RESEARCH ON THE RETENTION OF MATHEMATICS CONCEPTS
Research on the Retention of Mathematics Concepts in Middle School:
Integers and Integer Operations
Emily Toler
University of Georgia
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RESEARCH ON THE RETENTION OF MATHEMATICS CONCEPTS
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Problem Statement
I.
Area of Focus
Retention of Mathematics Concepts
II.
Research Problem
Often, students arrive in my math classroom at the beginning of the year with many
gaps in their foundational knowledge that will negatively impact them throughout the
upcoming school year. Many students do not get the opportunity to truly understand
concepts before the pacing guide tells us that it is time to move on to the next topic in
order to complete the entire curriculum by the end of the year. It is my belief that this
cycle of low achievement and moving on to new content quickly is increasing the
number of students that are below grade level with little understanding of
mathematical concepts. Students are often unable to master foundational concepts
related to number sense with integers. This is taught at the very beginning of the
seventh-grade curriculum and resurfaces throughout all other units later in the
school year. When students do not master these concepts, they in turn will struggle
with later concepts such as writing and solving equations and inequalities. Not only
does this negatively impact them in the seventh-grade curriculum, it will set them up
to be behind in eighth grade and beyond because future mathematics concepts
assume that students know and understand integer operations.
III.
Theoretical Beliefs About the Teaching & Learning of Mathematics
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I believe that mathematics should be a tool for students to apply in the real-world in
ways that are relevant to them, personally. I believe that real world application is the
ultimate goal of teaching students any mathematics concept. Unfortunately, I believe
that mathematics is instead being taught in a way to have students take state tests and
do well on paper. While sometimes this means that a student can apply mathematics
concepts to real-world situations, often students are simply following procedures with
no real knowledge or understanding of the concept or why it works. My philosophy of
teaching is that instruction should be relevant to students and be a tool that can be
used no matter what path a student goes down later in life. I do not believe in
teaching to a test or teaching procedures rather than teaching conceptually.
IV.
Significance of Research Problem
This research is significant for several reasons. Particularly, the purpose of
schooling should be to produce students who can be productive citizens. Students
should be taught the importance of follow through and putting forth effort until
reaching the desired outcome and how to persevere in tough situations. Students
continuously feel discouraged when repeatedly failing assessments, damaging their
feelings of self-worth. Students are failing assessments because they are not
understanding concepts and materials because they are not given the opportunity or
time needed to process new information and apply it in a variety of ways and realworld situations. Therefore, the knowledge of how to help students retain information
long term can be beneficial to the teaching and learning of mathematics.
RESEARCH ON THE RETENTION OF MATHEMATICS CONCEPTS
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This is also significant to students’ education after seventh grade. Students will be
expected to know and understand integers operations in eighth grade math and in
high school courses. Without this foundational knowledge, students will be unable to
keep up with the curriculum and expectations and will quickly fall behind grade level.
Integers are a mathematics concept that is seen in everyday situations. Often
students will attempt to justify the fact that they are struggling with a concept because
they feel that the particular concept is not relevant to them in everyday life. Integers,
however, can be related to temperature and other everyday things, most importantly
money. Money is something that all people will encounter and are expected to
understand many times in their life. This is significant for students to grow into
productive citizens in the future and to be able to take care of themselves.
Problem Statement
As an educator, I believe it is my purpose to educate students in a way that is relevant and
in a way that each student can see the importance and purpose of the concepts being taught. Also
as an educator, I am expected to follow specific guidelines, curriculum, pacing, and standards
that are decided by my school, district, and state. I often notice that my teaching philosophy
doesn’t always align with the expectations of me as an educator. For example, when I am
teaching a particular concept to my students and it is obvious to me that my students would
benefit from some hand-on, real-world, activities in order to better understand the concept. When
following a pacing guide, it is sometimes impossible to fit these activities in to a week-long
lesson plan divided up into 45 minute blocks. I know that certain information is expected to be
taught and then tested on and I know that I could simply teach my students standard algorithms
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in order to take and maybe pass a test. However, I also know that these students will not see the
importance or relevance of the concept or retain the information if it is not taught conceptually to
them. My problem is that when I am following district guidelines for pacing my curriculum, I am
also giving up instructional time that could be used to teach my students conceptually and how
these concepts are related to them outside of the classroom, which would result in long-term
retention of knowledge for students.
Literature Review
Retention of mathematics concepts and knowledge is important for students’ future
application of concepts in higher levels of mathematics. Retention is also important for students’
ability to navigate and solve everyday problems. This literature review will investigate why
students do not retain mathematics concepts and factors that contribute to lack of understanding.
