Mastery Learning of Fractions Research Project
Reid Foxwell
Stevenson University
Introduction
My
research question is what impact would mastery learning assessments and
techniques have on student’s mathematical foundations of fractions? I chose
this as my research topic because of my love of mathematics as well as noticing
the struggles of college students in tutoring sessions with fractions whose
mathematical rules seem fairly logical to follow even for students who are not
adept in mathematics (Peck & Jencks, 1981). There is no current plans to
adjust the learning process of fractions when there is an obvious and
significant struggle among students in the comprehension of fractions (Brown
& Quinn, 2007). I feel that if students are in a mathematics class and are
told of the rules of fractions and given practice with them, then this material
should not be difficult, yet this is not the case for most students.
Literature Review
Brown
& Quinn (2007) discuss the research behind teaching fractions and whether
fraction should be taught later in a student’s educational career as well as
what manner it should be taught in. For instance, this article cites the
psychological processing research of Piaget and formal operational thinking and
how this does not appear in most adolescents until 11 or 12. This article
states that this process of thinking is the same mental process used to perform
tasks related to fractions. Moreover, this article has found that there is no
current plan of action to correct this issue of mathematics education in
curriculums. One plan of curriculum improvement simply suggests adding more
time to be spent on the teaching and practice with fractions.
The
article summary by Mills (2011) looks at how students start with the
organization of whole numbers or discrete numbers and then they develop
fractions and decimals or continuous numbers. This mental organization of
learning for students takes time to develop and grow. This article looks at the
idea of implementing movement into the learning of fractions where students
work in group and pose their arms and bodies in different ways to create
fractions. This lesson was performed with students between their fifth and
eighth year of school and seemed to add discussion between groups and arguing
which fueled a more student led learning experience until an agreement was
made.
The
article by Odeh & Qareen (2009) examines the use of online learning tools
to help involve students more when compared to typical school lectures. The
results of using e-learning methods for fraction suggested that students did
better in this method of learning and showed more positive attitudes towards
fractions. This suggests that the options of e-learning provide teachers with a
new option to integrate classroom lessons while individualizing the learning
approach for each student. The study looks at traditional lectures and many
students felt that they enjoyed e-learning and learned more while working on
the computer.
This
article by Peck & Jencks (1981) looks at the interviews of students and
asking them questions related to fractions and various mathematical operations.
20 sixth grade students were interviewed while their answers were compared
against the answers provided by over hundreds of prior interviews of the same
type. From the interviews, it was observed that less than 10% of students held
a firm understanding of fractions even with 3-5 years of exposure to fractions
already. This suggests that a deeper
foundation and focus on fractions may be necessary to reach learning goals. This
article shows how even in 1981 there was a noticeable issue with the
foundations of students’ knowledge of fractions.
This
article by Yujing & Yong-Di (2006) looks at the bias of students who can
comprehend counting numbers yet struggle with fractions and other rational
numbers. Many students struggle with fractions because unlike whole numbers,
when the number for a fraction gets larger, the value gets smaller which some
students struggle to remember. This article found that much more time needs to
be dedicated to lessons involving fractions and other continuous number
processes because of the tendency of students to struggle with this new
concept. In addition, this study used a linear model to demonstrate negative
numbers which could be helpful in explaining continuous numbers and fractions.
Research Study Design
The solution for
improvement that I would like to propose would involve the teaching of
fractions to students with mastery learning assessments and an emphasis on
growth mindset as well. This would involve the use of online quizzes like Khan
Academy so students can continue to practice on their own and formative
assessments that are given to students without the punishment of failure, but
the reward of concept mastery as the ultimate goal (Odeh & Qaraeen, 2009).
Currently, I would go
about this study by examining studies on student’s struggles with fractions and
what issues they typically run into and how these issues can be fixed through
different engaging lessons or simply more practice (Yujing & Yong-Di, 2005). I would use several groups of either middle school
or high school age students in algebra and give them an initial assessment of
their abilities with fractions as a baseline. Then, I would present the created
lessons on fractions and provide the additional practice exercises to the
students. Throughout the lessons, I would provide students with multiple
opportunities to take the end formative assessment and receive a grade of
either “Mastered, Sufficient, Almost there, or Needs additional attention”.
Students who receive a grade of mastered or sufficient have learned the
material to the level necessary for mathematical success. Students could
practice drills in groups together so students who have mastered the material can
help the students who need additional attention (Mills, 2011).
I would allow the students to work on this progress for the same amount of time
that average teacher spends on fractions with their students in a typical
algebra class by taking a sample of at least 30 teachers. Then I would compare students’
results from the initial test to a final test to see which group performed
better and I would need a large sample size for data validation and significant
conclusions.
