lists grades as the confirmation that they are ‘good’ at math: “I consider myself good at math. I

think I’m good at math because I usually always get grades of high 80’s and sometimes 90’s”

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(s35). Students also expressed that they feel pressure from their parents who seem to have similar

ideas that their child’s grade is the element of learning which should be the focus.

In almost in all of my state test I always got a 2 or a 3 and my mom tells me to always get

a 4. iTS NEARLY Imposibe to get a 4 because you have to try hard and do everything i

had bad and good expieiences in math a good expirence in math was that in 7 grade i got

a 3 in the math state exam i never believed that i was that good in math [all grammatical

and spelling errors in original] (s17).

It seems this grade, on a singular state exam, has changed how this student sees themself in

relation to mathematics. It is almost unbelievable to them that they could do well. I wonder what

happens next to this student if the grade on the subsequent test is not as good? Do they again

change how they see their mathematical ability, or will they be more resilient after this success?

For another student, their grade seems not tied to their knowledge or work, nor are they

an entity that is predictable, or over which they have power.

As a student I have never really had any problems with math during class but when it

comes to tests my grade can go from a 90 to a 75 in a blink of an hour. It does not matter

what subject of math, if it is a test or a quiz it always seems to hurt me. But otherwise I

consider myself to be pretty good at math (s55).

This student has separated tests from the day to day of math class. Although they seem to

routinely not perform as well on tests as homework, their belief that they are ‘good’ at math is

unfazed. Writing that the test ‘hurt’ them, by decreasing their overall grade. The use of the word

hurt here is visceral, and although it means a decrease in a grade, can also be interpreted as

possible emotional harm. This harm is instant with the decrease of his grade but also to his

continued education. He does not seem to have developed an understanding that a grade should

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be a reflection of his understanding. Without this connection he is powerless to affect change to

this meaningful symbol.

Grades are so pervasive throughout school culture that even a student who had never

been assigned a number or letter grade before has strong feelings about what grade would be

acceptable.

[I]n terms of grades, I don’t know what to expect. This is because in my teeny-tiny

middle school, we had personal narratives instead of grades, so I don’t know what my

average is. What I do know is that if I get below a ninety, I will be angry with myself

(s2).

This student, who has never been assigned a grade previously, will be angry with himself if he

does not meet this rather arbitrary acceptable level. This level has not been determined based on

previous experience with grades, but what the culture of grades deems a level of excellence.

Grades are motivating for some students, often those who score well on exams, even if it is not

all exams. Others are disheartened over and over, and come to define themselves as ‘bad’ at math

and continue to integrate this impression into their narrative.

A student’s grade is held up to such a level that students are willing to cheat to achieve a

higher mark.

I was so desperate to pass a test and was caught cheating. It was the worst time in my

life. My teachers didn’t call my parents and just gave me warning. I was so surprised that

they gave me a chance, from that point I tried my best to pass all my tests but I never

got passed a 90 in all my math tests grades (s56).

This experience is remembered as the ‘worst time in my life’; this is what this student risked just

to achieve a passing grade on one test. The notion that a grade is supposed to express their

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mathematical understanding and ability is secondary to the grade itself. This statement suggests

that the grade achieved is more valuable for this student then what a high grade is intended to

represent, a high level of understanding. As an educator, I believe the grade achieved in math

class is almost completely irrelevant in one’s day-to-day life, but not developing the

mathematical understandings that allows access to cultural capital can be very detrimental. Yet,

the grade is not connected to understanding for so many students.

Reference to grades were expressed in response to all three of their new teacher’s

questions, from if you consider yourself to be good or bad at math, to what you want to

accomplish this year, to even acting as an answer to what are your good and bad experiences

with math. Grades on tests, as well as the ‘all defining’ final grade are powerful memories in

these student’s reflections. These students introduce themselves as their previous grade, as

though this is the measure that says the most about them in relation to their mathematics

education and one thing about themselves that they expect the teacher to want to know. Of all the

topics that students wrote about in this data set, writing about their grades was the most

prevalent.

The responses from students who separate grade from understanding need to be

contrasted with the narratives of other students who reflected on their grades. One student notes

that, “A bad memory was when I started slacking and did not study for a test and I got a C. That

was my wake up call to do better and try hard” (s1). Although this student, like the multiple

students responses previously discussed, uses an experience of receiving a grade as a powerful

negative experience in math class, they do not merely state the grade as the proof of ‘good’ or

‘bad’. The grade was the result of actions. They mention that before the test they were ‘slaking

off’ and instead use the experience to consider how to improve on future assessments. “The

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highs of my math career is when I achieved the highest score I have ever gotten which was a 90.

[I] think that I received this grade because my motivation level was higher then ever before”

(s12). These two students seem to understand that they have the ability to affect their experience.

Mathematics learning is not something that happens to them, which they are either good or bad

at. Rather, they see themselves as active participants who can affect their experience. By

understanding their ability to affect their skill, these students have more power. They do not view

their grades as final and defining; their grades are merely a signpost on a longer journey.

When these remembrances of experiences are considered in relation to ahimsa, does how

these students interpret the meaning of their grade decrease harm in their educational

experience? Does it allow them to take action? Does it demonstrate love in this complex system

we call mathematics class? Using ahimsa as a framework through which to consider implications

of education does not clearly define what is done as right or wrong. To use ahimsa as a

framework is to question, to rely on critical thinking, and love. It is also important to remember

that (any) harm is not to be justified under ahimsa, but it can be understood or excused if it leads

to decreased harm overall (Parekh, 1988).

Jackson & Leffingwell (1999) found in their study that grades nine to eleven are a cluster

for increased anxiety when pre-service teachers were asked to describe their worst, or most

challenging, mathematics classroom experience from kindergarten through college. The data that

I am analyzing comes from students who are just starting this period of their mathematics

education. Those students who wrote about their grades not as a static definition of their ability,

but as malleable and based on their changeable action seem better prepared to enter this anxiety

cluster. Even if students are currently pleased with their grades, without a connection to how a

grade was achieved, how can they be able to overcome the possible setbacks that they may face

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over the coming years? The ability to persevere though challenges and ‘bad’ grades is a practical

skill. Although students may not believe themselves to be strong at math, they can still possess

resilient dispositions.