It will also investigate strategies teachers can use in their individual classrooms to promote
retention of mathematics concepts
I.
Factors That Inhibit Students’ Ability to Retain Mathematics Concepts
a. Student Engagement
Student engagement with school can be described as a commitment to,
valuing of, and connection with the people, educational goals, and outcomes
promoted by a school (Finn, 1989). If students are engaged in academic tasks,
retention should be increased over a period of time. Students who are unengaged
during class activities, when new concepts are being taught, or when completing
assignments are more likely to have negative feelings towards the content and feel
that the task is unimportant. When students develop these feelings about
mathematics, it can result in something called “motivated forgetting”. Motivated
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forgetting is the result of students that experience a high-degree of stress in
mathematics courses (Ramirez, McDonough, & Jin, 2017). This phenomenon is
described as students having a higher occurrence of avoidant thinking regarding
mathematics. These students have negative experiences with mathematics, and
thus avoid the topic and their retention of content is lower compared to other
students.
b. Pacing of Curriculum
In efforts to ensure that teachers are keeping pace, many school systems
promote the use of curriculum guides. Often, these guides direct teachers on how
the curriculum should be taught and gives the time allotted to teach each concept.
This constraint inhibits teachers’ ability to flexibly plan and teach concepts based
on students’ academic ability and in a time frame that makes sense for a particular
group of students to gain a full understanding of content. For students to be taught
a concept and have time to fully master the skill or strategy takes time for practice
and implementation in multiple ways. If students are unable to complete this
process of learning, they are less likely to retain the information learned for a long
period of time.
There are some positives to the use of curriculum guides. These include
consistency across different schools, structure for educators, and it can provide
teachers with resources and time for self-reflection.
c. Low Achieving Students
Students come to us at the beginning of the school year at all different levels.
Some students are on-grade level or above and fully ready to learn new content
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that builds on their previous foundational knowledge. However, we often have
students that come to us that are not on grade level. These students are missing
key, foundational skills that are needed in order to be successful when learning
new content. These students are already behind, therefore when new content is
being taught, they may be able to memorize an algorithm but they have no real
understanding of why the algorithm works or how it relates to other areas of
mathematics. These students may be able to learn enough content to pass a
classroom test, but will not retain this information long term.
II.
Strategies That Promote Retention of Mathematics Knowledge
Earlier theories of knowledge acquisition (Anderson, 1982; Fitts, 1964; Rasmussen,
1986; VanLehn, 1996) can be consolidated into a three-phase process model of learning
(Kim, Ritter & Koubek, 2013). This model has these stages: (1) obtain declarative and
procedural knowledge, (2) combine and solidify the acquired knowledge and (3) finetune the knowledge through overlearning. During stage one, students learn basic facts and
concepts that support those facts. In state two, students’ knowledge is applied to a
procedure. Stage three, or the fine-tuning stage, students practice which allows them to
speed up their application of procedures from stage two. The last stage both creates
greater fluency of knowledge recall and when practiced over multiple days will increase
long-term knowledge retention (Wang & Beck, 2012). This confirms that a wide spacing
of practice provides increasing retention and less loss of knowledge later. The following
strategies were suggested to increase long-term retention of learning: (1) short
appropriately spaces practice sessions; (2) avoiding long, intensive study, using short
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bursts of intensive study instead; (3) review of concepts previous studied periodically;
and (4) regular assessing through tests or quizzes.
It has also been shown that more interactive methods of instruction have positive
effects on students’ knowledge retention. A study of college students (Narli, 2011) found
that students who were taught with an interactive methodology rather than a traditional
lecture retained more mathematics information 14 months later than a control group.
Another student of ninth grade students (Guvercin, Cilavdaroglu, & Savas, 2014) showed
that students who engaged in mathematical problem posing had better attitudes,
achievement, and retention of knowledge 40 days after instruction. A third study of
kindergarteners through college students compared the usage of concrete manipulatives
to a traditional instruction using abstract symbols (Carbonneau, Marley, & Selig, 2013).
This showed a positive effect of knowledge retention of students who used concrete
manipulatives.
Methods of interactive learning that has proven to be effective include: computer
games, traditional games, collaborative grouping, virtual manipulatives, concrete
manipulatives, online lecture videos, interactive diagrams, visuals and diagrams, and peer
discussions. These methods of interactive learning increase student engagement by
creating a multisensory learning environment. This increase of student engagement in the
key to increase long-term retention.
Research Question
To what extent will instruction that promotes CRA impact student’s ability to reason about
integer operations and conceptually understand integer operations?
