Teaching and Learning Patterns in School Mathematics: Psychological and Pedagogical Considerations
Abstract
This book synthesizes research findings on patterns in the last twenty years or so in order to argue for a theory of graded representations in pattern generalization. While research results drawn from investigations conducted with different age-level groups have sufficiently demonstrated varying shifts in structural awareness and competence, which influence the eventual shape of an intended generalization, such shifts, however, are not necessarily permanent due to other pertinent factors such as the complexity of patterning tasks. The book proposes an alternative view of pattern generalization, that is, one that is not about shifts or transition phases but graded depending on individual experiences with target patterns. The theory of graded representations involving pattern generalization offers a much more robust understanding of differences in patterning competence since it is sensitive to varying levels of entry into generalization. Empirical evidence will be provided to demonstrate this alternative view, which is drawn from the author's longitudinal work with elementary and middle school children, including several investigations conducted with preservice elementary majors. Two chapters of the book will be devoted to extending pattern generalization activity to arithmetic and algebraic learning of concepts and processes. The concluding chapter addresses the pedagogical significance of pattern learning in the school mathematics curriculum. © 2013 Springer Science+Business Media Dordrecht. All rights are reserved.
Chapters (6)
In this chapter, we discuss issues of depth that are relevant to the concept and process of generalization. We clarify the following useful terms that are now commonly used in patterns research: abduction; induction; near generalization and far generalization; and deduction. We also explore nuances in the meaning of generalization that have been used in different contexts in the school mathematics curriculum. In the closing section, we begin to discuss implications of the findings in the chapter on our proposal of a theory of graded pattern generalization that we explore in some detail in Chap. 4.
In this chapter, we synthesize at least 20 years of research studies on pattern generalization that have been conducted with younger and older students in different parts of the globe. Central to pattern generalization are the inferential processes of abduction, induction, and deduction that we discussed in some detail in Chaps. 1 and 2 and now take as given in this chapter. Here we explore the other equally important (and overlapping) dimensions of pattern generalization, namely: natures and sources of generalization; types of structures; ways of attending to structures; and modes of representing and understanding generalizations. In this chapter we remain consistent as before in articulating the complexity of pattern generalization due to differences in, and the simultaneous layering of, processes relevant to constructing, expressing, and justifying interpreted structures.
In this chapter, we initially clarify what we mean by an emergent structure from a parallel distributed processing (PDP) point of view. Then we contrast an emergent structure from other well-known points of view of structures in cognitive science, in particular, symbol structures, theory-theory structures, and probabilistic structures. We also expound on the theory of PDP in semantic cognition in some detail and close the chapter with a discussion of the implications of the PDP theory on pattern generalization processes that matter to mathematical learning. In the closing discussion we discuss the need to modify some of the elements in the original PDP model based on cognitive factors that bear on pattern generalization processes involving school mathematical patterns. Further, we demonstrate the usefulness of a PDP network structure primarily as a thinking model that enables us to describe the complexity of students’ pattern generalization processes not in terms of transitions from, say, arithmetical to algebraic generalizations, but as parallel and graded, adaptive, and fundamentally distributed among, and dependent on, a variety of cognitive and extracognitive sources.
In this chapter, we focus on pattern generalization studies that have been conducted with elementary school children from Grades 1 through 5 (ages 6 through 10 years) in different contexts. Our contribution to the current research based on elementary students’ understanding of patterns involves extrapolating the graded nature of their pattern generalization schemes on the basis of their constructed structures, incipient generalizations, and the use of various representational forms such as gestures, words, and arithmetical symbols in conveying their expressions of generality. The gradedness condition foregrounds the dynamic emergence of parallel types of pattern generalization processing that is sensitive to a complex of factors (cognitive, sociocultural, neural, constraints in curriculum content, nature and type of tasks, etc.), where progression is seen not in linear terms but as states that continually evolve based on more learning. In a graded pattern generalization processing view, there are no prescribed stages or fixed rules but only states of conceptual coalescences and coherent covariations that change with more experiences. The chapter addresses different aspects of pattern generalization processing that matter to elementary school children. We also explore approximate and exact pattern generalizations along three dimensions, namely: whole number knowledge, shape sensitivity, and figural competence. We further discuss the representational modes that elementary students oftentimes use to capture their emergent structures and incipient generalizations. These modes include gestural, pictorial, verbal, and numerical. In another section, we address grade-level appropriate use and understanding of variables via the notions of intuited and tacit variables. We close the section with an analysis of the relationship between elementary children’s structural incipient generalizations and the natural emergence of their understanding of functions.
In this chapter, we explore the graded pattern generalization processing of older children and adults. Graded patterning processing occurs along several routes depending on the nature and complexity of a task being analyzed. That is, students’ graded pattern generalization processing and conversion can change in emphasis from manipulating objects to relationships (and possibly back to objects) in numerical or figural contexts (and possibly both). Toward the end of the chapter, we discuss how older students’ understanding of (linear) functions as an instance of generalizing extensions emerged from their experiences in pattern generalization activity.
In this concluding chapter, we discuss ways in which algebra can be grounded in patterning activity. As a consequence, the development of algebraic generalization is also graded from nonsymbolic, to pre-symbolic, and finally to symbolic, reflective of the conceptual changes that occurred in the history of the subject. Over the course of four sections, we clarify the following points: (1) the different contexts of patterning activity and the kinds of algebraic generalizations they generate; (2) the relationship between arithmetical thinking and context-based structural thinking; (3) the grounding of algebra, functions, and models in nonsymbolic and pre-symbolic algebraic contexts; and (4) the graded vs. transitional nature of pattern generalization.
... Flera forskare (se t.ex. Ainley et al., 2003;Lee, 1996;Mason, 1996;Rivera, 2013) hävdar att användning av mönsteruppgifter i matematikundervisningen är en av flera vägar att introducera algebra för elever. Proportionalitetsresonemang anses vara en av fem grundläggande idéer (Blanton et al., 2015) för tidig algebraundervisning där ekvationer jämte proportionella och lineära samband bildar kärnan i innehållet. ...
... Flera länder, bland andra Portugal, Taiwan, Thailand och Libanon, har infört aktiviteter med mönster i sina läroplaner (Rivera, 2013). Likaså finns mönster med i den svenska läroplanen från 2011 (Skolverket, 2011b). ...
... Ett matematiskt mönster kan beskrivas som en förutbestämd regelbunden struktur, konsekvent uppbyggd av element, som har ett numeriskt, spatialt, eller annat samband (Mulligan & Mitchelmore, 2009). I skolans matematikundervisning möter elever från årskurs 1-3 geometriska mönster inom algebra (Skolverket, 2018(Skolverket, , 2021a som är av spatial typ, se Rivera (2013) för flera exempel på mönstertyper. Växande geometriska mönster beskrivs som en följd av geometriska konfigurationer som utvecklas likformigt enligt en fix procedur 22 (Strømskag Måsøval, 2011). ...
The purpose of this study is to shed light on how static and dynamic proportionality is treated by authors of teaching materials, on national tests and by teachers and students in the classroom, as well as how students encounter mathematics tasks where proportional reasoning is an option. The research is based on two sets of empirical data. In concrete terms, the thesis includes three studies examining three themes that relate to proportionality in classroom interaction and in texts. The first study analyses how proportionality is presented in some Swedish textbooks, in curricular texts and national course tests in mathematics for students in upper secondary school. The second study is a case study of how a teacher instructs and explains a task in a class in Grade 6, where proportional reasoning is a possible solution technique. Finally, the third study concerns how students in Grade 6 handle proportional reasoning when they encounter a patterning task involving proportional relationships. The analyses of textbooks and national course tests show that proportionality is handled differently in these two settings in the context of “Mathematics A” at the upper secondary school. About a quarter of the tasks in the textbooks and the national course tests involved proportionality tasks of one specific kind (missing value). Other types of proportionality tasks were infrequent. The results of the classroom studies show that students are able to engage in early forms of proportional reasoning before being taught about proportionality as a mathematical concept. The concept of learning trajectory is used to identify situations in the learning process during instruction where students meet obstacles and need scaffolding and teacher support. It is shown how a teacher dealing with a mathematical task involving mixtures of liquids encounters a task that has the possibility of making proportional reasoning visible for the students, and how she struggles to make the modelling required intelligible to herself and to the students. The instructional strategy of using everyday problems as a basis for learning implies that the initial modelling phase becomes crucial, and the students have to be aware of the conditions and limitations under which proportional reasoning is applicable. In conclusion, students engage in early forms of proportional reasoning well ahead of formal instruction. The difficulties they experience as they are to develop their proficiency, and where they require support from the teacher, concern how to model the familiar, everyday situations they encounter in exercises in mathematically precise and productive ways. In addition, in textbooks and national course tests proportionality is presented in a standardized, and rather simplified, form, and it is not sufficiently connected to the various areas of mathematics teaching and learning where it is applicable.
... Flertalet forskare ser ut att vara överens om att mönstergeneraliseringar handlar om att jämföra matematiska aspekter av mönster och sedan med utgångspunkt i jämförelserna göra generaliseringar (jfr Rivera, 2013). Med andra ord beskrivs det som att elever identifierar likheter och olikheter och med utgångspunkt i dem söker det generella (Radford, 2006;Rivera, 2013). Det handlar till exempel om att sätta enskilda elements matematiska aspekter i relation till varandra (Rivera, 2013). ...
... Med andra ord beskrivs det som att elever identifierar likheter och olikheter och med utgångspunkt i dem söker det generella (Radford, 2006;Rivera, 2013). Det handlar till exempel om att sätta enskilda elements matematiska aspekter i relation till varandra (Rivera, 2013). En algebraisk mönstergeneralisering beskrivs som en process i tre steg, (1) att urskilja några givna elements gemensamhet, (2) att utvidga eller generalisera denna gemensamhet till andra element och (3) att använda gemensamheten för att tillhandahålla ett generellt uttryck (verbalt, med gester eller med matematiska symbolspråket) (Radford, 2008). ...
... När undervisningen tar sin utgångspunkt i mönster, som föreliggande forskningsprojekt, handlar det om att elever ska ges möjlighet att urskilja mönsters strukturella egenskaper och sedan kunna abstrahera dessa att gälla godtyckliga element i mönstret. Elevers utforskande av mönsters underliggande matematiska strukturer beskrivs ta form genom att elever jämför matematiska aspekter av mönster och sedan med utgångspunkt i de jämförelserna uttrycker generaliseringar (jfr Rivera, 2013). ...
The purpose of this licentiate thesis is to study the aspects of the teaching that enable students of younger ages to be engaged in algebraic work. Learning study has been used as the method to produce data. A research team consisting of two primary school teachers in mathematics and a teacher researcher worked collaboratively, designing interventions iteratively during the learning study process. In the design as well as analysis, Davydov's learning activity theory, Variation theory and Radford's definition of algebraic pattern generalizations have been used as theoretical starting points. The empirical data consists of (1) video-recorded interviews with eight students as well as transcriptions thereof; (2) video recordings of three research lessons; (3) lesson plans; (4) synopsis of video recordings of three research lessons; (5) transcriptions of parts of video recorded research lessons.Results consists of three identified critical aspects that students may need to discern in order to express and justify for a pattern generalization algebraically: (a) to discern the relationship between the position of an element and the number of components; (b) to discern how to use the relationship between the position of an element and the number of components to predict an arbitrary element in the pattern; (c) to discern the constant (the component that does not change but is the same in all elements) in the pattern.Results give examples of what functions the theoretical principles of Davydov´s learning activity, problem situation, learning model and contradictions, may have for algebraic work to be established and maintained. Furthermore, the results may contribute to a deepened understanding of what it means to be able to express and justify for pattern generalizations algebraically at younger ages. The results may also contribute to knowledge that can be used by teachers to stage and carry out a teaching within the frame of early algebra.
