Prospective secondary mathematics teacherspedagogical knowledge for teaching the estimationof length measurements

Karthigeyan Subramaniam

Springer Science+Business Media Dordrecht 2013

Abstract Prospective secondary mathematics teachers pedagogical knowledge forteaching the estimation of length measurements was investigated by examining their

personal benchmarks for measurement estimation. Benchmarks for measurement estima-

tion are the meaningful representations of units that serve to increase ones understanding

of measurement and ones ability to estimate measurements. Data included electronic

journal responses, observation and verbal data, and work samples. Thematic analysis

revealed that prospective teachers possessed various benchmarks for measurement esti-

mation that enabled them to estimate length measurements, but these benchmarks for

measurement estimation were not evident in participants pedagogical knowledge for

teaching the estimation of length measurements. Participants pedagogical knowledge for

teaching the estimation of length measurements was instead based on the belief that hands-

on activities were the only way to teach the estimation of length measurements.

Keywords Pedagogical knowledge Measurement estimation Prospectiveteachers Benchmarks

Introduction

Measurement estimation is one of the three key quantitative estimation processes taught in

primary and secondary classrooms and forms the foundation for the learning of physical

measurement (length, volume, and mass) (Hogan and Brezinski 2003; Joram et al. 1998,

2005), numeracy skills (Joram et al. 1998; Sarama and Clements 2009; Siegler and Booth

2005), mathematical power (Crites 1993), measurement sense (Clements 1999), and

number sense (and spatial sense) (Lang 2001) and thus is identified as an important

K. Subramaniam (&)Department of Teacher Education and Administration, University of North Texas, 1155 Union Circle#310740, Denton, TX 76203-5017, USAe-mail: Karthigeyan.Subramaniam@unt.edu

123

J Math Teacher EducDOI 10.1007/s10857-013-9255-2

component in developing mathematical understanding in primary and secondary class-

rooms (Clements 2003; NCTM 2000; Towers and Hunter 2010).

Measurement estimation, in this study, refers to the process of determining an

approximate measure, the estimate, of an objects length, volume, mass, etc., using mental

and visual information, and without the use of measuring instruments and/or without

making an exact measurement (Forrester and Pike 1998; Joram et al. 2005; Van de Walle

et al. 2010). Within this theoretical construction of measurement estimation, the term

estimate refers to a number that is a suitable approximation for an exact number given

the particular context (Van de Walle et al. 2010, p. 241), for example, the need to

approximate a measure of an objects length, volume, mass, etc. The aforementioned

description distinguishes the process of estimation from guessing, as the process of esti-

mation is underpinned by some form of reasoning (Van de Walle et al. 2010, p. 241)

such as the need to provide a suitable approximation within a well-defined context without

an exact measurement or use of measuring instruments.

Accordingly, literature states that instruction of measurement estimation in primary and

secondary classrooms (1) provides the building blocks for mathematical concepts that

students will use in their future courses of study in mathematics, science, and higher

education (Baturo and Nason 1996; Sarama and Clements 2009; Siegler and Booth 2005),

(2) develops mathematical abilities that contribute to adaptive problem-solving abilities of

children and adults for various daily practical applications (Sarama and Clements 2009;

Siegler and Booth 2005), and (3) equips adults with the required knowledge and skills in

many fields of employment (Baturo and Nason 1996; Siegler and Booth 2005).

Even though the literature emphasizes the importance of measurement estimation in

primary and secondary classrooms, the literature also points out that very few studies have

investigated how measurement estimation is taught by teachers in these classrooms (Hogan

and Brezinski 2003; Siegler and Booth 2005). Similarly, there have been far fewer studies

or no studies about prospective teachers and their pedagogical knowledge for teaching the

estimation of length measurements. This area of research is important for three reasons:

First, length measurement estimation is a topic that is taught and developed throughout

the school years (NCTM 2000, p. 47) and thus is a topic included in the prospective

teachers future teaching practices. Second, research on subject-specific topics such as the

estimation of length measurements and how it is conceptualized for teaching and learning

by prospective teachers contributes to the knowledge base on building prospective

teachers conceptual and procedural understanding of subject matter to effectuate student

learning (Kinach 2002; Tsamir 2005). Third, research on subject-specific topics such as the

estimation of length measurements also provides a window into prospective teachers

habitual ways of thinking about subject matter teaching (Kinach 2002, p. 52) and how

these habitual ways impact their future practices.

Collectively, an understanding of prospective teachers conceptualizations and habitual

ways of thinking about the estimation of length measurements contributes to the ongoing

research on how prospective teachers represent the mathematics content as instructional

and learning tasks using conceptual and meaning-making goals (Philpp et al. 2007,

p. 439) instead of teaching mathematics as a prescribed set of algorithms (Ball et al. 2008;

Philpp et al. 2007). Hence, this area of research provides infrastructure to restructure

prospective teachers coursework in mathematics methods courses (Hill et al. 2004, 2005;

Philpp et al. 2007). The study presented in this article aimed to build and contribute to the

knowledge base for comprehending prospective secondary mathematics teachers peda-

gogical knowledge for teaching the estimation of length measurements.

K. Subramaniam

123

To situate this study of prospective teachers conceptualization of and habitual ways of

thinking about the estimation of length measurements for teaching and learning within the

larger research context, perspectives from the literature on pedagogical knowledge for

teaching and prospective teachers beliefs about mathematics and mathematics teaching

were used as constructs to build an analytical framework. Central to the perspective of

pedagogical knowledge for teaching is how specialized knowledge for a mathematics topic

is transformed from one symbolic representation into another through active and social

interactive relationships between the pedagogical knowledge, teacher, and students. This

construct of transformation coheres with the current trends of teaching and learning

measurement estimation. That is, measurement estimation as a complex domain, relying

on teachers and students capacities to integrate measurement concepts and estimation

capabilities through logical reasoning processes (Towers and Hunter 2010, p. 26).

The knowledge from the literature on prospective teachers beliefs about mathematics

and mathematics teaching was important for comprehending prospective secondary

mathematics teachers pedagogical knowledge for teaching measurement estimation

because the key contention within this area of research is that any study of prospective

teachers pedagogical knowledge must include an examination of their beliefs. In this

study, the aforementioned contention was taken into account, and additionally, the nature

of how prospective teachers beliefs about mathematics and mathematics teaching function

as filters to delineate student learning was investigated. This was especially important

because pedagogical knowledge for teaching in this study was conceptualized as an

interactive relationship between the teacher, students, and the mathematics content per-

taining to measurement estimation and not as a successful participation in the interaction

process with fixed and pre-given descriptions like beliefs.

In addition, the literature on measurement estimation, though heavy on K-12 students

knowledge of measurement estimation, provided the knowledge that numerate adults like

prospective secondary mathematics teachers have benchmarks for measurement estimation

readily available for representing measurement units to school-aged children when

teaching measurement estimation. The perspective of benchmarks for measurement esti-

mation also served as a construct for this study.

The key research question that guided this study was What is prospective secondary

mathematics teachers pedagogical knowledge for teaching the estimation of length

measurements? The framework for answering this question consisted of the following

organizing elements: (1) Pedagogical knowledge for teaching the estimation of length

measurement is inherent within the prospective secondary mathematics teachers con-

ceptualization and habitual ways of thinking about how to teach the estimation of length

measurements. (2) Benchmarks for the estimation of length measurements were assumed

to be one of the pedagogical means that prospective teachers deploy to interactively

construct and socially develop mathematical content pertaining to estimation of length

measurements. (3) Prospective teachers beliefs about teaching and learning the estimation

of length measurements served to determine the coaction between the teachers peda-

gogical knowledge and teaching activities, students learning activities, and mathematical

content pertaining to estimation of length measurements influencing and thus effectuating

learning as an interactive process.

This qualitative study aimed to build and contribute to the knowledge base for com-

prehending prospective secondary mathematics teachers pedagogical knowledge for

teaching the estimation of length measurements. Next, the paper provides a review of the

literature on these three areas of organizing elements to better situate the study within the

extant (and limited) knowledge base on the teaching of measurement estimation.

