How Teacher Knowledge and Perceptions in Representations
of Linear Functions Translate into Their Classroom Teaching
Center for STEM Education, University of North Carolina at Charlotte
Charlotte, USA
Shagufta Raja, sraja1@uncc.edu
David K. Pugalee, David.Pugalee@uncc.edu
Alisa Wickliff, abwickli@uncc.edu
Abstract
Teachers’ content knowledge and perceptions about representations of linear
functions is a major factor that affects students’ concept building and learning
processes. This paper reports students’ understanding of linear functions in
contextual and graphical representations. Students’ misconceptions and limited
understanding of linear functions in real life situations are discussed. The findings
highlight the social and cognitive aspects of students learning in STEM fields.
Background and Perspectives
Lots of attention directed to multi-representations of linear functions, such as
graphs and symbols, carry great significance in various contexts, including
medicine, business, and engineering. For many years the concept of linear
functions and their representations have been taught in middle schools and
algebra classrooms. After being added to high schools’ graduation requirements,
particularly for all college-bound students, algebra has gained momentum in its
importance as a mathematics course (McGraner, 2011). Algebra scores are also
now used as indicators of success for students pursuing higher education in
Science, Technology, Engineering, and Mathematics (STEM) related fields (Fey,
1989). Likewise, the Scholastic Aptitude Test (SAT), a highly reliable
standardized measurement of college readiness, is used in the admissions
process at nearly all four-year undergraduate colleges and universities in the
United States (College Board, 2011). The SAT is recognized as an objective
measure of mathematics and English skills. It is considered to be uniform across
all test takers, a fair- and valid-predictor of college success for students from
varied backgrounds. Unfortunately, over the past ten years, averaged SAT scores
have not been satisfactory as compared to global Algebra standards. Despite the
greater number of students taking the SAT, the overall scores have shown little or
no improvement each year since 2011. One possible reason for incorrect
responses on SAT math sections may be attributed to the lack of ability to fully
understand algebraic problems in linear functions (Welder, 2006). Performance
on algebra questions is highly dependent on how the students perceive or
understand the graphic representation of functions (Yerushalmy, 2000). Warren
and Pierce (2004) elucidate that although comprehension of linear functions and
value of graphs has presented serious challenges in teaching algebra content in
high school, algebra curriculums have not changed much in the past century. The
difficulty in recognizing linear function vocabulary is considered another major
issue for student understanding of linear functions and their representations.
(Brenner, 1997). Teachers play a major role for motivating students and
developing conceptual understanding. Successful math teachers can provide
opportunities for students to avoid misconceptions and enhance their learning
experiences in real life STEM (science, technology, engineering and
mathematics) perspectives.
Research Background
In the context of teacher preparation and student achievement, Monk (1994)
recorded that there is a positive impact of mathematics teacher education on
student knowledge. Rowan, Chiang, and Miller (1997) found significant effects of
teacher knowledge on student achievement, with the greatest effects occurring in
combination with teacher preparation (majoring in mathematics) and in low
performing schools. Previous research also has provided evidence that there is a
deep connection among teachers knowledge and students achievement in
mathematics. This indeed is a complex effect instead of being direct that is
mediated through the student teacher interactions in classroom discourse and
instructional techniques used in classroom Monk(1994). Markovits, Eylon, &
Bruckheimer (1983) reported that the student hold misconceptions about linear
functions and their graphs to be prototype models and images of functions. The
language of mathematics such as use of notations and symbols, sometimes pose
difficulty for students to comprehend the meaning of mathematical terms. For
example Booth (1984), found a student may read the notation f(x) for function as
“f times x” instead of “f of x” representing a function unless the convention is
explained during instructions.