... Thus, it is worth noting that algebraic thinking involves identifying and analysing patterns, studying and representing relationships and changes, and generalisation. In more detail, generalisation might be a generalising number, shape, or figural properties (9). ...
... In this step, they really implemented Pólya's suggestions to solve a related problem, which is more accessible or more general (12). Furthermore, the whole discussion of the first problem covered all the generalisation processes presented by (9), which start from concrete numerical experience and continue to a near or far member of the sequence until they reach the general term. The next problem proposed by AF is also connected to generalisation as it gave them the opportunity to find the general form for the area. ...
... During this step, they also tested the consistency of growth relative to the same pattern. Finally, it is hoped that the entire lesson will contribute to dealing with the unresolved issue in pattern generalisation as mentioned by (9), namely, how to make students see generalisation as both a process (a method of calculating) and a concept (conveying an interpreted structure of a collection of stages). ...
Algebraic thinking has been central to the mathematics curriculum. Its development is regarded as critical for solving mathematical problems. Considering that algebraic thinking among Indonesian students and prospective mathematics teachers has been found less satisfying, this research aims to depict a mathematical lesson using a problem-posing approach, resulting in generated problems that potentially promote algebraic thinking. By involving three Indonesian prospective teachers, this study leads to answering the question: What does instruction using a problem-posing approach look like, and how does the self-proposed problem promote algebraic thinking? The data came from a single episode of an active learning workshop intended to introduce active learning strategies. One of the strategies is to use problem posing as a pedagogical tool. In the lesson, the participants were asked to pose problems based on the given picture, choose one of the posed problems, and discuss the solution. The lesson was analysed qualitatively in nature, which revealed that in addition to prospective teachers who could compose mathematical tasks, mathematical problem posing promotes the emergence of manifestation stimulating algebraic thinking. The implications of these findings are discussed in detail.
... In fact, patterns are a concrete representation of functional relationships. Therefore, FPG have a unique capacity to promote functional thinking (Markworth, 2010;Rivera, 2013) as a gateway into algebraic thinking (Carraher & Schliemann, 2019). Thus, studying and exploring the mental constructs of students in FPG may help to facilitate this promotion. ...
... There are also a few studies that examined the generalization of patterns with the APOS framework and adapted the Action-Process-Object-Schema stages to the pattern generalization (Sutarto et al, 2016;Sutarto et al., 2018;Yuniati et al., 2020). Rivera (2013) synthesized at least 20 years of research studies on pattern generalization and organized a framework that took into account various aspects of pattern generalization. One aspect that Rivera has mentioned related to the types of structures. ...
... FPG has a unique capacity to enhance functional thinking (Markworth, 2010). Rivera (2013), after studying 11-year-old students longitudinally, states that students were able to solve modeling/function questions after working with pattern generalization tasks without being trained in modeling and functions. So, without any doubt, attention to figural pattern generalization will promote functional thinking. ...
Figural patterns have a unique capacity to promote functional thinking. This study aimed to identify the mental constructs of 7th-grade students in Figural Pattern Generalization (FPG) by using the Action, Process, Object, Schema (APOS) theory in order to develop a framework for evaluating students' understanding of FPG. A sample of 220 students completed a test designed based on the APOS framework and 19 students participated in a semi-structured interview. Results showed that there are emergent and partial action levels before the action stage and pre-emergent, and partial process/object levels before the process/object stage.
... Being aware that lower secondary students are in a transitional phase from arithmetic to algebraic thinking, we wanted to examine their way of thinking. We were intrigued by the book of Rivera (2013), according to which, in the process of problem-solving, secondary students, even adults are more likely to provide an empirical (numerical or visual) than a formal explanation. As Rivera, we also argue that more patterning activities might provide an opportunity to improve formal explanations and deepen students' algebraic thinking. ...
... We formulated our research question based on Rivera's (2013) findings. We wanted to determine whether 7 th -grade students reason more algebraically or arithmetically when posing and solving patterning problems. ...
... Patterning or pattern recognition is the search for regularities and structures (Clements & Sarama, 2009). Many different kinds of patterns in the school mathematics curriculum can be represented numerically or figurally (Rivera, 2013) and can be used in the classroom to promote problem-solving and posing. Pehkonen (1997) regards problem-posing as a particular type of problem-solving, indicating that problem-posing and problem-solving are deeply connected. ...
Based on the literature problem-posing is one possible way to observe students’ thinking. We (the author collaborating with two university experts) designed a chapter from the curriculum suitable for online learning that includes problem-solving and problem-posing. Sixty-one seventh-grade students were asked to solve patterning problems and pose problems based on model problems. This paper aims to analyze the characteristics of students’ problem-posing concerning patterning activities.
... Dos seis participantes, P18 é o único que não evidencia qual seria a regularidade a ser abordada, ao mencionar que "queremos que os alunos vão em busca de alguma relação para encontrar o padrão". Já os participantes P2, P3, P7, P9 e P10 acabam evidenciando apreensões da regularidade que vão na direção da ideia do que varia e do que não varia (Rivera, 2013) quando apontam os seguintes aspectos: "número de quadrados aumenta em uma razão 3 a cada etapa" (P2), "Percebam que em todas as operações haverá um valor um valor fixo e o outro se altera (...)" (P3), "todos os pontos da circunferência estão equidistantes do centro" (P7), "direcionar os alunos a observarem que a operação utilizada foi multiplicar por 2 o número de mesas e depois a adição de mais 2" (P9) e "perceber que estamos trabalhando com potências de base 2" (P10). ...
... Os cinco participantes buscam envolver os alunos na percepção de uma regularidade (Radford, 2006) como "fazê-los observar a relação que há entre as maneiras e a quantidade de formandos" (P15). No entanto, os participantes P1, P5 e P8 estão mais atentos às apreensões da regularidade que vão na direção da ideia do que varia e do que não varia (Rivera, 2013) Padrão algébrico. O participante P5 apresenta apenas a seguinte expressão/fórmula genérica: f(x) = ax + b. ...
... O participante P5 apresenta apenas a seguinte expressão/fórmula genérica: f(x) = ax + b. Neste caso, não foi evidenciado qual simbologia varia e qual não varia (Rivera, 2013 Em seguida, o professor pode discutir com os grupos para juntos formalizarem o conteúdo: f(x) = ax + b. ...
Resumo. O objetivo do presente estudo é compreender o planejamento de propostas de ensino de futuros professores de matemática para abordar a generalização de padrões algébricos no EAMvRP, direcionadas a alunos do ensino médio. Adotamos os pressupostos da pesquisa qualitativa, baseados na vertente descritiva e interpretativa que gerou a análise dos dados, obtidos das propostas de ensino de 18 licenciandos em matemática para abordar conteúdos do ensino médio. Os resultados mostraram que as escolhas das situações de Matemática envolveram a apresentação de casos particulares, a obtenção de solução por meio da construção de expressões matemáticas e a previsão de estratégias de busca de padrões algébricos por parte dos futuros professores. Verificam-se dificuldades na condução do processo de generalização algébrica com base no uso de casos particulares, de discutir a busca de regularidades e na articulação das expressões matemáticas obtidas com base nas simbologias dos contextos das situações de Matemática. Concluímos que utilizar o EAMvRP para abordar a generalização de padrões algébricos contribui para delimitar aspectos que direcionam os futuros professores de Matemática a aprenderem a ensinar e a se desenvolverem profissionalmente. Palavras-chave: ensino de matemática; pensamento algébrico; formação inicial de professores. Abstract. The aim of the present study is to understand the planning of teaching proposals of preservice mathematics teachers to address the generalization of algebraic patterns in MTLvPS, directed to high school students. We adopted the assumptions of qualitative research, based on the descriptive and interpretive strand that generated the data analysis, obtained from the teaching proposals of 18 mathematics undergraduates to address high school contents. The results showed that the choices of mathematical situations involved the presentation of particular cases, obtaining solutions by constructing mathematical expressions, and foreseeing strategies for finding algebraic patterns on the part of future teachers. There are difficulties in conducting the process of algebraic
... Con base en esto se construye y valida una regla para el término +1 que depende del término anterior (campo inferencial). Este proceso es útil para determinar términos "cercanos" de una sucesión (Radford, 2010;Rivera 2013;Vergel, 2015). La generalización algebraica se evidencia cuando se construye una regla o expresión verbal, basada en los términos del campo perceptual o en la regla recursiva, que determina una relación de dependencia entre y el término de la secuencia (sin la necesidad de obtener el término anterior −1 ), para ℕ. Ésta funciona para determinar valores "cercanos" y "lejanos" (Radford, 2010;Rivera 2013;Vergel, 2015). ...
... Este proceso es útil para determinar términos "cercanos" de una sucesión (Radford, 2010;Rivera 2013;Vergel, 2015). La generalización algebraica se evidencia cuando se construye una regla o expresión verbal, basada en los términos del campo perceptual o en la regla recursiva, que determina una relación de dependencia entre y el término de la secuencia (sin la necesidad de obtener el término anterior −1 ), para ℕ. Ésta funciona para determinar valores "cercanos" y "lejanos" (Radford, 2010;Rivera 2013;Vergel, 2015). Finalmente, la teoría que sustenta la es el Análisis Matemático. ...
... Es decir, constituyen un caso representativo de estudio (Thomas, 2015). Figura 6. Tareas de la situación 3 presentadas en la cuarta sesión del CM Esta situación, implementada en la tercera sesión, es una adaptación de un problema propuesto en Rivera (2013), para ejemplificar tres tipos de razonamiento inferencial: abductivo, inductivo y deductivo, durante el proceso de generalización y permite a los estudiantes tener un primer acercamiento a la formulación de expresiones cuadráticas del tipo 2 + + , para , , , ∈ ℕ . Con la Tarea 1 se pretende que los estudiantes construyan técnicas que podrían ser implementadas en la Tarea 2. Ambas tareas consisten en determinar una regla algebraica que permita obtener cualquier término, cercano o lejano de la secuencia (por ejemplo, = 10, = 20, = 500). ...
Se presenta un modelo teórico para el estudio del talento matemático, fundamentado en la Teoría Antropológica de lo Didáctico y la noción de creatividad. En dicho modelo se proponen dos componentes de la actividad matemática creativa: la Componente Matemática, que sustenta las técnicas matemáticas; y la Componente Creativa, definida por cuatro funciones: producir técnicas nuevas, optimizar técnicas, considerar tareas desde diversos ángulos y adaptar una técnica. Con base en los modelos Teórico y Epistemológico de Referencia sobre sucesiones infinitas, se genera un diseño didáctico conformado por seis situaciones problemáticas y se implementa en una institución creada para potenciar el talento matemático. El análisis de dos tareas realizadas por una pareja de niños constituye un estudio de caso, que permite ilustrar que enfrentar tareas retadoras de un mismo tipo, bajo condiciones institucionales propicias, posibilita el desarrollo del talento matemático.
