SOUTHEAST ASIAN MATHEMATICS EDUCATION JOURNAL
SOUTHEAST ASIAN MATHEMATICS EDUCATION JOURNAL
VOLUME 2, NO. 2 NOVEMBER 2012 ISSN 2089-4716
Southeast Asian Ministers of Education Organization (SEAMEO)
Regional Centre for Quality Improvement of Teachers and Education Personnel (QITEP) in Mathematics
Yogyakarta, Indonesia
Southeast Asian Mathematics Education Journal 2012, Vol. 2 No. 2
Southeast Asian Mathematics Education Journal (SEAMEJ) is an academic journal devoted to reflect a variety of research ideas and methods in the field of mathematics education. SEAMEJ aims to stimulate discussions at all levels of mathematics education through significant and innovative research studies. The Journal welcomes articles highlighting empirical as well as theoretical research studies, which have a perspective wider than local or national interest. All papers submitted to the Journal are peer reviewed before publication.
The SEAMEJ (ISSN: 2089-4716) is the official journal of SEAMEO QITEP in Mathematics and published yearly in December. Readers wishing to submit manuscripts for publication should refer to instruction notes which can be found on the inside back cover.
All correspondence including comments, suggestions, contributions or other related inquiries should be addressed to:
The Director SEAMEO QITEP in Mathematics Jl. Kaliurang Km. 6, Sambisari, Condongcatur, Depok Sleman, Yogyakarta, Indonesia. Phone: +62(274)889987 Fax: +62(274)887222 Email: qitepinmath@yahoo.com
Southeast Asian Mathematics Education Journal 2012, Vol.2 No.2
International Advisory Panels Mohan Chinnapan
University of South Australia
Philip Clarkson
Australian Catholic University
Lim Chap Sam
Universiti Sains Malaysia
Cheah Ui Hock
SEAMEO RECSAM Malaysia
Noraini Idris Universiti Pendidikan Sultan Idris, Malaysia Paul White
Australian Catholic University
Parmjit Singh
Universiti Technology Mara Malaysia
Michael Cavanagh
MacQuarie University Australia
Jaguthsingh Dindyal
Nanyang University Singapore
Chair Fadjar Shadiq
SEAMEO QITEP in Mathematics
Chief Reviewer Allan Leslie White
University of Western Sydney, Australia
Editorial Board Members Subanar
SEAMEO QITEP in Mathematics
Widodo PPPPTK Matematika Yogyakarta Ganung Anggraeni
PPPPTK Matematika Yogyakarta
Wahyudi
SEAMEO QITEP in Mathematics
Pujiati
SEAMEO QITEP in Mathematics
Sahid
SEAMEO QITEP in Mathematics
Anna Tri Lestari
SEAMEO QITEP in Mathematics
Punang Amaripuja
SEAMEO QITEP in Mathematics
Manuscript Editors Sriyanti
SEAMEO QITEP in Mathematics
Siti Khamimah
SEAMEO QITEP in Mathematics
Rachma Noviani
SEAMEO QITEP in Mathematics
Marfuah
SEAMEO QITEP in Mathematics
Luqmanul Hakim
SEAMEO QITEP in Mathematics
Mutiatul Hasanah
SEAMEO QITEP in Mathematics
Tri Budi Wijayanto
SEAMEO QITEP in Mathematics
Wahyu Kharina Praptiwi
SEAMEO QITEP in Mathematics
Administrative Assistants Rina Kusumayanti
SEAMEO QITEP in Mathematics
Rini Handayani
SEAMEO QITEP in Mathematics
Cover Design & Typesetting Joko Setiyono
SEAMEO QITEP in Mathematics
Suhananto
SEAMEO QITEP in Mathematics
Eko Nugroho
SEAMEO QITEP in Mathematics
Tika Setyawati
SEAMEO QITEP in Mathematics
Febriarto Cahyo Nugroho
SEAMEO QITEP in Mathematics
Southeast Asian Mathematics Education Journal ISSN 2089-4716 2012, Vol. 2 No. 2
CONTENTS
Wahyudi & Allan Leslie White
1 Editorial Xingfeng Huang, Jinglei Yang, 3 An Experienced Chinese Teacher’s
Bingxing Tang, Lingmei Gong, Zhong Strategies In Teaching Mathematics:
Translation of Quadratic Functions Paul White, Sue Wilson, & Michael
Tian
11 Teaching for Abstraction: Teacher
Mitchelmore
Learning Catherine Attard 31 Transition from Primary to Secondary
School
Mathematics: Students’
Perceptions
Sue Wilson & Steve Thornton
45 Bibliotherapy: A Framework for Understanding Pre-Service Primary
Teachers’ Affective Responses to Learning and Teaching Mathematics
S. Kanageswari Suppiah Shanmugam 61 IntroducingComputer Adaptive Testing & LeongChee Kin
to a Cohort of Mathematics Teachers: The Case of Concerto
Allan Leslie White 75 What Does Brain Research Say about Teaching and Learning Mathematics?
