Duncan Samson - Academia.edu (original) (raw)
Papers by Duncan Samson
Learning and Teaching Mathematics, 2015
INTRODUCTION During a recent Grade 11 geometry lesson the class engaging with the theorem that an... more INTRODUCTION During a recent Grade 11 geometry lesson the class engaging with the theorem that angles subtended by a chord in the same segment of a circle are equal. As part of the discussion we touched on one of the corollaries of this theorem, namely that angles subtended by chords of equal length in a given circle are equal.
Learning and Teaching Mathematics, 2016
Learning and Teaching Mathematics, 2015
INTRODUCTION In this article we explore three different visually engaging methods of carrying out... more INTRODUCTION In this article we explore three different visually engaging methods of carrying out long multiplication. The standard algorithmic approach to long multiplication, as typically taught at primary school level, is generally along the lines of that illustrated in Figure 1.
Learning and Teaching Mathematics, 2012
Viviani's theorem, named after the Italian mathematician and geometer Vincenzo Viviani (1622 ... more Viviani's theorem, named after the Italian mathematician and geometer Vincenzo Viviani (1622 - 1703), is a simple yet delightfully intriguing geometry theorem with a number of interesting extensions. Viviani's theorem states that for any point inside an equilateral triangle the sum of the perpendiculars from that point to the sides of the triangle is equal to the length of the triangle's altitude.
This paper uses a theoretical backdrop of enactivism and knowledge objectification to explore the... more This paper uses a theoretical backdrop of enactivism and knowledge objectification to explore the role of rhythm in the generalisation of pictorial patterns. Through an analysis of two vignettes, rhythm is shown not only to be an indicator of the conscious or unconscious perception of structure, but it is also shown that rhythm can be an artefact born out of specific counting processes, an artefact which in turn can lead to structural awareness.
In einer Konzeptstudie des VITALmathsLIC-Projekts der Rhodes University und der PH FHNW wurden 2 ... more In einer Konzeptstudie des VITALmathsLIC-Projekts der Rhodes University und der PH FHNW wurden 2 Varianten eines Videoclips erstellt, die sich lediglich durch die Einbettung in eine Dialogsituation in einer der beiden Varianten unterscheiden2. Anhand dieser beiden Varianten soll exemplarisch das Potential mathematischer Dialogsituationen in Videoclips veranschaulicht, analysiert und diskutiert werden.
Learning and Teaching Mathematics, Dec 1, 2015
Especially now that Euclidean Geometry is once again a component of the Grade 11 and 12 curriculu... more Especially now that Euclidean Geometry is once again a component of the Grade 11 and 12 curriculum, it is important for us to provide pupils with opportunities to engage with suitable geometric contexts in explorative and flexible ways. We should be exposing our pupils to scenarios that require them to think creatively, and with minimal directional guidance. The beauty of such scenarios is that students are exposed to a variety of approaches and strategies, as different students are likely to tackle such questions in different ways. A single simple scenario can often lead to a wealth of exploration and generalisation.
Routledge eBooks, Mar 7, 2023
2011b). The research associated with the latter focussed mainly on the de-velopment of a bank of ... more 2011b). The research associated with the latter focussed mainly on the de-velopment of a bank of online videos clips and the opportunities they of-fered for the mathematics teacher as interesting and appropriate teaching devices and tools to be used in conjunction with computers and mobile technologies such as tablets and mobile phones (cf. YouTube: “VITAL-maths ” and “linnemath”; facebook: “VITALmaths”). In this new project we would like to shift the research emphasis of the original VITALmaths project from the development and teaching of the videoclips, to the learn-ing process that the video clips can support and enhance. The new VI-TALmathsLIC project envisages foregrounding how learning can take place in different learning and contextual spaces with particular reference to communication and language. In conjunction with this we would also like to develop additional video clips in such a way that they utilize accom-panying resources such as worksheets and manipulatives. In order...
