Construction of Concepts Images from Mathematical Models Obtained with the Tracker Software

This work aims to investigate the potential of combining the Mathematical Modeling methodology with the use of the software Tracker with undergraduate students in Mathematics in the reconstruction of physical models for the construction of function concepts. For this, a qualitative and exploratory study of bibliographic and experimental character was carried out, involving field activities. Our results point to a greater understanding in the construction of the models, since the mathematical models can be recreated from the correlation to physical models already validated. We also verified that the Tracker software enables an excellent visual perception of different concept images.

The Language of “Rate of Change” in Mathematics

Language is an essential aspect of teaching and learning mathematics. It is necessary for communication, the transmission of concepts and ideas, and the formation of meaning of mathematical concepts. In mathematics, besides symbols, which are usually common among different languages, words and expressions are used, which may invoke different concept images to students in various languages. Some words are used in mathematics and in everyday language with different meanings, while others are used only in mathematics or in mathematics and other disciplines in similar but non-identical ways. In Mathematical Analysis, the used vocabulary is gradually enhanced, and the concepts are defined in a more formal way. In the current study, the language used regarding mathematics of change is examined, focusing on “rate of change”, and its relation to misconceptions among students.

The Language of “Rate of Change”

Language is an essential aspect of teaching and learning mathematics. It is necessary for communicating, transmission of concepts and ideas, and formation of meaning of mathematical concepts. In mathematics, besides symbols, which are usually common in different languages, words and expressions are used, which may invoke different concept images to students in various languages. Some words are used in mathematics and in everyday language with different meanings, while others are used only in mathematics or in mathematics and other disciplines in similar but non-identical ways. In Mathematical Analysis, the used vocabulary is gradually enhanced, and the concepts are defined in a more formal way. In the current study, the language used regarding mathematics of change is examined, focusing on rate of change and in relation to misconceptions of students.

Mathematics education undergraduates’ personal definitions of the notion of angle of contiguity in Kinematics

The concept of a mathematical definition causes severe difficulties among students during problem solving and proving activities. Students’ difficulties with the use of mathematical definitions often arise from the fact that students are often given those definitions instead of constructing them. With the aim of developing an understanding of the kinds of student teachers evoked concept images of the notion of angle of contiguity, a qualitative case study was conducted at one state university in Zimbabwe. Purposive sampling was used to select 28 mathematics undergraduate student teachers who responded to a test item. Qualitative data analysis was guided by ideas drawn from the theoretical framework of Abstraction in Context and idea of imperative features of a mathematical definition. Student teachers written responses revealed that student teachers personal concept definitions consisted of ambiguous and irrelevant formulations that did not capture the essence of the idea of the angle of contiguity. In some cases their responses were not consistent with the definition of the angle of contiguity. Although there were a few instances of adequate descriptions of the concept, (8 out of 32) these and the inadequate descriptors elicited can contribute significantly towards efforts intended to improve mathematics instruction. Improved mathematics instruction will lead to enhanced conceptualizations of mathematics concepts.

Teacher's Specialized Content Knowledge on the Concept of Square: A Vignette Approach

In learning geometry, the discussion about the definition of quadrilateral is a material that is difficult and not easily taught by the teacher. This study aims to explore the teacher's specialized content knowledge about square. This is a descriptive-qualitative research. The process of selecting subjects begins with searching prospective subject data according to the level of the teacher through a portfolio of 82 teachers in South East of Sulawesi: (33 First Teachers, 33 Young Teachers, and 16 Intermediate Teachers). The research subjects consisted of three teachers, namely: First Teacher, Young Teacher dan Intermediate Teacher with score > 50. Data were taken using vignette. The results show that there is a difference when the teacher is asked to define a square with when given a definition of a square. First Teacher is accurate when given a square definition with the symmetry and diagonal axis attributes; the side attribute is not accurate in giving arguments to the square definition. Young Teacher is inaccurate when given the definition of a square with side and angle attributes; accurate with symmetry and diagonal axis attributes; but it is not accurate when given a square definition. Regarding attributes of side; Intermediate Teacher revealed that the side and angle attributes are inaccurate but accurate with the symmetry and diagonal axis attributes but do not appear / are not used when asked to define a square. Specialized content knowledge First Teacher is better because it has been able to reconstruct concepts from a square, but Young Teacher and Intermediate Teacher are still influenced by concept images and figural concepts.

A semi-automated quantitative comparison of metal artifact reduction in photon-counting computed tomography by energy-selective thresholding

An evaluation of energy thresholding and acquisition mode for metal artifact reduction in Photon-counting detector CT (PCD-CT) compared to conventional energy-integrating detector CT (EID-CT) was performed. Images of a hip prosthesis phantom placed in a water bath were acquired on a scanner with PCD-CT and EID-CT (tube potentials: 100, 120 and 140 kVp) and energy thresholds (above 55–75 keV) in Macro and Chess mode. Only high-energy threshold images (HTI) were used. Metal artifacts were quantified by a semi-automated segmentation algorithm, calculating artifact volumes, means and standard deviations of CT numbers. Images of a human cadaver with hip prosthesis were acquired on the PCD-CT in Macro mode as proof-of-concept. Images at 140 kVp showed less metal artifacts than 120 kVp or 100 kVp. HTI (70, 75 keV) had fewer artifacts than low energy thresholds (55, 60, 65 keV). Fewer artifacts were observed in the Macro-HTI (8.9–13.3%) for cortical bone compared to Chess-HTI (9.4–19.1%) and EID-CT (10.7–19.0%) whereas in bone marrow Chess-HTI (19.9–45.1%) showed less artifacts compared to Macro-HTI (21.9–38.3%) and EID-CT (36.4–54.9%). Noise for PCD-CT (56–81 HU) was higher than EID-CT (33–36 HU) irrespective of tube potential. High-energy thresholding could be used for metal artifact reduction in PCD-CT, but further investigation of acquisition modes depending on target structure is required.