2011. Vol.2, No.3, 264-269
Copyright © 2011 SciRes. DOI:10.4236/ce.2011.23035
Primary and Secondary Teachers’ Knowledge, Interpretation,
and Approaches to Students Errors about Ratio and
Elena Fabiola Ruiz Ledesma
Graduate Department, School of Computer Sciences of the National Polytechnical Institute of Mexico,
Gustavo A. Madero, Mexico.
Received July 15th, 2011; revised July 29th, 2011; accepted August 2nd, 2011.
This study investigated elementary and secondary teachers’ understanding and pedagogical strategies applied to
students making errors in finding a missing length in similar rectangles. It was revealed that secondary teachers
had better understanding of ratio and proportion in similar rectangles than elementary teachers. While all secon-
dary teachers solved the similar rectangles problems correctly, a large portion of elementary teacher struggled
with the problem. In explaining their solution strategies, and even though similar strategies appeared both from
elementary teachers and secondary teachers, a majority of secondary teachers pointed out the underlying idea of
similarity, whereas less than half of the elementary teachers explained their reasoning for using ratios and pro-
portion. This article is derived from the research project registered under number 20110343 (Ruiz, 2011), and
developed in Escuela Superior de Cómputo del Instituto Politécnico Nacional (IPN) (School of Computer Sci-
ences of the National Poly-technical Institute of Mexico)
Keywords: Primary Teachers, Secondary Teachers, Knowledge, Ratio, Proportion’ Problems
The development of proportional thinking is important since
basic educational levels, as from it depend that children can be
able to comprehend and face everyday situations which are
linked with the proportion concept. At the same time, as it is
established by Ruiz and Lupiañez, (2009) so the student of
basic level can give sense and meaning to the proportion con-
cept it is very important to develop their proportional thinking
such as the qualitative as the quantitative. That is to say, for the
development of proportional thinking it is required, among
others that the subject can build the concept of proportion.
The NCTM document (2000) describes proportionality to be
“of such great importance that it merits whatever time and ef-
fort that must be expended to assure its careful development” (p.
82). Very often multiplication and division tasks in the lower
grades are presented in unit-rate form, which is a special form
of ratio and proportion. In the middle grades, word problems
involving equivalent fractions and fraction comparisons can also
be thought of as ratio and proportion situations (NCTM, 2000).
The ability to recognize structural similarity and multiplicative
comparisons illustrated in such proportional reasoning processes
are the cornerstone of algebra and more advanced mathematics
(Kilpatrick, Swafford, & Findell, 2001). Nevertheless, research
has consistently shown that many students have difficulty with
developing proportional reasoning. Hart (1984), for example,
reported that less than 42% of students in grade 7 succeeded in
solving simple problems of enlargement, a most common error is
additive reasoning (Lamon, 2007). According to these studies,
students tend to focus on the difference between the given quan-
tities rather than proportionality illustrated in given contexts.
The Problem and the Research Questions
In this study I set out to investigate elementary and secon-
dary teachers’ reasoning, their responses to student errors on
the topic of ratio and proportion, and the relationship between
their knowledge and approaches. I used aforementioned student
error in exploring teachers’ reasoning and approaches. I was
curious about how teachers would interpret and respond to
student errors in finding a missing length in similar rectangles,
and how their approaches related to their mathematical know-
ledge. Although a growing body of research has focused on
teachers’ treatment of student errors (e.g., Schleppenbach, Fle-
vares, Sims, & Perry, 2007; Ruiz, 1997; Ruiz, 2002), teach-
-ers’ responses and their strategies have received limited atten-
tion in the research literature. If teachers are called to use stu-
dent errors as springboards for inquiry into mathematical con-
cepts, it is important to explore teachers’ responses and strate-
gies to student errors, and prepare them to make better use of
student errors through teacher education programs.
