mark jacobs | CPUT - Academia.edu (original) (raw)

Papers by mark jacobs

Abstract The National Senior Certificate examination is the most important school examination in ... more Abstract
The National Senior Certificate examination is the most important school examination in South Africa. Analyses of learners’ performance in Mathematics in this examination is normally done and presented in terms of percentage learners’ achievement in the different bands of achievement. In some cases item difficulties are presented – item refers to the subsection of each examination question. Very little attention is paid to other diagnostic statistics such as the discrimination indices and item difficulties taking into consideration partial scores examinees achieve on items. In this article we report on a study which, in addition to the usual item difficulties, includes a discrimination index of item difficulties taking into account partial scores examinees achieved. The items, considered individually, are analysed in relation to the other items on the test. The focus is on the topic sequences and series and the data were obtained from a stratified sample of the marked scripts of the candidates who wrote the National Senior Certificate examination in Mathematics in November 2010. Rasch procedures were used for the analyses. The findings indicate that learners perform differently on sub-sections of topics, herein referred to as items, and that focussing on scores for full topics potentially mask these differences. Mathematical explanations are attempted to account for difficulties learners exhibited in these sub-sections, using a hierarchy of scale. The findings and our analysis indicate that a form of measurement driven testing could have beneficial results for teaching. Also, for some items the difficulty obtained from the work of examinees run counter to the commonly perceived wisdom that an examination ought to be structured in such a way that the less difficult items are at the start of a topic. An explanatory device anchored around the construct of ‘familiarity with problem types through repeated productive practice’ is used to account for the manifested hierarchy of difficulty of the items.

The focus of this study is to investigate how teachers use productive questions to promote discou... more The focus of this study is to investigate how teachers use productive questions to promote discourse in the mathematics classroom. Teachers are faced with a myriad of challenges and make use of productive questions that are geared towards stimulating and promoting mathematics classroom discourses and seeking solutions to problems. Teachers also require a deep understanding of both knowledge of mathematics and how students learn mathematics. The video-recorded footage of one lesson of a high school mathematics teacher was the basis for this study. This video-recorded lesson demonstrating classroom discourse involving student-teacher interactions through questions and answers was transcribed verbatim and analysed interpretively. An analysis of the classroom discourse in the recorded video serves not only as an analytic tool to study classroom interactions, but also as a resource for discussions and reflections on the teacher’s continuous professional development (CPD) activities. The findings of the study show that the teacher made use of a range of productive questions to direct students towards solutions; he also involved students in completing procedural processes, provided opportunities for students to demonstrate solutions to their peers, made visible their thinking processes, and encouraged them to assist one another, through peer-to-peer interaction, to find solutions.

Student enrolment trends in Mathematical Literacy and Mathematics since the inception of the NSC ... more Student enrolment trends in Mathematical Literacy and Mathematics since the inception of the NSC Mathematical Literacy is a focus of this paper. We found that enrolments for Mathematics largely lagged behind those for Mathematical Literacy. While the statistics was not fixed, we found a greater emphasis on trying to prove the uptake and success in Mathematics than Mathematical Literacy. We also explored and developed the critique of Mathematical Literacy assessment standards and compared those standards against an external education authority that was the reference for a benchmarking process by the Department Of Basic Education in 2013. We determined that the South African programme compared unfavourably against that of New South Wales. We also showed that the external body included algebraic modelling and financial mathematics which seemed to align better with our Mathematics curriculum than Mathematical Literacy. Lastly, we analysed the environment of Mathematical Literacy tasks and the cognitive loading of those tasks, and proposed, by way of illustration, ways in which similar tasks could be improved. These illustrative tasks we called authentic tasks. Our conclusion is that Mathematical Literacy in its present form offers a weakened version of a mathematically based subject for " making sense " in the world and should be improved.

This paper is concerned with the field of continuous professional development of mathematics teac... more This paper is concerned with the field of continuous professional development of mathematics teachers, specifically focussing on classroom teaching analysis. In particular we describe and analyse one teacher's single lesson in a classroom and her interaction with her learners. We viewed this lesson through three specific lenses: communication, classroom management and the mathematical content of the lesson. The findings show that the dominant teaching style used was the expository form. But as the analysis shows, the use of this form is not simple, but complicated. When Brendefur & Frykholm (2000)'s four communication types were used to analyse communication between the teacher and the learners in the classroom, we found a predominance of two kinds of communication, namely uni-directional and contributive. The mathematical content of the lesson was clearly outlined and enacted, however, we also found that the learners were mainly passive in during the lesson and that the lesson was driven by the teacher.

In making the transition from arithmetic to algebra students encounter continuities and discontin... more In making the transition from arithmetic to algebra students encounter continuities and discontinuities in the way in which the two branches of mathematics function. In this paper it is argued that how they adapt to this transition has implications for how they will learn pre calculus and calculus topics at tertiary level. Notions such as troublesome knowledge and threshold concepts are incorporated to uncover the area of transition which is most problematic. The suggestion is made that students require a psychological shift to make the transition to more abstract ways of thinking about mathematics more successful. It is argued that student comfort with arithmetic rules act both as an aid and as a barrier to learning algebra and that this disjuncture influences their ability and psychological preparedness to conceptualise and work with pre calculus and calculus topics.

