Inquiry Based Learning from the Learner’s Point of View: A Teacher Candidate’s Success Story
Caswell, C. J. and LaBrie, D. J. "Inquiry Based Learning from the Learner's Point of View: A Teacher Candidate's Success Story," Journal
of Humanistic Mathematics
Inquir y Based Learning from the Learner's Point of View: A Teacher Candidate's Success Stor y
Caroline Johnson Caswell 0
Rhode Island College 0
Derek J. LaBrie 0
Rhode Island Colleg 0
0 Rhode Island College , Providence, RI , USA dlabrie
Follow this and additional works at: http://scholarship.claremont.edu/jhm Part of the Educational Methods Commons, Mathematics Commons, and the Science and Mathematics Education Commons Recommended Citation

Inquiry Based Learning
from the Learner's Point of View:
A Teacher Candidate's Success Story
Caroline Johnson Caswell
Departments of Mathematics & Computer Science and Educational Studies,
Rhode Island College, Providence, RI, USA
Derek LaBrie
1. Introduction
Inquirybased learning (IBL) has been employed extensively in science for
many years. It is widely accepted that learning science means investigating
and exploring when developing a hypothesis. However, such has not always
been the case in mathematics [
32
]. Students have often been encouraged to
get \the right answer" and practice fundamentals over creation,
investigation, and discovery. Teachers of mathematics promise students they will use
the skills someday [34]. Few teachers use modeling or real world situations
to make mathematics relevant, even though such approaches can positively
impact student motivation [
25, 32, 34
].
Students need to take responsibility for their learning. They should construct
meaning that is \powerful, memorable and e ective" [25, page 4]. When
students are invested in their education, they are more engaged [
25, 32
]. Even
group work, which should be exciting and challenging, is often structured to
solve problems that many students can solve on their own, and so ends up
leading to boredom. According to Blair, \Inquiry is compatible with the
learning of mathematics, leads to higher levels of motivation and results in
a deeper understanding of concepts and of how to learn those concepts" [3,
page 32].
In this article we rst review studies which explore the bene ts of IBL over
traditional instruction methods. Then we aim to capture IBL challenges and
bene ts by documenting one student's journey in an advanced, collegiate
mathematics course delivered with IBL techniques. Our unique qualitative
approach seeks to capture a richer insight than survey data collection
methods which dominate existing research.
2. De nition of IBL
IBL is an instructional practice where students explore content by posing,
investigating, and answering questions. Students are at the center of the
learning experience and take ownership of their own learning [
4
]. They often
work independently and in small collaborative groups.
As Mahavier et al. state, in an IBL classroom, \the instructor plays the role of
coach, mentor, collaborator, guide, and occasional cheerleader" [20, page 1].
More speci cally, the teacher's role in IBL is to guide students and promote
thinking and curiosity [
32
]. This takes purposeful planning to manage
multiple student investigations simultaneously. Teachers monitor the progress of
each student and provide immediate feedback [
12, 17
]. IBL does not indicate
less guidance from the teacher, but rather delivers instruction in such a way
that the student constructs their own meaning [25]. The teacher serves as the
facilitator who plans, instigates, and observes the student learning process.
Currently, there are many de nitions of IBL and a variety of approaches. The
Academy of InquiryBased Learning states that IBL engages students and
requires them to: solve problems, conjecture, experiment, explore, create,
and communicate [
1
].
3. Theoretical Framework
Marshall and Horton use the four E's to describe IBL: Engage, Explore,
Explain, and Extend. The learner needs to be \perturbed" and challenged
to think critically and analytically [21, page 92]. In a similar vein, for the
Center for Inspired Teaching [
4
], the \Wonder Experiment Learning Cycle"
represents the student learning process, see Figure 1 on the next page. In
the gure, at the center of the cycle, students achieve with the 4 i's:
intellect, inquiry, imagination, and integrity. The outer cycle includes ve \core
elements of inspired teaching": 1) mutual respect, 2) student as expert, 3)
purpose, persistence, and action, 4) joy, and 5) evidence of learning.
Constructivists believe that individuals create their own meaning and
knowledge of the world through experiences and re ecting on said experiences [8].
Social constructivists further this idea that learning occurs through our
interaction with others. Vygotsky's [
29
] zone of proximal development is described
as, \ . . . learning awakens a variety of internal developmental processes that
are able to operate only when a child is interacting with people in his
environment and in cooperation with his peers" [29, page 90]. Learning is an
active process, not a passive one.