Measurement estimation

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Review of literature

Pedagogical knowledge for teaching

Within the North American context of mathematics education research, pedagogical

knowledge for teaching refers to the specialized knowledge teachers identify and possess

as a way or ways to teach specific concepts to aid students learning of a particular

mathematics construct (Ball et al. 2008; Hill et al. 2008; Leikin and Levav-Waynberg

2007; Stacey 2008). Pedagogical knowledge for teaching in the North American context is

described as (1) the appropriate, precise, and accurate symbolic representationsmean-

ingful rules, procedures, pictures, diagrams, representations, examples, everyday language,

and contextual and participation structuresthat teachers deploy to teach a particular

mathematics construct (Ball et al. 2008; Davis and Simmt 2006; Hill et al. 2004), (2) the

judgments that teachers apply to reduce mathematical complexity of that particular

mathematics content during instructional and learning tasks, respectively (Davis and

Simmt 2006), and (3) the connections they make between the mathematical content being

taught with already taught mathematical content (Ball et al. 2005). Collectively, this

knowledge assists the teacher to manage the mathematics content, instructional tasks, and

learning tasks and thus make the mathematics content accessible to students.

Also, some studies claim that this pedagogical knowledge for teaching is not mastered

as formal constructs within mathematics courses or mathematics methods courses but is

developed during instruction (Davis and Simmt 2006; Sullivan 2008). For example, a

number of researchers contend that pedagogical knowledge for teaching is a collection of

attributes of what the teacher knows about mathematics and of what the teacher knows

about students and curriculum developed and displayed during instruction (Davis and

Simmt 2006; Tsamir 2005; Stacey 2008), while others claim that pedagogical knowledge

for teaching is a product of the K-12 experiences (Forgasz and Leder 2008). Accordingly,

these conceptualizations of pedagogical knowledge for teaching construe this knowledge

as highly contextual, tacit, and logical in nature (Davis and Simmt 2006).

Conversely, within the German tradition of mathematics education research, the current

trend of moving from the Stoffdidaktik basis of pedagogical knowledge for teaching to

that of the evolving perception of pedagogical knowledge for teaching as a developmental

aspect (Steinbring 1998, 2008; Straesser 2007) provides an expansive conceptualization of

this knowledge. Instead of just descriptions and attributes, as in the North American

context of mathematics education research, the German tradition of mathematics education

research explicates pedagogical knowledge for teaching as a reciprocal process and as the

coaction between the teachers pedagogical knowledge and teaching activities, students

learning activities, and mathematical content influencing and thus effectuating learning as

an interactive process.

Moreover, within the aforementioned perspective of pedagogical knowledge for

teaching, the construction of content is perceived as an interaction process during

instruction wherein content exists as symbolic relational structures and are coded by

means of signs and symbols that can be combined logically in mathematical operations

(Steinbring 2008, p. 310). Thus, within the German tradition of mathematics education

research, pedagogical knowledge for teaching is conceptualized as an interactive rela-

tionship between the teacher, students, and the content and not as a successful participation

in the interaction process with fixed and pre-given descriptions. The interactive relation-

ship is a key to the active and social transformation of specific mathematics concepts

(symbolic representations) into meaningful symbolic representations by teachers and

K. Subramaniam

123

students during instruction. This perspective on pedagogical knowledge provides a com-

plex view of how content is constructed and how the constructs of pedagogical knowledge

for teaching, the teacher, students, content, symbolic representations, judgments, and

connections interact together for teaching and learning. Steinbrings (2008) captures the

essence of this complex view, he states:

Mathematical knowledge is interactively constructed by the participants on the basis

of specific epistemological conditions of mathematical knowledge, which are

effective also within instructional learning processes, and which in this teaching

learning context lead to a socially developed epistemology of (school) mathematical

knowledge (p. 314).

To sum up, the perspectives put forth by the scholars from the North American context,

particularly Ball et al. (2005, 2008), and from the German context, particularly Steinbring

(1998, 2008), are closely linked as both perspectives specifically focus on how teachers

manage the mathematics content, instructional tasks, and learning tasks, making the

mathematics content accessible to students. Both groups of scholars contend that in making

the mathematics content accessible to students, teachers transform mathematics content

into symbolic representations that then function as meaningful relational structures

effectuating learning as an interactive process between students, teachers, and the math-

ematics content. Also, there is a need to link the perspectives advocated by both groups of

scholars because they explicate the pedagogical knowledge of teaching mathematics

content as an interactive process instead of just advocating fixed descriptions and attributes

for instructional and learning tasks.

Most importantly, the contentions of Ball et al. (2005, 2008) and Steinbring (1998,

2008) resonate with Scardamalias (2002) arguments that teachers need to exercise col-

lective cognitive responsibility in their classroom teaching. Scardamalia describes this

collective cognitive responsibility as teachers knowledge that learning can be prob-

lematic and may require strategic moves to bring it about (2002, p. 71) and thus the need

to share these strategic moves with their students so that they engage cognitively with the

tasks and activities in the classroom instead of just overtly being responsible for com-

pleting classroom tasks and activities. Specifically, the aforementioned perspective coheres

with the need to move away from fixed descriptions and attributes for instructional and

learning tasks and toward meaningful symbolic relational structures (Ball et al. 2005, 2008;

Steinbring 1998, 2008) that effectuate learning.

Prospective teachers beliefs

As mentioned, another area of literature that guided this study was the literature on pro-

spective teachers beliefs about mathematics and mathematics teaching. A key contention

within this area of research is that any study of prospective teachers pedagogical

knowledge of teaching and learning must include an examination of their beliefs (Borko

et al. 2000; Cady et al. 2006; Charalambous et al. 2009; Frykholm 1999; Macnab and

Payne 2003; Philpp et al. 2007; Smith 2001, 2005; Wilson and Cooney 2002). For

example, some studies have indicated that prospective teachers beliefs about teaching are

closely related to, and inseparable from, their beliefs about how students learn (Frykholm

1999; Wilson and Cooney 2002), while others indicate that prospective teachers beliefs

about pedagogical knowledge are self-contained and not strongly linked to students

lives (Forgasz and Leder 2008 p. 182). On the other hand, some scholars contend that

exploring and explaining beliefs and/or espoused beliefs within the context of pedagogical

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knowledge for teaching reveals the logic behind teaching (Davis and Simmt 2006), the

unsophisticated understandings that underscore beliefs (Cooney 1999), and inconsistencies

between beliefs and teaching actions (Leatham 2006).

Despite what is known about prospective teachers beliefs about mathematics and

mathematics teaching, the nature of how these beliefs function to delineate student learning

is limited. This is especially important in this study as pedagogical knowledge for teaching

is conceptualized as an interactive relationship between the teacher, students, and the

content and not as a successful participation in the interaction process with fixed and pre-

given descriptions like beliefs. Notwithstanding, some studies have indicated that pro-

spective teachers beliefs act as filters through which knowledge about teaching and

learning are screened for viable instructional and learning tasks and for determining which

instructional and learning tasks are to be selected, stored, or discarded (Ahtee and Johnson

2006; Da-Silva et al. 2006). Thus, beliefs act as the conditioning elements (Da-Silva

et al. 2006, p. 463) validating or rejecting teaching actions and determining the viability of

instructional and learning tasks.

On the other hand, Boaler (2002) claims that the links connecting pedagogical

knowledge to beliefs and actions leading to the mastery and the appropriation of mathe-

matics knowledge in the classroom rests upon the teachers development of productive

relationship (p. 11). That is, she contends that teachers are able to effectuate the students

learning of mathematics because of their belief in acknowledging active relationships

between their own knowledge of mathematics, and the teaching practices they believe are

conducive for students learning of mathematics. Thus, if teachers believe this productive

relationship as appropriation of knowledge, the relationship between the mathematics

content and the resulting teaching practices will be one of fixed descriptions and attributes

for instructional and learning tasks. Conversely, if teachers believe this productive rela-

tionship as inter-relationships between mathematics content and teaching practices, their

classroom instruction will cohere with Ball et al.s (2005, 2008) and Steinbrings (1998,

2008) contention that mathematics content will need to be transformed into symbolic

meaningful relational structures between students, themselves, and the mathematics con-

tent. Additionally, this belief resonates with Scardamalias (2002) perspective of collective

cognitive responsibility where instructional and learning tasks are related to meaningful

symbolic relational structures (Ball et al. 2005, 2008; Steinbring 1998, 2008) to bring about

student learning instead of limiting learning to worksheets and activities.

