1. MATHEMATICS TUTORING PHILOSOPHY
IVAN RODRIGUEZ
ABSTRACT. The aim of this article is to factor out and expand upon the integral ingre-
dients towards tutor development and resulting tuition as a function of three prime con-
stituents: training sessions, assigned readings, and anecdotal experience.
When I began working at the THINK TANK, I had a distinct idea of what it meant to
be a tutor; this has, since then, immensely changed. This shift in mindset was spurred
primarily due to three components: the concepts encountered during training sessions,
these sessions’ assigned readings, and my experiences as a tutor. In this article, these three
elements will be analyzed according to how they influenced my tutoring modus operandi.
We begin first with my inceptive prejudices as a tutoring neophyte. Then, the training
sessions and their impact will be addressed followed by their respective assigned readings.
Lastly, the investigation will conclude with developmental anecdotes.
Initially, I was primarily eager as a tutor to devote my time and energy towards academ-
ically assisting fellow Wildcats; however, there was another central motivation: I wanted
to become a tutor as a response to my own substandard experience with math courses at
the University of Arizona—call that vengeance, if you will. I had this idea of becoming a
surrogate instructor; I wanted to be an alternative for students who also did not particularly
benefit from the University’s math classes and/or respective professors. After devoting my
verve to tutoring for a few semesters, however, I have slowly realized that this is not my
place. Specifically, I have come to terms with my new identity: resource. Rather than
trying to form myself into a professor, I now believe that what makes the THINK TANK
successful is the marriage of tutors’ prior experience and course successes coupled with
tutors’ ability to envision and articulate the bigger picture in a given domain of knowledge.
We exist to tie it all together, fill in any gaps, encourage beneficial learning strategies,
and/or provide different perspectives. To begin tracing these gradual realizations back to
the source, we first revisit the tutor training sessions.
Without a doubt, the most eye-opening sessions for me were “What’s It Like?” and
“Learning Challenges” within level-one tutoring. In the former, I recall interacting fre-
quently and without reservation as my inchoate tutor naïveté made me nervous and anxious
about my new role as a tutor. Before this session, I perceived tutoring as a one-way inter-
action: I divulge the requested information, the tutee makes the effort to understand it, this
back-and-forth is repeated until the lesson is learned, and we eventually part ways. Within
my first few hours of tutoring, however, I learned that this was not the case. I gradually
observed that every student had unique abilities and shortcomings, the identification and
incorporation of which would be beneficial in order to catalyze the tutee’s success.
When it came to more extreme cases of this, the “Learning Challenges” session proved
to be exceptionally useful. At the least, I became aware that my own strengths as a scholar
Date: April 14, 2016 and, in revised form, April 18, 2016.
Key words and phrases. Mathematics, tutoring, philosophy, purpose, motivation, statement, synthesis.
1

2. 2 IVAN RODRIGUEZ
would not guarantee success during each tutor-tutee session; rather, my ability to intellectu-
ally dance with the student—ideally while minimizing foot-stepping—became paramount.
Then, through the “Problem Solving” session, I further became cognizant of the cre-
atively different ways in which to assist a student. Admittedly, I have never encountered
a situation in which the tutor toolbox props seemed more promising than words, gestures,
and white-board defacing. Perhaps my own learning style is to blame, but I found the
toolbox to be superfluous. Nonetheless, the idea that this session championed was mean-
ingful: there are many ways in which to assist the student, the limit to this being the tutor’s
creativity.
This notion would be revisited during the “Problem Solving” session; here, I became
conscious of how innovative other tutors could be in their problem-solving and how crucial
it is for me to build a repertoire of techniques. I then understood that the THINK TANK
seeks to continually build a more dynamic environment, not a monochrome setting like
I had originally imagined. Indeed, I now perceive the THINK TANK as a potpourri of
perspectives that allows it to reap unprecedented success.
With regard to “Non Native Speakers,” I did not glean much from this session due
to my own background as a minority. Since Spanish was my first language, I naturally
understood and anticipated the impediments that these students would likely experience. I
did, however, keep this session in mind by working towards being more lucid, articulate,
and cogent with students. My biggest hurdle in this respect, I discovered, was my pacing;
I was loquacious to the point of hurting my throat during each shift. I have, since then,
learned my lesson. I am now markedly comfortable and collected in the way in which I
vocalize my ideas. Having established the lessons obtained from level-one training, we
now move on to level-two insights.
To begin, my scatter-brained self benefited tremendously from the “Scaffolding” work-
shop; even in short drop-in sessions, I found myself outlining a plan of attack before work-
ing with the student that seemed more effective than merely plunging straight into action.
