Generalization and Justification: The Challenge of Introducing Algebraic Reasoning Through Patterning Activities

Lannin, John K.

doi:10.1207/s15327833mtl0703_3

Cited by 136 publications

(179 citation statements)

References 25 publications

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“…According to [9], the strategy used by subject 1 can be categorized as whole-object strategy, in which students use multiplication in the previous stage to calculate the number of stickers needed. Students do not consider that if cube is glued, the stickers are not counted.…”

Section: Resultsmentioning

confidence: 99%

“…The failure in generalizing of patterns happens because students do not recognize the relationship between the patterns [8]. Therefore, recommendations are provided regarding the learning of algebra [9]: begin the learning of algebra early (in part, by building on students' informal knowledge), integrate the learning of algebra with the learning of other subject matter (by extending and applying mathematical knowledge), include several different forms of algebra thinking (by applying mathematical knowledge), build students' naturally occuring linguistic and cognitive powers (encouraging them at the same time to reflect on what they learn and to articulate what they know), encourage active learning (and the construction of relationships) that puts a premium on sense-making and understanding.…”

Section: Introductionmentioning

confidence: 99%

“…The importance of establishing the students' understanding of algebraic ideas through informal knowledge that students brought to school includes the notion of pattern [9]. The kindergarten students aged 5-6 years reveal a unique fact.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pattern Generalization by Elementary Students

Rusdiana¹,

Suriaty²,

Sutawidjaja³

et al. 2017

Proceedings of the 5th SEA-DR (South East Asia Development Research) International Conference 2017 (SEADRIC 2017)

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Abstract-This paper presents the strategies used by fifth and sixth grade elementary students when they faced problem on pattern generalization. The samples were 3 elementary school students from grade 5 and 6. Students were given a problem of pattern and asked to make generalizations on the given pattern. The data were analyzed to describe the strategies used by students in solving the problem of generalizing patterns. The results indicated that the students were able to generalize on a pattern and start using mathematical symbols in solving problem. The results also revealed that the students used effective strategies to solve the problem of generalizing patterns. Although pattern generalization is not a new concept for elementary school students, it is considered important because pattern generalization can help students learn arithmetic thinking to algebraic thinking so that students are able to produce rules that can be used to determine any value of the pattern.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pattern Generalization by Elementary Students

Rusdiana¹,

Suriaty²,

Sutawidjaja³

et al. 2017

Proceedings of the 5th SEA-DR (South East Asia Development Research) International Conference 2017 (SEADRIC 2017)

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show abstract

“…Researchers have used a number of frameworks to categorize students' proof productions and describe the progression from inductive or empirical justifications toward deductive arguments (Balacheff, 1987;Harel & Sowder, 1998;Lannin, 2005). The proof scheme taxonomy developed by Harel and Sowder (1998; is one of the most comprehensive and frequently cited in mathematics education research.…”

Section: Pedagogical Content Knowledge For Teaching Proofmentioning

confidence: 99%

Investigating Mathematical Knowledge for Teaching Proof in Professional Development

Lesseig

2016

IJRES

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Research documenting teachers' fragile understanding of proof and how it is advanced suggests that enhancing the role of proof in school mathematics will demand substantial teacher learning. To date, there is little research detailing what mathematical knowledge might support the teaching of proof or how professional development (PD) might afford such learning. This paper advances a framework for Mathematical Knowledge for Teaching Proof (MKT for Proof) that specifies proof knowledge across subject matter and pedagogical domains. To explore the utility of the framework, this paper examines data from four different PD settings in which teachers and leaders worked on the same proof-related task. Analysis of discussions across these four settings revealed similarities and differences in knowledge of proof made available in each setting. These findings provide a means to detail the MKT for Proof framework and demonstrate its usefulness as a tool for designing, analyzing and evaluating proof experiences in PD.