Math is by far my favourite subject but it’s not my strongest. I consider myself to

struggle sometimes but I will pull through or work through. Even though my answers

aren’t right sometimes at least I gave it a try. I’m hoping I will try harder to get my

grades up this year it’s very important to me. Especially this year (s33).

Developing a resilient disposition indicates less harm and more love being experienced.

Reasonable struggle is not avoided at any cost; it can be viewed as an integral part of the learning

process. Being able to encounter and overcome challenges can make experiences in the

mathematics classroom more positive.

On SpeedIn addition to grades, which were the most common themes of the narratives, numerous

students also discussed the speed at which they are able to finish classroom tasks, such as

worksheets and tests, as evidence of their mathematical ability. One student commented, “I say I

am good at math because I am one of the first 3 people to be finished in class when we do a

worksheet of have a quiz or test, depending on the type of math we are doing” (s16). Such

students show that they have come to understand that the faster they complete a task, the better

they are at that task. In mathematics the ability to work through problems at a reasonable speed

has value, but so too is the level of correctness, and your ability to approach novel mathematical

situations. Just as in speaking another language, fluency rather than speed contributes to

communicating with others. Does valuing speed without connecting it to these other important

traits allow students to develop a mindset that is productive, not just competitive?

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Speed was not only used as evidence of ability but also of quality of experience in

mathematics class. Being one of the fastest student’s in their class was noted as some of their

best experiences in math class. “Some good experiences I had in math were with fractions and

graphing. Each time I have homework, I always finish my math homework the quickest” (s8).

These positive feelings of being the fastest are often conflated with the idea of being one of the

best. This interpretation of speed can be motivating, and this student goes on to write how they

think they are “pretty good in math” (s8). Beyond knowing you are one of the first to complete a

task, which is motivating without explicit teacher validation, another student enjoyed being

rewarded by their teacher for their speed. “I’d have to say the best times I encountered with math

was in eighth grade when I was always one of the first people to complete math worksheets and

received bonus points for that” (s6). This student, who only considers themselves “ok at math”

(s6), is recalling being the fastest and receiving bonus points as a highlight of eight years of math

education up to this point.

The importance placed on speed is also entwined with competitiveness—knowing you

are fast is contingent on having someone to whom you can compare yourself. For some students

(normally ones who consider themselves ‘good’ at math) this competition can be very

motivating.

I guess I started loving math when I was in elementary school. I was great, and some

students in my class were really good too, so there was always a little competition. We

would always get multiplication sheets and I was always finishing them quickly, along

with some of the other proficient math students. In a way, those kinds of competitions

were fantastic for me. I was motivated to do well on them. The students also pushed me

to be great even if they didn’t realize it (s11).

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But what happens to these students if they find themselves losing their status as the fastest? Does

this kind of competition build mathematical resiliency? Without the ability to overcome set

backs, both large and small, in the mathematics classroom the act of doing math starts to become

anxiety ridden and would be less likely to try something in case they might fail.

Competition does not exist without students comparing themselves, or being compared

to, their classmates. Although students at this level of education are not awarded grades based on

their performance in relation to others, comparing themselves to those around them is an

extremely common practice. “Math was my favourite because I was the best in my class. I would

always help out the kids in my class that were confused and I always understood stuff right

away” (s5).

For many students in this study being the ‘best’, or one of the ‘best’, is determined by

their grade and the speed at which they achieve their grade. Any teacher who has handed back an

assignment or exam knows what many of the students will do first. After quickly looking at their

grade, if one has been given, they will turn to those around them and ask ‘What did you get?’

This is not done to help deepen their own, or others’, understanding, it is done to compare

themselves on a linear measure, be it from 0 to 100, or from 1 to 4. Hopefully their next move is

to then look through their tests to see where they went wrong, and again lean over to their friend

not to compare himself or herself, but to learn from them.

Similar to finding students who were reflective about their grades, some students who

wrote about their speed were more self-aware than others. They not only stated how fast they

were but also discussed characteristics that help them learn and do math quickly. “I think I’m

good at math. Some of my [best] qualities in math is being able to understand things quickly. I

can make connections to things I’ve learned in the past and I can create shortcuts to get the same

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exact answer, but just a bit quicker” (s42). This student shows a developing understanding of

himself or herself as a learner as they value the ability to tie what they are studying to what has

come before. Another student writes, “I find myself mostly good at math, but not like really good

at it. I am not one of those math genius, but I am willing to learn from my mistakes so next time

I’ll get a one and two zeroes” (s29). I wonder what a math genius looks like to this student, as I

know the willingness to learn from previous mistakes is a characteristic of excellent math

students.

In considering these remembrances of the speed at which students finish mathematical

tasks in relation to ahimsa, placing value on speed is both harmful and loving. Like most things,

to some they are harmful and to others it brings out something pleasing. “For Gandhi ahimsa

meant both passive and active love, refraining from causing harm and destruction to living

beings and positively promoting their well-being” (Parekh, 1988). Valuing speed in doing

mathematics is not intrinsically harmful; it can allow students feelings of accomplishment and is

thus motivating. Focusing on speed however can become harmful when it is considered an end in

and of itself. When being ‘good’ at mathematics becomes inescapably tied to being fast at

mathematics. The student can become inflexible, or inadaptable, as mathematical challenges

increase in complexity and cannot be done quickly, but require perseverance and problem

solving. If a goal of learning mathematics is to “enable individuals to see through the inequalities

and exaggerations that are so often cloaked in a mathematics rhetoric” then learning a

meaningful, powerful, mathematics must be prized (Davis, 2001, p. 18).

Grades and speed are ways of quantifying understanding I have named measured

performance. These measures can be used to report, and reflect on what understanding has

developed, but in isolation, without reference to the work that has taken place, or other skills of

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value, grades and speed can seem like ends in and of themselves. It is an overly simplistic view

of learning makes education a quest to collect points, not to grow one’s mind.

