Assessing Algebraic Solving Ability Of Form Four Students

Volume 1, Number 1, October 2006 www.iejme.com
ASSESSING ALGEBRAIC SOLVING ABILITY OF FORM FOUR STUDENTS
Lim Hooi Lian and Noraini Idris
.
INTRODUCTION
Background
Several mathematics researchers and educators studied the investigation into the
introduction and development of algebraic solving abilities which can be viewed from different
approaches such as generalization, modeling and functional.
Copyright © 2006 by GOKKUSAGI
ISSN: 1306-3030
International Electronic Journal of
Mathematics Education
ABSTRACT. Mathematics researchers generally agree that algebra is a tool for problem solving, a method
of expressing relationship, analyzing and representing patterns, and exploring mathematical properties in a
variety of problem situations. Thus, several mathematics researchers and educators have focused on
investigating the introduction and the development of algebraic solving abilities. However research works
on assessing students' algebraic solving ability is sparse in literature. The purpose of this study was to use
the SOLO model as a theoretical framework for assessing Form Four students' algebraic solving abilities
in using linear equation. The content domains incorporated in this framework were linear pattern (pictorial),
direct variations, concepts of function and arithmetic sequence. This study was divided into two phases. In
the first phase, students were given a pencil-and-paper test. The test comprised of eight superitems of four
items each. Results were analyzed using a Partial Credit model. In the second phase, clinical interviews
were conducted to seek the clarification of the students' algebraic solving processes. Results of the study
indicated that 62% of the students have less than 50% probability of success at relational level. The majority
of the students in this study could be classified into unistructural and multistructural. Generally, most of the
students encountered difficulties in generalizing their arithmetic thinking through the use of algebraic
symbols. The qualitative data analysis found that the high ability students seemed to be more able to seek
the recurring linear pattern and identify the linear relationship between variables. They were able to co-
ordinate all the information given in the question to form the algebraic expression and linear equations.
Whereas, the low ability students showed an ability more on drawing and counting method. They lacked
understanding of algebraic concepts to express the relationship between the variables. The results of this
study provided evidence on the significance of SOLO model in assessing algebraic solving ability in the
upper secondary school level.
KEYWORDS. Algebraic Solving Ability, Assessment, Linear Equation, Partial Credit Scoring, SOLO,
Upper Secondary School Students.

The important role played by generalization approach in the introduction of algebraic
solving abilities cannot be denied. In this approach, algebraic solving ability can be laid when
students engage in the investigative processes: i) finding a pattern ii) generalizing a formula by
using algebraic symbols, and iii) applying the formula to solve the problem (Fernandez &
Anhalt, 2001; Friedlander & Hershkowitz,1997; Herbert & Brown, 1997; Mason, 1996).
Ferrucci, Yeap and Carter (2003) have established modeling approach as a fundamental of
algebraic solving abilities. Essentially, this approach uses the pictorial representation to analyze
the relationship among the quantities in a problem. According to them, modeling approach
consists of two phases, the first phase involves the investigation some key relationships
between variables in the situation. The second phase comprises a series of mathematical
transformations or operations that lead to a model expressed such as symbolic expressions,
graphs or tables (Ferrucci, Yeap & Carter, 2003). In a functional approach, the emergence of
algebraic solving abilities involves the representation of variables as quantities with changing
values and an exploration the graphical and numerical representation that highlight the changes
for different function rules (Thornton, 2001).
Obviously, the nature of algebraic solving ability inherits in each approach is sufficient
to generate powerful algebraic solving ability in the classroom. Many recommendations have
been made to transform algebra from a skill-drilling sequence of practice into a meaningful topic
that can be approached through these different approaches. According to Bishop, Otto and
Lubinski (2001), Carey (1992), Herbert and Brown (1997), the application of these approaches
to introduce algebra provide concrete model and concrete experience that enable students to
experience algebra in the real world. In this manner, students will be able to construct a better
understanding of the algebra concept and connect the concrete experience with the abstract
symbolic algebra.
However, the question of how to assess algebraic solving ability through these
approaches may still be problematic for many teachers. Thus, in this study, SOLO model, known
as the Structure of the Observed Learning Outcome, developed by Biggs and Collis (1982) was
used to assess students' algebraic solving ability. It is a cognitive psychology model which
emphasizes more on the internal process and more interested in investigating how a problem is
handled by students rather than whether their answers are correct. It had provided the basic
theoretical underpinning for developing the technique for assessing students' cognitive
attainment. In the area of algebra, SOLO model have been used to describe students' elementary
equation solving (Biggs & Collis, 1982) and made the comparison with various learning theories
in describing development of algebraic ideas (Pegg, 2001) but there is no coherent description
of students' algebraic solving ability sufficient to inform instruction decision. Thus, in this study,
we claimed that the proposed framework enable upper secondary school students' algebraic
solving ability to be described across four levels of SOLO model.
56 Lian and Idris