... With growing pattern generalization, differing types of pictorial patterns, task representation, and prior experience, have elicited various approaches and levels of performance. More recently, it has been found that much younger students can notice the functional relationship between variables in a growing pattern (e.g., Brizuela, Blanton, Sawrey, Newman-Owens, & Murphy Gardiner, 2015) and that certain types of pictorial patterns can support a correspondence approach leading to explicit generalization (e.g., Rivera, 2013;Wilkie, 2016). ...
... Accumulated research has focused on students' approaches to generalizing functional relationships, varying in the types of patterns, the ages of students studied and their approaches, and subsequently in the level of generalization they present (e.g., Carraher & Schliemann, 2007;Dörfler, 2008;Radford, 2006;Rivera, 2013;Stacey, 1989). This research has highlighted students' difficulties in reaching algebraic generalizations (e.g., Orton et al., 1999;Stacey, 1989;Steele, 2008). ...
... With figural growing patterns, Kaput (2008) suggested that if the independent variable (the item or stage number) is not clearly visible alongside each growing pattern figure, the functional nature of the pattern is hidden, and students will continue to rely on recursive rather than explicit approaches to generalization. Rivera (2013) emphasized that figural growing patterns need to be chosen carefully so that their visual structure can be perceived in multiple ways and helps students to notice the indeterminate relationship between the dependent and independent variables. ...
Researching students’ responses to tasks at different year levels and in varied
curriculum contexts can provide insights that relate their understandings to
prior learning experiences and teaching approaches. In this article, we discuss
evidence of students from three curriculum contexts (English, Australian, and
Israeli) (n = 350) ways of attending to the functional relationship between
two variables at different levels of secondary school in their responses to two
linear functions tasks. We found that the students from an English national
curriculum context were more likely than the other cohorts to focus only on
the dependent variable when presented with a table of values. With a figural
pattern generalization task, no examples of invalid proportional reasoning
were found among student responses from the Israeli curriculum context,
unlike those in the English and Australian curriculum contexts. A high percentage
of the Year seven-eighths responses from the Israeli curriculum
context evidenced symbolic generalization whereas several responses from
the Australian curriculum indicated explicit descriptive, but not symbolic,
generalization. Twenty teachers also participated in the study. Analysis of
national curriculum content and task examples from each context, along with
the teachers’ expectations of their students’ responses to the tasks and
described teaching approaches for linear functions, led to our exploration
of possible reasons for differences and similarities found among the
responses. We suggest possible implications for teaching and learning linear
functions and avenues for further research.
... Semiotic representations not only register what students understand but also help students structure and extend their thinking, with different representations highlighting different aspects of a mathematical object. Indeed, research on pattern generalization has shown that learners' generalizing activities are shaped by the representation (e.g., figural, tabula, or numeral) of a task, the mathematical relationship observed in the representation, and the representation used to express and justify the observed generality (e.g., Steele, 2008;Rivera, 2013;Wilkie, 2016). This implies that semiotic representations mobilized in the process of generalizing mediate what is generalized and how generalization is produced, expressed, and validated. ...
... Eric's formula ( − 2) × !(!#$) & − ( − 3) for p-sided polygonal numbers was developed by structuring a polygon array into triangular arrays. According to Rivera (2013), and in line with Mason et al. (2009), the development of structural thinking in pattern generalization involves recognizing a relationship of similarity or difference in a situation, followed by perceiving the discerned relationship as a general property that characterizes the objects being analyzed, and then by reasoning based on an observed property. These two ways in which representation conversions supported transition from empirical to structural generalizations correspond to different developmental stages of structural thinking. ...
Pattern generalization can be either empirical or structural. A considerable body of research on pattern generalization has shown that learners of different ages tend to generalize based on numerical evidence or visual perception alone rather than an underlying structure of a pattern. Although there is evidence that learners can start with empirical investigation and then shift their attention to the structural aspects of a problem, mechanism of such a transition remains underexplored. Relying on Duval’s theory of registers of semiotic representations, this study used data from task-based interviews to examine the extent to which transformations of semiotic representations can support transition from empirical to structural generalization in figural pattern generalization tasks. This study identified structurally useful treatment and mathematically significant conversion as two types of representation transformations that supported transition from empirical to structural generalization. In contrast to structurally useful treatment, mathematically significant conversions often lead to generalizations that explains why the generalizations are the right ones. Moreover, this study identified two ways in which representation conversions supported transition from empirical to structural generalization: (1) by supporting addition of a structural insight to an empirical generalization; (2) by supporting reasoning based on an observed property. These results contribute to our understanding of the role of representation transformations in transition from empirical to structural generalization.
... Dealing with growing figural patterns is particularly important for the initiation and development of algebraic thinking (Rivera, 2010a;Warren & Cooper, 2008). Here the focus is on the generalization of mathematical relationships describing the pattern's structure (Carraher et al., 2008;Rivera, 2013) which, for example, can be used to continue the pattern figure by figure, to determine more distant figures directly (far generalization; Stacey, 1989), or to formulate algebraic generalizations (Radford, 2006). Generalizing is considered very challenging, especially for younger students (Rivera, 2010b;Stacey, 1989). ...
... (Assmus, 2018;Fritzlar & Karpinski-Siebold, 2012;Käpnick, 1998). Connections between mathematical ability or giftedness and success in pattern generalization tasks are obvious because pattern generalization requires a variety of competencies in dealing with numbers and number relationships, with shapes and similarity, and with figural properties (Rivera, 2013(Rivera, , 2018). ...
Relationships between mathematical giftedness and mathematical creativity have been widely studied, but few studies are
available for primary school age. For an investigation in this age group, it seems appropriate to use a content area that not
only has high relevance for mathematics and special potentials for creativity, but also requires only a little knowledge and is
easily accessible. We therefore investigated whether mathematically gifted primary school students differ from non-gifted
ones in high creativity in dealing with mathematical patterns and structures. This question was explored in an interview
study in which 24 third graders were asked to invent as many different figural patterns as possible, which enabled creative
mathematical activity also by combining arithmetic and geometric aspects. A detailed qualitative analysis of the data revealed
among other results several types of flexibility concerning the invention of patterns. The selection of students ensured that all
participants performed well to very well in regular mathematics classes and that 14 of them could additionally be assumed
to be mathematically gifted based on a specific test. This allowed a comparison of both subgroups. Results indicate a high
correspondence between mathematical giftedness and mathematical creativity concerning the invention of figural patterns.
... So much so that different psychological tests involve a variety of patterning tasks as part of their study area (e.g., Kaufman & Kaufman, 1983;Raven et al., 1992;Wechsler, 1991). With regard to mathematics development, the literature documents a similar situation, with studies on number patterns (Baroody et al.,2015;Zazkis & Liljedahl, 2002), growing/shrinking patterns (English & Warren, 1998;Papic, 2015), spatial/geometric patterns (Hendricks et al., 2006), repeating patterns (Rittle-Johnson et al., 2019;Threlfall, 1999), structural generalizations (determining pattern rules, Callejo et al., 2019;Economopoulos, 1998;Rivera, 2013;Zazkis & Liljedahl, 2002) or abstract language patterns . ...
... Tanto es así, que diferentes test psicológicos incluyen una amplia variedad de tareas de reconocimiento de patrones de repetición como parte de su evaluación (por ejemplo, Kaufman & Kaufman, 1983;Raven et al., 1992;Wechsler, 1991). Con respecto al desarrollo matemático, la literatura documenta una situación similar, con estudios sobre patrones numéricos (Baroody et al.,2015;Zazkis & Liljedahl, 2002), patrones de crecimiento/decrecimiento (English & Warren, 1998;Papic, 2015), patrones espaciales/ geométricos (Hendricks et al., 2006), patrones de repetición (Rittle-Johnson et al., 2019;Threlfall, 1999), estructuras generalizables (determinación de reglas en patrones, Callejo et al., 2019;Economopoulos, 1998;Rivera, 2013;Zazkis & Liljedahl, 2002) o patrones con lenguaje abstracto . La investigación empírica que se está llevando a cabo actualmente está proporcionando evidencias que indican que el conocimiento relacionado con los patrones de repetición puede desempeñar un papel fundamental en el rendimiento matemático y académico de los estudiantes. ...
Patterning, as a component of early mathematic knowledge, is a common activity carried out at elementary levels in which children are not equally successful. This study aimed to measure different variables affecting performance on patterning tasks in early childhood. For this purpose, the success of Pre-K (N = 33), K (N = 31) and first-grade (N = 33) children when solving 14 repeating-pattern tasks, which varied in complexity, was analysed. The results revealed no significant differences between core-2 and core-4 length patterns, and greater success with patterns involving size. The study also introduces distractors, as a novel factor, in the patterning activities related to the presence of contradictory or surplus data. The impact of each factor and the relations between complexity and difficulty on the patterning tasks are discussed in order to contribute to the design of teaching itineraries.
... Términos clave: Estructuras matemáticas; Generalización de patrones; Pensamiento algebraico; Pensamiento matemático; Vista distribuida del procesamiento de la generalización de patrones Pattern generalization ability involves the proficiency to construct and justify an interpreted well-defined structure from a constrained set of initial cues (Rivera, 2013). Such structure is mathematical in which case it refers to "a mental construct that satisfies a collection of explicit formal rules on which mathematical reasoning can be carried out" (National Research Council, 2013, p. 29). ...
... Consider the following interview episode below with a US second-grade student named Skype (S; age 7 years) who was asked to obtain a PG for the Beam pattern task shown in Figure 1 in a clinical interview setting that took place after a one-week teaching experiment on growing figural patterns (Rivera, 2013). Dur-ing the experiment, the students in Skype's class explored linear patterns to begin to develop the habit of noticing and paying attention to parts in figural stages that appeared to them as being common and shared across the given stages. ...
Drawing on a review of recent work conducted in the area of pattern generalization (PG), this paper makes a case for a distributed view of PG, which basically situates processing ability in terms of convergences among several different factors that influence PG. Consequently, the distributed nature leads to different types of PG that depend on the nature of a given PG task and a host of cognitive, sociocultural, classroom-related, and unexplored factors. Individual learners draw on a complex net of parallel choices, where every choice depends on the strength of ongoing training and connections among factors, with some factors appearing to be more predictable than others.La naturaleza distribuida de la generalización de patronesSobre la base de una revisión de trabajos recientes en el área de generalización de patrones (PG), este artículo aboga por una visión distribuida de PG, que básicamente sitúa la capacidad de procesamiento en términos de convergencias entre diferentes factores que influyen en PG. En consecuencia, la naturaleza distribuida conduce a diferentes tipos de PG que dependen de la naturaleza de una tarea PG dada y una serie de factores cognitivos, socioculturales, inexplorados y relacionadas con el aula. Alumnos individuales se basan en una compleja red de opciones paralelas, donde cada elección depende de la fortaleza de la formación continua y las conexiones entre los factores, con algunos factores más predecibles que otros.Handle: http://hdl.handle.net/10481/34989WOS-ESCINº de citas en WOS (2017): 1 (Citas de 2º orden, 1)
... Patterning activities with repeating patterns are supposed to develop general mathematical concepts in children such as ordering, comparing, sequencing, classification, abstracting and generalizing rules and making predictions (see e.g., PME 2014 Threlfall, 1999). These concepts then lead to the development of mathematical reasoning in young children (English, 2004;Mulligan, & Mitchelmore, 2009, 2013. It is mostly in the area of algebra (or pre-algebra) that repeating pattern work is seen as a conceptual stepping stone (Threlfall, 1999). ...
