Making Connections: Elementary Teachers' Construction of Division Word Problems and Representations

If teachers make few connections among multiple representations of division, supporting students in using representations to develop operation sense demanded by national standards will not occur. Studies have investigated how prospective and practicing teachers use representations to develop knowledge of fraction division. However, few studies examined primary (K‐3) teachers' learning of contextual division problems, making connections among representations of division, and resolving the ambiguity of representing quotients with remainders. A written post‐course assessment provided evidence that most teachers created partitive division word problems, used a set model without splitting the remainder, and wrote equations with limited success. Post‐course written reflections demonstrated that many teachers developed pedagogical knowledge for helping students make connections among multiple representations, and mathematical knowledge of unit fractions. These findings suggest two areas that have implications for mathematics teacher educators who design professional development courses to facilitate teachers' learning of mathematical content and pedagogical knowledge of division and fraction relationships.     

Student achievement effects of technology-supported remediation of understanding of fractions

Students have difficulty learning fractions, and problems in understanding fractions persist into adulthood, with moderate to severe consequences for everyday and occupational decision-making. Remediation of student misconceptions is hampered by deficiencies in teachers’ knowledge of the discipline and pedagogical content knowledge. We theorized that a technology resource could provide the sequencing and scaffolding that teachers might have difficulty providing. Five sets of learning objects, called CLIPS (Critical Learning Instructional Paths Supports), were developed to provide remediation on fraction concepts. In this article, we describe one stage in a research program to develop, implement and evaluate CLIPS. Two studies were conducted. In Study 1, 14 grade 7–10 classrooms were randomly assigned, within schools, to early and late treatment conditions. A pre-post, delayed treatment design found that CLIPS had no effect on achievement for the Early Treatment group due to unforeseen implementation problems. These hardware and software issues were mitigated in the late treatment in which CLIPS contributed to student achievement (Cohen's d = 0.30). Study 2 was a pre-post, single group replication involving 18 grade 7 classrooms. The independent variable was the number of CLIPS completed. Completion of all five CLIPS contributed to higher student achievement: Cohen's d = 0.53, compared to students who completed none (d = 0.00) or 1–4 CLIPS (d = 0.02). The two studies indicate that a research-based set of learning objects is effective when the full program is implemented. Incomplete sequences deprive students of instruction in one or more constructs linked to other key ideas in the conceptual map and reduce the amount of practice required to remediate student misconceptions.     

Preparing K‐8 Preservice Teachers in a Content Course for Standards‐Based Mathematics Pedagogy

Many K–8 preservice teachers have not experienced learning mathematics in a standards‐based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards‐based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Furthermore, findings are presented from a study on this course that describe students' pre‐ and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards‐based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Reflective Tutoring: Insights into Preservice Teacher Learning

An undergraduate seminar was designed to help preservice teachers focus on students' learning. Preservice teachers planned and conducted weekly tutoring sessions with fourth graders and discussed their experiences in weekly discussions. The author investigated what preservice teachers learned about teaching mathematics from their focus on students' learning of mathematics. The author examined the tasks that preservice teachers posed to children, the questions they asked of children, and the reflections they wrote about their experiences. The article describes what the preservice teachers learned from their experiences and provides insights into their knowledge and skills for developing children's mathematical power.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.     

Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

A process of students and their instructor developing a final closed-book mathematics exam

This article describes a study, from a Canadian technical institute's upgrading mathematics course, where students played a role in developing the final closed-book exam that they sat. The study involved a process where students developed practice exams and solutions keys, students sat each other's practice exams, students evaluated classmates' solutions to the practice exams, and finally the instructor used questions from the practice exams to develop the ‘live’ final exam. Phenomenography is used to analyse interview data and report students' experiences. Through the results, claims are made that students experienced deep approaches to learning and worked as partners in learning, teaching and assessment during the process of developing the final exam with their instructor.     