The exploration of number patterns as a pedagogical approach to intro-ducing algebra has been adv... more The exploration of number patterns as a pedagogical approach to intro-ducing algebra has been advocated by many mathematics educators. French (2002) comments that the introduction of algebra through what is potentially a wide range of pattern generalisation activities may be effective in assisting pupils to see algebra as both meaningful and purposeful right
Pattern generalisation has become an important feature of mathematics classrooms around the globe... more Pattern generalisation has become an important feature of mathematics classrooms around the globe. Sometimes these activities focus purely on given numerical terms, but the use of pictorial or figural patterns is now becoming part of the standard repertoire for such generalisation exercises. From a pedagogic point of view, the investigation of pictorial patterns potentially allows for a meaningful way of arriving at and exploring algebraically equivalent expressions of generality. A typical approach to presenting such a patterning task is shown in Figure 1. Students are generally required to determine (a) the number of dots in the next few terms, (b) the number of dots in one or two terms further along in the sequence, for example the 10th or 50th terms, and (c) an expression for the number of dots in the nth term, i.e., an algebraic expression of generality.
Learning and Teaching Mathematics, 2013
One of the aspects of mathematics that I particularly enjoy is how a simple idea can often lead t... more One of the aspects of mathematics that I particularly enjoy is how a simple idea can often lead to a wealth of mathematical exploration. One such idea is the classic 'difference of two squares'. Within the school curriculum the difference of two squares is usually introduced as a specific form of factorisation.
Learning and Teaching Mathematics, 2015
Quartiles are useful for describing the distribution of a data set as they split the distribution... more Quartiles are useful for describing the distribution of a data set as they split the distribution into four equal parts, each part containing one quarter of the total. As such, quartiles form a critical component of the "five-number summary" used for drawing box-and-whisker plots. Calculating the upper and lower quartiles of a data set should be a relatively straightforward procedure. However, while the basic concept of a quartile is simple enough, there are many different accepted methods for their calculation, with different methods producing slightly different numerical values. In this article I explore those particular methods that I have seen employed at school level in South Africa.
Learning and Teaching Mathematics, 2013
The method of completing the square is a useful algebraic technique. Within the South African sch... more The method of completing the square is a useful algebraic technique. Within the South African school context, "completing the square" is most often used for three specific purposes: (i) solving quadratic equations, (ii) writing parabola equations in turning point format, and (iii) writing circle equations in centre-radius format. In my experience, learners who have good algebraic skills master the technique of completing the square fairly quickly. Learners who are less algebraically confident take a little longer to acquire the skill but are nonetheless able to master the technique with sufficient practice. However, for most learners the method of completing the square is little more than an arcane process of algebraic manipulation accomplished somewhat mechanically through the use of a guiding algorithm or mantra such as halve it, square it, add it to both sides. As such, most learners leave school without really understanding the conceptual basis of the technique. This i...
Almost 20 years ago, Cuoco, Goldenberg and Mark wrote a seminal paper for the Journal of Mathemat... more Almost 20 years ago, Cuoco, Goldenberg and Mark wrote a seminal paper for the Journal of Mathematical Behavior entitled 'Habits of Mind: An Organizing Principle for Mathematics Curricula' (Cuoco et al., 1996). The article remains as relevant today as when it was originally published. The premise of their paper is that mathematical habits of mind such as searching for patterns, creating, experimenting, describing, visualising and conjecturing are far more important considerations than specific mathematical content, and that providing students with meaningful opportunities to engage in such habits of mind is a powerful means of exposing them to genuine experiences of mathematical activity.
The research associated with the latter focussed mainly on the development of a bank of online vi... more The research associated with the latter focussed mainly on the development of a bank of online videos clips and the opportunities they offered for the mathematics teacher as interesting and appropriate teaching devices and tools to be used in conjunction with computers and mobile technologies such as tablets and mobile phones (cf. YouTube: "VITALmaths" and "linnemath"; facebook: "VITALmaths"). In this new project we would like to shift the research emphasis of the original VITALmaths project from the development and teaching of the videoclips, to the learning process that the video clips can support and enhance. The new VI-TALmathsLIC project envisages foregrounding how learning can take place in different learning and contextual spaces with particular reference to communication and language. In conjunction with this we would also like to develop additional video clips in such a way that they utilize accompanying resources such as worksheets and manipulatives.
INTRODUCTION In a previous article (Samson, 2015) I highlighted the importance of providing pupil... more INTRODUCTION In a previous article (Samson, 2015) I highlighted the importance of providing pupils with opportunities to engage with geometric contexts in explorative and flexible ways. Such geometric scenarios should be posed with minimal directional guidance, thereby encouraging creative mathematical thinking. The beauty of such contexts is that they allow pupils to tackle the question in different ways. As the final question in our November 2017 Grade 11 Paper 2 examination we included the following:
Learning and Teaching Mathematics, 2015
INTRODUCTION During a recent Grade 11 geometry lesson the class engaging with the theorem that an... more INTRODUCTION During a recent Grade 11 geometry lesson the class engaging with the theorem that angles subtended by a chord in the same segment of a circle are equal. As part of the discussion we touched on one of the corollaries of this theorem, namely that angles subtended by chords of equal length in a given circle are equal.