Approaches to Learning

On Memory

The students who discussed their memory, or memorizing steps, when remembering key

mathematical experiences highlight different ways of viewing what mathematics is, and what it

means to do math. In writing about how they approach their learning much can be known about

how these students think developing understanding occurs. Some of their reflections point

towards seeing mathematics as a finished set of procedures and formulas to be memorized,

which leads to an understanding that mathematics knowledge is un-challengeable, constructed,

and static. However, others see mathematics as more complex, useable, powerful, accessible, and

do-able. This orientation allows for critical thinking and the opportunity to view one’s world and

the pursuit for the development of knowledge as changeable, not static.

In many recollections the students wrote about the importance of being given clear steps

or a formula in allowing them to excel mathematically.

I like to see math as just multiple steps that happen in order, and not when there is

uncertainty about the answer. I guess I feel more confident where there is only one

possible answer to a problem instead of multiple answers or choices. If I know the exact

steps in an equation to solve for the answer, I can easily get the work done without

breaking a sweat…All I need to excel in math is a clear understanding of the content we

are learning about, along with knowing how to solve any problem given, and a focused

mindset that doesn’t get distracted (s49).

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This student sees math as a clearly defined set of rules that merely need to be memorized to get

good grades. Mathematical thinking should not require them to ‘break a sweat.’ They need only

find the ability to focus so as to remember all the steps that they will need to succeed. Innovative

thinking or problem solving do not seem to be part of their definition of doing mathematics, yet

these are part of the powerful set of skills that should be being developed if students are to

become mathematically literate.

Another student starts by describing math as a puzzle, a game or toy designed to test

ingenuity, yet quickly moves from the type of problem solving thinking that would be required to

solve a novel puzzle, to a clearly defined set of rules. “Math, to me, is simply a puzzle. The only

problem is remembering the rules... I want rules, that need to be followed to lead you to a result”

(s40). This student has perhaps heard someone describe doing math problems as solving a

puzzle, but have not integrated the ideas of what it takes to solve a puzzle with what it means to

do math. They see math as a number of algorithms that need to be memorized, and then drawn

upon in test situations to produce correct answers. Mathematics does rely on and use algorithms,

but this is only part of what makes up mathematical knowing. The issue with students

approaching math by trying to memorize a set of rules is that when they come to a question that

does not quite fit the rule, they will get it wrong. They have not developed any mathematical

flexibility in their understanding and are not adaptable to a new situation without a new set of

rules. A powerful way of doing mathematics allows students to take their understanding and

adapt it to novel situation through problem solving.

This ties to what many students wrote about grades, the ability to easily memorize

algorithms leads to high grades, which seems to be the goal for many. These students’ definition

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of what it means to ‘do’ mathematics misses out on developing ingenuity or being

mathematically flexible; a clearly defined path to achieving a high score is of most value.

Many students in the study focused on the memorization of rules in math as this allows

for a clearly defined path to success. Instead of learning being an on-going process, these

students ‘request’ the rules to be clearly laid out allowing for memorization. All they believe that

they require from their teacher’s is to clearly outline these steps so they can reach their goals.

At sixth grade I didn’t do well on a test because I didn’t really know how to do the math.

Unlike my eighth grade teacher, she didn’t explain the steps in detail. For me, that’s how

I learn math better. I expect myself to do well on tests and get a high grade that my

parents will be proud of me (s22).

These steps and this kind of mathematics does not lead to continued success as mathematical

problems become more complex or if it necessitates problem solving. I could be interpreting this

student’s statement incorrectly, for it is beneficial for a teacher to explain why mathematical

choices are made, their meaning, or how they arrive at an approach, perhaps this is what is

meant. However, this student’s use of the phrase ‘explain the steps’ leads me to interpret them to

be focusing on learning to memorize algorithms composed of steps, over developing flexible

understanding. For many students their approach to learning is to take the clearest, most direct,

path to the highest grade by memorizing clearly laid out procedures.

Other students define math in the opposite way, having little to do with memorization.

“[M]ath has always been my favorite subject since 6th grade and I think it’s easy to understand.

It’s not like other subjects where you have to memorize facts and dates you just learn how to do

it” (s47). This student in contrast, who most likely went through similar educational experiences

as those above up till this point, has such dramatically different understandings of what it takes

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to ‘do’ math. Some view doing mathematics as memorizing steps, and others do not think they

are memorizing anything.

When considered in relation to ahimsa, students in this study who wrote about

approaching learning as memorizing algorithms were harmed when their teachers did not present

the information in a way conducive to this strategy. They felt love when they were able to use

this strategy to achieve high grades, and praised these teachers for structuring their learning in

this fashion. However, harm can also be noticed when students focus on memorization as the

main way to approach learning. They have a diminished opportunity to take up math in a

meaningful way that will allow for continued success. These students who approach their

learning as memorization would be harmed if a teacher organized their classroom to value

critical thinking and problem solving, but beyond the harm of adapting their learning approach, if

they are able, they would develop more powerful approaches to learning that would read as love

as they grow into more critical thinkers.

Other than the many students who view math as a set of procedures that need to be

memorized, many others blame their memories for previously not living up to the goals they and

their parents had set for them. They view mathematical learning, as something that should

happen instantly and then, with no further labor, should never be forgotten. “The problem I have

with math is that I have a very hard time trying to remember what I have learned. Even if I took

notes I would not have remembered much unless I looked back at all the notes I have taken”

(s58). It seems that this student has not yet realized that understanding is something that happens

over time, with continued practice and review. That part of the purpose for taking notes is to

have a document to reference in the future, as this new learning has not yet been committed to

longer-term memory. Even if they have started to view their learning as something that takes

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review, they believe they are somehow not learning correctly as it takes more time than a single

class for them to understand something. “I overall don’t consider myself to be bad at math but

I’m not really good because I have a problem of learning concepts quickly and sticking them in

my mind. When a concept is taught to me it takes me a couple days to actually get it and then

I’m fine if I study it” (s54). Should this not be what how every student approaches developing

their understanding? Instead they think that something is wrong with them, as they did not learn

it instantly.

