Making sense of double number lines in professional development: exploring teachers’ understandings of proportional relationships

Orrill, Chandra Hawley; Brown, Rachael Eriksen

doi:10.1007/s10857-012-9218-z

Cited by 40 publications

(19 citation statements)

References 16 publications

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“…Various studies identified several characteristics of knowledge in pieces in teachers' pedagogical knowledge: (a) Context sensitivity: teachers were sometimes unable to rely on knowledge available in other situations (e.g., Carraher et al, 1985; Kali et al, 2011; Orrill and Eriksen Brown, 2012). (b) Incoherence: teachers' thoughts about one specific teaching situation contradicted those about another specific situation (Kane et al, 2002; Eley, 2006; Postareff et al, 2008).…”

Section: Introductionmentioning

confidence: 99%

Preparing learners with partly incorrect intuitive prior knowledge for learning

Ohst¹,

Fondu²,

Glogger³

et al. 2014

Front. Psychol.

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Learners sometimes have incoherent and fragmented intuitive prior knowledge that is (partly) “incompatible” with the to-be-learned contents. Such knowledge in pieces can cause conceptual disorientation and cognitive overload while learning. We hypothesized that a pre-training intervention providing a generalized schema as a structuring framework for such knowledge in pieces would support (re)organizing-processes of prior knowledge and thus reduce unnecessary cognitive load during subsequent learning. Fifty-six student teachers participated in the experiment. A framework group underwent a pre-training intervention providing a generalized, categorical schema for categorizing primary learning strategies and related but different strategies as a cognitive framework for (re-)organizing their prior knowledge. Our control group received comparable factual information but no framework. Afterwards, all participants learned about primary learning strategies. The framework group claimed to possess higher levels of interest and self-efficacy, achieved higher learning outcomes, and learned more efficiently. Hence, providing a categorical framework can help overcome the barrier of incorrect prior knowledge in pieces.

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Section: Introductionmentioning

confidence: 99%

Preparing learners with partly incorrect intuitive prior knowledge for learning

Ohst¹,

Fondu²,

Glogger³

et al. 2014

Front. Psychol.

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“…The study makes at least two contributions. First, very little research has been conducted on teachers' reasoning with regard to proportional relationships (e.g., Orrill & Brown, 2012), and the results of this study demonstrate that they have difficulties in proportional reasoning based on their use of additive vs. multiplicative relationships in forming ratios. Second, no research has been reported about the effects of additive and multiplicative relationships on reasoning proportionally from the two perspectives on ratios, and the results of this study suggest that pre-service teachers' ability to use the two perspectives on ratios with appropriate reasoning, drawings, and words depends on how much they relied on multiplicative and additive relationships.…”

Section: Introductionmentioning

confidence: 80%

“…Orrill and Brown (2012) found in their study that middle grades teachers mostly relied on addition and subtraction rather than multiplication in tasks with ratios and proportions. In this study, while two of the six pre-service teachers used only multiplicative relationships, the remaining four resorted to additive relationships.…”

Section: Conclusion and Educational Implicationsmentioning

confidence: 94%

“…In these relatively few studies, teachers were found to perform poorly on proportional relationship tasks and to have difficulties similar to those of students (e.g., Harel & Behr, 1995;Pitta-Pantazi & Christou, 2011;Sowder, Philipp, Armstrong, & Scappelle, 1998). Just as students do, teachers might rely on rote computation procedures, such as cross-multiplication, and apply it inappropriately (e.g., Riley, 2010); might not focus on the invariance relationship between two co-varying quantities (e.g., Lim, 2009); and, might have difficulty coordinating two quantities in a proportional relationship (e.g., Orrill & Brown, 2012).…”

Section: Introductionmentioning

confidence: 99%

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Two Distinct Perspectives on Ratios: Additive and Multiplicative Relationships between Quantities