IBL in mathematics is often associated with \the Moore Method," which
was developed by Robert Lee Moore during the early 1940s. Moore expected
his students to discover mathematics on their own presenting their work to
their peers in class [
20, 24
]. Moore believed that \[t]he student that is taught
the best is told the least" [24, page 390]. Over eighty years after Moore, his
method is still not widely known, adapted, or adopted.1
The Moore Method includes several crucial elements that lead to student
inquiry, such as the student struggling on their own and presenting their
ndings to their peers. However, the peer interaction is important in IBL
and this was missing in Moore's practice. Modern IBL researchers Laursen
et al. [
17
] describe the \twin pillars" of IBL to include students' deep mental
1Moore's association with racism and other discriminatory ideals and practices probably
accounts for at least some of the reluctance from the mainstream mathematics
community. Today many IBL practitioners within mathematics recognize that it is important to
acknowledge this past: \It is time that we in the community of mathematics educators
who teach with inquiry similarly acknowledge that the legacy of RL Moore is a symbol
that includes racism and misogyny  even if every member of this community comes to
it because they nd value in teaching with inquiry and proactively rejects the bigoted
elements of this symbol" [
14
]. We must therefore emphasize that even though the Moore
Method was probably the rst step to IBL in mathematics instruction at the college level,
today the community of IBL practitioners does not see Moore as a representative of their
values and ideals.
engagement with challenging mathematical tasks and peertopeer
collaboration whether whole class or in small group. Through their research, they
realized that both pillars are necessary since students saw both engagement
and collaboration as integral to their learning. \When students engaged
deeply with the mathematics on their own, they came to class wellequipped
to contribute meaningfully to group work. In communicating their ideas and
listening to others, they deepened and crystallized their own knowledge" [17,
page 137].
We synthesized the above frameworks by using some aspect of each of them.
Our consolidated framework de nes IBL as an instructional practice that
actively engages students collaboratively through challenging tasks whereby
students take ownership of their learning while respecting one another in
intellectual exploration. Further, students develop conceptual meaning through
verbal communication, both written and oral, with their peers. As they
reect upon their learning, new deeper understanding is achieved and increased
selfe cacy is possible.
4. Literature Review
Research has shown that students who participate more in classroom
discussions obtain higher scores on standardized testing. Talking more means
higher scores, in general, regardless of race, poverty, and/or gender di
erences. In fact, traditional gaps of subgroups is signi cantly reduced when
students are required to participate in active learning exercises [
5
]. Cohen
and Lotan [
5
] suggest that teachers must be explicit that everyone
participates and everyone assists each other. The quality of the discussion predicts
nal performance on individual and group assignments [
6
]. When students
share ideas and selfassess individually and as a group, the product is
enhanced [
5
].
A number of studies seek to compare IBL and nonIBL (e.g., traditional
lecture) classrooms [
11, 16, 17, 18, 35
]. Laursen et al. [
17
] prepared a
report for the Educational Advancement Foundation and the IBL
Mathematics Centers Assessment & Evaluation Center for InquiryBased Learning in
Mathematics in which four university mathematics departments sponsoring
IBL Mathematics Centers were studied during 2008. Classroom observations,
surveys, and student interviews were conducted to gather data. Laursen et
al. [
17
] reported that undergraduate students that take IBLtaught
mathematics courses enroll in more future mathematics courses than students in
traditional mathematics classes, also see [
11, 16, 18
]. Further the study
reports that, \IBL students reported higher gains in understanding concepts,
mathematical thinking, con dence in doing and communicating about
mathematics, persistence, and positive attitude about mathematics learning" ([
11,
374
] and [17, page 49]). This study also revealed that women, in particular,
bene ted the most from IBL methods [
11, 375
], as compared to all other
groups (nonIBL men, IBL men, and nonIBL women) on all cognitive and
a ective domains (see Figure 2).
Furthermore, low achieving students earn higher average grades in future
mathematics courses when compared to their nonIBL peers. This indicates
that IBL instruction may have lasting e ects. These ndings could have
important implications for the way undergraduate mathematics courses are
structured and taught.
In the same study, Kogan and Laursen stated, \On average, about 60% of
class time in IBL courses was spent on studentcentered activities such as
small group work, student presentation of problems at the board, or
wholeclass discussion, while in nonIBL courses over 85% of class time consisted
of the instructor talking" ([16, page 185], also see [17, page 163]). Both
male and female IBL students pursued further IBL courses at higher rates.