... However, it is important to note that these assumptions have been mainly derived from either observation, a experience, or are theoretical considerations. From the empirical perspective, in the last decade there is a substantial body of research, mainly qualitative studies, focusing on patterning strategies and looking at the level of students' awareness of or attention to pattern and structure (see e.g., Mulligan, & Mitchelmore, 2009;Papic, Mulligan, & Mitchelmore, 2011;Radford, 2010;Rivera, 2013;Warren, & Cooper 2006. Few studies however have tried to quantitatively measure the significance of patterning abilities in the early years for later mathematics learning. ...
Recent studies in early mathematics education highlight the importance of patterning abilities and their influence on mathematics learning and the development of mathematical reasoning in young children. This paper focuses on young children's repeating patterning abilities and reports results from an ongoing four-year longitudinal study that investigates the development of early numeracy understanding of 408 children from one year prior to school until the end of grade 2. The analyses in this paper reveal a significant influence of young children's repeating patterning abilities one year prior to school on their mathematical competencies at the end of grade one.
... Nessa concepção, "é natural pensar as variáveis como generalizadoras de modelos" (Usiskin, 1994, p. 13). Rivera (2014) argumenta que a visualização permite que os estudantes construam uma estrutura mental clara de padrões, relacionando-a com expressões matemáticas generalizadas. Isso facilita a identificação da regularidade e da estrutura ao longo da construção de uma sequência. ...
O estudo investigou como futuros professores, brasileiros e portugueses, em formação inicial, resolviam problemas de generalização de padrões matemáticos, com ênfase nas múltiplas resoluções e na criatividade. A pesquisa adotou um paradigma qualitativo, exploratório, com foco na caracterização do desempenho dos participantes na resolução de tarefas envolvendo padrões figurativos, assim como nas dimensões da criatividade que emergiram nessas resoluções e nas variações entre os dois contextos. O estudo destaca a importância de integrar a visualização e a exploração de múltiplas representações na formação de futuros professores. Foi possível identificar que o foco na visualização em Portugal promove uma compreensão mais robusta das regularidades e generalizações algébricas e, no Brasil, o uso da notação simbólica favorece o desenvolvimento de habilidades formais.
... Sayma ve geometri ile ilişkili olduğundan müzik, sanat ve giysilerde bulunabilir; bu ilişkiler, ritimleri, tekrarları, sıralamaları ve sınıflamayı anlamayı desteklerken, sayıların düşünme yeteneklerini geliştirmesi ve geometri aracılığıyla fiziksel dünyayı sınıflandırmamıza yardımcı olması açısından da önemlidir (Akman, 2002). Örüntü aktiviteleri, çocukların genellemeler yapmasını (Burns, 2007;Threlfall, 1999;Rivera & Becker, 2007), ilişkileri anlamasını (Burns, 2007), matematiğin düzen ve mantığını çözümlemesini (Burns, 2007) ve cebirsel düşünme becerilerini geliştirmesini sağlar (Battista & Van Auken Borrow, 1998;Lin vd., 2004;Nathan & Kim, 2007;Rivera, 2013;Tanışlı & Özdaş, 2009;. Örüntüleri tanıyan öğrenciler cebirsel problemleri çözme ve anlamada gelişim gösterir (Battista & Van Auken Borrow, 1998). ...
Örüntüler, matematiksel kavramların öğrenilmesinde önemli bir yere sahiptir. Örüntüler matematik eğitiminde kural oluşturma, genelleme yapma ve cebirsel düşünme becerilerinin geliştirilmesinde etkilidir. Bu süreçte bireylerin bilişsel becerilerinin yanı sıra üstbilişsel farkındalıklarını kullanarak süreci planlamaları, yönetmeleri ve değerlendirmeleri büyük önem taşır. Bu araştırmada, sınıf öğretmeni adaylarının örüntüleri genelleme becerilerinin üstbilişsel farkındalık düzeylerine göre incelenmesi amaçlanmaktadır. Araştırma, Kuzey Anadolu’da bulunan bir üniversitenin Eğitim Fakültesi Sınıf Öğretmenliği Bölümünde öğrenim gören 122 öğretmen adayı ile gerçekleştirilmiştir. Nicel araştırma yöntemlerinden tarama modeli kullanılan çalışmada, veri toplama araçları olarak Bilişötesi Farkındalık Envanteri ve Örüntü Genelleme Testi kullanılmıştır. Verilerin analizi betimsel istatistikler, Mann Whitney U testi ve içerik analizi ile yapılmıştır. Araştırmanın sonuçlarına göre öğretmen adaylarının üstbilişsel farkındalık düzeyleri ile örüntü genelleme becerilerinin yüksek olduğu tespit edilmiştir. Öğretmen adayları örüntülerin matematiği günlük yaşamla ilişkilendirme, pratiklik kazandırma, problem çözme, formül oluşturma, genelleme yapma, mantıksal düşünme, zihinsel gelişim ve matematiksel becerilerin gelişimine önemli katkılar sunduğunu belirtmişlerdir. Ancak öğretmen adaylarının üstbilişsel farkındalık düzeylerine göre örüntü genelleme becerileri arasında anlamlı fark bulunamamıştır.
... Exploring patterns is a key opportunity to develop reasoning or thinking as the pattern with variety for children to develop recognition strategies and supports children's mental representation development (Taylor & Harris, 2014). Furthermore, Rivera (2013) describes if children can perform pattern generalization, which means they can combine their perceptual and symbolic inferential abilities to build and reasoning their structure or pattern. ...
This research aims to explore the mechanism of making sense number patterns at home, because of local pandemic COVID-19 outbreaks during the enforcing of the conditional motion control order (CMCO). The new mechanism framework is based on the blending of Skemp's understanding property, Tall’s idea of compression, and Chin's supportive and problematic conceptions framework. The researchers employed a qualitative case analysis study design, and a purposive sampling method was used to obtain a sample of one kindergarten child and one parent. Data were gathered in semi-structured parent interviews, direct observations, and analysis of documents such as worksheets and journal entries were performed to gain a comprehensive picture of a child’s making sense of number patterns. Results demonstrate that the mechanism was potential successful for helping the kindergarten child in making sense the number patterns. The child was able to make sense and recognise various number patterns at the end of this study.
... Both numerical and figural patterning have been used as a means of sensitizing young students to the generalizing inherent to algebraic activity, with many studies centering on growing figural patterns (Rivera 2013). With respect to numerical patterning, some studies (e.g., Mason et al. 2009) have focused on pattern recognition and detecting underlying structures, such as 121, 1221, and 12,221. ...
... In previous studies, the most common tasks were based on activities that show the first terms of a pattern and then ask the resolvers to copy, expand, complete, describe or generalise the pattern (e.g. Fujita & Yamamoto, 2011;Lüken, 2012;Morales et al., 2017;Papic et al., 2011;Rivera, 2013;Wijns et al., 2021). Dörfler (2008) noted that, primarily, this type of task provides strong guidance; the figurative clues and their layout practically exclude trying out other generalisations or continuations; furthermore, they show and describe a previously prescribed general structure, therefore implying there is only one way of continuing the sequence. ...
... However, researchers such as Warren (2005) and Wilkie (2016) have suggested that growing shape patterns can be used to improve students' conceptual learning in algebra, their skills in expressing generalisations symbolically, and their reasoning. Because shape patterns support students to generalisation and associate different representations from an early age (Amit & Neria, 2008;Rivera, 2013). Therefore, this study, in which a shape pattern task is used, is aimed to present how secondary school students can support their generalisation processes by allowing them to use multiple representations flexibly through GeoGebra software, which is a graphic visualisation tool. ...
The aim of this study is to show how the GeoGebra software-supported collaborative learning environment contributes to a group of students using multiple representations and reaching generalisations while solving problems collectively. Blending individual and collaborative learning activities were carried out with a learning method called ACODESA (from French: Apprentissage Collaboratif, Débat Scientifique, Auto-réflexion). Within this context, four 8th-grade students participating in the study were asked to solve the pattern problem by using the GeoGebra software and following the stages of the ACODESA method. The data were collected via a pattern problem developed by the researcher, students’ audio records and screen recordings, GeoGebra files and written productions. The data were analysed using Duval’s theory of registers of semiotic representation and Toulmin’s model. The Toulmin model was used to analyse students’ argument structures, that is, to determine their reasoning structures. The data analysis revealed that a GeoGebra software-supported collaborative learning environment supported secondary school students in using multiple representations, making algebraic reasoning and generalisation. The discussions in this process contributed to the students’ transformation of representations and the evolution of their representations. Providing students with the opportunity to express and examine the pattern with multiple representations, GeoGebra software helped students to obtain data, test their conjectures and thus reach generalisation.
... La vignette présentée dans cette section est issue de la deuxième journée de formation durant laquelle les chercheurs souhaitaient faire réfléchir les enseignants sur les types d'aide à fournir aux élèves lorsqu'ils s'engagent dans une généralisation s'appuyant sur une analyse incorrecte de la suite. Du point de vue des chercheurs, ce choix de thématique se justifie par le fait que la littérature de recherche montre que les enseignants éprouvent des difficultés à analyser des stratégies incorrectes et ont tendance, dans leurs interventions, à engager les élèves dans la voie qu'ils ont euxmêmes utilisées pour résoudre la tâche, plutôt que d'encourager les élèves à réguler leur propre démarche (Rivera, 2013). ...
Based on the framework of meta-didactic transposition analysis (Arzarello et al., 2014) and specifically the concepts of brokering and boundary object, this paper studies how knowledge to teach algebra is exchanged during a training program involving nine mathematics teachers and two researchers specialised in algebra teaching and learning. Organized in three half-day sessions, this program is based on a problem pointed out in the research literature as particularly rich to develop algebraic thinking. In addition, the materials used in training are come directly from the classes of the teachers participating in the program. In this sense, the program values knowledge that makes sense in both research and teaching practice. The analysis of interactions between researchers and teachers highlights three types of collaborative activities between the two groups and thus questions the potential of such a mechanism to foster integration of research results by teachers.
... The second level of our classification of algebraic generalisation is noticing and describing regularities, where students are asked to notice regularities among a sequence of particular cases. In these cases, students attend to quantities that stay fixed and those that vary (Radford, 2006;Rivera, 2013) within the context of the task. This is an important level as the next three algebraic generalisation levels rely upon being able to notice regularities. ...
This symposium will draw on the evidenced-based learning progressions for multiplicative thinking, algebraic reasoning, geometrical reasoning, and statistical reasoning presented at previous MERGA conferences (see references by symposium authors in the papers that follow). The four papers will consider key shifts in thinking identified within each progression, without which students’ progress may be seriously constrained.
... In fact, distinguishing between what is invariant and what it is that is varying constitutes a crucial first stage in the activity of patterning (Kieran, 2006;Mason, 2005). According to Rivera (2013), and in line with Mason et al. (2009), the development of attention to structural aspects within patterning activity involves the recognition of relationships of similarity and difference within a structure, followed by the perceiving of properties that characterize the objects being analyzed, and then by reasoning on the basis of the identified properties. And for the case of figural patterns, Radford (2011, p. 19) argues that: ...
