Journal of Mathematics Education © Education for All

March 2018, Vol. 11, No. 1, pp. 62-80 https://doi.org/10.26711/007577152790021

Investigate the Effects of Flipped

Learning on Understanding of
Mathematics for Secondary Students
Dunia Zeineddine
Long Beach Unified School District, California

The purpose of this study was to examine the effect of the flipped classroom on
learning outcomes and increased student participation in problem solving
with reasoning in the high school precalculus classroom. Twenty three
students in a precalculus class were given a traditional lecture and assigned
homework to be completed away from class; twenty students were taught
using flipped learning where they read and viewed lecture videos outside of
class with focus on problem-solving in class. Three worksheets were collected
to measure student participation in problem-solving with reasoning. The
results were analyzed quantitatively. Results of this study showed that
participation in problem solving increased for students who did not regularly
complete homework as a result of implementing the flip.

Keywords: Flipped classroom, traditional classroom, problem solving,

cognition

California Schools are implementing the Common Core State

Standards and students are expected to achieve deep understanding of
mathematical concepts and consequently perform well on the Smarter
Balanced test (Smarter Balanced, 2012).
Although the traditional teaching methods have been successful in
helping students achieve mathematical skills with less than adequate levels of
understanding on average, there is a need to try another teaching method in an
attempt to increase the deep understanding of concepts and applying them to
problem solving (Freeman, Eddy, McDonough, Smith, Okoroafor, Jordt, &
Wenderoth, 2014). The traditional teacher-centered lectures are ineffective in
teaching students with understanding. The problem is not that lecturing is bad
for students, but that there is not enough time for both lecture and in-class
problem solving group sessions (Schwerdt & Wupperman, 2010). The lack of
deep understanding is attributed mostly to lack of time invested in problem
solving since the teacher is usually rushed to complete a vast curriculum
within an inadequate amount of time. This is especially true for higher level
classes like precalculus where students are expected to achieve fluency in
complex algebraic skills with a variety of solution strategies in a variety of
topics. Excluding between-teacher variations which could influence the way
Zeineddine 63

any teaching style is implemented, and therefore produce different results,

when students are given more time to work on complex mathematical
problems with peer and teacher support, students will benefit. If we want
teachers, administrators, and school districts to buy into the flipped model,
more research is needed to prove the effectiveness of the flipped model in
creating an interactive learning environment that promotes student
understanding in the secondary school.
This study is an attempt at finding a better way to teach with
understanding through the Flipped Classroom model. Students are introduced
to the lecture materials through a combination of online notes and a pre-
recorded lecture by their teacher or other sources, they complete an
accompanying assignment to encourage students to view the videos, and then
students will have a cognitively challenging, predesigned peer-group activity
to apply the lesson in class with teacher guidance.

The Purpose of Study and Research Questions

The purpose of this research study was to investigate the benefits of
the Flipped Learning model in increasing student test scores and student
participation in solving application problems with explanation and reasoning
in the high school precalculus classroom. This research study investigated the
answer to two main questions:
(1) Will the flipped classroom model result in higher test scores in
comparison to the traditional classroom model in the high school precalculus
class?
(2) Will students participate in application problem solving and
explain their work with mathematical reasoning to a greater extent with the
flipped classroom compared to the traditional teacher-centered classroom?

Theoretical Framework

Two high school Chemistry teachers, Johnathan Bergmann and Aaron

Sams started recording videos as lectures in 2007 to free more class time to
work on meaningful and engaging assignments. It later came to be called the
Flipped Learning or Flipped Classroom (Noonoo, 2012). The definition of
the flipped classroom used in the context of this literature review will include
any form of the flip model as long as the main idea is: students will be
exposed to the lecture entirely or partially outside of the classroom using
video lectures or a combination of video and reading lectures, then they will
use class time to work on meaningful challenging problems in a group setting
with teacher guidance. It could be completely student driven with teacher as a
facilitator, or the teacher may interject mini lectures as needed (Honeycutt,
2014.) However, there seems to be a consensus on the lack of research to
prove the efficacy of the flipped classroom model. The research conducted in
higher learning institutions and research that requires the purchase of a
64 Flipped Learning

particular resource is beyond the scope of this literature review. Higher

education research does not necessarily apply to younger students in the
secondary schools who lack the maturity and experience their college
counterparts have. And, since this teaching model requires a large investment
of time and money by teachers and school districts, further research is needed
to support the theory of flipped classrooms as a way to promote higher
cognition and student engagement.