For other students, not being able to access what they have previously ‘learned’ causes

them to struggle on tests and exams. “My weaknesses in math are Geometry, word problems, and

sometimes, I get mental blocks that keep me from doing really easy problems, even though I

know how to do them” (s34). Although these mental blocks can arise from having high anxiety

about test performance, this can also arise from students having a misunderstanding of how well

they grasp the material. They seem to believe that if they have solved a kind of problem once

they should be able to apply that to all further encounters with that kind of problem without

incident. “The problem resurfaces when test or quizzes come along. I suddenly forgot everything

I have ever learned in math and I have to struggle to remember things” (s7). This student has a

view that at some point they ‘learned’ a topic, and that their understanding is complete.

Therefore, it should be easily accessible to solve future problems. “I can be studying math for 3

hours and the next day, I will sometimes forget what I learned or what I have to do to solve the

math problem” (s23). Time spent studying does not mean that topics have been mastered, yet for

some students this time spent ‘studying’ should equate to ‘knowing how to do it’.

It would help alleviate some of these student anxieties if they viewed learning as a

process, not something that happens instantly, where you only have one chance to understand. It

37


is okay to struggle, and that not understanding in an instant does not mean that they cannot

understand or that they are bad at math. “My worst experience would have been learning how to

divide and multiply fractional expressions in eighth grade. It was pretty straightforward at the

beginning, but as the expressions grew more difficult, I found myself struggling. Fractions were

confusing enough on their own. I also hated whenever we did a unit on word problems, I feel like

the words get me even more confused” (s39). Struggling should not be seen as a failure. It is part

of the process of learning; if you are not, at times, struggling you are not improving.

It is less harmful to have an understanding that learning is recursive, and not fixed. That

meaningful learning takes revisiting a topic numerous times to master an idea; even then

understanding can be continually improved. Learning does not have a clear beginning and end.

Student twenty-three goes on to say, “What I hope to accomplish this year in math is to be able

to get great grades and not be able to forget what I study or learn.” Not forgetting is not entwined

with what steps they might take to better remember what is worked on in class, it reads as a wish

that may or may not be granted by the math gods. The student is limited in their control over the

outcome. Learning math is seen as something innate, that they are either fortunate enough to

possess, or not. As student thirty-five wrote, “Math is a subject that many people either excel or

struggle on. People either are good at it or bad at it.”

Students who view learning as a process that should happen instantly and should then

remain the same feel no love from education. They feel as though there is something wrong with

them, since their brains do not work like computers, that it takes work and study to come to

understand something. Even if they feel they have understanding, tests betray them and they feel

harm while sitting looking at a problem that looks familiar and thus they assume they ‘know how

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to do it’, they have merely forgotten. There is no love to be gleaned from this approach to

learning, only repetitive harm.

To approach learning as memorization, where learning should be accomplished instantly

and should endure until it is tested, does not allow for lasting knowing, or becoming truly

mathematically literate. When considering this view doing math, through the framework of

ahimsa, it may allow for some initial positive feelings, but in the long term it is harmful. It

allows students to feel they are succeeding as they memorize algorithms, which they replicate on

tests. Although learning mathematics does employ the use of acquiring of algorithms, they are

not the entirety of what is means to become mathematically literate. It makes students feel smart

and accomplished as they look at the high grade they were able to achieve. However approaching

learning in this way does not allow for continued success and lasting understanding. Some might

be learning to be mathematically flexible enough to adapt in the future, but many will not. They

will become frustrated when their learning approach no longer scores them high grades, and they

will struggle to keep up. To approach education with ahimsa and not do harm, although an

impossible goal, is a good guiding principle. For to live is to cause harm, it is inescapable.

However, what lasting harm is crueler, to allow students to not develop lasting mathematical

understanding, but have high grades, or to develop lasting understanding, but need to struggle

more to get there?

On Awareness of LearningOther students wrote about their approaches to their learning in a way that contrasted the

above students’ focus of memorization. These students displayed more sophisticated

metacognition, an awareness of their process for learning and understanding. One student

defined themself as good at math because they are one of the first to complete tasks, and then

wrote about their last year’s grades as a product of their work ethic, and of their view that their

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previous teacher wanted them to succeed. They finish by setting goals for themselves for the

coming year;

I say I am good at math because I am one of the first 3 people to be finished in class like

when we do a worksheet of have a quiz or test depending on the type of math we are

doing. Some math questions might be challenging to me, but I still try my best to try and

solve it...I think the highest grade I’ve got in math was a B…My main goal this year to

get at least one A in math. I am going to try my best by paying attention in class, doing

my homework, doing my classwork, but mostly by respecting the teachers because

with out them, I won’t be able to succeed in this class (s16).

This student’s entry discuses speed and grades, but instead of focusing on the memorization of

algorithms as the method to improve these areas, they are aware of areas of growth and focus

that could help them achieve their goals. They know they have control over their learning, and

that by changing habits they will be more likely to have a positive outcome.

Having a more developed idea of what factors can be manipulated or changed to improve

a result allows these students to know themselves better as learners. Allowing them to view their

learning as something they have power over. Such as the following student, “You can’t just

make me do the work without showing me how and you can’t just show me how and not let me

try it out and get comfortable” (s14). This student sees learning as something that takes time and

effort to develop understanding. They see their teacher as having a role, but they see their own

role as well. This understanding would allow them to be more resilient when they are faced with

a challenging problem, or situation in math class. This student went on to set two goals for

themself for the coming school year.

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I want to improve my understanding of certain subjects in math such as geometry, and I

want to work on memorizing formulas, and definitions of certain math terms and words. I

would also like to improve on focus by removing myself from the few close friends I

have in my class and not talking to anyone distracting... And I will try not to distract

myself in the process (s14).

Like many of the students he discusses memorization, but it is not memorizing how to solve a

problem. It is to have formulas and terminology memorized so they can focus on the problem. As

well, it shows great responsibility as a learner to disclose to your teacher that you do not work

well with your close friends while in class. This student is taking ownership over their learning.

Another student discussed that over the past few years, different teachers had caused

them to be more or less in interested in math, but the main focus of their entry was on how they

could develop more advantageous study skills.

I can solve problems quickly (its not always organized though) ... I would like to think

I’m a good mathematician. I can usually keep up with what’s going on, but if I cant I just

re-read the notes until I know what were learning (s4).