SOLO model was used to construct items which reflect the four levels of SOLO model:
unistructural, multistructural, relational and extended abstract. The levels are in hierarchical
manner which are increasingly complex. Students' algebraic solving ability were assessed
through their performance in using linear equation to solve the problem situations across the four
content domains, these include linear pattern (pictorial), direct variation, concept of function and
arithmetic sequence. Based on Malaysian Integrated Curriculum for Secondary Schools
syllabus, linear equation is a prerequisite to the learning of more complex topics at the upper
secondary school such as straight line; gradient and area under graph; index and logarithm;
matrix; variation; graph of function and quadratic equation (Teng, 2002). Therefore, linear
equation became the focus of this study.
Statement Of The Problem
Clements (1999), Stacey and Macgregor (1999a), and Murphy (1999) stated that at the
present time, the assessment in algebra is still focusing on getting a correct answer, symbol
manipulations, rote skill and little or no application of algebraic concepts in the problem
situation. Teng (2002), Tall and Razali (1993) have noted that symbol manipulation and
procedural skill practice in algebra class among the secondary school students might serve to
prolong the interpretation that algebra is a 'menagerie' of disconnected rules to deal with
different contexts. It, therefore exhibits the poor understanding of the basic concept and
cognitive obstacles among the students as this practice to algebra relies almost exclusively on
written symbolic forms as the tool to make representation, generalization and interpretation to
the applied problem. In this regard, it is impossible to inculcate students with algebraic solving
ability if the assessment procedure is not changed. As Stacey and MacGregor (1999a) viewed
that although the students have apparently learned algebra, in reality they find algebra difficult
and do not know how to apply it. They still can't see the way to use what they learnt about
algebra and it is still seen separately. Therefore, more complex algebraic problem items that
demonstrating the efficiency and power of algebraic solving ability should be constructed and
frequently used in examination and classroom practice.
Purpose Of The Study
This study aimed to assess the Form Four students' levels of algebraic solving ability in
using linear equation. In order to capture the manifold nature of algebraic solving ability in using
linear equation to solve the problem situations, the framework of this study incorporated four
content domains of linear equation. Further, this study sought to the breath and depth of
students' algebraic solving processes in using linear equation through interview method.
57
International Electronic Journal of Mathematics Education / Vol.1 No.1, October 2006

Research Questions
In this study, two research questions addressed are the following:
a. What are Form Four students' levels of algebraic solving ability (according to SOLO model)
with regard to the use of linear equation to solve a series of tasks across the four content domains
(linear pattern, direct variation, concept of function and arithmetic sequence)?
b. How do Form Four students solve the four levels items (according to SOLO model) which are
constructed in assessing students' algebraic solving ability in the process of:
i. investigating the pattern?
ii. representing and generalizing the pattern?
iii. applying the rule to the related situation?
iv. generating an alternative solution for the new situation?
Significance of The Study
The results of this study might provide evidence on the significance of SOLO model in
assessing algebraic solving ability in the upper secondary level. It provided a guideline for
teachers who want to identify the level and the process of algebraic solving ability among their
students in using linear equation across the four content domains.
Subsequently, the instrument of the study might also be used as a diagnostic assessment
tool to evaluate the strengths and weaknesses of the students' conceptual understanding about
linear equation.
The results of this study also provided an evidence whether SOLO model can be used
as an alternative method of assessment in the upper secondary level. Thus, the findings might
provide useful information to the assessment developer particularly in the subject of
mathematics.
Perhaps and even more significant is that, there were very few researches conducted
concerning the assessment of algebra. So far, there had been four studies concerning the learning
of algebra in Malaysia (Cheah & Malone, 1996; Heng and Norbisham, 2002; Ong, 2000; Teng,
2002). They investigated and identified the conceptual understanding problems in learning and
teaching of algebra. These studies however did not attempt to develop a model to assess students'
algebraic solving ability in the problem situations. This lack of research suggested that an
assessment of algebraic solving ability in mathematics classroom would be an added value to
the research on algebra.
58 Lian and Idris

