How can CAPA capitalize on its students’ creativity to improve math ability?
How can CAPA capitalize on its students’ creativity to improve mathematics ability?
Research Proposal
Jeremy Wright
University of Pennsylvania
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How can CAPA capitalize on its students’ creativity to improve math ability?
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Introduction and Statement of Question
At the Philadelphia High School for Creative and Performing Arts (CAPA),
students all have one commonality; a passion for one or more artistic forms. To attend
CAPA, all students go through a two-step application process before they are offered
acceptance. The first step is to pass an academic and behavior screening where the
faculty reviews students’ grades, PSSA scores, attendance, and behavior records. As long
as students meet a certain criteria, one of which is attaining an advanced or proficient
level in English and Mathematics, CAPA will then offer students an invitation to audition
in one of our seven artistic areas: creative writing, dance, instrumental music, media and
television production, theater, visual art and vocal music.
Although students have met a certain academic standing prior to admission, it
does not mean that we have success in academic achievement during their time at CAPA.
In the current world of high-stakes testing in education, CAPA falls behind in certain
areas. On the Keystone Exam particularly, Pennsylvania’s most recent standardized test
that measures proficiency in Mathematics, English and Biology, most students do not
show the same proficiency level in all of these contents. At CAPA specifically, 60.7% of
our students are proficient in Mathematics but close to 40% are not. This poor rating now
affects teacher’s ratings, the rating of the school, and for the class of 2017, whether they
can graduate from high school.
As I sit considering this problem and how to address the issue, I am reminded of a
few different commonalities. One is that almost all of our students are very dedicated to
being the best “scholar artists” they can be. Another is that they have all shown an
aptitude to an artistic area given the structure of the school. Given these facts and
How can CAPA capitalize on its students’ creativity to improve math ability?
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knowledge of prior proficiency in mathematics as well as my position as the lead
mathematics teacher in the building, the mathematics achievement scores are the focus of
this study. Specifically, the question I would like to consider is how can CAPA capitalize
on its students’ creativity to improve mathematical ability?
Impact on the Educational Community as a Whole
Although this research is specifically to CAPA, many other schools have a similar
problem of how do they reach students and increase their students’ abilities in
mathematics. One question that other schools have to answer is, “What does it take to
create a mathematical mind?” Having a certain mindset, which can be fostered in a
creative world, can open up students to have more success in mathematics. Another area
that impacts other schools in many areas of the country is, “What does mathematics
instruction look like?” The old “chalk and talk” format of instructional delivery is no
longer sufficient to reach all students. Using creative elements in mathematical
instruction can lead to better comprehension as well as more engagement of students. In
addition to looking at instruction, teachers and administrators should also consider
looking at assessments that allow for more creative student responses.
Literature Review
At CAPA and other schools, mathematics is viewed in a very old, systematic
instruction. With the new development of the Common Core State Standards (CCSS),
students are required to think in a more concentrated way using higher-ordered thinking
skills. However, teachers tend to be stuck in a fashion where they emphasize algorithmic
mathematics. This is due to their own High School education (Bolden, D. S., Harries, T.
V., & Newton, D. P., 2010). With a focus on algorithmic solving a problem, many
How can CAPA capitalize on its students’ creativity to improve math ability?
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students lose the connection with the algorithmic process of solving a problem and why
they are solving it. In addition, Haycock (1987) describes “algorithmic fixation” where
one finds success in some algorithmic way of analyzing a problem and sticks to it despite
its inappropriateness for the particular problem. Assessments created by teachers also
lack creativity and focus mostly on speed and accuracy as opposed to connecting
assessment to something tangible or test creativity (Mann, 2006). Whitcombe has an
interesting view of mathematics as consisting of three tiers: algorithms, beauty and
creativity. He cites history as a place where we can look back at see that mathematics and
logic go back to a point where its advancement came from imagination, creativity and
invention. He argues that it has stagnated now due to the most arbitrary and useless part
of mathematics, which is the algorithmic piece of the puzzle that can now be done mostly
by computers (Whitcombe, 1988).
There are a number of solutions that can lead CAPA teachers to moving away
from this “algorithmic fixation” and towards an environment that fosters more creativity.