This student goes on to make a goal to “be more organized with my work instead of just writing

on the page to get the answer” (s4). There is nuance to this student’s entry. They understand that

factors external to them will influence their mathematics experience, but at the same time, they

have control over how they approach their own learning. They can identify when they need to

review more, and see value in improving the clarity of their communication.

How students approach what is means to do mathematics impacts their experience.

Students who have developed metacognitive skills can better analyze their current practices and

can then set goals on how to change how they approach their learning to improve the outcome.

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For these students their mathematical experience is filled with ahimsa, both in the short term, and

these positive experiences will foster success for them in the future. In contrast, those who are

less sophisticated metacognitive awareness see their learning as either happening, or not,

memorized, or not, are harmed by their approach to learning as they do not see how they could

change their approach. They hope for a better outcome, but how to reach the goal is not clear.

Many of them have come to see learning math as memorizing algorithms, which should be

instantly understood and committed to memory. This harm is decreased when student approach

their learning with metacognition. They know more about themselves as learners, and have

multiple tools to rely on to help them learn.

On Not UnderstandingThere is a difference in the approach to learning involved in realizing the difference

between not remembering and not understanding. To answer the question “What have been your

good and bad experiences with math?” many of the negative moments mentioned in their posts

were based on not understanding. Either not understanding because they had missed classed,

classes move too fast for them, or are not understanding because their approach to learning.

Many of these students are aware enough of their learning to know that something has been

missed, but did not express sophisticated knowledge about their own thinking to name possible

resolutions for when or if they have similar issues this year.

One student reflected on missing the first two days of a new chapter. “Everyone had

grabbed the concept of it by the time I got back to school. Because I was two days behind

everyone else I didn’t understand a thing and I just barely past the test and quiz on the subject”

(s3). This negative experience stands alone in her post. She did not write about the impact, or

anything that she or her teacher did to improve her understanding. Although this experience was

negative, she still defines herself as good at mathematics, and has meaningful goals for the

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coming year. For another student, missing school had larger consequences; they wrote that they

do not think they have reached their full potential mathematically.

[W]e had a geometry unit and I got severely sick, to the point of having constant

hospital visits, so I missed the introduction class which was always a full period of

…[explanation]. I missed three days and fell completely behind dropping my grade from

an 87 to about a 72. I lost hope, and dropped even further into doodoo, because of stupid

mistakes I made in work habits and in school mindset. I want to improve my

understanding of certain subjects in math such as geometry, and I want to work on

memorizing formulas, and definitions of certain math terms and words. I would also like

to improve on focus by removing myself from the few close friends I have in my class

and not talking to anyone distracting... And I will try not to distract myself in the process

(s14).

This student was absent because of illness often during the year, and falling behind the rest of the

class caused him to change how he viewed and approached learning. Although he had a difficult

year missing so much class, luckily he had positive experiences from previous years to fall back

on. That although he did not learn as much as he could have while sick, he still believed that he

can be successful and had laid out positive, meaningful goals for the coming year.

Other students did not have a concrete reason, like being absent, to attribute for not

understanding. Many wrote about when they were ‘lost’ and did not know how to proceed.

My experiences with math haven’t been good ones in middle school. I didn’t understand

things and was confused most of the time. I also didn’t do much of my homework

because I would just give up if I didn’t know what it was about. I’m hoping this year will

be easier to understand (s5).

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This student’s entry is upsetting. He has developed a coping strategy of just giving up when he

does not understand. This is a student who, like all students, would like to do well, but does not

have the metacognitive understanding to analyze their approach, or the view that this can change

the outcome of their learning experience. He just hopes that this year will be easier. It seems this

student has experienced harm in the math class repeatedly, leaving him demoralized. Few

resilience or metacognitive skills have been developed to overcome as it seems that challenges

have been much more present than successes. This entry clearly shows the harm a student can

feel in a mathematics classroom.

Unfortunately, students are constrained by the system in which they learn, where each

year they move on to a different grade, where it is expected that they have some understanding of

the topics covered in previous years. This system is not overly adaptable to meet different

student needs and understanding. Differentiation within a class only deals with this issue to a

limited amount. “Sometimes in math class I will begin to understand something and then the

class will switch to learning something different or a different strategy and I find myself to be

lost yet again” (s32). This student seems to be more confident then student five; however, they

are also restricted by the system in which they are learning. Many students wrote about times

when they felt that their class was moving too fast for them or they needed more time to develop

their understanding before moving on to a new idea.

My bad experience with math was when I went into the eighth grade.… [E]veryday we

would have a new lesson, by the middle of the year we were already in Algebra 3 going

onto Trig. The class moved way too fast for me to have time to understand and we had a

six page test almost every week on the topics that we covered. It was rough and I got

lower grades (44).

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Learning about a topic is restricted to the program of study for the student’s grade, but it is also

designed to fit into the specific number of math minutes each week. This system does not allow

for major adaptation to individual needs and thus has the potential to harm students each year.

Even when students are aware enough of their understanding to realize that they need more time,

the system itself lacks responsivity. The structure in which they learn is restrictive to students

trying to develop meaningful understanding, while at the same time making them feel not

capable of learning.

After writing my literature review I originally thought that I would be, at least partially,

writing about systemic social inequality in education. Yet, in reading the student’s reflections

they did not reference things that might be associated with social inequality, such as

socioeconomic status, race, or gender. This perhaps shows how invasive systems of inequality

can be in institutions like schools The school system can seem to be static and unchangeable,

therefore unquestionable. The lack of evidence in students’ writing of social inequality could

also possibly steam from the students in this data group having similar backgrounds and

educational experiences. If so, they may be less aware of how these factors influence their lives.

Student responses do however shine light on structural inequality, unequal results built

into institutions that will produce inequality, in school. The restrictive timetable and grade levels

set very strict boundaries on what students need to know, and how long they have to learn it.

Even if, as the students above show, realize that they do not ‘understand’ to an acceptable level,

the structure of the system wins out, as they must move along to the next topic and the next

grade. Year after year, topic after topic, these students are being left behind, continuing to not

gain access to cultural capital only available to the mathematically literate. Year after year,

feeling like they are not ‘smart’ as their learning could not fit into a predefined system. In my

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experience in the Alberta school system, it is not until high school that they can even retake a

course to allow for deeper understanding. During the seminal kindergarten to grade nine years,

there are no second attempts, or taking more time. No wonder student five just hoped that this

coming year would just be easier. Perhaps they do understand, that the system is engineered

against them. Grade levels and timetables are not practices based in ahimsa, they are founded on

notions of efficiency and control.