Theoretical Framework
The SOLO taxonomy is designed mainly as a mean to assess students' cognitive ability
in school learning context (Biggs & Collis, 1982; Collis & Romberg, 1991; Chick, 1988; Collis,
Romberg & Jurdak, 1986; Reading, 1998; Vallecillos & Moreno, 2002; Watson, Chick & Collis,
1988). It had been used to analyze the structure response of student's problem solving ability,
mathematical thinking ability and understanding of mathematical concepts over a wide
educational span from primary to tertiary levels (e.g Chick, 1988; Collis, Romberg & Jurdak,
1986; Lam & Foong, 1998; Reading, 1999; Vallecillos & Moreno, 2002; Watson, Chick &
Collis, 1988; Wilson & Iventosch, 1988). SOLO provides a framework to classify the quality of
response which can be inferred from the structure of the answer to a stimulus. According to
SOLO model, coding a student's response depends on two features. The first is a series of five
modes of cognitive development and the second is a series of levels of response.
In SOLO model, mode is closely related to the existing notion of Piaget's stage of
cognitive development which proposes a number of developmental stages demonstrating
increasing abstraction form sensori-motor (infancy), ikonic (early childhood of preschool),
concrete-symbolic (childhood to adolescene), formal (early adulthood) through to postformal
(adulthood) (Biggs & Collis, 1982).
Although the sequence of five modes followed from simple to complex, it is common
knowledge that students do not always operate at the same level as their developmental age
suggests they should, nor do they perform consistently (Biggs & Collis, 1982; Biggs & Teller,
1987; Collis & Romberg, 1991; Romberg, Zarinnia & Collis,1990). For example, a formal mode
response in mathematics given by a student might be followed by a series of concrete-symbolic
mode response in mathematics. Further, concrete-symbolic mode response in mathematics given
by a student on this week might be followed by formal mode response on the next week. Was
that particular student at the formal mode or concrete-symbolic mode? According to SOLO
model , this kind of difficulty can be solved by shifting the label from the student to his response
to a particular task (Biggs & Collis, 1982). In other words, SOLO model improves this
phenomenon by describing the complexity of the structure response to a particular task within a
mode.
According to SOLO model, the structure responses in solving algebra problems can be
classified into four levels which include unistructural, multistructural, relational and extended
abstract. Researcher hypothesized that Form Four students could exhibit four levels of algebraic
solving ability. The theoretical framework had been developed along with the expected students'
algebraic solving ability across the four levels for each of the four content domains. Table 1
showed the framework on the characteristic of students' algebraic solving ability incorporate four
content domains of linear equation and Figure 1 represented the theoretical framework of the
study.
59

Figure 1 represented the theoretical framework of this study. Based on Friedlander and
hershkowitz's (1997) and Swafford and Langrall's (2000) views, the ability of using equation to
solve and represent the problem situation involves a number of algebraic processes which consist
of three phases: i) investigating the pattern by collecting the numerical data; ii) representing and
generalizing the pattern into table and equation; iii) interpreting and applying the equation to the
related or new situation. This study suggested that the algebraic solving ability can be assessed
based on these three phases. Eight linear equation problem situations in the form of superitem
were to be assigned to the students allowing them to display such abilities. The problem
situations represented the following four content domains of linear equation: linear pattern
(pictorial), direct variation, concept of function and arithmetic sequence. There were four items
in each superitem which represent four level of SOLO model: uinstructural, multistructural,
relational and extended abstract. The correct response of one of the item would indicate an ability
to respond to the problem at least at the level reflected in the SOLO structure of that item.
Figure 1. Theoretical framework of the study
60 Lian and Idris
Problem situations represent:
* four content domains of linear equation
1. linear pattern (pictorial)
2. direct variation
3. concept of function
4. arithmetic sequence
1. Unistructural
2. Multistructural
3. Relational
4. Extended Abstract
*characterize the algebraic
solving ability across three
processes within four levels
of SOLO model
investigate
pattern
interpret&
apply the
finding
represent &
generalize
pattern
Algebraic solution
algebraic
solving
ability
solve
assessed
by
across
3 processes
solve