This begins with looking at the curriculum CAPA offers and infuse more aspects of
intuitive teaching that will stimulate the creative minds of its students. Jacobs (2009)
states, “Curriculum should not only focus on the tools necessary to develop reasoned and
logical construction of new knowledge in our various fields of study, but also should
aggressively cultivate a culture that nurtures creativity in all of our learners” (p. 284).
With creativity in instruction, teachers also need to look at assessment in creative ways.
Many teachers and instructional leaders need to initially think about what does it
take to create mathematical minds. Aiken defines two types of mathematical minds:
logical and intuitive. Following that, creative mathematicians, he states, are more
How can CAPA capitalize on its students’ creativity to improve math ability?
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intuitive in their thought process that is different from the algorithmic mentality towards
teaching recently. Aiken also states a critical thinking process for students: preparation
and involvement in the problem, incubation where the problem is set-aside for a while,
illumination or insight into the problem, and finally verification where we ensure the
insight was correct (Aiken, 1973). One key part of this process in all mathematics is the
verification and ability to state why a solution fits the problem.
At CAPA, there is a stigma that follows students. Since students are creative, they
are not “naturally” good at math. In fact, there is a lot of research contrary to that idea.
Aiken describes a number of personalities of mathematicians, including creative
mathematicians, and describes the notion that creative people have a lot of independence,
a greater openness to experience, less defensiveness, a high energy level, and “willing to
take risks which may help account for their success in discovering novel solutions”
(Aiken, 1973).
Balka describes the need for students to think in more divergent ways when it
comes to mathematical thinking whereas prior most thinking went down a convergent
path (Balka, 1974). Haylock (1987) describes divergent production in mathematics as:
problem-solving situations, problem posing situations, and redefinition. This threeproduction model should be incorporated in mathematics instruction, which will be
discussed later. Mann (2006) describes how enjoyment is central in creative students
learning and peaking their interests while also developing their talent. He also goes as far
to say “the essence of mathematics is thinking creatively; not simply arriving at the
correct answer” (p. 239).
How can CAPA capitalize on its students’ creativity to improve math ability?
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Instruction
Mann (2006) also describes what instruction should look like in a creative
fashion: “Encouraging a child to reach beyond the familiar and probe deeper into the
relationships and structures of a problem is the essence of teaching mathematics
creatively” (p. 253). He also states that creativity looks at problems, searching for
solutions, testing results and communicating results (p. 238). There are many models that
currently exist to foster this idea that Mann has to delve deeper into problems and one is
project-based learning.
Project-based learning focuses on students’ learning through an in-depth project
that has many different facets to its intention and focuses on applying knowledge to a
tangible, comprehensible outcome. Mann (2006) states, “The fundamental nature of such
authentic high-end learning creates an environment in which students apply relevant
knowledge and skills to the solving of real world problems” (p. 241). Students need more
time to understand concepts beyond the practice exercises as what they are accustomed to
in a logical, algorithmic way. Mann cites a study that shows greater aptitude for a concept
taught in an algebra class where the algebra class allowed for the transference of concepts
easier than in physics where the math was restricted to one type of problem. Although
physics is a form of applied math, the application of math is limited to a certain type of
problem to solve a certain type of solution.
Project-based learning is conducive to arts education as well as where teachers
“develop students’ higher order thinking skills as they investigate ill-defined problems
drawn form real life situations including aesthetic inquiry that is explicitly included in art
curriculum” (Bequette, J. W., & Bequette, M. B., 2012, p. 44). Tapping into this idea of
How can CAPA capitalize on its students’ creativity to improve math ability?
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aesthetic inquiry may be a very powerful for students but is hard for some mathematics
teachers to determine strategies for how this could be incorporated. Munakata (2012)
uses photography as an example where students use a real life, unaltered photograph to
view potential problems that can arise from what is in the project.
Freire (1968) describes problem-based education versus “banking” education. He
states that teachers should view students as critical thinkers when it comes to problembased education and also states that there is a need for students and teachers to be
cognitive and reflective (p. 61-2). Freire states that in “banking” education, there is a
dichotomy between what teachers think and what students think. He continues to say that
knowledge emerges only through invention and re-invention, through the restless,
impatient, continuing, hopeful inquiry human beings pursue in the world, with the world,
and with each other” (p. 53).