Language of Mathematics: Use of Notations and Symbols
Mathematics has its own rich vocabulary and language consisting of patterns,
diagrams, symbols, representations, notations, algebraic equations, formulas and
functional relationships. Conceptual understanding of algebra and in particular
linear functions involves the use of the language of mathematics with emphasis
on manipulating symbols and graphic representations. The use of various types
of treatment such as mathematical notation and vocabulary is dependent on the
type of the problem at hand. Kieran (1992) considers that in our schools the
algebra problems are solved using either the “procedural” or “structural”
operations. Where procedural “refers to arithmetic operations carried out on
numbers to yield numbers”, and structural refers to “a different set of operations
that are carried out, not on numbers, but on algebraic expressions.” (p. 392).
NCTM standards for schools ( 2000) consider modeling as an essential category
for learning algebra using “identifying and selecting relevant features of a realworld situation, representing those features symbolically, analyzing and
reasoning about the model and the characteristics of the situation, and
considering the accuracy and limitations of the model” (NCTM, 2000, p. 303). The
syntactic knowledge (that deals with the formal properties of signs and symbols.)
includes the knowledge of algebra terms such as equal, equivalent, simplify, and
solve, variable and functions. If any of the algebra terms are defined ambiguously
or not consistently used across teaching materials or may be conflated with
students’ everyday language, cause confusion and difficulty in long term
conceptual understanding in algebra especially linear functions. Therefore it is
very important to address this issue in teachers’ training. Besides knowing the
conventional language of mathematics, the teachers must also be able to analyze
alternate definitions.
In United States school teachers commonly use the acronyms to describe steps
and pre-determined methods for solving algebra problems including linear
functions. For example students are directed to use “PEMDAS” (parenthesis,
multiplication, division, addition, subtraction) for the order of operations, FOIL
(first, outside, inside, last) for multiplying two binomial expressions and STAR
(search, translate, answer, review) for word problems. The use of acronyms acts
as short cuts to help the students for memorizing the method for problem solving
written in a certain form. The students get confused if the problem is stated in a
different form.
Students learn the standard equation of line for making graphs, learning about
slope intercept forms but in many cases do not consider it to be the algebraic
representation of linear function. Conventional notations for representing
algebraic terms 2x and x2 are not recognized by students and they are often
confused for the exponential notation as x2 instead of x2 or confuse among
superscript and subscripts ( x2 and x2 ).
Student Misconceptions about Linear Functions and their Representations
Some examples for students misconceptions retrieved from a recent research
study (Understanding of Multi-Representation of Linear Functions for Pre-College
students in STEM Context) are briefly discussed and illustrated below in diagram
that is a copy of students’ work.
Figure 1: An example of student’s lack of understanding of functional notations
The given task required student to look at the specified figures and find a
rule/functional relationship for the patterns. The student found the rule in words
as “adding three to each figure to get the next”. In order to complete the pattern,
he followed the rule and wrote 6+3 and generalized for pattern 2 as x(6) + 3.
Interestingly, for pattern 6, he generalized the rule as (x (5)) + 3; then the student
substituted the value of x = 6 and found 6(5) +3 =33 and for pattern 10, he
obtained (10(5)) +3, generalizing his rule as 5n +3 =53. The researcher asked the
student to make the figure/pattern 6, representing his results from calculations for
pattern 6. He built it and found it should have been 24 squares not 33. This
discussion clearly supports the research results from Booth (1984) mentioned
above, the misconception was clearly because of student not understanding the
functional notation and using algebraic operation of multiplication represented by
parenthesis after a numerical values.
Figure 2: Another example student lack of understanding of graphical
representation of linear functions
The task in figure 2 was to write an algebraic equation representing the given line
graph and then make a graph for the algebraic representation for the function y=
3x -4. The student work sample shown in figure 2 displays that the student first
wrote y= 5x for the graph and then used his memory to write the definition y2- y1
/x2 –x1 and ∆y/∆x without calling it slope or rate of change. After that he wrote
the equation y= ____ x +5 and final equation as y = -4x +5. It is noted that the
student used the intercept from the given equation to write as slope for the given
graph and added 5 (which was used by him as the coefficient of x in the equation
y= 5x). Furthermore, the students drew the graph for the given equation as an
exponential representation. In response to the question from researcher to
explain his drawing, the student indicated in his explanation that “a function is
either quadratic or exponential” represented algebraically as y = a x bx and y= mx
+ b that is the equation of line. Clearly this misconception is due to lack of
conceptual understanding of functions and their representations.