Learning and Teaching Mathematics, 2016
Learning and Teaching Mathematics, 2015
INTRODUCTION In this article we explore three different visually engaging methods of carrying out... more INTRODUCTION In this article we explore three different visually engaging methods of carrying out long multiplication. The standard algorithmic approach to long multiplication, as typically taught at primary school level, is generally along the lines of that illustrated in Figure 1.
Learning and Teaching Mathematics, 2012
Viviani's theorem, named after the Italian mathematician and geometer Vincenzo Viviani (1622 ... more Viviani's theorem, named after the Italian mathematician and geometer Vincenzo Viviani (1622 - 1703), is a simple yet delightfully intriguing geometry theorem with a number of interesting extensions. Viviani's theorem states that for any point inside an equilateral triangle the sum of the perpendiculars from that point to the sides of the triangle is equal to the length of the triangle's altitude.
This paper uses a theoretical backdrop of enactivism and knowledge objectification to explore the... more This paper uses a theoretical backdrop of enactivism and knowledge objectification to explore the role of rhythm in the generalisation of pictorial patterns. Through an analysis of two vignettes, rhythm is shown not only to be an indicator of the conscious or unconscious perception of structure, but it is also shown that rhythm can be an artefact born out of specific counting processes, an artefact which in turn can lead to structural awareness.
In einer Konzeptstudie des VITALmathsLIC-Projekts der Rhodes University und der PH FHNW wurden 2 ... more In einer Konzeptstudie des VITALmathsLIC-Projekts der Rhodes University und der PH FHNW wurden 2 Varianten eines Videoclips erstellt, die sich lediglich durch die Einbettung in eine Dialogsituation in einer der beiden Varianten unterscheiden2. Anhand dieser beiden Varianten soll exemplarisch das Potential mathematischer Dialogsituationen in Videoclips veranschaulicht, analysiert und diskutiert werden.
Learning and Teaching Mathematics, Dec 1, 2015
Especially now that Euclidean Geometry is once again a component of the Grade 11 and 12 curriculu... more Especially now that Euclidean Geometry is once again a component of the Grade 11 and 12 curriculum, it is important for us to provide pupils with opportunities to engage with suitable geometric contexts in explorative and flexible ways. We should be exposing our pupils to scenarios that require them to think creatively, and with minimal directional guidance. The beauty of such scenarios is that students are exposed to a variety of approaches and strategies, as different students are likely to tackle such questions in different ways. A single simple scenario can often lead to a wealth of exploration and generalisation.
Routledge eBooks, Mar 7, 2023
2011b). The research associated with the latter focussed mainly on the de-velopment of a bank of ... more 2011b). The research associated with the latter focussed mainly on the de-velopment of a bank of online videos clips and the opportunities they of-fered for the mathematics teacher as interesting and appropriate teaching devices and tools to be used in conjunction with computers and mobile technologies such as tablets and mobile phones (cf. YouTube: “VITAL-maths ” and “linnemath”; facebook: “VITALmaths”). In this new project we would like to shift the research emphasis of the original VITALmaths project from the development and teaching of the videoclips, to the learn-ing process that the video clips can support and enhance. The new VI-TALmathsLIC project envisages foregrounding how learning can take place in different learning and contextual spaces with particular reference to communication and language. In conjunction with this we would also like to develop additional video clips in such a way that they utilize accom-panying resources such as worksheets and manipulatives. In order...