Another type of instructional strategy is multiple solutions tasks. With multiplesolutions tasks, students strive for answers that do not have only one answer. LevavWaybberg (2012) states how having multiple solutions for students to see increases the
quality of mathematics lessons and also leads to discovering different routes in students’
mathematical knowledge. How it relates to creative students is artists, designers, and
those in the design industry all have concepts that they want to portray and they use a
design process to reach an outcome. “Interestingly, problem-solving and reconciling
multiple solutions by conducting aesthetic inquiry are common undertakings of designers
and artists too” (Bequette, J. W., & Bequette, M. B., 2012, p. 43).
Yet another instructional strategy is discovery learning is a method of learning
where students participate in activities where they create conjectures by looking at
How can CAPA capitalize on its students’ creativity to improve math ability?
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patterns to gain mathematical understanding. Aiken (1973) states that discovery learning
yielded better results than those taught in a more expository fashion. In addition, Aiken
states that students are encouraged to explore and discover themselves. One item that
students struggle on the most outlined above is the inability for students to expand on a
basic algorithm or when to use a particular algorithm. Bolden, Harries, and Newton
(2010) state that creativity in teaching uses resources and applications to everyday
examples and creative learning as students undertaking practical activities and
investigations as well as developing computational flexibility. The discovery activities
can have a variety of different views and activities associated with them. Giving
mathematics a performance lens allows for emotional, aesthetic, and imaginative
inclinations to come out and have some value. Look at mathematical surprise, emotional
mathematical moments, and mathematical beauty as an emphasis for activities and
mathematical teaching (Gadanidis, G., Hughes, J., & Borba, M. C., 2008).
Curriculum
Teachers play a large role when considering how a curriculum is designed,
enacted, and assessed. Creative teachers have a variety of different characteristics. Aiken
(1973) states that a creative teacher does the following: arranging order of topics, posing
problems, asking questions, encouraging discussion, and providing opportunities to try
and explore. Tabac (1998) argues that teachers should encourage students to look at the
different solutions that their peers came up with and “look back” at the solutions and how
they relate to the problem. There is a large amount of creativity that can go into solving
problems and to see different ways of solving them can open up perception and
How can CAPA capitalize on its students’ creativity to improve math ability?
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understanding of mathematical concepts that may have been overlooked by the
individual.
In addition, teachers need to emphasize certain problem solving strategies. Some
problem solving suggestions include encouraging questions, verbalization, and
hypothesis testing; constructing models and using heuristics, stressing relationships;
shifting attention to break up incorrect mental sets; and being patient and friendly (Aiken,
1973). Teachers also need to consider the types of questions asked. Aldous (2005)
described the need for more novel and real life problems and assist in probing the visual
spatial and non-conscious elements of the self. He also states that there are three elements
in the formation of a conceptual framework of creative problem solving: visual-spatial
and linguistic circuits within the brain, conscious and non-conscious mental activity, and
the generation of feeling in listening to the ‘self’ including that of intuition (p. 52).
Besides teachers, there are many other resources that are needed. Although
written in 1973 and cited often here, Aiken discusses the need for textbooks to be
redesigned to incorporate real world problems but now also updated to enhance discovery
learning, multiple-solutions tasks, and project based learning. Lonergan (2007) described
the need to ensure projects were themed and also they had time to meet as a faculty
together to make sure they could see what was working and constantly change projects to
try and make them better. This idea, also referred to as common planning time, is a vital
resource that CAPA currently does not have the resources to accommodate.
Assessments
Teachers also need to consider how they design and create assessments. Balka
(1974) and Levav-Waynberg (2012) describe assessing creativity as divided into three
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parts: fluency, flexibility and originality. Aiken describes creative assessment as solving
a problem where they were graded on how many different responses they had, the
different kinds of responses, and originality of responses. Mann (2006) states the need for
the movement away from an emphasis on accuracy because it discourages risk taking by
the student and use creativity to make original strides to solving a problem.
Mann continues to say that assessments that allow students to think and create
problems that have many solutions tends to create more problems then were posed
otherwise and are calculated more accurately and arrived at correct results. Haylock
(1987) states that the ability “to overcome fixations in the mathematical problem-solving
and the ability for divergent production in mathematical situations might constitute
important components of any assessment of mathematical creativity” (p. 72). Teachers
should consider types of assessment that allow students to create their own questions as
well as have multiple answers, which will yield more accurate results.