IBL women took, on average, a full elective course beyond their nonIBL
female peers. Medium and high achievers earned subsequent grades similar
to those of their nonIBL peers, but low achievers from IBL sections earned
consistently higher grades than their lowachieving peers in nonIBL sections.
Lower achieving students' grades improved in later courses.
Another study found that IBL students succeeded at least as well as their
peers in later courses, challenging concerns that the slower pace of IBL,
through omitted material or other structural di erences may hinder student
success in the future [
35
]. Blair [
3
] reported that teachers use IBL less often
due to lack of time with the pressure of standardized exams. Some teachers
also believe that students do not have the skills necessary to do
independent work. These individuals feel that students need clear guidelines and
instructions to stay on task and behave [
28
].
Freeman et al. [10] reported that active learning yielded increases in
examination performance over traditional lecturing. Additionally, lecture formats
yielded 55% higher failure rates. These increases in achievement held across
all of the STEM disciplines and occurred in all class sizes, course types, and
course levels. Active learning was found to be particularly bene cial in small
classes and yielded increased performance on concept inventories.
English Language Learners (ELLs) exhibited higher pro ciency on
standardized tests in science, mathematics, English language, and reading the longer
they were enrolled in IBL science classes [
2
]. One reason is that students
are able to construct context with hands on activities. They also developed
a positive attitude toward learning while engaging in classroom discourse.
Group work provides students, particularly ELLs, the opportunity to hear
the directions multiple times, ask questions, practice what to say, and
communicate, at times, in their native language thus gaining access to the task
and meaningful discourse [
5
].
According to the MidAtlantic Equity Center [23], minority students, for
example, perform best in mathematics classes when the content is related to
their previous experience. Activitybased programs have been demonstrated
to signi cantly improve minority student performance in mathematics. In
Virginia, at Hollin Meadows School, the Standards of Learning (SOL) test
scores improved, the achievement gap diminished, and qualitative
assessments showed that students were developing identity and con dence in their
educational achievements. Teachers used an interdisciplinary approach that
was inquirybased. Students outscored division and state performance in
mathematics on the 2010 Virginia SOL tests (State 87%, our division 84%,
Hollin Meadows 95%). The passing rate for all students in mathematics
increased from 81% in 2007 to 95% in 2010. Of particular note was that
black students' scores increased from 76% to 88%; Hispanic students' scores
increased from 72% to 96%; white students' scores increased from 93% to
100%). In 2010, there was no gap between lowincome students' scores and
scores of other students [
25
].
Kahle et al. [
13
] o er further support. Their study showed improvement
of test scores among black middle school students in classes where teachers
received inquiry teacher training versus students in traditional settings. They
further determined that the achievement gap narrowed between male and
female students participating in IBL. Teachers also reported that students
were more interested in the content.
Li et al. found higher scores and understanding of material in IBL groups
[19, page 11]. Qualitative evidence in responses of teachers and students
supported students overall higher degree of understanding and favorable
attitudes for IBL [19, pages 1215].
Based on prior research of science courses using concept inventories, Epstein
[9] conducted research in college calculus classes that implemented traditional
instruction versus Interactive Engagement (IE) using a concept inventory.
The lowest gain score of any group of students at the University of Michigan
was 0.21 in IE classes compared to traditional instructional methods.2 Most
strikingly, the highest gain score in the traditional sections was equal to the
lowest gain score in the IE sections.
Despite such studies that promote IBL over traditional instructional
approaches, there are also critics of IBL. They claim that IBL has minimal
2\The class performance is measured by the normalized gain, de ned to be
hgi = 10f0
0 ;
0
where 0 is the mean score of the class (not of individuals, hence the angle brackets) at
the start and f is the mean score at the end (in percent correct). This measures the
gain in the class's performance on the [concept inventory] as a fraction of the maximum
possible gain." (p.10191020)" [9, pages 1019{1020].
teacher guidance during instruction. \The past halfcentury of empirical
research on this issue has provided overwhelming and unambiguous evidence
that minimal guidance during instruction is signi cantly less e ective and
e cient than guidance speci cally designed to support the cognitive
processing necessary for learning" [15, page 76]. Prior knowledge is applied to new
problems. The more experience and expertise teachers have, the more the
students will be able to solve new problems. Such critics write:
The goal of instruction is rarely simply to search for or discover
information. The goal is to give learners speci c guidance about
how to cognitively manipulate information in ways that are
consistent with a learning goal and to store the result in longterm
memory. [15, page 77]
Their research shows that students learn better when guided [
15
]. Kirschner
et al. [
15
], clearly not proponents of IBL, state that tests performed over
the decades seeking data on IBL students show that overall e ectiveness is
less than students who were taught in traditional ways. Posttesting scores
are lower for IBL.They further purport that with more than fty years of
investigation, there is no body of research supporting the minimal instruction
technique. They argue that research almost uniformly supports direct, strong
instructional guidance rather than constructivist based minimal guidance
during the instruction of novice to intermediate learners [
15
].
