Mathematical connections – a growing construct

1. Introduction

Making mathematical connections has been an important issue in mathematics education for many years. That is partly because establishing connections between mathematical concepts, ideas or procedures enables students to know how to use them in variety of contexts, as well as to explain why they work (Kilpatrick et al., 2001). Mathematical connections can emerge, for example, when students undertake mathematical tasks or solve mathematical problems (García-García & Dolores-Flores, 2018). How they can be brought the surface has been an interest of many researchers. As a result, models or taxonomies have attempted to capture mathematical connections in student (or teacher) written texts or in the oral arguments that they generate.

A model originated by Businskas (2008) describing fine-grained mathematical connections contained in quadratic functions and equations has been used in many investigations in this field. An adaptation of this model is presented in Table 1; the types of mathematical connections referred to are cited throughout.

Table 1. Type of mathematical connections and examples.

	Description
Connection type	These connections:	Example
Different representations	are identified when one represents a mathematical concept by using its alternate or equivalent representations. An equivalent representation is the transformation of representation made in the same representation domain (symbolic-symbolic). An alternate representation is the translation of representation made across representation domains (symbolic-graphical).	Equivalent representations: y = (x + 2) × (x-2) is an equivalent representation of y = x^2-4.Alternate representations: The graph of y = x^2-4, for instance, is an alternate representation of this function.
Implication or if-then	appear when a mathematical concept leads to another mathematical concept using a logical relationship.	In equation x^2-4 = 0, second degree root implies there are two solutions, namely x = -2 and x = 2.
Part-whole relationships	are manifested in two ways: inclusion and generalisation. Inclusion is present when a mathematical concept is contained within another, and generalisation appears when a mathematical concept or idea is the generalisation of a particular case.	Inclusion: Functions are specific types of relations, and all functions are relations. Generalisation: y = ax^2-c is a generalisation of y = x^2-4, or y = x^2-4 is a specific case of y = ax^2-c.
Procedural	are identified when the person uses rules, formulae or algorithms to complete a mathematical task.	Using the methods of elimination or substitution in solving simultaneous equations.
Analogies	occur when a conceptual relationship between a familiar source domain and an abstract target domain is established.	A function is like a machine with inputs and outputs. An equation is like a balance scale.
Feature/property	occur when characteristics of a mathematical concept are depicted, or properties of the concept are described in terms of what makes it similar or different to others.	A rectangle has two sets of parallel sides and four 90-degree angles.
Instruction-oriented connections	refer to incorporating the relations among mathematical ideas in the teaching of mathematics. They can be manifested both when a new concept is linked to prior knowledge, and when an understanding of a new concept is dependent on the understanding of other concepts or prerequisites. An additional connection (revisiting taught knowledge) occurs when students are reminded about what they have been taught or learnt previously.	Prerequisite: Factors and multiples are concepts that must be known to understand working with fractions. Links with prior knowledge: Linking the concept of function with the concept of relations. Revisiting taught knowledge: Reminding ninth-grade students that 2^3 is not equal to 3^2 as they are taught in earlier grades.

Note: Reproduced from Hatisaru (2023).

This Article Collection brings together 8 papers, whose authors have contributed to the development of the model for mathematical connections originally proposed by Businskas (2008) and extended the model to Extended Theory of Mathematical Connections (ETMC; Rodríguez-Nieto, Moll, et al., 2022). The papers in the Collection provide comprehensive empirical support for the validity of ETMC and its potential to investigate the associations between a teacher’s mathematical knowledge for teaching a concept and their ability to establish connections in teaching that concept (Hatisaru, 2023; Mhlolo, 2013).

I hope that, if the conceptualisations presented in these papers are new to you, you will find them useful in your research or teaching, and that they bring insight to the work that you do.

2. Collection papers

Dolores-Flores et al. (2019) proposed four tasks to the 33 pre-university students who participated in their study. The central concept of the first task is slope, while the last three tasks contain concepts like velocity, speed and acceleration. Task-based interviews were conducted to collect data, which were later analysed with thematic analysis.

The results showed that most of the students made mathematical connections of the procedural type. The mathematical connections of the common features type were made in smaller quantities and the mathematical connections of the generalisation type were scarcely made. Furthermore, students considered slope as a concept disconnected from velocity, speed and acceleration.

This research provides insights into how students make connections between different mathematical concepts and procedures. It also highlights areas where further instruction may be needed to help students better understand and apply these concepts.