Benchmarks for measurement estimation

A review of literature indicates that the often mentioned strategy to teach measurement

process in classrooms is to estimate and then measure particular attributes of everyday

objects (Castle and Needham 2007; Joram et al. 1998, 2005; Lang 2001). The general

didactical character of measurement estimation in the literature is that of teachers using or

modeling the following measurement estimation strategies: unit iteration strategy,

benchmark estimation strategy (Crites 1992, 1993; Joram 2003; Joram et al. 1998, 2005;

Lang 2001), decompositionrecomposition strategy (Crites 1993), guess-and-check pro-

cedure/practice with feedback procedure, flowchart model, and combination strategies

model (Joram et al. 1998, 2005). Basically, all these measurement estimation strategies

involve the estimator in a process of physical measurement without tools and without the

aid of physical reminders, but with knowledge about the principles of measurement (Joram

2003; Joram et al. 1998, 2005).

K. Subramaniam

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All of the mentioned estimation strategies are dependent on the teacher to assist students

to (1) comprehend the practicality and credibility of the estimation strategy, (2) adapt

thinking skills toward deciding on the credibility and practicality of the estimation strategy

used, (3) compare and contrast the exact physical measurement attribute with the estimate

and then adjusting initial estimates through an understanding of the relationship between

the exact physical measurement attribute and the estimate, and (4) decide on the physical

measurement attributes based on uncertainty, validity, and measurement principles

(Trafton 1986).

Of these estimating strategies, the benchmark estimation strategy seems to have a

greater scope as a teaching strategy for estimating length measurements (Joram et al. 2005;

Sarama and Clements 2009), especially since the literature claims that this strategy is

prevalent in numerate adults (Joram 2003; Sarama and Clements 2009). Both groups of

scholars, Joram et al. (2005) and Sarama and Clements (2009), share the view that esti-

mators who use the benchmark estimation strategy for estimating length measurements

possess mental rulers or conceptual rulers and that they use these rulers for the mental

estimation of length by projecting an image onto present or imagined objects (Sarama and

Clements 2009, p. 292). The mental estimation is characterized as the possession of an

internal measurement tool, the mental or conceptual ruler, which operates by the mental

partitioning or segmenting of a length into a nonverbal or perceived magnitude that rep-

resents a numerical magnitude. Estimation of the length is then achieved by scaling either

the related nonverbal or perceived magnitude to numerical magnitudes or counting the

perceived segments to create the numerical magnitude (Joram et al. 2005; Sarama and

Clements 2009). The following examples exemplify how the conceptual ruler operates by

mental partitioning using a scaling factor and by segmenting using counting and connected

lengths, respectively: (1) Four times (the scaling factor) around a high school track used by

an estimator to provide an estimate of the physical magnitude of a kilometer illustrates how

a benchmark, the nonstandard unit (high school track), is used to symbolically represent

the estimated length of a standard unit (a kilometer) using scaling (Joram et al. 2005) and

(2) I imagine one meter stick after another along the edge of the room. Thats how I

estimated the rooms length is nine meters (Sarama and Clements 2009, p. 292) illustrates

segmenting followed by counting.

The perspective of benchmarks for the estimation of length measurements served as the

construct in understanding prospective teachers pedagogical knowledge for teaching

the estimation of length measurements for the following reasons. First, the literature on

the pedagogical knowledge for teaching claims that a key component of this knowledge

is the teachers use of symbolic representations to transform specific mathematics concepts

from one representation to another to aid students learning of a particular mathematics

construct (Davis and Simmt 2006; Steinbring 1998, 2008). Correspondingly, the particular

domain of benchmarks for measurement estimation involves the use of units of measure

in a mental way without the aid of measuring tools and underscored by conceptual

prerequisites of logical reasoning and knowledge of specific measurement concepts

(Towers and Hunter 2010, p. 26). This also coheres with the standard ways in which

teachers develop students measurement estimation strategies (Castle and Needham 2007;

Joram et al. 1998, 2005; Lang 2001; Montague and van Garderen 2003; Trafton 1986).

Second, benchmarks for measurement estimation consist of meaningful and familiar

objects that provide symbolic representations of standard units that help students generate

estimates about unfamiliar quantities (Joram 2003). Literature claims that benchmarks act

as meaningful symbolic representations of units and serve to increase students under-

standing of measurement by comparing the to-be-estimated object to an object whose

Measurement estimation

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physical measurements are known through mental imagery (Castle and Needham 2007;

Joram 2003; Joram et al. 1998, 2005; Montague and van Garderen 2003). In doing so,

students are able to improve their ability to estimate measurements (Castle and Needham

2007; Joram 2003; Joram et al. 1998, 2005) by developing associated cognitive attributes

for measurement estimation like the construction of accessible and meaningful nonverbal

magnitudes to mentally iterate the physical attributes of the to-be-estimated object (Castle

and Needham 2007; Crites 1992, 1993; Joram 2003; Joram et al. 1998, 2005).

Third, as a key teaching strategy for teaching measurement estimation, benchmarks for

measurement estimation strategy are supported by research literature as an acceptable

teaching strategy (Joram 2003; Joram et al. 1998, 2005). Gleaning this literature on

benchmarks for measurement estimation revealed that the transformation of a nonstandard

unit into a standard unit using symbolic representations involves the teacher introducing

benchmarks as nonstandard units that students use to make measurements of familiar

objects. Following this, the teacher uses the benchmarks as visual images of familiar

objects to represent the nonstandard units providing students with a standard to refer to

when estimating measurements. As a last step, the teacher helps students to understand

how benchmarks, as individual objects of a given magnitude, may fit into a measurement

system. Accordingly, this perspective on benchmarks for measurement estimation corre-

sponds with the coaction and interactivity suggested by Steinbring (2008) as the specific

epistemological conditions of pedagogical knowledge, which are effective within

instructional learning processes, and which leads to a socially developed content.

Synthesis: toward an organizing framework

Based on the shared and common perspectives gathered from the review of the literature

(Ball et al. 2005, 2008; Boaler 2002; Joram et al. 2005; Sarama and Clements 2009;

Scardamalia 2002; Steinbring 1998, 2008), the organizing framework that served as the

lens to delve into student teachers pedagogical knowledge for teaching estimation of

length measurements was as follows: Pedagogical knowledge for teaching estimation of

length measurement is inherent within the prospective teachers conceptualization and

habitual ways of thinking about how to teach measurement estimation (Ball et al. 2005,

2008; Steinbring 1998, 2008). The conceptualization consists of prospective teachers

knowledge of measurement estimation, student teachers instructional strategies, and their

students learning activities. In this study, benchmarks for estimating length measurement

were assumed as the symbolic representations that prospective teachers deploy to inter-

actively construct and socially develop the mathematical content pertaining to measure-

ment estimation (Joram et al. 2005; Sarama and Clements 2009). This is so because

prospective teachers as numerate adults have benchmarks for measurement estimation

readily available for teaching students about measurement estimation and also benchmarks

for measurement estimation is an empirically supported strategy for teaching measurement

estimation (Joram et al. 2005; Sarama and Clements 2009).

Correspondingly, prospective teachers beliefs about teaching and learning measure-

ment estimation serve to determine the coaction of their pedagogical knowledge and

teaching activities, students learning activities, and their benchmarks for measurement

estimation influencing and thus effectuating learning as an interactive process (Boaler

2002) and as a collective cognitive responsibility (Scardamalia 2002). In this way, peda-

gogical knowledge for teaching estimation is not reduced to the consensus of fixed and pre-

given descriptions of conditions afforded by the teacher, but is seen as complex symbolic

relational structures coded by means of signs and symbols that are combined logically in

K. Subramaniam

123

mathematical operations by the teachers and students leading to students construction of

content knowledge pertaining to measurement estimation.