Table 1. Algebraic Solving Ability Framework
Unistructural Multistructural Relational Extended abstract
Linear
pattern
(pictorial)
• investigate the
pictorial pattern and
extend the next term
of it by referring
directly the
information and
diagram given in the
stem.
• ability to see the
given pattern as
successive process.
That is, identify the
recursive relationship
between the terms in
the sequence.
• ability to understand
and use the given
information serially to
compute some specific
cases and represent the
data in table.
• generalize the linear
relationship of the pattern
symbolically based on all
information given
.• all the information given
need to integrate to generate
an algebraic expression and
a rule for the pictorial
pattern.
• apply the rule to solve
the related situation.
• ability to analyze the
linear pattern across a
wider range of cases. That
is, use of linear
relationship, shape and
perimeter concept to form
a rule for the new linear
pictorial pattern.
• in forming the new rule
for the new pattern,
attempt to make
conjecture and verify the
conjecture deductively.
Direct
variation
• investigate the form
of direct variation by
identifying the
constant.
• use only one or
relevant aspect of the
information to find the
answer.
• ability to use more
than one numerical
operation (such as
addition operation and
multiplication
operation) to give the
response.
• make generalization by
using algebraic expression
and linear equation.
• all the necessary
information given is
integrated to make the such
generalization.
• application of the formula
to solve the related problem.
• ability to extract the
abstract concept from the
information given and
apply the related concept
(percentage) into a more
abstract situation.
Concept
of
function
• ability to use one
aspect of the available
information (first stage
of the process input-
output which involved
only one operation) to
find the output.
• notice the numerical
relationship of
variables, forming the
arithmetic expression
to compute the values
of dependent variable
and represent it in
table.
• all the information
given is used as
sequence to find the
values of output that
involved more than one
operations.
• ability to generalize the
input-output process in
terms of a linear
relationship between the
variables.
• ability to inter-relate all
the available information to
form an algebraic
expression and linear
equation to represent the
situation.
• working backward which
requires the application of
the formula.
• inter-relate all the
available information and
test it against appropriate
abstract general principle.
(functional relationship
between the variables) in
order to generate
alternative solution.
•.ability to examine the
structure and consider the
possibility of more than
one answer to the problem
situation.
arithmetic
sequence
• investigate the
pattern that comes
next to it. It requires
the understanding of
the sequence of
number pattern by
referring directly the
information given in
the stem.
• identify the numerical
relationship of the
dependent and
independent variables
by computing some
specific cases and
represent the data in
table.• the information
given in the stem is still
used as sequence.
• make a generalization by
integrating all the
information given in order
to formulate an algebraic
expression and formula.•
involve working backward
which requires the
application of the rule.
• ability to use an
hypothesis or an abstract
general principle (concept
of arithmetic sequence) to
form a possible alternative
solution.
• ability to use of logical
skill to account for the
possibility. For instance,
work with variables to
conceive the possibility in
a range of number.
61

METHODOLOGY
This study used quantitative and qualitative approach to assess the students' algebraic
solving ability based on SOLO model. The rational of researcher chose the quantitative method
was to assess the students' levels of algebraic solving ability. The dataset was submitted to partial
credit analysis. Subsequently, the qualitative method was used to seek the clarification of the
students' algebraic solving processes. Thus, this study was divided into two phases. In the first
phase, a test was given to the students. There were eight questions which were designed
according to the SOLO superitem format. Each superitem consisted of a situation or story (the
stem) and four items related to it. The items represented four levels of reasoning defined by
SOLO (unistructural, multistructural, relational, extended abstract). In the second phase,
clinical interview was conducted for the qualitative method. The clinical interview session was
carried out after the pencil-and-paper test.
Participants
In Malaysia, basic topic of algebra is taught during lower secondary school and the topic
of linear equation is taught in Form Two and Form Three (grade 8 and grade 9). Thus, the
construction of an assessment to assess algebraic solving ability amongst Form Four students
was important as the teachers could gain a greater awareness of students' algebraic solving
abilities about this topic before they learnt the more complex topics which were required the
basic algebra knowledge. The participants of this study consisted of 40 Form Four students from
a secondary school. Six out of the 40 samples were selected for the clinical interview (two
subjects from unistructural level, relational level and extended abstract level respectively).
Instrumentation
In this study, the instrument of data collection consisted of eight superitems. All of the
superitems were open-ended questions. Two superitems were constructed for each content
domain to be assessed. The following is an example of a superitem (superitem 1: linear pattern
pictorial) designed for this study.
Pictures are hung in a line. The pictures that are hung next to each other share two tacks
as shown below.
62 Lian and Idris

Level 1: Unistructural
How many tacks are needed to hang four pictures by this way?
Answer: 10 tacks
Note: The item requires the response based on referring the concrete information (given terms in
the diagram) to find the next term for the given sequence. This item can be answered most simply
by drawing and counting the number of tacks in the diagram to come up with the answer of 10.
Level 2: Multistructural
How many tacks are needed to hang 10 pictures, 16 pictures, and 20 pictures? Represent your
answers in a table.
Answers:
Note: This item requires the given information is handled serially. That is, identify the
recursive relationship between the terms in the sequence in order to compute the specific cases
and represent them in table.
Level 3: Relational
i) If you have y pictures, how many tacks are needed?
ii) Write a linear equation for finding a number of tacks for any number of pictures. Let t
represent the number of tacks and p represent the number of pictures.
iii) How many pictures can be hung if the number of tacks is 92? Try to apply linear equation
to solve it.
Answers: i) 2 +2y
ii) t = 2 + 2p
iii) 92 = 2 + 2p
90 = 2p
45 = p
(There are 45 pictures)
Note: This item requires the response that integrates all the information to make generalization
for the pattern. In order to response, the student must identify not only the two tacks per picture
but also the need for two more to hang the last picture in the series (e.g. t = 2 + 2p). If the student
provided this response, it would demonstrate his/her algebraic solving ability in identifying the
linear relationship between variables and applying algebraic symbols to make the representation.
Besides, the student may involve working backward which requires the application of rule.
the number of pictures the number of tacks
10 22
16 34
20 42
63

