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Web-Based Video Clips: A Supplemental
Resource for Supporting Pre-service
Elementary Mathematics Teachers
Ann LeSage
Abstract Teacher understanding and confidence with rational numbers are
important factors contributing to student success with this foundational concept.
The challenge facing many Ontario elementary mathematics teacher educators is
finding the time, within a 1-year teacher education program, to provide opportunities
for elementary pre-service teachers to re-learn rational number concepts in ways
they are required to teach. In an effort to address this challenge, web-based video
clips were created as an accessible learning resource to support the needs of pre-
service elementary teachers. This chapter describes how and why the videos were
incorporated into the program and describes the reflections of elementary pre-
service teachers after viewing selected videos. The reflections reveal the influence
of web-based videos on pre-service teachers’ perceived understanding of and
confidence with rational numbers.
Keywords Elementary mathematics teacher education • Knowledge for teaching
mathematics • Rational number understanding • Web-based videos • Teacher
efficacy • Instructional design
Introduction
Effective mathematics instruction is based on mathematical and pedagogical knowledge
and understanding of students’ mathematical development (Ontario Ministry of Education,
2011,p.6).
A. LeSage ()
Faculty of Education, University of Ontario Institute of Technology (UOIT), Oshawa, ON L1H
7K4 Canada
e-mail: ann.lesage@uoit.ca
D. Martinovic et al. (eds.), Visual Mathematics and Cyberlearning, Mathematics
Education in the Digital Era 1, DOI 10.1007/978-94-007-2321-4 8,
© Springer ScienceCBusiness Media Dordrecht 2013
187
188 A. LeSage
Inherent to this statement is the presumed breadth and depth of teachers’
mathematical content and pedagogical content knowledge. Research highlights a
significant relationship between student achievement and teachers’ understanding
of the mathematics they teach (Burton, Daane, & Giesen, 2008; Hill, Rowan,
& Ball, 2005;Ma,1999). More specifically, two seminal studies on the effect
of teacher mathematical knowledge on student achievement conclude that the
combined influence of teachers’ mathematical content knowledge and pedagogical
content knowledge more strongly correlate with student achievement than any
other moderating factor, including socioeconomic and language status (Darling-
Hammond, 2000; Hill et al., 2005).
Given this assertion, faculties of education are obliged to modify the structure and
content of their programs to provide prospective elementary teachers opportunities
to re-learn mathematics in ways they are required to teach. Regrettably, this mandate
is particularly challenging for Ontario elementary mathematics teacher educators as
instructional time devoted to mathematics methods courses is generally restricted to
36-h (18 h per semester). Consequently, with limited face-to-face instructional time,
many pre-service teachers become frustrated and more anxious as they struggle to
re-learn mathematics in new ways.
In an effort to address these issues, the author, an elementary mathematics teacher
educator, designed a technology-enhanced elective course to support pre-service
teachers’ understanding of mathematics content and nurture their confidence as
elementary mathematics teachers. The Math4Teachers course was introduced as
an elective offered in the first semester of the program (9 weeks2 h/week). In
addition to the technologies utilized during the face-to-face component of the course
(e.g., interactive whiteboard, virtual manipulatives, interactive applets/software),
web-based video clips were created as virtual resources to support learning beyond
the physical classroom environment.
This chapter describes how and why web-based videos were integrated into
the structure and content of a face-to-face elementary mathematics course. More
specifically, the chapter summarizes the current literature on teaching and learning
mathematics as well as integrating technology in teacher education; it describes how
the web-based video clips (WBVCs) were incorporated into the course; conveys the
reflections of pre-service teachers on the perceived impact of the videos on their
understanding of and confidence with mathematics; and outlines some of the issues
and implications of incorporating WBVCs into an elementary pre-service program.
Literature Review
Four bodies of literature were influential in guiding the development of the course
and the WBVCs: knowledge for teaching mathematics, knowledge of rational num-
bers, web-based learning tools, and integrating digital video in teacher education.
Analysis of the literature provides the theoretical lens for modifying the course
structure and designing the web-based video clips (WBVCs).
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 189
Knowledge for Teaching Elementary Mathematics
Research highlights a direct correlation between student achievement in mathe-
matics and their teachers’ understanding of the mathematics content they teach
(Burton et al., 2008; Hill et al., 2005;Ma,1999; Schmidt, Houang, & Cogan, 2002).