García-García and Dolores-Flores (2021) focused on the mathematical connections that pre-university students made when they solved problems involving derivatives and integrals. The authors define a mathematical connection as a relationship between two or more mathematical ideas, concepts, definitions, theorems or meanings. The research was conducted using task-based interviews that included four application problems and the data were collected from 25 students and analysed using thematic analysis.

The research identified five types of intra-mathematical connections: procedural, different representations, part-whole, feature and reversibility. These connections are dependent on each other and form systems of mathematical connections around the reversibility connection between the derivative and the integral.

The authors found that the students seldom used visualisation to solve graphical tasks. They suggested that future research should develop classroom intervention proposals to promote the use of visualisation and the development of the ability to make mathematical connections, which could improve students’ mathematical understanding.

This research contributes to the understanding of the types of mathematical connections that students are able to make. It provides a preliminary theoretical framework to study mathematical connections in Calculus in future research.

García-García and Dolores-Flores (2018) explored the intra-mathematical connections that high school students make when they solve Calculus tasks. The authors consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. The research particularly focuses on tasks involving the derivative and the integral.

The data were collected through task-based interviews and analysed using thematic analysis. The analysis of the productions of the 25 participants led to the identification of 223 intra-mathematical connections.

These connections allowed the researchers to establish a mathematical connections system which contributes to the understanding of higher concepts, in this case, the Fundamental Theorem of Calculus. The types of mathematical connections found include different representations, procedural, features, reversibility and meaning as a connection.

Dogan et al. (2022) investigated the understanding of the linear independence concept in linear algebra, based on the type and nature of connections. The authors conducted interviews with seven non-mathematics majors and asked them a set of open-ended questions. The responses were analysed qualitatively to understand how these students perceived and connected the concept of linear independence.

Through this analysis, six categories of frequently displayed connections were identified. These categories provide insights into how students understand and apply the concept of linear independence in different mathematical contexts.

This research contributes to the field by offering a new perspective on understanding mathematical concepts through connections. It also provides valuable insights for educators to enhance their teaching strategies in conveying complex mathematical concepts like linear independence.

Rodríguez-Nieto, Rodríguez-Vásquez, et al. (2022) made a theoretical reflection about connections that contributes to the development of the model for mathematical connections originally proposed by Businskas (2008) and extended with the contributions of other researchers.

The context for the reflection was the transcription of several episodes of a lesson on the derivative. The authors first analysed the connections given in this transcription grounded in a qualitative methodology based on a-priori categories given by the theoretical model. They then selected some paragraphs where the model proved to be ambiguous, and finally, argued the need to clarify or extend the categories in the model to deepen the study of the connections in the transcript. As a result of this analysis, they proposed metaphorical connections as a new type of connection.

Metaphorical connections were evident in an investigation conducted by Hatisaru (2022). The author examined teacher-made analogies to function by content analysing of a sample of 26 secondary mathematics teachers’ responses to an open-ended questionnaire where they were prompted to define and exemplify the concept of function. Among the total of 224 responses, there were 61 instances where the teachers made an analogy between a function and real-life situation. The most popular analogies were the conventional ‘machine’ and ‘factory’ analogies, followed by a novel ‘child-mother linkage’ analogy. In all these analogies, connections are established through the comparison of the concept of function with analogous real-world scenarios such as biological linkage between a child and their mother, factory and machine, fostering a deeper conceptual understanding through relatable contexts.

In their later work, Rodríguez-Nieto, Moll, et al. (2022) explored the mathematical connections established by a teacher when teaching the derivative. The authors used the ETMC and the Onto-Semiotic Approach to analyse mathematical connections by relating the graphs of f and f’. This research contributes to understanding how mathematical connections are established and how they can be analysed using different theoretical frameworks. Rodríguez-Nieto and his colleagues use ETMC in several other investigations, and sometimes networking ETMC with other theories.

Mhlolo (2013) developed an analytical tool that can be used to determine the quality of following the connections made in practice: different representations, part-whole, if–then, procedural and instructional-oriented connections. A Likert scale from Level 0 (connection made was mathematically erroneous) to Level 3 (connection made was mathematically acceptable and justified) was employed. Pilot testing of the tool on 20 lessons delivered by four Grade 11 teachers revealed that these connections could be content-dependent, that teacher content knowledge can impact the quality of connections that they make, and the analytical tool developed has potential to identify the strengths and weaknesses of mathematical connections made in a lesson.

Acknowledgements

I thank Constantine Lozanovski for his assistance. I also thank the editor of IJMEST for his editing of the earlier draft.

Disclosure statement

No potential conflict of interest was reported by the author.