As one considers the sources listed here, there is much to relate to CAPA.
Mathematics teachers should relate their instruction to a variety of instructional and
assessment techniques, which cater to the intuitive nature of mathematical problem
solving those creative students possess. In addition, teachers at CAPA should provide
students that relate mathematics with what they plan on doing for the rest of their lives.
This is a necessary piece as Lonergan (2007) states that students need to be focused more
on what are they going to create as opposed to when are they ever going to use this in real
life.
Research Methods
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Further questions that the literature that can provoke this study are how arts
teachers view mathematics, how it is used in their contents, and how they view their
connection to mathematics. In addition, I want to ask some specific questions to a large
group of students about their perceptions of mathematics and what successes and failures
they have had. Finally, I want to have a variety of students with varying artistic
backgrounds form a focus group (or groups) to get ideas from a student perspective on
how teachers at CAPA can help their students increase their mathematical abilities.
For research purposes, I will incorporate three types of data collection. The first
one is a survey to students to get some initial baseline perceptions of students with a wide
range of mathematical ability and artistic background. Menter (2011) describes survey
questionnaires can gather a wide range of information and study attitudes, values, beliefs,
and past behaviors. Further, Menter mentions limitations of surveys relating to length and
the lack of complicated questions that one can ask. This is why I want to gather current
perceptions and get some background information as a use of the survey as opposed to
asking long and complicated questions.
I plan to arrange a focus group, or multiple groups, of students to tackle the
research question as well as some sub-questions. I also plan on following up many of the
perceptions given in the survey with further dialogue about those perceptions. Menter
states “we can better understand why people report certain motives or behaviors. The
information provided in this way can ‘fill the gaps’ in survey data when ambiguities
arise, when it is clear that a particular question did not fully work in eliciting meaningful
data.”
How can CAPA capitalize on its students’ creativity to improve math ability?
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Lastly, I am going to use interviews of the CAPA staff members, specifically
those of the math teachers and the arts teachers, to see their perceptions and thoughts of
teaching creative students. I am very interested to see if the math teachers use any of the
strategies outlined in the literature review and ask about their perceptions of using
creativity in their teaching as well as their impressions of teaching creative students. I
believe the arts teachers have a great view of their students’ abilities, the connection
between what they do in their classes (how their curricula ties to mathematics) and
connections that could be made in their classes or math classes to increase mathematical
ability. Meehan states that some advantages of interviews include teachers using their
own terminology, have contextualized information and adapt questions. Due to the large
diversity of majors at CAPA, I would like to vary the interview questions I use tailored to
their artistic area.
Timeline for Proposed Research
1. March 2 and 3 – Survey to Students in four math classes
2. March 4-7 – Analyze results of surveys and adapt the questions if needed for
interviews and focus groups; extend interview invitations; extend focus group
invitations
3. March 9-18 – Hold Teacher interviews
4. March 24 – Hold focus group for selected students
5. March 26 – Hold additional focus group for students if needed
How can CAPA capitalize on its students’ creativity to improve math ability?
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Bibliography
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Appendix
Protocol and Questions for Survey: The survey will be given to four Algebra 2 classes
which consist of about 50% sophomores and 50% juniors. Two classes are honors level
and two are not. The majors of the students represented are all varied and have a good
distribution:
Good Morning, this survey is being used to answer a research study conducted by Mr.
Wright for his master’s degree. Please answer the questions below to the best of your
ability and thank you very much for participating.
Grade __________ Major ____________________________ Gender ____________
1. What concepts in math do you feel students you work with at CAPA struggle
with?
2. What concepts in math do you feel students you work with at CAPA excel at and
why?
3. How can math be related to your particular art area(s)?
4. What is it about creative students that allows for potential success in mathematics
classes?
5. What is it about creative students that does not allow for potential success in
mathematics?
6. How could mathematics classes be more interesting?
7. What could be added to mathematical instruction that would allow the class to be
more creative and have you express your creative side?