Although this connection seems self-evident, the paucity of empirical evidence on
effective ways to develop knowledge for teaching mathematics has been highlighted
in recent research (Berk & Hiebert, 2009; Burton et al., 2008; Hill & Ball, 2009;
Kilpatrick, Swafford, & Findell, 2001). One explanation for the scarcity of this
empirical research may be the lack of consensus on the nature and depth of
knowledge required for teaching mathematics. Initially, Shulman (1986) described
the knowledge required for teaching as the interconnection between subject-matter
knowledge, pedagogical content knowledge and curricular knowledge. Yet, a decade
later, Schifter (1998) bemoaned the absence of research on this topic. She urged re-
searchers and mathematics teacher educators to pursue the question: “What kinds of
understandings are required of teachers working to enact the n ew pedagogy?” (p. 57)
Since Schifter’s (1998) call for research, the breadth and depth of teachers’
mathematical knowledge has been explored largely by Ball and her colleagues
(Ball, Hill, & Bass, 2005; Ball, Thames, & Phelps, 2004; Hill et al., 2008; Hill &
Ball, 2009; Hill, Schilling, & Ball, 2004; Hill et al., 2005). Ball and her colleagues
have focused on a complex dimension of teacher knowledge, namely, mathematical
knowledge for teaching. At the core of mathematical knowledge for teaching is a
deep understanding of mathematics content. Specifically, mathematical knowledge
for teaching assumes an understanding of “common” mathematics knowledge,
which is the content knowledge “that any well-educated adult should have” (Ball
et al., 2005, p. 22). Unfortunately, many elementary teachers lack this common
knowledge and, therefore do not have the foundation to build their mathematical
knowledge for teaching.
The collective effect of insufficient content knowledge and high levels of
mathematics anxiety can overwhelm novice teachers as they begin their careers as
elementary mathematics educators. Thus, the impetus for designing the WBVCs
were two-fold: to foster an affinity for mathematics such that pre-service teachers
can engage their students and get them excited about mathematics; and to contribute
to the research on how to nurture pre-service elementary teachers’ mathematical
knowledge for teaching, and their efficacy as mathematics teachers and learners.
Knowledge of Rational Numbers
In decomposing the depth of content knowledge required for teaching elementary
mathematics it becomes apparent that an understanding of rational numbers is
central in the upper elementary curriculum (e.g. Grades 4–6). Gersten et al.
(2009) acknowledge this content focus in their report on interventions that best
190 A. LeSage
support students struggling with mathematics. Specifically, the authors make five
recommendations, of which one includes focusing “intensely on in-depth treatment
of rational numbers in grades 4 through 8” (p. 18). Gersten et al. advocate focusing
the curriculum content on “understandingthe meaning of fractions, decimals, ratios,
and percents, using visual representations, and solving problems with fractions,
decimals, ratios, and percents” (p. 19).
Nurturing students’ understanding of rational numbers is often deemed one
of the most challenging aspects of teaching elementary mathematics (Gould,
Outhred, & Mitchelmore, 2006;Li&Kulm,2008). This pedagogical challenge is
complicated by teachers’ own conceptual misunderstandings of rational numbers
(Hill et al., 2005; Jones Newton, 2009;Li&Kulm,2008;Ma,1999; McLeman &
Cavell, 2009). Regrettably, teachers’ misunderstandings inevitably lead to students’
misunderstandings; which often follow children into adulthood (Lipkus, Samsa, &
Rimer, 2001; Reyna & Brainerd, 2007) and further intensify should these adults
pursue careers as elementary teachers (Ball et al., 2005;Ma,1999; Menon, 2008;
Yeping, 2008).
In spite of this pessimistic perspective, research also reveals that it may be
possible to disrupt this cycle of conceptual misunderstanding by targeting teacher
education. For example, Siegler et al. (2010) conducted an extensive review of
research published over the past 20 years on the effects of instructional interventions
on student understanding of rational numbers. The authors put forth five recommen-
dations, of which one recommendation recognized the significant impact of teacher
knowledge on student learning. Specifically, Siegler et al., believe it is critical for
preservice teacher education and professional development programs to “place a
high priority on improving teachers’ understanding of fractions and of how to teach
them” (p. 42).
As a mathematics teacher educator, I embrace this recommendation by providing
opportunities for pre-service elementary teachers to re-form their mathematics
content knowledge and re-learn mathematics in ways they are required to teach. To
this end, the instructional design of the Math4Teachers elective course and WBVCs,
adhere to research rooted in effective teaching strategies that support students
struggling with mathematics and effective professional development models for
teachers of mathematics. Specifically, research highlights positive student achieve-
ment outcomes for mathematics interventions which: (1) combine manipulatives
and pictorial representations to model abstract concepts (Butler, Miller, Crehan,
Babbitt, & Pierce, 2003; Siegler et al., 2010); (2) incorporate a mixed model
of instruction which blends principles of explicit instruction including teacher
modeling, guided practice, and corrective feedback (Baker, Gersten, & Lee, 2002;
Flores & Kaylor, 2007; Gersten et al., 2009; Kroesbergen & Van Luit, 2003); and
(3) feature ample time for discussion, including student-focused discussions which
provide alternative solution strategies expressed in students’ language (Grouws,
2004; Shellard, 2004).
