Over the past few decades, Piaget’s forms of abstraction have proved productive for developing explanatory models of student and teacher knowledge, yet the broader applicability of his abstraction forms to mathematics education remains an open question. In this paper, we adopt the Piagetian forms of abstraction to accomplish two interrelated goals. Firstly, we analyze instructional tasks to develop hypothetical accounts of the abstractions that might occur during students’ engagement with them. Secondly, we draw on middle- and secondary-grades classroom data to discuss the abstractions that occurred during the implementation of those instructional tasks. Because this paper represents an initial attempt at extending the applicability of Piagetian forms of abstraction, we close with potential implications of such use and possible avenues for future research. Most notably, we highlight the complexities involved in supporting abstraction through curriculum and instruction.

Generalizing is a critical aspect of mathematics learning, with researchers and policy documents highlighting generalizing as a core mathematical practice. It can also be challenging to foster in class settings, and teachers need access to better resources to teach generalizing, including an understanding of effective forms of instruction. This article proposes Classroom Supports for Generalizing (CSGs), investigating how multiple elements-such as tasks, teacher moves, student interactions, and representations-interact to meaningfully foster student generalizing. Drawing on class video data from a middle school teacher and two high school teachers, we present the CSG Framework, which identifies three categories of supports: Interactions for Generalizing, Structures for Generalizing, and Routines for Generalizing.

Over the most recent several decades, researchers have argued the importance of quantitative and covariational reasoning for students’ learning. These same researchers have illustrated the importance of these reasoning processes with respect to local and longitudinal development. In both grain sizes, researchers are detailed in their descriptions of the intended topics or reasoning processes. There is, however, a lack of specificity of generalized criteria for concept construction from a quantitative reasoning perspective. In this chapter, we introduce such criteria through the construct of an abstracted quantitative structure, which has its roots in quantitative reasoning, covariational reasoning, and various Piagetian notions. In introducing the construct, we focus on ideas informing its development and its criteria, and we use it to characterize examples of student actions. We close with comments regarding implications for both teaching and research.

In this study, based on the analysis of a teaching experiment with middle school students, we propose a framework for describing meanings of a point represented on a plane in terms of multiplicative objects in the context of graphing. We classify those meanings as representing (i) non-multiplicative objects, (ii) quantitative multiplicative objects (Type-1 and Type 2), and (iii) spatial multiplicative objects. We then discuss implications of these meanings with respect to students’ graphing activities.

This paper introduces a new mode of variational and covariational reasoning, which we call scaling-continuous reasoning. Scaling-continuous reasoning entails (a) imagining a variable taking on all values on the continuum at any scale, (b) understanding that there is no scale at which the continuum becomes discrete, and (c) re-scaling to any arbitrarily small increment for x and coordinating that scaling with associated values for y. Based on the analysis of a 15-h teaching experiment with two 12-year-old pre-algebra students, we present evidence of scaling-continuous reasoning and identify two implications for students’ understanding of rates of change: seeing constant rate as an equivalence class of ratios, and viewing instantaneous rate of change as a potential rate. We argue that scaling-continuous reasoning can support a robust understanding of function and rates of change.

In this paper, we illustrate and discuss two undergraduate students’ reasoning about quantities’ magnitudes. One student identified regularities regarding the relationship between two quantities by focusing on successive amounts of change of one quantity (i.e., a pattern) while the other attended to relative amounts of changes in both quantities (i.e., a relationship). We illustrate that although reasoning about amounts of change is useful for making sense of the rate of change in quantities, reasoning about relative changes in identifying a relationship between quantities’ magnitudes is likely more productive in developing the concept of rate of change.

In this report, we present an analysis of two prospective secondary mathematics teachers’ generalizing actions in quantitative contexts. Specifically, we draw from a teaching experiment to report how Lydia and Emma engaged in different generalizing processes for the same task. Based on these differences, we found Lydia’s generalizing actions (i.e., coordinating quantities) to be a more productive generalization than Emma’s (i.e., slopes of tangent lines). The former was extendable to a new case, instance, or situation, whereas the latter was constrained to a specific representational system.