Pre-Service Teacher training and Knowledge
Students’ learning is very much influenced by pre-service training and knowledge
in subject area and the complexity background issues both inside and outside the
math classroom. Teacher knowledge in mathematics is considered to be derived
from three categories: content knowledge of subject area, pedagogical content
knowledge, and effective content knowledge. The academic gain in knowledge
for students is not a direct outcome of the influence from the teachers’ advanced
degrees. The content knowledge is more focused on the understanding of the
connection between big ideas thus supporting conceptual notions in the subject
content area. The pedagogical content knowledge transferred to students in the
form of activities and personal experiences help in providing mathematical
proficiency leading to a conceptual understanding of topics in high school math,
especially in algebra. Effective knowledge is implemented through motivational
teaching learning strategies. The effective teaching knowledge can only be
acquired if the prospective teachers develop a mathematical taste and passion.
That subsists in them beyond college graduation and they can see through the
thick walls of suggested curriculum; what to emphasize in their instructions for
enhancing students learning experiences instead of just teaching skills practices.
This helps the students thinking process and how the mathematical learning
experiences can be sequenced. The effective teaching knowledge helps the
students with the ability to flexibly move among representations, work and
manipulate algebraic relationships and functions. It also strengthens the students’
understanding of basic operations, notations and representations such as tabular
and graphical forms in a real-life perspective.
Conclusions
Student’s work in example 2 reflects that they had some background information
about the ‘rate of change and slope’ but he was unable to make connections
among graphic representation and rate of change and slope of linear functions.
The student relied on his visual memory and arithmetic skills for working with
sequence by adding a constant number to the previous value and get the next
value of the function in the sequence. Learning about linear function is a
continuous progression that begins in elementary school math curriculum: e.g.,
making patterns and bar graphs. In middle school the experience of patterns
leads to the concept of ratios and proportions, laying the foundations for high
school algebra with the advancement to one to one correspondence,
development of functional notation and mapping of domain into range and more.
While example 1 indicated that student was unable to relate patterns to an
algebraic rule and understand the use of functional notations. It is due to the lack
of building concepts of linear functions at different stages in school math with
some incomplete concepts’ development which resulted in confusion and nonsustainability of concepts for future applications in advanced algebra courses.
Lot of previous research had been published about students understanding of
linear functions and their representations. In most of the research articles the
emphasis is only on one-type of representation. Students are taught linear
functions as a concept in class and the text books, followed by the drill pattern
using algebraic rules and problem solving skills. Each representation is taught
separately without making a connection among them or understanding the
dynamic behavior or linear function in action. Referring to example 2, the student
could not recognize linear functions because the concept of function is often
discussed in the context of quadratic, exponential and not in the unit for linear
functions. Linear functions are discussed in graphic representations without
connecting it or comparing to other representations and real-life situations.
There can be several reasons for student’s perceptions difficulties to learn, such
as student background due to prior schooling or teacher’s perceptions,
instructional practices, content knowledge or professional development. In this
research paper we will discuss how teacher knowledge and perceptions in
representations of linear functions translate into their classroom teaching. This
lack of student conceptual knowledge is often due to inadequate teachers’
professional development for effective instructional strategies. The aspect of
students’ in-depth understanding of graphs, written and verbal representations is
a reason for the students’ inability to characterize linear functions. The students
not familiar with relevance of linear functions and their representations in real-life
situations and functional notations face difficulty in making connections and
developing critical thinking skills. Almost all process in real-life situations is
simultaneous and it is often hard to separate them without restricting some of the
variables in the ongoing processes. The findings of example-1 and example-2
imply that the students had difficulties with language of mathematics and
vocabulary; which resulted in the students’ inability to see linear functions in a
bigger picture. The students were given mathematical definitions and formulas
such as slope m = (y2 – y1)/(x2 –x1), and provided demonstrations for how to
make graphs, pictures and use technology instead of following up on the ideas
and allowing students an opportunity to learn the concepts using multirepresentations for linear functions. Some students’ serious misconceptions
about types of functions, their characteristic, real-life applications and inferences
from graphic representations surfaces concerns about the teachers’ up-to-date
content knowledge and training to teach 21st century students in schools. A
possible reason for this supposition is that students are taught linear functions
strictly to focus on individual representation and from the perspective of being
successful in the course of study instead of a broader perspective of success in
life careers. Learning is a cognitive- and social-activity that must be viewed in
broad perspective. As mentioned in the previous paragraphs the students’
comments were simply reflecting their abstract learning of math concepts. It was
very difficult for them to analyze the situation for problem solving in relevant or
real-life perspectives.