Ida Karnasih & Wahyudi 89 Exploring Student Perceptions on Teacher-Students Interaction
and Classrooms Learning Environments in Indonesian Mathematics Classrooms
Southeast Asian Mathematics Education Journal 2012, Vol. 2 No. 2
Editorial
This is the second edition of the South East Asia Mathematics Education Journal (SEAMEJ) which is an academic journal devoted to publishing a variety of research studies and theoretical papers in the field of mathematics education. SEAMEJ seeks to stimulate discussion at all levels of the mathematics education community. SEAMEJ aims to eventually publish an edition twice a year, in June and December.
SEAMEJ is supported by the Southeast Asian Ministers of Education Organization (SEAMEO), Centre for Quality Improvement of Teachers and Education Personnel (QITEP) in Mathematics situated in Yogyakarta Indonesia. Launched on July 13, 2009, there are now three QITEP SEAMEO Centres for Quality Improvement of Teachers and Education Personnel in Indonesia. One centre is in Mathematics (Yogyakarta), one in Science (Bandung) and one in Languages (English - Jakarta).
The first edition was produced using revised papers from the first International Symposium of QITEP Mathematics in November 2011, where a number of paper presenters were approached to submit their reworked papers to this journal. In this issue we are proud to state, there are papers that have been submitted by researchers from a number of countries. We hope that trend this will continue and swell as the journal becomes widely read and enable us to meet our aim of two editions in one year.
In this issue we begin with a paper that provides some insights into the mathematics teaching in Shanghai China. The paper describes the struggle of a teacher and tends to concentrate more upon the teaching issues and less upon the research issues. While researchers may not find all the information they would like, nevertheless, as this journal seeks to serve both teachers and researchers, this paper deserves to be widely read. The papers that follow cover a wide range of issues and perspectives and include research into: translating concrete understanding to the abstract by students and how to help teacher achieve this end; a further elaboration of a longitudinal study of Australian transition years and school student engagement; a further elaboration of bibliotherapy with a framework for use with pre- service teachers; a report on a professional learning workshop using a computer adaptive assessment program; the implications that brain research has for the teaching and learning of mathematics, and finishing with a study exploring on psychosocial learning environment in Indonesian mathematics classrooms. We are very thankful for this early support.
As this is only the second edition we are still refining our processes and so we wish to apologise if we have made errors or omissions. We welcome feedback and suggestions for improvement, but most of all, we welcome paper contributions.
The Journal seeks articles highlighting empirical as well as theoretical research studies, particularly those that have a perspective wider than local or national interests. All contributions to SEAMEJ will be peer reviewed and we are indebted to those on the International Advisory Panel for their support.
Wahyudi Allan L White
Southeast Asian Mathematics Education Journal 2012, Vol. 2 No. 2, 3 - 10
An Experienced Chinese Teacher’s Strategies In Teaching Mathematics: Translation of Quadratic Functions
Xingfeng Huang, Jinglei Yang, Bingxing Tang, Lingmei Gong, Zhong Tian Department of Mathematics, Changshu Institute of Technology, Jiangsu, China. <xfhuang0729@gmail.com>
Abstract
The study selected the topic of translation of quadratic functions. In order to explore some effective instructional strategies to help students understand this topic, an experienced teacher was chosen for a case study. Based on lesson observation and semi-structure interviews, this study found that the teacher employed various strategies to facilitate students understanding of translations of quadratic functions.
Keywords experienced Chinese teacher; strategies; translation of quadratic function
Introduction
With the background of Chinese curriculum innovation, how do mathematics teachers apply teaching strategies to classroom practice? This issue aroused researchers’ interest. A lot of mathematics public lessons (in Chinese 公开课 ) attracted their research (Huang, 2009; Huang & Fan, 2009; Li & Li, 2009, Huang & Li, 2009). The purposes of these studies were not only to promote policy-makers to understand the implementation of curriculum innovation, but also to encourage teachers to have opportunities to learn while reflecting on their own teaching (Yu, 2002; Wong, 2009). However, there has been criticism of the model lessons. Some people argued that public lessons looked like certain shows which are pretty but not practical and that these lessons did not represent the real matter (Qian, 2007). With this in mind, it became important to pay special attention to teachers’ strategies in their routine classroom practice. This study focuses on a junior school teacher with 10 years of teaching experience, and explores his teaching strategies in his routine classroom.
Focus On A Challenging Topic
As the evaluation of the teacher, as well as the evaluation of classroom teaching strategies, it is often rewarding if he completes a challenging task. It is a good opportunity for the teacher to exhibit his instructional wisdom, when he is given challenging tasks.
