Content uploaded by John Mason
Author content
All content in this area was uploaded by John Mason
Content may be subject to copyright.
A preview of the PDF is not available
Seeing an Exercise as a Single Mathematical Object: Using Variation to Structure Sense-Making
Abstract
In this theoretical paper we take an exercise to be a collection of procedural questions or tasks. It can be useful to treat such an exercise as a single object, with individual questions seen as elements in a mathematically and pedagogically structured set. We use the notions of ‘dimensions of possible variation’ and ‘range of permissible change’, derived from Ference Marton, to discuss affordances and constraints of some sample exercises. This gives insight into the potential pedagogical role of exercises, and shows how exercise analysis and design might contribute to hypotheses about learning trajectories. We argue that learners’ response to an exercise has something in common with modeling that we might call ‘micro-modeling’, but we resort to a more inclusive description of mathematical thinking to describe learners’ possible responses to a well-planned exercise. Finally we indicate how dimensions of possible variation inform the design and use of an exercise.
... According to Yerushalmy et al. (2017), asking students to generate different examples may encourage them to develop their mathematical conceptualisation and enrich their PES. According to Watson and Mason (2006), personal example space refers to the collection of examples that an individual can draw upon when thinking about a particular mathematical concept. This space is unique to each person and is shaped by the experiences, understanding, and contexts in which they encounter the concept. ...
... Asking the students to choose multiple examples allows them to distinguish between essential and non-essential aspects, which may enhance their perception of 'deep structures' (Sun, 2011) and improve their outcomes (Watson & Thompson, 2015). The richness of students' example space can serve as an indicator of their mathematical understanding (Watson & Mason, 2006;Zazkis & Leikin, 2007). ...
Feedback is most effective when learners actively engage in self-reflection on their learning processes. A key challenge for formative assessment platforms is developing tasks that, through automated assessment, stimulate such self-reflection. This study presents a design pattern featuring complementary example-eliciting tasks (EETs), developed using multiple modes of automated feedback, interactive diagrams, and inquiry logic principles. Focused on equivalent fractions, the study involved 75 7th grade students. Using a design-based research approach, the study aimed to refine the platform and stimulate metacognitive actions (MCAs). We concentrated on one student to assess the platform's effectiveness in encouraging metacognitive actions during interaction with its elements, with a focus on how these tasks and assessments could enrich the student's personal example space (PES). Data was collected through triangulation of observation methods, including a task-based protocol and automated assessment reports generated by the platform. A qualitative analysis, grounded in metacognitive action theory, was employed to examine the students' metacognitive actions, while descriptive analysis explored changes in PES. The results revealed that engaging with the platform's elements successfully stimulated the student's metacognitive actions, leading to enhanced self-reflection and learning. This research contributes to understanding how automated feedback within inquiry EETs can improve learning outcomes.
... Variation theory presumes that individuals see and understand phenomena from their own perspective (Xu & Pedder, 2015), and that learning takes place when the way of understanding a phenomenon, or Object of Learning, has changed (Lo, 2012;Marton & Booth, 1997). For this learning to occur, some aspects of the Object of Learning must vary while other aspects remain constant (Marton, 2015;Marton & Booth, 1997;Watson & Mason, 2006;Xu & Pedder, 2015). If a learner understands an Object of Learning in terms of its critical aspects, a learner has gained relational understanding of the Object of Learning (Marton, 2015, p. 14-15). ...
... Third, the design of the lesson, with use of systematic differences. Through carefully constructing a set of tasks in which (non-)critical aspects are systematically varied or kept invariant, learners are provided the opportunity to discern what is critical and what is not (Marton, 2015;Sun, 2011;Watson & Mason, 2006). For instance, for the OL "recognizing a figure as a square," learners first experience (standard) examples and non-examples of squares (like triangles, rectangles, circles, etc.), to discern what makes a square (its critical aspects are varied in the non-examples; non-critical aspects are kept the same where possible). ...