Level 4: Extended Abstract
'I don't have enough tacks to hang the such a lot of pictures by this way!', Said Lisa. Try to
create a new linear equation which represent the number of tacks (t) for any number of
pictures(p) to help Lisa.
Answer: t = p + 1 or
t = p or
t = 2p
Note: This level represents the highest level of algebraic solving ability. The response shows
an ability to extend the application of the given information in the new situation (new pattern)
and recognize an alternative approach which is distinguishable by the abstract features (linear
relationship).
The clinical interview session was carried out after the pencil-and-paper test had been
conducted. Clinical interview which is also known as 'flexible interviewing' is available for the
purpose of assessing any kind of mathematical solving ability. It is flexible, responsive and open-
ended in nature. The samples who were involved in the clinical interview session were selected
based on their performance in written assessment. Before the interview started, the samples were
shown their test papers to enable them to recall the methods and procedures they used to solve
the items. The interview questions were structured based on the items in the test.
Data Analysis
The data analysis had been done based on the findings from pencil-and-paper test and
interview. The data analysis procedures were categorized into two levels.
Level 1: The test paper results were analyzed by using Partial Credit Model. Partial
Credit Model (Wright & Masters, 1982) is a statistical model specifically incorporates the
possibility of having different number of steps or levels for the items in a test ( Bond & Fox,
2001). For example, the ordered values 0, 1, and 2 might be applied to an item which has three
ordered performance category levels as follow: 0 = totally wrong, 1 = partially correct and 2 =
completely correct. In this study, the ordered values 0, 1, 2, 3, 4, 5 and 6 might be applied to the
superitem as follow: 0 = totally wrong or no response, 1 = unistructural level, 2 = multistructural
level, 3 = lower relational level, 4 = relational level, 5 = higher relational level and 6 = extended
abstract level. Codes 0, 1, 2, 3, 4, 5, 6 covered all the response possibilities in the test. Code 0
as the code for the lowest possible response level and 6 as the code for the highest possible
response level in each superitem.
WINSTEP software program was used to run the analysis. It computed probability of
each response pattern in which took into account the ability of the learner and the difficulty of
the questions. The purpose for this computer analyze was also estimating the value of validity,
64 Lian and Idris

the reliability index, difficulty of the items and the levels achieved by the students based on the
different content domain items.
Level 2: The information from the clinical interview session was transcribed into writing
form. Six students were selected to be interviewed. Each interview was audiotaped and lasted
between 30 minutes to an hour.
RESULTS
Quantitative Results
In the Partial Credit model, reliability is estimated both for person and for items. The
item reliability index indicates the replicability of item placements along the pathway if these
same items are given to another sample with comparable ability levels. Person reliability index
indicates the replicability of person ordering that could expect if this sample are given another
set of items measuring the same construct. In this analysis, the item reliability index and person
reliability index were 0.91 and 0.73. The values fall within the acceptable range.
Valid is referred to the reliability index and the success of this evaluation to fit. If the
fit statistics (infit and outfit) of an item is acceptable, the expected value of the mean square
(variation in the observed data) is shown between 0.7 and 1.3. In the analysis, the infit and outfit
mean square for each superitem fall within the acceptable range. Besides, the means for all the
infit mean square and outfit mean square were considered reasonably well: 1.06 ( mean for infit
mean square) and 0.98 (mean for outfit mean square). (see Table 2)
Table 2. Partial Credit Analysis Of SOLO Item Dataset
Superitem Infit MNSQ Outfit MNSQ
1 0.99 0.94
2 1.31 1.30
3 0.75 0.83
4 0.84 0.88
5 1.16 1.15
6 1.14 1.10
7 0.88 0.82
8 0.99 0.87
Mean 1.06 0.98
65