On Asking QuestionsOne of the key tools that students have to assist in their learning is the ability to ask

questions. In this study students showed that they knew that asking questions would help with

their understanding, but for different reasons they did not take advantage of this tool. Many made

one of their goals for the year to ask more questions.

One reason to not ask questions is that it shows that they do not understand, and thus the

act of asking questions allows others to know this about them as well.

I do not like asking the teacher for help because it makes me feel extremely stupid,

especially in front of the entire class. For example if a teacher has just explained an entire

math problem for the whole class and everyone else gets it but me I wouldn’t feel

comfortable asking a question because it makes it seem like I wasn’t paying attention and

personally I don’t like that (s7).

Student seven did go to on to make a goal to improve her ‘study ethic’, but did not discuss a goal

of trying to ask more questions. She, like many, when faced with a task that makes them feel less

than an other, expresses that it seems easier to just give up. “My bad experience of math is when

I can’t solve a problem and usually I just would give up on the problem instead of asking for

continuous help” (s30). This student seems to be able to ask questions, but feels that there is a

limit, because asking too many questions would indicate they are struggling more than their

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classmates. They would rather give up, knowing that they do not know, than make it explicitly

known to their class and teacher that they are not understanding.

Others recognized that they should be taking advantage of asking questions and although

they have instead given up in the past, they want this year will be different.

When I am taught a new lesson and don’t fully understand it right away I sometimes get

frustrated and don’t try my hardest. I do ask for help sometimes but still won’t understand

it. This year I want to fully understand every lesson by asking the teacher for help (s43).

Even these students who have not considered themselves successful in mathematics in the past,

would like that to change. They would like to know what it feels like to learn and understand. To

not be afraid to ask the questions that they know they have. They know that staying quiet does

not benefit them over time. “Some bad experiences I have had is when I don’t understand

something the teacher explained and I stay quiet it doesn’t benefit for me, cause then I do bad on

that type of problem” (s20). These students have the metacognitive awareness to know that

asking more questions would help with their understanding. “I know I will ask for help if I need

it instead of acting like I know what I am doing (in seventh grade, that was a really big

problem)” (s11). They are aware that they do not understand the mathematics, but unlike some

students in the previous section, they have an idea of how they could change this, however,

social pressures to appear ‘smart’ are harming them.

To sit in a class everyday knowing that you do not know what is happening, and feeling

social pressure to appear that you do, is harmful both to the students’ self-worth daily, and

reduces their opportunity to work towards powerful mathematical literacy. To reduce harm,

asking questions should not been seen negatively, but positively. Asking questions should be

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celebrated as the student is able to identify an area they need more assistance in, and are willing

to try.

As well, the interactions that arise out of questions are opportunities for students to not

only build their mathematical understanding but also their relationship with their teacher.

Through being asked questions teachers can demonstrate a loving pedagogy and show that they

care. Not asking questions not only harms a student’s mathematical understanding but also

diminishes the possibility of developing a caring and constructive teacher-student relationship.

Direct Teacher InteractionsWhen reading though the students reflections, the most discussed topics were their

purpose for learning, grades and understanding, and their approaches to learning, memorization

and metacognition. Yet, some student had experiences that they recalled as their best and worst

memories that resonated with my own experiences, both as a student and a teacher. These

recollections speak directly to the love and harm in mathematics education. Of these students’

eight plus years of learning math, these memories rise to the top. These are the lasting memories

that they have created. These are the stories that they tell as adults when they meet a math

teacher at party, or to their own children.

The following student had a feeling of wanting to give up, when they were ask about

something they were not clear about in front of the group.

I remember in 7th grade we would be called on even if our hands weren’t raised. I do not

like speaking to the class or even a group. Being called on would make me shy and doubt

myself. This made me want to give up, especially when I got the answer wrong. That is

one of my bad math experiences (s32).

This is the harm that teachers have direct control over. Unlike the harm caused by structural

inequality like grade levels and timetables, which teachers have little direct control over, and can

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only attempt to deal with the consequences, teachers decide how to ask questions in their

classroom. Teachers knowing their students and having thoughtful questioning practices in place

can mitigate this harm.

Whereas some students remembered the good grades they got on tests, or how fast they

were able to complete tasks, other students remembered their teachers. The following students do

not name specific interactions that occurred for them to feel that their teacher wanted them to

succeed, but in their memories of their positive experiences in math class these memories rose to

the top. “I think the highest grade I’ve got in math was a B. The reason I have got B’s was

because my math teachers cared about my [sic] and wanted me to succeed” (s16). They felt their

teacher wanted for them to succeed, or through patience allowed them to feel success.

“Throughout my school years, my math teachers have always been kind and patient to me

because I don’t always get stuff on the first try. They never gave up on me, and that’s probably

why I enjoy learning math” (s6). The act of noticing and reacting left students feeling love. “My

teachers were always nice and helped me when they noticed I was struggling” (s7).

Feeling like their teacher cares, and wants them to succeed is connected to how these

students view how they were able to be successful. Loreman (2011) feels that learning happens

best with relationships that are intimate, safe, caring, and warm and that by “using love as

pedagogy is an antidote to superficial learning” (p. 8). So much of the harm that has been noticed

in this study are occurrences that diminish the capacity of the students to take up mathematics in

a meaningful way. The above students have overcome at least some of these barriers by having a

student-teacher relationship centered in caring. They felt cared for from their teacher noticing

and responding to their learning. Here ahimsa is palpable; in a caring teacher-student

relationship.

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ConclusionAfter teaching for several years, mostly mathematics classes, I began a master’s degree at

the University of Alberta with many questions. During my program I learned to ask questions

differently and to value different kinds of answers. One idea which resonated with me during this

time was Gandhi’s notion of ahimsa, an active love, — doing no harm in word, thought, or deed.