My control group would be
several classes where teachers are simply asked to administer an initial test
and the final test after they have covered the material on fractions that they
want with their students.
In order to have a valid
study, a sample size of at least 30 is usually required to make any sort of
significant conclusions, but in this case, I feel that a much larger sample
would be more accurate and provide a lower level of error within the control
and test groups scores. In my study, I would prefer to have sample sizes of at
least 100 for both the control and test groups, so a total study of students of
at least 200 would be ideal. These numbers are very high for data collection,
but as an applied mathematics major, I feel that the large sample size is a
necessity for accuracy in the realm of teaching improvement plans.
Data Collection Plan
The data I would need to
collect would include students’ scores on a pre-lesson assessment, the amount
of time in hours that the students in the different classes spent on the
lessons, and the post-lesson assessment scores.
I would use excel to post
each student’s pre and post assessment scores as well the time they spent on
the lessons for fractions. I would display the data using scatterplots to
compare pre and post assessment scores. I would display the data using scatterplots
to compare post assessment scores with the amount of time spent on fraction
lessons. I would also display data using scatterplots to compare the average
difference of assessment improvement from pre to post assessment as compared to
the time they spent on lessons about fractions. I would display scatterplots to
compare tests scores for the traditional methods versus the mastery method of
learning fractions that I provide teachers to implement.
In my study, I would look
at typical issues that students face when learning fractions as the conceptual
framework. I would emphasis that teachers expose their students to these
issues. I would include examples of these problems on both the pre and post
assessment to see if they students were able to overcome this typical issues.
Also, I would check to see whether or not the new mastery method lead to more
students with higher scores and the overall improvement of students compared to
students typical improvement under traditional lessons on fractions.
While collecting the data for my project,
I would pick teachers on a basis that they have not covered fractions in class
yet and that they have class of students who are not labeled as honors or
remedial to test the average population of students who are learning fractions.
I would use at least 10 teachers for my study to ensure that there is a large
enough sample of students to study significant results.
As teachers are
presenting their lessons, I would ask for feedback from teachers for separate
data collection testimonials. I would ask teachers from both the control and
the test groups to give their students a prepared survey that I e-mailed to
them to give to their students after the lessons on fractions and before the
post assessment to gauge how much they felt they learned and how well they feel
prepared for the post assessment and whether or not they liked the lessons from
the traditional method compared to the test groups under mastery learning
methods.
I would collect the data
for my project by e-mailing each of the teachers involved in the study an
outline of an excel document so that the teachers could input students names
into one column, pre assessment scores, the hours of time on fraction lessons,
and the post assessment scores. I would tell teachers to e-mail their class
data after their class has completed the study only. I would e-mail or speak to
the teachers in person about the methods that I would like them to teach their
lessons on fractions. I would speak to the teachers about mastery learning about
websites and sources of data to provide to their students during their lessons
on fractions.
Expected Improvement &
Conclusion
From
my research, I would expect the students who are usually quick learners and
strong in mathematics to have the same success still, however, the students who
take longer to learn or who are weaker in mathematics, I would expect more of
these students to learn more through mastery learning than if they were to
receive a traditional lesson on fractions. In order to determine the validity
of these test results I would compare the assessment scores from the pre-test
and post-test. If the post test results are statistically significant at a 99%
confidence level, then I would consider this new learning method to be worth
implementing into more schools for further research and development. Hopefully
mastery learning could be added to entire curriculums for students that way they
develop strong foundations in mathematics.
References
Brown, G., & Quinn, R. J.
(2007). Fraction Proficiency and Success in Algebra: What Does Research Say?.
Australian Mathematics Teacher, 63(3), 23-30.
Mills, J. (2011). Body Fractions: A
Physical Approach to Fraction Learning. Australian Primary Mathematics
Classroom, 16(2), 17-22.
Odeh, S. s., & Qaraeen, O. q.
(2009). Designing Multimodal User-Interfaces for Effective E-Learning in the
School Primary Stages Applied on Real Fractions. International Journal Of
Emerging Technologies In Learning, 4(2), 39-47. doi:10.3991/ijet.v4i2.274
Peck, D. M., & Jencks, S. M.
(1981). Conceptual Issues in the Teaching and Learning of Fractions. Journal
for Research in Mathematics Education, 12(5), 339.
Yujing, N., & Yong-Di, Z.
(2005). Teaching and Learning Fraction and Rational Numbers: The Origins and
Implications of Whole Number Bias. Educational Psychologist, 40(1), 27-52.
doi:10.1207/s15326985ep4001_3
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