Primary and Secondary Teachers’ Knowledge, Interpretation, and Approaches to Students Errors about Ratio and Proportion Topics

doi:10.4236/ce.2011.23035

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Creative Education

2011. Vol.2, No.3, 264-269

Primary and Secondary Teachers’ Knowledge, Interpretation,

and Approaches to Students Errors about Ratio and

Proportion Topics

Elena Fabiola Ruiz Ledesma

Graduate Department, School of Computer Sciences of the National Polytechnical Institute of Mexico,

Gustavo A. Madero, Mexico.

Email: [email protected]

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

Introduction

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

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?

Background

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.

Methodology

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).

Sample

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.

Task

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

strategies.

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.

Figure 1.

Contains the problem. Ruiz, 2002.

E. F. R. LEDESMA

266

Summary of Results proportional reasoning.

How Do Teachers Identify Lilith’s Learning

Difficulties?

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

as follows:

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.

Table 1.

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

rectangle. x6104=

410



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:

10.

10 1

6,62,1231

5

Table 2.

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)

of sides.

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

procedure-based.

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-

dural-oriented.

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

Table 3.

Teachers’ Identification of Lilith’s error.

Type of Identification Subcategory # of response Total

Similarity-based

Meaning of similarity

Recognizing the similarity by a constant ratio

Comparing lengths between rectangles

Visualization of enlargement in similar figures

12*

19(34%)

Procedure-based

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

13*

30(53%)

Misdiagnosed Additive reasoning or incorrect identification 8 8(13%)

Total 57 57

Table 4.

Type of strategy employed by Lilith.

ote: * represents the most frequently referred category.

Type of Strategy Subcategory # of response Total

Similarity-based

Recognizing the similarity by a constant ratio

Comparing lengths between figures

Idea of enlargement in similar figures

10*

21(37%)

Procedure-based

Finding a ratio proportion in calculation

Scaling up by a scale factor in calculation

Comparing lengths within a figure in finding a proportion

22*

33(58%)

Misdiagnosed Additive reasoning or incorrect identification 3 3(5%)

Total 57 57

E. F. R. LEDESMA

268

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-

the-slave-boy approach.

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-

sualization.

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

original draw.

Figure 2.

Contains the problem. Ruiz, 2002.

Figure 3.

Task 2. Ruiz, 2002.

Figure 4.

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.

Conclusion

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,

1997.

References

Fisher, L. (1998). Strategies used by secondary mathematics teachers to

solve proportion problems. Journal for Research in Mathematics

Education, 19, 157-168. doi:10.2307/749409

Hart, K. (1984). Ratio: Children’s strategies and errors. Windsor,

England: NFER-Nelson.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up:

Helping children learn mathematics. Washington D.C.: The National

Academies Press.

Lamon, S. J. (2007). Rational numbers and proportional reasoning:

Toward a theoretical framework. In F. Lester (Ed.), Second hand-

book of research on mathematics teaching and learning (pp.

629-668). Charlotte, NC: Information Age Publishing.

Lim, K. H. (2009). Burning the candle at just one end: Using nonpro-

portional examples helps students determine when proportional

strategies apply. Mathematics Teaching in the Middle School, 14,

492-500.

NCTM (2000). Principles and standards for school mathematics.

Reston, VA: National Council of Teachers of Mathematics.

Ruiz, E. F. (2011) Indicadores teóricos para la Constricción de

conceptos del cálculo diferencial. Proyecto de investigación

registrado en la Secretaría de Investigación y Posgrado (SIP), del

IPN con núm. de registro CGPI 20110343. México. IPN,

Ruiz, E. F. (1997). Uso de las calculadoras math explorer y ti-92 en la

resolución de problemas: Una experiencia con profesores de los

niveles básico y medio. Memorias del Seminario Nacional de

Calculadoras y Computadoras en Educación Matemática, 25-35.

Ruiz, E. F. (2000) Study of solving strategies and proposal for the

teaching of ratio and proportion. Proceedings of the Twenty-Second

Annual Meeting North American Chapter of the International Group

For The Psychology of Mathematics Education, 2, 395-396.

Ruiz, E. F. (2002). Estudio de estrategias de solución y una propuesta

de enseñanza de razón y proporción. Tesis Doctoral. Cinvestav-IPN.

México. 17-344.

Ruiz, E. F., & Lupiañez, J. L. (2009). Detección de obstáculos

psicopedagógicos en la enseñanza y el aprendizaje de los tópicos de

razón y proporción en alumnos de sexto grado de educación primaria.

Revista Electrónica de Investigación Psicoeducativa, 7, 397-427.

Schleppenbach, M., Flevares, L. M., Sims, L. M., & Perry, M. (2007).

Teachers’ responses to student mistakes in Chinese and U.S. mathe-

matics classrooms. The Elem e n ta r y S c h o o l J o u r nal, 108, 131-147.

doi:10.1086/525551