Analysis of Levels and Difficulty of Superitem
Figure 2 showed that the number of students who have 50% and above probability of
success at six levels. From the findings, the difference of the number of students who have 50%
and above probability of success between the higher relational level and the highest level
(extended abstract) were only 4, compared with the difference of the number of students who
have 50% and above probability of success between the unistructural level and multistructural
level, there were 10. Besides, the difference of the number of students who have 50% and above
probability of success between the multstructural and lower relational level were 8. These
findings indicated that 62% of the students have less than 50% probability of success at
relational level. Majority of the students in this study could be classified into lower level of
algebraic solving ability especially unistructural level and multistructural level. Generally, most
of them were only able to numerically solve a variety of problems involving specific cases. They
encountered difficulties in generalizing the arithmetic through the use of algebraic symbols.
Figure 3 depicted that the number of students who have the 50% and above probability of
success at each superitem. Findings from this study showed that superitem 3 and 4 (direct
variation) were the easiest to respond. There were 34 and 29 students who have 50% and above
chance to respond successfully to superitem 3 and 4 respectively. However, there was some
range in the difficulty of the same content domain of superitems, especially superitem 1 and 2
(linear pattern pictorial).
Qualitative Results
Six subjects were interviewed. They were selected from three different levels of
algebraic solving ability: 1) Subjects from Unistructural Level (Sul1 and Sul2), 2) Subjects
from Relational Level (Srl1 and Srl2) 3) Subjects from Extended Abstract (Seal1 and Seal2).
The four processes of algebraic solving ability in solving superitem 1 (linear pattern
pictorial) were investigated as below :
66 Lian and Idris
1.Sul1 and Sul2: Samples who were only able to answer most of the unistructural level
of items successfully. The item responses include representation of one specific value
that comes next for the pattern.
2. Srl1 and Srl2: Samples who were able to answer most of the relational level items
successfully. The item responses include making connection between the information
given to form an algebraic expression and linear equation.
3. Seal1 and Seal2: Samples who were able to answer most of the extended abstract level
items successfully. The item responses include extracting the abstract general principle
from the information given to form an alternative rule for the new situation.

1a) Investigating the numerical pattern that comes next to it
Six subjects were able to solve and respond correctly by seeking out the sequence of
pattern which comes next to it or extend the sequence by referring directly to the diagram and
information given. The following dialogues depicted how the three subjects responded to item
1a: ( In the following dialogues, R denotes researcher)
R : Please tell me how did you get the answer for item1a?
Sul1 : (counting the number of tacks that he drew in the diagram given). I draw to get the
answer for the number of tacks.
R : How did you get 10 tacks for item 1a?
Srl1 : 1, 2, 3…10. (pointing to the diagram given)
R : How did you solve the item 1a?
Seal2 : (pointing to the diagram). I count from here. 1, 2, 3…10.
Figure 2. Difficulty of SOLO levels and ability levels
67
Level Person
0.65 x
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25 x
0.20
EA • 0.15
0.10 xxx
HR • 0.05 x
0 x
-0.05 x
-0.10 x
R • -0.15 xx
-0.20 xxx
LR • -0.25 x
-0.30 xxx
-0.35
-0.40 xxx
M • -0.45 xx
-0.50 xxxxxx
-0.55 xxxx
U • -0.60
-0.65 x
-0.70
-0.75
-0.80
-0.85 xx
-0.90 x
-0.95 x
-1.00 x
-1.05
-1.10
-1.15
-1.20 x
easy logits scale least able
U-unistrucrtural HR - higher relational
M-multisturctural EA - extended abstract
LR- lower relational
R-relational

Figure 3. Difficulty of superitems and ability levels
1b) Investigate the pattern by computing specific cases
When subjects were confronted with various number tasks as a way to assess and refine
their understanding of the pattern, subjects began to notice the pattern and understand the linear
relationship involving an arithmetic operation. Subject Srl1, Srl2, Seal1 and Seal2 did not use
the manipulative to get the solution instead of substituted the specific values into the arithmetic
expression. These were shown in the extracts below:
R : How did you solve the item 1b?
Srl1 : (Recalling) I multiply the number of pictures by two then I add two to it.
R : Can you try to explain the method you used?
Srl1 : There are at least two tacks needed for each pictures. So, I multiply the number of
pictures by 2. There are two tacks in the end. So, I add two to it. (see Figure 4)
68 Lian and Idris
Level Person
0.65 x
0.60
0.55
0.50
0.45
0.40
0.35
8 • 0.30
0.25 x
0.20
0.15
7 • 0.10 xxx
0.05 x
0 x
-0.05 x
2 • -0.10 x
-0.15 xx
-0.20 xxx
-0.25 x
6 • -0.30 xxx
-0.35
5 & 1 • • -0.40 xxx
-0.45 xx
4 • -0.50 xxxxxx
-0.55 xxxx
-0.60
-0.65 x
3 • -0.70
-0.75
-0.80
-0.85 xx
-0.90 x
-0.95 x
-1.00 x
-1.05
-1.10
-1.15
-1.20 x
easy logits scale least able