As a result, we would typically both explore why it is that something works the way it does,
and, if time allowed, we would go beyond the exercise to consider bigger questions. I have
also noticed that, through successful cases of scaffolding, I tend to speak less than the tutee
who then focuses, in turn, on constructing thoughtful ideas and questions. Moreover, this
session was crucial for overcoming my fear of one-on-ones. I went from being intimidated
by these quasi-perennial hour-long sessions to being eager to interact and engage with the
student.
Next, the “Round Table Discussion” was refreshing and inspiring because, as adept as
we tutors may become, I was allayed to hear that we will all likely fall on our faces at
some point in our time at the THINK TANK. I was definitely no exception. What this
session provided for me was conviction in myself and, thus, a gradual cure for my self-
imposed impostor syndrome. More than being reassured that we had all erred at least
once as tutors, what unequivocally motivated me was listening to how tutors handled their
seeming failures. I observed that a common stratagem involved seeking the assistance of
other tutors. I had originally been afraid to ask for help because of pride, most of which
was derived from the fact that I was—and still am—a math major. How could a math
major be asking for help on a math problem? This persistent thought prevented me from
reaching out to other tutors when I most needed it; thanks to this session, therefore, I let go
of this stubborn, vitiating pride and I have experienced more success as a result. We are,
I was reminded, a powerful community; this is one feature that makes the THINK TANK
stand out in terms of instructive efficacy.

3. MATHEMATICS TUTORING PHILOSOPHY 3
With respect to the “Brain Compatible Learning” session, I did not substantially hone
my tutoring acumen because of how bemusing the concepts were to me. Regardless, at this
second level of training, I believe that most tutors already have an experiential understand-
ing of how to tailor a session towards the student’s learning style. I do not think that it is
quintessential to be able to label these styles, per se, which was particularly emphasized
in this session. I found out that what works best for me is simply asking the student how
they prefer to learn a topic and then changing my approach if necessary. Additionally, if
I am being ineffective with an approach I am trying, the students are usually quick to ex-
plicitly bring it to my attention or otherwise provide discernible cues. Now that we have
covered the level-one and -two training sessions, we shift to two assigned readings that
further ameliorated my tutoring sapience.
I found the article by Lipson[1]
to be a stellar source in my development. In this text,
the author describes and reflects on a difficult scenario within a children’s book. Herein,
the vociferous Dodecahedron baffles careful Milo, celeritous and often incorrect Humbug,
and slow yet steady Tock by devising onerous questions for the trio to entertain. I could
instantly see the connections between this tale and my role as a math tutor.
I reflected on my tutor-tutee encounters after reading this article and I eventually ac-
knowledged that I often brought out either the Milo or the Humbug in my tutees. After
this realization, I worked first on disciplining myself in order to establish a calm and open
environment for the tutee. For example, I would slow the cadence of my speech and allow
adequate pauses to follow after posing a question. After improving myself in this manner,
I observed that students notably became more comfortable around me. Thereafter, I wit-
nessed that tutees generally became more deliberate and thoughtful, further expediting the
learning process.
However, what I found notably interesting in Lipson’s[1]
work is her perception of skep-
tical Milo: “In many ways, Milo’s view is the enlightened one, the preferred one.” This
character, before commencing work on the problem, cautiously considered first the an-
swer’s practicality if it existed; this was in stark contrast with Tock who does not question
the question itself and proceeds to procure an answer. Upon reflection, however, I began
to understand why Milo’s behavior was laudable according to the author: his desire to
understand why the outcome matters and its impact on obtaining meaningful insight.
This also routinely manifests in numerous math students; it can be summed up as the
‘What’s the point?’ mentality. Without solid answers to this inquiry, students become
wont to developing apathy towards math. Rather than lashing out at these questions like
the Dodecahedron, I now applaud this behavior and encourage that they unapologetically
demand answers to it. It has, thus, become a priority for me to be able to explain why
the subject is important beyond simply passing the class. As THINK TANK tutors, I
believe that our job does not end after the material is absorbed by the student. This is
the bare minimum. Our superlative mission involves, I believe, actively trying to inspire
passion and wonder within the student with respect to the subject they may study. With the
Lipson[1]
text in mind, we now proceed to another article that helped shape my ambitions
and purpose as a tutor.
As the old adage goes, “Give a man a fish and you feed him for a day. Teach a man to
fish and you feed him for a lifetime;” this was precisely what I learned from Valkenburg[2]
.
I was initially unsure whether or not the ‘Show—don’t tell.’ mentality applied to tutoring.
In my experience, the answer seems to be context-dependent, and the author agrees with
this sentiment.

4. 4 IVAN RODRIGUEZ
Valkenburg[2]
expressed that explicitly solving a problem completely for the student
can sometimes be apt: “The tutor solves the problem in order to demonstrate the proper
method to the student. This serves as the foundational learning platform for students.”
After reading this article, I definitely started to understand when it is best to work out
an example for the student. This was helpful because I assumed wrongfully that tutors
were never encouraged to do this for tutees, but I now grasp why this, too, has its place.