History of Flipped Learning

According to the definition presented by the Flipped Learning Network
(founded by Sams & Bergmann, 2012):
Flipped Learning is defined as a “pedagogical approach in which direct
instruction moves from the group learning space to the individual learning
space, and the resulting group space is transformed into a dynamic,
interactive learning environment where the educator guides students as they
apply concepts and engage creatively in the subject matter (Flipped Learning
Network, 2014). By moving from a flipped class to actively engaging in
Flipped Learning, teachers are able to implement new or various
methodologies into their classrooms. It frees up class time, allowing for more
individual and small group instruction.” (Yarbro, Arfstrom, McKnight, &
McKnight, 2014, p. 5)

Bishop and Verleger (2013) stated in a paper they presented at a conference in

Daytona Beach, Florida, a comparison between two models. The first is
merely a description of reordering the events of what takes place in a
traditional classroom. It offered video lectures away from the classroom, and a
combination of practice exercise and problem solving in class. The second
offered Video Lectures with closed-ended quizzes and practice exercises away
from class, and group-based plus open-ended problem solving in class. The
latter example seems to be more like the Flipped Learning Network’s
definition. Bishop and Verleger (2013) agreed also that the second model is
what educators call the flipped classroom citing student-centered learning
theories based on the works of Piaget(1967) and Vygotsky(1978) (as cited in
Bishop & Verleger, 2013).
But flipped learning started earlier at the university level. The Harvard
University Physics professor Mazor started a version of this model in the early
1990s (as cited in Moore, Gillett, & Steele, 2014). More recently, when
Salman Khan (an MIT graduate) started the Khan Academy producing
instructional videos on You Tube in 2006 and others followed in his footsteps,
it made using the flipped classroom more possible for other high school
teachers.
Zeineddine 65

Technology and Equity: a Limitation

Equal access for all students to computers (or tablets or smart phones)
and internet is a major concern we as educators are faced with when
contemplating the use of the flipped classroom. As one survey conducted by
the National Education Association in collaboration with the American
Federation of Teachers (2008) reveals that teachers and students in public
schools have some access to internet and computers, but not enough access to
use them as effective learning and teaching tools. Furthermore, educators are
concerned about the disparity in the levels of accessibility for students in
different community types or levels (National Education Association, 2008).
A teacher has to make sure students have equal access to the technology
needed for the implementation of the flipped model. The new trend in
education is to supply students with tablets to make the playing field equitable
for all students. But, this is not an immediate solution.

Cognition and the Flipped Classroom Improve Student Understanding

Another component of the effectiveness of technology away from the
classroom as a learning tool is what happens in the classroom and how
learning takes place. Lave and Wagner (1990, 1991) described situated
cognition as the learning that happens in context and culture. Therefore, it is
important to know how learning takes place and not only where it happens (as
cited in Szymanski & Morrell, 2009). When the learner (student) develops
skills in authentic activities, the goal is to extend this acquired knowledge for
future problem solving (Szymanski & Morrell, 2009). The student will learn
and retain the learning more readily in the classroom with the support of peers
and the teacher. This is another compelling reason to devote more class time
for problem solving rather than pure lecture. To further support this theory,
four benefits of situated cognition were discussed by Collins (as cited in Brill,
2001) as a basis for student learning. The first is students learn about the
application of knowledge. The second is students are provided a dynamic
social environment to encourage invention and creativity in problem-solving.
The third is students are more likely to recognize and find from teacher and
peers the implications of what they learn. And the fourth is that students are
learning with teacher and peer support to insure immediate feedback. This
leads to learning the appropriate and correct methods for solving problems.
The social and diverse context promotes lasting transferable knowledge that
can be applied in future situations. The flipped classroom model promotes
student-centered learning and is one of the applications based in part on
situated cognition; which leads us to the next section.

Benefits of Active Learning in Problem Solving

Freeman, Eddy, McDonough, Smith, Okoroafor, Jordt, and Wenderoth
(2014), authors of Active Learning Increases Student Performance in Science,
Engineering, and Mathematics claimed that this study was the largest to date
66 Flipped Learning

on undergraduate STEM education. The study is an analysis of the benefits of

active learning over traditional learning in the STEM education for
undergraduate students. The authors of this article conducted a survey and as
a result arrived at this definition of active learning: “Active learning engages
students in the process of learning through activities and/or discussion in
class, as opposed to passively listening to an expert. It emphasizes higher-
order thinking and often involves group work.” And they used this definition
of traditional lecture: “…continuous exposition by the teacher” (Freeman,
Eddy, McDonough, Smith, Okoroafor, Jordt, & Wenderoth (2014, p.8413-
14).

Method

Site
The school site is a High School located in an inner city in southern
California. It is one of the 10 largest urban schools in the country, well known
for its athletics and academics. The school implements the Smaller Learning
Communities system to promote academic achievement and equity for such a
large high school. Students are placed into a Smaller Learning Community
which is managed by one counselor. Students within the smaller community
usually have the same teachers. The exception is electives and mathematics
classes which include students from different Smaller Learning Communities.
The most recent statistics for school enrollment by grade level (2013-2014)
shows a total enrollment of 4497 students for grade levels 9-12 with about
24% African American, 20% Asian, 35% Latino, 10% white, and 11% of other
ethnicities. Also, about 59% are socioeconomically disadvantaged, 13% are
English learners, and about 9% are students with disabilities.