R : How did you find the number of tacks for the different number of pictures (item 1b)?
Seal2 : (Pointing to the solution to find the number of tacks for 10 pictures). 2(10) + 2 = 20
+ 2 = 22. '2' means two tacks, each pictures at least has two tacks, '10' means the
number of pictures and ' + 2' means two tacks in the end. The same method I use to
get the answer for 16 and 20 pictures.
R : Can you explain how you solve the item 16?
Seal1 : (16-1)2 + 4 = 34. (16-1) means I subtract the first pictures. The first pictures has 4
tacks so (+4). The others pictures have 2 tacks each, so (15 x 2).
Sul1 and Sul2 were unable to notice the linear pattern in the diagram. They solved the
items by counting method. They drew the pictures to count the number of tacks. They noticed
the pattern by working with manipulative to get the answers. They relied on the drawing and
counting. Their explanation were shown in the following dialogue:
R : How about item 1b? How did you solve it?
Sul1 : I draw the number of pictures that the question asked and count the number of tacks
are needed.
R : Can you show me how you get the answer for 10 pictures?
Sul2 : 1,2,….22 (counting from the pictures).
Figure 4. SRL1's arithmetic expression and algebraic expression
2a) Representing the data into table
Subjects represented their data in a table and classified it into categories which reflected
the different ways of solving ability about the pattern. Subject Srl1, Srl2, Seal1 and Seal2 who
were able to comprehend the pattern. They used the arithmetic rule to compute the table.
Whereas subject Sul1 and Sul2 who failed to comprehend the pattern and the linear relationship
between the variables, made the computational errors along the way he represented the findings
in the table.
69

2b) Representing the unknown value with the use of letter
Subject Srl1, Srl2, Seal1 and Seal2 were able to gather the information and findings at
the concrete level into more abstract level. In other words, they were able to transfer the
meaning from arithmetic expression to an abstract conjecture. They linked their interpretation
and mathematical findings to connect the counting action with an accurate symbolic
representation in the form of algebraic expression. The following dialogues shown Srl1, Seal1
and Seal2 explained how the algebraic expression was formulated:
R : How many tacks are needed for y pictures (item 1b)? Try to explain your answer.
Srl1 : 2(y) + 2 (referring to the table). I substitute the figure with y.
R : What is y?
Srl1 : y is an unknown… can be a certain number.
R : For y pictures (item 1b), how many tacks are needed?
Seal2 : 2y add to 2. (2y + 2).
R : How did you form this expression? Which part of information did you refer?
Seal2 : Based on the data in the table (pointing to the table). I substitute the figure with y,
an unknown. So, 2(y) + 2.
R : What is unknown?
Seal2 : (Thinking) …Eh… represent a quantity, a number but don't know how many.
R : For y pictures, how many tacks are needed?
Seal1 : (y-1)2 + 4. I refer this (pointing to arithmetic expression)
The problem about lack understanding of algebraic concepts such as unknown and
misconception that 'letter has a unique value' as opposed to an ability to make transition from
arithmetic method to the application of algebraic symbols . For example, Sul1 and Sul2 showed
that they were unable to use algebraic concept such as algebra expression to express or describe
the numerical relationship that existed in the pattern into the abstract situation. The problem
about lack understanding of unknown and misconception can be seen in the extract below:
R : For item 1b, if y pictures, how many tacks are needed? How did you get 52?
Sul1 : I simply write the answers.
R : What does it mean ' y pictures'?
Sul1 : I don't understand actually.
R : If y pictures, how many tacks are needed?
Sul2 : If y is 8, the number of tacks is 18 (counting the number of tacks).
70 Lian and Idris

2c) Writing linear equation to make generalization of pattern
Srl1, Srl2, Seal1 and Seal2 were able to generate the linear equation for the problem
situation based on the information given and the data from their table. Srl1 and Seal2's
explanation were shown in the following extracts:
R : Try to explain your equation for item 1c(i)? How did you get it?
Srl1 : t = p x 2 + 2, t = 2p + 2. I got it from the findings in question b (pointing to the
table).
R : Ok, for item 1c(i), try to write a linear equation to represent the
situation?
Seal2 : p(2) + 2 = t, p multiply by 2 and then add to 2 equal to the number of tacks (t).
R : How did you form this equation?
Seal2 : I got it based on the answer of item 1a and 1b.
3) Application of the rule to solve the related problem
The equation was applied to represent the relationship between dependent variable and
independent variable of the problem. Srl1, Srl2, Seal1 and Seal2 were able to analyze the
problem into the rule that they formed. In the following extract, Seal2 tried to explain the
application of linear equation:
R : If you have 92 tacks, how many pictures can be hung (item 1cii: linear pattern)? How
did you solve the problem?
Seal2 : I use equation to solve it.
R : How did you solve it?
Seal2 : 92 = 2p + 2. 2p =90, so p = 45. I find p to get the number of pictures.
R : Any method that you can use?
Seal2 : I think this is the fastest. (see Figure 5)
Figure 5. SEAL2's use of linear equation to solve problem
71