For secondary students, function is a difficult mathematics concept (Leinhardt, Zaslavsky, & Stein, 1990; Sajka, 2003; Vinner & Dreyfus, 1989). From grade 7 to 9, the curriculum introduces the specific function of three types: linear function, inverse function and quadratic function (Ministry of Education, 2001). The results show that students encounter a lot of cognitive obstacles in learning quadratic functions (Zaslavsky, 1997). Textbooks of grade 7 to 9 often include this topic, which is finally introduced in the algebraic
field. In the chapter of quadratic function, translation of y=ax 2 graph is an important
An Experienced Chinese Teacher’s Strategies In Teaching Mathematics: Translation of Quadratic Functions
component. The translation of quadratic functions includes vertical, horizontal and compound ones. In order to understand translation of quadratic function, first we must understand the concept of quadratic function. Because any translation of quadratic function is, by nature, an operation, students should think of quadratic function as an operational object. The order of instruction also brings obstacles to students’ learning because they are usually taught vertical translation ahead of horizontal translation (Zazics, Liljedahl, & Gadowsky, 2003). Vertical translation of quadratic functions coincides with students’ intuition. For example, students are
asked to translate y=x 2 to y=x 2 +1. Comparing the two function expressions, if they add a positive one to x 2 , the graph of y=x 2 will be moved up one unit in the positive direction along axis y. However, horizontal translation is against intuition. If y=x 2 is supposed to be translated to y=(x+1) 2 , x in the former expression is also added into a positive one, then the graph moved left toward in the negative direction along the axis x. Since it is so different from vertical translation, it would be quite difficult for students to understand the concept of horizontal translation. A study by Eisenberg and Dreyfus (1994) shows it would be hard for students to understand horizontal translation. Baker, Hemenway, and Trigueros (2000) used APOS theory to explain students’ learning difficulty. The vertical action is operated directly on a quadratic function, while the actions in horizontal are different, in which two operations are included, first on the variable, and then on the function.
Research Question
Recently some studies began to investigate Chinese curriculum innovation influences on mathematics classroom teaching (Huang, 2009; Huang & Fan, 2009; Li & Li, 2009, Huang & Li, 2009). The main concern of these studies was to observe some excellent mathematics lessons, which were usually isolated, in which teachers were not observed in a structured way. In order to overcome the deficiencies of these studies, this study was grounded in curriculum innovation and conducted a series of successive observations of routine lessons, in which the focus was on the experienced teacher’s strategies. In order to promote an understanding of the teacher’s strategies, we chose a challenging topic of translation of quadratic functions. The research question was: What strategies does the experienced teacher use to help students understand the concept of translation of quadratic functions?
Methodology
Participant
Mr S graduated from the Department of Mathematics of a Normal university in 1999. He has 10 years teaching experience in Qin Chuan Junior School. He studied part-time for his Education Master Degree, and now he is preparing his dissertation. Qin Chuan Junior School covers about 66 acres, located in the west of Yucheng, which is a small city in the southern
Xingfeng Huang, Jinglei Yang, Bingxing Tang, Lingmei Gong, Zhong, Tian
region of Jiangsu. This school has about 2,500 students divided into three grades from seven to nine. Each grade has 16 classes, and each class has 45 to 50 students mostly from local families.
We invited Mr S to participate in this study, because on the one hand he is an experienced teacher, and on the other hand, he and the researchers have a good relationship so as he can help us to complete the study successfully.
Data collection
From October to November 2009, Mr S’s class of grade 9 was studied. The content of translation of quadratic functions was divided into three lessons, each of which has about 40minutes. In each mathematics classroom, a researcher videoed, and another wrote field notes. After each lesson, the lesson plan was copied, and the teacher was interviewed by the researcher using a semi-structured process where the researcher wrote notes. At the same time, the researcher who had videoed the lesson also recorded the whole process of the interview. Videos of the lessons and interviews were translated into scripts by four assistants.
In the next section a description is presented of how the experienced teacher used effective strategies to help students understand vertical translation of quadratic functions. His strategies have been identified to be similar with the other two lessons, the result of which is set aside for an appropriate occasion.
Results
Review and foreshadowing (in Chinese 铺垫 )
Before the lesson, Mr S copied a mathematics task on the blackboard. The task was: If the quadratic function y=ax 2 and the line y=x-1 have only one common point, then how many points of intersection do the function y=4ax2 and the line y=x+3 have? Mr S pointed out that the quadratic function could be denoted by an algebraic expression, but also could be represented by a graph. Therefore students could access solutions using two perspectives. He said: “We have learned features of function graph for some days, so I hope that you can use the graph features to solve this problem.” Next, the problem solving process was completed under the control of him. He asked his students to draw the two possible graphs of the quadratic function y=ax2. At the same time he drew the two graphs of concave up and down on the blackboard.
The following are the teacher-student interactions. On the blackboard Mr S drew the graph, in which y = ax 2 and y = x-1have only one common point. Then Mr S asked students to stretch the graph of y = ax 2 , and translate the graph of y = x-1 in the same system, so that they constructed the graph of y = 4ax 2 and y = x+3. Finally, the teacher guided his students to solve this problem by setting up an equation system. In this teaching episode, based on the task, the class reviewed the relation between the opening direction and size of a parabola and its coefficient ‘a’ contained in the expression. On
An Experienced Chinese Teacher’s Strategies In Teaching Mathematics: Translation of Quadratic Functions
the other hand, translation of the linear function was employed to foreshadow the translation of the quadratic function.