Subjects
Participants in this research study were two 11th grade precalculus
classes of one teacher. Students were of mixed ethnic and socio-economic
backgrounds reflecting the demographics of the school. They were designated
as honors level with mixed levels of mathematical background. Students did
not have similar past experiences in learning mathematics as they had
different teachers in previous grades, and in many cases came from different
middle or high schools. One class consisted of 14 girls and 9 boys, and the
other class consisted of 11 girls and 9 boys. Most students were involved in
extra-curricular activities which took them away from the classroom
occasionally. This affects the classroom experiences of students who miss the
problem-solving group sessions since the activities could not be recreated to
make up for missed work.
Zeineddine 67

Procedure
The first unit taught to both classes for the duration of 3 weeks was on
sequences and series, the Binomial Theorem, counting, and probability. One
precalculus class (group 1) was taught the traditional way of lecture,
examples, guided practice, and homework with some group work. Students in
the other class (group 2) were taught the same unit implementing the Flipped
Learning method for the same duration. Students of both groups 1 and 2 were
given the same pretest composed of 20 questions (chosen from the precalculus
text book test bank used at the school) before the unit was taught. The
traditional, teacher centered approach for group1 was a lecture with examples,
a guided practice, and then homework was assigned. The next day students
had a limited portion of class time to ask questions, teacher answered the
questions, retaught where necessary, and then the process was repeated for the
next part of the unit. As teacher lectured, students were asked questions to
connect to prior learning, teacher checked for understanding, and students
practiced with examples. Students in the second group were given a short 5-10
minute lecture to introduce a section of the unit and they were asked to take
notes on the new concepts using online resources and a note-taking guide as
homework. Then they were asked to attempt the corresponding homework
assignment. The next day, teacher checked for understanding of key concepts
using questions and homework problems for a few minutes. Teacher also
provided students with any needed clarification on the lesson in the form of a
whole group discussion. They checked and corrected their homework with
help from peers and teacher as needed. When the teacher was satisfied that
students understood the basic concepts, they were asked to work in small
groups on carefully selected complex application problems. Then they shared
results as a class with teacher guidance to insure accuracy of responses and
solutions. Students of both groups (1) and (2) were given a short quiz after
each section which they corrected in class for immediate feedback. Students in
the second group consistently did more group work and problem solving
sessions than group one.
Another unit on conic sections was taught using a traditional teacher centered
model for group (1) and the flipped learning method for group (2). A similar
instructional process was implemented as with the previous unit. This unit
consisted of four sections taught over a period of three weeks.
Also for this unit, they were given 3 MSAR (model-strategy-application with
reasoning) work sheets (An & Wu, 2014) where they independently had to
model, explain, solve, and justify their work with mathematical reasoning.
One worksheet was given after each of the first three sections. Students had a
few minutes to ask questions on the MSAR sheets before turning them in.

Instruments and Data Collection

At the conclusion of the first unit, both study groups were given the
same 20-question pretest which was administered prior to teaching this unit as
68 Flipped Learning

a post test. It consisted of problems chosen from the test bank of the text book
(Larson, Hostetler, & Edwards, 2015) used in part for teaching the class. The
test given for the second unit consisted of a total of 10 problems in the form of
matching, True/False, multiple-choice, short response, extended response, and
a performance task. The first 9 questions were based on problems from the
same test bank mentioned above, but were slightly modified by the teacher.
Another part of the same test was given on another day as a multi-step
“performance task” style problem created by the teacher, based loosely on a
problem from the test bank. Students were given enough time to complete this
problem as needed.
The extent of student participation in problem solving was measured
quantitatively using problems given in MSAR format (An & Wu, 2014).
Three problems in the form of MSAR worksheets were assigned right after the
respective section, then collected and graded only for participation and
showing work with reasoning, and not for accuracy. One MSAR was on
circles, the second one was on parabolas, and the third one was on Ellipses
(see Appendix A.)
A rubric by Wu and An (2016) was used to score and code the MSAR
work sheets (see Appendix A.) The scores of the worksheets were analyzed
for Modeling, Strategy, and Reasoning, omitting the Application part for time
constraints.

Data Analysis
Independent T-Tests were used to measure the difference in
performance between the traditionally taught group and the flipped learning
group.
In order to minimize the threat to validity, the test scores were
analyzed on improvement of the posttest scores over the pretest scores. The
pretest/ posttest results were compared using an independent t-test to measure
the difference in test scores between group (1) and group (2).
The test for unit 2 was analyzed using descriptive statistics and an
independent t-test. The two parts of the test were analyzed separately The
results show the varying degrees of achievement among the two groups of
students and the impact the Flipped Classroom model has made on student
learning as opposed to traditional teaching.

