Middle School Students’ Reasoning in Nonlinear Proportional Problems in Geometry

In this study, we investigate sixth, seventh, and eighth grade students’ achievement in nonlinear (quadratic or cubic) proportional problems regarding length, area, and volume of enlarged figures. In addition, we examine students’ solution strategies for the problems and obstacles that prevent students from answering the problems correctly by using a mixed method research design. A total of 935 middle school students were given a paper-pencil test and 12 of them were interviewed. Findings indicated that achievement of the participants were low and that students used a limited number of strategies for solving the problems. In addition, these strategies were found to have lacked the argument of the linear proportional and nonlinear proportional relationships among length, area, and volume concepts for most of the participants’ answers. Moreover, analysis revealed that the confusion of linear proportional and nonlinear proportional relationships and misinterpretation of additive and multiplicative relationships were serious obstacles while solving the nonlinear proportional problems related to the area and volume of enlarged figures.     

Preservice Middle and High School Mathematics Teachers’ Strategies when Solving Proportion Problems

The purpose of this study was to investigate eight preservice middle and high school mathematics teachers’ solution strategies when solving single and multiple proportion problems. Real-world missing-value word problems were used in an interview setting to collect information about preservice teachers’ (PSTs) reasoning about proportional relationships. An explanatory case study methodology with multiple cases was used to make comparisons within and across cases. Analysis of the semi-structured interviews with each PST revealed that using practical problems, in which plastic gears and a mini balance system were provided, and multiple proportion problems facilitated the PSTs’ recognition of the proportional relationships in their solutions. Therefore, they avoided using cross-multiplication and erroneous strategies in those problems. Among the strategies that the PSTs used in solving single and multiple proportion problems, the ratio table strategy was the most frequent and effective strategy. The ratio table strategy enabled the PSTs to recognize the constant ratio and product relationships more than the other strategies. The results of this study illuminate how PSTs reason about proportional relationships when they cannot rely on computation methods like cross-multiplication.     

Elementary School Teachers’ Use of Technology During Mathematics Teaching

Various educational technologies have been advanced as potential vehicles to transform teaching and learning. Still, research studies have documented that primary school teachers struggle to integrate technology in meaningful ways. This article presents the findings of a year-long study in which the author frequently observed three primary school teachers’ enactments of technology into their mathematics teaching. Each teacher was observed between 25 and 30 times during the school year. The types of technologies used as well as the types of mathematical tasks and problems that participants posed while teaching with technology were inductively analyzed. Inductive qualitative analyses indicated that participants’ technology use focused on presentation technologies such as the document camera or interactive whiteboard more than computer-based technologies or interactive activities. Further, teachers varied widely in their enacted pedagogies while integrating technology, and two participants demonstrated more frequent enactments of learner-centered pedagogies toward the end of the school year. Implications for researching teachers’ use of technology in the future are also shared.     

Yup’ik Cosmology to School Mathematics: The Power of Symmetry and Proportional Measuring

This article shows how Yup’ik cosmology, epistemology, and everyday practice have implications for the teaching of school mathematics. Math in a Cultural Context (MCC) has a long–term collaborative relationship with Yup’ik elders and experienced Yup’ik teachers. Because of this long–term ethnographically–oriented relationship, the authors – both insiders and an outsider – have been able to understand the mathematical implications of everyday Yup’ik practice. As the article demonstrates, body proportional measuring and symmetry/splitting are two generative solution strategies used by Yup’ik elders in solving everyday problems. We argue that proportional measuring coupled with symmetry/splitting can provide school mathematics with an alternative pathway to the teaching of some aspects of geometry and rational number reasoning.     

Collaborative Effort to Develop Middle School Preservice Teachers’ Mathematical Knowledge

Abstract

This article describes reflections of two mathematicians and a mathematics teacher educator who collaborated on the development and implementation of courses (probability and statistics connections, number concept connections, and middle school mathematics methods) for middle school mathematics preservice teachers. The instructors of the courses, two in mathematics and one in mathematics education, worked together to more explicitly link course materials, assignments, and the pedagogical approaches. Collectively, the courses were designed to address the five components of preservice teachers’ mathematical knowledge for teaching (PT-MKT), and to model effective teaching practices. Using their collective experiences co-planning and implementing these course adjustments were made in the subsequent year. The instructors were pleased by their implementation and student outcomes in all three courses.

We describe how each component of the PT-MKT framework was approached in these courses and discuss challenges experienced by the instructors, who were part of a larger effort to develop and implement a middle school teacher preparation program. The information shared is based on data collected as part of a program evaluation effort, and is bolstered by the instructors’ recollection of events. Overall, the instructors enhanced the curricula and their instructional practices and found that the attention placed on developing PT-MKT support the mathematical development of middle school mathematics preservice teachers.     

Teachers’ Sickness Absence in Primary Schools,School Climate and Teachers’ Sense of Efficacy

Training programmes may compromise governors' ability to offer an authentic lay’ input. The article reports the views of governors interviewed about their perceptions of the need for a degree of expert status. It is tentatively suggested that training courses may be contributing both to a sense of inadequacy and to the tendency to become ‘incorporated’ into an institutionalised role. Some models of governor training are considered in relation to their possible effects.     