Actually the teacher attempted to highlight the importance of the graph solution, although he guided his students to solve this problem by algebraic solution later. In the lesson, he gave a cue to his students: "We have learned the graph of a function, so I hope you can use graph features to solve this problem first of all." In the interviews, Mr S emphasized repeatedly that students lacked the idea of combining figures with graph (in Chinese数形结合). It is hard for students to associate the features of a function graph with its algebraic expression, and they felt it was difficult to learn quadratic functions. He also made it clear that he would always stress the idea of combining figures with graphs in teaching quadratic functions. He is convinced that if students generated this idea, they would be assured of understanding the content of quadratic functions, and be able to solve the problems related to it.
Transition
Mr S made a transit to the theme of this lesson, vertical translation of quadratic functions. He asked: "What is the relationship between the quadratic functions y = ax 2 and y = ax 2 +bx+c?" After a moment in silence, he told the students: “Special versus General". Then, Mr S asked his students to classify y = ax 2 +bx+c according to its coefficients whether it is zero or not. Then, he raised a question: "What is the relationship between the
graph of y = ax 2 and y = ax 2 +bx+c?"
In fact, it was found that the classification of the quadratic function was too difficult for the students. Although he gave his students a lot of cues, and gave them time to explore it, they did not finally succeed. The students’ performance in the lesson surprised the teacher. After the lesson he said:
I have spent too much time dealing with classification, which should not be stressed in this lesson. My intention was to enable students to realize the algebraic relation between y = ax 2 and y = ax 2 +c, and pilot them to consider the graph relation between them.
His comment implies that he dealt with the classification of quadratic functions because of two points. Firstly, considering the pedagogy, he wanted to strengthen the coherence of the classroom instruction. Secondly, he emphasized the connection among
y=ax 2 +bx+c, y=ax 2 and y=ax 2 +c, in order to encourage his students to understand the concept of quadratic functions.
Core concept
Conjecture. The instruction focuses on the core concept of the lesson. Mr S posed the problem which should be solved in the lesson. He said: “We have learned the function of y=ax 2 . Today we will explore the features of the function y=ax 2 +c. These look quite alike. As to how obtain y=ax 2 +c, just add c to y=ax 2 . What changes on the graph? ” His students
Xingfeng Huang, Jinglei Yang, Bingxing Tang, Lingmei Gong, Zhong, Tian
answered in unison: “Translate it upward or downward”. It looked like a very obvious matter for them. Mr S invited a student to make the explanation. He asked: “Yun, can you explain?”
Yun: Just like the translation on linear function. Mr S: Can you give us an example? Yun: y=2x, y=2x+1
Yun made an analogy translation of linear function to translation of a quadratic one. Perhaps, the foreshadowing at the beginning of the lesson had an effect on their thinking. So the student could easily connect two type of translation with. Mr S said would like to hear other different answers.
Feng: Take a point on y=ax 2 . Its abscissa is x, while its ordinate is ax 2 .
Mr S: Let（x 1 ,y 1 ）.
Feng: Substitute x 1 in y=ax 2 +c, then get y 1 +c.
Mr S: What is the position relation between the points of (x 1 ,y 1 ）and（x 1 ,y 1 +c）? Feng: Translate upward or downward.
In fact, the teacher emphasized the explanation of the graphical features on the algebraic perspective. In the previous lesson, he had always taken the trouble to lead his students to use point-coordinates to interpret the symmetry of y=ax 2 . In the interview, he said:
Algebraic explanation is the most powerful. When we see a function expression, we should think of its graphical features; when looking at function graph, we should explain it on the algebraic perspective.
It also shows that Mr S always highlighted the idea of combining figures with graphs. In this episode, he first attempted to give students an overall impression on the
translation of quadratic function. He expected that students would preliminarily perceive the translation of a parabola before accessing details. This strategy which is consistent with the view of Gestalt could facilitate students to connect existing cognitive conceptions (translation of linear function) with the overall impression so as to establish the structure which can assimilate specific knowledge.
Depicting points to draw a parabola. Mr S requested his students to complete the
2 mathematical task: Depicting points to draw parabolas of y = ½ x 1 , 2 +1 1 , and 2 y = x y = x -1
2 2 at the same coordinate system, then discussing their features. In the first step, the teacher listed a table on the blackboard, took x =- 3, -2, -1, 0, 1, 2,
3 symmetrically, and found the corresponding value of
y 2 = x in order. He said: “Do you
1 2 think the value of y = x +1 must be calculated in this way? ... Yes, don’t calculate any
longer, as long as the latter’s corresponding value of the y 2 = x plus one. ” When they were
An Experienced Chinese Teacher’s Strategies In Teaching Mathematics: Translation of Quadratic Functions
to find the corresponding value of function y 2 = x -1 , he also asked the students to focus
their attention on the feature: as long as the last function’s value is same from the function
1 2 y = x but add negative one.
2 In the tabulation process, Mr S asked his students repeatedly to pay attention to the
numerical relation among the three functions’ values, in which the same number was taken from. In this way, he intended to help his students to understand the relation among the function graphs.
