Middle School Students’ Intuitive Approaches to Algebraic Word Problems,

With the Guidance of Dr. Virginia Horak

by
Michelle Roehler

Fall 2003

How do students approach word problems? Are students intuitively drawn to algebraic methods, or do their approaches differ from these commonly emphasized techniques? In this study, I am exploring middle school students’ methods for solving word problems and their utilization of algebra for problem solving. In many classrooms, students are taught to tackle word problems with specific algorithms for each problem type, and they often develop a reliance on cookie-cutter equations without fully comprehending the underlying problem and the algebra used to solve it. Since many students do not understand the concepts behind these methods, it is important to explore a student’s inherent approaches to problem solving and algebra. This study provides insights into middle school students’ grasp of variables and their ability to develop equations with or without previous formal algebraic experience. It also reveals which techniques are more intuitive to a student and might indicate ways to present algebraic methods that build upon these skills.

Since this study requires interaction with middle school students at a local Tucson school, I have had to go through an extensive application and approval process to obtain permission to conduct the study. Thus, this research project remains in progress, and my concrete data at this point is limited. Much of the research has been focused on previously published studies in addition to the arduous application processes that I will detail later in this paper.

I was finally granted final permission to conduct my actual research in the middle of November. Thus, my interaction with the students and teachers has been limited to observation in the final weeks of the semester. When I observe the classroom, I am able to identify classroom dynamics, study the teacher’s approach and methods for teaching mathematics, and watch the students’ responses. My observations also allow the students to become more comfortable with my presence in the classroom, which will hopefully make them more talkative and at ease during any one-on-one sessions with me in the future.

To obtain my research data, I will
be working with students individually in a problem-solving session. My study includes five 6^{th} grade
students with little or no previous formal algebraic exposure and five 8^{th}
grade students who have had formal experience with algebra. I am working with one 6^{th} grade
and one 8^{th} grade teacher, and they have selected the students for
the study from their own classrooms.
The 6^{th} grade group includes a mixture of mathematical
proficiencies, while the 8^{th} grade group is composed of students in
the advanced algebra class.

I will be conducting a problem-solving session with each individual student in a quiet area free from distraction. First, I will ask the student some questions to “break the ice,” such as their feelings about math or their favorite subject. Then, I have three word problems for the student to solve one at a time. I will present him or her with two word problems that can be represented algebraically and one geometry problem that can be solved using algebraic methods. Each student will have available materials to aid them in solving the problem such as tiles, rulers, graphing paper, and a calculator. The students will be free to use these materials instead of utilizing an algebraic approach if they desire. After going through the problem requirements and ensuring the student understands what is expected, the student will work on each problem alone. I will observe how each student sets up and finds the solution to the problem, paying close attention to the tools utilized. When the problem is completed or the student decides that they are finished, I will have the student explain their problem solving strategies, elaborating on what they are doing at each step and why they chose each method. In the end, I will be able to use the data collected to identify students’ intuitive approaches to algebra and their ability to grasp algebraic concepts such as equations and variables. Each session will be recorded on audiotape, and I will be taking notes and observing throughout the process. I anticipate approximately 30 minutes per session. For exact details on the methodology, see the Application Process section.

**DEVELOPMENT OF MY RESEARCH
FRAMEWORK**

In developing my methodology and defining details for the
problem-solving sessions, I utilized past studies and examples from various
textbooks, my own priorities for research, and the experience and knowledge of
my advisor, Professor Virginia Horak. I
am not alone in pursuing this style of research, and I have gained much insight
from others’ research on similar topics.
Articles such as those mentioned below have provided extremely helpful
ideas and strategies for conducting the problem-solving sessions, predicting
results, and analyzing my observations.
For instance, my research on the use of word problems to define
intuitive algebraic understanding led me to use story problems with familiar,
easy to understand contexts in my research.
Also, I was able to use past studies and the school district’s textbooks
to formulate word problems for the sessions.
I observed which styles were most indicative of algebraic understanding,
and built from examples in textbooks that took into account the 6^{th}
grade student’s basic mathematical knowledge.
Thus, I have three problems that build upon different algebraic skills
at different levels of varying levels of understanding (see selected word problems).