The purpose of this study is not to add to the collection of
studies documenting teacher weakness, but rather to inform the
design of teacher education in this area. Exploration of teach-
ers’ interpretations of and responses to student ideas and, in
particular, student errors will help enrich a dialogue among
reformers, educators, and professional developers in ways they
could help teachers learn to teach math to promote student un-
derstanding. More specifically, the research questions guiding
the study included:
E. F. R. LEDESMA 265
What instructional support do teachers need to provide?
How do teachers need to use student errors in instruction?
Teacher Knowledge, Approaches and Its Relationship
Although many researchers have devoted considerable atten-
tion over the last two decades to what teachers should know
and be able to do, there are still large gaps in this research, in
particular, research studies on teachers’ knowledge of ratio and
proportion. Fisher (1998), for example, investigated secondary
mathematics teachers’ understanding and strategy usage in ratio
and proportion problems. He used word problems involving
direct proportional reasoning and inverse proportional reason-
ing. He reported that, overall, teachers were more successful
with direct proportional reasoning problems (i.e., y = kx) than
with inverse proportional problems (i.e., xy = k). However, the
strategy chosen by teachers varied with the type of problem. In
terms of teaching approaches, he reported that most teachers
said they would use the same strategies they used while teach-
ing the problems tested. Lim (2009) investigated twenty-eight
preservice teachers with four types of invariance in miss-value
problems: ratio, sum, product, and difference. He found that
preservice teachers had different levels of understanding de-
pending on the type of problems. While teachers had less diffi-
culty with the first and second problems (ratio and sum), they
showed greater difficulty with a missing value task involving
product and difference. Ruiz, 1997, 2000 and 2002, reported
that teachers generally do not pay close attention to the mean-
ing of ratios when they set up a proportion to solve a missing
value problem, indicating that teachers tend to use the same
approach for a missing value problem regardless of different
contexts. Although the findings from these studies help us un-
derstand teachers’ understanding and strategies, there remains a
need for research to unfold how the teachers’ own understand-
ings impact their interpretations of the students’ misconceptions.
An investigation of this important topic is described here.
Strategies in Ratio and Proportion problems
In analyzing teachers’ knowledge, interpretation, and ap-
proaches to student error(s), I was guided by the studies of Ruiz,
2000, and 2002, who distinguished between the use of proce-
dural and conceptual knowledge. I also referred to the studies
of Fisher (1999) and Lamon (2007), who articulated different
levels of understanding of ratio and proportion. Ruiz defined
conceptual knowledge as explicit or implicit understanding of
the principles that govern a domain and of the interrelations
between pieces of knowledge in a domain. They defined pro-
cedural knowledge as action sequences for solving problems.
These two types of knowledge lie on a continuum and cannot
always be separated; however, the two ends of the continuum
represent two different types of knowledge. More detailed ex-
amples will follow of these two forms of knowledge regarding
ratio and proportion in similar rectangles as I discuss the find-
ings of the study.
This study investigated teachers’ interpretation of and re-
sponses to student error(s) through a classroom scenario in
which an imaginary student incorrectly solved a similar rectan-
gles problem. The study also examined how teachers’ peda-
gogical strategies used to address the student errors are related
to their content knowledge of ratio and proportion.
One task (content knowledge and pedagogical content know-
ledge) was developed based on Ruiz, (2002). The content
knowledge task was aimed at assessing teachers’ understanding
of ratio and proportion. The first question, called similar rec-
tangles problem, required teachers to find the missing side of a
rectangle given the condition the two rectangles are similar, and
to explain their solution. The pedagogical content knowledge
followed the content knowledge task. After completing the
similar rectangles problem, they were asked to interpret and
respond to a student’s incorrect solution. According to Ruiz,
2000, 2002, and Ruiz and Lupiañez, 2009, the most common
incorrect strategy in finding the length of the missing side in
similar rectangle is to use additive reasoning (A – B = C – B)
focusing on the difference between the given length in similar
rectangles, rather than focusing on proportional relationship
between two figures (A/B = C/D). I based my exploration of
teachers’ pedagogical content knowledge (PCK) on this com-
mon incorrect strategy. The teachers were asked to identify one
fictitious student’s (Lilith) error(s) and then to provide a written
description of how they would respond to her (see Figure 1).