Transforming the mathematical practices of learners and teachers through digital technology*

ABSTRACT

This article argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: (1) dynamic and graphical tools; (2) tools that outsource processing power; (3) tools that offer new representational infrastructures for mathematics; (4) tools that help to bridge the gap between school mathematics and the students’ world; (5) tools that exploit high-bandwidth connectivity to support mathematics learning; and (6) tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies. Following exemplification of each category, the article ends with some reflections on the progress of research in this area and identifies some remaining challenges.     

Enhancing pre-service teachers’ fraction knowledge through open approach instruction

This study explores whether using the open approach instruction in teaching mathematics has a positive effect for enhancing pre-service teachers’ fraction knowledge. The test consisted of 32 items that were designed to examine pre-service teachers’ procedural and conceptual knowledge of fractions before and after receiving open approach instruction. The study was undertaken among students in four mathematics content and methods courses for the Elementary School Education program in a mid-western public university. The findings show that most of the teachers achieved improved learning outcomes through the open approach instruction.     

Re‐creating a Recipe for Science Instructional Programs: Adding Learning Progressions,Scaffolding, and a Dash of Reading Variety

The purpose of this article is to focus on the development and refinement of a science instructional design program arguing for the feasibility and usability of integrated reading and science instruction as implemented in third‐ and fourth‐grade science classrooms to meet the learning needs of diverse learners. These instructional components are easily inserted into existing programs that build students' science background knowledge and abilities to apply learning through scaffolded activities focused on (1) providing structured opportunities for students to engage in hands‐on activities; (2) increasing vocabulary knowledge and understanding of concept‐laden terms, and (3) reading paired narrative and informational science texts. Extensive research shows that as students transition from third to fourth grade and beyond, they are often challenged in science by new vocabulary coupled with new concepts. Active ingredients of our reconceptualized science instructional design program are narrative informational texts, hands‐on science activities, and science textbook(s).     

Mathematical modelling in university education. An experiment at the University of Oldenburg

This paper suggests that mathematics teacher educators should listen carefully to what their students are saying. More specifically, it demonstrates how from one pre-teacher's non-traditional geometric representation of a unit fraction, a variety of learning environments that lead to the enrichment of mathematics for teaching can be developed. The paper shows how new knowledge may be generated through an attempt to validate an intuitive idea; in other words, how the quest for rigour may serve as a catalyst for the growth of mathematical concepts in the context of K-16 mathematics.     

An Integrated Research on Children's Construction of Meaningful,Symbolic, Partitioning-related Conceptions and the Teacher's Role in Fostering That Learning

A teaching experiment was conducted with two fourth graders to study the co-emergence of teaching and children's construction of fraction knowledge. The children's learning, i.e., modifications in their fraction schemes, was fostered through working on tasks in a computer microworld. The children advanced from thinking about a unit fraction as one of several equal parts in a whole (the equipartitioning scheme) to operating with a unit fraction as a symbolized, iterable part the magnitude of which is based on the numerosity of the partitioned whole (the partitive fraction scheme). The paper interweaves an analysis of children's construction of partitioning-related symbolic conceptions of fractions with an analysis of the teaching—planning and using tasks—that fosters such an advancement by introducing fraction words and numerals in the context of the children's partitioning activities.     

Promoting Percent as a Proportion in Eighth‐Grade Mathematics

The literature provides many and varied suggestions for promoting conceptual understanding of percent and performing percent calculations. The diversity of ideas provides a wide selection but offers little clarity on the true nature of percent. From the premise that percent is fundamentally a proportion, this study incorporated a proportional approach for percent problem solving within an instructional program on percent. Classroom research with eighth‐grade students indicated that the method was readily adopted by students and helped them experience success in percent problem solving, with percent problem solving proficiency maintained over a delayed period. It is hypothesized that the method has the potential to promote students' conceptual knowledge of percent as a proportion and the multiplicative structure of percent, as well as to build proportional knowledge.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

Where do Fractions Encounter their Equivalents? – Can this Encounter Take Place in Elementary-School?

The concept of equivalence class plays a significant role in the structure of Rational Numbers. Piaget taught that in order to help elementary school children develop mathematical concepts, concrete objects and concrete reflection-enhancing-activities are needed. The “Shemesh” software was specially designed for learning equivalence-classes of fractions. The software offers concrete representations of such classes, as well as activities which cannot be constructed without a computer. In a discrete Cartesian system students construct points on the grid and learn to identify each such point as a fraction-numeral (a denominator-numerator pair). The children then learn to construct sets of such points, all of which are located on a line through the origin point. They learn to identify the line with the set of its constituent equivalent fractions. Subsequently, they investigate other phenomena and constructions in such systems, developing these constructions into additional fraction concepts. These concrete constructions can be used in solving traditional fraction problems as well as in broadening the scope of fraction meaning. Fifth-graders who used “Shemesh” in their learning activities were clinically interviewed several months after the learning sessions ended. These interviews revealed evidence indicating initial actual development of the desired mathematical concepts. This revised version was published online in July 2006 with corrections to the Cover Date.     

STUDENT TEACHERS’ UNDERSTANDING OF RATIONAL NUMBER: PART-WHOLE AND NUMERICAL CONSTRUCTS

Previous research has shown that secondary school students’ understanding of fractions is dominated by the part-whole concept to the possible detriment of their understanding of a fraction as a number in its own right. The present paper reports on an investigation into the understanding of intending primary teachers in this area. Four representatives of a cohort of sixty students on a PGCE course specialising in the lower primary age range were asked detailed questions probing their knowledge of fractions. The conclusion was that the part-whole concept dominates. All of the students were familiar with the numerical concept from their work on the PGCE course, but they reverted to the more familiar part-whole ideas in attempting to solve problems.     

A cross-national comparison of reform curricula in Korea and the US in terms of cognitive complexity: the case of fraction addition and subtraction

The overall level of conceptual understanding and mathematical proficiency of students has been a matter of increasing national interest in South Korea. Recently, a new edition of mathematics textbooks aligned with the amendment of the 7th national mathematics curriculum has become available for all elementary grade levels. To characterize the current reform efforts in South Korea, this study examined the quality of the mathematical problems in the current version of the Korean reform textbooks (KM 2) compared with the previous version (KM 1) and one representative US reform curriculum text (EM). Webb’s (Research monograph No. 18: Alignment of science and mathematics standards and assessments in four states. National Institute for Science Education, Madison, 1999) depth of knowledge framework and Son and Senk’s (Educ Stud Math 74(2):117–142, 2010) cognitive expectation feature were employed to examine the kind and level of students’ opportunities to learn along with the type of word problems presented in the three sets of materials. Analysis revealed that the KM 2 provided better opportunities for students to learn fraction addition and subtraction than the KM 1 in terms of the depth and breadth of cognitive complexity. However, there was little difference in addressing and developing the meaning of fraction addition and subtraction through word problems. Moreover, compared with the US reform curriculum materials, the KM 2 provided more problems requiring lower depth of knowledge levels than the US counterpart. Implications of these findings for curriculum developers, textbook and learning materials developers, teachers and future researchers are discussed.     

A framework to categorize students as learners based on their cognitive practices while learning differential equations and related concepts

There are numerous theories that offer cognitive processes of students of mathematics, all documenting various ways to describe knowledge acquisition leading to successful transitions from one stage to another, be it characterized by Dubinsky's encapsulation, Sfard's reification or Piaget's equilibration. We however are interested in the following question. Who succeeds at making the leap and can we describe the attributes that set them apart from the ones that do not? In this article, we offer a framework to categorize students as learners based on their individual approaches towards learning concepts in differential equations and related concepts – as demonstrated by their efforts to resolve a conflict, conserve and rebuild their cognitive structures.