Developing Pre-Service Teachers’ Understanding in Division of Fractions by
Using TPACK
Hsing-Wen Hu
College of Education
University of Alaska Anchorage
USA
hhu2@uaa.alaska.edu
Wei-Ying Hsiao
College of Education
University of Alaska Anchorage
whsiao@uaa.alaska.edu
Abstract: Although previous studies have demonstrated pre-service elementary school teachers
(PSTs) lack of understanding of the division of fraction, little is known about PSTs’ conceptual
understanding toward this content area. The research on the TPACK (technological, pedagogical,
and content knowledge) model is used to explicate PSTs’ capacity to learn and teach specific
content such as division of fractions with technology. A case study approach was used in this study
to look at PSTs’ TPACK development levels in a framework, and that framework facilitates
description of their conceptions and their difficulties in this integrating processes.
Introduction
Research in mathematics education has demonstrated that understanding the division of fractions and
teaching this concept has been one of the most problematic areas for elementary pre-service teachers (Ball, 1990;
Schwartz, 2008; Tirosh, 2000). Little research has investigated the reasons for such difficulties or the various ways
that pre-service teachers (PSTs) understand the division of fractions. In many studies, researchers have gained
significant insights in how technology assists learning and teaching mathematics (Harper, Schirack, Stohl, &
Garofalo, 2001; Mishra & Koehler, 2006) and adopting Mathematical Habits of Mind into the mathematics
curriculum (Goldenberg, Shteingold, Feurzeig, 2002). There has been little focus on how to integrate technological
tools (such as interactive whiteboards) and pedagogical practices like the Mathematical Habits of Mind into the
content area of dividing fractions. Without such analyses, researchers in mathematics education have had to make
less reliable inferences to investigate pre-service teachers’ teaching in conceptual understanding in division of
fractions and to improve the teaching. This case study examines the pre-service teachers’ TPACK (Technological
pedagogical content knowledge) in one elementary mathematics method course as they attempted to learn and teach
the division of fractions by using an interactive whiteboard. The main research questions were: (1) Does TPACK
help Pre-service teachers’ conceptual understanding in learning division of fractions? (2) Does TPACK help preservice teachers teach more effectively when they teach the division of fractions? Grounded theory (Glaser &
Strauss, 1967) was used to construct a model for answering these questions.
Related Research
Many studies have revealed pre-service teachers have a difficulty to work on division problems with the
way of conceptual understanding (Ball, 1990; Huang, 2009; Simon, 1993; Triosh, 2000). The invert-and-multiply
algorithm has become the pre-service teachers’ main method of dealing with division problem of fractions instead of
understanding and explaining the concept with meaning (Ball, 1990; Behr, Khoury, Harel, Post, & Lesh, 1997;
Hiebert & Lefevre, 1986; Schwartz, 2008; Tirosh, 2000). In order to teach division of fractions effectively, preservice teachers need to create multiple representations for abstract concepts (National Research Council, 2000).
Furthermore, they should conceive a system of relations between fractional symbols, concepts, operations, activities,
and situations (Niemi, 1996).
Besides building their own conceptual understanding toward division of fractions, pre-service teachers
should also be gaining knowledge that can improve elementary students’ conceptual understanding in division of
fractions. A sequence of teaching activity has been developed to help learners interpret situations involving division
of fractions which can lead to better understanding in the algorithmic procedures. The sequence of the activity
incorporates the “common-denominator” algorithm with four types of division problems of fractions (Schwartz,
2008). This increases the conceptual understanding which leads to success in problem-solving.
Another way for pre-service teachers to develop students’ conceptual understanding is the Mathematical
Habits of mind. It may help learners understand, control, modify, and make decisions when they perform
mathematical tasks. Through practicing mathematics, the following habits of mind are created: patterning,
experimenting, describing, tinkering, inventing, visualizing, conjecturing, and guessing (Cuoco, Goldenberg, &
Mark, 1996).