Early algebraic thinking is the reasoning engaged in by 5- to 12-year-olds as they build meaning for the objects and ways of thinking to be encountered within the later study of secondary school algebra. Ever since the 1990s when interest in developing algebraic thinking in the earlier grades began to emerge, there has been a steady growth in the research devoted to exploring ways of fostering this thinking. While in its early days this research had to grapple with the question of what kinds of algebraic thinking might be feasible for the younger student, the evolution of the field over the past 30 years has led to an ever-increasing range of activity that is truly multi-dimensional. In this survey paper, I have framed the multi-dimensionality of early algebraic thinking according to three overarching types, namely, that of analytic thinking, structural thinking, and functional thinking, with generalizing being the scarlet thread that runs through all three. The first part of the paper looks back to the history of the notion of early algebra and the initial research efforts aimed at characterizing early algebraic thinking. The second part delineates the three overarching theoretical dimensions of early algebraic thinking, presents a sampling of past empirical findings, and points to some of the more recent work in the field, including the contributions to this Special Issue. The paper concludes by highlighting the new directions of this domain of research and offering suggestions for further research.
... Haverty et al. (2000) argue that pattern finding is a "fundamental type" of inductive reasoning. In patterning activity, there is a difference between near generalization (e.g., a pattern allowing one to determine the next term in a sequence), and far generalization (e.g., the construction of a general rule for the terms of a sequence) (Rivera 2013). At a basic level, both types of generalizations involve identifying similarities and differences among successive given terms, with the purpose of building analogies. ...
While patterning was commonly seen as evidence of mathematical thinking, interdisciplinary interest has recently increased due to pattern-recognition applications in artificial intelligence. Within two empirical studies, we analyze the analogical-transfer capability of primary school students when completing three types of bi-dimensional patterns, namely, numerical, discrete geometric, and continuous geometric. We found that the mechanisms involved in analogical transfer for continuing sequential patterns are based on two complementary cognitive processes: decoding and adapting. In addition, at a basic level of processing, students activate one of the operational tools of shape recognition or counting, and based on it, they find a surface analogy that leads them to use isometric transformations or one-dimensional development for continuing the given pattern. At a more complex level of processing, students activate both shape recognition and counting and are thus able to apply a filter of processing that uncovers a deep-structure analogy, which allows cognitive framing of the problem and leads to coherent 2D developments within the understood conceptual frame. At a more advanced level of processing, students can use a refined filter not only to uncover a deep-structure analogy but also to use an external language to verbalize that analogy, and consequently, to find 2D developments that trigger changes in cognitive framing, showing that pattern generation is a creative activity. Teaching and learning implications are discussed.
... Φαίνεται ότι η έννοια της μεταβλητής και η εισαγωγή της στη διδασκαλία μπορεί να αποτελέσει ένα σημείο επικοινωνίας μεταξύ των εκπαιδευτικών των δύο βαθμίδων. Τελικά, μέσω των εφαρμογών αναδείχτηκαν εννοιολογικές δυσκολίες των μαθητών αναφορικά με την έννοια της μεταβλητής όπως αυτές έχουν αποτυπωθεί στην υπάρχουσα βιβλιογραφία (Ferrara & Sinclair, 2016;Radford, 2010;Rivera, 2013). ...
Στο παρόν άρθρο παρουσιάζεται έρευνα που στοχεύει στη μελέτη του διδακτικού σχεδιασμού και της πρακτικής μιας εκπαιδευτικού της πρωτοβάθμιας εκπαίδευσης (της Ελένης) μέσα από το θεωρητικό πλαίσιο της Θεωρίας Διδακτικής Τεκμηρίωσης. Η έρευνα έλαβε χώρα εντός του ερευνητικού προγράμματος PREMaTT, το οποίο εστίαζε στον συνεργατικό σχεδιασμό πόρων μιας ομάδας εκπαιδευτικών πρωτοβάθμιας και δευτεροβάθμιας εκπαίδευσης για την εισαγωγή της άλγεβρας στη διδασκαλία μέσα από τον κύκλο συναντήσεις επαγγελματικής ανάπτυξης-σχεδιασμός-εφαρμογή-αναστοχασμός. Παράλληλα, μελετήθηκαν οι συζητήσεις που έλαβαν χώρα στο πλαίσιο των συναντήσεων της ομάδας ώστε να εντοπιστούν τα κύρια ζητήματα που αναδύθηκαν και να εξεταστεί η πιθανή επιρροή τους στον διδακτικό σχεδιασμό και την πρακτική της Ελένης αλλά και ενδείξεις επαγγελματικής μάθησης. Η ανάλυση ανέδειξε ότι τα κύρια ζητήματα των συναντήσεων αυτών, όπως για παράδειγμα η έννοια της μεταβλητής και ο τρόπος εισαγωγής της στη διδασκαλία, επηρέασαν τη διαδικασία διδακτικής τεκμηρίωσης της Ελένης εντός και εκτός τάξης φέρνοντας στην επιφάνεια την πολυπλοκότητα της διδακτικής αξιοποίησης της ψηφιακής τεχνολογίας στη διδασκαλία της άλγεβρας. Στα αποτελέσματα καταγράφονται επίσης ενδείξεις επαγγελματικής μάθησης της εκπαιδευτικού αναφορικά με την εισαγωγή της άλγεβρας στην πρωτοβάθμια εκπαίδευση.
... The opening question associated with this task, ATRNS1 (see Figure 1), gives the number of wheels for Train sizes 1 and 2 and asks students to complete a table for Train sizes 3, 4, 5 and 6. 2. Noticing and describing Regularities. These cases are important because they assist students to see regularities, in other words, to attend to those quantities that remain fixed and those that vary (Radford, 2006;Rivera, 2013). Regularity can be represented using a table of values or some other summary representation. ...
Generalisation is a key feature of learning algebra, requiring all four proficiency strands of the Australian Curriculum: Mathematics (AC:M): Understanding, Fluency, Problem Solving and Reasoning. From a review of the literature, we propose a learning progression for algebraic generalisation consisting of five levels. Our learning progression is then elaborated and validated by reference to a large range of assessment tasks acquired from a previous project Reframing Mathematical Futures II (RMFII). In the RMFII project, Rasch modelling of the responses of over 5000 high school students (Years 7–10) to algebra tasks led to the development of a Learning Progression for Algebraic Reasoning (LPAR). Our learning progression in generalisation is more specific than the LPAR, more coherent regarding algebraic generalisation, and enabling teachers to locate students’ performances within the progression and to target their teaching. In addition, a selection of appropriate teaching resources and marking rubrics used in the RMFII project is provided for each level of the learning progression.
... Visualization appears to be a useful approach in attracting students' attention, motivating the student, making it meaningful by concretizing learning, organizing the student's knowledge, and connecting with concrete and abstract expressions of concepts (Işık & Konyalıoğlu, 2005). Additionally, it is stated by many researchers (Presmeg, 2006;Rivera, 2013;Van Garderen, Scheuermann, & Poch, 2014) that visualization is important for teaching mathematics by going beyond understanding a subject in terms of visual representations. Besides, some researchers (Abdullah, Zakaria, & Halim, 2012;Alcock & Simpson, 2004;Stylianou & Dubinsky, 1999) have associated visual reasoning with an in depth understanding of mathematical concepts. ...
Reasoning is handled as a basic process skill in mathematics teaching. When the literature was examined, it was seen that many types of reasoning related to mathematics education were mentioned. In the present study, it was focused on visual reasoning, which is one of the types of reasoning and also used in different research areas. The purpose of the study was to propose a conceptual framework for what visual reasoning is and what its components are. The conceptual framework constructed consists of three components as visual representation using, visualization, and transition to mathematical thinking. In this framework, a clear distinction was made between the concepts of visual reasoning and visualization, which are thought to be intertwined with each other in the literature. At the same time, we tried to explain where visualization will take place in visual reasoning. Additionally, how visual reasoning will relate to mathematical thinking also distinguishes the framework from other frameworks.
... In addition, numerical representation is a useful bridge to find relationships more than isolated computations; this type of representation provides the opportunity to enrich the students' mathematical learning. Third, and as other authors point out (e.g., Rivera, 2013), the use of visual pattern in functional thinking problems is an important path which helps students in their first generalizations, favoring the interaction between visual, numerical, and general relationships. ...
This study describes how 24 third graders (8–9 years old) relate and represent the relationships between variables when working with a functional thinking problem. This aspect contributes to providing insights about how elementary school students attend properties and relationships between covarying quantities rather than isolated computations. From a functional approach to early algebra, we describe written students’ answers when working with a problem that involves a function, which includes questions for specific values and to generalize. Design research guidelines, specifically those set out for Classroom Teaching Experiment were followed. This study addresses the fourth and last Classroom Teaching Experiment session, which involved a function of the type y = ax + b and students had not previously worked it. Students primarily evidenced correspondence relationship, using natural language and numerical representation to express this functional relationship. Our findings let us to state that (a) although students were not used to working with these types of problems, eleven of them go beyond arithmetic computations, finding relationships that relate the variables; and (b) three students generalized using natural language as a useful vehicle, while there are other students who perceived the same regularity for different specific values but they are unable to represent generalization clearly.
... The generalization is regarded as one of the important activities in learning (Rivera, 2014). However, since this generalization process is usually not trained in the school, it is difficult for us to put in the habit of generalizing words every time we learn with example sentences. ...
When we translate Japanese sentence into English, sometimes several English words become the candidates. However, the usage situation of these candidate words is not the same. In order to choose appropriate words from them, we need to understand the usage situation for each candidate words. Usage situation of the words can be inferred by co-existing words in their example sentences. Co-existing words in example sentences are not always the same, so in order to understand usage situation, we need to generalize co-existing words from several example sentences. However, some of us who do not consciously generalize the co-existing words do not acquire the usage situation. This paper proposes the system which provides the environment where we can explicitly generalize co-existing words (keywords) in the example sentences to acquire the usage situation of the target words. This system also has a generalization support mechanism to provide concepts of words acquired through WordNet as hints. According to the experimental results, participants who used the system in learning English words reduced the number of incorrectly choosing the words and promoted to derive the own understanding of the usage situation.
... Similar to the unit of repeat for repeating patterns, recognizing the functional rule of growing patterns is believed mathematically important, because it is necessary to generalize growing patterns (i.e., predict any element of the pattern). This means children have to move from describing the rule of a growing pattern in a recursive way (i.e., as a function of previous terms; e.g., previous term + 2) to describing it in a functional way (i.e., as a function of position; e.g., position x 2), which is a first step towards functions and algebra (Moss & McNab, 2011;Rivera, 2013;Wilkie, 2014). Some activities are assumed not to require the recognition of the underlying rule and therefore to be less relevant for the development of children's algebraic insight, for example copying, extending, or interpolating. ...
In this study, we aimed to address two gaps in research on early mathematical patterning, namely the lack of attention (1) to growing patterns and (2) to the association between different aspects of patterning and numerical ability. Participants were 400 four-year olds from a wide range of socioeconomic backgrounds. Children's patterning and numerical ability were assessed by means of individual tasks. The patterning tasks assessed their performance on three patterning activities (i.e., extending, translating, and identifying the pattern's structure) for two types of patterns (i.e., repeating and growing). The numerical measure included a set of eight well-known numerical tasks. We additionally controlled for individual differences in spatial ability and visuospatial working memory. Results indicated an effect of both activity and patterning type on children's patterning performance, as well as an interaction between both. Furthermore, children's performance on four out of six patterning tasks uniquely contributed to their numerical ability above age, spatial ability, and visuospatial working memory. These findings support the importance of specific pattern types and patterning activities in the early stage of children's mathematical development and give directions for further educational practices.