More recent research seems to imply consistently that student performance
bene ts from the IBL approach. However given contradicting work such
as [
15
], one might consider other tangible outcomes besides performance.
Within the past ten years, a growing body of research has suggested that IBL
has numerous bene ts to students in addition to improved performance. In
particular, recent research has emphasized attitudinal or behavioral changes
in the students as a result of taking an IBL course. Davidson [7] found that
students reported greater con dence in their mathematical abilities. They
also reported a preference for inclass activity as opposed to lecture. In
Davidson's study, the professor noticed that the students were more engaged
and energized. Matthews and Hodge [
22
] also studied IBL implementation
in a history of mathematics college course for preservice teachers and found
that students reported higher levels of con dence. Additionally, students
were inspired by the experience.
In a deviation from much of the existing research, these studies address
qualitative student a ects stemming from IBL techniques. In an attempt
to augment research on student bene ts, Caroline, a mathematics education
researcher, approached Derek, a graduate preservice educator, about his rst
experience with IBL in a collegiate, linear algebra course. The rest of the
article reports on the collaboration that resulted.
In what follows, we speci cally emphasize the student experience. Students
have strong feelings about their instructional experiences and we believe their
voices need to be a part of our investigations. There have been a number
of studies on the e cacy of teachers, but few on the e cacy of students. In
the following we work with the principle that if we ask them, they will tell
us what they learn and how they learn best. It is our conviction that asking
students to tell their stories has power and meaning, that test scores and
assessments do not tell the whole story, and that a students journey is of
value to the learning process.
5. The Study
The coauthors in this study are advisee (Derek) and advisor (Caroline) in the
Mathematics and Computer Science Department. Derek is a graduate
student earning an MAT in Mathematics while obtaining his secondary teaching
licensure. Caroline is a faculty member who teaches practicum and student
teaching courses for secondary mathematics teacher candidates. During an
advising session, the coauthors decided to examine the implementation of
inquirybased instruction in an upper level mathematics course, linear
algebra, from his perspective.
In what follows, Derek writes about his experiences throughout the semester
(Fall 2015) and re ects a year later (July{August 2016) on his overall
impressions. To supplement the student perceptions, the Linear Algebra course
instructor was interviewed and student feedback forms were used to validate
the student's ndings.
The class met four (4) hours per week for fourteen weeks. There were two
(2) exams, ve (5) problem sets, weekly in class problems and homework
culminating to a nal project with presentation.3 The problem sets were
graded, required proof writing, and a deep understanding of the material
that was being covered in class. The course was held at Rhode Island College
in Providence, RI.
The student, Derek, was enrolled in the course with six other matriculating
students. The majority of the students were Mathematics majors. Two were
Secondary Mathematics majors; of which both were second bachelor's degree
candidates, which means they had both earned a prior undergraduate degree,
not in mathematics.
6. A Student's Re ection of IBL
Throughout his studies, Derek had noticed a dichotomy between the
education courses and the mathematics courses he was taking. In general,
his mathematics classes were all predictable in their pedagogical processes.
Each class was teacher centered and students were expected to write notes,
by copying what the teacher dictated and recorded on the board. Education
courses meanwhile employed ideas about techniques such as \Understanding
by Design" [
33
] and IBL.
Education instructors expected students to write lesson plans that
encouraged their pupils to think critically and work cooperatively to develop
solutions that are meaningful. He liked the academic and professional support
from his instructors. From his educational experience, now pursuing a
master's in teaching mathematics, the education department seemed to be on
the right track. Yet he observed that the pedagogy he was learning via the
education department was not realized in his mathematics courses. He did
not see these best practices being implemented to increase students' learning.
He was constantly wondering why he was not seeing these more innovative
IBL lessons in his mathematics courses. Why this dichotomy?
It was during his fth semester that he was scheduled to take linear algebra.