Ölmez¹

2015

İlköğretim Online

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ABSTRACT. The purpose of this study was to investigate how pre-service teachers' formation of additive and multiplicative relationships support and constrain their understandings of ratios and proportional relationships in terms of quantities. Six pre-service teachers were selected purposefully based on their performances in a previous course. An explanatory case study with multiple cases was used to make comparisons within and across cases. A semi-structured interview was conducted with pairs of pre-service teachers. The results revealed that pre-service teachers' heavy reliance on additive or multiplicative relationships critically shaped their reasoning about ratios from the two perspectives. Pre-service teachers who only attended to multiplicative relationships were found to have a robust understanding of proportional relationships. Pre-service teachers, who did attend to additive relationships, even if they used multiplicative relationships, struggled to form appropriate proportional relationships from the two perspectives. Keywords: Ratio and Proportion, Addition and Multiplication, Pre-service Teachers ÖZ. Bu çalışmada, öğretmen adaylarının nicelikler arasında oluşturduğu toplamsal ve çarpımsal ilişkilerin onların oranlar ve orantısal ilişkiler konusundaki anlayışlarını nasıl desteklediği ya da sınırlandırdığı araştırılmıştır. Çalışmaya dâhil edilen altı öğretmen adayı bir önceki lisans dersindeki performanslarına göre amaçlı bir şekilde belirlenmiştir. Durum içi ve durumlar arası karşılaştırmalar yapabilmek için çoklu durumla açıklayıcı durum çalışması kullanılmıştır. İkili gruplara ayrılan öğretmen adaylarıyla yarı-yapılandırılmış birer görüşme yapılmıştır. Sonuçlara göre, öğretmen adaylarının nicelikler arasındaki toplamsal ya da çarpımsal ilişkilere odaklanması, onların oranlarla ilgili iki yaklaşımı kullanarak akıl yürütmelerini kritik bir şekilde etkilemiştir. Nicelikler arasında yalnızca çarpımsal ilişkiler kuran öğretmen adaylarının orantısal ilişkilerle ilgili sağlam bir anlayışa sahip olduğu bulunmuştur. Diğer taraftan, nicelikler arasındaki toplamsal ilişkilere odaklanan öğretmen adayları, çarpımsal ilişkiler de kursalar, oranlarla ilgili iki yaklaşımı kullanarak orantısal ilişkiler oluşturmada zorluk yaşamışlardır. Anahtar Kelimeler: Oran ve Orantı, Toplama ve Çarpma, Öğretmen Adayları ÖZET Amaç ve Önem: Öğrencilerin orantısal akıl yürütmelerini araştıran geniş bir alanyazına kıyasla (Karplus, Pulos ve Stage, 1983;Noelting, 1980aNoelting, , 1980b detaylı bir inceleme için, Lamon, 2007), öğretmenlerin bu konudaki anlayışlarını sorgulayan göreceli olarak çok az çalışma vardır. Konuyla ilgili var olan bu sınırlı sayıdaki çalışmalarda, orantısal akıl yürütme konusunda öğretmenlerin öğrencilerle benzer zorluklar yaşadıkları belirlenmiştir (Harel ve Behr, 1995; PittaPantazi ve Christou, 2011; Sowder, Philipp, Armstrong ve Scappelle, 1998). Öğrencilerde gözlemlendiği gibi, öğretmenlerin de orantısal ilişki problemlerini çözerken içler-dışlar çarpımı gibi ezber odaklı yöntemlere başvurabilecekleri (Ri...

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“…(p. 189). In other studies, teachers have (a) had difficulty distinguishing missingvalue problems that describe proportional relationships from ones that do not (e.g., Cramer et al 1993;Fisher 1988;Lim 2009;Riley 2010); (b) had trouble coordinating two quantities in a proportional relationship (e.g., Orrill and Brown 2012); (c) made inappropriate additive comparisons (e.g., Canada et al 2008;Lim 2009;Son 2010); and (d) had trouble conceiving of a ratio as a measure of a physical attribute, such as steepness or speed (Akar 2010;Simon and Blume 1994;. With respect to problem-solving strategies, teachers have relied heavily on cross multiplication or other formal methods (e.g., Fisher 1988;Harel and Behr 1995;Orrill and Brown 2012;Riley 2010), guessed at operations (Harel and Behr 1995), and searched for key words (Harel and Behr 1995).…”

Section: Appendix: Released Itemsmentioning

confidence: 99%