A related body of research on effective professional development advocates
providing teachers with opportunities to develop their pedagogical content knowl-
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 191
edge and deepen their conceptual understanding of mathematics through actively
engaging in the learning process (Hill, 2004; Manouchehri & Goodman, 2000;
Ross, 1999; Spillane, 2000). The NCTM Professional Teaching Standards (1991)
extends this recommendation stating that teacher education focus on “mathematical
concepts and procedures and the connections among them; ::: [as well as] multiple
representations of mathematical concepts and procedures” (p. 132). Consequently,
embedded in the Math4Teachers course design as well as the design of the WBVCs
are opportunities for pre-service teachers to do similar tasks as their students (Saxe,
Gearhart, & Nasir, 2001; Siegler et al., 2010), to explore multiple representations of
concepts to discuss the nature of the mathematics and mathematics pedagogy, and
to reflect on their learning experiences (Li & Kulm, 2008;Saxeetal.,2001; Tirosh,
2000).
Web-Based Learning Tools
Web-based learning tools (WBLTs), such as web-based video clips (WBVCs)
evolved from a need for accessible, affordable and flexible learning via the Internet
(Ally, 2004;Downes,2004). WBLT are distinct from other digital resources in that
“instructional design theory ::: must play a large role in the application of WBLTs
if they are to succeed in facilitating learning” (Wiley, 2000, p. 9). Kay and Knaack’s
(2005) review of the WBLT literature, categorized the many WBLT definitions as
either technology-focused or learning-focused. They, in turn, defined WBLTs as
“reusable, interactive web-based tools that support the learning of specific concepts
by enhancing, amplifying, and guiding the cognitive processes of learners” (p. 231).
The WBVCs discussed in this paper adhere to Kay and Knaack’s definition of
WBLTs.
Given the ubiquitouscomputing environmentat this lap-top university, the devel-
opment and use of WBLTs have been the focus of various research studies within our
Faculty of Education (Kay & Kletskin, 2010; Kay & Knaack, 2009a,2009b;Kay,
Knaack, & Muirhead, 2009). Consequently, the design and implementation of the
WBVCs discussed in this chapter are groundedin sound theory and practice both in
the relevant literature and in other examples of implementation within the university.
For example, the WBVC format and design were guided by Kay and Knaack’s
(2007) findings on conditions that most benefit student learning via technology,
including student perceptions of the usefulness of the content, clear instructions,
and visual appeal.
Thus, the WBVCs describedin this chapter are brief (2–3 min), easy to navigate,
have visual appeal and explore content considered useful to the end-user. More
importantly, the WBVCs begin to address the challenge of limited face-to-face
instructional time common in pre-service education programs.
192 A. LeSage
Digital Videos and Teacher Education
In 2000, the National Council of Teachers of Mathematics (NCTM) established
the Technology Principle as a component of their Principles and Standards of
School Mathematics The Principle states that the “existence, versatility, and power
of technology make it possible and necessary to re-examine what mathematics
students should learn as well as how they can best learn it” (p. 25). Consequently,
teachers need to re-examine how they might integrate technology into their existing
mathematics curriculum to meet the diverse needs of their students. Similarly,
teacher educators are obliged to prepare pre-service teachers for the task of guiding
students in exploring mathematics supported by technology.
A variety of technologies can be useful for exploring elementary school
mathematics, including: interactive whiteboards, interactive applets and software,
classroom response systems, virtual manipulatives, dynamic geometry tools (i.e.,
Geometer’s Sketchpad®), and exploratory data analysis software (i.e., TinkerPlots®
&Fathom®). These technologies not only provide multiple representations of
mathematical concepts; but they allow students to explore mathematics in more
dynamic ways.
Niess and Walker (2010) advocate integrating digital videos as another viable
technology tool. Specifically, they assert that digital videos provide “students with
a different and often more engaging way for communicating what they know
and understand” (p. 103). However, if teachers are to integrate digital videos into
their teaching practices; they need opportunities to explore learning through this
medium. Thus, the challenge for mathematics teacher educators is to restructure
courses to prepared teachers to effectively incorporate “digital videos in ways that
provide exciting, effective, and rigorous mathematics learning opportunities for K-
12 students” (Niess & Walker, p. 104).
Kellogg and Kersaint (2004) advocate restructuring pre-service elementary
mathematics methods courses to include videos which “help teachers examine
mathematics teaching and learning” (p. 25). For that reason, they incorporated
ready-made digital videos (i.e., accessible on the Internet) demonstrating reform-
oriented mathematics teaching into their mathematics methods courses. Kellogg
and Kersaint conclude that integrating these videos helped pre-service teachers
“:::appreciate alternatives to traditional methods for learning and presenting
mathematics ideas” (p. 32).
Although an appreciation for alternative pedagogies is important, teachers can
only provide “exciting, effective and rigorous mathematics learning opportunities
for their students” (Niess & Walker, 2010, p. 104) if they possess a deep conceptual
understanding of the mathematics they are expected to teach. Consequently, Gawlik
(2009) recommends developing digital video tutorials to support elementary pre-
service teachers’ understanding of mathematical concepts. Specifically, she asserts
that digital videos are a particularly valuable instructional tool for auditory or visual
learners. Furthermore, Gawlik concludes that the length of the videos, the inclusion
of step-by-step explanations and visual demonstrations were key components that
impacted students’ perceptions of the usefulness of the video tutorials.