Methodology
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The participants that I will be using in this study are seventh grade students in a diverse
middle school located in the Atlanta metropolitan area. These students will be placed in an onlevel seventh grade math class and will be typical, meaning they do not receive accommodations.
I selected this type of participant in order to get a general idea for how my interventions will help
a typical seventh grade student and in order to identify areas where a typical seventh grade
student may have misconceptions.
The setting where this research will be conducted is in my own classroom located in a
middle school in the Atlanta metropolitan area. This is a general education classroom with
approximately 23 students. I have each group of students for 46 minutes per day, 5 days a week.
This intervention will be implemented between 2 and 3 days each week for a total of 9 weeks.
Assessments will be given periodically after the intervention has been provided to assess student
retention. These particular assessments will be given 2 weeks after the end of the intervention, 4
weeks after, and 8 weeks after.
The intervention that I will be using with my participants is CRA, or ConcreteRepresentative-Abstract learning. I will be modelling how to use manipulatives in order for
students to be able to create concrete representations of integer related problems. Students will
have their own manipulatives to use and follow along as well. We will gradually work towards
using representative depictions of problems, and eventually solve problems abstractly. I will
always model all three ways for students and encourage them to do the same. The intervention
will be implemented in a whole group setting, but student participants will be pulled for one-onone interviews for assessment of progression and retention.
I will be collecting both quantitative and qualitative data from students during my
interventions. Students will take assessments that include both multiple choice questions
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(qualitative) and short response questions (quantitative). I will not only be looking for correct
answers but will be looking for the strategies that are being used by each student. When
analyzing short response questions, I will be looking for correct use of academic vocabulary and
also the student’s description of their strategy and how they solved the problem. I also plan to
conduct student interviews which will also serve as qualitative data. These interviews will give
insight to students’ thinking and give a clear picture into their understanding of the material and
the strategy that they chose to use. Assessments will be given 2 weeks, 4 weeks, and 8 weeks
after the end of the intervention in order to assess retention of integer concepts.
Plan
Week
Intervention
Plan
1
CRA
Compare integers on a number line, as represented by counters, and
as written numerals
2
CRA
Add integers with the same signs - on a number line, as represented
by counters, and as an abstract addition problem
3
CRA
Add integers with different signs - on a number line, as represented
by counters, and as an abstract addition problem
4
CRA
Subtract integers with the same signs - on a number line, as
represented by counters, and as an abstract subtraction problem
5
CRA
Subtract integers with different signs - on a number line, as
represented by counters, and as an abstract subtraction problem
RESEARCH ON THE RETENTION OF MATHEMATICS CONCEPTS
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CRA
Multiply integers with the same signs - use of pictorials and as an
abstract multiplication problem
7
CRA
Multiply integers with different signs - use of pictorials and as an
abstract multiplication problem
8
CRA
Divide integers with the same signs - use of pictorials and as an
abstract division problem
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CRA
Divide integers with different signs - use of pictorials and as an
abstract division problem
Day
Topic
Activity
1
All Integer
Operations
Pre-Assessment
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Comparing
Compare integers on a number line and as represented by counters
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Comparing
Compare numeral integers using greater than and less than signs
6
Adding –
same signs
Adding on a number line
8
Adding –
same signs
Adding using counters
10
Adding –
same signs
Adding “naked” problems and word problems
11
Adding –
different
signs
Adding practice using CRA
13
Adding –
different
signs
Adding practice using CRA – One-on-one interview
11
RESEARCH ON THE RETENTION OF MATHEMATICS CONCEPTS
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Adding - all
Written assessment
16
Subtracting
– same signs
Subtracting on a number line
18
Subtracting
– same signs
Subtracting using counters
20
Subtracting
– same signs
Subtracting “naked” problems and word problems
21
Subtracting
– different
signs
Subtracting practice using CRA
Subtracting
– different
signs
Subtracting practice using CRA – One-on-one interview
25
Subtracting all
Written assessment
26
Multiplying
– same signs
Multiplying using counters and pictorials representation
28
Multiplying
– same signs
Multiplying with word problems
30
Multiplying
– all
Multiplying “naked” problems
31
Multiplying
- all
Multiplying practice using CRA
33
Multiplying
- all
One-on-one interview
35
Multiplying
- all
Written assessment
36
Dividing –
same signs
Dividing using counters and pictorial representation
23
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Dividing –
same signs
Dividing with word problems
40
Dividing all
Dividing “naked” problems
41
Dividing all
Dividing practice using CRA
43
Dividing all
One-on-one interview
45
Dividing all
Written assessment
13
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Fitts, P. M. (1964). Perceptual-motor skill learning. Categories of human learning, 47, 381-391.
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