This case study explores the impact of a Learning Study (LgS) course on pre-service teachers' (PSTs) Mathematical Knowledge for Teaching (MKT), and its central domains Specialized Content Knowledge (SCK), Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT) in particular. Research suggests that PSTs may benefit from participating in an LgS – a Lesson Study guided by variation theory. However, little is known about how to integrate variation theory within the design of an LgS to foster PSTs’ development. To address this issue, we designed and implemented an LgS course with two teams of seven PSTs in total. The research questions focus on whether the participants’ MKT does develop, and which design elements foster this development. Data included interviews, written reflections and artefacts such as lesson plans. The results show that participants’ MKT did develop, and that analyzing their students’ pre-tests and observing of and reflecting on research lessons were most important in enhancing their KCS and KCT. Additionally, reasoning about (students' and own) (mis)conceptions was most helpful in increasing their SCK. However, not all design elements were as useful as expected. Further research might zoom in on enhancing the less effective parts of the LgS course.
... Pengalaman pembelajar bergantung pada ciri-ciri kritis dari objek yang menjadi tujuan kesadaran mereka. Konsekuensinya, merancang urutan tugas memerlukan analisis kemungkinan variasi sehingga pembelajar "dapat mengamati keteraturan dan perbedaan, mengembangkan ekspektasi, membuat perbandingan, membuat kejutan, menguji, menyesuaikan, dan mengonfirmasi dugaan mereka dalam latihan" (Watson and Mason, 2006). Analisis ruang variasi, pola dalam pengalaman pembelajar, dan bagaimana pola ini cocok dengan objek pembelajaran yang dimaksud adalah elemen kunci dalam kerangka TV tingkat menengah. ...
Kurikulum Merdeka merupakan upaya pemerintah dalam mendukung visi
pendidikan Indonesia dan pemulihan proses belajar mengajar setelah
Indonesia dilanda pandemi. Kurikulum Merdeka dikembangkan sebagai
kerangka kurikulum yang lebih fleksibel, sekaligus berfokus pada materi
esensial dan pengembangan karakter dan kompetensi peserta didik.
Efektivitas proses belajar dapat tercapai dengan adanya desain
pembelajaran yang baik diseluruh mata pelajaran sehingga dapat
memaksimalkan kompetensi peserta didik berdasarkan minat dan bakat di
dibidang tersebut. Hal inilah yang menjadi dasar pentingnya Desain
Pembelajaran Pendidikan Menengah Yang Efektif dalam Kurikulum
Merdeka.
Lebih detail buku ini membahas tentang :
Bab 1 Pengantar Kurikulum Merdeka
Bab 2 Desain Pembelajaran Agama yang Efektif
Bab 3 Desain Pembelajaran Bahasa Indonesia yang Efektif
Bab 4 Desain Pembelajaran Matematika yang Efektif
Bab 5 Desain Pembelajaran IPA yang Efektif
Bab 6 Desain Pembelajaran IPS yang Efektif
Bab 7 Hakikat Pembelajaran PJOK
Bab 8 Desain Pembelajaran Muatan Lokal yang Efektif
Buku ini dapat diwujudkan karena partisipasi, dukungan, kerjasama,
bimbingan dan kritik dari berbagai pihak, sehingga tulisan ini
diterbitkan. Kritik maupun saran dari pembaca demi perbaikan dan
kelengkapan buku ini sungguh diharapkan semoga buku ini dapat
memberikan manfaat bagi para pembaca, sekaligus memberi sumbangsih
bagi pengembangan ilmu pengetahuan terkhusus dalam bidang kurikulum
dan pendidikan. Akhir kata, penulis mengucapkan terimakasih kepada
semua pihak yang telah berperan serta dalam penyusunan buku ini.
Semoga Tuhan Yang Maha Esa senantiasa memberkati kita. Amin.