Abstract The National Senior Certificate examination is the most important school examination in ... more Abstract
The National Senior Certificate examination is the most important school examination in South Africa. Analyses of learners’ performance in Mathematics in this examination is normally done and presented in terms of percentage learners’ achievement in the different bands of achievement. In some cases item difficulties are presented – item refers to the subsection of each examination question. Very little attention is paid to other diagnostic statistics such as the discrimination indices and item difficulties taking into consideration partial scores examinees achieve on items. In this article we report on a study which, in addition to the usual item difficulties, includes a discrimination index of item difficulties taking into account partial scores examinees achieved. The items, considered individually, are analysed in relation to the other items on the test. The focus is on the topic sequences and series and the data were obtained from a stratified sample of the marked scripts of the candidates who wrote the National Senior Certificate examination in Mathematics in November 2010. Rasch procedures were used for the analyses. The findings indicate that learners perform differently on sub-sections of topics, herein referred to as items, and that focussing on scores for full topics potentially mask these differences. Mathematical explanations are attempted to account for difficulties learners exhibited in these sub-sections, using a hierarchy of scale. The findings and our analysis indicate that a form of measurement driven testing could have beneficial results for teaching. Also, for some items the difficulty obtained from the work of examinees run counter to the commonly perceived wisdom that an examination ought to be structured in such a way that the less difficult items are at the start of a topic. An explanatory device anchored around the construct of ‘familiarity with problem types through repeated productive practice’ is used to account for the manifested hierarchy of difficulty of the items.

The focus of this study is to investigate how teachers use productive questions to promote discou... more The focus of this study is to investigate how teachers use productive questions to promote discourse in the mathematics classroom. Teachers are faced with a myriad of challenges and make use of productive questions that are geared towards stimulating and promoting mathematics classroom discourses and seeking solutions to problems. Teachers also require a deep understanding of both knowledge of mathematics and how students learn mathematics. The video-recorded footage of one lesson of a high school mathematics teacher was the basis for this study. This video-recorded lesson demonstrating classroom discourse involving student-teacher interactions through questions and answers was transcribed verbatim and analysed interpretively. An analysis of the classroom discourse in the recorded video serves not only as an analytic tool to study classroom interactions, but also as a resource for discussions and reflections on the teacher’s continuous professional development (CPD) activities. The findings of the study show that the teacher made use of a range of productive questions to direct students towards solutions; he also involved students in completing procedural processes, provided opportunities for students to demonstrate solutions to their peers, made visible their thinking processes, and encouraged them to assist one another, through peer-to-peer interaction, to find solutions.

Student enrolment trends in Mathematical Literacy and Mathematics since the inception of the NSC ... more Student enrolment trends in Mathematical Literacy and Mathematics since the inception of the NSC Mathematical Literacy is a focus of this paper. We found that enrolments for Mathematics largely lagged behind those for Mathematical Literacy. While the statistics was not fixed, we found a greater emphasis on trying to prove the uptake and success in Mathematics than Mathematical Literacy. We also explored and developed the critique of Mathematical Literacy assessment standards and compared those standards against an external education authority that was the reference for a benchmarking process by the Department Of Basic Education in 2013. We determined that the South African programme compared unfavourably against that of New South Wales. We also showed that the external body included algebraic modelling and financial mathematics which seemed to align better with our Mathematics curriculum than Mathematical Literacy. Lastly, we analysed the environment of Mathematical Literacy tasks and the cognitive loading of those tasks, and proposed, by way of illustration, ways in which similar tasks could be improved. These illustrative tasks we called authentic tasks. Our conclusion is that Mathematical Literacy in its present form offers a weakened version of a mathematically based subject for " making sense " in the world and should be improved.

This paper is concerned with the field of continuous professional development of mathematics teac... more This paper is concerned with the field of continuous professional development of mathematics teachers, specifically focussing on classroom teaching analysis. In particular we describe and analyse one teacher's single lesson in a classroom and her interaction with her learners. We viewed this lesson through three specific lenses: communication, classroom management and the mathematical content of the lesson. The findings show that the dominant teaching style used was the expository form. But as the analysis shows, the use of this form is not simple, but complicated. When Brendefur & Frykholm (2000)'s four communication types were used to analyse communication between the teacher and the learners in the classroom, we found a predominance of two kinds of communication, namely uni-directional and contributive. The mathematical content of the lesson was clearly outlined and enacted, however, we also found that the learners were mainly passive in during the lesson and that the lesson was driven by the teacher.

In making the transition from arithmetic to algebra students encounter continuities and discontin... more In making the transition from arithmetic to algebra students encounter continuities and discontinuities in the way in which the two branches of mathematics function. In this paper it is argued that how they adapt to this transition has implications for how they will learn pre calculus and calculus topics at tertiary level. Notions such as troublesome knowledge and threshold concepts are incorporated to uncover the area of transition which is most problematic. The suggestion is made that students require a psychological shift to make the transition to more abstract ways of thinking about mathematics more successful. It is argued that student comfort with arithmetic rules act both as an aid and as a barrier to learning algebra and that this disjuncture influences their ability and psychological preparedness to conceptualise and work with pre calculus and calculus topics.