From Professor Horak’s educational research experiences and my own
background research, we built a methodology that should put the students at
ease and truly give insight into their algebraic problem solving
abilities. My interview questions are
based upon other related research experiences and my own investigational
needs. I tried to pose them in a way
that will encourage conversation and identify each student’s strategies and
possible uses of algebra. When I have
obtained data from the problem-solving sessions, I will be able to identify
intuitive algebraic techniques, if any, and apply them to possible teaching
strategies.

The most significant portion of my time this semester has been spent gaining approval to conduct the study. This process included a training program for working with human subjects, an approval process through the University of Arizona to conduct the research, an approval process through the school district, a proposal and interview with the principal of the school I am working with, and, finally, I had to gain permission from the teachers. For those interested in conducting a similar study, I will detail the application process.

Since I will be working with middle
school students for my research, it is necessary to obtain approval from the
University of Arizona’s Institutional Review Board. To ensure the protection of all research subjects, the Vice President for Research to the Human
Subjects Committee must approve all aspects of the research process. Thus, there is significant paperwork and
training to accomplish before an investigator can have any contact with the
human subjects. First, all
investigators must complete the Human Subjects Training. The university provides a training manual
that gives detailed explanations of the methods and bureaucracy required to
perform research with human subjects, policies and regulations, how to gain and
maintain approval for your research, and other topics. Upon completion of the manual, the
investigator must pass the Human Subjects test with at least an 85%. Once Human Subjects Training is completed,
you move into the Project Approval process.

The next step is to obtain approval from the
Institutional Review Board at the University of Arizona. In the Project Approval process, the
investigator must explicitly state every aspect of the proposed research. This includes all methodology, potential interview questions,
a description of all observations, and precise details such as the number of
subjects and the rationale for their selection. You must also include a copy of your proposed consent form for
the parents
and students
and clearly state how consent is to be obtained and honored throughout the
project. In addition, I had to submit a
recruitment
script for presenting the project to the selected students. The necessary paperwork is then sent through
a preliminary critique, and the proposals are usually sent back repeatedly for
corrections or additions. Every detail
is thoroughly scrutinized. I was
fortunate to only be required to do one extensive revision. When your application is deemed as perfect
as possible, it goes to committee for final approval. I gained approval for my project from the Review Board on
November 4. To view the final
application and detailed methodology, click here.

I also had to undergo an approval process through the school district’s Office of Accountability and Research before I could enter a classroom or approach the principal or teachers. The district requires its own forms, but they are much less severe and detailed than the University’s. To view the approved school district application, click here.

After receiving approval from the University and school district, Professor Horak and I met with the principal at the middle school where we hoped to conduct the research. The principal was very supportive and offered suggestions that might aid us in the problem-solving sessions. After obtaining the principal’s permission, we approached the two teachers. Each teacher has shown great interest in and support for our project, and they have been extremely helpful and pleasant to work with thus far.

Much of my research has involved examining related research from other sources. These findings influenced the framework of my research approach, gave me insight into the current mathematical standards and pertinent issues in teaching, and suggested possible results or factors to identify while working with the students and processing my data. I will summarize some of the important results from previous studies that have impacted my research.

Past studies have revealed that
children can grasp algebraic concepts at an early age if algebra problems are
presented in intuitive ways. In his
work “Algebraic Problem Solving in the Primary Grades,” Robert Fermiano
presents the theory that “even though primary-grade students may lack the
formal level of thinking required to ‘efficiently’ solve equations, algebraic
reasoning is still possible when approached in less sterile and more practical
ways” (Fermiano 1). Studying his own
classroom of first through third grades, he proposed that if equations are put
into concrete situations, children can more easily grasp the problem and use
their intuitions to find a solution. He
takes almost every aspect of mathematics in his classroom and puts it into a
problem-solving setting (Fermiano 1).
When problems are put into a story format, children are more apt to
understand the problem, and their informal, intuitive approaches provide the
basis for understanding the fundamental concepts of algebra. When an algebra problem is presented in
symbolic form, students are often intimidated and have no real instinct for
solving the problem without the memorized methods available to older students
(Fermiano 2). Fermiano suggests that
algebraic problem solving is crucial for a child’s development of mathematical
skills, and the process of learning algebraic techniques must be built from the
child’s “discovery” and intuitive concepts of algebra (Fermiano 4). Thus, using word problems may provide a much
more insightful view of the students’ inherent understanding of algebra and its
fundamental concepts since they cater to a child’s innate mathematical
responses.

On a similar note, Mitchell Nathan
and Kenneth Koedinger suggest that contrary to many curriculums, students
develop verbal problem solving skills, i.e. the ability to work with word
problems, before they can comprehend symbolic problems (Nathan 2). In their study, teachers repeatedly ranked
word problems as the most difficult, but students showed surprising skill with
these verbal problems, indicating that perhaps these skills develop before
“symbolic reasoning” (Nathan 3). These
results have a huge impact on teaching strategies for algebra. Nathan and Koedinger recommend teaching
strategies that build more upon the student’s informal methods for solutions
and the student’s verbal skills (Nathan 3).
When students build their own methods, they have a more in-depth
understanding of the concepts. Formal
methods and symbolic representations should build upon these informal
methods. Alternative strategies, such
as guess-and-check, help students discover fundamental algebraic concepts on
their own, and building on the student’s inherent understanding will greatly enhance
their mathematical success.

Lesley R. Booth further explored
some of the problems at the core of algebra learning in her article,
“Children’s Difficulties in Beginning Algebra” (Booth 20). Instead of focusing on a student’s innate
understanding of the algebra problem, Booth presents common misconceptions
about algebra that appear frequently throughout middle and high school. Her study revealed some of the students’
struggles regarding the significance of a variable and its value in relation to
arithmetic. Students often do not feel
that a problem has a solution if it contains variables; there is no concrete
“final answer” for them to achieve (Booth 23).
Many students explained that, unless explicitly stated, a variable has
no value, and many difficulties arise when students attempt to apply this
confusing value to arithmetic. In a
different way, several students replaced a letter variable with its position in
the alphabet or adamantly stated that two variables cannot have the same value
if they are represented by different variables; there was also an insistence
that the variable must start with the same letter as the unknown quantity (Booth 22).
This coincides with the arguments in the previous articles that students
often struggle with symbolic representation in general, and I have read several
studies that reveal difficulties in algebra arising from these symbolic
misconceptions.

Though she admits that many children utilize informal methods to
understand algebra concepts, Booth was wary about allowing students to rely on
these methods. There must be a balance
between visualizing the problem and defining a procedure. She proposed that “if students are to learn
(and use) the more formal procedures, they must first see the need for them”
(Booth 31). She suggested that teachers
should be aware of possible informal methods used by the students, and there
should be discussion regarding the usefulness and limitations of the informal
methods (Booth 31). Thus, this article
indicated a need for a combination of formal and informal teaching methods to
obtain maximum algebraic understanding for beginning algebra students.

Harold L. Schoen, in his work on “Teaching Elementary Algebra with a
Word Problem Focus,” presented his belief that “it is possible to focus on
interesting applications and word problems in the teaching of first-year
algebra without deleting important topics” (Schoen 120). In his article, Schoen presented several
recommendations for teaching a first-year algebra course. First, he recommended that teachers should
“build new learning on students’ existing knowledge and understanding” (Shoen
120). Students enter an algebra course
with a significant amount of prior mathematical knowledge. He suggested that basic topics such as areas
and percentages should be re-introduced with an algebraic focus to build on
concepts the students already know (Schoen 121). Second, Schoen urged that teaching should “lead gradually from
verbalization to algebraic symbolism” (Schoen 121). By using word problems and verbal representations, teachers can
better connect the underlying concept to the symbolic representation. In addition, he recommends that teachers
“introduce algebraic topics with applications” and teach these topics “from the
perspective of how they can be applied” (Schoen 122,123). This does not only enhance understanding of
how the concept works; it also shows the student its usefulness in real
life. Overall, Shoen argued that the
use of word problems in the classroom is beneficial to developing an
understanding of symbolic representations.
He proposed the use of word problems to provide application to algebraic
methods, and insisted that implementing
a word problem focus in the mathematics classroom is extremely feasible.

In her article entitled “Developing
Algebraic Reasoning in the Elementary Grades,” Jinfa Cai used a cross-national
comparison to point out the need for increased development of algebraic
strategies at the elementary level to give students a better grasp of problem
solving (Cai 1). In a comparison between
the U.S., Japan, and China, fourth and sixth graders from the U.S. repeatedly
avoided algebraic techniques, while students of the same age from China and
Japan used algebra much more frequently in problem solving (Cai 2). The fact that any students of this age would
use algebra reveals that algebraic concepts can be taught and understood at the
elementary level. Thus, to enhance
their problem solving abilities, students should be well exposed to algebra
before middle or high school.

Cai explained the differences in problem-solving techniques by citing
different teaching methods. In the
U.S., there is an emphasis on concrete methods and examples, while Chinese
educators focus on a student’s understanding of the concept, not necessarily
the visual example. Chinese students
are asked to solve a problem both arithmetically and algebraically, and then
discuss the differences and similarities (Cai 3). According to Cai, this method helps students to gain a deeper
comprehension of the quantitative relationships, enhances their thinking skills
and dexterity with different problem solving approaches, and helps students
discover the similarities and differences between methods on their own (Cai
3). Overall, Cai seemed to advocate the
introduction of algebraic concepts at the elementary level with a shift from
the traditional concrete visual representations of algebra to more abstract
conceptual representations. Again,
there is a push to introduce algebra at a younger level to enhance a student’s
future mathematical abilities. To
become more successful at problem solving, algebraic skills are a necessity,
but letting students explore alternate methods will further enhance their
conceptual understanding.

The most influential and helpful past research for this project thus
far comes from Jane Swafford and Cynthia Langrall’s article entitled “Grade 6
Students’ Preinstructional Use of Equations to Describe and Represent Problem
Situations.” Swafford and Langrall
studied 6^{th} grade students’ ability to use equations to “describe
and represent problem situations prior to formal instruction in algebra”
(Swafford 1). In my research, I am
focusing on the students’ intuitive techniques, while this study uses directed
questions that have the child construct equations and identify how they obtain
a solution. First, the investigators
had the students express the general case verbally. Then, the students were asked to use variables to represent the
relationships in the problem. Lastly,
the researchers determined if and how the students would use these symbolic
representations to obtain their solution (Swafford 2). The students were given six word problems
that represented different mathematical concepts, such as linear relationships and
proportionality (Swafford 3). Similar to our research, Langrall and
Swafford utilized an audiotape recording, the students’ written work, and
interview notes for analysis. In their
research, the investigator walked through several tasks with the students. I will simply observe the students on their
own, and then I will have them explain their methodology when they are
finished.

Swafford and Langrall discovered that “sixth-grade students in this study showed a remarkable ability to generalize problem situations by describing relationships and writing appropriate equations using variables” (Swafford 6). On the other hand, students were more able to represent the relationships verbally than symbolically, and few used their equations, even if correct, to obtain a solution (Swafford 6). These results coincide with Nathan and Koedinger’s belief that the ability to represent a problem verbally is more inherent than the symbolic representations. Also, Swafford and Langall observed that students were very able to generalize familiar arithmetic situations, and proposed that “middle school students would benefit from more experiences with a rich variety of multiplicative situations” (Swafford 9). Similar to Cai, they suggested that teachers should focus more on changing values within the same context instead of giving examples of several different problems in different contexts (Swafford 9). In Swafford and Langall’s study, the children had difficulty seeing their equations as “mathematical objects,” and they were not confident while working with variables (Swafford 10). Thus, there seems to be a definite intuitive understanding of algebraic relationships and equations, but students have difficulties applying the symbolic representations to their problem-solving methods. This indicates that curricular changes might help bridge this gap between students’ inherent comprehension of algebra and the formal symbolic representations and concepts.

With the
limited time remaining in the semester after gaining approval to enter the classroom,
I have only had the opportunity to observe the 6^{th} grade class. There is a casual feel to the classroom, and
the desks are grouped into clusters instead of rows. Colorful prints adorn the walls, and an expansion of pi runs along
the top of the chalkboard.

The 6^{th} grade teacher
utilizes more “untraditional” teaching methods. In each class period I observed, there was little or no
lecture. Every concept was introduced
in a hands-on format, and the teacher presented each mathematical concept with
some sort of visual aid. Some tasks
demonstrated the basics of algebra, but they were disguised as fun, exploratory
activities.

In the 3^{rd} period class,
the student sitting next to me explained each activity. For instance, there is an activity called
“Hands-on Equations” that uses a picture of a scale, different colored game
pieces, and number tiles. The game
pieces represent “unknowns” and the number tiles represent values that will be
added to each side of the scale. The
goal is to “balance” the equation with tiles on one side and game pieces on the
other. The student I worked with
declared, “This is a fun one!” and was eager to finish all of her
problems. For the most part, there is
considerable class participation and plenty of volunteers to answer
questions. The students get excited
when they discover the answer, and the teacher intermingles written work with
the hands-on examples.

I expect that this classroom will
differ greatly from the 8^{th} grade algebra students. From the principal’s comments, I learned
that the algebra teacher utilizes more “traditional” methods, and the classroom
has a more orderly feel. I look forward
to working with the 8^{th} grade class early in the spring semester.

It will be interesting to identify
any differences that these mathematical backgrounds may produce in the
problem-solving sessions. It is
probable that the 6^{th} grade students will use alternative methods
that model algebraic techniques, while the 8^{th} grade students might
incline toward a more procedural approach.
Regardless, the 6^{th} grade students have had no formal
algebraic instruction, and the diverse classroom situations may not affect the
results. I predict that the 8^{th}
grade students will utilize standardized methods similar to those from my own
traditional mathematical upbringing. Overall,
the problem-solving sessions will reveal which techniques are more inherent to
a beginning algebra student and those that become engrained after formal
algebraic exposure. Since each student
must explain his or her methods to me after they are finished with each
problem, I will gain significant insight into his or her true understanding.

By the time spring semester starts at the University, the excitement of winter break should have died down at the middle school. I hope to be in the classrooms by the second week of the semester resuming my observations. As soon as possible, the teachers will be contacting the students’ parents and sending home consent forms for their participation in the research project. When the consent forms are signed and collected, I will begin the individual problem-solving sessions. The teachers have given me permission to excuse each student from class for a 30-minute session. Since there are ten students total, these sessions will probably be scattered over several days. I will work with each teacher to find times when the impact of lost class time will be minimized. As each problem-solving session is completed, I can begin to transcribe the interviews and analyze the data. I hope to continue the classroom observations throughout the semester to obtain a better understanding of the teaching methods and environments. The arduous application process delayed my opportunity to begin collecting significant data, and I am very eager to resume interacting with the students next semester.

Booth, Leslie R. “Children’s
Difficulties in Beginning Algebra.” __The
Ideas of Algebra, K-12__.

National Council of Mathematics Teachers, 1988.

Cai, Jinfa.
“Developing Algebraic Reasoning in the Elementary Grades”. __Teaching Children __

__Mathematics__ 5:4, December
1998.

Femiano, Robert B.
“Algebraic Problem Solving in the Primary Grades.” __Teaching Children __

__Mathematics__ 9:8, April 2003.

Nathan, Mitchell J., and Kenneth R. Koedinger. “Teachers’ and Researchers’ Beliefs About the

Development of Algebraic
Reasoning.” __Mathematics Teacher__
93:3, March 2000.

Schoen, Harold. “Teaching
Elementary Algebra with a Word Problem Focus.”
__The Ideas of __

__Algebra, K-12__. National Council of Mathematics Teachers,
1988.

Swafford, Jane O., and Cynthia W. Langrall. “Grade 6 Students’ Preinstructional Use of

Equations to Describe and Represent
Problem Situations.” __Journal for
Research in __

__Mathematics Education__ 31:1,
January 2000.