Middle School Students’
Intuitive Techniques for Solving Algebraic Word Problems,

Final Report, May 5, 2004

by Michelle Roehler

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 explored 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 provided insights into middle school students’ grasp of variables and their ability to develop equations with or without previous formal algebraic experience. It also revealed which techniques are more intuitive to a student and indicated ways to present algebraic methods that build upon these skills.

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 used three problems that build upon different algebraic skills
at different levels of varying levels of understanding (see attached problems).

From Professor Horak’s educational research experiences and my own
background research, we built a methodology that helped put the students at
ease and gave me insight into their algebraic problem solving abilities. My interview questions
were based upon other related research experiences and my own investigational
needs. I tried to pose them in a way
that wouldl encourage conversation and identify each student’s strategies and
possible uses of algebra. After
obtaining my data from the problem-solving sessions, I was able to identify
intuitive algebraic techniques and apply these to possible teaching strategies.

To obtain my research data, I
worked with students individually in a problem-solving session. My study included five 6^{th} grade
students with no previous formal algebraic exposure and five 8^{th}
grade students who have had formal experience with algebra. I worked with one 6^{th} grade and
one 8^{th} grade teacher, and they selected the students for the study
from their own classrooms. The 6^{th}
grade group included a mixture of mathematical proficiencies, while the 8^{th}
grade group was composed of students in the advanced algebra class with high
B’s or A’s in the class.

Ideally, I wanted to conduct the problem-solving sessions with each individual student in a quiet area free from distraction. In reality, the public schools are overcrowded and “quiet areas” are scarce. My settings ranged from the hallway to the art display steps by the office and even the cafeteria. Though my audio recordings contained the clanking of cafeteria tables and the sounds of students changing classes, the interviews were very successful.

First, I asked the student some questions to “break the ice,” such as their feelings about math and their favorite subject. Then, I had three word problems for the student to solve one at a time. I presented him or her with two word problems that could be represented algebraically and one geometry problem that could be solved using algebraic methods. Each student had the following available materials to aid them in solving the problem: tiles, rulers, graphing paper, scratch paper, a formula sheet, nickels and dimes, and a calculator. The students were free to use these materials instead of utilizing an algebraic approach if they desired.

After working through the problem requirements and ensuring the student understood what was expected, he or she worked on each problem alone. I observed how each student set up and found the solution to the problem, paying close attention to the tools utilized. When the problem was completed or the student decided that they were finished, I encouraged the student to explain their problem solving strategies, elaborating on what they were doing at each step and why they chose each method. I am now working with the collected data to identify students’ intuitive approaches to algebra and their abilities to grasp algebraic concepts such as equations and variables. Each session was recorded on audiotape, and I took notes and observed throughout the process. Each session ranged from 15-35 minutes, depending on the student’s success on the problems.

To conduct this research, I spent much of the Fall 2003 semester working to gain approval for the project from the University Research Board and the school district. To learn more about the approval process, click here.

Much of my research 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.

In this study, I worked with one 6^{th}
grade teacher and one 8^{th} grade teacher. The 6^{th} grade teacher emphasizes hands-on mathematical
learning. Her activities include
“balancing equations” with a real scale and demonstrating multiplication using
different sized rectangles that fit together.
The students see many of the activities as fun, and the teacher rarely
uses a lecture-style approach. In fact,
the textbook was only used as threatened punishment while I was observing her
classroom. There was continual class
participation, and most students were actively involved in each classroom
activity.

The 6^{th} grade sample
consisted of five students of varying ability levels and backgrounds. The sample included one African American
female, one Hispanic male, two Caucasian females, and one male of mixed
Hispanic origin. The students
represented all levels of achievement in the class. Some were more shy, but each student was very cooperative and
eager to help.

The 8^{th} grade classroom
embodies a more traditional mathematical teaching approach. The teacher uses more of an interactive
lecture style. There is an emphasis on
homework, but the class goes through the homework together each day. In general, there seems to be more focus on
the textbook-style learning of mathematics, but there was significant class
participation during the homework review.