Bu çalışmada, matematik öğretmen adaylarının sahip olduğu analitik, geometrik ve harmonik düşünme yapıları belirlenip, bu yapıların matematiksel modelleme becerilerini nasıl etkilediğinin ortaya çıkarılması amaçlanmıştır. Özel durum çalışması olan bu araştırmada çoklu yöntem kullanılmıştır. Çalışma grubu, bir devlet üniversitesinin, tezsiz yüksek lisans programında öğrenim gören 75 matematik öğretmen adayından oluşmaktadır. Öğretmen adaylarının düşünme yapılarını belirlemek için Matematiksel Süreç Aracı, modelleme becerilerini belirlemek için Matematiksel Modelleme Testi uygulanmıştır. Matematik öğretmen adaylarının %12’sinin geometrik, %31’inin analitik, geriye kalan % 57’sinin ise harmonik düşünme yapılarına sahip olduğu bulgusuna ulaşılmıştır. Öğretmen adaylarının modelleme testinde yer alan soruların tümüne toplamda verdikleri doğru cevap oranı %40 olarak belirlenmiştir. Modelleme testinin toplamda cevaplanma oranına bakıldığında düşünme yapılarının modelleme testindeki başarıyı etkileyen bir faktör olmadığı sonucuna ulaşılmıştır. Fakat her biri farklı bir beceriyi ölçmeye yönelik olan modelleme testindeki sorulara verilen doğru cevap oranları düşünme yapılarına göre farklılık göstermektedir. Bu da düşünme yapılarına göre öğretim programının, ders kitaplarının, ders planlarının hedefe ve kazanıma yönelik belirlenmesi ve tertiplenmesi, öğrencilerin performanslarının daha da artırılabileceği hususunda fikir vermektedir.

Bu calismayla, matematik ogretmen adaylarina modelleme etkinlikleri uygulanip, bu etkinliklerin bireysel ve grup seklinde calisildiginda performansi ve sureci nasil etkilediginin ortaya cikarilmasi amaclanmistir. Ozel durum calismasi olan bu arastirmada nitel yontem kullanilmistir. Calisma grubu olarak bir devlet universitesinin, tezsiz yuksek lisans programinda ogrenim goren 75 matematik ogretmen adayi secilmistir. Bireysel ve grup calismasinin modelleme surecine ve performansa etkisinin incelenmesi amaciyla, modelleme yaklasimina uygun tarzda problem cozme etkinlikleri yaptirilmistir. Etkinlik kâgitlari dagitilarak ogrencilerin etkinlikler uzerinde bir sure calismalari saglandiktan sonra calisma kâgitlari toplanmistir. Hemen ardindan gruplar olusturularak her bir gruba etkinlik kâgitlari tekrar dagitilarak grup calismasi yapilmistir. Butun etkinliklerde grup calismalarindaki dogru cevaplanma orani bireysel calismalardaki dogru cevaplanma oraninin yaklasik bir bucuk kati olmustur. Grup calismalarinda yanlis cevaplanma orani bireysel calismalara gore neredeyse yari yariya azalmistir. Yine ayni sekilde grup calismalarinda, kismi cevap ve cevap yok kategorilerinde bireysel calismalara gore azalma gorulmustur. Ayni zamanda grup calismalarinda gercekci sekil ve model kullanimi bireysel calismalara gore daha fazla kullanilmistir. Grup calismasinin yer aldigi sosyal yapilandirmaci ogrenme ortaminin, rutin olmayan problemlerin dogru cozulme oranini artirdigi bilinmektedir. Arastirmamizda da rutin olmayan, alisilmamis problem tarzinda olan modelleme etkinlik ve sorularin cozumunde grup calismasinin daha etkili oldugu gozlenmektedir.

This study aims to establish the analytic, geometric and harmonic thinking structures of Mathematics teacher trainees, and to see how these structures determine their mathematical modelling skills. The research has been carried out as a special case study and uses multi-method approach. The study group consists of 75 teacher trainees who attend a post graduate programme at a state university. The programme does not require the submission of a dissertation. Mathematical Process Instrument has been used to assess the thinking structures of the Mathematics teacher trainees, and to determine their modelling skills Mathematical Modelling Tests have been carried out. It has been found that %12 of the Mathematics teacher trainees have geometric, %31 analytic and the remaining %57 have harmonic thinking structures. The overall correct answer score of the teacher trainees in the modelling tests is %40. On the basis of these percentages, it has been concluded that thinking structures do not determine the overall success rates in the modelling test. However, in the modelling test, the percentage of the correct answers given to the questions each of which is designed to test a different skill varies according to the thinking structures. This result gives an indication that the performance of the students can be further improved if the thinking structures are taken into account when teaching programmes, text books and lesson plans are decided and organised in line with aims and objectives.

"An analysis of pre-service mathematics teachers' performance in modelling tasks in terms of spatial visualisation ability." Research in Mathematics Education, 14(3), pp. 297–298 Acknowledgements This study is part of a project (project number EGT-C-YLP-040310-0058) funded by the Marmara University Scientific Research Projects Board.