1 1 The teacher required each student to depict the graphs of
2 , y 2 = x y = x +1 , and
1 2 y = x -1 on the cross-section paper. A few students drew exotic graphs: (1) The three
2 parabolas’ vertices coincided at the origin; (2) Parabolas intersected; (3) Parabolas extended in different directions, but did not intersect. Mr S saw several students drawing intersected parabolas, then required all students to watch the parabolas which he had drawn on the blackboard. He explained: “It is impossible for these parabolas to intersect, because each
1 2 point on the graph of y = x is moved upward one unit, so as to get the graph of y = x +1 .
2 Can the original graph intersect with the new one? No, they cannot. ”
After the lesson, we told Mr S the other errors in drawing the parabolas. He felt surprised that these students drew the three parabolas with the same vertex. He said:
If so, students still have no idea of translation? I already elicited students to conjecture. The purpose is to promote them to generate a overall impression on the translation. And I stressed repeatedly the change of the values in the table. Perhaps their impression on the graph of 2 y = ax is too deep -vertex must be at the origin.
Mr S had confidence that what he had done could enhance students’ understanding of translation. He just contributed a few students’ errors to their prior experience. So we can understand why the teacher employed similar strategies in the other two lessons on translation.
Because the teacher did not consider these errors in drawing before the lesson, his interpretation was confusing. In fact, a graph may intersect with the moved one. For example,
a circle may intersect with the moved one. It was hard for him to make effective strategies to deal with unexpected accidents in classroom. Depicting points in order to draw a parabola can be seen as the first stage of learning as Bruner (1967), namely the enactive mode. With concrete operations, students can take a quadratic function as an object, and construct the concept of translation based on their understanding.
Xingfeng Huang, Jinglei Yang, Bingxing Tang, Lingmei Gong, Zhong, Tian
1 1 Sketch . The teacher required the students to sketch graphs of
2 y 2 = - x , y = - x +1 ,
1 and
- y 2 = x -1 . As far as the teaching arrangement was involved, he explained:
At first, students can familiarize with the knowledge that they have just learned. At second, they have learned the parabolas of up-opening, and now they do an exercise on down- opening. Furthermore, in solving problems we usually sketch rather than depict points.
The teacher’s intention was obvious. On the one hand, he tried to make students understand the concept of translation with variation. On the other hand, he trained students’ sketch skills, which could be used to solve problems. The teacher has a further understanding of the sketch value. He believed that conceiving a graph in the mind is more important than depicting points to draw it. Here, the former means the graph of a quadratic function operated in the mind, which is the second stage of Bruner’s iconic mode, semi-concrete operation. The stage of iconic operation is important for students to understand the abstract concept of translation, the symbolic mode, which is the third learning stage defined by Brunner.
Summary
In classroom interactions, students observed similarities and differences between the graphs of y = ax 2 and y = ax 2 +c , and then generalized the relation between them, so that they completed the third learning stage of symbolic mode. In the instructional process of the core concept, the teacher firstly gave his students an overall impression of the translation process, and then accessed details. Students experienced the specific--semi-specific and semi-abstract-- abstract stage, in which they depicted points to draw a graph, or sketch, and generalized vertical translation of quadratic functions.
After that, there was no time to do other exercises in the class. However, the teacher’s plan had not been completed. His intention before the lesson was to consolidate students’ concept of the translation with some mathematics exercises.
Conclusion
In the lesson, the teacher emphasized the idea of combining figures with graphs, and highlighted the strategy of connecting algebraic representation with visualization. To be specific: (1) A mathematical task was selected to review what the students have learned, and prepare for learning new conception; (2) Foreshadowing and transit ion added to the lesson fluency, and different mathematics structures were connected; (3) the teacher gave students an overall impression of the core concept first, then accessed details; (4) The students learned the conception in a specific--semi-specific and semi-abstract-- abstract process; (5) The teacher intended to consolidate the concept with some mathematics exercises.
An Experienced Chinese Teacher’s Strategies In Teaching Mathematics: Translation of Quadratic Functions
Acknowledgement
This research was supported by Ministry of Education (China) under GOA107014.
Reference
Baker, B., Hemenway, C., & Trigueros, M. (2000). On transformations of basic functions. In:
H. Chick, K. Stacey, & J. Vincent (Eds.), Proceedings of the 12 th ICMI Study Conference on the Future of the Teaching and Learning of Algebra (pp. 41-47). University of Melbourne.
Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press. Huang, X. F. (2009). Research on Mathematics Classroom in Shanghai. Nanning: Guangxi Education Publishing House. Huang, X. F., & Fan, L. H. (2009). Instructional practice in mathematics classroom driven by curriculum reform: a case study of model lesson from the Shanghai Two Round Curriculum Reform. Journal of Mathematics Education, 3, 25-30.
Huang, R., & Li, Y. (2009).Pursuing excellence in mathematics classroom instruction through exemplary lesson development in China: a case study.ZDM-The International Journal on Mathematics Education , 41, 297-309.
Eisenberg, T., & Dreyfus, T. (1994). On understanding how students learn to visualize function transformations. In: E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education 1 , (pp. 45-68). Providence, RI: American Mathematical Society.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Function, graphs, and graphing: Tasks, learning, and teaching. Review of educational Research, 60, 1-64. Li, Y., &Li, J. (2009). Mathematics classroom instruction excellence through the platform of teaching contests. ZDM-The International Journal on Mathematics Education, 41, 263- 277.