The 8^{th} grade sample
consisted of five high-achieving students selected by the teacher. There was not a distinct cultural mix, but I
was uncertain of their heritages. The
sample consisted of four females and one male.
This is the highest-level math class offered, so these are exceptional
math students. Each one in my sample
maintained a high B or A in the class, and two of the females held the top
positions in the class. There was a
general feeling of repulsion toward the subject of word problems in the
classroom, even when they were told that they were the same problems given to
the 6^{th} graders. Except for
the typical 8^{th} grade attitude problem, the students were very
willing to help.

To see a chart that details each student’s response, click here.

For this section, I will examine each word problem individually. I will present each problem in the same order that I gave them to the students. Incidentally, the method of systems of equations was taught early in the spring semester and the problem-solving sessions were completed toward the middle of the semester.

__Problem #1____:__

Madison has a pocket full of nickels and dimes. She has 4 more dimes than nickels. The total value of the dimes and nickels is $1.15. How many dimes and nickels does she have?

This problem, surprisingly, showed the most significant results.

__6 ^{th} Grade Responses__:

Each 6^{th} grade student utilized a guess and check
method. To organize their thoughts, two
students were successful using charts, one drew pictures/diagrams, one found
the coins useful, and one student was unable to make any progress on the problem
despite using the coins.

In general, the students’ approaches were very clear and demonstrated an excellent understanding of the problem. Consider one student’s use of pictures to represent the problem below:

This student decided to think of the total $1.15 in terms of
quarters. She drew a picture to clarify
her thoughts. She knew there are two
dimes and one nickel in every quarter, and there are 4 quarters in a
dollar. Thus, she had only one nickel
and one dime left after she obtained enough for a dollar. Counting up the number of *d*’s and *n*’s,
she successfully answered the problem quickly and with an excellent
explanation.

The next example is an admirable use of a chart to solve this problem.

Though this student made a mistake on the first line, her logic was perfect. She started out with one nickel, which meant she needed 5 dimes (4 more). By multiplying the number of nickels by 5 and adding this value to the product of the number of dimes and 10, she was able to obtain her total. Using guess and check, she continued to increment her values for nickels until she arrived at a total of $1.15.

Four of the 6^{th} graders obtained the correct
solution to this problem. Each was able
to explain his or her work well, and their strategies remained fairly
consistent throughout the problem-solving sessions.

__8 ^{th} Grade Responses:__

__ __

To my surprise, the 8^{th} graders showed the most
difficulty with this problem. Each
student started out using systems of equations to model the problem. Four of the five algebra students developed
a version of the following system:

*N = nickels, D =
dimes*

*D = N + 4*

*N + D = 115*

Problem #1 was consistently the most difficult and time
consuming for the 8^{th} graders, partly because of their stubborn
efforts to force the problem to work.
When I asked one stumped student what her equations meant, she replied,
“I don’t know! I just put them
down!” These students could not explain
their work well. To provide insight
into the struggles on this problem, consider this 8^{th} grade (who
maintains an A in the class) student’s work:

From the beginning, she incorrectly set up her systems by defining the equation:

*D + N = 1.15*.

Her problems only became worse. Using the method of substitution, she replaced *D* with *(N-4)*,
only to realize that she had a decimal value on one side of the equation and a
whole number on the other side. She
realized that this was a problem, but simply added a decimal in front of the
4. She had no concept of what these
values represented or even what her equations meant. Though she repeated her calculations several times, she could not
get a satisfactory result. So she tried
a different approach:

She knew that the value of a dime is equal to two
nickels. Thus, she replaced her *N =
D+4* equation with *D=2N*. It
seems clear that she also has little understanding of her variable
meanings. In the same system, *D*
stands for a monetary value $0.10 as well as the number of dimes. Once again, she could not explain her work
well, and in the end obtained the correct answer using guess and check methods.

The other students displayed similar difficulties, although one of the four eventually remedied her mistake. Three of the five total students reverted to guess and check to obtain their final solution.

** Problem #2**:

*1.
**Elinor loves to paint.
She paints pictures to sell in her mother’s store. She earns $3.00 for her small paintings and
$5.00 for her large paintings. *

*a.
**How much money would Elinor earn if she sold 6 large
paintings and 11 small paintings?*

*b.
**One Monday, Elinor sold only small paintings. She earned $36. How many small paintings did she sell?*

*c.
**On Tuesday, Elinor sold 3 more large paintings than
small paintings. In total, she earned
$39. How many of each size painting did
she sell?*

* *

This example produced results very similar to the first
problem. Each student was able to solve
parts (*a)* and *(b)* successfully or with only insignificant
arithmetic errors. Part (c) proved the
most challenging.