... As opposed to the previous section, the function examples here are neither continuous nor defined for all real numbers, yet the domain consists of an infinite and unbounded set of numbers. Moreover, these examples demonstrate a recognized human tendency of "continuing the pattern" (e.g., Rivera, 2013), that is, assigning the same rule of multiplication by 3 to all integers. In this sense, the assignment of the same rule to a restricted domain demonstrates the arbitrary choice of the domain in the function concept, though not the arbitrary choice of correspondence between the domain and codomain. ...
In this study we focus on example spaces for the concept of a function provided by prospective secondary school teachers in an undergraduate program. This is examined via responses to a scripting task-a task in which participants are presented with the beginning of a dialogue between a teacher and students, and are asked to write a script in which this dialogue is extended. The examples for functions fulfilling certain constraints provide a lens for examining the participants' concept images of a function and the associated range of permissible change. The analysis extends previous research findings by providing refinement of students' ideas related to functions and the concept of the function domain.
... As opposed to the previous section, the function examples here are neither continuous nor defined for all real numbers, yet the domain consists of an infinite and unbounded set of numbers. Moreover, these examples demonstrate a recognized human tendency of Bcontinuing the pattern^(e.g., Rivera 2013), that is, assigning the same rule of multiplication by 3 to all integers. In this sense, the assignment of the same rule to a restricted domain demonstrates the arbitrary choice of the domain in the function concept, though not the arbitrary choice of correspondence between the domain and codomain. ...
In this paper, we highlight a further use of scripting tasks in the work of researchers-teacher- educators, which we present in two parts. In Part 1, we study prospective secondary school mathematics teachers’ responses to a scripting task on the topic of functions. The generated examples for a function that satisfies certain constraints provide a lens for examining participants’ example spaces and concept images of a function. In Part 2, we demonstrate how the script analysis is subsequently used by teacher educators as a springboard for broadening prospective teachers’ personal example spaces for a function, strengthening their understanding of the concept, and extending connections between tertiary and school mathematics.
Bu çalışmanın amacı, matematikte akademik başarısı yüksek olan öğrencilerin örüntü genelleme görevlerinde kullandıkları stratejileri belirlemektir. Bu amaca uygun olarak nitel araştırma yöntemlerinden durum çalışması yöntemi tercih edilmiştir. Çalışmaya matematik ders notları 4 veya 5 olan 10 sekizinci sınıf öğrencisi dahil edilmiştir. Veriler, 2022-2023 eğitim-öğretim yılı güz döneminde şekil örüntü görevlerinden oluşan bir çalışma yaprağı ile toplanmıştır. Verilerin analizi betimsel analiz yöntemiyle yapılmış ve MAXQDA programından yararlanılmıştır. Öğrencilerin kullandıkları stratejilerin kodlanmasında alanyazında belirtilen stratejiler dikkate alınmıştır. Elde edilen bulgular, öğrencilerin örüntünün yakın adımı ile ilgili görevlerde yalnızca yinelemeli stratejiyi kullandıklarını göstermiştir. Ancak, örüntünün uzak adımının belirlenmesinin istendiği görevlerde, öğrencilerin ağırlıklı olarak kural temelli çözümleri tercih ettikleri tespit edilmiştir.
This study focused on analyzing how students' understanding of the structure of patterns improved after the lessons of finding rules were reorganized and taught to second graders to emphasize the structure of patterns. Students from one class in each of the two schools were selected as research participants and given a pre-test to assess their initial understanding of patterns and a post-test after the reorganized lesson were administered. One additional class from each school was selected as a comparison group, and the pre-and post-test results of the experimental and comparison groups were compared and analyzed. The results of the study showed that the reorganized lesson was effective on the following items: finding multiple components, arrangements, and changes in patterns; comparing the structures between patterns; guessing the specific terms in patterns: and explaining why. Based on these results, instructional implications for helping lower elementary students understand the structure of patterns through the finding rules unit were discussed.
There is international emphasis on the right that all individuals should have to comprehensive education with learning opportunities tailored to their educational needs, and Colombia is no exception. Thus, the work reported here aims to (a) propose a curricular structure that allows addressing diversity in mathematics class, enabling flexibility and adaptation according to students' particularities and (b) construct didactic designs of mathematics adjusted to a flexible and adaptable curricular structure, addressing diversity in the mathematics classroom in Colombia. This article partially addresses these objectives by exploring the question: What conceptual elements need to be considered to construct didactic designs of mathematics that address diversity in the classroom? Consequently, the study presents elements of a curricular proposal based on universal design for learning (UDL) to address diversity in mathematics classes. This is exemplified through a didactic design created for the study of sequences and patterns, promoting, in basic and middle education, the development of algebraic thinking through activities involving generalization and the study of patterns.
Pattern recognition is an important skill in computational thinking. In this study, an equation puzzle game was developed by combining pattern recognition with algebraic reasoning, and scaffolding was designed to support learners' learning. Sixty participants were enrolled in this study, divided into a control group and an experimental group to compare the results and differences in game achievement, flow, anxiety, and motivation of participants with and without algebraic reasoning scaffolding. The results of the study showed that the participants in both groups had positive flow and motivation during the game, did not feel over-anxious, and there was no significant difference in the game achievement of the two groups. In addition, the game with the scaffolding may have the potential to make a positive correlation between game achievement and psychological status. The results of this study indicated that the game did not cause too much anxiety to the participants. The scaffolding-based design achieves the intended effect on the participants' assistance and facilitates the participants' engagement in pattern recognition problem solving. And as learners became more focused and engaged, they could also perform better in the game. This game mechanism can be used as a reference for designing pattern recognition games.
Given the significance of functional thinking at the elementary school level, there has been growing interest in teachers who play a major role in enhancing students’ functional thinking. This study surveyed 119 elementary school teachers in Korea to investigate their knowledge for teaching functional thinking. A written assessment was developed for this study regarding the knowledge of mathematical tasks, instructional strategies, and mathematical discourse to teach functional thinking. The results of this study showed that many teachers could design mathematical tasks corresponding to simple relationships of two quantities but some of them had difficulties in constructing the task for [Formula: see text]. Teachers could explain the affordances of key instructional strategies to foster functional thinking and identify students’ typical errors in recognizing or representing a functional relationship but some explanations included a superficial understanding of the core ideas of functional thinking. Based on these results this article closes with a discussion of several implications regarding the aspects needed for elementary school teachers to further promote students’ functional thinking.
Given the significance of functional thinking as one of the main routes to foster early algebraic thinking, in this paper we describe how a current elementary mathematics textbook’s unit titled “Pattern and Correspondence” was developed in the Korean context. The following four key instructional elements for fostering functional thinking were extracted from the literature review: (a) correspondence relationships in real-life contexts; (b) various pattern tasks; (c) exploration for a correspondence relationship; and (d) symbol variables to represent a correspondence relationship. These elements were employed to analyze the strengths and weaknesses of the unit covering pattern and correspondence in the previous mathematics textbook. Based on this analysis, a new draft for the pattern and correspondence unit was designed and implemented in an elementary school to test its appropriateness and ability to enhance students’ functional thinking. Students’ responses using the new draft were analyzed in terms of the four elements. Performance between the students with the new draft and their counterparts with the previous textbook was compared. The students with the new draft were more successful than their counterparts in exploring and representing the relationship between two quantities. The final version of the unit was developed and has been applied to all schools. The whole process of developing the unit can be an example of how to foster functional thinking through changes to curricular resources.
The current study investigated creativity-directed problem-solving processing that explicitly requires solving pattern generalization problems in multiple ways. To examine mathematical creativity, we employed a multiple solution tasks approach, asking participants explicitly to solve pattern generalization problems in multiple ways. The participants were engaged with pattern generalization-multiple solution tasks in which patterns were presented numerically, diagrammatically, or contextually. The study aimed to examine whether success and creativity associated with solving pattern generalization-multiple solution tasks are affected (I) by expertise in mathematics (EM) and general giftedness (G) and (II) by task complexity and representation. Two hundred and ninety-eight participants from four research groups that differed in levels of general giftedness (G) and mathematical expertise participated in the study. We found that G affected statistically significant all the components of creativity when solving pattern generalization-multiple solution tasks, independently of task representation. Surprisingly, the effect of EM was lower and dependent on the type of task representation. Also surprising, we found statistically significant interactions between G and EM factors when solving pattern generalization-multiple solution tasks. The findings demonstrate a significant effect of task representation on students’ levels of success and creativity when engaged in solving pattern generalization-multiple solution tasks: contextual representations were approached with the highest level of success and creativity, whereas numerical representations appeared to be tackled with the lowest level of success and creativity. We suggest that mathematical instruction should integrate pattern generalization-multiple solution tasks to promote students’ mathematical expertise and creativity.
Proof, a key topic in advanced mathematics, also forms an essential part of the formal learning experience at all levels of education. The reason is that the argumentation involved calls for pondering ideas in depth, organizing knowledge, and comparing different points of view. Geometry, characterized by the interaction between the visual appearance of geometric elements and the conceptual understanding of their meaning required to generate precise explanations, is one of the foremost areas for research on proof and argumentation. In this qualitative analysis of the arguments formulated by participants in an extracurricular mathematics stimulus program, we categorized students’ replies on the grounds of reasoning styles, representations used, and levels of generality. The problems were proposed in a lesson on a quotient set based on the similarity among triangles created with Geogebra and the responses were gathered through a Google form. By means a content analysis, the results inform about the reasoning style, the scope of the argumentation, and the representation used. The findings show that neither reasoning styles nor the representations used conditioned the level of generality, although higher levels of argumentation were favored by harmonic and analytical reasoning and the use of algebraic representations.
Traditionally, the teaching and learning of algebra has been addressed at the beginning of secondary education with a methodological approach that broke traumatically into a mathematical universe until now represented by numbers, with bad consequences. It is important, then, to find methodological alternatives that allow the parallel development of arithmetical and algebraic thinking from the first years of learning. This article begins with a review of a series of theoretical foundations that support a methodological proposal based on the use of specific manipulative materials that foster a deep knowledge of the decimal number system, while verbalizing and representing quantitative situations that underline numerical relationships and properties and patterns of numbers. Developing and illustrating this approach is the main purpose of this paper. The proposal has been implemented in a group of 25 pupils in the first year of primary school. Some observed milestones are presented and analyzed. In the light of the results, this well-planned early intervention contains key elements to initiate algebraic thinking through the development of number sense, naturally enhancing the translation of purely arithmetical situations into the symbolic language characteristic of algebraic thinking.
Students who have learned arithmetic in elementary mathematics would learn algebra in middle schools in earnest. So, the transition from arithmetical thinking to algebraic thinking is an important success factor for algebra learning. Therefore, in this study, 64 sixth graders solved the problems of near generalization and far generalization concerning the generalization of some figure patterns. Subsequently we analyzed their problem solving. As a result of the analysis, several cases of students having difficulties in performing algebraic generalization were identified. Through the discussion of the research results, several pedagogical implications were drawn to support the transition from arithmetical thinking to algebraic thinking in elementary school students.