... Schoenfeld (2013) as well as Burkhardt and Bell (2007) condemn those school texts mostly filled with Mathematics questions that require learners to prioritise understanding of rule-based mathematical process above problem-solving procedures. Although it may seem difficult to select such mathematical thinking problems for use, researchers in Mathematics Education intentionally emphasise the essence of the learners working with such problems (Lupahla, 2014;Ofori-Kusi, 2017;Sullivan, Clarke & Clarke, 2013;Watson & Mason, 2006). The present study, 55 through its 8Ps problem-solving learning model, also hopes to provide the learners with the critical-thinking learning strategies needed for solving non-routine Mathematics problems. ...
The thesis centers on the effect of utilizing the 8Ps learning model (newly developed by us, researchers of this inquiry) on Grade 12 students' mathematical problem-solving skills....
... Although mathematical concepts are considered central in mathematics education, there is no unanimous definition of the notion. Nevertheless, a mathematical concept typically refers to an abstract idea (Wedman, 2020), such as a formal theoretical definition (Sfard, 1991;Tall & Vinner, 1981;Vinner, 2020), or an individual construction (Watson & Mason, 2006). Simon (2017) elaborates on the notion and defines a mathematical concept as "a researcher's articulation of intended or inferred student knowledge of the logical necessity involved in a particular mathematical relationship" (p. ...
... We demarcated episodes based on the teacher announcing a transition to a new task with a new numerical example. The rationale for using episodes instead of, for example, the whole lesson as the unit of analysis was to permit the micro-analysis of each task in terms of the variation made possible to experience within and between tasks (Watson & Mason, 2006). The first episode in each lesson was an introduction that did not specifically address subtraction, while the other episodes consisted of one subtraction task addressed in each episode. ...
Not all students in early grades develop efficient strategies for solving subtraction tasks. In this paper, we examine subtraction teaching in the 1-20 number range. We analyzed two first-grade lessons addressing similar subtraction tasks, using variation theory to identify what aspects of the content were foregrounded in the teaching. The analysis showed that both lessons supported the discernment of aspects of subtraction. However, the learning opportunities differed depending on what aspects the students were enabled to discern, the recurrence and variation of the elicited aspects, and how the teachers made the aspects visible by means of representations. The findings highlight how teaching can make it more likely for students to experience aspects of learning necessary to solve subtraction tasks such as 13-5 = _. Additionally, the findings show how subtraction can be taught to enhance strategies using number relations that are also useful in higher number ranges. ARTICLE HISTORY
... När lärare undervisar elever om bråk måste de ta hänsyn flera aspekter, såsom val av representationer (Sveider, 2021), exempel (Sun, 2015;Watson & Mason, 2006) och hur elevsvar hanteras (Maunula, 2018;Karlsson & Wennergren, 2014). Tidigare forskning visar att lärare använder olika representationer, som areamodeller och längd-eller linjemodeller (tallinjen), för att fördjupa elevernas förståelse av bråk (Van de Walle m.fl., 2018). ...
Syftet med studien är att bidra med kunskap om vilka möjligheter elever får för att lära sig förstå likvärdiga bråk vid olika lektionsdesigner. För att besvara detta syfte genomfördes en Learning study i tre cykler i årskurs 5 med 58 elever. Elevernas möjligheter till lärande analyserades både kvalitativt genom observation av undervisningen och kvantitativt genom test där eleverna fick visa sina kunskaper om likvärdiga bråk före och efter lektionerna. Lektionerna designades med stöd av variationsteori och matematikdidaktisk forskning. Resultaten visar att eleverna utvecklade sin förmåga i alla tre cykler, särskilt i den sista. Framgångsfaktorer i undervisningen inkluderade lärarens användning av kontraster, tallinjen och ett strukturerat sätt att hantera elevernas svar. Dessa faktorer visade sig vara effektiva för att främja elevernas lärande. Studiens resultat kan användas som underlag för lärare och forskare för att ytterligare öka kunskapen om hur undervisningen kan möjliggöra att elever lär sig förstå likvärdiga bråk. Teaching that develops students' ability to understand equivalent fractions The aim of the study is to contribute knowledge about the opportunities students get to learn to understand equivalent fractions through various enacted objects of learning in different lesson designs. To address this, a Learning study was conducted in three cycles in grade 5 with 58 students. Students learning opportunities were analyzed both qualitatively through observations and quantitatively through tests where students demonstrated their knowledge of equivalent fractions before and after the lessons. Lessons were designed with the support of variation theory and mathematics education research. Results show that students improved their ability in all three cycles, especially the last one. Success factors included the teacher's use of contrasts, the number line, and a structured approach to student responses. These factors proved effective in promoting students' learning. Teachers and researchers can use the study's results to increase understanding of how teaching can enable students to learn equivalent fractions.