__6 ^{th} Grade Responses__:

Again, the 6^{th} grade students utilized guess and
check methods to solve this problem.
Several used charts to organize their work, one would write an answer
and erase it until he found the correct solution, and one student was unable to
solve the problem. The techniques were
very similar to the first problem.

__8 ^{th} Grade Responses__:

Again, only one algebra student successfully used algebra to solve part (c). The other students set up the system with the same error:

*S = Small, L =
Large*

*S + L = 39*

*L = S + 3*

These students eventually solved the problem using guess and check methods.

__Problem #3__:

Arwen wants to plant a tulip garden. To keep out the rabbits, she must set up a fence around the flowerbed. She has 24 feet of fencing.

*a.
**If Arwen makes a square flowerbed with her fencing, how
long will each side be? What is the
area of this flowerbed?*

*b.
**In the square flowerbed, Arwen wants to plant 2 tulips
for every square foot inside the flowerbed.
How many tulips will she plant in the flowerbed from part(a)?*

*c.
**Arwen decided to make the flowerbed a rectangle. She made one side 4 feet longer than the
other side. How long is each side of
her flowerbed?*

This problem is geometry-based but can be solved algebraically. Surprisingly, only half the students drew a diagram while working on this problem. Most students had little trouble with parts (a) and (b) outside of an occasional lack of conceptual understanding of area and perimeter. Again, the most interesting part of Problem #3 was part (c).

Each student that drew a picture also employed the guess and
check method. Most of these students
used charts; one student drew the rectangles to scale using a ruler and 1 inch
= 1 foot. One 6^{th} grade
student was quick to solve this style of problem in the previous two algebraic
examples, but had difficulty seeing any relationship between those problems and
this geometry problem. In the end, she
was successful, but could not see the same algebraic relationships that she had
used in the previous problems.

Only two of the students missed this problem. The first was the 6^{th} grade
student who was unable to understand any of the previous similar
situations. Ironically, the second was
the only student who set up the problem correctly using algebraic methods. Each of the other students used guess and
check methods to obtain their solutions.
The algebra students again set up the system incorrectly, and one did
not even attempt to use algebra by this point.

This is an example of the correct system of equations used
by an 8^{th} grade student:

She set up her first equation correctly as *l – w = 4*,
but made a mistake in her elimination method by failing to multiply the right
side by 2. She never doubted her
answer, even though the problem statement requires the length to be 4 feet
longer than the width. The 8^{th}
graders tended to use their methods blindly without understanding the reasoning
behind them; thus, this student trusted her solution without question. Each problem is a technique or routine. There is little demand for understanding
relationships or underlying motives.

Furthermore, the 8^{th} graders often could not see
the relationship between the problem-solving methods they were using. For example, one perplexed 8^{th}
grader had failed on Problem #1 because of her poor choice in equations. Using the method of substitution, her
solution was completely unreasonable.
All of sudden she declared “I know what I was doing wrong!” and proceeded
to solve the same system using the technique of elimination. She did not understand that these methods
indeed produce the exact same results.

In general, most of the 6^{th} grade students showed
a much better grasp for the relationships and logic within each problem. They were able to better visualize each
situation and make it rational. Each
utilized some form of guess and check, and some students were more orderly than
others. One 6^{th} grade
student had an excellent grasp of the important parts of Problem #1:

*”The most important part is knowing I have this many nickels and how many
cents would it equal and the same with dimes.”*

This student was able to see a deeper meaning and relationship than the common response,
“I have four more dimes than nickels and a total of $1.15.” If some of the 8^{th} graders had
explored beyond this stripped reading of the problem, they might have a better
understanding of the relationships between the variables and within their own
systems of equations.