Mathematical Habits of Mind has become a useful tool in helping learners develop the conceptual
understanding needed for learning mathematics effectively (Goldenberg, Shteingold, & Feurzeig, 2003). There are
no studies that directly use Mathematical Habits of Mind in teaching performance, such as lesson designs and
instructional processes. Therefore, the researchers have extended the contents of “Models-Strategies-Applications”
(Wu & An, 2008) by integrating Mathematical Habits of Mind. This study has made a teaching sequence which may
supply pre-service teachers or in-service teachers to integrate Mathematical Habits of Mind into the processes of
conceptual understanding (see Figure 1).
a. Pattering Result Model
b. Patterning-Inventing Result
Model
Figure 1. Two models of integrating Mathematical Habits of Mind into mathematical learning Patterning Result Model
starts from visualizing problems or contexts to conceptualizing mathematical knowledge (patterning) by applying the
habits of describing, experimenting, and tinkering. When students are not able to form concept by using the Pattering
Results Model, they need to apply other habits, such as conjecturing and guessing. Eventually, they may find new
patterns (concepts) from inventions which are the results of applying conjecturing and guessing into the processes of
solving problems. (WPA = word problem application; CU = conceptual understanding; PF = procedural fluency)
Recently, the TPACK has been developed for satisfying the current needs in mathematics education
because TPACK requires to integrate technology, pedagogy, and content knowledge when teachers design their
instruction (Garofalo, Drier, Harper, Timmerman, & Shockey, 2000; Niess, 2008; Thompson & Mishra, 2007).
Furthermore, TPACK provides a framework for evaluating teachers’ knowledge in curriculum and instruction
(Niess, Ronau, Shafer, Driskell, Harper, Johnston, Browning, Özgün-Koca, & Kersaint, 2009). Figure 2 portrays the
integrations in which pre-service teachers engage when they develop their knowledge and understandings in ways
that combine multiple knowledge bases ― technology, content, and pedagogy (Niess, al., 2009). On the left side of
the graphic, the figure highlights the intersection between pedagogy (Mathematical Habits of Mind) and content
(division of fractions). When knowledge of technology (SmartBoard) begins to intersect with Mathematical Habits
of Mind (pedagogy) and division of fractions (content knowledge), the pre-service teachers’ TPACK is designated
so that the pre-service teachers may properly use SmartBoard’s functions to help elementary students learn
mathematical content effectively.
PCK
TPACK
Content
Content
(Divsion of
fractions)
(Division of
fractions)
Pedagogy
(Mathematical
Habits of
Mind)
Pedagogy
(Mathematic
al Habits of
Mind)
Technology
(SmartBoard)
Technology
(SmartBoard)
Recognizing
Accepting
Adapting
Exploring
Advancing
Figure 2. Pre-Service Teachers’ levels as they think and understand toward the interconnection and integration
identified by TPACK. Adapted from “Mathematics teacher TPACK standards and development model” by M. L.
Niess, R. N. Ronau, K. G. Shafer, S. O. Driskell, S. R. Harper, C. Johnston, C. Browning, S. A. Özgün-Koca, and G.
Kersaint, 2009, Contemporary Issues in Technology and Teacher Education. 9(1), p 10. Copyright 2009 by the
Center for Technology and Teacher Education.
Methods
Participants
The participants, from a comprehensive state university in Alaska, were enrolled in an elementary
education initial certification program. Fifteen students participated in this study. Two participants majored in the
sciences, and four students majored in the social studies. Six students majored in English and literature, and the rest
majored in the fine arts. Most of the students have taken two math content courses before they entered the
mathematics method course. For in-depth understanding about the impact of TPACK for these pre-service teachers’
learning and teaching in division of fractions, this study randomly chose a focus group of four pre-service teachers.
Observation Tasks within the workshop
The teaching content for the observation was developed by the researchers, piloted, and then modified. The
workshops were designed to study the PSTs’ knowledge and capacity in performing TPACK in the following
settings: reviewing previous knowledge for learning division of fractions, using SmartBoard to teach Type I & II
division problems of fractions, introducing Mathematical Habits of Mind, using SmartBoard to teach Type III & IV
division problems of fractions, and extending SmartBoard’s function. Participants had a sixty-minute workshop with
every lesson.
Data collection
Each pre-service teacher participated in six workshops within two months. All sessions were videorecorded and transcribed. In the workshop, each PST completed a common collection of tasks and also taught
conceptual materials. With the customized TPACK teaching which integrated SmartBoard and Mathematical Habits
of Mind into the content teaching, PSTs were encouraged to work on in-depth explanations and reflections. The
information enabled researchers to make inferences about the conceptions of division of fractions underlying the
observation and the explanation.
This study analyzed the first section of observation to get an initial understanding of the PSTs’ conceptions
of TPACK in dealing with division of fraction and to identify areas for further investigation and workshop
preparation. In order to test the initial model of PST’s conception, the second section of observation was designed to
explore those areas, clarify points, and develop deeper conception in TPACK. The final observation focused on
PST’s holistic growth in teaching division of fraction with TPACK. This study analyzed the transcripts of these
observations for a more complete description of PSTs’ TPACK.
Analysis
Data analysis began as soon as data was collected and transcribed. This study analyzed each case or event
by examining the PSTs’ actions, explanations, or illustrations in the context of the TPACK workshop. The analysis
focused on the PSTs’ capacity of integrating Mathematical Habits of Mind and SmartBoard into the math
curriculum as well as the PSTs’ development in conceptual understanding of the division of fractions. Throughout
the workshop, the researchers constantly developed and assessed capacity of the PSTs’ conceptual understanding
and TPACK application towards division of fractions. After several sections had been observed and transcribed,
categories emerged from PSTs’ conceptual understanding and TPACK statement. After completing all the
observation sessions, the researchers reviewed and organized sessions and refined initial categories until all sessions
were assigned.
Overall, the analysis was executed through an iterative process in which the researchers constructed and
refined the categories through the procedures of description, categorization, and connection. To establish intercoder
reliability, an independent researcher coded all the data after the framework had been established, and all
observation sections had been categorized, agreeing with 83% of the original codes. The difference in the coding
was resolved through discussion.
Results
Learning concepts
This study found that prior to the workshop, PSTs’ concepts regarding fractions and fractional operation
were not clear. In addition, PSTs preferred or only knew to use the algorithm of invert-multiplication to solve the
division problems of fractions. However, they had difficulty explaining the procedures of the algorithm, such as why
there is a need to change sign, invert the divisor, or multiply the dividend and divisor. During the workshop,
participants improved in making connections between models and mathematical sentences (conceptual
understanding) by integrating Mathematical Habits of Mind to solve type I and II problems. However, creating
various models to establish the meaning of division of fractions was more challenging for them. At the end of
workshop, pre-service teachers started to learn type III & IV problems. The study found that participants needed
more time to solve this kind of problem because it requires the habits of conjecturing, guessing, and inventing to
solve.
Teaching concepts
Initially, PSTs tended to use the SmartBoard as the white board to teach the division of fractions
(technology functioning as replacement). After learning how to use the SMART MATH tools in SmartBoard
through workshops, PSTs were able to create models effectively for teaching division of fraction (technology
functioning as amplification). In the final project, PSTs demonstrated to work on the invented strategies for teaching
division of fractions with a SmartBoard (technology functioning as transformation). Through evaluating the results
of the final project, PSTs’ levels in TPACK (Recognizing, accepting, adapting, exploring, advancing) were decided.
Significant of the study
This study supplies another way to teach division of fractions for math educators by applying TPACK
development Model. According to the outcomes of this study, integrating Mathematical Habits of Mind and the
SmartBoard into the curriculum of division of fractions is helpful to pre-service teachers. The model assists in the
organization of lessons or curriculum in a way that educators can guide their students to understand mathematical
problems by conceptual understanding. Based on our findings, we confirmed that SMART MATH Tools is a
powerful tool for pre-service teachers to create accurate and diverse models. The effectiveness of the models
depends on the teachers’ previous knowledge in the division of fractions. This study also suggests that learning and
teaching Mathematical Habits of Mind improves learners’ conceptualization. Mathematical Habits of Mind
promotes the development of learners’ conceptual understanding by habits such as patterning, describing,
visualizing, tinkering, etc., during the problem solving process. Working together with these habits, the SmartBoard
may properly function as replacement, amplification, and transformation.
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