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 193
Given the extant research on the positive effects of integrating digital videos
into pre-service mathematics course, the videos described in this chapter extend the
research by focusing on the particular mathematics concept of rational numbers.
More specifically, the videos described in this chapter were designed to meet
the needs of pre-service elementary teachers who lack the capacity to identify,
understand and engage in the mathematics they are required to teach.
In the subsequent section I describe how current research informed the devel-
opment of the video clips, and explain how the videos were incorporated into the
pre-service mathematics program.
From Theory to Practice
In August 2009, a technology-enhanced elective course, titled Math4Teachers,
was introduced to provide additional support for elementary pre-service teachers
struggling with the mathematics they were expected to teach. However, as the course
ended (November 2009), many of the teachers requested additional resources be
provided that would support their continued developmentas elementary mathemat-
ics teachers. Consequently, the WBVCs evolved in an effort to provide this support
in the most cost-effective manner.
Development of the WBVCs began in April 2010 with the intent of providing
targeted instruction on specific mathematics concepts deemed problematic for
elementary teachers. Previous research confirmed my experiences as an educator
of elementary pre-service teachers; that strengthening teachers’ conceptual under-
standing of rational numbers was critical (Hill et al., 2005; Jones Newton, 2009;Li
&Kulm,2008;Ma,1999; McLeman & Cavell, 2009; Newton, 2008). Beyond the
content focus, previous research on web-based learning objects (Kay & Knaack,
2007; Lim, Lee, & Richards, 2006; Wiley, 2000) and online videos in teacher
education (Gawlik, 2009; Kellogg & Kersaint, 2004;Niess&Walker,2010) guided
the design of the video clips; while research on instructional practices supporting
students struggling in mathematics guided the pedagogicalfocus (Baker et al., 2002;
Butler et al. 2003; Gersten et al., 2004, 2009; Kroesbergen & Van Luit, 2003;Siegler
et al., 2010). Consequently, each 2–4 min video explicitly demonstrates one aspect
in an instructional sequence supporting the progressive development of rational
number understanding.
Instructional Sequence DDevelopmental Continuum
The initial WBVC instructional sequence was modeled on three principal resources:
(i) grade level curriculum expectations from the Ontario Mathematics Curriculum,
Grades 1–8 (Ontario Ministry of Education, 2005); (ii) research by Moss and
Case (1999) on teaching rational numbers; and (iii) Professional Resources and
194 A. LeSage
Instruction for Mathematics Educators, PRIME, an empirically validated teaching
resource (Small, McDougall, Ross, & Ben Jaafar, 2006) which supports the
progressive development of mathematical understanding.
In particular, the Canadian developed resource PRIME, details a continuum of
developmental phases that students’ progress through as they develop conceptual
understandings in mathematics. PRIME also describes appropriate instructional
strategies to support students’ movement along the learning continuum (Small,
2005a,2005b,2005c). The earlier work of Canadian researchers, Moss and Case
(1999), also highlighted the significance of exploring rational numbers in a specific
teaching – learning sequence. Moss and Case advocate for teaching fractions and
decimals using a lesson sequence which builds on students’ understanding of bench-
mark percentages. Thus, their lesson sequence begins with visual representations of
benchmark percentages, and then progresses to connecting percentage to decimal
representations and finally connecting decimals to fractional representations.
In knowing that the WBVCs would not be viewed by pre-service teachers until
they had completed at least 2 weeks of the Math4Teachers elective course, initial
explorations of rational numbers began in a face-to-face environment. Thus, prior to
viewing the WBVCs, pre-service teachers were introduced to benchmark numbers,
including percentage values which could be represented using concrete or virtual
manipulatives (e.g., 10 10 Geoboards, Base Ten Blocks, and money). By limiting
exploration of percentages to only those that can be represented as terminating
decimals (e.g., 25, 50, 90%, etc.), it ensures that each quantity can easily be modeled
concretely. Consequently, the learner associates a visual model or concrete represen-
tation to an abstract concept. The face-to-face instructional sequence transitioned
from representing benchmark percentages to concrete representations of decimals
using the same virtual and concrete manipulatives (e.g., 1010 Geoboards, Base
Ten Blocks, and money).
Approximately half of the face-to-face instructional time (10 h) was allocated
to representing, comparing and decomposing decimals using manipulatives; and
then connecting the concrete models to pictorial representations; and finally, to
the abstract representation. The balance of the face-to-face instructional time was
dedicated to the exploration of fractions. The subsequent fraction lessons progressed
from representing tenths and hundredths using area models (e.g., Geoboards, Base
Ten Blocks) to representing and comparing unit fractions and then simple fractions
using area models (e.g., Geoboards, Tangrams, pattern blocks) and measured
models (e.g., fraction strips, linking cubes, Cuisenaire Rods™). The face-to-face
instructional sequence concluded with an introduction to addition and subtraction
of fractions by extending the comparison of fractions lessons. For example, pre-
service teachers were asked to use fraction strips to compare 1/2 and 1/3. Next, they
were asked: “Which fraction is greater?” followed by “How much greater?” By
creating a concrete or virtual model, the pre-service teachers were able to visualize
the missing 1/6 fraction piece and then create an addition and subtraction sentence
based on the concrete representation.