Student Views of Computer-Based Mathematics in the Middle Years: Does Gender Make a Difference?

How the use of computers in mathematics classrooms was viewed by students in two middle years mathematics classrooms was the focus of the research described in this paper. The primary data sources consisted of questionnaires, classroom observations supported by videotaping of mathematics lessons, and interviews with two girls and two boys from each class. Thus both qualitative and quantitative methods were used. Girls viewed the computer-based lessons less favourably than did boys. In general, the boys were likely to believe that computers contributed to their experiencing pleasure in these lessons, and to making mathematics more relevant to them. Girls were typically more concerned about whether computers facilitated learning and enabled success in mathematics. The attitudes of students to computer-based mathematics were related to their views of computers.     

Manipulative Use and Elementary School Students’ Mathematics Learning

Using data from the Early Childhood Longitudinal Study (ECLS) 1998/1999, we examine the relationship between elementary students’ (K–5) manipulative use and mathematics learning. Using a cross-sectional correlational analysis, we found no relationship between manipulative use and student mathematics achievement. However, using a longitudinal analysis, we documented a positive relationship between manipulative use and student mathematics learning during their elementary school years (K–5). From a teaching and learning perspective, these findings provide important evidence of the influence of long-term manipulative use on students’ overall learning. From policy and methodological perspectives, these findings provide evidence for the importance of modeling student learning (as opposed to achievement) when studying the effectiveness of instructional strategies.     

The Development of Diploma in Education Student Teachers’ Mathematics Pedagogical Content Knowledge

Mathematics pedagogical content knowledge (MPCK) of teachers cannot be easily defined but is a complex concept integrating generic pedagogical knowledge, mathematics teaching methodology as well as knowledge of the discipline of mathematics. It is the objective of mathematics methodology courses in teacher preparation programmes to begin the development of MPCK in their pre-service teachers. As part of a research study on the development of MPCK in primary school beginning teachers, a 16-item instrument was developed to measure some aspects of the MPCK for teaching mathematics at primary level. The instrument was administered to the 2003 Intake of the Diploma in Education student teachers at the National Institute of Education, Singapore just at the beginning of their programme. As they completed their methodology course in February 2005, the instrument was administered again. This paper will discuss the findings concerning their performance in these two tests, with reference to the overall performance as well as topic-specific and MPCK construct-specific performance. The findings indicate that student teachers at the beginning of their programmes are generally quite weak in their mathematics pedagogical content knowledge, as might be expected. There was significant improvement in some aspects of their MPCK on completion of their mathematics pedagogy course.     

Middle School Science Teachers’ Confidence and Pedagogical Practice of New Literacies

Due to the rapid advancements of information and communication technologies (ICTs), educational researchers argue that multimodal and new literacies should become common practices in schools. As new ICTs emerge and evolve, students need the new literacies skills and practices to successfully participate fully in the civic life of a global community. Are teachers prepared to integrate ICTs in the classroom to develop students’ new literacies skills? The purpose of this study is to suggest a new literacies framework that guides ICTs integration and supports scientific inquiry, as well as investigate middle school teachers’ confidence to practice new literacies in science classrooms. The study adopted mixed-methodology design, surveyed 32 middle school science teachers’ ICTs and new literacies skills, and randomly observed 15 teachers’ new literacies practices in the classrooms. The results revealed that even though teachers have high confidence in using ICTs, the meaningful technology integration and new literacies practices were scarcely observed in their classroom practices.     

Reasonable and Reasonableness of Answers: Kuwaiti Middle School Teachers’ Perspectives

This study described mathematics teachers’ views on the significance of reasonable answers and how they address this concept in their classrooms. Data were gathered from thirteen eighth-grade Kuwaiti mathematics teachers in two middle schools. All teachers in Kuwait use the national textbook and follow the accompanying instructional plans provided by the Ministry of Education. The results revealed that the overwhelming majority of Kuwaiti teachers reported that an answer needed to be exact to be reasonable. Only three teachers reported discussing reasonable answers and their discussions were in response to student mistakes. None of the teachers addressed the concept proactively, considering reasonableness when planning their lessons or preparing activities or exercises. They defended their lack of attention to this concept by noting that the notion of reasonableness is not included in the Kuwaiti national curriculum.     

An Investigation of Pre-service Middle School Mathematics Teachers’ Ability to Conduct Valid Proofs,Methods Used,and Reasons for Invalid Arguments

The purposes of this study are to investigate Turkish pre-service middle school mathematics teachers’ ability in conducting valid proofs for statements regarding numbers and algebra in terms of their year of enrollment in a teacher education program, to determine the proof methods used in their valid proofs, and to examine the reasons for their invalid arguments. A proof questionnaire containing three proof statements was administered to 115 pre-service middle school mathematics teachers in a large state university in Ankara, Turkey. The results showed that more than half of the pre-service teachers were able to conduct valid proofs for the given statements. In terms of year levels, it was seen that the seniors were the least successful group in conducting valid proofs for each statement. When pre-service teachers’ valid proofs were analyzed, it was concluded that mathematical induction and direct proof were the mostly used methods for the given statements. When pre-service teachers’ invalid arguments were analyzed, it was seen that “inserting numbers to verify the given statement” and “rewriting the givens in the statement” were the common reasons for stating invalid arguments.     

