Re-thinking ‘Concrete to Abstract’ in Mathematics Education: Towards the Use of Symbolically Structured Environments

Coles, Alf; Sinclair, Nathalie

doi:10.1007/s42330-019-00068-4

Cited by 21 publications

(12 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…I have long been fascinated in the similarity between this starting point and Gattegno's curriculum (1974), which starts with play with Cuisenaire rods, first codified in terms of precisely those relations, with a focus on length. Venenciano et al (2020) analyse the activities of students within the Measure Up programme, partly drawing on work I have done with Nathalie Sinclair, in relation to the notion of a symbolically structured environment (Coles & Sinclair, 2019). One of the features of symbolically structured environments is that symbols are introduced to represent actions or relationships.…”

Section: Transposing a Curriculummentioning

confidence: 99%

Commentary on a special issue: Davydov’s approach in the XXI century

Coles

2021

Educ Stud Math

Self Cite

View full text Add to dashboard Cite

The articles in this special issue, collectively, provide overwhelming evidence that a curriculum which has as its primary basis the counting of discrete objects (and hence which introduces numbers as discrete) is not an effective or rational organisation. In this commentary, I discuss the contribution of each article to an understanding of Davydov’s ontology and epistemology, issues around the transposition of a pedagogy or curriculum from one country to another, early algebra and proportional or multiplicative reasoning. From Davydov’s own research and the writing in these articles, we know children are able to understand abstract structures from an early age. The thinking in this special issue provides tools to investigate and question any context in which such understanding is not routinely taking hold.

show abstract

Section: Transposing a Curriculummentioning

confidence: 99%

Commentary on a special issue: Davydov’s approach in the XXI century

Coles

2021

Educ Stud Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…This recollection of details, I am sad to say, has not been maintained as I became involved with more and more articles. Case in point: The very next issue, 16(4), includes four articles focused on mathematics education whose publication I had overseen as editor-Zazkis (2016), Smith et al (2016), Chorney et al (2016), and Rodney et al (2016)-but I am unable to recall many of the details of their submission-to-publication journey. In other words, the sensitization I was experiencing was slowly getting desensitized.…”

Section: Sincerely Eganmentioning

confidence: 99%

“…By the end of the year, I had handled 36 more. Of those 36 manuscripts, a few made their way into the pages of issue 19 (3)—e.g., Takeuchi et al ( 2019 )—and issue 19 (4)—e.g., Voutsina ( 2019 ), and Coles and Sinclair ( 2019 ). The accepted articles, I remember; I would not be able to recall details about the 33 manuscripts that were rejected that year quite as quickly.…”

Section: The Springer Era (2018–present)mentioning

confidence: 99%

On the Occasion of an Anniversary, Eh: Confessions of a Canadian Math Ed Editor

Chernoff

2020

Can. J. Sci. Math. Techn. Educ.

View full text Add to dashboard Cite

As its title suggests, this commentary utilizes the 20th year of publication of the Canadian Journal of Science, Mathematics, and Technology Education as an opportunity for me, the current English language mathematics education editor, to confess. I confess to imposter syndrome and editorial naiveté. I confess to evolving from putting the pathetic in empathetic to near-total emotional desensitization. I confess to having stumbled upon the notions of form letters and desk rejections. I confess that my former French teachers would be disappointed in me. I confess to having never forgotten my first, to seeing ghosts, and to attempting to handle multiple concurrent timelines based on the geologic time scale. Lastly, most importantly, and utilizing an entirely different meaning of the word confess, I confess to giving credit where credit is due. In other words, what follows are the absolutely true confessions of a Canadian mathematics education editor. Résumé Comme son titre l'indique, ce commentaire profite de la 20 e année de publication du Canadian Journal of Science, Mathematics and Technology Education pour me donner l'occasion, en tant qu'actuel rédacteur anglophone pour l'enseignement des mathématiques, de faire des aveux. J'avoue avoir été atteint du syndrome de l'imposteur et avoir fait preuve de naïveté éditoriale. J'avoue avoir évolué d'une empathie pathétique à une désensibilisation émotionnelle quasi totale. J'avoue être tombé sur des esquisses de lettres types et des manuscrits rejetés par le pupitre. J'avoue que mes anciens professeurs de français seraient déçus de moi. J'avoue n'avoir jamais oublié ma première fois, avoir vu des fantômes et avoir tenté de gérer de multiples chronologies simultanées basées sur l'échelle de temps géologique. Enfin, le plus important, et en utilisant un sens complètement différent du mot « avouer », j'avoue donner à chacun le crédit qui lui revient. Autrement dit, les lignes qui suivent présentent les aveux tout à fait vrais d'un rédacteur pour l'enseignement des mathématiques canadien.

show abstract

“…Abstraction in context is an activity or process of reorganization of previous mathematical knowledge into new mathematical structures that incorporate the context motivating it. Coles and Sinclair (2018) critique the assumption that learning should begin with the concrete and familiar while abstraction arrives later. They argue for a relational view according to which number learning is, from the very beginning, rooted in abstractions that are already part of young children's lives.…”

Section: Introductionmentioning

confidence: 98%

Body motion, early algebra, and the colours of abstraction

et al. 2020

View full text Add to dashboard Cite

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector-WiiGraph-with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A-B = C. We examine the notion of abstraction on its own, without assuming a dualism abstract-concrete according to which more of one is less of the other. We propose a distinct path for the attainment of abstraction, which involves navigating a surplus of sensible qualities. The work described in this paper belongs to early algebra, we suggest, because it involves the elementary symbolic treatment of unknowns and generals. More broadly, it advances a perspective on the nature of mathematical abstraction.

show abstract

Re-thinking ‘Concrete to Abstract’ in Mathematics Education: Towards the Use of Symbolically Structured Environments

Cited by 21 publications

References 28 publications

Commentary on a special issue: Davydov’s approach in the XXI century

Commentary on a special issue: Davydov’s approach in the XXI century

On the Occasion of an Anniversary, Eh: Confessions of a Canadian Math Ed Editor

Body motion, early algebra, and the colours of abstraction

Contact Info

Product

Resources

About