Method

Since the study was exploring prospective secondary mathematics teachers pedagogical

knowledge for teaching the estimation of length measurements, a qualitative approach with

its emphasis on capturing the meanings akin to situational contexts and subjectivities of

these prospective teachers seemed appropriate to this study. The adoption of a thematic

analysis which focuses on identifiable themes and patterns of intentions and behaviors

(Boyatzis 1998) cohered with the aim of the study. That is, the exploration and identifi-

cation of prospective teachers conceptualization and habitual ways of thinking about how

to teach the estimation of length measurement examined through the particular domain of

benchmarks for measurement estimation.

Context

The site for this study was in a secondary mathematics methods course conducted within

the teacher education program in the USA. This particular site was chosen because the

author was a teacher education faculty member of the teacher education program within

which this mathematics methods course was situated and thus, this provided the author

access to both the methods course and participants.

The mathematics methods course was taught by a full-time faculty with a background in

mathematics education methodology and methods. The methods course is the last course in

the sequence of three courses taken before student teaching. The methods course was

designed to provide an in-depth concentration of content and pedagogical content

knowledge required for teaching the secondary mathematics curriculum. Included within

the methods course curricula were the topics that dealt specifically with the concept of

measurement as recommended by the National Council of Teachers of Mathematics

(NCTM 2000). Topics included developing teacher candidates knowledge and skills to

choose and use appropriate tools and units for measurement in various contexts, the his-

torical development of measurement and measuring system. The aforementioned emphasis

on developing teacher candidates knowledge and skills of measurement and measuring

system was also an important reason for selecting this particular site for the study.

Benchmarks for measurement estimation were not explicitly addressed in this methods

course.

Participants and role of researcher

The three male participants and three female participants in this study were teacher can-

didates enrolled in a certification program for teaching grades 712/secondary mathe-

matics. Their teacher preparation coursework included mathematics content requirements

(32 credit hours) and education course requirements (35 credit hours). Also, these six

participants were opting to become teachers as a second career choice and previously had

diverse work backgrounds that required them to be numerate adults (Joram 2003; Sarama

and Clements 2009) doing or applying mathematical knowledge in various contexts

(Stacey 2008). These work experiences included working as substitute teachers in sec-

ondary schools (Laura, Samy, and Jackie), as a football coach (Zaul), as an engineer

Measurement estimation

123

(Horus), and as being part of the army reserve (Aaron). All six participants who took part

in the study were given information sheets about the study and consent forms approved by

the universitys Institutional Review Board committee. All six participants were informed

that their participation in the study was voluntary and they had the option to withdraw from

the study at any time without any disadvantage to themselves of any kind. Participants

chose their own pseudonyms, and these were used in all data collection methods. Further

information on participants life/work experiences, their ages, and other information are

listed in Table 1.

The authors role in this study was that of a qualitative researcher who worked in unison

with his role as observer of the settings within which participants carried out the mea-

surement estimation tasks during the measurement estimation activity. In addition, the

author also took on the role of surveyor of participants beliefs and implicit ideas sur-

rounding the teaching and learning of measurement estimation through the analysis of their

pre- and post-electronic journal responses and other data.

Data collection

Four types of data were collected in this study: electronic journal responses to pre- and

post-interview questions collected at the onset of the study and at the end of the study,

observations and anecdotal data of participants engaged in a measurement estimation

activity conducted between the pre- and post-interviews, verbal data transcribed from the

measurement estimation activity, and participants work samples collected after the

measurement estimation activity. Participants electronic journal responses consisted of the

same pre- and post-interview questions. These questions included the following:

1. How do you plan to teach measurement estimation in your secondary classroom?

2. When you think of a kilometer, a kilogram, or a liter what do you think of?

3. How do you plan to help students estimate measurements of everyday objects?

4. What are your thoughts about teaching a unit on measurement estimation in your

secondary classroom?

Pre-interview questions were administered at the beginning of the methods course, and

post-interview questions were administered after the measurement estimation activity. The

pre- and post-questions served to promote a reflective discourse on the topic of mea-

surement estimation among participants. In addition, the pre- and post-questions were the

method of choice for ascertaining participants beliefs about the measurement estimation

process and also to corroborate participants beliefs about measurement estimation and the

enactment of these beliefs-to-actions within the measurement estimation activity.

Table 1 Demographics of participants

Participant Gender Age Work experience Major in college Teaching experience(years)

Laura Female 25 Substitute teacher Mathematics 3

Samy Female 31 Substitute teacher Mathematics 2

Jackie Female 25 Substitute teacher Mathematics 3

Aaron Male 34 Army reserve Mathematics 0

Zaul Male 27 Football coach Physical education mathematics 2

Horus Male 27 Engineer Engineering mathematics (Minor) 0

K. Subramaniam

123

In this study, electronic journals were chosen as the platform for interviews because of

the constraints of time, scheduling, and geographical issues (Nicholson and Bond 2003,

p. 261) resulting from participants work schedules and field observation commitments, the

author could not arrange for individual interviews with participants, also the electronic

journals and discussion boards provided for immediate communication between partici-

pants (Edens 2000; Levin 1999; Nicholson and Bond 2003). That is, it enabled support

among participants, and it created a place and time away from class where participants

could talk and help one another and collaboratively reflect on their responses about

measurement estimation and the practice of teaching measurement estimation.

Observations were conducted by the author in the mathematics education methods

course as participants were engaged in the measurement estimation activity. Observations

served two purposes: (1) to record the associated verbal data of participants subject matter

knowledge of measurement estimation and pedagogical understanding of learning to teach

measurement estimation and (2) to record and corroborate participants beliefs about

measurement estimation and the enactment of these beliefs-to-actions within the mea-

surement estimation activity. Anecdotal data (Bean 2005; Brabant and Kalich 2008) often

in the form of spontaneous conversations by the author with the participants during and

after the measurement estimation activity were also documented by the author for analysis.

These conversations provided information of participants knowledge and skills of mea-

surement estimation during the measurement estimation activity and also provided clari-

fication of and further insights into electronic journal responses collected prior to the

measurement estimation activity.

The final data collected were participants work samples, resulting from the measure-

ment estimation activity (NCTM 2000). These data served as communicative and rep-

resentational (Hodder 2003, p. 160) objects of participants knowledge and meanings of

measurement estimations, and strategies for estimating measurements. In summary, four

types of data were collected, electronic journal responses, observation notes (which also

included anecdotal data), verbal data, and work samples.

Measurement estimation activity

The activity required participants to identify common units of measurement, and to esti-

mate and measure various objects in a room. Although the measurement estimation activity

required participants to estimate area, volume, and mass, only the measurement estimation

activity dealing with the estimation of lengths is reported in this study. The measurement

estimation activity dealt specifically with the concept of measurement as recommended by

the National Council of Teachers of Mathematics (NCTM 2000) and included measure-

ment estimation activities dealing with participants knowledge and skills to choose and

use appropriate strategies for measurement estimations in various contexts. For example,

participants were grouped into pairs and were engaged in estimating the length of the floor,

height of a door, length of a window, and length of a signboard and then record the

estimates. Each pair made four estimations (length of the floor, height of a door, length of a

window, and length of a signboard), and overall, there were 12 estimations of length

measurements in total. No measuring tools were used for this part of the activity. Fol-

lowing this, they were required to take measurements with measuring tools and record the

measurement. At the end of the activity, participants computed the difference between the

estimations and measurements. The activity was selected because it was similar to an

activity the participants were expected to teach in a secondary classroom. This activity

served as a window into observing how participants subject matter knowledge of

Measurement estimation

123

measurement estimation and pedagogical knowledge for teaching interact during the

process of learning to teach measurement estimation. Moreover, the activity also provided

a platform to observe and record, and corroborate participants beliefs about measurement

estimation and the enactment of these beliefs-to-actions within the activity of learning to

teach measurement estimation.