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“…Genelleme olmaksızın matematiksel düşünmeden söz edilemeyeceği gibi (örn., Mason, 1996), cebirsel düşünmenin gelişiminde de genellemenin önemi büyüktür. Genelleme, öğrencilerin aritmetikten cebire geçiş yapmalarına yardımcı olan önemli bir yaklaşımdır (Blanton ve Kaput, 2011;Kaput, 1999;Lee, 1996;Lannin, 2005). Doğrulama ise matematiksel bilginin oluşumunda, gelişiminde ve iletilmesinde gerekli olan temel kavramlardan biridir.…”

Section: Introductionunclassified

Preservice Mathematics Teachers’ Knowledge of Generalization and Justification about Patterns

Tanışlı¹,

Köse²,

Camci³

2017

ENAD

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Öz. Bu çalışmanın amacı ortaokul matematik öğretmen adaylarının örüntüler bağlamında genellemelerini incelemek, genellemeler için ortaya koydukları doğrulamalarını keşfetmek ve yapılan genelleme ile doğrulama arasındaki ilişkiyi saptamaktır. Temel nitel araştırma yaklaşımının benimsendiği ve sekiz öğretmen adayı ile gerçekleştirilen bu çalışmada, verilerin toplanması klinik görüşme tekniği aracılığıyla gerçekleştirilmiş ve veri toplama aracı olarak doğrusal ilişki içeren ve içermeyen örüntü görevleri kullanılmıştır. Klinik görüşme verileri tematik analiz yöntemi kullanılarak analiz edilmiştir. Çalışma bulguları öğretmen adaylarının örüntüleri genelleme sürecinde belirgin ve belirgin olmayan muhakemeleri kullandıklarını ve doğrulama sürecinde ise tümdengelim ve tümevarım yöntemlerini benimsediklerini ortaya koymuştur. Belirgin muhakemeyi kullanan adayların doğrulama sürecinde genel olarak tümdengelim ve tümevarım yöntemlerini kullanırken, belirgin olmayan muhakemede bulunan adayların ise sadece tümevarım yöntemini kullandıkları bazı adayların da doğrulamalarının deneysel deliller ve dış otoriteden destek alma düzeyinde kaldıklarını göstermiştir. Çalışmada önemli görülen iki sonucun şekil aracılığıyla verilen örüntü görevlerinin genelleme ve doğrulama süreçlerini desteklemesi ve doğrulama yapmanın genelleme yapmayı katkı sağlaması olduğu söylenebilir. Öğretmen eğitiminde alan bilgisi ve alan öğretimine yönelik derslerde genelleme ve doğrulama süreçleri üzerinde adayların daha derin bir anlayış kazanmaları sağlanması önerilebilir.Anahtar Kelimeler: Genelleme, doğrulama, örüntüler, matematik öğretmen adayları Abstract. The purpose of this study is to examine preservice elementary school mathematics teachers' knowledge of generalization within the context of patterns, to elaborate their justifications for generalizations and to identify the relationships between the generalizations and justifications. In this study, which was conducted with eight preservice teachers using the basic qualitative research design, the research data were collected with the clinical interview technique. As the data collection tool, linear and non-linear pattern tasks were used. The data coming from clinical interview were analyzed through thematic analysis method. The research findings revealed that the preservice teachers used explicit and implicit reasoning in the process of generalizing the patterns, and applied deductive and inductive methods in the process of justifying their reasoning. In the study, the preservice teachers who prefer explicit reasoning generally used deductive and inductive methods in the justification process, while those also used implicit reasoning applied only the inductive method. It was also found that some of the preservice teachers' justifications were limited to using empirical evidence and taking support from external authority. One important result obtained in the study was that the given figural pattern tasks supported the processes of generalization and justification. In addition, another important resul...

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Generalization and Justification: The Challenge of Introducing Algebraic Reasoning Through Patterning Activities

Cited by 136 publications

References 25 publications

Pattern Generalization by Elementary Students

Pattern Generalization by Elementary Students

Investigating Mathematical Knowledge for Teaching Proof in Professional Development

Preservice Mathematics Teachers’ Knowledge of Generalization and Justification about Patterns

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