Findings

The following are tables of the SPSS analysis of data collected on the
two tests and three MSAR worksheets. Keeping in mind that group (1) was
mathematically stronger than group (2) since more students from group (1)
came from accelerated Algebra 2 while more students in group (2) came from
the regular Intermediate Algebra class which was not designated as an honors
class. The other reason group1 students are stronger is they generally do
Zeineddine 69

homework more consistently. The Flipped Classroom model was a natural

consequence for the group (2) students since they needed to work in class with
peer and teacher support they were not successfully completing assigned
homework.
Table 1 of the descriptive group statistics shows that class 1 of 23
students had a mean pretest score of M=26.96, SD=15.28 and class 2 of 20
students had a mean score of M=24.25, SD=13.01. The posttest score of class
1 of 22 students (due to absence) had a mean score of M=75.36, SD=15.14
and class 2 of 19 students (one absence) had a mean test score of M=81.16,
SD=11.04. Ch 9 test for class 1 had a mean test score M=69.46, SD=16.53,
class 2 with a mean test score M=76.81, SD=12.25. Ch 9 performance task
mean score of class one (23 students) was M=10.04, SD=3.59 and the mean
score of class 2(20 students) was M=13.40, SD=3.89.

Table 1
Group Statistics for the Pretest and Posttest for Two Classes

Group Statistics
Group# N Mean SD
pretest 1 Class 1 23 26.96 15.281
2 Class 2 20 24.25 13.006
post test 1 Class 1 22 75.36 15.136
2 Class 2 19 81.16 11.037
ch 9 total percent 1 Class 1 23 69.45 16.530
2 Class 2 20 76.81 12.250
ch9 perf 1 Class 1 23 10.04 3.586
2 Class 2 20 13.40 3.885

Notes: pretest refers to the test on sequences and series; posttest is the same
test given after the conclusion of the unit. Percent ch 9 refers to the test on
conics consisting of 9 questions graded from 60 points changed into a
percentage score. Ch 9 perf refers to the performance task graded from 20
points. Class1 and class2 refer to group1 and group2.

Table 2 shows the results of the independent sample t-test analysis.

Table 2 shows there is no significant difference in mean scores for the pre-
test between two groups (t(41)=.62, p=.54); also for the post-test there is no
significance in mean scores between two groups (t(39)=1.38, p=.175); there is
no significant difference in mean scores for the chapter 9 test scores between
two groups (t(41)=1.64, p=.109); For the performance task, there is significant
difference in the mean between two groups (t(41)=2.95, p=.005)
70 Flipped Learning

Table 2
Independent Samples Test for 2 Classes- Group1 and Group2 (pre-test
and post-test)
Levene's Test t-test for Equality of Means

F Sig. t df Sig.

pretest Equal variances

.610 .439 .620 41 .539
assumed

posttest Equal variances

1.423 .240 -1.38 39 .175
assumed

percent ch 9 Equal variances

.918 .344 -1.63 41 .109
total percent assumed

ch9perf Equal variances

.099 .754 -2.94 41 .005
assumed

Table 3 displays the descriptive statistics of the modeling part of the 3

MSAR worksheets. Modeling refers to the visual representation of the
problem to be solved. The mean scores of class 1 of 21 students and class 2 of
18 students(not all students turned in the worksheets) for the first 2
worksheets show very small difference at M=3.86, SD=.36 and M=3.94,
SD=.24 for the first worksheet, and M=3.52, SD=.60 and M=3.67, SD=.46 for
the second worksheet. The mean scores of the last worksheet show more of a
difference between class 1(M=3.67, SD=.48) and class 2(M=3.94, SD=.24.)

Table 3
Descriptive Statistics for MSAR 1, 2, and 3 for “Modeling”
Group Statistics
Group# N Mean Std. Deviation
MSAR1modeling 1 Class 1 21 3.86 .359
2 Class 2 18 3.94 .236
MSAR2modeling 1 Class 1 21 3.52 .602
2 Class 2 18 3.67 .485
MSAR3modeling 1 Class 1 21 3.67 .483
2 Class 2 18 3.94 .236
Notes: Table 3 shows the results of the modeling part only of the MSAR
worksheet.

Table 4 displays the results of the t-test. This is comparing the

worksheet scores of the modeling part of all 3 worksheets for the two groups
being studied, class1 (control group) and class2 (flipped class.) Worksheet 1
was on circles, the first section of the conics unit. Table 4 shows there is no
significant difference in mean scores between two groups (t(37)=.88, p=.38),
Zeineddine 71

Worksheet 2 was on parabolas, also the first section of the conics unit.
Table 4 shows there is no significant difference in mean scores between two
groups (t(37)=.81, p=.43),
Worksheet 3 was on ellipses, the second section of the conics unit.
Table 4 shows there is significant difference in mean scores between two
groups (t(37)=29.4, p=.027).

Table 4
Independent samples test for MSAR 1, 2, and 3 for “modeling”
Levene’s Test t-test for Equality of Means
Sig. (2-
F Sig. t df tailed)
MSAR1modeling Equal variances
3.413 .073 -.882 37 .384
assumed
MSAR2modeling Equal variances
2.440 .127 -.807 37 .425
assumed
MSAR3modeling Equal variances
32.683 .000 -2.221 37 .033
assumed

Table 5 displays the descriptive statistics of the strategy part of the 3

MSAR worksheets. Strategy refers to the solution strategy of setting up
mathematical equations and such to arrive at a solution of the given problem.
The mean scores of class 1 of 21 students and class 2 of 18 students(not all
students turned in the worksheets) for the strategy part of the 3 worksheets
show very small difference at M=3.90 and M=3.89 for worksheet 1, M=4.00
and M=3.89 for worksheet 2, and M=3.86 and M=3.67 for worksheet 3.

Table 5
MSAR Group Statistics for “Strategy”
Group Statistics
Group# N Mean Std. Deviation
MSAR1strategy MSAR 1-strategy 1 Class 1 21 3.90 .301
2 Class 2 18 3.89 .471
MSAR2strategy MSAR 2-strategy 1 Class 1 21 4.00 .000
2 Class 2 18 3.89 .323
MSAR3strategy MSAR 3-strategy 1 Class 1 21 3.86 .478
2 Class 2 18 3.67 .686

Table 6 displays the results of the t-test. This is comparing the

worksheet scores of the strategy part of all 3 worksheets for the two groups
being studied, class 1 and class 2. Worksheet 1 was on circles. Table 6
shows there is no significant difference in mean scores between two groups
(t(37)=.127, p=.899)
Worksheet 2 was on parabolas, also the first section of the conics unit
Table 6 shows there is no significant difference in mean scores between two
groups (t(37)=1.578, p=.123.)
72 Flipped Learning

Worksheet 3 was on ellipses, the second section of the conics unit

Table 6 shows there is no significant difference in mean scores between two
groups (t(37)=1.017, p=.316.)

Table 6
Independent samples test for MSAR 1, 2, and 3 for “Strategy”
Levene's Test t-test for Equality of Means
Sig. (2-
F Sig. T df tailed)
MSAR1 Equal variances
strategy assumed .121 .730 .127 37 .899

MSAR 2 Equal variances

strategy assumed 13.011 .001 1.578 37 .123

MSAR3strategy Equal variances

assumed 3.831 .058 1.017 37 .316

Table 7 displays the descriptive group statistics of the reasoning part of the 3
MSAR worksheets. Reasoning refers to the explanation of the visual model,
the solution strategy, and the interpretation of the answer of the given
problem. The mean scores of class1 of 21 students and class2 of 18 students
(not all students turned in the worksheets) for the reasoning part of the 3
worksheets show a difference at M=3.8, SD=.401 and M=3.61, SD=.61 for
worksheet 1, M=4.00, SD=0 and M=3.89, SD=.32 for worksheet 2, and class
1 of 22 students with M=3.45, SD=.51 and class 2 of 18 students with
M=3.39, SD=.78 for worksheet 3.

Table 7
MSAR Group Statistics for “Reasoning”
Group Statistics
Group# N Mean Std. Deviation
MSAR1 reasoning msar 1- 1 Class 1 21 3.81 .402
reasoning 2 Class 2 18 3.61 .608
MSAR2 reasoning msar 2- 1 Class 1 21 4.00 .000
reasoning 2 Class 2 18 3.89 .323
MSAR3 reasoning msar 3- 1 Class 1 22 3.45 .510
reasoning 2 Class 2 18 3.39 .778

Table 8 displays the results of the t-test for reasoning of MSAR 1, 2,

and 3. This is comparing the worksheet scores of the extent to which students
explained and justified their visual model and solution strategy/answer in
writing.
Worksheet 1 was on circles. Table 8 shows there is no significant
difference in mean scores between two groups (t(37)=.020, p=.231)
Zeineddine 73

Worksheet 2 was on parabolas, also the first section of the conics unit.
Table 8 shows there is no significant difference in mean scores between two
groups (t(37)=1.578, p=.123)
Worksheet 3 was on ellipses, the second section of the conics unit
Table 8 shows there is no significant difference in mean scores between two
groups (t(38)=.321, p=.750)

Table 8
Independent samples t-test for MSAR 1, 2, and 3 for “reasoning”
t-test for Equality
Levene's Test of Means

F Sig. t df Sig.
MSAR1reasoning Equal
variances 5.910 .020 1.218 37 .231
assumed
MSAR2 Equal
reasoning variances 13.011 .001 1.578 37 .123
assumed
MSAR3 Equal
reasoning variances 1.388 .246 .321 38 .750
assumed

Summary of Major Findings

The results of the analysis for the pre-test and post-test between the
two groups shows a significant difference between the results of the two
groups with the flipped learning group scoring higher overall on the post-test.
This is remarkable given that the control group was at a higher level of skills
to begin with and they generally do homework more consistently. This result
is encouraging for the use of the flipped approach.
The analysis results of the first part of the chapter 9 test show
significance between the 2 groups. Although better results were expected for
the control group since they were at a higher skills level and they did more
homework which meant they would remember formulas and rules more than
the flipped group. Since the first part of the test was geared more toward
application of the rules and formulas, it stands to reason that doing more
homework will produce better results. But, again the flipped group performed
better overall proving that in-class learning combined with homework is more
effective than traditional learning.
The analysis results for the performance task were in favor of the
flipped learning again. The performance task required remembering and
integrating prior knowledge into solving the problem. Higher scores for the
flipped learning group were expected since students had more practice in
problem solving and were more willing to try. The expectation was met;
students who were considered at a “higher level” in the control group tried
only what they knew was correct and did not even try to approach the parts
74 Flipped Learning

they were not sure of. Most students in the flipped group tried all sections of
the task. Many of them did all parts correctly although they were not graded
for correct responses, but were graded only for trying and explaining
mathematically. The results of the performance task as part of the test were a
way of monitoring the effects of flipped learning on independent problem
solving since most of the problem solving was done within groups and with
teacher support. The results were extremely satisfactory.
The analysis for the MSAR worksheets shows different results from
the test results. The control group did slightly better than the flipped group.
One of the reasons could be that the student population of the control group
generally does better on completing assignments than the flipped class
students. Since homework and classwork grades were not as consequential to
the grade as test grades many students of the flipped class chose not to spend
very much time on elaborating and explaining and were satisfied with the
minimum credit. It is significant to mention that those students would have
most likely chosen not to even try prior to the flipped treatment.
While the performance task was independent work and part of a test,
the MSAR worksheets were given in class and students had the option of
collaboration. The results of the performance task show clearly the advantage
of the flipped over the traditional teaching. For the traditional class, no student
was able to do all parts correctly, and most students completely skipped one or
more parts of the problem.

Discussion

The idea for this action research on the flipped classroom is directly
related to the need of an effective method of teaching mathematics for
understanding. Existing literature on flipped learning or the flipped classroom
showed potential for this teaching pedagogy in accomplishing this goal. Since
there was not enough research done at the high school level, it was necessary
to find ways to implement this flipped classroom using ideas from research
done at the college level in combination with existing literature on cognition
and the way students learn. According to Lave and Wagner (1990, 1991)
situated cognition is the learning that happens in context and culture. The
goal is to extend this acquired knowledge from skills learned for future
problem solving (Szymanski & Morrell, 2009). The idea of learning with
peer support and teacher support as a more effective process for application
and retention of skills and concepts leads to the necessity of devoting
classroom time for problem solving. Since classroom time is limited, a
compromise has to be reached to strike a good balance between problem
solving sessions and lectures. The flipped classroom seems to be one way to
create this balance without compromising the quality of student learning; but
with the lack of existing literature on flipping the high school mathematics
class, the need for this action research was evident. What applies to college
Zeineddine 75

students learning mathematics might not be suitable for high school students.
High school students have more demands on their time with packed academic
and extra-curricular activities, and they do not enjoy the flexibility that college
students might have in scheduling their classes. More importantly, high school
students might not have developed enough skills as college students to learn
independently from online lectures.
Although the results of the data analysis did not show significant
difference in test scores in every case between the flipped and traditional
classrooms, the flipped classroom was of benefit for the group of students
who practiced it. Students of the flipped classroom were exposed to learning
in a supportive group environment with rigorous and complex problem
solving sessions. Often, there were whole class discussions as a direct result of
the group sessions which constitute active learning. Active learning increases
student engagement (Freeman et al., 2014) and when students are engaged
they stand a better chance of understanding and retaining. Performance task
scores of both tests show a significant difference in favor of the flipped
method over the traditional method. Teachers might find that change from
repeatedly modeling examples for students is necessary if students are to learn
to be independent problem solvers. The question remains as to how each
teacher will accomplish this task. Specifically, the result of the performance
task is evidence that students who would have not even attempted the task, not
only attempted it, but most students arrived at logical and viable answers. The
performance task was given on a test without peer support; this is what makes
it significant. All the work done in groups leading up to the test produced the
desired effect. This finding could impact the future of teaching mathematics
in a high school classroom.

Recommendations and Conclusions

In this study, all students had been asked to do the MSAR worksheets
on Chapter 9 Conic Sections concepts of circle, parabola, and ellipse several
times throughout the school year leading up to that point. Therefore, all
students had equal time to work on MSAR worksheets and they all understood
what was expected of them. Although the analysis was slightly in favor of the
traditional classroom in case of the MSAR results, in reality- the flipped
learning students were the ones who traditionally would not have tried doing
the MSAR type problems before the flipped treatment. The control group
students are the ones who went beyond the required standard to complete
assignments to the best of their ability. Since all students were consistently
required to show and explain all work for the months preceding the
assessments, all students came through and showed improvement in
discussing and showing work. Students needed the constant exposure to real
world application problems in order to become comfortable with them. They
also needed to work in a safe environment where they could ask questions,
76 Flipped Learning

make mistakes, and get answers knowing that making mistakes is part of the
learning process (Furner & Gonzalez, 2011).
This study was conducted over one semester although preparations for
it started the previous semester. The teacher/researcher set the plan in motion
by starting the flipped learning very slowly for the first semester. Students
needed time to adjust to the change in the learning and teaching style. For
most students, the change was a relief from having to do difficult mathematics
as homework, but for a few students, it was still a difficult transition and they
were not willing to accept the change. Fewer students still considered this
method as the teacher “not teaching” since they were not watching endless
examples worked out for them to emulate while doing homework. The flip
was modified by offering short lectures and a few examples to model basic
skills since high school students might not be ready to learn a new concept
independently. At home, students were asked to work for an average of one
hour on a particular assignment including note-taking and problem solving.
By doing so, the idea was they will have well thought out questions which-
when answered in class- will help clarify concepts for each individual as
needed. The questions might be answered by teacher or peers depending on
the complexity and depth of the required explanation.
This study has many implications for future teaching. Since it was in
the experimentation stages for the first school- year, two tests and 3 MSAR’s
were analyzed. Further study is needed to measure the effect of the flip over a
whole school-year. Focus could be more on doing class work and less
assigned homework. This could prove beneficial for younger students who
need more teacher input. To avoid misinterpretation, class work needs to be
in the form of complex and rigorous problems or projects and not practice of
exercises and simple skills applications. The simple skills applications must
be part of what students do at home to save class time for the more difficult
problem solving and concept explorations. Student mathematical strengths
and weaknesses are assessed the first few weeks of school through class work
and teacher monitoring. Students are introduced to the rigor and complexity
expected of them through class work which helps create a safe work
environment rather than create anxiety with difficult homework exercises. A
slow and easy transition to the flip must be almost seamless so as to reduce
student resistance to the change in the status quo of years of learning in the
traditional classroom. The next question to be answered would be: how
effective is the flip in promoting accuracy in solving problems and
mathematical reasoning?
The results of the study were favorable. Student engagement and
understanding increased more with the flipped classroom over the traditional
classroom. The results are encouraging and carry implications for further
implementation of the flip as a success in the high school mathematics
classroom. Caution is warranted since the study was done with a small
number of precalculus students who were designated as honors students. The
Zeineddine 77

results may not be generalized for high school students in lower grades or of
average to lower than average mathematical abilities. If the flip is to be
implemented with a different grade level and/or different mathematical
abilities, adjustments and changes in the implementation process must be
considered. The flip in some form might still be an option with any level class
given the potential benefit for increase in student mathematical understanding
and participation in problem solving. Also the MSARs were not graded on
accuracy to encourage students to participate in showing and explaining their
work. For future studies, grading for accuracy in the second semester when
students have had enough practice in showing work could be a goal.
This study is significant to high school teachers who are considering
the Flipped Classroom as a teaching model. There are not enough studies
conducted on flipping the mathematics classroom at the high school level.
Research done on college math classes might not be applicable to high school
students for at least two reasons. The first: mathematics students in college
have made the choice to be in those classes, and therefore are motivated to
succeed. High school students are still at an exploratory stage in their math
education. They may or may not choose a college path which requires
advanced math skills. The second: most college students are mature enough to
take the responsibility for their own learning and would most likely do the
online assignments and be prepared for the classroom discussions, while very
few high school students would accept the responsibility of learning any
concepts at home without initial teacher input. This study is also significant
for this teacher/researcher, since it is an investigation of the possible benefits
of flipping the classroom for the Precalculus students who were the subject of
the study and how to apply it to future classes. The teacher’s job is to prepare
Precalculus students for the rigor and complexity of an Advanced Placement
Calculus course. Most Algebra teachers focus on simpler procedures and skills
and do not train students in complex problem solving, may be due to time
restrictions. If the time restriction is the reason teachers do not engage
students in enriching problem solving sessions, then the Flipped Learning
model could be an answer. As students become confident in their ability to
solve math problems with reasoning, they are more likely to choose a math
related field of study. There is a national need for students who choose a
STEM field of study identified by several educational, business, and
government entities. Therefore, this research is significant in influencing
future policy decisions in the form of financial support for technology and
teacher training.
It was evident that students had to accept the change in the teaching-
learning style for the method to be effective. Administrators and parents need
to be educated on the flipped classroom through research and literature which
may result in support for the teachers willing to implement it. In a Doctoral
dissertation on flipped learning Overmyer stated that Byron high school in
Minnesota reported increased scores in mathematics for students after
78 Flipped Learning