Data analysis

Data analysis was done in two steps: First, the electronic journal responses (individual and

group) were analyzed for thematic content (Boyatzis 1998), that is, these data were read

and reread to look for participants conceptualization and habitual ways of thinking about

how to teach measurement estimation, specifically focusing on how participants concep-

tualized the mathematics content, instructional tasks, and learning tasks for making the

mathematics content accessible to their future students (Ball et al. 2005, 2008; Steinbring

1998, 2008). Recurring themes inherent within this identified set of data were highlighted

and preliminarily coded. For example, a recurring theme was participants belief that

hands-on activities were the sole means by which the estimation of length measurements

was taught in classrooms. On identification of this theme, Ethnograph software was used to

code the word hands-on in all pre-electronic journal responses. Furthermore, the lines

above and below the word hands-on within the responses were also copied and coded.

This was done to preserve the context in which the word hands-on was encountered, and

to also identify the function the word hands-on conveyed in context, that is, to pinpoint

participants (1) descriptions of the coaction and interactivity between pedagogical

knowledge, the teacher, and students (Ball et al. 2005, 2008; Steinbring 1998; 2008); (2)

indications of collective cognitive responsibility (Scardamalia 2002); and, (3) descriptions

of productive relationships (Boaler 2002) as appropriation of knowledge and/or as inter-

active processes underscored by meaningful relational structures between students, the

participants as teachers, and the mathematics content.

Additionally, the aforementioned analysis strategy helped the author to sort and group

the similar associated participants conceptualizations of teaching actions that provided

the context and elaboration for hands-on. In doing so, the author was able to cate-

gorize hands-on and the conceptualizations of teaching actions as a preliminary code

and apply this code to post-electronic journal responses for hands-on and the corre-

sponding contextual and participants elaborated conceptualizations of teaching actions.

This also allowed for a negative case analysis (Seale 1999), which is to track for the

existence or nonexistence over time for associated descriptive teaching actions. This

analysis procedure was then applied to other data: observations of participants engaged

in the measurement estimation activity, verbal data transcribed from the measurement

estimation activity, and participants work samples collected after the measurement

estimation activity.

Next, each of the identified themes of participants conceptualizations of mathematics

content, instructional tasks, and learning tasks for making the mathematics content

accessible teaching was further analyzed to identify how participants conceptualized the

mathematics content and made it accessible to their future students. Particularly, partici-

pants descriptions of how they transformed mathematics content into symbolic repre-

sentations to function as meaningful relational structures effectuating learning as an

interactive process between students, teachers, and the mathematics content were identified

and coded (Ball et al. 2005, 2008; Steinbring 1998, 2008).

K. Subramaniam

123

Limitations

The findings of this study are specific to the six participants of this study and to the premise

of using benchmarks for estimation as a pedagogical means for teaching measurement

estimation. In addition, the small number of participants in this study and the collection of

data through a single unit of a measurement estimation activity are major limitations of this

study. In view of this, both negative case analysis (Seale 1999) and triangulation (Flick

1998) were used to corroborate emerging themes from data. For example, themes emerging

from pre- and post-interview responses were corroborated with observation and verbal data

recorded during the measurement estimation activity. Furthermore, using electronic pre-

and post-interview questions that were administered at two different times of the study

helped to alleviate the issue of participants crafting fictional and deceptive responses as

indicted by the literature on electronic interviewing (Fontana and Frey 2005). Moreover,

the collection of anecdotal data (Bean 2005; Brabant and Kalich 2008) in the form of

spontaneous conversations by the author with the participants during and after the mea-

surement estimation activity provided clarification and corroboration of participants

electronic journal responses collected prior to the measurement estimation activity.

Findings

Participants habitual ways of thinking about teaching the estimation of length

measurements

Analysis of electronic journal response data collected prior to the onset of the measurement

estimation activity revealed both participants benchmarks for representing length and their

strategies for how they plan to teach the estimation of length measurements. Participants

personal benchmarks for representing length consisted of two features. First, benchmarks

for length were nonstandard units. Second, the benchmark for length was represented by

the number of times around a running track. Participants did differ among themselves in

their personal benchmarks for how many times around the track was a kilometer: a kilo-

meter as three times (Aaron, Samy, and Horus)/three and a half times (Jackie)/four times

(Laura and Zaul) around a track. Table 2 shows the other benchmarks for representing

mass and volume that participants also possessed and indicated that they have benchmarks

for measurement estimation readily available for representing measurement units.

Participants strategy for teaching the estimation of length measurements, as revealed

by the analysis of electronic journal response data collected prior to the onset of the

measurement estimation activity, was centered on the use of hands-on activities. These

Table 2 Nature of participants benchmarks

Participant Length (1 km) Mass (1 kg) Volume (1 L)

Laura Four times around a track 4 quarter pounders A bottle of soda

Samy Three times around a track A bag of dried beans A bottle of Pepsi

Jackie Three and a half times around a track A packet of meat A bottle of soda

Aaron Three times around a track 4 quarter pounders A bottle of Coke

Zaul Four times around a track A basket ball A bottle of Coke

Horus Three times around a track Laptop computer Personal water bottle

Measurement estimation

123

hands-on activities were described by participants as activities that involved the

determination of a dimension and/or capacity of the to-be-estimated objects such as dif-

ferent solid geometrical shapes and/or manipulatives and real-life examples (estimating the

length of shadows, estimating the volume of soda bottles, etc.) without the aid of a

measuring tool and involved descriptions, visualizations, and some basic calculations. For

example, the following quote retrieved from a response to interview questions collected

prior to the measurement estimation activity and verified and corroborated by anecdotal

data exemplified participants habitual thinking of the steps for teaching measurement

estimations for classroom teaching:

Measurement estimation is important topic to learn in any mathematics classroom. It

is based on the steps we use. For example, it includes making an estimate and then

recording the actual measurement and comparing the estimate with the actual

measurement. Teachers need to set up these activities for students to try them out for

themselves. Its more like I give them the definitions and descriptions of measure-

ment estimation and then they go and do the hands-on activities. This helps them and

gives them something to visualize about measurement estimation (Saul, electronic

journal response collected before measurement estimation activity).

The hands-on activities were also supported by the need for the teacher to define

measurement estimation, to provide explanations describing the importance of measure-

ment estimation in real-life situations, and the need to design or use worksheets that

required students to record their estimation data. For example, Lauras quote below

retrieved from a response to interview questions collected prior to the measurement esti-

mation activity exemplified all of the participants strategy for using hands-on activities

to help students learn measurement estimation.

Measurement estimation should be taught with different investigations, like hands-on

activities. Students should be able to use demonstrations to associate with certain

measurement estimations. Students can learn how to estimate by performing various

hands-on activities that will require them to estimate the measurements of various

items and relate them to standards of measurements (electronic journal response

collected before measurement estimation activity).

Correspondingly, analysis of electronic journal response data also indicated that par-

ticipants strategies for using hands-on activities to teach the estimation of length

measurements was centered on the belief that the descriptions, visualizations, and calcu-

lations enabled the estimator to assign a numerical value to the to-be-estimated object. This

belief among participants was a result of their own experiences with learning measurement

estimation in primary and secondary classrooms. For example, the following quote was

exemplary of this belief: My most memorable experience was when I was in primary

school and our teacher brought in different props and we had to make estimations and then

take measurements (Samy, Electronic Journal Response).

Participants knowledge of estimating length during the measurement estimation

activity

Analysis of observation data and verbal data collected during the measurement estimation

activity revealed that participants strategies for estimating length measurements were

centered on the use of their personal benchmarks for estimating length measurements. The

use of benchmarks for estimating length measurements during the measurement estimation

K. Subramaniam

123

activity (length of a floor, height of a door, width of a window, and length of a signboard)

included the use of mental rulers or conceptual rulers to mentally estimate the lengths by

projecting benchmarks onto the to-be-estimated lengths. From the observation of three

pairs of participants involved in estimating lengths, it was evident that participants pos-

sessed the operation of mentally partitioning the to-be-estimated length into a nonverbal or

perceived magnitude followed by either scaling or counting the partitions to represent a

numerical magnitude. Each of the three pairs estimated the length of a floor, height of a

door, length of a window, and length of a signboard separately, and in total there were 12

observations. Within these 12 sets of estimations of lengths, there were eight instances of

mental partitioning followed by scaling the related nonverbal or perceived magnitude to

numerical magnitudes and four instances of counting the perceived partition to create the

magnitude of connected lengths (refer to Table 3).

The following extract from observation, anecdotal, and verbal data details how par-

ticipants used scaling of the perceived magnitude (height of Zaul) to the numerical

magnitude (length of the floor as eight meters: four times the height of Zaul).

Laura Lets use your height as a means to estimate the length (refers to floor)

Zaul 1.8 m tall, thats how tall I am

Laura If I lay you across the floor, I will need four of you to lay head to toe. Thats about four clones ofyou from one end to the other

Zaul Well 8 m in length

In the extract that follows, estimation of the length is achieved first by segmenting a

length, the length of the floor, into a perceived magnitude (each rectangle tile) and then

counting the perceived segments to create the magnitude of connected lengths, the entire

length of the floor.

Jackie The floor is very big, but it is made up of rectangle tiles

Aaron We can measure the length of each rectangle tile first and record its dimensions

Jackie Then we can use that dimension as a ruler to visualize the length of the floor

Aaron I think we have to first estimate the length of one floor tile, we cant use a ruler

Jackie Do you agree each tile is the length of my forearm?

Aaron Okay! We will use that estimate to make the overall estimate of the floor. My forearm is about20 cm long, so each tile is about 20 cm long

Jackie There are a total of 17 tiles. So that makes it 17 forearm lengths

Table 3 Participants benchmarks for measurement estimation activity

Jackie and Aaron Laura and Zaul Samy and Horus

Length of floor 17 Aaron forearm lengths 4 times Zaulsheight

25 times the lengthof Horuss notebook

Height of door 2 times the height of the chair 3 times Laura shoe 6 notebook lengths

Length of window 2 times the length of Aaronsforearm

2 times Zaulsforearm

3 times the lengthof Horuss notebook

Length of signboard 2 notebook lengths 2 times Zauls foreman 2 notebook lengths

Measurement estimation

123

Participants knowledge of estimating length after the measurement estimation activity

Findings from the analysis of electronic journal response data collected after the com-

pletion of the measurement estimation activity revealed that both participants benchmarks

for representing length and their strategies for how they plan to teach the estimation of

length measurements were markedly similar to the benchmarks and their strategies prior to

the onset of the measurement estimation activity. That is, participants benchmarks for how

many times around the track was a kilometer were the same as their benchmarks collected

prior to the onset of the measurement estimation activity: a kilometer as three times

(Aaron, Samy, and Horus)/three and a half times (Jackie)/four times (Laura and Zaul)

around a track.

Participants strategy for teaching the estimation of length measurements, as revealed

by the analysis of electronic journal response data collected after the measurement esti-

mation activity, was again centered on the use of hands-on activities. For instance, the

following quotes from Samy collected prior to and after the onset of the measurement

estimation activity exemplified participants habitual ways of thinking about teaching the

estimation of length measurements: Students learn from hands-on activities that allow

them to make connections. For example, students measure each others arm lengths and

compare it to other objects in the room that might have the same lengths as their arms

(electronic journal response collected before measurement estimation activity), and In

math, students could estimate their shoe size and then their arm length and compare the

two (electronic journal response collected after measurement estimation activity).

Also, participants still described hands-on activities as the determination of a dimension

and/or capacity of different to-be-estimated objects without the aid of a measuring tool. For

example, the following quotes from Zaul collected prior to and after the onset of the

measurement estimation activity exemplified participants habitual ways of thinking about

how they plan to teach the estimation of length measurements: Demonstrating is better

than explaining. Students will optimally learn when they can visualize concepts and

allowing them ample time to explore and discover (electronic journal response collected

before measurement estimation activity), and Demonstrating is better than explaining.

Well done is better than well said. Hands-on activities (electronic journal response col-

lected after measurement estimation activity).

The belief that hands-on activities were the sole teaching strategy for teaching the

estimation of length measurements was also a persistent and recurrent theme in the

electronic journal responses collected after the measurement estimation activity. This was

similar to participants belief as revealed from the analysis of electronic journal response

data collected prior to the onset of the measurement estimation activity. For example:

I believe students learn to measure by using real life situations and hands-on

activities. As teachers we should take measurement out of the theoretical and into the

practical first and then go from the practical to the theory so that they understand why

measurement works like it does (electronic journal response collected before mea-

surement estimation activity).

I believe students learn by trial and error. They learn mostly by doing, especially

with a concept like measuring. I learned mostly by doing. I did projects that required

measurement of length, width and height. I did a project of cutting a carpet for a

room. This also required a lot of measuring (electronic journal response collected

after measurement estimation activity).

K. Subramaniam

123

To summarize, participants in this study had conceptualized various personally mean-

ingful representations of nonstandard units that enabled them to estimate length. The

nature of the transformation process involved using the nonstandard unit as a mentally

represented familiar object to refer to when thinking about the estimation of length

measurements. This transformation process involving conceptual rulers and associated

processes of scaling and counting segments was evident in participants measurement

estimation strategies but was not evident in their beliefs about teaching measurement

estimation to their students. On the other hand, data from the pre- and post-electronic

journal responses to interview questions revealed that participants favored the use of

hands-on activities as the key teaching and learning strategy for estimating length mea-

surements and did not mention the use of benchmarks.

Discussion

The premise of this study was that participants as numerate adults (Joram 2003; Sarama

and Clements 2009) will have meaningful benchmarks for measurement estimation and

that these benchmarks will be part of participants pedagogical knowledge for teaching the

estimation of length measurements. Findings from this study revealed that participants

benchmarks for the estimation of length measurements were meaningful and familiar

objects (Castle and Needham 2007; Crites 1992, 1993; Joram 2003; Montague and van

Garderen 2003; Towers and Hunter, 2010) (refer Table 3). Correspondingly, participants

used benchmarks as a strategy to estimate length measurements of a to-be-estimated object

without the aid of a measuring tool during the measurement estimation activity (Joram

2003; Joram et al. 1998, 2005). This involved the retrieval of a preconceived benchmark

from memory, and the superimposing of the preconceived benchmark onto the to-be-

estimated object (Castle and Needham 2007; Crites 1992, 1993; Joram 2003; Montague

and van Garderen 2003; Towers and Hunter 2010). That is, participants were using their

benchmarks of measurement estimation in a mental way without the aid of measuring tools

and underscored by conceptual prerequisites of logical reasoning and knowledge of spe-

cific measurement concepts (Towers and Hunter 2010).

This study also showed that participants used their benchmarks of measurement esti-

mation in a mental way (Towers and Hunter 2010) that was similar to mental rulers (Joram

et al. 2005) or conceptual rulers (Sarama and Clements 2009). That is, participants pro-

jected their benchmarks, for example, the Zauls height and Jackies forearm length, to

estimate the length of the floor. This mental estimation operated as the mental partitioning

of a length into perceived magnitude to represent a numerical magnitude. Estimation of the

length was then achieved by scaling either the related nonverbal or perceived magnitude to

numerical magnitudes or counting the perceived segments to create the magnitude of

connected lengths (Sarama and Clements 2009).

Although participants use of benchmarks of measurement estimation as tools was

evident in the measurement estimation activity, participants did not include benchmarks

for measurement estimation as specialized pedagogical knowledge for teaching the esti-

mation of length measurements (Ball et al. 2008; Hill et al. 2008; Leikin and Levav-

Waynberg 2007; Stacey 2008). Instead, findings indicated that participants pedagogical

knowledge for teaching the estimation of length measurements rested upon the use of

hands-on activities (together with definitions, explanations, and worksheets), as a key

strategy for teaching the estimation of length measurements. This finding coheres with the

research literature on pedagogical knowledge for teaching as the need for rules,

Measurement estimation

123

procedures, and teachers explanations (Ball et al. 2008; Hill et al. 2008; Leikin and Levav-

Waynberg 2007).

Even though participants in this study were using symbolic representations (Ball et al.

2008; Davis and Simmt 2006; Hill et al. 2004; 2008; Steinbring 1998, 2008) (benchmarks

for measurement estimation) that enabled them to make estimations during the measure-

ment estimation activity, these same participants were unable to connect this knowledge

in creating a meaningful way to teach the estimation of length measurements. Clearly,

participants were unaware how to use symbolic representations as an element of peda-

gogical knowledge during the coaction and interactivity required (Steinbring 2008) when

teaching students to construct meaning for the estimation of length measurements.

Furthermore, participants were conceptualizing hands-on activities as a way to teach the

estimation of length measurements because it allowed for student interaction/participation

with pedagogical knowledge of measurement estimation presented by the teacher and the

activities. This linear flow of knowledge for teaching the estimation of length measure-

ments from the planned hands-on activities and definitions, descriptions, explanations, and

worksheets provided by the teacher to students was their perception of a socially con-

structed mathematical content pertaining to measurement estimation. Steinbring (2008) has

stated that this linear approach to teaching mathematics is simplistic and thus does not

include reciprocity, coaction, and interaction between teachers symbolic representations

of the mathematics content and its role in socially developing the content with students

during instruction.

Moreover, the findings of this study showed that participants knowledge for teaching

the estimation of length measurements was predisposed to the level of habitual thinking

which rested heavily on their beliefs about teaching (Charalambous et al. 2009; Frykholm

1999; Macnab and Payne 2003; Philpp et al. 2007; Smith 2001, 2005). That is, participants

believe that hands-on activities based on their own K-12 learning experiences of mea-

surement estimation seemed to be the choice of instruction for teaching measurement

estimation, even though they had no explanations for how this helps their future students

construct knowledge and skills in making estimations of physical measurements. It can be

argued that this belief was acting in tandem to reduce the complexity of teaching (Davis

and Simmt 2006) the estimation of length measurements with the belief that hands-on

activities mean real-life experiences and examples of to-be-estimated objects. Also, par-

ticipants belief that hands-on activities were the choice of instruction for teaching mea-

surement estimation showed that their beliefs were acting as filters and/or as conditioning

elements (Ahtee and Johnson 2006; Da-Silva et al. 2006) for determining instructional and

learning tasks.

This discussion highlights that participants in this study were (1) conceptualizing the

collective cognitive responsibility (Scardamalia 2002) of teaching measurement estimation

to the hands-on tasks and activities they planned to use in their future classrooms and (2)

not transforming mathematics content into symbolic representations because they were

unaware that learning can be problematic and may require strategic teaching actions

(Scardamalia 2002), such as the use of benchmarks for measurement estimation, to bring

about meaningful learning as supported by the literature (Joram 2003; Joram et al. 2005;

Sarama and Clements 2009). Most importantly, this study highlights that participants

pedagogical knowledge for teaching the estimation of length measurements was not

influenced by their engagement in the measurement estimation activity because the mea-

surement estimation activity perpetuated the interactive process between students, teach-

ers, and the mathematics content or the productive relationship (Boaler 2002) as

appropriation of knowledge and this cohered with their belief about hands-on teaching as

K. Subramaniam

123

the best approach to teaching the estimation of length measurements. Also, there were no

attempts by the course instructor to relate participants own benchmarks for measurement

estimation as tools to estimate length measurements, that is, the inter-relationships between

knowledge and teaching practice (Boaler 2002) was not emphasized.

Conclusion and implications

The findings of the study indicated that participants pedagogical knowledge for teaching

the estimation of length measurements was influenced by their belief that hands-on

activities are the choice of instruction for teaching measurement estimation even though

participants themselves used benchmarks for estimating length measurements. Another

pertinent outcome of this study is that it showed that participants did possess mental or

conceptual rulers which operated with their benchmarks estimating length measurements,

but this knowledge was not transferred to their ideas about teaching the estimation of

length measurements. Also important to this study and its findings was the adoption of a

German perspective (Steinbring 1998, 2008) on pedagogical knowledge for teaching. This

perspectives emphasis on coaction and interaction between the symbolic representations

of content, the teacher, and students as the key elements for social construction of content

provided a complex and epistemological organization to frame participants pedagogical

knowledge for teaching the estimation of length measurements.

Because the sample of participants is small (n = 6), the author makes no claims to

generalize beyond the scope of this study. But findings in this study, especially concerning

numerate adults, their benchmarks, and conceptual rulers for estimating length measure-

ments, are consistent with other studies (Joram et al. 2005; Sarama and Clements 2009)

reinforcing the idea of benchmarks and conceptual rulers as one means for teaching

measurement estimation.

Implications for mathematics teacher educators include (1) the need to make visible the

measurement estimation strategies and associated benchmarks that prospective secondary

mathematics teachers possess and use in measurement estimation tasks and (2) the need to

implement measurement estimation activities in mathematics teacher preparation programs

that allow prospective secondary mathematics teachers to examine their teaching strategies

for measurement estimation and the associated underlying beliefs or habitual ways of

thinking. By doing so, prospective secondary mathematics teachers involvement in mea-

surement situations might help them to make sense of the tasks, build and test conjec-

tures, and explore the reasonableness of their measurement estimation strategies and help

build their knowledge that interactive relationship is a key to the active and social trans-

formation of specific mathematics concepts (symbolic representations) into meaningful

symbolic representations by teachers and students during instruction. Also, these activities

may help prospective secondary mathematics teachers see how meaningful tasks can help

build their students mathematical reasoning and content.

References

Ahtee, M., & Johnson, J. (2006). Primary prospective teachers ideas about teaching a physics topic.Scandinavian Journal of Educational Research, 50(2), 207219.

Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, R. J., Schmid, W., & Schaar, R. (2005). Reaching forcommon ground in K-12 mathematics education. Notices of the American Mathematics Society, 52(9),10551058.

Measurement estimation

123

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?Journal of Teacher Education, 59, 389407.

Baturo, A., & Nason, R. (1996). Prospective teachers subject matter knowledge within the domain of areameasurement. Educational Studies in Mathematics, 31, 235268.

Bean, G. (2005). Does feedback from postgraduate students align with the recommendations of academicauditors? Quality in Higher Education, 11(3), 261272.

Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice, and identity inmathematics classrooms. For the Learning of Mathematics, 22(1), 4247.

Borko, H., Peressini, D., Romagnano, L., Knuth, E., Willis-Yorker, C., Wooley, C., et al. (2000). Teachereducation does matter: A situative view of learning to teach secondary mathematics. EducationalPsychologist, 35(3), 193206.

Boyatzis, R. E. (1998). Transforming qualitative information: Thematic analysis and code development (2nded.). Thousand Oaks, CA: Sage.

Brabant, S., & Kalich, D. (2008). Who enrolls in college death education courses? A longitudinal study.Omega: Journal of Death and Dying, 58(1), 118.

Cady, J., Meier, S., & Lubinski, C. (2006). Developing mathematics teachers: The transition from preserviceto experienced teacher. Journal of Educational Research, 99(5), 295305.

Castle, K., & Needham, J. (2007). First graders understanding of measurement. Early Childhood EducationJournal, 35, 215221.

Charalambous, C., Panaoura, A., & Philippou, G. (2009). Using the history of mathematics to inducechanges in preservice teachers beliefs and attitudes: Insights from evaluating a teacher educationprogram. Educational Studies in Mathematics, 71(2), 161180.

Clements, D. H. (1999). Teaching length measurement: Research challenges. School Science and Mathe-matics, 99(1), 511.

Clements, D. (Ed.). (2003). Learning and teaching measurement. National Council of Teachers of Math-ematics Yearbook. Reston, VA: National Council of Teachers of Mathematics.

Cooney, T. (1999). Conceptualizing teachers ways of knowing. Educational Studies in Mathematics,38(13), 163187.

Crites, T. W. (1992). Skilled and less skilled estimators. Strategies for estimating discrete quantities.Elementary School Journal, 92, 601619.

Crites, T. W. (1993). Strategies for estimating discrete quantities. The Arithmetic Teacher, 41(2), 106108.Da-Silva, C., Mellado, V., Ruiz, C., & Porlan, R. (2006). Evolution of the conceptions of a secondary

education biology teacher: Longitudinal analysis using cognitive maps. Science Education, 91,461491.

Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics thatteachers (need to) know. Educational Studies in Mathematics, 61, 293319.

Edens, K. M. (2000). Promoting communication, inquiry and reflection in an early practicum experience viaan on-line discussion group. Action in Teacher Education, 22(2A), 1423.

Flick, U. (1998). An introduction to qualitative research: Theory, method, and applications. London: Sage.Fontana, A., & Frey, J. H. (2005). The interview: From neutral to political involvement. In N. K. Denzin &

Y. S. Lincoln (Eds.), The Sage handbook of qualitative research (3rd ed., pp. 695728). ThousandOaks: SAGE Publications.

Forgasz, H. J., & Leder, G. C. (2008). Beliefs about mathematics and mathematics teaching. In P. Sullivan& T. Wood (Eds.), The international handbook of mathematics teacher education (pp. 173192).Rotterdam: Sense.

Forrester, M. A., & Pike, C. D. (1998). Learning to estimate in the mathematics classroom: A conversation-analytic approach. Journal for Research in Mathematics Education, 29(3), 334356.

Frykholm, J. A. (1999). The impact of reform: Challenges for mathematics teacher preparation. Journal ofMathematics Teacher Education, 2, 79105.

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptu-alizing and measuring teachers topic-specific knowledge of students. Journal for Research inMathematics Education, 39(4), 372400.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers mathematical knowledge for teaching onstudent achievement. American Educational Research Journal, 42(2), 371406.

Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measure of teachers mathematics knowledgefor teaching. The Elementary School Journal, 105(1), 1130.

Hodder, I. (2003). The interpretation of documents and material culture. In N. K. Denzin & Y. S. Lincoln(Eds.), Collecting and interpreting qualitative materials (2nd ed., pp. 155175). Thousand Oaks: SagePublications Inc.

K. Subramaniam

123

Hogan, T. P., & Brezinski, K. L. (2003). Quantitative estimation: One, two, or three abilities? MathematicalThinking and Learning, 5(4), 259280.

Joram, E. (2003). Benchmark as tools for developing measurement sense. In D. H. Clements & G. Bright(Eds.), Learning and teaching measurement 2003 yearbook (pp. 5767). Reston, VA: The NationalCouncil of Teachers of Mathematics Inc.

Joram, E., Gabriele, A. J., Bertheau, M., Gelman, R., & Subrahmanyam, K. (2005). Childrens use of thereference point strategy for measurement estimation. Journal for Research in Mathematics Education,36(1), 423.

Joram, E., Subrahmanyam, K., & Gelman, R. (1998). Measurement estimation: Learning to map the routefrom number to quantity and back. Review of Educational Research, 68(4), 413449.

Kinach, B. M. (2002). A cognitive strategy for developing pedagogical content knowledge in the secondarymathematics methods course: Toward a model of effective practice. Teaching and Teacher Education,18, 5171.

Lang, F. K. (2001). What is a good guess anyway? Estimation in early childhood. Teaching ChildrenMathematics, 7(8), 462466.

Leatham, K. R. (2006). Viewing mathematics teachers beliefs as sensible systems. Journal of MathematicsTeacher Education, 9, 91102.

Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gapbetween theory-based recommendations and school practice in the use of connecting tasks. Educa-tional Studies in Mathematics, 66, 349371.

Levin, B. B. (1999). Analysis of the content and purpose of four different kinds of electronic communi-cations among preservice teachers. Journal of Research on Computing in Education, 32(1), 139156.

Macnab, D., & Payne, F. (2003). Beliefs, attitudes and practices in mathematics teaching: Perceptions ofScottish primary school prospective teachers. Journal of Education for Teaching, 29(1), 55.

Montague, M., & van Garderen, D. (2003). A cross-sectional study of mathematics achievement, estimationskills, and academic self-perception in students of varying ability. Journal of Learning Disabilities,36(5), 437448.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for schoolmathematics. Reston, VA: The National Council of Teachers of Mathematics Inc.

Nicholson, S. A., & Bond, N. (2003). Collaborative reflection and professional community building: Ananalysis of preservice teachers use of an electronic discussion board. Journal of Technology andTeacher Education, 11(2), 259279.

Philpp, R. A., Ambrose, R., Lamb, L. L. C., Sowder, J. T., Schappelle, B. P., Sowder, L., et al. (2007).Effects of early field experiences on the mathematical content knowledge and beliefs of prospectiveelementary school teachers: An experimental study. Journal for Research in Mathematics Education,38(5), 438476.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning tra-jectories for young children. New York: Routledge.

Scardamalia, M. (2002). Collective cognitive responsibility for the advancement of knowledge. In B. Smith(Ed.), Liberal education in a knowledge society (pp. 6798). Chicago: Open Court.

Seale, C. (1999). The quality of qualitative research. London: Sage Publications.Siegler, R. S., & Booth, J. L. (2005). Development of numerical estimation: A review. In J. I. D. Campbell

(Ed.), Handbook of mathematical cognition (pp. 197212). New York: Psychology Press.Smith, J. D. N. (2001). Mathematics prospective teachers responses to influences and beliefs. Research in

Mathematics Education, 3(1), 115138.Smith, J. D. N. (2005). Understanding the beliefs, concerns and priorities of trainee teachers: A multi-

disciplinary approach. Mentoring and Tutoring, 13(2), 205219.Stacey, K. (2008). Mathematics for secondary teaching: Four components of discipline knowledge for a

changing teacher workforce. In P. Sullivan & T. Wood (Eds.), The international handbook of math-ematics teacher education (pp. 87113). Rotterdam: Sense.

Steinbring, H. (1998). Stoff Didaktik to social interactions: An evolution of approaches to the study oflanguage and communication in German mathematics education research. In H. Steinbring, M. Bart-olini-Bussi, & A. Sierprinska (Eds.), Language and communication in the mathematics classroom (pp.102119). Reston, VA: National Council of Teachers of Mathematics.

Steinbring, H. (2008). Changed views on mathematical knowledge in course of didactical theory ofdevelopment-independent corpus of scientific knowledge or result of social constructions? ZDM, 40,303316.

Straesser, R. (2007). Didactics of mathematics: More than mathematics and school! ZDM, 39, 165171.Sullivan, P. (2008). Knowledge for teaching mathematics: An introduction. In P. Sullivan & T. Wood (Eds.),

The international handbook of mathematics teacher education (pp. 19). Rotterdam: Sense.

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123

Towers, J., & Hunter, K. (2010). An ecological reading of mathematical language in Grade 3 classroom: Acase of learning and teaching measurement estimation. The Journal of Mathematical Behavior, 29,2540.

Trafton, P. (1986). Teaching computational estimation: Establishing an estimation mindset. In H. L. Schoen& M. J. Zweng (Eds.), Estimation and mental computation: 1986 yearbook (pp. 1630). Reston, VA:National Council of Teachers of Mathematics.

Tsamir, P. (2005). Enhancing prospective teachers knowledge of learners intuitive conceptions: The caseof same A-same B. Journal of Mathematics Teacher Education, 8, 469497.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathe-matics: Teaching developmentally. USA: Pearson.

Wilson, M., & Cooney, T. (2002). Mathematics teacher change and development. In G. C. Leder, E.Pehkonen, & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 127147).Dordrecht: The Netherlands Kluwer Academic Publishers.

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Prospective secondary mathematics teachers pedagogical knowledge for teaching the estimation of length measurementsAbstractIntroductionReview of literaturePedagogical knowledge for teachingProspective teachers beliefsBenchmarks for measurement estimationSynthesis: toward an organizing framework

MethodContextParticipants and role of researcherData collectionMeasurement estimation activityData analysis

LimitationsFindingsParticipants habitual ways of thinking about teaching the estimation of length measurementsParticipants knowledge of estimating length during the measurement estimation activityParticipants knowledge of estimating length after the measurement estimation activity

DiscussionConclusion and implicationsReferences