On the rst day, the professor announced to the class that they would be
learning the course content through IBL methods. The instructor explained
that everyone would be learning the material for the most part on their
3L. Pinheiro, course syllabus, 2015.
own and would be responsible for preparing material for class discussions
on a given topic each day. At rst, Derek was really apprehensive because
he knew that a higher level mathematics course would be rigorous. Even
though he had always been a top performing student in the past, he did
not feel con dent that he could excel in an IBL classroom in this particular
course. Derek felt like he was going to fail without the professor's direct
guidance and support. In other words, even though he knew of the value of
the particular pedagogy from prior experience, Derek himself noted that he
was hesitant and had qualms about its practice in a mathematics class.
Higher level mathematics courses provide a challenge for any student. As
mentioned earlier, the class was small, less than 10 students. It met twice a
week for two hours. From the very beginning, the students were expected to
read the book and do the homework on their own without being shown any
previous examples. As many would agree, reading a mathematics text book,
especially one on linear algebra, is not an easy task.
Nonetheless Derek took on the challenge. He spent multiple hours preparing
for each class. Most of the time he felt like he had no idea what he was
doing. Unfortunately, he did not understand how to use the textbook to
help him. Thankfully the expectations for accuracy on proof and calculations
that were done for homework were extremely relaxed. Students were allowed
to leave their thoughts and comments blank when the answers eluded them.
At the beginning, he consistently left questions unanswered. Deciphering
mathematical text and coming up with proofs for concepts he had never seen
was quite challenging and often seemed impossible.
The cohort reviewed homework together and all the problems were discussed
as a group. Encouraging students to work together not only deepens the
\teaching" student's comprehension of the content, but a ords the peer with
an alternate explanation [7].
The professor discussed his approach as one that did not give the students
solutions to problems. He would steer the students in the right direction if
something was missing or if their logic was unsound. From the professor's
perspective, \I would ll in the gaps."4
4L. Pinheiro, personal communication, 2016. See Appendix B.
Eventually the students began to ask each other for assistance instead of the
professor. It was reassuring for Derek to know that other students were also
experiencing di culties. As the group went through the assigned problems,
they were asked to mark any comments or corrections in colored pen so the
professor could see the progression of thinking. The assignment was passed
in at the end of class. Derek and other students (as could be seen from the
student feedback forms from that semester) expressed value in this group
approach to learning.
In one class, for example, the assignment required students to show that the
zero vector, 0 of a vector space V is unique. Derek attempted to solve the
question based on his prior knowledge of algebra, see Figure 3 below.
When he presented his proof to the class, the professor said that more
justication was necessary (shown by the arrow in the gure). A classmate then
presented a solution which justi ed the proof. As required by the instructor,
Derek wrote this down in pen, see Figure 4 on the next page.
Over the rst couple of months, Derek went to the professor during o ce
hours and met with classmates for whatever insights he could muster. He
constantly felt defeated by the subject matter. But as time went on, he
noticed that his knowledge of the subject slowly improved. He recognized
that he was building strategies for proof writing and problem solving and
actually starting to understand the material.
It should be noted that Derek's extraverted nature facilitated the IBL
process. He was not afraid to go to the board to present a proof that was
un nished. He also felt very comfortable engaging with other students even
though he often had very little knowledge of the subject to o er when
learning something new.
This content in this class was extremely challenging. Derek and the other
students in the class (as seen in the student feedback forms from that semester)
spent countless hours solving for every problem they were given. It was
through this determination to nd the answer that he noticed a positive
change in his ability to think critically about new concepts. As he became
better versed in the language of linear algebra, he found himself able to
look at solutions from a variety of standpoints. Derek became a much more
exible thinker in mathematics.
Looking back a year later, Derek still remembers and understands the
content. According to the instructor, the students may not have covered as
much as a typical linear algebra class, but the students understanding went
much deeper than a traditional mathematics course. This can be attributed
to a student centered learning environment in which participants dictate the
pace. Smaller content scope combined with a student managed pace
enhanced retention of the material. In fact, Derek reported recall of content
more than one year later for the majority of topics that were covered in class
(i.e. vector spaces, linear transformations, determinants, linear independence
and dependence, bases, span, eigenvectors, and kernel).
In contrast to a traditional lecture approach, the students in Derek's class
spent considerably less time taking notes, memorizing theorems and
algorithms, and taking exams. Derek's overall perception was that this
experience was a strong impetus for him to implement IBL in his future classroom.
Finally, experiencing it personally and participating in its structure within
the mathematics classroom was an extremely bene cial experience.