Fifty-seven teachers participated in this study. Thirty-one
were of the elementary school and twenty six were of secon-
dary school in Mexico City.
The tasks went through multiple phases of revisions and
were pilot tested with two volunteers who were then inter-
viewed to check for possible misunderstandings. The final ver-
sion of the task was then administered as an in-class survey in
three mathematics methods course sections, two elementary and
the other secondary, towards the end of the semester. This study
only includes data from teachers who signed the study’s con-
sent form. For data analysis, responses to the content know-
ledge task were first identified based on correctness and then
solution strategies were identified. The final phase of analysis
consisted of the identification of emergent patterns in teaching
You are teaching 6th graders. You asked the students to find the length
of the missing side in similar rectangles shown below. After a few
minutes, you asked Lilith, one of your students, to explain how to solve
the problem. Lilith explained that the side would be 12 cm long be-
cause 4 + 2 = 6.
Contains the problem. Ruiz, 2002.
E. F. R. LEDESMA
Summary of Results proportional reasoning.
How Do Teachers Identify Lilith’s Learning
How Do Teac her s Unde rstand Rati o and Proportion?
The findings from the content knowledge task were helpful
in providing an initial framework for analyzing the PCK task.
In the first problem, teachers were asked to find a missing
length in similar rectangles where a rectangle that began as a 4
cm by 6 cm was enlarged to a rectangle with a short side of 10
cm. All secondary teachers answered correctly and 73% of
elementary teachers (22 out of 31) answered correctly. Among
the teachers who provided incorrect answers, 6 teachers used a
common strategy, additive reasoning, by focusing on the dif-
rences between the given quantities. One typical response was
In a classroom setting, our student, Lilith was asked to solve
the same problem teachers completed--find a missing length in
similar rectangles. Lilith concluded that the long side is 12 cm
since 4 cm + 2 cm = 6 cm. As I addressed earlier, finding a
missing length in similar rectangles involves at least four big
ideas: 1) understanding the concept of similarity, 2) determine-
ing a between ratio, within ratios, or a scale factor, 3) setting up
a proportion, and 4) carrying out calculations correctly. In Li-
lith’s case, she did not understand the concept of similarity-the
lengths of the corresponding sides in similar rectangles increase
(or decrease) by a constant ratio. As such, she focused on the
difference, in particular, within difference, by comparing the
difference between the length and the width within a rectangle.
Although she carried out the calculation correctly based on
additive reasoning, she was not able to find the correct missing
length in similar rectangles. The fundamental error in Lilith’s
case results from not understanding the concept of similarity.
Teacher 1. “I think that the missing side is 12 cm because the
difference between 4 cm and 6 cm is 2 therefore the difference
between the 2 sides of the larger rectangle would also be 2. 10
cm + 2 = 12 cm”.
Among teachers who provided correct answers, three differ-
ent solution approaches were used—within ratios, between
ratios, and unit-rate (or scale factors) as addressed in Table 1. In analyzing the responses of the teacher participants, three
categories of interpretation were identified and are illustrated
below. The first interpretation was to classify different ways of
identifying Lilith’s learning difficulties using a conceptual ap-
proach—in this case, focusing on the meaning of similarity of
rectangles, in which the following hold: two figures are similar
if 1) the measures of their corresponding angles are equal, 2)
the lengths of their corresponding sides increase (or decrease)
by the same factor, called the scale factor, and 3) the perimeter
from one rectangle to another rectangle also increases by the
same scale factor. The conceptual approaches also included the
idea of enlargement or reduction in similar rectangles. I call this
type of identification of Lilith’s error as similarity-based. One
typical response was as follows:
Table 2 shows the frequency of each type of solution strate-
gies by teachers. While the most frequently used strategy is the
within ratio approach among elementary teachers, the between
ratio approach is used most often in secondary teachers.