8. What was a time that you felt that you were challenged in math class but really
understood the material afterwards?
a. What was the nature of the assignment?
b. How did the teacher teach the material?
c. What made you feel like you were successful?
How can CAPA capitalize on its students’ creativity to improve math ability?
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think – head, feel – self, believe – in btw
Protocol and Questions for Teacher Interview: The audience for this would be the arts
teachers at CAPA. There are two sets of questions, one for arts teachers and one for math
teachers. The math teacher explanation may turn into a focus group.
Good Afternoon, first off thank you for participating in this interview. It should take
about 30-45 minutes and afterwards, I have a little gift card to Dunkin Donuts as a thank
you for your participation. During the interview, I will have an audio recorder going so I
can go back and ensure that I do not miss anything. The research question that I am trying
to answer is “How can CAPA capitalize on its students’ creativity to improve
mathematical ability?”
Arts Teachers:
1. What is your name and what do you teach at CAPA? How long have you taught at
CAPA?
2. What were your perceptions of math class when you were in high school?
a. Were you a successful math student or did you struggle?
b. What do you use math for outside of teaching?
c. Were their specific areas or contents that you did particularly well in? Bad
in?
3. What is it about creative students that allows for potential success in mathematics
classes?
4. What is it about creative students that does not allow for potential success in
mathematics?
5. How could mathematics classes be more interesting?
6. How can math be related to your particular art area(s)?
7. What could be added to mathematical instruction that would allow the class to be
more creative and have students express their creative side?
8. Why do you think CAPA students do not have as high a proficiency level in
mathematics as we do in Literature?
9. Are there any other comments you would like to add to the connection of
creativity and arts with mathematics?
Math Teachers:
1. What is your name and what do you teach at CAPA? How long have you taught at
CAPA?
2. What were your perceptions of math class when you were in high school?
a. Were you a successful math student or did you struggle?
b. What do you use math for outside of teaching?
How can CAPA capitalize on its students’ creativity to improve math ability?
18
c. Were their specific areas or contents that you did particularly well in? Bad
in?
3. What is it about creative students that allows for potential success in mathematics
classes?
4. What is it about creative students that does not allow for potential success in
mathematics?
5. How could mathematics classes be more interesting?
6. What could be added to mathematical instruction that would allow the class to be
more creative and have students express their creative side?
7. Do you incorporate project-based learning in your teaching? Can you provide an
example of what you did?
8. Do you incorporate examples of multiple solutions tasks in your assignments?
Can you provide an example of what you did?
9. To what extent to you incorporate discovery learning when you perform
instruction?
10. Why do you think CAPA students do not have as high a proficiency level in
mathematics as they have in Literature?
11. Are there any other comments you would like to add to the connection of
creativity and arts with mathematics?
How can CAPA capitalize on its students’ creativity to improve math ability?
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Focus group for Students: After collecting data from surveys and interviews, I would like
to revisit students and ask them for additional clarification on responses from the other
mediums. I plan on inviting seven students to be part of the focus group (one from each
major) with varying ability levels, grade levels, genders, and race.
Good _______________________ everyone, If you didn’t know, I am currently working
on my master’s degree at Penn. For this, I am doing some research on all of you as
scholar artists and your connection to math.
If you look around, there are samples from 10th-12th grade in all majors and all ability
levels in math to get an idea of differences between you all and also some commonalities.
Please feel free to respond to each other’s comments but make sure you respect each
other’s opinion. The question I am trying to answer in my research is “How can CAPA
capitalize on its students’ creativity to improve mathematical ability?”
Questions:
1. What are your general feelings and impressions about mathematics?
2. What areas of math did you enjoy the most?
a. Explain any specific activities that made it enjoyable or helped you learn
the most?
3. What areas of math do you not enjoy?
4. Explain any connection with your major and mathematics?
5. To what extent is there a stigma with creative students and their ability level in
math?
6. How could math classes be more interesting?
7. What was a time that you felt that you were challenged in math class but really
understood the material afterwards?
a. What was the nature of the assignment?
b. How did the teacher teach the material?
c. What made you feel like you were successful?
8. What should CAPA teachers do differently to ensure academic success in
mathematics?
9. What should CAPA teachers do the same to ensure academic success in
mathematics?
10. What can be done to better show the connection between math and the “real
world”?