7. Discussion
When students learn conceptually, a more signi cant emotional and
intellectual engagement occurs. According to Epstein [9] most students leave
introductory courses without a solid grasp of the basic [mathematical] concepts.
Wathall [
30
] explains that conceptual understanding is more important than
the memorizing facts and formulas. One way to approach conceptual learning
is through IBL. Derek's experience with IBL was both emotionally and
intellectually challenging. He felt that he learned the mathematics more deeply
by rst struggling with the content on his own and then with his classmates.
When developing inquiry tasks, the stimulus should be just above the
current knowledge of the student, according to Vygotsky's Zone of Proximal
Development construct [
3, 26, 28
]. The task is meant to stimulate interest
and spark curiosity. It should evoke some level of frustration, but not to the
point where the student gives up. In the context of an IBL classroom, it
is common for students to be frustrated in the beginning.5 Matthews and
Hodge [
22
] reported students were challenged and felt out of their \comfort
5L. Pinheiro, personal communication, June 2, 2016. See Appendix B.
zone" [22, page 35] at rst. Derek also reported feeling lost at rst and
extremely frustrated. Over time, his con dence increased so he could achieve
success beyond what he believed he was capable of.
According to Blair, \[i]nquiry is not about independent discovery, but about
a collaborative process in which the teacher introduces mathematically valid
forms of reasoning at a point when they are necessary and relevant" [3,
page 35]. Inquiry leads to higher levels of motivation and deeper levels of
conceptual understanding [
3, 30, 32
]. Derek's own experience in working with
his peers solidi ed his own conceptual understanding and strengthened his
ability to write proofs. His professor infused key concepts as necessary to
propel the group forward.
Some teachers believe that there is not enough class time to implement IBL
practices; we do know, however, that research such as [
3
] disagrees. Other
teachers assert that some students are not able to handle this type of rigor
and will not be successful. The studies we referenced above refute that claim.
Furthermore in our study, the professor found that the students were
successful in spite of his reservations. As educators, we need to expect more
of our students by providing them with challenging inquiry tasks,
encouraging them to collaborate with classmates, and supporting them to achieve
amazing results. Marshall and Horton [
21
] report that students who sit and
listen to lectures and participate in rote learning are less likely to be involved
in critical thinking activities. Their study showed higherlevel thinking and
learning takes place when traditional class time is substituted with student
exploration activities in mathematics or science classes. Upon observing
over a hundred middlelevel classrooms, the researchers found students that
were involved in IBL activities were required to \verify, justify, develop, and
formulate" [21, page 99] during exploration activities deepening their
understanding of conceptual mathematics involving higher levels of critical
thinking. Furthermore, students typically enjoy the challenge. Derek reported
enjoying the inquiry tasks while deepening his conceptual learning.
According to Scruggs and Mastropieri [
27
], when using IBL in place of a
textbook approach in two science units, high school special needs students
yielded signi cantly higher achievement than their peers who used a
textbook. Ninetysix percent enjoyed using the IBL approach, while 80% of the
students found IBL activities to be more helpful in the learning process and
were more motivated. Pre and posttests revealed that when students were
taught with IBL, they learned more and remembered more than their peers
taught through traditional instruction [
27
]. This nding is consistent with
Derek's own experience with IBL. Derek believes that the IBL instructional
method is the reason he has retained the mathematical content for more than
one year. As a future teacher, he wants to employ instructional strategies
that yield long term bene ts for students.
Derek learned many lessons while enrolled in an IBL course. Some of the
lessons were about mathematics and some were about him. He learned to be
persistent in working through challenging content. Another lesson learned
was that he does not have to do it all alone. Solving challenging tasks
may require support from peers. There are bene ts to thinking and talking
through problems together. In the end, his acknowledgement of owning his
learning with longterm recall and increased selfe cacy was life changing.
Instructors should be aware that students need to be prepared and instructed
to learn in new ways.6 This takes time and patience on the part of the
teacher. The instructor should realize that, at times, direct instruction may
be useful, particularly when a group of students needs support on a particular
concept; this was noted in the student feedback forms from the semester. A
common behavior, however, is to relapse into old comfortable habits of direct
instruction. Students may themselves welcome the traditional approach as
it is familiar and predictable, but instructors should persevere.
Content is often \learned" for the test and then forgotten. If we want our
students to become lifelong learners, then we need to give them challenging
content where they learn to think for themselves. Students will then develop
the ability to learn content with a greater sense of responsibility, autonomy,
and con dence.