Ministry of Education. (2001). Mathematics Curriculum Standards in Compulsory Education . Beijing: Beijing Normal University publishing House. Qian, W. W. (2007). Commentary of several contentious issues about open class.Shanghai Research On Education , 7, 34-37. Sajka, M. (2003). A secondary school students’ understanding of the concept of function: a case study. Educational Studies in Mathematic, 53, 229-254. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.Journal for Research in Mathematics Education, 20 , 356-366. Wong, N. (2009). Exemplary mathematics lessons: What lessons we can learn from them? ZDM -The International Journal on Mathematics Education, 41, 379-384. Yu, P. (2002). Briefly on dual-function of publicly-given class. Shanghai Research On Education1 , 31-33. Zaslavsky, O. (1997). Conceptual obstacles in the learning of quadratic functions.Focus on Learning Problems in Mathematics, 19 , 20–44. Zazics, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of function translation:
obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450.
Southeast Asian Mathematics Education Journal 2012, Vol. 2 No. 2, 11 - 30
Teaching for Abstraction: Teacher Learning
Paul White, Sue Wilson Australian Catholic University <paul.white@acu.edu.au; sue.wilson@acu.edu.au>
Michael Mitchelmore Macquarie University <mike.mitchelmore@mq.edu.au>
Working collaboratively with the researchers, a small team of teachers developed and taught two Grade 6 mathematics lessons based on the Teaching for Abstraction model (White & Mitchelmore, 2010). This paper reports how one teacher learned about the model and implemented it in practice. It was found that she assimilated several key features of the model, such as starting with several embodiments of the target concept and guiding students to look for similarities between them. However, it was more difficult for her to help students abstract and reify the target concept and link it to other mathematical concepts. It was concluded that teachers also need to abstract Teaching for Abstraction, and need more embodiments of it before they can reify and implement an effective model.
Keywords : learning by abstraction, Lesson Study, primary school education, professional growth, teacher learning
Over the past decade, the first and last author have been developing a mathematics teaching model called Teaching for Abstraction which is based on a specific theory of learning mathematics. However, we have repeatedly found that teachers’ ability to comprehend the underlying theory has been a barrier to effective implementation. This paper represents an attempt to identify more closely what aspects of the Teaching for Abstraction model teachers find particularly difficult and to design more effective professional development.
Background and Rationale
Learning by abstraction
One theory of mathematics learning (Dienes, 1963) holds that all elementary concepts are the result of abstraction and generalisation from common experiences. For example, children experience many objects that are basically linear such as the edge of a table and a line drawn with a ruler. They see that two edges of a table top meet at a corner and they mark points on a drawn line to indicate a particular length. However, they also experience objects that are too wide, curved or indistinct to be called a line (such as a path across a field) and other lines that do not meet at a point (such as railway tracks). By recognising the underlying similarity among the first category of experiences, and differences with the second category,
Teaching for Abstraction: Teacher Learning
children may abstract the concepts of line, point and intersection and generalise that two lines never meet in more than one point.
The role of abstraction in this theory of learning was well described by Skemp (1986) as follows:
Abstracting is an activity by which we become aware of similarities ... among our experiences. Classifying means collecting together our experiences on the basis of these similarities. An abstraction is some kind of lasting change, the result of abstracting, which enables us to recognise new experiences as having the similarities of an already formed class. ... To distinguish between abstracting as an activity and abstraction as its end- product, we shall ... call the latter a concept. (p. 21, italics in original)
We call the formation of basic mathematical concepts in this manner empirical abstraction because it is based on experience. Empirical abstraction occurs both inside and outside the classroom. For example, parents often help young children recognise similarities in their everyday experiences (e.g., by teaching them the names of particular classes of objects) and school children frequently look for patterns that could simplify their learning (“Just tell us the rule, miss!”). Such empirical abstraction is almost always superficial. Everyday concepts such as “red” and “building” are based on surface appearances anyway, so they are necessarily superficial. But mathematical concepts are based on deeper similarities and it is important that these deeper similarities be learned (Mitchelmore, 2000) . For example, students must go beyond thinking that a fraction is “one number over another”.
Empirical abstraction is to be contrasted with theoretical abstraction, which is the construction of concepts to fit into a specific mathematical theory (Davidov, 1990). For example, the mathematical concepts of line and point are theoretical abstractions; their only existence is in terms of axioms (such as “two lines never meet in more than one point”) that define relations between them and other geometrical concepts. Reasoning about theoretical points and lines must be based on such precise axioms, and no appeal can be made to the imprecise points and lines of experience. Theoretical abstraction plays an extremely important role in mathematics.
We have argued (Mitchelmore & White, 2004) that most abstract mathematical theories are constructed to model empirical abstractions. For example, theoretical lines and points only represent real linear objects and points ⎯there is no such things in our world as a perfectly straight line or a dimensionless dot. It is not until one reaches the refined atmosphere of research mathematics that theories are invented for their own sake, but even they can be traced back through a succession of theories to some aspect of our experience.