implementing the flip and students reported a preference for the flip over the
traditional learning method (Overmyer, 2014). This study shows that students
who were just above average in study skills were more willing to go along
with this learning approach than students who had excellent study skills.
Students who completed all homework ahead felt they did not need the
problem solving session; instead they wanted the teacher to give them a quick
answer to their question just the way they have been learning mathematics all
along. All students can benefit from flipped learning as the results of the
performance task show, and it is important to find ways to convince them to
try. Forcing students into a particular method of learning might not be
productive.
This research showed that the flipped classroom is a viable
pedagogical option to use at the high school level. Students who are confident
in their math abilities are more likely to major in STEM fields of study.
Therefore, it is necessary to conduct more research to further determine how
the flipped learning might be implemented for different levels of high school
math students.

References

An, S., & Wu, Z. (2014). Using the evidence-based MSA approach to enhance
teacher knowledge in student mathematics learning and assessment.
Journal of Mathematics Education, 7(2), 108-129.
Bergmann, J., & Sams, A. (2012). Flip your classroom: reach every student in
every class every day. International Society for Technology in
Education. Eugene, Or. : Alexandria, VA.: International
Society for Technology in Education
Bishop, J. L., & Verleger, M. A. (2013, June). The flipped classroom: a
survey of the research. In ASEE National Conference Proceedings,
Atlanta, GA.
Common Core State Standards Initiative. (2010). Common core state
standards for mathematics. Washington, DC: National Governors
Association Center for Best Practices and the Council of Chief State
School Officers.
Van Roekel, N. P. D. (2008). Technology in schools: the ongoing challenge of
access, adequacy and equity. National Education Association,
Washington DC.
Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N.,
Jordt, H., & Wenderoth, M. P. (2014). Active learning increases
student performance in science, engineering, and mathematics.
Proceedings of the National Academy of Sciences, 201319030
Zeineddine 79

Furner, J. M., & Gonzalez-DeHass, A. (2011). How do students’ mastery and

performance goals relate to math anxiety. Eurasia Journal of
Mathematics, Science & Technology Education, 7(4), 227-242.
Hamdan, N., McKnight, P., McKnight, K., & Arfstrom, K. (2013). The
flipped learning model: a white paper based on the literature
review titled a review of flipped learning. Arlington, VA: Flipped
Learning Network.
Honeycutt, B., & Garrett, J. (2014). Expanding the definition of a flipped
learning environment. Faculty Focus. Magna publications
Moore, A. J., Gillett, M. R., & Steele, M. D. (2014). Fostering student
engagement with the flip. Mathematics Teacher, 107(6), 420-425.
Overmyer, G. R. (2014). The flipped classroom model for college algebra:
Effects on student achievement (Doctoral dissertation, Colorado State
University).
Schwerdt, G., & Wupperman, A.C. (2011). Is traditional teaching really all
that bad? A within-student between- subject approach. Economics of
Education Review, 30(2), 365-379.
Smarter Balanced Assessment Consortium. (2012). Smarter balanced
assessments. Retrieved from: http://www.smarterbalanced.org/
Szymanski, M. & Morrell, P. (2009). Situated cognition and technology. The
International Journal of Learning, 15(12) 55-56.
Talbert, R. (2014). Inverting the linear algebra classroom. Primus, 24(5), 361-
374. DOI: 10.1080/10511970.2014.883457
Watkins, J., & Mazur, E. (2013). Retaining students in science, technology,
engineering, and mathematics (STEM) majors. Journal of College
Science Teaching, 42(5), 36-41.
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w%20June%202014.pdf.

Author:

Dunia Zeineddine
Long Beach Unified School District, California
Email: Dunia.zeineddine@gmail.com

Appendix A
 80 Flipped Learning

Rubric for MSAR worksheets:

Level Modeling Strategies of Computation Reasoning

Model used highly efficient and No computational errors and The process of MSA
meaningful, revealing used a flexible or creative explained with
Level 4 comprehensive understanding strategy in computation, academic language and
revealing complete deep understanding.
understanding of solving
Appropriate model used, and the No computational errors, but Appropriate
Level 3 process of modeling solved problem by routine way explanation.
demonstrated or only by trial and error
Appropriate model used, but Only few computational errors, Incomplete reasoning.
either not fully demonstrated, or but followed rules and formulas
Level 2 possibly based the operation on computations (routine way),
only, did not show the process of or only by trial and error
conceptual developing
Either no model or model Either missing computation or Nothing or
Level 1
completely inappropriate many computational errors inappropriate.
MSAR Rubric (Wu & An 2016)

MSAR 1-student work sample # 1