Having an IBL textbook is optimal for ensuring that the tasks assigned to
students are of IBL scope.7 Today the IBL community in mathematics,
organized around the Special Interest Group of the Mathematical Association
of America on IBL (IBL SIGMAA) and the Journal of InquiryBased
Learning in Mathematics, is thriving. As a result, there are resources available to
instructors at any level, and it is possible to use IBL for any mathematics
course; see Appendix A, for instance, for a list of IBL Calculus texts.
6L. Pinheiro, personal communication, June 2, 2016. See Appendix B. 7L. Pinheiro, personal communication, June 2, 2016. See Appendix B.
8. Limitations
This paper incorporates one student's perspective of one speci c college class
with a small group of students. Because all of the students were mathematics
or mathematics education majors, this is hardly a random sample.
Additionally, at a class size of n=7, these observations are not valid for statistical
inference. The class was comprised of a homogeneous group of students. Nor
it is particularly generalizable for all students. However, to date, little work
has been done on a student's re ection on his/her learning with the IBL
instructional methodology. We hope that our study will encourage others to
explore student experiences in similar ways in the future.
9. Conclusion
Current research documents the bene ts of active, group learning over
traditional teaching methods across multiple disciplines [
32
]. IBL is one method
that employs active group learning [
21
]. IBL reduces the gap between
subgroups of students (gender, race, socioeconomic status, learning needs, and
language learners). IBL also improves student engagement, motivation, and
selfcon dence [
17
].
However, existing research seldom explores qualitative data on IBL
techniques, ignoring a dimension of measuring its e cacy. This study examined
key impacts of IBL through documenting a student's personal experience
with a graduate course in linear algebra.
Derek's personal experience, which supports and augments existing research,
indicated that bene ts of IBL include greater content retention, con dence
in mathematical reasoning, and increased group and personal responsibility
for material. Additionally, Derek observed that his desire to engage in
challenging tasks independently and with peers was increased. It is particularly
interesting to note that Derek is a mathematics educator. The ndings of
the Laursen et al. study [
17
] did not support mathematics educators to have
signi cant a ective changes. Yet, Derek and the students in his class
experienced greater feelings of e cacy. Perhaps this is due to the fact that
mathematics educators take similar courses to Bachelor of Arts mathematics
majors so their mathematics content exposure is comparable.
The signi cance of this study's ndings is twofold: 1) in the IBL context,
the students took responsibility of their learning, which yielded longterm
bene ts in content retention, and 2) this teacher candidate is determined to
integrate inquirybased learning in his future classroom instruction.
Unfortunately, IBL is not yet widely applied in mathematics instruction. The
existing literature in the eld of IBL, coupled with the unique approach of our
study to measuring the improved learning outcomes in the IBL setting, should
encourage its broader application. Our results will hopefully be deemed
worthy of consideration for mathematics faculty and others interested in the
types of pedagogy where students take ownership of their own learning.
[7] Denise D. Davidson, \Using peer instruction and the ipped classroom
to teach reliability and validity," Syllabus, Volume 4 Number 2 (2015),
pages 1{5.
[8] John Dewey, Democracy and Education: An Introduction to the
Philosophy of Education, Macmillan, New York, 1916.
[9] Jerome Epstein, \The calculus concept inventory { Measurement of the
e ect of teaching methodology in mathematics," Notices of the
American Mathematical Society, Volume 60 Number 8 (September 2013),
pages 1018{1026.
[10] Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith,
Nnadozie Okoroafora, Hannah Jordta, and Mary Pat Wenderoth,
\Active learning increases student performance in science, engineering, and
mathematics", Proceedings of the National Academy of Sciences of the
United States of America, Volume 11 Number 23 (June 2014), pages
8411{8415.
http://scholarship.claremont.edu/jhm/vol6/iss2/4, accessed on
June 9, 2017.
[23] MidAtlantic Equity Center, Opening up the Mathematics and Science
Filters: Our School Did It, So Can Yours, booklet by Deanna Banks
Beane, The American University, Washington DC, 1992.
[24] John Parker, R. L. Moore: Mathematician and Teacher, Mathematical
Association of America, Washington DC, 2005.
[34] Brian Winkel, \Computers have taken us to the brink in mathematics
. . . and we have balked," Computers in the Schools, Volume 30 (2013),
pages 148{171.
A. Open Source Calculus Textbooks for IBL
Below is a list of some Calculus I open source textbooks available. All URLs
made available were last accessed on June 9, 2017.