Considering mathematical experience through the framwork of Gandhi’s ahimsa become the

focus of my final project.

I found one hundred and twelve student written reflections online, which I read through a

framework of ahimsa, focusing on love and harm. In their blog posts the students were asked to

answer three questions, 1) Do you consider yourself to be good or bad at math? 2) What have

been your good and bad experiences with math? 3) What do you hope to accomplish this year in

your math class? I read through these posts, then using the program NVivo, coded the data. After

coding forty entries new codes did not seem to be continuing to emerge. I then coded twenty

more, and did not create any new codes during that time, so I stopped coding new entries. I then

organized the codes into themes that seemed to be related and transferred some quotes from each

area to a word document. I printed and cut out these sections so I could easily change their

arrangement and analyzed them for insights.

One of the first insights of the study was how often, and in many different ways students

referenced grades. None of the questions directly asked students to write about their previous

grades, but almost every post referenced previous grade(s) at least once. For many, a previous

grade was named as their ‘best’ or ‘worst’ experiences in math class. Students did not elaborate

beyond the grade about what lead up to this grade, or how they might alter or replicate this

memory in the future. Others used their grade as their proof for how ‘good’ or bad’ they consider

themselves to be at math. Again, nothing more than the grade was stated, there was little further

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reflection on any thing that lead up to these grades. Some students wrote about their parent’s

grade expectations, and others how their grades on individual assessments always bring their

grades down. There was one student who had never been assigned a grade in their previous

school who had developed a very specific idea of what grade he thought was an acceptable for

the coming year. These students showed that they were highly motivated by grades, but did not

always show an understood connection between their grades and what happens in class. One

discussed cheating, how they were willing to cheat to achieve what they considered an

appropriate grade, never reflecting that a high grade should be a reflection of a high level of

understanding. The grade number itself was more valuable to them then the understanding it

represents. Other students talked of their grades in relation to preceding actions. Such as having a

high level of interest and receiving a high grade, or ‘slacking off’ and getting a low grade. These

students tied their grade to more then just a number, it was the result of actions which they have

control over. Others wrote about grades to make it seem that they did not consider grades to

represent their level of understanding. Their goal was a high grade, not a deep understanding.

This is the harmful side to grades, when the number has more meaning then what it represents.

Students wrote their goals for grades over the coming school year, but many did not demonstrate

an understanding of what steps could be taken for this to goal to come to fruition.

Another measured performance that students wrote of often was the speed at which they

were able to complete mathematical tasks. Their speed, like grades, was given as the proof of

being ‘good’ or ‘bad’ at math. Many showed they had come to understand that being fast at math

meant they are ‘good’ at math. However, in my experience, powerful mathematical literacy

requires much more then speed, such as; fluency, problem solving ability and persistence.

Multiple students, named being the first to finish as their best experience in math class. It was

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very motivating for them. Valuing speed is tied up in competition; you cannot be the fastest if

there are not others to compare yourself to. Competition can be motivating and harmful to

different learners. Especially harmful to those who take more time, but who also believe that

speed at math means being ‘good’ at math. Some students, as with grades, discussed their speed

but put it into context as to why they are faster or slower. One student believes that they are fast

as they are able to tie their current learning to past learning. Another reflected that they are

normally slower but are proud that they always learn from their mistakes. Fixating on speed can

become harmful when it is considered an end in and of itself. When being ‘good’ at mathematics

becomes inescapably tied to being fast at mathematics. This can cause students to become

inflexible, or inadaptable, as mathematical challenges increase in difficulty.

Another common theme in the student’s reflections was memory. Many wrote about how

they value teachers who break down the math into easy to follow steps, and then they memorize

these steps and thus do well on assessments. Although mathematics does rely on some

algorithms, memorizing algorithms is not all that learning math entails. Students also have to

solve problems, and be innovative. These reflections show how these students view what it

means to do math and how they view what it takes to learn math. By focusing on memorizing

algorithms they miss out on other aspects of mathematics that make it powerful and useful in the

long term. If they only have algorithms memorized they cannot solve problems that do not fit the

exact parameters. Viewing leaning in this way can lead to high grades and positive feelings in

the short term and in particular situations. However, to view mathematics like this over an entire

school experience leads to students struggling and experiences of harm as problems become

more complex.

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Connected to memorization, other students wrote about how their memory fails them

during assessments. In their reflections they wrote about how during assessments they forget

‘everything they have learned’. They feel deceived by their mind. Their writings show that these

students view learning as something that happens instantly and then should be lasting. Another

student wrote about how she is ‘not good a math’ as it takes her a few days and studying to learn

new concepts. Showing the belief that if you are ‘good’ at learning it should happen

instantaneously. These students are harmed every time they are assessed as they assume that

learning happened at some point and should therefore be accessible. When it is not, they feel less

than, and that they are ‘bad’ at math. It would help alleviate this harm is these students were to

view learning as an ongoing process, that does not have clearly defined beginning and end

points. It takes review and possibly some struggle to be successful. Struggle does not have to

mean failure; it is key part of the learning process.

In contrast to approaching learning as memorization, other students showed sophisticated

understanding of their learning through metacognitive statements. These students often wrote

about their grades and/or the speed at which they finish tasks, but they also discussed skills and

approaches they rely on for learning beyond memorization. They showed an understanding that

they have control over their learning experience. That learning takes time and effort. These

students seem as though they would be more resilient in challenging learning situations. Their

goals for the coming year went beyond setting a grade they would like to achieve. They also

discussed the steps and strategies they would rely on to achieve their goal(s). How students view

what learning is, and what it looks like, greatly impact their learning experience. Students who

have developed metacognitive awareness can examine their existing practices and can then set

meaningful goals on how to improve future outcome. For these students their mathematical

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experience is filled with positive moments. While those who see their learning as either it

happens, or not, either they remember it, or not are harmed by their approach to learning. They

hope for a better outcome, but how to reach their goal(s) is not clear.

In addition to the students who are harmed by ‘not remembering’ their learning, other

students discussed the harm they felt never having developed understanding. Some attributed

being absent for multiple or for important days as to why they were unable to reach their full

potential for understanding a concept. These seemed to be students who felt that when in

attendance their understanding developed consistently. Other students felt they did not ever

really understand and this had led to them to give up. Others attributed the pacing of the class as

to the reason they did not understand. They felt as if they had had more time to develop and work

on a topic before moving on they could have been successful. The structure of modern schools is

broken into grade levels, with specific topics to be covered and understood during that year.