... As mentioned by Anghileri (2006) and Watson and Mason (2006), teachers now are taking care in choosing the right contexts that cannot distract students from the mathematical purpose tasks. Keeping mathematical subject connections and goals as short as possible can support those learners who are just focusing on some context issues at the expense of mathematics. ...
This qualitative study aimed to propose a model for teaching metacognition in solving mathematical word problems, utilizing a Multiple Case Study Method. The research explored how teachers employ metacognitive strategies, focusing on two components: knowledge of cognition and regulation of cognition. The findings suggest that metacognitive instructional techniques enhance students' mathematical knowledge and problem-solving abilities. Teachers who incorporate various metacognitive strategies help students develop their own learning skills and create conditions for meaningful learning. The study concludes that connecting metacognitive teaching approaches makes math problem-solving more significant. The proposed model allows teachers flexibility in applying strategies based on their circumstances and students' needs. Importantly, the research emphasizes that mathematics teachers must have a thorough understanding of mathematical concepts to effectively implement metacognitive methods.
... Kullberg et al., 2016b;Pang and Lo, 2012;Holmqvist, 2011). Studies have explicitly shown that in the collaborative work in LearS, principles from VT can guide teachers in planning and teaching mathematics tasks with specific patterns of variation and in that way draw students' attention to specific critical aspects, for instance, juxtaposing and comparing, thus making a contrast (Pillay et al., 2022), or sequencing and pairing examples with systematic variation within as well as between examples (Watson and Mason, 2006). ...
Purpose
The purpose of this paper is to explore whether and how principles from variation theory can contribute to the planning of teaching and learning beyond learning study.
Design/methodology/approach
We study whether and how principles from variation theory contributed to a group of teachers’ planning of teaching and learning about decimal numbers in Grades 4 to 7 working in Subject Didactic Groups – a collaborative arrangement suited to daily teaching. A theoretical thematic analysis approach was used when analyzing eight audio-recorded meetings and written documents.
Findings
The study shows that variation theory principles contributed to the teachers’ planning of teaching and learning. Two themes were identified: the theory contributed to the teachers being able to (1) specify what their students needed to learn and (2) design tasks that they anticipated would afford the opportunity to learn what was identified as being necessary to learn.
Originality/value
The paper demonstrates how variation theory can contribute to teachers’ planning of teaching and learning when used in a collaborative arrangement other than learning study. This leads into a discussion about variation theory being used separately from learning study and the benefits and limitations this other collaborative arrangement can have for gaining knowledge of what is to be learned and taught.
