Fostering a student's abstraction of the relationship between parallelogram and trapezoid within quadrilateral hierarchy

This article provides an analysis of a data set coming from a two-phase qualitative study that focused on fostering primary students’ abstraction of interrelations among quadrilaterals (squares, rectangles, parallelograms, rhombuses, trapezoids). The pilot study consisted of work with eight primary students operating at van Hiele level 2 (e.g., understanding quadrilaterals without interrelations). We benefitted from the teaching experiment methodology to develop a task sequence applied in a dynamic geometry environment and a paper-pencil environment to help each participant develop quadrilateral hierarchy at van Hiele level 3 (e.g., understanding quadrilaterals with their interrelations). The main study used a case study approach to investigate two primary students’ progress (Efe and Ayla, age 10). After the pre-interviews, each participant was taught the developed task sequence individually during seven up-to-one-hour teaching sessions, followed by a post-interview. This article only details Efe’s case as he worked on developing the relationship between parallelogram and trapezoid. We analyzed Efe’s data (from the pre-interview, Teaching Session-7, and the post-interview) to describe how the different parts of the task sequence fostered his abstraction of the interrelation between parallelogram and trapezoid as he moved from van Hiele Level 2 to 3. This article provides initial evidence for the classification process

Teaching place value conceptually – Part II

No abstract available

A micro-curricular analysis of unified mathematics curricula in Turkey

This article aims to give a detailed micro-level curricular analysis of the extent to which the intended mathematics curriculum matches the potentially implemented curriculum using the case of Turkey. The article makes inferences about what it means to have a match or mismatch between these two types of curricula. As a result, it is clear that even though there is a close match between the intended and the potentially implemented mathematics curricula, such a match does not seem to be enough to help students to have a solid understanding of targeted mathematical concepts outlined in the overall Turkish curricular standards

An alternative route to teaching fraction division: Abstraction of common denominator algorithm

From a curricular stand point, the traditional invert and multiply algorithm for division of fractions provides few affordances for linking to a rich understanding of fractions. On the other hand, an alternative algorithm, called common denominator algorithm, has many such affordances. The current study serves as an argument for shifting curriculum for fraction division from use of invert and multiply algorithm as a basis to the use of common denominator algorithm as a basis. This was accomplished with the analysis of learning of two prospective elementary teachers being an illustration of how to realize those conceptual affordances. In doing so, the article proposes an instructional sequence and details it by referring to both the (mathematical and pedagogical) advantages and the disadvantages. As a result, this algorithm has a conceptual basis depending on basic operations of partitioning, unitizing, and counting, which make it accessible to learners. Also, when participants are encouraged to construct this algorithm based on their work with diagrams, common denominator algorithm formalizes the work that they do with diagrams

An Alternative Route to Teaching Fraction Division: Abstraction of Common Denominator Algorithm

From a curricular stand point, the traditional invert and multiply algorithm for division of fractions provides few affordances for linking to a rich understanding of fractions. On the other hand, an alternative algorithm, called common denominator algorithm, has many such affordances. The current study serves as an argument for shifting curriculum for fraction division from use of invert and multiply algorithm as a basis to the use of common denominator algorithm as a basis. This was accomplished with the analysis of learning of two prospective elementary teachers being an illustration of how to realize those conceptual affordances. In doing so, the article proposes an instructional sequence and details it by referring to both the (mathematical and pedagogical) advantages and the disadvantages. As a result, this algorithm has a conceptual basis depending on basic operations of partitioning, unitizing, and counting, which make it accessible to learners. Also, when participants are encouraged to construct this algorithm based on their work with diagrams, common denominator algorithm formalizes the work that they do with diagrams

Conceptualization and appreciation of the meaning of assimilation principle by prospective elementary teachers

This study investigated the main tenets of constructivism which is one of the hot topics in education that has recently taken considerable attention in Turkey. It explicated on assimilation principle and offered an instructional design for mental construction of this principle through an application of an action research including the work of 45 prospective elementary teachers. In the article, the designed instructional sequence, its in-class application, and how such instruction affected teachers’ previously gained ideas about teaching and learning were investigated. As a result, helping teachers develop a sense for assimilation principle and its importance in teaching seems to be positively affected by three main activities: (1) Engaging learners in a situation in which they have to feel a need to go back to their prior knowledge and reflect on that situation, (2) Helping learners distinguish between different types of knowledge (social, physical, logicomathematical) and reflection over them, (3) Having learners analyze instructional designs (and their applications) that do not take into consideration the assimilation principle

Working on the same problem - concepts; with the usual subjects - prospective elementary teachers

The purpose of current study was to shed light on what concepts or ideas prospective elementary teachers drew on (and how) when they reasoned about problems related to division of fractions. The participants were 153 senior prospective elementary teachers who registered for a mathematics-methods course either during the 2004-2005 or 2005-2006 academic year. The participants were from a metropolitan city university located in the central part of Turkey. The study was based on an open-ended written assessment item administered during the very first class of the semester before any teaching. One of the applied open-ended questions was: “Write a real-world problem that would be solved or modeled by 2 1/2 ÷ 3/4.” A question like this requires a detailed analysis of the concepts of division and fractions as opposed to getting a single numeric result. Such a question, to some extent, provided an opportunity to investigate these teachers’ reasoning about division and fractions. The results showed that prospective elementary teachers had significant difficulties in thinking about fractions, division, and units

Conceptualising Place Value with Different Bases

Place value plays a crucial role in understanding numbers and operations throughout primary/secondary school mathematics. Pupils mostly use rote procedures to make sense of place value, numbers, and operations. One way to teach place value is to use different number systems rather than limiting pupils' experiences to base ten. This workshop aims to engage participants in a curricular piece that uses the "grouping" idea in the context of different base systems. The audience will have the opportunity to investigate the place value concept (using physical or virtual sticks and unit cubes) and reflect on its use for arithmetic operations

Conceptualization and appreciation of the meaning of assimilation principle by prospective elementary teachers

This study investigated the main tenets of constructivism which is one of the hot topics in education that has recently taken considerable attention in Turkey. It explicated on assimilation principle and offered an instructional design for mental construction of this principle through an application of an action research including the work of 45 prospective elementary teachers. In the article, the designed instructional sequence, its in-class application, and how such instruction affected teachers’ previously gained ideas about teaching and learning were investigated. As a result, helping teachers develop a sense for assimilation principle and its importance in teaching seems to be positively affected by three main activities: (1) Engaging learners in a situation in which they have to feel a need to go back to their prior knowledge and reflect on that situation, (2) Helping learners distinguish between different types of knowledge (social, physical, logicomathematical) and reflection over them, (3) Having learners analyze instructional designs (and their applications) that do not take into consideration the assimilation principle