Overall, the training sessions along with two meaningful texts have been explored; this
article will now investigate how the synthesis of this preparation has manifested itself in
terms of tutoring anecdotes.
I would like first begin with my most disheartening experience. At the time, I was a
fledgling tutor. One day, as I was struggling to successfully assist a student, she suddenly
and candidly said, “You’re not very helpful.” Although her unexpected frankness took me
aback and shook my confidence, I managed to thank her for being honest. To make matters
worse, this student was a regular. Id est, I would have to inevitably assist her again in the
future. This experience, without a doubt, had a profound and disconcerting mental effect
on me. It was the first time that I questioned my qualifications as a tutor; I felt that I was
no longer fit to be a THINK TANK tutor.
After sharing this story and having my morale boosted during the “Round Table Discus-
sion,” I decided to revamp my strategy: I would pay more attention to the THINK TANK
emails regarding the material being covered that week and do my best to work out some
sample problems before encountering them at work. Additionally, I would pay more at-
tention to befuddling concepts and take or design some sample problems to work out in
my own time. I fostered a deep-seated habit of working out problems on the white-boards
during slower hours and taking home formidable exercises to try to conquer.
Because I failed this student, this event inspired humility by causing me to solemnly
admit and reflect on my drawbacks. To improve myself, I began to request the assistance
of my fellow tutors with greater frequency whenever I would need it. They became an
invaluable resource throughout this period of personal insecurity. Many semesters later, as
I am now reviewing the incident, I have not had one similar issue. The cycle, it seems, has
gone full circle: I now find myself predominantly being a reliable resource for new tutors
whenever they confront a challenging exercise. I now even look forward to these exercises
that once caused me great anxiety and distress. With this monumental experience covered,
we now proceed to two occasions that brought me great joy as a tutor.
One of these fond memories includes a time when an important exam was coming up
and I found myself catering impromptu to four to six students. At the time, I was fairly
familiar with the material in this course, so I was confident helping out students on a case-
by-case basis.
On this busy day, however, I decided to try something different, inspired partly by a
session where grouping students was discussed. I tried this, and, much to my surprise,
this was absolutely successful. I found myself basically becoming like a moderator at
a conference panel whose role is to motivate discussion by forming thoughtful questions.
What happened during this time was like observing lights going off left and right—students
bouncing ideas off of each other in what seemed to be a mathematical collective stream
of consciousness, my role suddenly becoming that of a referee who additionally inspired
momentum whenever it waned.
What was notably difficult during this group session was the range of the students’
grasp of the material; there were one to two students who were sharp and quick with their
answers to my prompts, leaving the rest of the pack to not gain much from the session.

5. MATHEMATICS TUTORING PHILOSOPHY 5
As a result, I decided to try selectively asking non-contributors for input before opening
the forum to the group en masse. This was mostly effective, although the pressure was
off-putting for some of the students.
Nonetheless, what I obtained from this experience was the opportunity to practice mo-
tivating a group discussion and bringing out the strengths of each individual student. In
addition, I discovered how enjoyable and efficient group tutoring could be; in fact, I felt
like I was hanging out and having a good time with this group of students more than any-
thing, and yet we were utterly besting the study guide. Lastly, I believe that both the
students and I learned how important the students can be for one another. I recall on this
day that this group began to collaborate more before seeking out a tutor, and whenever
they unanimously agreed to request help, their questions were less conceptual and more
fastidious—how and when to round numbers and how to type an answer on WebAssign,
for instance. This, to me, was testament to their radical improvements. Setting this positive
anecdote aside, we now move on to another.
A last example of how I have been able to apply the fruit of my training and experiences
towards successful tutoring involves watching the maturation of two students in particular.
There are often regulars at the THINK TANK who I get to promptly know on a first-name
basis. Two of these, in particular, began as minority students who were between a rock and
a hard place in their math classes.
However, they sought out help just in time. They would religiously frequent the THINK
TANK at least once a week even if they did not yet have any questions in mind. Watching
them develop over time, I am simply bewildered by their drastic improvements. In fact, one
student is now considering applying to become a THINK TANK tutor as well, which made
me overjoyed for her sake. The other student, incredibly, is heavily considering declaring
a math major.
What I have contributed to help these students gain this conviction in themselves is,
mostly, helping them realize that they were more than capable of success. I would, for
example, make it a point to acknowledge that, by the sheer fact that they were bold enough
to seek assistance, they clearly displayed the necessary discipline and willpower to conquer
their math courses. I would admit to them that I was terrible at this in comparison due
to pride and insecurity. Then, throughout my time assisting these two students, I would
especially do my best not just provide a superficial understanding of the material; I was
adamant in presenting thorny questions that went beyond memorization and regurgitation
of facts so that these students actually understood the material and the why.