These outcomes are to be covered in a set amount of minutes. This is an extremely restrictive

system that is not adaptive to individuals. This system produces, year after year, topic after topic,

students that do not have an understanding of the intended outcomes, who are moved onto the

next grade to continue to feel ‘lost’ and like they are not smart. These students are harmed as

they do not gain access to the cultural capital only available to the mathematically literate and

may never come to see themselves as mathematically capable.

A number of students felt that they should be asking more questions, but for multiple

reasons did not want to. In their reflections it was made clear that they understood that in the

long term staying quiet would not benefit them, but to the desire to appear like they understood

won out. They felt that asking questions would show their teacher and other students that they do

not understand, and felt more confortable giving up. Some students wrote about wanting to

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change this behaviour, making asking questions a goal for the coming year. While others knew

not asking questions was holding them back but made no explicit goals to change this behaviour.

The social pressure to appear ‘smart’ was too powerful. Feeling like asking questions makes

them less than, and is harmful both to the students view of themselves and their ability to take up

mathematics in a powerful way.

Many students spoke vaguely about their previous teachers while some discussed them

more directly in response to the questions asked. One student recalled a teacher’s practice of

calling on students when their hand was not raised. This made the student feel shy and doubt

himself or herself. Unlike the harm felt from an education structure like the timetable, this is an

area of harm that teachers have direct control over to decrease student harm. A few other

students discussed previous mathematical accomplishments, attributing the positive outcomes to

their teachers wanting them to succeed. That by their teaches noticing and responding to their

learning they felt love. So much of the harm that has been noticed in this study entails events that

reduce the ability of students to take up mathematics in a significant way. However, these

students have overcome, at least some possible obstacles, by experiencing a student-teacher

relationship centered in caring.

Love is not the antonym of harm. A loving pedagogy, based in ahimsa, works to respond

to harm that exists in the world and in education. There will never be a classroom without harm.

“Although the Euclidean triangle is impossible to draw, we know how to approximate it; we also

ascribe its properties to actual, but imperfect triangles, and use them to guide our practice”

(Parekh, 1988, p. 215). Therefore, to teach with ahimsa is to honestly workout for oneself how to

decrease harm in one sphere, to continue to reach for the perfect even while knowing it is not

attainable. This study showed that students were harmed in numerous ways. Being intensely

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focused on grades, having conflated being the ‘fastest’ and being the ‘best’, viewing learning

math as memorizing many sets of steps, believing that they have learned a topic but merely

‘forget’ it on assessments, not feeling confident to ask questions, being acutely aware they are

not learning at the current pacing, and being singled out. It also showed students benefiting from

tying grades and speed to more than just end goals in and of themselves, setting goals with

metacognitive awareness, and by feeling that their teacher both wants them to and works towards

them succeeding. “Gandhi often remarked that it was not his job to tell his followers how to

behave, and the each of them must sincerely work out for himself how best to practice ahimsa”

(Parekh, 1988, p. 216). This study focuses on the first half of a pedagogy based in ahimsa,

reflecting on the current condition. The next step is to address noticed areas of harm. To teach

math with ahimsa, teachers must truly work out for themselves how to decrease the harm in their

classroom. This study has afforded me such time and reflection by reading through student

voices of their experience.

ImplicationsBased on this study a pedagogy based in ahimsa would need to approach performance

measures carefully. The role of grades in learning should be explicitly discussed. These

discussions or activities should also attempt to connect grades to an awareness of the learning

process, refining goals, and adapting learning strategies. Using formative assessments to help

improve understanding can assist students in developing the idea that they are not just doing

work to earn a grade, that their main goal should be to be improving their understanding.

Assessments can be a powerful tool to improve understanding by providing feedback and areas

for growth.

To teach with ahimsa would mean explicitly valuing and discussing skills and

characteristics that allow for the development of resiliency and productive dispositions. For

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example, instead of only valuing speed, give problems that require problem solving and

perseverance. As Mason, Burton and Stacey (2010) wrote about learning mathematics, “Probably

the single most important lesson to be learned is that being stuck is an honourable state and an

essential part of improving thinking” (p. viii). It is important to value perseverance in your

classroom. In addition, making sure that given problem sets do not reinforce the idea that math is

merely memorizing algorithms. These problems should require at least some adaptation to build

mathematical flexibility, not merely repetition. It can be valuable to struggle at times. To teach in

such a way that shows that mathematical understanding is neither instantaneous, nor permanent,

and requires review and practice.

These changes in focus are centred in assisting students to develop their metacognitive

awareness, in my view an act of ahisma. The students who wrote more sophisticatedly about

their goals for the year, by referencing metacognitive skills seem more aware of what it means to

learn mathematics. By focusing more on these skills, and not only on the outcomes, perhaps

students could develop greater metacognitive skills. By explicitly focusing on metacognitive

skills, and challenging misconceptions of what it means to learn mathematics, students could feel

more confident to ask questions. Conceivably, if learning is not viewed as something that should

happen instantly, asking questions would not produce the same amount of fear in students trying

to hide their not understanding.

It is easy to recognize the harm that structured grade levels, and timetables cause.

However, it is not a part of education that teachers can easily change or challenge. This study

found a number of students who attributed their success to their teacher’s attitude towards them.

They felt their teachers wanted them to succeed; they noticed their struggles, and worked with

them with patience to improve their ability. Perhaps, these students can find some success

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through caring relationships with their teachers. As Noddings (2013) writes, “It may be that

much of what is most valuable in the teaching-learning relationship cannot be specified and

certainly not pre-specified. The attitude characteristic of caring comes through in acquaintance”

(p. 20). This is where a teacher has the most power over the experience of students in their care.

This is the central goal of developing a pedagogy based in ahimsa. To teach with love is to

reflect on ones practices and to work to actively decrease harm for ones students.

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Davis, B. (2001). Why teach mathematics to all students. For the Learning of Mathematics,

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