Paul White, Sue Wilson, & Michael Mitchelmore
Therefore, if students are to appreciate the value of theoretical mathematics and be able to apply abstract theory to concrete situations, they need to have a sound understanding of the empirical abstractions on which the theoretical abstractions are based.
Teaching For Abstraction
Teaching for Abstraction (White & Mitchelmore, 2010) attempts to build on children’s natural tendency to seek similarities and make rules in order to assist them to abstract mathematical concepts. The Teaching for Abstraction model consists of four phases:
1. Familiarity. Students explore a variety of contexts where a concept arises, in order to form generalisations about individual contexts and thus become familiar with the underlying structure of each context.
2. Similarity. Teaching then focuses on helping students recognise the similarities and differences between the underlying structures of these various contexts.
3. Reification. The general principles underpinning the identified similarities are drawn out, and students are supported to abstract the desired concept into a mental object that can be operated on in its own right.
4. Application. Students are then directed to new situations where they can use the concept.
In this model, teachers start by carefully selecting situations known to the students that embody a significant mathematical concept and ensure that students understand the mathematics within each situation. They then deliberately focus students’ attention on the underlying similarity between those situations and help them formulate that similarity in abstract terms. Finally, they help students learn how to operate with those abstract concepts and apply them to solve problems in other situations that embody that concept. Several examples of specific skills and understandings at each level of the model are given in White and Mitchelmore (2010).
The model is the reverse of the traditional ABC (Abstract Before Concrete) method of teaching mathematics (Mitchelmore & White, 2000). As one teacher put it, “It is the opposite of what we’re doing in school now. It is starting with the blurred and being revealed. Backwards. Not doing specific instructions first, more a thinking thing.”The model has many similarities to several approaches advanced in the mathematics education literature over the past 20 years (encapsulated in such slogans as realistic learning, communities of discourse, and teaching for understanding) in its emphasis on drawing mathematics out of familiar contexts through teacher-guided exploration and reflection. The major difference lies in the
Teaching for Abstraction: Teacher Learning
relation posited between abstract ideas and familiar experience: Concepts are seen as representing what is common among several experiences, rather than as ideas that can be explained or justified through specific experiences. As a result, Teaching for Abstraction uses multiple contexts for each concept, focuses on the relation between them, and constantly links abstract ideas to the several contexts from which they were derived.
Our earliest experiments with the Teaching for Abstraction model involved teaching the concept of angle in Years 3 and 4 (Mitchelmore & White, 2002a, 2002b). Familiar contexts included corners, scissors, body joints, doors, clock hands, and slopes. The first lessons concentrated on ensuring that students understood each of these contexts to the extent that they could represent them graphically (Phase 1: Familiarity). The next lessons involved a variety of matching exercises, both within and between contexts, leading students to recognise that an angle (wherever it occurred) had two arms meeting at a point and that each angle had a particular size (Phase 2: Similarity). Students then completed various activities to develop facility in interpreting and using abstract angle diagrams (Stage 3: Reification) and using them in new situations (Phase 4: Application). There was evidence that many students developed a quite sophisticated concept of angle by as early as age 10.
Subsequent experiments have investigated the teaching of decimals, percentage and ratio by abstraction. Summaries of these experiments and their effectiveness are given in White and Mitchelmore (2010). The investigations have confirmed our conviction that the Teaching for Abstraction model has promise in terms of student learning of key concepts and generalisations. They have also consistently shown the vital significance of teacher learning.
Teacher learning
In the teaching studies we have undertaken, we have provided teachers with detailed lesson outlines but many teachers have found it difficult to implement the Teaching for Abstraction model. Because the model is so different from common teaching practice, we have found it difficult to communicate to teachers despite revising the professional learning activities several times. As a result, many teachers have either followed the materials provided to the letter, not being in a position to adapt them to their particular classroom situation, or they have subverted the whole process and reverted to the ABC method. As a result, the model was not implemented faithfully in several of the classrooms we studied. Nevertheless, most teachers have spoken favourably of the results in their classrooms, and some have expressed the wish for further materials.
Paul White, Sue Wilson, & Michael Mitchelmore
A possible reason for the poor implementation of the model in previous studies is that the teachers were not involved in planning the unit or developing the teaching materials. The literature on teacher learning repeatedly cites the importance of active teacher involvement if innovative ideas are to be accepted (Pegg & Panizzon, 2008). In our investigations, the researchers did all the development and simply presented lesson plans to the teachers. Consequently, the teachers had no real ownership of the experiments. They may even have considered us as outsiders with no knowledge of classroom realities (Jaworski, 2004).
We therefore decided to investigate a different method of implementing Teaching for Abstraction, one in which, instead of presenting teachers with completed teaching materials, we would work collaboratively with teachers to develop Teaching for Abstraction lessons that would better fit their own classroom situations. We hypothesised that such a procedure would lead to greater ownership and thus more faithful implementation, deeper student learning and, most importantly, greater teacher learning than the previous method.
The present study
As a theoretical framework for assessing the impact of this new method of professional development, we call on the Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002). This model contains four domains and various interactions between them, as shown in Figure 1. Note that most interactions are bidirectional. For example, teachers enact a new idea, belief or practice in their classroom and then reflection provides (positive or negative) feedback on the critical features of that innovation and its value to them.