I was also curious to know if teachers pointed out the lying
idea of similarity of rectangles, the concept of similarity, when
explaining their solution approaches because finding a missing
length in similar rectangles involves not only understanding the
concept of similarity but also procedural knowledge of setting
up a proportion and performing calculations.
Among elementary teachers who provided correct solution
strategies, 48% of elementary teachers (11 out of 23) referred to
the concept of similarity while 81% of secondary teachers (21
out of 26) pointed out the property of similarity of figure. This
indicates that a smaller percentage of secondary teachers car-
ried out three methods as a rote procedure that requires little
Lilith does not understand that similar means proportion or
she may not understand what proportional means.
Example of different methods used by teachers.
Between Ratios Within Ratios Scale factor method
If the rectangles are similar, their sides are at a
constant ratio. Thus you can compare the ratio
between the width and length and use this propor-
tion to find missing length from the width of same
Assuming the 4cm side is similar to the 10cm
side I must find a relationship between 6 cm
and x. I create a fraction with 6cm and 4cm and
set it equal to x and 10 cm. Cross multiply and
solve for x
Ratio of smaller to bigger similar rectangle is 4:
Solution Strategy used in finding a missing length by t eachers.
Category Elem. (N = 31) Second. (N = 26) Total (N = 57)
Incorrect Additive 7 (22%) 0 (0%) 7 (12%)
Correct Within ratio 14 (45%) 7 (27%) 21 (36%)
Between ratio 8 (25%) 15 (58%) 23 (40%)
Scale factor methods 3 (1%) 4 (15%) 7 (12%)
E. F. R. LEDESMA 267
A proportion is a ratio of two numbers, where Lilith looked
at the sum (or difference, depending on how you think about it)
The procedural approach involves finding the missing value
in a proportion, which relates to big ideas 2 through 4 outlined
above (i.e., determining ratios, setting up a proportion, and
carrying out the calculation). In this approach, teachers also
indicated the need for a ratio, proportion, or a scale factor for
calculation. This type of identification of Lilith’s error is called
Lilith did not calculate the ratio of corresponding sides, i.e., 4
cm/10 cm = ratio of sides. What Lilith did was 6 − 4 = 2 cm
difference then added 10 cm + 2 cm = 12 cm.
In addition to these two categories of interpretation, a third
category involved responses indicating misdiagnosis of Lilith’s
error based either on additive reasoning or incorrect focus. In
most cases of additive reasoning, teachers indicated Lilith’s
errors stemmed from not comparing the difference between
rectangles. The following are examples:
Lilith is explaining the relationship between the sides 4 and 6
rather than first comparing sides 4 and 10 then 6 and x. So she
should be looking at how side 4 is related to side 10, then use
that same relation with side 6 to get side x.
Table 3 shows subcategories of each approach and the dis-
tribution of response in terms of similarity-oriented vs. proce-
Four subcategories were devised with respect to the similar-
ity-oriented approach. While teachers in category one did not
provide a definition of similarity, teachers in category two
stated specifically that Lilith did not see the relationship be-
tween two similar rectangles as a constant ratio. Although
teachers in category three and four stated Lilith’s limited un-
derstanding of the concept of similarity, teachers in category
three pointed out specifically that Lilith’s errors came from not
comparing the lengths between the two rectangles. In the case
of category four, teachers indicated Lilith’s difficulty as not
visualizing enlargement of the second rectangle. In terms of the
procedural approaches, four subcategories were devised as well.
The major difference between the conceptual vs. procedural
approach lies in the focus of indication of Lilith’s errors. Inter-
estingly, when teachers were asked to identify Lilith’s errors,
they tended to rephrase Lilith’s method by pointing out the use
of difference in calculation, which is coded into the first cate-
gory in the procedural approach. Table 3 shows that, although
Lilith’s errors came from her limited understanding of similar-
ity rather than from procedural knowledge of setting up an
equation, teachers in this study tended to identify her errors
more from a procedural perspective.
How Do Teachers Respond to Lilith’s Work?