The face-to-face instructional sequence was used to create the parallel series of
WBVCs. Consequently, the specific pedagogical sequence of the videos reinforces
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 195
the progressive development of rational number understanding. Although the
WBVCs’ instructional sequence evolved from current research and my experiences
teaching the concepts face-to-face; as the videos were created it became apparent
that the sequence required refinement. Thus, multiple video clips were often
necessary to demonstrate the micro-components within a single concept and ensure
each video was within the 2–4 min time frame. Moreover, it was important to
“:::attend to potential cognitive overload caused by too much information being
presented too quickly” (Bell & Bull, 2010,p.2).
For example, the Using Geoboards to Represent Decimals video explores
fraction and decimal representations to tenths and hundredths. However, upon
creating the original version of this video clip, it became apparent that additional
videos were required to introduce basic decimal vocabulary, and represent decimals
to tenths and hundredths individually prior to representing them in a single WBVC.
Additional video clips are currently under construction to address other conceptual
gaps within the existing instructional sequence.
The next section summarizes the WBVCs pre-service teachers viewed for this
research project, and illustrates a sample instruction sequence modeled on the video
clips.
Development of the WBVCs
Development of the WBVCs began in April 2010 with the intent of providing
targeted instruction on specific mathematics concepts deemed problematic for
elementary teachers. The following WBVCs were viewed by pre-service teachers
participating in this study:
• Decimal Vocabulary (4:55 min)
• Comparing Decimals (5:32 min)
• Exploring Tenths (3:24 min)
• Exploring Hundredth using Geoboards (2:55 min)
• Using Geoboards to Represent Decimals (3:28 min)
• Representing Fractions: Area Model (3:41 min)
• Representing Fractions: Measured Model (3:05 min)
• Representing Fractions: Set Model (3:26 min)
• Representing Mixed Fractions: Area Model (2:21 min)
• Representing Mixed Fractions: Measured Model (2:13 min)
The WBVCs posted online represent the first phase of a longer term research
project (available at http://lesage.blogs.uoit.ca/?page id=30). The clips are inten-
tionally designed to build on the pre-service teachers’ strengths, existing knowledge,
and learning needs. The video clips allow for some user interaction by allowing the
user to control the speed at which s/he views the demonstration (e.g., play, pause,
stop, fast-forward and rewind). However, the long term goal is to expand the format
of each WBVC to include: interactive tasks, extension problems, and an on-line
196 A. LeSage
self-assessment. That said, Bell and Bull (2010) acknowledge that “evidence is still
evolving regarding the types of video and associated pedagogical methods that are
most effective for teaching specific curricular topics” (p. 4).
Although the video clips to date include only instructor demonstration through
guided discovery, the mathematics content within each clip is supported by current
research on scaffolding rational number understanding (Moss & Case, 1999; Small,
2005a,2005b,2005c). The instructor guides the user through a series of tasks as she
demonstrates the connection between the abstract concept (e.g., decimal numbers)
and the concrete representation (e.g., Geoboard).
For illustrative purposes, the details of one WBVC are described here:
The WBVC on Using Geoboards to Represent Decimals is a 3:28 min video clip
modelling how a 10 10 Geoboard can be used to represent decimal equivalents.
The previous clips in this developmental video sequence demonstrate represent-
ing simpler decimal numbers, including decimals to the tenth and hundredth
place value. Consequently, if the user views the clips in the sequence they are
posted, the clips should extend the user’s existing knowledge incrementally.
Three primary components of the instructional sequence demonstrated in this
WBVC are highlighted in Table 1including screen shots supplemented by a
summary of the instructor’s explanations.
Methodology
The goal of this qualitative study is to describe the experiences of elementary pre-
service teachers in their journey toward understanding rational numbers assisted by
online digital videos. The research participants are 40 elementary pre-service teach-
ers who completed the Math4Teachers elective course from August to November
2010. The participants are a purposivesample of pre-service teachers who agreed to
participate in the study.
The dual purpose of this study is to describe the experience of using WBVCs
as a tool for developing rational number understanding;and to evaluate the efficacy
of the WBVCs as established by pre-service teachers’ perceptions of cognitive and
affective gains.
Data Collection
“Qualitative research uses narrative, descriptive approaches to data collection to un-
derstand the way things are and what it means from the perspectives of the research
participants,” (Mills, 2003, p. 4). The qualitative data collection instruments include:
narrative reflections from two homework assignments completed during the course;
voluntary post-coursefeedback concerning the value of the WBVCs; and the official
course evaluations completed at the end of the term.
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 197
Tab l e 1 Sample WBVC instructional sequence
Step 1: Divide the Geoboard into “4 equal pieces”; name the fractional piece: 1/4
Step 2: Scaffold content from previous clip. Prove 1 square represents 1/100 of the Geoboard
Step 3: Count each coloured square by 1/100 to demonstrate 1/4 D25/100 of the “whole” Geoboard
Step 4: Include the decimal representation, stating that 0.25 is “twenty-five hundredths”
The instructor advises the viewer to periodically pause the clip to complete a specific task using the Geoboard paper provided in class
Step 1: Divide the Geoboard into “5 equal pieces”; name the fractional piece: 1/5
Step 2: Scaffold content from previous example in the same clip
Step 3: Count pairs of coloured square by 2/100 demonstrating1/5 D20/100 of the Geoboard
Step 4: Include the decimal representation, stating that 0.20 is “twenty hundredths”0.20D20/100
Step 5: Remind viewers that 0.2 is “two-tenths” which is the decimal equivalent of 0.20
Step 1: The viewer is instructed to divide the Geoboard into “2 equal pieces”; and then record the decimal equivalents
prior to viewing the demonstration
Step 2: Demonstrate that 1/2 the Geoboard is covered by 50 small coloured squares or 50/100
Step 3: Include the decimal representation, stating that 0.50 is “fifty hundredths”
Step 4: Demonstrate that 1/2 the Geoboard is also covered by 5 rectangular pieces or 5/10
Step 5: Include the decimal representation, stating that 0.5 is “five tenths” which is the decimal equivalent of 0.50
Step 6: Review decimal equivalents presented in the clip
198 A. LeSage
The homework reflections required pre-service teachers to view the five Repre-
senting Decimals WBVCs (total viewing time of the 5 WBVCs D20:14 min), and
then provide a descriptive reflection of their significant learning and their enduring
questions. At the end of the term, 40 of the 57 pre-service teachers voluntarily re-
submit their reflective assignments and provided post-course feedback on the value
of the WBVCs to be used for research purposes.
Data Analysis
Given the purpose of the study is to describe the phenomenon of using WBVCs as
learning tools; qualitative content analysis was used to analyse the collected data.
All data were transcribed into the qualitative data analysis program, ATLAS.ti™,
which was used to process the data, create codes, and analyze and interpret codes by
searching for common words, phrases, themes and patterns. ATLAS.ti™ provided
an exploratory approach through which to build complex queries, and begin to
develop a comprehensive understanding of the data.
Data analysis began by creating first level codes generated from extent literature
on: knowledge for teaching mathematics; knowledge of rational numbers; and
digital videos in teacher education. The first level codes were then entered into
ATLAS.ti™. Next, all data were read repeatedly to achieve immersion and obtain a
sense of the whole (Tesch, 1990). Eachdocument in ATLAS.ti™ was then reviewed,
both line by line and as paragraphs or chunks of information. As relevant data was
encountered, the text was highlighted, creating a quotation to which a code was
attached. Each quotation was automatically identified by ATLAS.ti™ and assigned
a display name based on the document number, the location of the quotation within
the document, the line numbers of the quotation and its first 20 letters. For example,
“2:13 (100:101) I did not know that we had to :::”identifies the quotation from
Document #2, the 13th quotation within that document which is located from line
100 to line 101. A code is then attached to the quotation using either open coding
(the researcher creates the code name) or in-vivo coding (the quotation text acts as
the code name).
As this process continued, codes were added, merged, and removed as new
insights emerged from the data (Miles & Huberman, 1994). Codes were then
organized into categories based on how they were related “to one another in
coherent, study-important ways” (Miles & Huberman, p. 62).
The selection of quotations and coding procedures denote the beginning of the
interpretation phase. Through the coding process I developed initial interpretations
and identified themes and patterns as they emerged from the data. Thus, as I began to
analyze the data, I had created memos and anecdotal notes highlighting patterns that
had become apparent during coding. These interpretations of the descriptive codes
marked the initial stages in the development of pattern-based themes from the data.
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 199
Summary of Findings
It is through interpretation that raw data evolves into organized information allowing
themes to emerge and inferences to develop. In this section, I present the themes that
evolved from the data analysis and describe the 40 participants’ experiences using
web-based video clips with examples from their narratives to illustrate the responses
of the group.
Based on the analyses of the participants’ experiences viewing and reflecting
upon the five Representing Decimals WBVCs, the following three shared themes
emerged which describe the perceived impact of the videos on the participants’
understanding of and confidence with mathematics:
• Development of rational number understandingCself-efficacy;
• Development of pedagogical content knowledgeCteacher efficacy; and
• Considerations for instruction design of WBVCs.
Each of the three themes is described in detail as follows:
Development of Rational Number
Understanding CSelf-Efficacy
The majority (n D34; 85%) of the pre-service elementary teachers indicated the
WBVCs influenced their understanding of rational numbers. Specifically, 21 of the
participants indicated that the WBVCs served as a “refresher” of previously “for-
gotten or never really understood” knowledge; while the remaining 13 participants
indicated that the videos presented new information that they had not previously
known or had misunderstood.
Within this theme, three sub-themes emerged highlighting specific domains
participants identified as significant in supporting their understanding of rational
numbers. These included: vocabulary of decimal numbers, comparing decimal
quantities, and face value versus place value of decimal numbers.
Vocabulary of Decimal Numbers
The majority (n D31; 78%) of the participants identified learning new vocabulary
specific to decimal numbers as an outcome of viewing the 5 min “Decimal
Vocabulary” video. In particular, participants misunderstood that the word “and”
denotes a decimal; while the term “point” should not be used when naming a
decimal (i.e., 3.2 should be read as “three and two-tenths” not “three point two”).
Additionally, participants indicated that, prior to viewing the video, they did not
realize that the place value of the digit furthest to the right dictates the name of a
decimal number (i.e., 3.04 is read “three and four hundredths” because the place
value of the 4 indicates hundredths).
200 A. LeSage
Comparing Decimal Quantities
Although a half of the participants (nD20) viewed the 5 min “Comparing Deci-
mals” video; most of them (n D14) focused their attention simply on the vocabulary
of the decimal numbers as opposed to the concept of decimal equivalence. The
principal focus of the video clip was to illustrate how one might think about the
comparison between two equivalent decimal numbers: 0.2 and 0.20. However, most
participants (n D14) seemed to focus their attention on what they had just learned
in the “Decimal Vocabulary” clip viewed previously.
As an example, one participant explained, “I would have pronounced these two
numbers the same, not counting the zero. Now I have learned the proper way is
pronouncing the second number as twenty-hundredths.”
Although many of the participants acquired a rudimentary understanding of
decimal comparisons, other participants (n D6) were able to assimilate their
previous knowledge of rational numbers to construct new understandings of the
relationship between fractions, decimals and visual representations. As an example,
one participant explained,
[The video] helped me understand fractions as well as decimals. The part of clip on 0.20
and 0.2 being hundredths and tenths finally made sense to me. I always knew this, but I
don’t think that I really understood why.
Similarly, another participant highlighted a similar learning outcome based on
the inclusion of a concrete model to represent the decimal numbers. She stated:
When I saw the clip on this, it was as if a light bulb went on in my head! I always knew that
0.20 and 0.2 were the same, but it really makes a difference when you can see that 0.20 is
twenty hundredths and 0.2 is two tenths on the Geoboard!
Face Value Versus Place Value of Decimal Numbers
Although only a few participants (n D6; 15%) explicitly indicated that the videos
helped them rectify their place value misunderstandings; this sub-theme is con-
nected to the two previous sub-themes. For example, participants indicated that
in viewing the “Comparing Decimals” video, they now understood that “it is the
place value of the digits that matter not the number of digits.” Similarly, another
participant explained, “I thought 0.0948 was greater than 0.13. Unlike whole
numbers, it is not how many digits you have (i.e., 948 versus 13), but their place
value.”
While another participant described her feelings of empowerment based on her
newly-discovered understanding, stating:
I could never picture a number written to three decimal places because I always said
‘point four three two’ (0.432). I was hearing the ‘hundred’ and perhaps picturing the whole
number; so it was difficult to visualize a hundredth decimal number. I was really confused
about where the ‘thousandths’ came in. ::: [By watching the video, it] has now fused my
imagined numeral with the verbal and visual representations.
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 201
Though the participants’ descriptions of their place value learning experiences
provide insight into the existing knowledge of pre-service elementary teachers; one
participant’s comment describes an undeveloped understanding of rational numbers.
Specifically, the participant stated, “I now know that the first position behind the
decimal is called ‘tenths’, I would have called it the ‘ones’ position.”
Although the majority of the participants (nD34; 85%) indicated that the
WBVCs substantively influenced their understanding of rational numbers; only a
few participants (n D7; 18%) explicitly cited influences on their mathematics self-
efficacy or improved confidence in their abilities to do mathematics. Participants
offered comments concerning their improved mathematics self-efficacy, making
statements such as; “After seeing these five short videos, I already feel more
confident in my understanding of decimals :::”
Finally, one participant’s comment is worthy of sharing as it illustrates the
potential impact of digital videos on pre-service teachers’ confidence and self-
efficacy:
I am really surprised at how well I am grasping decimals. I remember this as one of my
worst mathematics experiences; which usually ended in a lot of tears. But, watching the
clips and using the manipulatives just made something click.
Development of Pedagogical Content Knowledge
CTeacher Efficacy
Perhaps not surprising, all 40 of the study participants indicated that the WBVCs
influenced their understanding of how to teach rational numbers. However, the
purpose of the WBVCs was to provide explicit, teacher directed, just-in-time
instruction on concepts deemed difficult for students struggling with mathematics.
Thus, although the sequence of five Representing Decimals WBVCs is developmen-
tally appropriate, illustrates decimal numbers using various manipulatives (e.g., Ten
frames, Base-Ten Blocks, Geoboards), and demonstrates multiple representations of
rational numbers (e.g., visual, verbal and symbolic); the WBVCs are not exemplary
models of reform oriented mathematics teaching practices. Unfortunately, some
participants (n D8; 20%) extrapolated the explicit instruction model demonstrated
in the videos as an example of exemplary classroom practice.
As an example, one participant concluded, “I will definitely be applying these
video clips to teach my future students, as it made [the content] very clear and
straightforward”. Another participant commented, “I think that watching the videos
was a good reminder to speak slowly when explaining a topic to children.”
In spite of this unanticipated learning outcome, the majority of the pre-service
teachers (n D32; 80%) seemed to have generalized beyond the explicit instruction
modeled in the WBVCs and gained new pedagogical insights into reform math-
ematics teaching practices. Specifically, participants emphasized the value of the
following:
202 A. LeSage
• encouraging multiple representations, including variability in the materials used
to explore the same concept;
• focusing instruction on progressive development of concepts by encouraging
concrete models, pictorial representations, verbal descriptions, and symbolic or
numeric representations;
• providing students with sufficient time to explore concepts using concrete
materials;
• utilizing diverse teaching strategies;
• incorporating technology (e.g., interactive whiteboard technology, virtual manip-
ulatives, WBVCs); and
• supporting continued professional development and access to on-line resources
(e.g., WBVCs).
As a final point, besides improved pedagogical content knowledge specific to
teaching decimal numbers, all 40 pre-service elementary teachers indicated that
the WBVCs influenced their teaching efficacy. The participants cited improved
confidence in how to teachrational numbers and integrate reform teaching strategies
into their classroom practice. As an example, one participant stated:
I now feel more confident in introducing decimals while paving the way for the young
learners to explore ways of interpreting and estimating decimals.
Considerations for the Instruction Design of WBVCs
In addition to the cognitive and affective outcomes described, the study participants
emphasized the significant influence of the WBVC design on the quality of their
learning experiences. Specifically, the participants highlighted the following instruc-
tional design components as contributing to their understanding of the concepts
presented in the WBVC:
• integration of the WBVCs into the Math4Teachers face-to-face course design;
• careful sequencing of the content presented in each WBVC (e.g., each clip
explores one component of a broader concept);
• clarity of the explanations including step-by-step explanations;
• combined use of visual models/virtual manipulativesCsymbolic representation
(numbers)Cclear verbal explanations;
• abbreviated viewing time (e.g., each clip was less than 5 min in length);
• slower pace than actual classroom lessons;
• ability to control the pace of the learning (e.g., pause to take notes, rewind to
review);
• the inclusion of practice questions; and
• the absence of judgement (e.g., pause or rewind the video as often as needed
without the judgement of others).
Web-Based Video Clips: A Supplemental Resource for Supporting. . . 203
Although the study participants deemed the WBVCs an effective tool for
improving their understanding of decimal numbers; there was a general consensus
that additional WBVCs are needed to support pre-service elementary teachers’
understanding of and confidence with rational numbers. The participants offered
the following ‘next-step’suggestions which will be considered in subsequent stages
of this research project:
• provide additional examples of each concept within each WBVC;
• include additional manipulatives for representing decimals (e.g., money, rela-
tional rods, fraction strips);
• create additional practice questions for each WBVC;
• include sample classroom vignettes for some of the WBVCs;
• discuss common misconceptions associated with teaching and learning rational
numbers; and
• create WBVCs modelling ineffective teaching strategies for teaching rational
numbers.
Issues and Implications
The WBVCs discussed in this chapter were developed to address challenging
content areas; provide pre-service elementary teachers with accessible and flexible
learning opportunities; and offer additional support for previewing or reviewing
important concepts addressed during face-to-face instruction. These instructional
design considerations allow pre-service teachers more control over their learning
experiences. For elementary teachers with a history of negative mathematics
experiences; being in control of mathematics is a novel yet welcome change. Thus,
providing on-line resources which can be accessed in a “just-in-time” manner seems
to be a promising strategy for supporting the individual learning needs of pre-service
elementary teachers.
Although the WBVCs were positively received by the pre-service teachers in
this study, the process of designing and creating this supplemental resource was
an arduous task. When designing video clips to support pre-service mathematics
teachers, careful consideration must be paid not only to the instructional design of
the learning tool, but also to effectively supporting the development of pre-service
teachers’ pedagogical content knowledge and content knowledge of mathematics.
Consequently, video clips must model effective teaching strategies; such as,
using concrete and pictorial representations to model rational numbers. However,
similar to face-to-face instructional tasks, the video clip activities should also “help
teachers understand how the representations relate to the concepts being taught”
(Siegler et al., 2010, p. 44). Equally important is the careful sequencing of the
video clip content. Conceptual understanding of mathematics is incremental and
develops over a life-time of learning experiences. The specific sequence of the
knowledge and skills developed while progressing along the mathematical learning
204 A. LeSage
continuum warrant further research and consideration from an adult learning/pre-
service teacher education perspective.
The research described in this chapter marks a small step toward better un-
derstanding the specific program components and lesson sequence which may
contribute to the development of pre-service elementary teachers’ conceptual
understanding of rational numbers. However, the ultimate goal for this research is
to: (1) contribute to the collective body of research in determining the breadth and
depth of mathematical knowledge necessary for teaching elementary mathematics;
and (2) design effective, open-access learning tools, such as WBVCs, to support
pre-service teachers in developing this knowledge.
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