This research aims to explore the Number Generalization Process of the winner students of Mathematics Olympiad. The Number Generalization Process in this study is based on APOS theory. This research is explorative research with a qualitative approach. The subjects of the study were the junior high school students who won the provincial mathematics Olympiad in South Sulawesi, Indonesia. Subjects were given instruments that had been developed, namely the number patterns generalization test. Data collection of this study is the number patterns generalization test and in-depth interviews. The data analysis process was reducing, describing, validating, and concluding. The results showed that the students who won the Olympics in the action stage determine the next term, if given a sequence by using a number pattern. At the stage of the process, action interiorizing by calculating the value of the next term repeatedly and explaining the process of determining the next term. At the object stage, encapsulating the process by showing that a number pattern has certain characteristics, de-encapsulating by assessing the observed pattern, and checking the number pattern found. At the schema stage, explaining the generalization properties of number sequence patterns by linking the actions, processes, objects of a concept with other concepts, doing thematization by linking the existing concept patterns to the general sequence.
This study was conducted from a perspective that adopts a broad vision of mathematical talent, defined as the potential that a subject manifests when confronting certain types of tasks, in a successful way, that generate creative mathematical activity. To analyse this, our study proposes a Praxeological Model of Mathematical Talent based on the Anthropological Theory of Didactics and the notion of mathematical creativity, which defines four technological functions: (1) producing new techniques; (2) optimizing those techniques (3); considering tasks from diverse angles; and (4) adapting techniques. Using this model, this study analyses the creative mathematical activity of students aged 10–12 years displayed as they sought to solve a series of infinite succession tasks proposed to encourage the construction of generalization processes. The setting is a Mathematics Club (a talent-promoting institution). The evaluation of results shows that the Praxeological Model of Mathematical Talent allows the emergence and analysis of mathematical creativity and, therefore, encourages the development of mathematical talent.
The aim of this study is to provide a deeper insight related to the features of mathematical activity in early childhood. In general, preschoolers are encouraged to get involved in tasks, games or situations with mathematical objects or content, e.g., to measure a distance, or to count a number of objects, to recognize shapes or find the next element in a pattern, in order to develop initial mathematical ideas. However, whether young children think or act mathematically or develop mathematical concepts depends on how thinking and acting mathematically are considered. In this study, we first attempt to specify the characteristics of genuine mathematical activity in early years and then to emphasize the importance of generalization as an essential component of this activity. Finally, we present examples and results of teaching practices supporting generalization. The study is based on research findings used to clarify suggestions, but it is not a research study.
This study aims to describe the Generalisation Patterns Strategy of junior high school students based on gender. The subject of this study is two junior high school students in Makassar City, one male, and one female. Data collection techniques are written tests and interviews which used to obtain data about the strategy of the generalization patterns of students. The validity used time triangulation. The analysis data consists of three stages: reducing data, presenting data, and daring conclusions. The results of this study show that the male subject used the strategy of Counting, Whole-object-No Adjustment, and Difference-Rate Adjustment. Meanwhile, the female subject used the strategy of Countin and Explicit in the generalization of patterns on numbers and pictures.
In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants’ approaches to identifying the nth term were recorded with eye-tracking technology. Our aims are to identify their strategies and to explore whether and how format influences these strategies. Qualitative analysis of eye-tracking data offers several strategies: Sequence of Differences, Building a Relationship, Known Formula, Linear Recursive and Initial Conjecture. Sequence of Differences was the most common strategy, but Building a Relationship was more likely to lead to the right formula. Building from Square and Factor Search were the most successful methods of Building a Relationship. Findings about the influence of format on the range of strategies were inconclusive but analysis indicated sporadic evidence of possible influences.
The concept of function is a fundamental base in the process of learning mathematics, which can not be acquired in one step. Formation of it is a long process.
Studies are done of function concept formation processes in Ukrainian and Hungarian curriculum and textbooks showed that the Hungarian curriculum and textbooks prescribe more targeted preparatory work from the elementary school to establish the concept of function. The investigations among one class of Hungarian and one class of Ukrainian students in 6th and 7th grades concluded the source of the concept incorporation into the teaching process strengthens the formation of the correct function schema. According to the results of our studies we prepared and implemented an improvement program on the preparation of the function concept among 6th grade students from Ukraine. Within the program, we aimed the concept formation with the help of concrete problem situations. At the same time we developed the skills necessary for problem solving: communication skills (development of the relational vocabulary, reading comprehension and interpretation of the text).
In our paper we are examining the effect of the improvement program on the formation of the function concept and on the development of the skills, mentioned above after 1 year (in 7th grade).
Much research on generalization of Algebra, but related to the generalization of the pattern is still lacking. In this study we characterizing middle school students generalization of pattern. The participants were 40 students grade 8 took the test with instruments that have been developed and analyzed students working. The findings indicate that students showed the two characterizing in generalization of patterns that: (1) Factual, (2) Symbolic. Possible reason are discussed and suggestions for teaching with generalization of patterns are presented.
In our recent book, we present a parallel distributed processing theory of the acquisition, representation and use of human semantic knowledge. The theory proposes that semantic abilities arise from the flow of activation amongst simple, neuron-like processing units, as governed by the strengths of interconnecting weights; and that acquisition of new semantic information involves the gradual adjustment of weights in the system in response to experience. These simple ideas explain a wide range of empirical phenomena from studies of categorization, lexical acquisition, and disordered semantic cognition. In this précis we focus on phenomena central to the reaction against similarity-based theories that arose in the 1980's and that subsequently motivated the "theory-theory" approach to semantic knowledge. Specifically, we consider i) how concepts differentiate in early development, ii) why some groupings of items seem to form "good" or coherent categories while others do not, iii) why different properties seem central or important to different concepts, iv) why children and adults sometimes attest to beliefs that seem to contradict their direct experience, v) how concepts reorganize between the ages of 4 and 10, and vi) the relationship between causal knowledge and semantic knowledge. The explanations for these phenomena are illustrated with reference to a simple feed-forward connectionist model; and the relationship between this simple model, the broader theory, and more general issues in cognitive science are discussed.
Observers represent only a tiny fraction of the total amount of information available at any given moment. This small amount of information has been quantified: throughout the lifespan we typically maintain only three or four visual items in working memory at a time. Yet we are also capable of impressive quantificational feats: we can count the objects in arrays containing hundreds, or estimate that a scene contains "about 100" people. Given the strict limits on working memory, how do observers accomplish this? Here I propose that although working memory is limited in the number of items it can store, it is also flexible in what counts as an item. At least three types of representations can serve as an item in working memory: an individual object, a set, and an ensemble. Shifting between these types of representations allows us to bypass some of the strict constraints imposed by WM, thereby empowering quantification.
This paper reports on a study of the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. The research tool comprised a series of 22 pencil-and-paper exercises based on linear generalisation tasks set in both numeric and two-dimensional pictorial contexts. The responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. The method or strategy adopted was analysed and classified into one of seven categories. In addition, a meta-analysis focused on the formula derived for the nth term in conjunction with its justification. The results of this study strongly support the notion that question design can play a critical role in influencing learners’ choice of strategy and level of attainment when solving pattern generalisation tasks. An understanding of the importance of appropriate question design has direct pedagogical application within the context of the mathematics classroom.
The National Statement on Mathematics for Australian Schools advises that algebra learning begins with the study of sequences and patterns leading to their description as algebraic rules relating dependent and independent variables. This study assessed the success of 512 students in seven schools at year levels 7 to 10 in· recognizing· and describing algebraic rules relating two variables. When given a relationship described by a table of values for two variables, the majority of these stUdents were unable to write an algebraic rule of the form y = ax + b. Comparison of the test results from schools using different approaches to algebra suggests that the pattern-based approach (as implemented at the schools taking part in the study) was no more helpful than traditional approaches. In A National Statement on Mathematics for Australian Schools (Australian Education Council, 1990), it is suggested that algebraic thinking should be based on initial experiences with patterns and sequences. It is expected that in the primary years children will investigate patterns and sequences and make generalizations . about them in everyday. lartguage. For example, a pattern of matches: could be described by children as "You start with four for the first square and then add on three for each extra square" (National Statement, p. 191). In the secondary school students learn how to express such generalizations mathematically using algebraic symbolism. Some of the first algebraic skills to be developed, accordinR to the Statement, are Ca) describing in words relationships and rules for generating elements in a pattern or sequence, (b) expressing one of the rules algebraically, and (c) generating further elements of a pattern from a given verbal or algebraic rule. Introducing algebra in this way as a language for expressing relationships between two . variables, which we will call the "pattern-based approach", represents a clear break with tradition. It seems likely to facilitate the later study of formulas and functions. It is aesthetically ple,asing to teachers, and students can make good use of concrete materials for building designs like the two-dimensional pattern above. As well as being endorsed in the National Statement, the pattern-based approach appears in the RIME Lesson Pack (Lowe and Lovitt, 1984), in the New South Wales mathematics syllabus (NSW Board of Secondary Education, 1989) and in some currently-used secondary school textbooks (e.g., Blane & Booth, 1991; Tomlinson, Ardley, Mottershead, Thompson and Wrightson, 1987).
Bringing Out the Algebraic Character of Arithmetic contributes to a growing body of research relevant to efforts to make algebra an integral part of early mathematics instruction, an area of studies that has come to be known as Early Algebra. It provides both a rationale for promoting algebraic reasoning in the elementary school curriculum and empirical data to support it.
In this article, the role of abductive reasoning within Peirce's diagrammatic reasoning is discussed. Both abduction and diagrammatic reasoning bring in elements of discovery but it is not clear if abduction should be a part of a fully developed diagrammatic system or not for Peirce. This relates to Peirce's way of interpreting abduction in his later writings. Iconicity and perceptual elements as a basis for discoveries are analyzed, both in deductive and abductive reasoning. At the end, the role of modern ideas of distributed cognition applied to the Peircean scheme is shortly delineated.
We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline.
In this article, we deal with students' algebraic generalizations set in the context of elementary geometric-numeric patterns. Drawing from Vygotsky's psychology, Leont'ev's Activity Theory, and Husserl's phenomenology, we focus on the various semiotic resources mobilized by students in their passage from the particular to the general. Two small groups of Grade 9 students are investigated through a four- dimensional analysis: video, audio, transcripts, and written material. The resulting qualitative analysis shows how discourse, gestures, actions, and rhythms orchestrate one another and how, through a complex and subtle coordination of them, the students objectify different aspects of their spatial-temporal mathematical experience. The analysis also suggests connections between the syntax of the students' algebraic formulas and the semiotic means of objectification through which the formulas were forged, thereby shedding some light on the meaning of students' algebraic expression. Some implications for the teaching and learning of mathematics are discussed.
involving linear patterns. Our research questions were: What enables/hinders students' abilities to generalize a linear pattern? What strategies do successful students use to develop an explicit generalization? How do students make use of visual and numerical cues in developing a generalization? Do students use different representations equally? Can students connect different representations of a pattern with fluency? Twenty-three different strategies were identified falling into three types, numerical, figural, and pragmatic, based on students' exhibited strategies, understanding of variables, and representational fluency. BACKGROUND In 1999, with a grant from the Noyce Foundation, San Jose State University and 30 school districts formed a Mathematics Assessment Collaborative (MAC) in an effort to balance state-sponsored multiple-choice tests and to provide multiple measures to evaluate students. The MAC exams are summative performance assessments in grades 3-10. The exams are hand scored using a point rubric and audited for reliability. Student papers are returned to teachers for further instruction and programmatic review. In developing this model system of performance assessment, the MAC spent a year writing Core Ideas for each grade level tested, adapting the National Council of Teachers of Mathematics Standards (NCTM, 2000). The assessments are written to match these Core Ideas. MARS results are correlated to state test results and analyzed by various demographic characteristics of students. In 2003, over 60,000 students were tested by the MAC. At the eighth and ninth grades, one of the Core Ideas tested is that of patterns, relations and functions. Students are asked: to generalize patterns using explicitly defined functions; and, understand relations and functions and select, convert flexibly among, and use various representations for them. Over the five years of MARS data collections, we have found a similar pattern; while students are quite successful in dealing with particular cases of patterns in visual and tabular form, they have considerable difficulty in using algebra to express relationships or to generalize to an explicit, closed formula for a linear pattern. Summary data are shown in Table 1. To gain more insights, we embarked on an in-depth study of a small number of 9th grade students to pinpoint more specifically why they have difficulties in forming
This study examines the perceptual agility and strategy use of 27 prospective secondary school teachers in Singapore when solving a quadratic generalising problem. The data showed that the teachers were very capable of employing a variety of strategies to visualise the same pattern in different ways, resulting in not only a diverse range of equivalent rules but also some creative visual representations of the pattern structure.