However, Srl1 was unable to apply the rule due to the misunderstanding of the use of
reverse operation between multiplication and division. This weakness had obstructed to the
progress of mapping the steps appropriately to the solution.
4) Making generalization for the new pattern or new situation by forming the alternative solution
Seal2 was able to consistently generalize the new pattern or new situation by forming
the alternative solution for each problem situation, with the exception of the concept of function
problem situation. Seal1 was able to solve most of the items in this level with the exception of
the direct variation problem situation. In solving the item 1d, Seal1 and Seal2 were able to
extract the abstract concept (linear pattern) from the information given to form the linear
equation to generalize and represent the new situation that they created. For example, Seal2
apply the concept of linear relationship to assess the causal and effect relationship between the
number of pictures and the number of tacks that he created. Then, he proved the decision by
designing a new rule to the new situation. The extract below demonstrated how Seal2 provided
an alternative solution to another problem situation:
R : How did you solve the item 1d ?
Seal2 : From item 1c, the equation is 2p + 2 = t. In order to reduce the number of tacks in
each picture, I divide 2p + 2 by 2, try to reduce half of the quantity of tacks.
R : How did you prove your equation?
Seal2 : For example, if there are two pictures t =
t = p + 1
t = 2 +1
t = 3
There are only 3 tacks needed compared with previous equation in which 6 tacks are
needed.
Six subjects gave the detailed responses in explaining their solving abilities within the
four processes. They added more to their initial responses when probed and prompted in an
interview situation but in many instances, their responses still at the same level rather than
increasing the level of the responses.
2
2
2 +
p
72 Lian and Idris

DISCUSSION AND CONCLUSION
Six levels of sophistication in algebraic solving ability can be found in learner's response
to the tasks: unistructural, multistructural, lower relational, relational, higher relational and
extended abstract. The quantitative data analysis demonstrated that majority of students (62%)
achieved algebraic solving ability at unistructural level and multistructural level. Most were able
to numerically solve a variety of problem involving specific cases. They encountered difficulties
in making generalization through the use of linear equation. This finding are consistent with
previous research findings (Swafford & Langrall, 2000; Orton & Orton, 1994) that majority of
the middle grade students were able to solve the problems involving specific cases, explain the
sequence of pattern only in terms of difference between successive terms and very few of them
able to generalize the problem into algebraic form. In the qualitative data analysis, researcher
found that the high ability students seemed to be more able to seek out the recurring linear
pattern and identify the linear relationship between the variables. They were able to co-ordinate
all the information given to generalize the pattern algebraically, (forming an algebraic
expression and linear equation), ability to use the linear pattern concept in more abstract situation
such as forming a rule for the new linear pattern that they created. Also, they used their methods
more consistently to find the solutions. Whereas the low ability students showed an ability more
on counting method in where the task given is performed and understood serially. They failed
to inter-relate the features of linear pattern given in the question due to the lack understanding of
algebraic concepts especially unknown and linear equation.
However, this study did not reveal the cognitive obstacles encountered in using linear
equation to solve the problem situation. Further research might address these issues so that the
framework will be more effective for supporting instructional programs that build on students'
prior knowledge, foster their solving ability and monitor their understanding. Further studies are
also needed to investigate whether the framework is appropriate for students in other grade levels
to determine the extent to which it can actually be used to inform instructional and assessment
programs in secondary school algebra.
As noted above, the framework not only suggest an item writing with the format of
superitem, it also can be used to score the item, can allow for crediting partial knowledge. Thus,
it provides teachers an indication of some of the levels of algebraic solving ability they can
expect to encounter in their classroom. Notwithstanding, this framework has the potential to
contribute to both instruction and assessment. In the instructional perspective, it would seem
prudent for teachers to use solving ability level descriptors as broad guidelines for organizing
instruction and building problem task. From an assessment perspective, it appears to be valuable
in providing teacher with useful background on students' initial solving ability, and in enabling
them to monitor general growth in algebraic solving ability.
73

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75

Authors: Lim Hooi Lian And Noraini Idris
Email: hllimusm@yahoo.com & noridris@um.edu.my
Address: Faculty if Education
University of Malaya
Malaysia
76 Lian and Idris

Assessing Algebraic Solving Ability Of Form Four Students

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