This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R³ and to their flexibility in the use of different representations.
Nadia Hardy has written an article called Students’ perceptions of institutional practices: the case of limits of functions in college level Calculus courses. The article has recently been published online in Educational Studies in Mathematics. Here is the abstract of her article:
This paper presents a study of instructors’ and students’ perceptions of the knowledge to be learned about limits of functions in a college level Calculus course, taught in a North American college institution. I modeled these perceptions using a theoretical framework that combines elements of the Anthropological Theory of the Didactic, developed in mathematics education, with a framework for the study of institutions developed in political science. While a model of the instructors’ perceptions could be formulated mostly in mathematical terms, a model of the students’ perceptions included an eclectic mixture of mathematical, social, cognitive, and didactic norms. I describe the models and illustrate them with examples from the empirical data on which they have been built.
- Mathematics of currency and exchange: arithmetic at the end of the thirteenth century, by Norman Biggs
- Formulating figurate numbers, by Janet L. Beery
- Mathematics goes ballistic: Benjamin Robins, Leonhard Euler, and the mathematical education of military engineers, by Janet Heine Barnett
- Bernt Michael Holmboe (1795–1850) and his mathematics textbooks, by Andreas Christiansen
Yesterday, I presented an article by three Swedish scholars about mathematical reasoning when using digital tools in the mathematics classroom, and today I follow up with an article about the Potential scenarios for Internet use in the mathematics classroom. The article is written by Marcelo C. Borba, and it was published online in ZDM on Friday. Here is the abstract of Borba's article:
Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.
Thanks to Alexander G. Rivadeneira for pointing me to this story :-)
Here is the abstract of Boyland's article:
This article focuses on the relationship between social justice, emotionality and mathematics teaching in the context of the education of prospective teachers of mathematics. A relational approach to social justice calls for giving attention to enacting socially just relationships in mathematics classrooms. Emotionality and social justice in teaching mathematics variously intersect, interrelate or interweave. An intervention, using creative action methods, with a cohort of prospective teachers addressing these issues is described to illustrate the connection between emotionality and social justice in the context of mathematics teacher education. Creative action methods involve a variety of dramatic, interactive and experiential tools that can promote personal and group engagement and embodied reflection. The intervention aimed to engage the prospective teachers with some key issues for social justice in mathematics education through dialogue about the emotionality of teaching and learning mathematics. Some of the possibilities and limits of using such methods are considered.
The general background of this study is an interest in how digital tools contribute to structuring learning activities. The specific interest is to explore how such tools co-determine students’ reasoning when solving word problems in mathematics, and what kind of learning that follows. Theoretically the research takes its point of departure in a sociocultural perspective on the role of cultural tools in thinking, and in a complementary interest in the role of the communicative framing of cognitive activities. Data have been collected through video documentation of classroom activities in secondary schools where multimedia tools are integrated into mathematics teaching. The focus of the analysis is on cases where the students encounter some kind of difficulty. The results show how the tool to a significant degree co-determines the meaning making practices of students. Thus, it is not a passive element in the situation; rather it invites certain types of activities, for instance iterative computations that do not necessarily rely on an analysis of the problems to be solved. For long periods of time the students’ activities are framed within the context of the tool, and they do not engage in discussing mathematics at all when solving the problems. It is argued that both from a practical and theoretical point of view it is important to scrutinize what competences students develop when using tools of this kind.
- Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions, by Ji-Won Son and Sandra Crespo. Abstract: Recognizing meaning in students’ mathematical ideas is challenging, especially when such ideas are different from standard mathematics. This study examined, through a teaching-scenario task, the reasoning and responses of prospective elementary and secondary teachers to a student’s non-traditional strategy for dividing fractions. Six categories of reasoning were constructed, making a distinction between deep and surface layers. The connections between the participants’ reasoning, their teaching response, and their beliefs about mathematics teaching were investigated. We found that there were not only differences but also similarities between the prospective elementary and secondary teachers’ reasoning and responses. We also found that those who unpacked the mathematical underpinning of the student’s non-traditional strategy tended to use what we call “teacher-focused” responses, whereas those doing less analysis work tended to construct “student-focused” responses. These results and their implications are discussed in relation to the influential factors the participants themselves identified to explain their approach to the given teaching-scenario task.