Figure 1 . The Interconnected Model of Professional Growth (reproduced by permission from Clarke & Hollingsworth, 2002).
Teaching for Abstraction: Teacher Learning
As an initial exploratory study, we worked with a small group of teachers in a small regional town to prepare two unconnected lessons. The teachers then taught these lessons in a kind of lesson study mode ( Hart, Alston, & Murata , 2011). In terms of Figure 1, the External Domain was the collaborative lesson preparation and study, the Personal Domain was teachers’ knowledge and acceptance of Teaching for Abstraction, the Domain of Practice was the lessons taught, and the Domain of Consequence consisted of student outcomes from the lessons taught.
Our research question was: Can collaborative lesson development lead to faithful implementation of the Teaching for Abstraction model? More specifically: Which parts of the model are easier or more difficult for teachers to implement? Can student learning be linked to the parts of the model that are implemented? To provide answers to these questions, data were collected on all four components of the model in Figure 1. We report a case study of one of the teachers involved.
Method
Participants
The study took place in Grade 6 at St Joseph’s 1 , a small primary school in regional New South Wales. Initially, three teachers volunteered to participate: David, Bridget, and Uarda. However, David was unwell during our visits and only participated marginally. Also, Bridget was a part-time teacher and had several other calls on her time. Only Uarda participated fully, and she is therefore the focus of this paper. Uarda had been teaching for 5 years, and was enrolled in a master’s degree at the time.
Procedures
The authors paid two 2-day visits to St Joseph’s. Both visits followed the same pattern: The research team (the authors, Uarda, and whichever of the other teachers were available) met for an initial discussion on the afternoon of Day 1 to decide on the aims for the following day. Early on Day 2, the team developed a lesson and prepared the necessary materials. Both sessions were guided by the principle that each participant should contribute their particular expertise in the lesson design. Thus, the authors shared their knowledge of the Teaching for Abstraction Model and their experience of its previous implementations while the teachers contributed their knowledge of the school curriculum and their students.
Paul White, Sue Wilson, & Michael Mitchelmore
Later in Day 2, Uarda taught the experimental lesson while the others observed, after which the team discussed the lesson and modified it as they felt appropriate. Bridget then taught the revised lesson while the others observed. On the second visit, an additional team session was devoted to identifying the strengths and weaknesses of the Teaching for Abstraction model and reflecting on the teacher learning that had occurred.
After the second visit, Uarda indicated that she would attempt to apply the model to her teaching about angles in the following month. After teaching this unit, which we did not observe, she provided the authors with written feedback.
Data collection and analysis
The research team’s discussion sessions were audio recorded and transcribed, but no recordings were made of the lessons. Instead, one member of the research team acted as a lesson recorder, taking detailed notes of the lessons that included time markers for the major transitions. The other members subsequently added their individual observations to these notes.
Because student outcomes have a significant feedback effect on teacher learning (Clusky, 2002), an attempt was also made to assess short-term student learning. The recorder noted students’ comments during the face-to-face teaching and all members of the team circulated and observed students during the small group work, occasionally interacting with them to clarify what they were attempting to do. In addition, the teachers administered a short quiz at the end of the first visit and a short questionnaire at the end of the second visit.
The analysis of how Uarda interpreted and applied the Teaching for Abstraction model, as well as its resulting effectiveness and potential, focussed on the four components shown in Figure 1. Firstly, each author formed an interpretation of each of these components on the basis of their own observations, notes and informal discussions during the site visits. The three authors then cross-validated and synthesised their separate interpretations during extensive discussions, frequently re-examining transcripts and field notes to reach consensus.
Results
The first visit
Initial discussions. At the first meeting of the research team, the authors outlined the Teaching for Abstraction model and gave some examples from their previous research. In particular, they provided the teachers with copies of the instructional materials developed for
a previous Grade 6 percentages investigation (White, Mitchelmore, Wilson, & Faragher,
Teaching for Abstraction: Teacher Learning
2008). They also explained the purpose of the study, and asked teachers for their reaction to the model and its potential in their situation.
The teachers expressed interest in experimenting with the model. Uarda indicated that she regularly trialled novel activities and approaches that she believed might be beneficial to her students. Teaching for Abstraction had a definite resonance for her because she was particularly keen on the use of realistic scenarios and always tried to embed the mathematics she was teaching in contexts that she felt would be familiar to students.
The teachers then outlined a number of topics where they felt their students were having most difficulty, and it was agreed that the next day’s lesson would focus on place value in decimals. It was also agreed that Uarda and Bridget would both teach the same lesson with their own classes, in that order, with time for discussion and revision between the two lessons.
Lesson planning. The teachers reported that students had been taught about decimals but some students were still having difficulty deciding, for example, whether 0.65 was bigger than 0.8. It was decided to focus the lesson on this topic, restricting the content to 1- and 2- place decimals. It was considered that this topic was sufficiently narrow for a single lesson, but that it could nevertheless be of significant value to students. The teachers could extend students’ understanding to other decimals later.