Preparing Teachers’ to Raise Students’ Mathematics Learning

The purpose of this study was to investigate the effectiveness of the ORIGO Stepping Stones program in a large suburban school district in the Midwestern U.S.. The sample included 11 elementary schools that implemented the program during the 2013–2016 school years. Findings are presented from teacher surveys, classroom observations, and analysis of standardized student mathematics achievement scores on the Northwest Evaluation Association Measurement of Academic Progress (NWEA MAP). Using multilevel models, the program did not demonstrate a detectably larger effect on mathematics achievement than the comparison. Classroom observations and views of teachers that support program implementation were explored. In terms of fidelity of implementation, findings suggested that the program was delivered differently from how it was intended. With respect to researchers and policymakers, the approach used to evaluating mathematics program effectiveness is informative and could be used as part of larger accountability systems.     

Coaching Teachers to Improve Students’ School Readiness Skills: Indirect Effects of Teacher–Student Interaction

This study examined indirect associations of MyTeachingPartner coaching with pre-K students’ (N = 1,570; 73% low income) school readiness, via improvements of teachers’ (N = 393; 47% Black; 41% Head Start) classroom interactions. Data were collected across 2008–2009 and 2009–2010 in 10 urban sites across the eastern United States. The number of completed coaching cycles was examined as a predictor of teacher–student interactions, as were direct or indirect associations with students’ literacy, receptive language, and working memory skills. Significant findings indicated that teachers engaged in more feedback cycles showed greater improvements in instructional interactions, in turn predicting greater increases in students’ early literacy and working memory. Results confirm the theory of change for coaching and an ecological-developmental conceptualization of school readiness.     

The Relationships among Teachers’ Perceptions of Student Behaviour,Teachers’ Characteristics,and Ratings of Students’ Emotional and Behavioural Problems

This study examined the relationships among teachers’ perceptions of students’ behavioural problems and their own efficacy and psychological symptoms. Findings suggested that teachers were more bothered by externalising (i.e., acting out) than internalising (i.e., withdrawal, depression) behavioural problems in their students, and believed that students had greater control over externalising behavioural problems. In addition teachers’ personal teaching efficacy contributed significantly to the prediction of how bothersome internalising behavioural problems were perceived as being. Further, teachers’ personal and general teaching efficacy both contributed significantly to the prediction of teachers’ perceptions of student control over externalising behavioural problems. These findings may have particular relevance to interventions addressing students’ behavioural problems in the classroom.     

Mathematics Teachers’ Approaches to Computer‐based Instruction

Schon (1987) has criticized the normative curriculum of professional schools including technical colleges: at first basic science (mathematics, physics, etc.), then applied science, and finally technical skills of day‐to‐day practice. In Schön's opinion, the tradition of professional education is an obstacle to changing teaching methods from drill and practice to open‐learning environments. Solomon (1986) presents the different approaches which teachers take in computer‐based instruction in stages from drill and practice to open‐learning environments. This paper reports a six‐year case study, inspired by the work of Schön and Solomon, on the impact of, and the teachers’ role in, the use of computers and Mathcad in the mathematics instruction of technical colleges. The teaching experiment was started with 51 first‐year students during the academic year 1988‐1989 at the Technical Institute of Jyvaskyld. The class of mechanical engineering had 162 and the class of electrical engineering 206 lessons of mathematics during the first experimental year. Two senior lecturers of mathematics taught the classes. Before the experiment, computers had never been used in mathematics instruction at the institute. The various ways of using Mathcad were classified on the basis of the forms filled and files saved by the teachers, as well as on the notes made by the researcher. The teachers’ ways of using computers in our experiment during the years 1988‐1995 clearly showed the existence of stages. Some of the stages were the same for both teachers, but there were also some differences. Both teachers started by fitting an open computer‐based environment into the traditional instruction.     

Determinants of Teachers’ Decision to Help Students with Behaviour Problems

The present study explored the link between teachers’ perceptions of students’ problem behaviours, their emotional reactions, and their likelihood to help such students. One hundred and thirty five teachers responded to four written vignettes which depicted students with either aggressive or withdrawal behaviours. The effects of teacher gender and the nature of problem behaviours in male and female students on teacher's willingness to help were also explored. Results showed that teachers were least sympathetic and most angry with aggressive students. These students were more likely to be held responsible for their behaviours. Teachers were, however, willing to help all students regardless of their problems. Female teachers were more willing to help students than male teachers. The results supported the “empathy‐altruism” hypothesis with regard to helping behaviour. The influence of role expectation of teachers in the context of Chinese culture was also discussed.