The terms generalization and abstraction are used with various shades of meaning by mathematicians and mathematics educators, each representing both a process and the product of that process. In this article we attempt to rationalize the use of these terms in a cognitive context, in a manner intended to shed some light on the different qualities of generalization in advanced mathematics. Using this analysis we will be able to suggest pedagogical principles designed to assist students'comprehension of advanced mathematical concepts.
Findings, insights, and issues drawn from a three-year study on patterns are intended to help teach prealgebra and algebra.
Directly or indirectly, The Fractions Project has launched several research programs in the area of students’ operational development. Research has not been restricted to fractions, but has branched out to proportional reasoning (e.g., Nabors 2003), multiplicative reasoning in general (e.g., Thompson and Saldanha 2003), and the development of early algebra concepts (e.g., Hackenberg accepted). This chapter summarizes current findings and future directions from the growing nexus of related articles and projects, which can be roughly divided into four categories. First, there is an abundance of research on students’ part-whole fraction schemes, much of which preceded The Fractions Project. The reorganization hypothesis contributes to such research by demonstrating how part-whole fraction schemes are based in part on students’ whole number concepts and operations.
Second, several researchers have noted the limitations of part-whole conceptions and have advocated for greater curricular and instructional focus on more advanced conceptions of fractions (Mack 2001; Olive and Vomvoridi 2006; Saenz-Ludlow 1994; Streefland 1991). The Fractions Project has elucidated these limitations while articulating how advancement can be realized through the construction of key schemes and operations that transcend part-whole conceptions. In particular – and deserving of its own (third) category – research on fraction schemes has highlighted the necessity and power of the splitting operation in students’ development of the more advanced fraction schemes, such as the reversible partitive fraction scheme and the iterative fraction scheme.
Considering algebra as a culture, this chapter looks at the introduction of algebra as an initiation process where generalization activities can be extremely effective. After a reflection on my own immersion into algebra and the evolution of attitudes toward the teaching of algebra, a teaching experiment using generalization activities is presented. Two generalizing activities are described in some detail, looking at the behavior of adults in the experimental group in the light of research results of high school students on tests and interviews involving the same activities. The paper concludes with a “cultural” reflection on the teaching experiment and a more general consideration of the role of generalization in the introduction of algebra.
The main purpose of this study is to determine the strategies of using the generalizing patterns of the primary fifth grade students. The practice of this research is conducted on twelve students, which have high, middle and low success levels. Task based interviews and students journals are used as the tools for data collection. For the analysis of the data, a classification method including "data reduction", "data display"and"drawing conclusion and verification" are used. At the end of the research, it is seen that the visual and numerical approaches are adopted in the generalization of patterns and the visual approach is made easy for generalization, as well. In generally, the present strategies in the generalizing of patterns are also taken into account of near or far generalizing. The recursive strategies are used in the near generalizing. However, the explicit strategies are determined in using far generalizing. © 2009 Eǧitim Danişmanliǧi ve Araştirmalan İletişim Hizmetlri Tic. Ltd. Şti.
Proof is considered to be central to the discipline of mathematics and the practice of mathematicians. Yet its role in secondary school mathematics has traditionally been peripheral at best; the only substantial treatment of proof is limited to geometry. According to Wu (1996, p. 228), however, the scarcity of proof outside of geometry is a misrepresentation of the nature of proof in mathematics.
Inductive ReasoningDeductive ReasoningConclusion
References
This Brief Report describes a secondary analysis of the solutions written by 306 second-year algebra students to four constructed-response items representative of content at this level. The type of solution (symbolic, graphical, or numerical) used most frequently varied by item. Curriculum effects were observed. Students studying from the second edition of the University of Chicago School Mathematics Project's (UCSMP) Advanced Algebra curriculum used a higher percentage of graphical and numerical strategies than comparison students. Achievement and choice of strategy were also related. Both UCSMP and non-UCSMP students who used symbolic or graphing strategies were generally successful on the quadratic comparison item; UCSMP students who used graphing strategies were also successful on items dealing with logarithm properties and a quadratic application.
This interdisciplinary work is a collection of major essays on reasoning: deductive, inductive, abductive, belief revision, defeasible (non-monotonic), cross cultural, conversational and argumentative. They are each oriented toward contemporary empirical studies. The book focuses on foundational issues, including paradoxes, fallacies, and debates about the nature of rationality; the traditional modes of reasoning, as well as counterfactual and causal reasoning, it also includes chapters on the interface between reasoning and other forms of thought in general, this last set of essays represents growth points in reasoning research, drawing connections to pragmatics, cross-cultural studies, emotion, and evolution. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The purpose of this study was to investigate 6th-grade students' use of equations to describe and represent problem situations prior to formal instruction in algebra. Ten students were presented with a series of similar tasks in 6 different problem contexts representing linear and nonlinear situations. The students in this study showed a remarkable ability to generalize the problem situations and to write equations using variables, often in nonstandard form. Although students were often able to write equations, they rarely used their equations to solve related problems. We describe students' preinstructional uses of equations to generalize problem situations and raise questions about the most appropriate curriculum for building on students' intuitive knowledge of algebra.
Numerous studies have claimed that eye movements are a critical or even obligatory part of explicit counting whereas here we will show that counting relies on a set of attention pointers that individuate targets of interest and specify their locations independently of eye movements. We demonstrate that explicit counting can proceed to very high numbers without error in afterimages where eye movements are not possible. Previous studies with afterimages had used displays too dense to allow individuation of items by attention: the displays suffered from crowding. We also show that explicit counting is defeated for displays of more than about six items in motion because there is no mechanism available to mark already-counted items and keep that marking linked to the items as they move. In this case, only the approximate number system can operate and, interestingly, this system shows fairly accurate estimates, rather than the underestimation typically seen for denser displays.
Th e main purpose of this study is to determine the strategies of using the generalizing patterns of the primary fifth grade students. Th e practice of this research is conducted on twelve students, which have high, middle and low success levels. Task-based interviews and students journals are used as the tools for data collection. For the analysis of the data, a classification method including "data reduction", "data display" and "drawing conclusi-on and verification" are used. At the end of the research, it is seen that the visual and nu-merical approaches are adopted in the generalization of patterns and the visual approach is made easy for generalization, as well. In generally, the present strategies in the generali-zing of patterns are also taken into account of near or far generalizing. Th e recursive stra-tegies are used in the near generalizing. However, the explicit strategies are determined in using far generalizing.
The paper presents results of the research focused on studying students' abilities to make generalisations in geometry. Students' activities in a classroom have been analysed through the evaluation of their inquiry work on different tasks that required deductive reasoning and non-routine approach to carry out possible generalisations. Cognitive processes regarding to different geometrical structures are described and analysed in detail. The special emphasis has been given to students' obstacles while making generalisations. Generalising activity has traditionally been given significant attention both in schools and in research. Within the literature, different types of generalisation are distinguished, e.g. empirical and theoretical generalisations (Davydov, 1990). At the same time there are many papers dealing with various aspects of generalising process in the different areas of mathematics education. Radford (2001) identifies three levels of generalisation in algebra (factual, contextual and symbolic generalizations). Ainley et al (2003) note that the importance of generalising as an algebraic activity is widely recognised within research on the learning and teaching of algebra. Undoubtedly, generalising activities in geometry are very important in research on the learning and teaching of geometry as well. Moreover, taking into account the great role of visualisation and perception in the learning geometry, investigation students' abilities to make generalisations of different concepts, definitions, properties and ways of their development are of significant interest for researchers in mathematics education.
The development of patterning strategies during the year prior to formal schooling was studied in 53 children from 2 similar preschools. One preschool implemented a 6-month intervention focusing on repeating and spatial patterns. Children from the intervention group demonstrated greater understanding of unit of repeat and spatial structuring, and most were also able to extend and explain growing patterns 1 year later. The findings indicate a fundamental link between patterning and multiplicative reasoning through the development of composite units.
The development of patterning strategies during the year prior to formal schooling was studied in 53 children from two similar preschools. One preschool implemented a 6-month intervention focussing on Repeating and Spatial patterns. An interview-based Early Mathematical Patterning Assessment (EMPA) was developed and administered pre- and post-intervention, and again following the first year of formal schooling. The intervention group outperformed the comparison group across a wide range of patterning tasks at the post and follow-up assessments, most also being able to extend and explain Growing Patterns which they had not previously experienced.
In this study, 23 seventh- and eighth-grade students were interviewed as they solved problems related to four linear geometric
number patterns involving perimeter and area. In particular, they developed symbolic expressions for the pattern relationships
and assessed the validity of given expressions. The strategies indicated in the responses suggest four levels of thinking
about linear geometric number patterns: (1) concrete modelling and counting, (2) inappropriate use of proportion, (3) focus
on recursive relationships, and (4) analysis of the functional relation between a perimeter or area and the shape number.
The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept
formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which
the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by
which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought
processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures
by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not
spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what
extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people
cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what
percentage of the population really cares about it.
The expectation that students be introduced to algebraic ideas at earlier grade levels places an increased burden on the classroom teacher to help students construct and justify generalizations. This study provides insight into the reasoning of 25 sixth-grade students as they approached patterning tasks in which they were required to develop and justify generalizations while using computer spreadsheets as an instructional tool. The students demonstrated both the potential and pitfalls of such activities. During whole-class discussions, students were generally able to provide appropriate generalizations and justify using generic examples. Students who used geometric schemes were more successful in providing general arguments and valid justifications. However, during small-group discussions, the students rarely justified their generalizations, with some students focusing more on particular values than on general relations. It is recommended that the various student strategies and justifications be brought to the forefront of classroom discussions so that students can examine the mathematical power and validity of the various strategies and justifications typically introduced by students.
This chapter briefly discusses the issues of symbols, meanings, and embodiment. It explains the solution to the symbol grounding problem. It illustrates the ingredients that are employed in the experiments about language emergence using a specific example of a color guessing game. It argues that these experiments show that there is an effective solution to the symbol grounding problem. The objective test for this claim is in the increased success of agents in the language games.