The exploration of number patterns as a pedagogical approach to intro-ducing algebra has been adv... more The exploration of number patterns as a pedagogical approach to intro-ducing algebra has been advocated by many mathematics educators. French (2002) comments that the introduction of algebra through what is potentially a wide range of pattern generalisation activities may be effective in assisting pupils to see algebra as both meaningful and purposeful right
Pattern generalisation has become an important feature of mathematics classrooms around the globe... more Pattern generalisation has become an important feature of mathematics classrooms around the globe. Sometimes these activities focus purely on given numerical terms, but the use of pictorial or figural patterns is now becoming part of the standard repertoire for such generalisation exercises. From a pedagogic point of view, the investigation of pictorial patterns potentially allows for a meaningful way of arriving at and exploring algebraically equivalent expressions of generality. A typical approach to presenting such a patterning task is shown in Figure 1. Students are generally required to determine (a) the number of dots in the next few terms, (b) the number of dots in one or two terms further along in the sequence, for example the 10th or 50th terms, and (c) an expression for the number of dots in the nth term, i.e., an algebraic expression of generality.
Learning and Teaching Mathematics, 2013
One of the aspects of mathematics that I particularly enjoy is how a simple idea can often lead t... more One of the aspects of mathematics that I particularly enjoy is how a simple idea can often lead to a wealth of mathematical exploration. One such idea is the classic 'difference of two squares'. Within the school curriculum the difference of two squares is usually introduced as a specific form of factorisation.
Learning and Teaching Mathematics, 2015
Quartiles are useful for describing the distribution of a data set as they split the distribution... more Quartiles are useful for describing the distribution of a data set as they split the distribution into four equal parts, each part containing one quarter of the total. As such, quartiles form a critical component of the "five-number summary" used for drawing box-and-whisker plots. Calculating the upper and lower quartiles of a data set should be a relatively straightforward procedure. However, while the basic concept of a quartile is simple enough, there are many different accepted methods for their calculation, with different methods producing slightly different numerical values. In this article I explore those particular methods that I have seen employed at school level in South Africa.
Learning and Teaching Mathematics, 2013
The method of completing the square is a useful algebraic technique. Within the South African sch... more The method of completing the square is a useful algebraic technique. Within the South African school context, "completing the square" is most often used for three specific purposes: (i) solving quadratic equations, (ii) writing parabola equations in turning point format, and (iii) writing circle equations in centre-radius format. In my experience, learners who have good algebraic skills master the technique of completing the square fairly quickly. Learners who are less algebraically confident take a little longer to acquire the skill but are nonetheless able to master the technique with sufficient practice. However, for most learners the method of completing the square is little more than an arcane process of algebraic manipulation accomplished somewhat mechanically through the use of a guiding algorithm or mantra such as halve it, square it, add it to both sides. As such, most learners leave school without really understanding the conceptual basis of the technique. This i...
Almost 20 years ago, Cuoco, Goldenberg and Mark wrote a seminal paper for the Journal of Mathemat... more Almost 20 years ago, Cuoco, Goldenberg and Mark wrote a seminal paper for the Journal of Mathematical Behavior entitled 'Habits of Mind: An Organizing Principle for Mathematics Curricula' (Cuoco et al., 1996). The article remains as relevant today as when it was originally published. The premise of their paper is that mathematical habits of mind such as searching for patterns, creating, experimenting, describing, visualising and conjecturing are far more important considerations than specific mathematical content, and that providing students with meaningful opportunities to engage in such habits of mind is a powerful means of exposing them to genuine experiences of mathematical activity.
The research associated with the latter focussed mainly on the development of a bank of online vi... more The research associated with the latter focussed mainly on the development of a bank of online videos clips and the opportunities they offered for the mathematics teacher as interesting and appropriate teaching devices and tools to be used in conjunction with computers and mobile technologies such as tablets and mobile phones (cf. YouTube: "VITALmaths" and "linnemath"; facebook: "VITALmaths"). In this new project we would like to shift the research emphasis of the original VITALmaths project from the development and teaching of the videoclips, to the learning process that the video clips can support and enhance. The new VI-TALmathsLIC project envisages foregrounding how learning can take place in different learning and contextual spaces with particular reference to communication and language. In conjunction with this we would also like to develop additional video clips in such a way that they utilize accompanying resources such as worksheets and manipulatives.
INTRODUCTION In a previous article (Samson, 2015) I highlighted the importance of providing pupil... more INTRODUCTION In a previous article (Samson, 2015) I highlighted the importance of providing pupils with opportunities to engage with geometric contexts in explorative and flexible ways. Such geometric scenarios should be posed with minimal directional guidance, thereby encouraging creative mathematical thinking. The beauty of such contexts is that they allow pupils to tackle the question in different ways. As the final question in our November 2017 Grade 11 Paper 2 examination we included the following: