María C. Cañadas

Utilización de un modelo para describir el razonamiento inductivo de los estudiantes en la resolución de problemas

Abstract Introduction . We present some aspects of a wider investigation (Canadas, 2007), whose main objective is to describe and characterize inductive reasoning used by Spanish students in years 9 and 10 when they work on problems that involved linear and quadratic sequences. Method. We produced a test composed of six problems with different characteristics related to sequences and gave it to 359 Secondary students to work on. The problems could be solved using inductive reasoning. We used an inductive reasoning model made up of seven steps (Canadas and Castro, 2007) in order to analyze students’ responses. Results. We present some results related to: (a) frequencies of the different steps performed by students, (b) relationships between the frequencies of steps depending on the characteris-tics of the problems, and (c) the study of the (in)dependence relationships among different steps of the model of inductive reasoning. Discussion. We can conclude that the inductive reasoning model was useful to describe stu-dents’ performance. In this paper, we emphasize that the model is not linear. For example, in some problems students reach the generalization step without passing through the previous steps. To describe how students reach more advanced steps without the previous ones, and to analyze whether accessing the intermediate steps could have been helpful for them, are tasks for future research.

Uso de representaciones y generalización de la regla del producto

Resumen Analizamos los protocolos de resolución de problemas de 50 estudiantes con el objetivo de profundizar en la construcción de la regla del producto como esquema básico de resolución de problemas de conteo. En particular, indagamos en el uso que se hace de la inducción y de diferentes representaciones para pasar de la enumeración exhaustiva y el recuento total de posibilidades a la generalización de dicha regla. El análisis ha puesto de manifiesto la existencia de dos procesos de generalización: sobre la dimensión del problema y sobre el número de elementos que intervienen en cada factor. Mostramos cómo ambos procesos se relacionan con el uso efectivo de los diagramas de árbol que los estudiantes generan de manera espontánea y apuntamos posibles implicaciones para la instrucción. Por otra parte, el análisis de los datos ha generado la necesidad de indagar en la conexión entre las representaciones textuales y otros tipos de representaciones, evaluando su funcionalidad.

Call for manuscripts: Early algebraic thinking / Convocatoria de presentación de manuscritos: Pensamiento algebraico temprano

Call for manuscripts: Early algebraic thinking / Convocatoria de presentación de manuscritos: Pensamiento algebraico temprano María C. Cañadas, Maria Blanton & Bárbara M. Brizuela To cite this article: María C. Cañadas, Maria Blanton & Bárbara M. Brizuela (2017) Call for manuscripts: Early algebraic thinking / Convocatoria de presentación de manuscritos: Pensamiento algebraico temprano, Infancia y Aprendizaje, 40:3, 657-660, DOI: 10.1080/02103702.2017.1357291 To link to this article: http://dx.doi.org/10.1080/02103702.2017.1357291

Book review of The Singapore model method for the learning of mathematics / Reseña del libro, The Singapore model method for the learning of mathematics

The central thesis of this monograph, entitled Symbolic instruments in early mathematical knowledge, is that mathematical knowledge is the result of a socio-cultural construction where representations, including symbolic representations, play an essential role. The various research articles and micro-reports presented show that children can access mathematics through many paths, and that depending on the socio-cultural context in which children participate and the representations they employ, they may use different ways to construct, express and reflect on their mathematical thinking. Given that the development of children’s mathematical thinking depends largely on the education they receive in school, understanding different approaches for the teaching of mathematics becomes relevant. The emphasis that this monograph places on representational aspects resulted in an interest for the Model Method for Learning Mathematics, known in some countries as the ‘Singapore Method’. The book The Singapore model method for the learning of mathematics (Kho, Yeo, & Lim, 2009), published by the Singapore Ministry of Education, is a valuable resource for readers interested in understanding the Model Method, both at a theoretical and practical level. This method was developed during the early 1980s by a team of curriculum design and mathematics pedagogy specialists from the Singapore Ministry of Education. The team was led by Dr Kho. Their objective was to develop pedagogic materials that would help improve Singaporean students’ performance in mathematics. During the 1990s, the method was applied in primary schools and in the first two years of secondary education. The Model Method became widely recognized throughout the world because of the excellent results obtained by Singapore students in the Trends in International Mathematics and Science Study (TIMSS) international

CERME7 Working Group 3: Algebraic thinking

The Algebraic Thinking Working Group is an established CERME theme, and WG3 continued the work carried out in previous conferences. 13 papers and 4 posters were presented, representing 13 countries. Four papers were focussed on the transition to algebraic symbolisation. Caspi and Sfard showed how 7th Grade Israeli students’ discourse contains some algebralike features, not normally found in everyday discourse. Dooley used epistemic actions to analyse and describe the development of algebraic reasoning amongst Irish pupils aged 9 11 years, and argued that the use of ‘vague’ language was central to such development. Gerhard exemplified the use of an analytic tool with secondary students in Germany, and highlighted the importance of focusing on the question of how algebraic knowledge interacts with arithmetic knowledge. Pytlak demonstrated how relatively sophisticated algebraic thinking can be achieved by children with geometric and numeric approaches, but without the use of symbols. A second theme concerned equations and symbolisation. Alexandrou-Leonidou and Philippou found that primary children in Cyprus were capable of developing the dual meaning of the equals sign. Through a teaching intervention, children were enabled to solve equations in multiple representation formats. Didiş, Baş and Erbaş examined students understandings and errors when solving quadratic equations. Their findings added further weight to the literature, highlighting the ubiquity and problems of a purely instrumental, or procedural, understanding. Other authors tackled technology. Chiappini demonstrated how AlNuSet software can enable students in Italy to overcome crucial epistemological obstacles with negative numbers and the equivalence of algebraic forms. Hewitt used the software Grid Algebra to analyse the activity of English students aged 9 10 years in order to examine the nature of algebraic activity. Maffei and Mariotti used Aplusix CAS in Italy to examine the interplay between different representations in algebra. They concluded that natural language has a dual role. Nobre, Amado, Carreira and da Ponte showed how Excel enabled three Grade 8 Portuguese students to engage with algebraic structure without the need for algebraic symbolisation. These students were able to model and solve a complex problem. Generalisation was the fourth theme. Barbosa analysed the strategies used by 54 Portuguese students in 6th Grade working on generalisation. Students achieved better results with near generalisation than with far generalisation problems.