When we have something in the form
ex
= ey
,
why is that we can immediately conclude that x = y? There is a hidden step behind the
memorization, the use of the natural exponential’s inverse—namely, the natural logarithm—
as an intermediate and hidden step:
ex
= ey
,
ln(ex
) = ln(ey
),
x = y.
Through the fundamental theorem of calculus, how is the definite integral of a function
related to the difference of an anti-derivative evaluated at the end-points? We can consider
a simple example such as f(x) = x. Then, we use a geometrical approach to observe that

6. 6 IVAN RODRIGUEZ
the area under the curve between x = 0 to x = 1 amounts to finding the area of a triangle
with base and height of 1, which comes out to 1/2. Lastly, we notice that
b
a
f(x) dx =
1
0
x dx
=
x2
2
1
0
=
(1)2
2
−
(0)2
2
=
1
2
as expected.
Why is it that the famous harmonic series,
∞
n=1
1
n
= 1 +
1
2
+
1
3
+ . . . ,
actually diverges while something like
∞
n=1
1
n2
= 1 +
1
4
+
1
9
+ . . .
converges? Well, consider a series with a smaller sum than the harmonic series—say,
S = 1 +
1
2
+
1
4
+
1
4
+
1
8
+
1
8
+
1
8
+
1
8
. . .
However,
S = 1 +
1
2
+
1
4
+
1
4
+
1
8
+
1
8
+
1
8
+
1
8
. . .
= 1 +
1
2
+
1
2
+
1
2
+ . . . ,
so S diverges. But, since S is a smaller sum than that of the harmonic series, it must follow
that the harmonic series diverges as well.
Through approaching concepts and developing questions in this way, I have challenged
my tutees in ways that require them to think differently about the problems they are given.
Though memorization and cramming may be all that students need to get their desired
grade, I believe that the THINK TANK stands for more than this baseline indicator of
success.
My role as a tutor has morphed into becoming someone who can delineate the bigger
picture in a subject and help the student fill in the rest to the best of my ability. These two
students are wonderful examples of this goal being exceeded; they now naturally ask all the
right questions and use me in all the right ways—“bigger questions, better answers” indeed.
My day is made whenever I guide students towards these epiphanies, big and small, that
leaves a lasting impression on them; furthermore, if all goes well, perhaps they might just
be inspired to fear math less and maybe—just maybe—perceive it with a newfound sense
of awe. We have now gained an understanding of my formative experiences as a tutor-in-
training as well as how these insights played out in practice; thus, this article concludes
with a recapitulation.
Reflecting on all of these influences involved in my progress as a tutor, I can safely ad-
mit that I have found more confidence in myself and a deep respect for the bigger message

7. MATHEMATICS TUTORING PHILOSOPHY 7
of what the THINK TANK stands for. All training sessions, at least to some degree, planted
some seed that would prove to be worthwhile in the long haul. To me, the most beneficial
sessions included “What’s It Like,” “Learning Challenges,” and “Scaffolding.” “What’s It
Like” helped me to overcome insecurity by recognizing that we all, as tutors, face unique
challenges. “Learning Challenges” made me aware of the myriad complications in any
tutor-tutee session. “Scaffolding,” in turn, probed me to strategize my sessions with stu-
dents, be they mere minutes or one hour in duration; this session was the most valuable to
me and my unsystematic self. Through Lipson[1]
and Valkenburg[2]
, I discovered that there
should be an emphasis on setting up an optimal environment for students as well as under-
standing select methods in which to guide a student towards lasting knowledge. Then, in
terms of anecdotes, I had a deflating experience with one student who inadvertently chal-
lenged me to reinvent myself, I mustered up the courage to lead a session of around five
students with satisfactory results, and I have notably influenced at least two students who
were struggling at first and are now alacritously contemplating excellent questions and re-
ceiving meaningful answers from our services. Through a careful evaluation of all of these
elements in my growth as a tutor, I now stand in a more secure frame of mind with a much
different intention than when I was first hired. I have thawed my initially selfish ambitions
and cultivated a sense of inspiring and motivating the student as a resource to supplement
their education. It is not about me and what I can do—it is about the student and what can
be done for them. My continuous objective is now to become the best supplement I can
be for anyone requesting our services. I look forward to only growing within this flourish-
ing community of students, tutors, and staff that make the THINK TANK the phenomenal
service it has become and will indeterminately continue to be.
REFERENCES
1. A. Lipson, The road to Digitopolis: Perils of problem solving, School Science and Mathematics 95 (1995),
282–289.
2. J. Valkenburg, Joining the conversation: Scaffolding and tutoring mathematics, Learning Assistance review
15 (2010), 33–41.
DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF ARIZONA, TUCSON, ARIZONA 85721
E-mail address: ivanrodriguez@email.arizona.edu