In the past century research, one can find the evidence for emphasis on algebraic
representation. Students and teachers were trained for skills- and fast- paced
learning practices such as how many algebra problems can a student’s solve
correctly in shortest time and the focus was just on the solution of problem. Such
practices and skill drills did improve students’ reflexes and challenged the brain
for quick reactions but did not help with critical thinking or reasoning processes
for long-term understandings. A student in the study remarked, that “I loved math
very much but not anymore because algebra is hard”. Another student said, “I am
very good in my math class”. The algebra classroom teacher (of the students’
sample work in example 1 and 2) shared that there are too many objectives in
algebra course to be covered within a year and the class-size is very large;
therefore, it is difficult for her to allow time for students to practice with multirepresentations for algebra problems.
As observed in this research paper, students were able to understand the
patterns but they were not able express until they wrote it on paper or draw the
figures. The student comments should not be ignored because they are ones who
claimed for being the best in their classes. It also indicated the problem of using
multiple choice test-items by teachers for determining student level of
accomplishments in math. The multiple choice test scores can be wrongly
representing student ability for problem solving (students could have made a
lucky guess for the answer). The two samples for students’ performance showed
that their first attempt for solving problems was to use visual and mental math
skills but due to misunderstanding or lack of conceptual understanding of linear
functions, they became confused. To overcome the difficulty, some of them used
calculators and tried to work with random numbers and operation for finding a
reasonable answer for them, others tried to guess and wrote previously
memorized facts or definitions to help solve the problem. This behavior leads to
check for teacher education and training for teaching to diverse students’
population. Questions like “was the teacher aware of the relationship among
fundamental concepts of ratios/proportions to linear functions?” Or “Did the
teacher teach students about the one-on-one correspondence and mapping,
concept of domain and range, etc. in class?” Did the student have firsthand
experience in working with multi-representations of linear functions?” and “how
the student learning was assessed?”. These are just few questions that can help
in finding about the difficulties faced by students and their progress in learning
linear functions.
With the introduction of common core standards, perhaps one can see the light at
the end of the tunnel that might help in the students’ global learning process. The
key shifts in curriculum if correctly forced and implemented can make it easy to
find the places and areas where the problem begins for each individual child.
Teachers’ content knowledge, pedagogy and up-to-date training in the use of
skills, technology and other community resources, along with adequate
evaluation methods is major task and hope for a successful implementation of
common core practices. Current research such as TPACK is focused on this goal
for students’ achievement directly linked with teachers’
professional
development.
References
Booth, L. (1984). Algebra: Children’s Strategies and Errors. Windsor, UK: NFERNelson.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws
(Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390419). Reston, VA: National Council of Teachers of Mathematics.
Monk, D. H.: (1994), 'Subject area preparation of secondary mathematics and
science teachers and student achievement'. Economics of Education Review,
13(2), 125-145.
Raja, S. Y. & Pugalee, D. K.,(2015) Multi-Representation of linear functions for
pre-college students in STEM context (Unpublished doctoral dissertation).
University of North Carolina, Charlotte, NC.
Rowan, B., Chiang, F.-S., & Miller, R.: (1997), 'Using research on employees'
performance to study the effects of teachers on students' achievement'.
Sociology of Education, 70(October), 256-284.