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.