I performed the same analysis on the responses the teachers
provided for the question asking them to describe how they
would respond to Lilith. Table 3 shows the distribution of re-
sponse in terms of concept-oriented vs. procedure-oriented.
Table 4 shows that more than half of the teachers provided
guidance from procedural aspects of similarity. I was intrigued
Teachers’ Identification of Lilith’s error.
Type of Identification Subcategory # of response Total
Meaning of similarity
Recognizing the similarity by a constant ratio
Comparing lengths between rectangles
Visualization of enlargement in similar figures
Use of addition or difference in calculation
Use of a ratio or proportion in calculation
Scaling up by a constant scale factor in calculation
Comparing lengths between figures in calculation
Misdiagnosed Additive reasoning or incorrect identification 8 8(13%)
Total 57 57
Type of strategy employed by Lilith.
ote: * represents the most frequently referred category.
Type of Strategy Subcategory # of response Total
Recognizing the similarity by a constant ratio
Comparing lengths between figures
Idea of enlargement in similar figures
Finding a ratio proportion in calculation
Scaling up by a scale factor in calculation
Comparing lengths within a figure in finding a proportion
Misdiagnosed Additive reasoning or incorrect identification 3 3(5%)
Total 57 57
E. F. R. LEDESMA
to find out how the teachers who recognized Lilith’s error in
terms of conceptual aspects of similarity would respond to Li-
lith. A comparison of Tables 3 and 4 shows that although the
frequency of focusing on the meaning of similarity slightly
increased from identifying the learning difficulties and re-
sponding to it, teachers tended to provide their intervention
based on the procedural aspects of similarity.
In addition to this approach, I explored two forms of ap-
proaches addressed by teachers to student errors—teacher-
focused vs. student-focused. While 81% of the secondary
teachers provided teacher-directed approaches of telling and
explaining, less than half of the elementary teachers used a
student-focused approach by engaging Lilith in activities or
questions. Furthermore, I also observed teachers’ using the
three different approaches to student errors identified by Ruiz
and Lupiañez (2009) in their examination of teachers’ peda-
gogical strategies to student errors in reflective symmetry: 1)
Generalization, 2) Return to the basics, and 3) A Plato-and-
Now I present an analysis of two tasks Lilith answered in-
correctly: task 1 and task 2.
In task 1, the drawing of a house was presented and the stu-
dent was required to select the correct reduced sketching of the
original drawing (see Figure 2). Lilith selected a sketching that
did not correspond to the original drawing and she argued that
her choice resembled best the original drawing of a house. I
observed that she almost did not use her common sense or vi-
About the task 2, the text is: Mr. Escalante has been asked to
make an amplification of the following original drawing (see
Figure 3). To the right, you can see a portion of the amplified
drawing. Complete that amplification keeping the form of the
Contains the problem. Ruiz, 2002.
Task 2. Ruiz, 2002.
Task 2 solved by Lilith. Ruiz, 2002.
As shown in Figure 4, Lilith completed the drawing but she did
not notice that she had amplified it twice and not thrice.
I worked with those notions by referring to concrete situa-
tions of the type of the experience of reproducing a drawing to
scale and of the idea of using a photocopier.
I would like to mention that Lilith was representative of
those students who had a lot of recourse to handling algorithms
that made no sense and who simultaneously exhibited few
elaborations in the qualitative context.
This study showed that, although a student’s error stemmed
from lack of understanding of the concept of similarity, a ma-
jority of elementary and secondary teachers identified the stu-
E. F. R. LEDESMA 269
dent’s errors from a procedural perspective of similarity. When
they responded to the student’s errors, they guided the student
by invoking procedural knowledge.
Another interesting finding was that, although secondary
teachers showed better understanding of the mathematical con-
cepts presented in this study than elementary teachers when
responding to student errors, secondary teachers tended to rely
on a teacher-focused approach of telling or explaining students’
errors whereas elementary teachers tended to use a student-
ocused approach by asking questions or providing related ac-
tivities to help students overcome conceptual misunderstanding.
These results are consistent with the findings reported by Ruiz,
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