Learning is here considered to have taken place when someone has developed the habit, propensity, and disposition to attend productively to things not previously noticed, and in ways not previously experienced, to do with some specific and particular content. ‘Active learning’ sounds like a tautology, but was introduced as a contrast to the more passive activity of students sitting in lectures listening and transcribing mathematics written on a board or screen onto their own paper. The stance taken here is that effective and efficient learning involves active engagement in activity, but includes enculturation through being in the presence of a relative expert ¹ who themselves is manifesting mathematical thinking, not simply passing on the records of the results of previous mathematical thought. Such ‘passivity’ does not necessarily require intention. Following Bennett 2 actions are here taken to involve three agents or impulses: initiating, responding, and reconciling or mediating. All three agents are thus active, but in different ways. Interactions intended to contribute to learning are considered to be actions, and so involve three agents: learner, teacher (in some manifestation), and mathematical content, all within a culture or ethos. Since there are six different ways in which the triple of agents can be assigned to the triple of impulses, six different modes are possible. Analysing these modes sheds light on different ways in which learning could be said to be ‘active’. Activity takes place within a mode of interaction. Again following Bennett, effective activity is here taken to require appropriate relationships among the gap between current state and intended goal, the resources available, and the tasks set.
¹ Vygotsky (1978) pointed out that ‘higher psychological processes’ are first encountered in others.
² Bennett (1993); see also Shantock Systematics Group (1975)
Perception of the mathematics classroom as an arena in which there are various opportunities to learn mathematics leads to a fine-grained focus on the structure of mathematical tasks. Each mathematical task affords engagement with mathematics in certain ways. Variation within a task is a major factor influencing learning. I was teaching a year 9 (13 year-olds) all-attainment 1 class. They had been working on this task: On a coordinate grid you are only allowed to move to the right or upwards. You can do this in any order you like. How may routes are there from the origin to the point (1,1)? How many routes to the points (1,2), (2,1), (1,3) , and so on…..? After about ten minutes I gathered all the students around the board and asked them what they had found. Silently at the back of the group sat Paul, who had been described to me as having special needs. When he had entered school he could not talk, and still at 13 he could neither read nor write. After several students had described how they had counted routes systematically, and deduced a sort of symmetry emerging, I challenged them to find a method which allowed them to work out how many routes there would be to get to any point, for example (6,7). Paul said immediately 'if I knew how many it would take to get to (5,7) and (6,6) I could add them to get (6,7)'. This reply would have been a pleasant surprise from any student, but from Paul it was doubly so because it was his first utterance in such a group. This was a turning point for me as a teacher, and for him as a learner. M y expectations of his mathematics were biased by what I had been told, and he was able to grasp spatial situations with an abnormally skilful level of generality and structure. 2 I would have to work on my expectations and Paul would have to work on mathematics through spatial representations. Shortly after this lesson I began to wonder about how teachers make judgments about their pupils' mathematics, and did some research about this (Watson, 1999). I found that teachers seemed to have no qualms about saying that their judgments came from 'knowing the child' or 'gut feeling' or 'professional judgment'. The latter phrase seemed to mean 'experienced judgment'. One of the 30 teachers I interviewed noticed, as she was talking to me, that she had inadequate strategies in place to ensure her judgments were fair. She realised that she depended on seeing certain facial expressions, but that many of her students, being young Muslim women, would keep their faces down and she would never see their expressions. As a result of this research (of which the observations above are only a glimpse) I drew the sad conclusion that, even with teachers' judgments contributing towards high-stakes assessment decisions, there was a lack of serious monitoring and professionalism about their impressions of students.
Updated and expanded, this second edition satisfies the same philosophical objective as the first -- to show the importance of problem posing. Although interest in mathematical problem solving increased during the past decade, problem posing remained relatively ignored. The Art of Problem Posing draws attention to this equally important act and is the innovator in the field. Special features include: •an exploration ofthe logical relationship between problem posing and problem solving •a special chapter devoted to teaching problem posing as a separate course •sketches, drawings, diagrams, and cartoons that illustrate the schemes proposed a special section on writing in mathematics. © 1990 by Stephen I. Brown and Marion I. Walter. All rights reserved.
Abilities are always abilities for a definite kind of activity ; they exist only in a person's specific activity. Therefore they can show up only on the basis of an analysis of a specific activity. Accordingly, a<span style=background-color: #ffff00;> mathematical ability exists only in a mathematical activity and should be manifested in it. (S. 66)