Four of the five 6^{th} graders were generally
successful with Problems 1, 2(c), and 3(c).
One student was not able to make any progress on these problems. Each student varied in methodology, especially
in their pattern for using guess and check.
Some were very methodical in their choices and utilized elaborate
charts. Others were a little more
haphazard (and some might say “lucky!”).
Consider one student’s interesting reasoning for Problem #2(c):

“I think I need to use a chart again this time… Since [the large paintings] go for $5, 5 goes into 39 six times… that would be a remainder of 9… because then I know it could go into 35 for 7, but 3 wouldn’t go into 5 very well and there wouldn’t be any small paintings, but if you did 39 then there’s 6 and there’s 9 left over and 3 goes into 9 three times and the total of it would be…. $39 and then 6 large is 3 more paintings than the small which is the answer.”

Ideally, algebraic methods will help this student organize
her intuitive reasoning and allow her to tackle much more complicated problems
with a more consistent approach.
Overall, the 6^{th} graders could give clear explanations of
each step and easily identified the important parts of the problem. Their grasp of the relationships between
quantities in the problem was excellent for the most part. They solved each problem using their
intuitive understanding of these relationships and let this intuition guide their
chosen methodology.

On the other hand, the 8^{th} grade students often
showed a stubborn need to use algebra.
When I suggested to one frustrated student that she could use any of the
tools on the table to help her, she indignantly replied, *“I don’t want to
use the coins!”* In another instance,
a student who could not remember the methods for solving systems of equations
stated,

*“I could use this [the coins]…but I could solve it
easier if I just used this [his algebraic system]…but I don’t remember how…”*

Alternative methods seemed inferior to some 8^{th}
grade students, and most of these students had difficulty thinking about the
problem outside of their own systems of equations. In general, each student with difficulties set up his or her
system without understanding its application.
He or she repeatedly made the same errors and several could not explain
what their equations or variables really meant. Consider the following explanation for Problem #1:

* Investigator:
“Which equation deals with money?”*

* Student
#9: “D + N = 115”*

* Investigator:
“What does D represent?”*

* Student
#9: “The dimes”*

* Investigator:
“What about the dimes?”*

*Student #9: “I guess how many there are… I guess then
it should be the number— what *

* it’s worth…I guess
I don’t know…I just can’t do these…I just don’t know*

* how many there are…”*

Later on, Student #9 above had difficulties with Problem #2(c):

* *

*“I remember doing these in class, but I don’t remember
the formulas we used… It’s like you
find one variable but I don’t remember what you’re supposed to plug in and
stuff… You probably do this one the
same as the other one, so if I could figure out this one it would be better…”*

This statement demonstrates a reliance on the memorization of formulas and methods without understanding the underlying purpose. The students who could remember the basic algebraic techniques often became frustrated when the dependable algebra methods did not produce a logical answer. Often students seemed to regurgitate methods without understanding why that method was useful. Repeatedly they could not explain their variables or systems very well. Consider this perplexed student’s reasoning for Problem #1:

*Student #4: “I’ll try the algebra different… [after
several minutes] “I always get the *

* same number!”*

* Investigator:
“What do your variables mean?”*

* Student
#4: The variables are how many dimes
and how many nickels.”*

* Investigator: “How did you use those in your equation?”*

* cents and a nickel is five cents… Was I supposed to put that in there?”*

Even when the students found success using guess and check, they often could not apply their solution’s logic to the symbolic representations. They could explain their reasoning for the guess and check strategy, but this explanation did not reveal their mistakes in their algebraic systems. Only one of the four students was able to remedy her mistakes after explaining her guess and check solution.

Guess and check was the most prevalent strategy used in the problem-solving sessions. It was used consistently by several students and showed general success. It seems to be the most intuitive approach. Using this method, the students seem more likely to give good explanations of their reasoning and techniques. Most of the students who used the guess and check method organized their thinking using charts.

Many educators promote the teaching of algebra using the “Continuum of Abstraction.” As a student develops algebraic reasoning, there should be a gradual increase in the level abstraction. These levels were demonstrated in the choice of strategies in the problem-solving sessions.

The least abstract strategy involves the use of
manipulatives. Several of the 6^{th}
grade students used the coins to solve or better understand Problem #1. This hands-on, concrete strategy allows
students to touch and visualize the relationships.

At the next level, students move on to figures or pictures. The student who drew pictures of nickels and dimes and used a ruler to sketch rectangles to scale demonstrated this second level of abstractness. The use of pictures helps to clarify and define an approach to each problem.

The highest level involves formal algebraic techniques. By this point, the students should have the ability
to represent algebraic relationships symbolically. In this project, the algebra students utilized systems of
equations paired with the substitution and elimination methods to find the
solution to each problem.
Unfortunately, most of the 8^{th} grade students were unable to
demonstrate an understanding of the underlying relationships and reasoning
behind their techniques. Somewhere
along this continuum, the 8^{th} graders lost their understanding of
the connection between each step. .

In general, the 6^{th}
grade students seem free from the constraints of algebra. They explore their own intuitive methods and
are not restricted to a single technique.
These students were much more open to thinking about the problem and
utilizing alternative strategies.
Overall, I believe some of the 6^{th} grade students
demonstrated an excellent foundation and intuitive understanding of algebraic
techniques and relationships. For
example, consider Student #7’s chart used to solve Problem #2(c):

This student had a clear
understanding of the relationships between the quantities involved in this
problem. She knew to multiply the value
of the nickel by the number of nickels and add this to the product of the value
of a dime and the number of dimes to obtain the total amount of money. Without any formal algebraic instruction,
this student has basically set up an equation that perfectly describes the
relationships in this problem. The 6^{th}
graders showed an excellent understanding of the definition and purpose of each
quantity, as well as the relationships between them. In general, the 6^{th} grade students reveal a fine
foundation for learning formal algebraic techniques.

Unfortunately, this fundamental
understanding seemed to be lost on many of the 8^{th} grade
students. Somewhere between 6^{th}
and 8^{th} grade, students are forgetting the underlying meanings and
relying on memorized equations and techniques to produce a solution. The 8^{th} grade students were
focused on algebraic techniques. Often,
they would not consider alternative methods even if they were repeatedly
unsuccessful. They continuously focused
on equations and procedures with no understanding of the context. From this study, it appears that curriculum
needs to somehow better connect the 6^{th} grade intuitive grasp to the
advanced symbolic manipulations.

From my research, it seems crucial
for teachers and mathematics curricula to continue to be open to and support
students’ intuitive approaches to algebra.
Students must be given the opportunity and encouragement to try other
methods, take risks, and learn by experimenting with algebraic relationships
and formulas. Algebra is inherently
procedural, and there is a definite need for the memorization of formulas and
methods. But students need to build
upon their intuitive grasp of the fundamental concepts to truly learn and
realize the potential of these algebraic techniques. Students are mostly tested on their ability to regurgitate
algebraic methods without any emphasis on explanation, and multiple-choice
tests cannot always determine an understanding of the underpinnings of each
technique. From the 8^{th}
grade students’ abilities to successfully use guess and check methods, it seems
the inherent understanding may be present.
Unfortunately, this comprehension is somehow lost in the memorization of
methods and transition to symbolic representation.

Teachers must take advantage of
students’ prior knowledge. The 6^{th}
grade students in my study demonstrated a wealth of potential for succeeding in
algebra. Somehow, teachers and
curricula need to build from intuitive understandings and methods, such as
guess and check, to ensure students comprehend the underlying purpose of algebraic
techniques. This should lead to a
deeper understanding of their work in algebra.
In the current 6^{th} grade classroom, the use of guess and
check methods is greatly encouraged.
Interestingly, several of the 8^{th} grade students had the same
6^{th} grade teacher that I worked with in the study, and students are
not connecting the early exploratory methods to what they learn in algebra.

In addition, there appears to be a need to contextualize the material and apply the algebra to familiar contexts. For example, many studies show that students develop the ability to represent algebra verbally before symbolically, contrary to many teaching approaches. Students are often presented with symbolic rules and methods, and these are then applied to verbal situations. Some studies suggest that word problems should be introduced first. In general, it seems clear that algebra must be applied to a situation that students can relate to and truly grasp.

With my limited mathematics
education experience, I cannot make any sweeping recommendations. It was exciting to observe that middle
school students often do possess an intuitive understanding of algebraic
concepts and relationships. It is
important for teachers to encourage and explore this inherent understanding,
perhaps by hands-on exploratory activities or allowing students to try to
discover the formulas on their own.
Somewhere between 6^{th} grade and the introduction of formal
algebraic instruction, students lose that connection between the methods,
equations, and variables and their underlying meanings. If this connection is maintained, students
may truly comprehend and appreciate the power of algebra.