- Working with mathematics teachers and immigrant students: an empowerment perspective, by Núria Planas and Marta Civil. Abstract: This article centers on a professional development project with a group of high school mathematics teachers in Barcelona. The eight participating teachers taught in low-income schools with a high percentage of immigrant students. Our model of professional development is based on the involvement of the teachers as co-researchers of their local contexts and practices. In this approach, our concept of social justice is tied to the notion of empowerment, both for teachers and for their immigrant students. Our analysis of data from twelve sessions with the teachers shows the development of a shared awareness of their local situation that leads to their questioning of their practices followed by a reconstruction of those. Teachers worked together to move from talking to action. Our analysis of data from the implementation of one lesson in a classroom shows that action, and illustrates signs of empowerment in the teacher and the students, such as students’ challenging of aspects of the task and taking on a more participatory role and the teacher’s reflection on the overall experience.
- Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge, by Jon D. Davis. Abstract: This study examines the influence of reading and planning from two differently organized mathematics textbooks on prospective high school mathematics teachers’ pedagogical content knowledge and content knowledge of exponential functions. The teachers completed a pretest and two posttests. On the pretest, the teachers possessed an incomplete understanding of content and pedagogical content knowledge related to exponential functions. The teachers’ understanding of how to translate from table to closed-form and recursive equations grew as a result of their use of the Mathematics: Modeling Our World textbook, while the Discovering Algebra textbook appeared to be more beneficial in terms of pedagogical content knowledge. Teachers read from the student lessons in both textbooks, but read differently from the sections of both textbooks intended for the teacher. They focused more on the purpose of the Mathematics: Modeling Our World lesson and more on the places where students might experience difficulties in the margins of the Discovering Algebra lesson. The teachers’ learning was influenced by their own personal characteristics (e.g., previous textbook experiences) as well as textbook qualities (e.g., organization).
In most teacher-preparation programs, the preservice students are faced with some sort of field experiences (in Norway we call this "practice", or "praksis" in Norwegian). A focus on the quality of field experiences received a lot of attention when U.S. teacher education was reformed in the 1980s, and there were several recommendations indicating that preservice teachers should have more "authentic experiences to prepare them to handle the complexity and challenges of the school context" (p. 124). Several teacher education programs, including ours at the University of Stavanger (Norway) have strong emphasis on field experiences. According to Santagata et al., two assumptions are underlying:
- "exposure to examples of teaching creates learning opportunities for prospective teachers"
- "through field experiences preservice teachers meld theory into practice" (ibid.).
The authors of this article propose the use of videos of classroom instruction as an alternative approach. Videos can be studied over and over, and this allows for a deeper and more reflected analysis than during ordinary "live observations". The authors argue: "Teaching is a cultural activity, and cultural routines are more easily unveiled when the teaching process is slowed down and critically analyzed" (p. 125).
The use of video is not only "a means to expose preservice teachers to specific behaviors to be imitated" but it is also (or is becoming) "a tool for the development of teachers' professional judgment" (p. 126). In the article they report from two studies in Italy, where videos from the TIMSS 1999 Video Study (report) have been used in preservice teacher education. I will not go into the results from these studies here, but I recommend reading the entire article for further information!
Santagata, R., Zannoni, C., and Stigler, J. (2007). The role of lesson analysis in pre-service teacher education: an empirical investigation of teacher learning from a virtual video-based field experience. Journal of Mathematics Teacher Education, 10(2):123-140.
A recent assessment of mathematics performance around the world ranked the United States twenty-eighth out of forty countries in the study. When the level of spending was taken into account, we sank to the very bottom of the list. We are falling rapidly behind the rest of the developed world when it comes to math education—and the consequences are dire.
In this straightforward and inspiring book, Jo Boaler, a professor of mathematics education at Stanford for nine years, outlines concrete solutions that can change things for the better, including classroom approaches, essential strategies for students, and advice for parents. This is a must-read for anyone who is interested in the mathematical and scientific future of our country.
THE MONTANA MATHEMATICS ENTHUSIAST
Vol.6, No.3, July 2009
TABLE OF CONTENTS
0. THE JOURNAL (WHEEL) KEEPS ON TURNING
Bharath Sriraman (USA)
1. TWO APPLICATIONS OF ART TO GEOMETRY
Viktor Blåsjö (Sweden/USA)
2. INTUITIONS OF "INFINITE NUMBERS": INFINITE MAGNITUDE VS. INFINITE REPRESENTATION
Ami Mamolo (Canada)
3. ON THE USE OF REALISTIC FERMI PROBLEMS FOR INTRODUCING MATHEMATICAL MODELLING IN SCHOOL
Jonas Bergman Ärlebäck (Sweden)
4. MATHEMATICAL BEAUTY AND ITS CHARACTERISTICS- A STUDY ON THE STUDENT'S POINT OF VIEW
Astrid Brinkmann (Germany)
5. AN APPLICATION OF GRÖBNER BASES
Shengxiang Xia and Gaoxiang Xia (China)
6. SMALL CHANGE - BIG DIFFERENCE
Ilana Lavy and Atara Shriki (Israel)
7. MATHEMATICAL CURIOSITIES ABOUT DIVISION OF INTEGERS
Jérôme Proulx and Mary Beisiegel (Canada)
8. HELPING TEACHERS UN-STRUCTURE: A PROMISING APPROACH
Eric Hsu, Judy Kysh, Katherine Ramage, and Diane Resek (USA)
9. WHO CAN SOLVE 2x=1? AN ANALYSIS OF COGNITIVE LOAD RELATED TO LEARNING LINEAR EQUATION SOLVING
Timo Tossavainen (Finland)
10. IF MATHEMATICS IS A LANGUAGE, HOW DO YOU SWEAR IN IT?
Dave Wagner (Canada)
11. FROM TRAPEZOIDS TO THE FUNDAMENTAL THEOREM OF CALCULUS
William Gratzer and Srilal Krishnan (USA)
12. GRAPH ISOMORPHISMS AND MATRIX SIMILARITY: SWITCHING BETWEEN REPRESENTATIONS
Thierry Dana-Picard (Israel)
13. SUM OF "N" CONSECUTIVE INTEGERS
Steve Humble (UK)
14. THE CONTRIBUTIONS OF COMPREHENSION TESTS TO COGNITIVE AND AFFECTIVE DEVELOPMENT OF PROSPECTIVE TEACHERS: A CASE STUDY
Yüksel Dede (Turkey)
15. CUBISM AND THE FOURTH DIMENSION
Elijah Bodish (Missoula, Montana)
16. WHAT'S ALL THE COMMOTION OVER COMMOGNITION? A REVIEW OF ANNA SFARD'S THINKING AS COMMUNICATING
For more information (the website is in Norwegian only) you can contact me (reidar.mosvold_AT_uis.no) or my colleague Raymond Bjuland (raymond.bjuland_AT_uis.no). Please note that the deadline for application is June 29, 2009!
As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.
Amy J. Hackenberga and Erik S. Tillema have written an article entitled Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The article was published online in The Journal of Mathematical Behavior on Thursday. Here is the abstract of their article:
This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students’ construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme. These findings contribute to previous research on students’ construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered.
Hackenberg, A. J., & Tillema, E. S. (n.d.). Students' whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, In Press, Corrected Proof. doi: 10.1016/j.jmathb.2009.04.004.
Huseyin Bahadir Yanika and Alfinio Flores have written an article called Understanding rigid geometric transformations: Jeff's learning path for translation. The article has recently been publishedin The Journal of Mathematical Behavior. Here is the abstract of their article:
This article describes the development of knowledge and understanding of translations of Jeff, a prospective elementary teacher, during a teaching experiment that also included other rigid transformations. His initial conceptions of translations and other rigid transformations were characterized as undefined motions of a single object. He conceived of transformations as movement and showed no indication about what defines a transformation. The results of the study indicate that the development of his thinking about translations and other rigid transformations followed an order of (1) transformations as undefined motions of a single object, (2) transformations as defined motions of a single object, and (3) transformations as defined motions of all points on the plane. The case of Jeff is part of a bigger study that included four prospective teachers and analyzed their development in understanding of rigid transformations. The other participants also showed a similar evolution.
Yanik, H. B., & Flores, A. (n.d.). Understanding rigid geometric transformations: Jeff's learning path for translation. The Journal of Mathematical Behavior, In Press, Corrected Proof. doi: 10.1016/j.jmathb.2009.04.003.
The aim of this study is to investigate the efficiency of cooperative learning on preschoolers’ verbal mathematics problem-solving abilities and to present the observational findings of the related processes and the teachers’ perspectives about the application of the program. Two experimental groups and one control group participated in the study. Results found that preschoolers in the experimental groups experienced larger improvements in their problem-solving abilities than those in the control group. Findings also revealed that the cooperative learning method can be successfully applied in teaching verbal mathematics problem-solving skills during the preschool period. The preschoolers’ skills regarding cooperation, sharing, listening to the speaker and fulfilling individual responsibilities in group work improved. The teachers’ points of view also supported these findings.
Tarim, K. (2009). The effects of cooperative learning on preschoolers’ mathematics problem-solving ability. Educational Studies in Mathematics. doi: 10.1007/s10649-009-9197-x.
This study investigates elementary school children’s flexible use of mental calculation strategies on additions and subtractions in the number domain 20–100. Sixty third-graders of three different mathematical achievement levels individually solved a series of 2-digit additions and subtractions in one choice and two no-choice conditions. In the choice condition, children could choose between the compensation (56 + 29 = ?; 56 + 30 = 86, 86 − 1 = 85) and jump strategy (56 + 29 = ?; 56 + 20 = 76, 76 + 9 = 85) on each item. In the two no-choice conditions, children had to solve each item with either the compensation or the jump strategy. The results demonstrated that children of all achievement levels spontaneously applied both the compensation and the jump strategy to solve the items from the choice condition. Furthermore, they all executed the compensation strategy equally accurately, but faster than the jump strategy in the no-choice conditions. Finally, children neither took into account the expected task nor individual strategy efficiency characteristics during the strategy choice process. Results are discussed in terms of recent models of adaptive strategy choices and instructional practices in the number domain 20–100.
Here is the abstract of their article:
Largely absent from the emerging literature on flexibility is a consideration of experts’ flexibility. Do experts exhibit strategy flexibility, as one might assume? If so, how do experts perceive that this capacity developed in themselves? Do experts feel that flexibility is an important instructional outcome in school mathematics? In this paper, we describe results from several interviews with experts to explore strategy flexibility for solving equations. We conducted interviews with eight content experts, where we asked a number of questions about flexibility and also engaged the experts in problem solving. Our analysis indicates that the experts that were interviewed did exhibit strategy flexibility in the domain of linear equation solving, but they did not consistently select the most efficient method for solving a given equation. However, regardless of whether these experts used the best method on a given problem, they nevertheless showed an awareness of and an appreciation of efficient and elegant problem solutions. The experts that we spoke to were capable of making subtle judgments about the most appropriate strategy for a given problem, based on factors including mental and rapid testing of strategies, the problem solver’s goals (e.g., efficiency, error-free execution, elegance) and familiarity with a given problem type. Implications for future research on flexibility and on mathematics instruction are discussed.
In an effort to determine the most efficacious manner to deliver professional development training to early childhood educators, this study investigated the effect of a 2-h workshop followed by side-by-side classroom coaching. Twelve early childhood educators with 4-year degrees teaching in a university child development center participated in the study. The twice weekly classroom observations were analyzed for the use of math mediated language. Results indicate a 56% increase of math mediated language following the professional development; however, the greatest increase (39% increase over professional development condition) occurred during the side-by-side coaching phase of the treatment. These results corroborate previous findings that implementation of teaching strategies presented in professional development trainings can be enhanced by coaching teachers on the use of the strategies.
Aligned with the enhanced international commitment to early childhood education, recognition of the importance of providing young children with opportunities to develop mathematical understandings and skills is increasing. While there is much research about effective mathematics pedagogy in the school sector, less research activity is evident within the early childhood sector. Focused on people, relationships and the learning environment, this article draws on a synthesis of research on effective pedagogical practices to describe effective learning communities that can enhance the development of young children's mathematical identities and competencies. Concerned that the wider synthesis noted limited cross-sector collaboration within the mathematics education community, this article aims to act as a bridge for researchers currently working within the preschool and school sectors. The authors argue that understandings of effective pedagogies that enhance young children's mathematics learning will benefit from more cross-sector research studies.
- Towards new documentation systems for mathematics teachers? by Ghislaine Gueudet and Luc Trouche
- Experiencing equivalence but organizing order, by Amir H. Asghari
- A categorization of the “whys” and “hows” of using history in mathematics education, by Uffe Thomas Jankvist
- Intuitive vs analytical thinking: four perspectives, by Uri Leron and Orit Hazzan
- Using graphing software to teach about algebraic forms: a study of technology-supported practice in secondary-school mathematics, by Kenneth Ruthven, Rosemary Deaney and Sara Hennessy
- Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes, by Robyn Pierce and Lynda Ball
- Learning opportunities from group discussions: Warrants become the objects of debate
K. Weber, C. Maher, A. Powell, & H. Lee
- Transitions among different symbolic generalizations by algebra beginners in a computer intensive environment
M. Tabach, A. Arcavi, & R. Hershkowitz
- Abstraction and consolidation of the limit precept by means of instrumented schemes: The complementary role of three different frameworks
- Signifying “students”, “teachers” and “mathematics”: A reading of a special issue
- On semiotics and subjectivity: A response to Tony Brown’s “signifying ‘students’, ‘teachers’ and ‘mathematics’: a reading of a special issue”.
- Cognitive styles, dynamic geometry and measurement performance.
D. Pitta-Pantazi & C. Christou
- Embodied design: Constructing means for constructing meaning
- Constructing competence: An analysis of student participation in the activity system of mathematics classrooms
M. Gresalfi, T. Martin, V. Hand, & J. Greeno
- Teachers’ perspectives on “authentic mathematics” and the two-column proof form
M. Weiss, P. Herbst, & C. Chen
- From arithmetical thought to algebraic thought: The role of the “variable”
E. Malisani & F. Spagnolo
This paper investigates the nature of the interaction between the teacher and students as they worked on different mathematics activities in a single classroom over a 10-month period. Sociocultural theories and the Vygotskian zone of proximal development provide the main framework for examining the teaching and learning processes and explaining the incorporation of a four-phase lesson plan as increasing participation of the teacher and students in the teaching and learning process. Drawing on the analyses of discourse from videotaped lessons and the interviews with the teacher and students, five different types of interactions that emphasized mathematical sense-making and justification of ideas and arguments were identified. Excerpts from transcriptions of such interactions are provided to illustrate the learning practices, either academic or non-academic, that students developed in response to these interactions.
- Exemplary mathematics instruction and its development in selected education systems in East Asia, by Yeping Li and Yoshinori Shimizu
- Mathematics classroom instruction excellence through the platform of teaching contests, by Yeping Li and Jun Li
- How a Chinese teacher improved classroom teaching in Teaching Research Group: a case study on Pythagoras theorem teaching in Shanghai, by Yudong Yang
- Pursuing excellence in mathematics classroom instruction through exemplary lesson development in China: a case study, by Rongjin Huang and Yeping Li
- Characterizing exemplary mathematics instruction in Japanese classrooms from the learner’s perspective, by Yoshinori Shimizu
- In search of an exemplary mathematics lesson in Hong Kong: an algebra lesson on factorization of polynomials, by Ida Ah Chee Mok
- Characteristics of good mathematics teaching in Singapore grade 8 classrooms: a juxtaposition of teachers’ practice and students’ perception, by Berinderjeet Kaur
- Good mathematics instruction in South Korea, by JeongSuk Pang
- Searching for good mathematics instruction at primary school level valued in Taiwan, by Pi-Jen Lin and Yeping Li
- Exemplary mathematics lessons: what lessons we can learn from them? by Ngai-Ying Wong
- Exemplary mathematics lessons: a view from the West, by Susie Groves
- Book review: Joan B. Garfield and Dani Ben-Zvi: Developing students’ statistical reasoning: connecting research and teaching practice, by Jane Watson
- Teaching for social justice: exploring the development of student agency through participation in the literacy practices of a mathematics classroom, by Raymond Brown
- Using social semiotics to prepare mathematics teachers to teach for social justice, by Elizabeth de Freitas and Betina Zolkower
- Mathematics in and through social justice: another misunderstood marriage? by Kathleen Nolan
- How to drag with a worn-out mouse? Searching for social justice through collaboration, by Miriam Godoy Penteado and Ole Skovsmose
- Comparative studies of mathematics teachers’ observable learning objectives: validating low inference codes, by Paul Andrews
- The role of contextual, conceptual and procedural knowledge in activating mathematical competencies (PISA), by César Sáenz
- Prospective elementary teachers’ motivation to participate in whole-class discussions during mathematics content courses for teachers, by Amanda Jansen
- Using the history of mathematics to induce changes in preservice teachers’ beliefs and attitudes: insights from evaluating a teacher education program, by Charalambos Y. Charalambous, Areti Panaoura and George Philippou
- Mathematical enculturation from the students’ perspective: shifts in problem-solving beliefs and behaviour during the bachelor programme, by Jacob Perrenet and Ruurd Taconis