Boelkins, M. (2015) Active Calculus: single variable. American Institute of
Mathematics. Available at http://scholarworks.gvsu.edu/books/10/ as
well as at http://faculty.gvsu.edu/boelkinm/Home/Active_Calculus.
html.
Mahavier, W. (April 2015). Calculus I, II, & III: A ProblemBased Approach
with Early Transcendentals, Journal of InquiryBased Learning in Math,
Issue 37. Available at http://www.jiblm.org/downloads/dlitem.php?id=
100&category=jiblmjournal.
Stallman, C. (September 2015). Calculus I: Notes and Problems, Journal of
InquiryBased Learning in Math, Issue 39. Available at http://www.jiblm.
org/downloads/dlitem.php?id=103&category=jiblmjournal.
B. Professor Interview  June 2, 2016
To supplement Derek's re ections that this study revolves around, Caroline
(CJC) also decided to interview Leo Pinheiro (LP), the assistant professor of
Mathematics who was teaching Derek's linear algebra course, about his rst
experience teaching using inquirybased learning (IBL).
CJC: What made you decide to teach using IBL?
LP: I was selected as a fellow in the Project NEXT program [http://www.
maa.org/programs/facultyanddepartments/projectnext] through the
Mathematics Association of America (MAA). The professional development
at the conferences during that year focused on IBL. So I thought I would
give it a try.
CJC: What is Project NEXT?
LP: It is a year long fellowship. There are three conferences during the year:
summer, winter, and summer.
CJC: What was your initial reaction to implementing IBL?
LP: This is not going to work. I was teaching a linear algebra course in the
fall of 2015 and did completely understand how the students would be able
to write proofs. I did not teach them how to [write proofs].
CJC: How did you introduce it to the students?
LP: I asked them questions like:
How does a person learn something new? What do you reasonably expect to remember from your courses in 20 years?
What is the value of making mistakes in the learning process? How do we create a safe environment where risk taking is encouraged and productive failure is valued?
CJC: Were there any bene ts to using IBL?
LP: It was an incentive for students to come to class. The teacher is no
longer the focus. The students actually learn by doing.
CJC: What were you most valuable resources?
LP: The Academy of Inquiry Based Learning was very helpful.
[Available at http://www.inquirybasedlearning.org/, last accessed on June
9, 2017.] I recommend Dana Ernst's book. [Available at https://zenodo.
org/record/29875#.WTs5G5Lyvcc, also check out Ernst's blog at http://
maamathedmatters.blogspot.com/2013/05/whatheckisibl.html, both
last accessed on June 9, 2017.]
CJC: What is di erent using IBL than the traditional direct instruction or
lecture?
LP: The communication is two way now and my role is di erent. I ll in
the [learning] gaps, you know, the fundamental things they would not have
learned.
CJC: What surprised you most by using IBL? LP: I had a family emergency one morning so I cancelled class. The next class, I found out that the class had met in my absence. They wanted to
continue the work. At that moment, I realized that it is less about the
professor teaching what they want. We are so used to hearing ourselves talk.
Now, I nd myself mediating class discussions, but I don't give them answers.
The students are actively learning every day.
CJC: What concerned you the most when the course was completed?
LP: I wondered how they would fare in Abstract Algebra, the next required
mathematics course.
CJC: Now that you have taught IBL for two semesters, what do you feel
strongly about?
LP: It was extremely helpful having an IBL textbook in the second semester.
It made such a di erence.
CJC: Is there any course that you would not consider using IBL?
LP: I cannot imagine how it would work in Precalculus or Calculus I. [See
Appendix A for open source IBL Calculus textbooks.]
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[11] MarjaLiisa Hassi , Marina Kogan, and Sandra Laursen, \ Student outcomes from inquirybased college mathematics courses: Bene ts of IBL for students from underserved groups," Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education , edited by S. Brown, S. Larsen, K. Marrongelle , and M. Oehrtman , Portland, Oregon, Volume 3 ( 2011 ), pages 73 { 77 .
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[17] Sandra Laursen , MarjaLiisa Hassi , Marina Kogan, AnneBarrie Hunter , and Tim Weston, Evaluation of the IBL Mathematics Project: Student and Instructor Outcomes of InquiryBased Learning in College Mathematics, Assessment & Evaluation Center for InquiryBased Learning in Mathematics , Boulder CO , 2011 . Report available at http://www.colorado.edu/eer/research/documents/ IBLmathReportALL_050211.pdf, accessed on June 9 , 2017 .
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