For over a decade now, I have aimed to understand the mathematical knowledge needed to teach mathematics effectively, and its impact on student learning (e.g., Hatisaru & Erbas, 2017). As continuation of these works, most recently, I investigated secondary mathematics teachers’ analogies – a type of mathematical connection – to describe the concept of function (Hatisaru, 2022).

Here’s what I found in my research. As shown in the diagram below, the teachers’ repertoire was rich in analogy, and among them, a ‘child-mother-linkage’ analogy was quite popular.

Figure 1.* Analogies made by the teachers for describing the concept of function*

**Why does this matter?**

Mathematics is important as a core skill for life and as a school subject, for attending STEM or non-STEM tertiary studies, and is a core skill for life. This makes mathematics learning and success imperative for both individuals and the general society.

As mathematical facts, ideas, concepts, or procedures are interconnected, mathematics needs to be taught in a connected way. Students who have such classroom experiences learn likely more mathematics than students who are introduced to mathematical content as discrete rules and formulas. More importantly, students who learn mathematics in a connected way find more opportunities to develop key mathematical competencies such as how to solve mathematical problems and how to think mathematically.

In reality, mathematics is usually taught as a series of unrelated facts, ideas, formulas, or procedures (OECD, 2016). Most students therefore view their mathematics lessons where generally routine and non-contextual questions are practised (e.g., solve the equation 2*x* + 1 = 13 for *x*) (Hatisaru, 2020a). Teacher capability is one of the several key factors contributing to views and learning outcomes of students (https://theconversation.com/optimising-the-future-with-mathematics-22122).

**How is a mathematical connection defined?**

Broadly defined, a mathematical connection is a relationship between two or more mathematical ideas, and it can be two types: *extra-mathematical connections* and *intra-mathematical connections*.

Whilst *extra-mathematical connections* refer to relationships between mathematical and non-mathematical contexts (e.g., musical rhythms and mathematical patterns), *intra-mathematical connections* are formed between ideas, concepts, procedures, or representations within mathematics (e.g., the graphical and algebraic forms of *y* = 2*x* + 1 represent the same relation).

Generally, there is an emphasis for mathematical connections in national curricula in many countries including the USA, Singapore, Australia, and England.

As a result of making mathematical connections students develop key competencies. They may use the connections across mathematical topics and see mathematics as an integrated whole and apply mathematical skills to solve problems that arise in other disciplines or in everyday life (see Hatisaru, 2022b).

An analogy can be a powerful cognitive tool for not only connecting mathematical content within mathematics but also connecting mathematical content to other disciplines or real life.

**What is an analogy?**

In their instruction, teachers sometimes use phrases such as ‘it is like’, ‘similarly’, or ‘just as’ to explain a new concept to a range of students who may have no prior knowledge or compare/contrast a familiar content with a less familiar or abstract content. In these cases, teachers in fact make analogies. Broadly speaking, an analogy is connecting an abstract or complex idea, concept, or situation with a more familiar or concrete idea to make that abstract idea more understandable.

When learning an unfamiliar or abstract mathematical content – such as the concept of function, students do not necessarily have the background knowledge to grasp that content. In the teaching and learning processes, analogies are used to make those unfamiliar concepts familiar.

By way of examples, sometimes balancing scales are used when teaching equations. The concept of function is frequently portrayed as a machine, and many teachers use function machines when introducing the idea of function to their students.

**The concept of function: ‘it is everywhere in real life’**

Many situations in the real world represent a function. These include the modelling and graphs we have frequently seen on news in the last couple of years showing the COVID patterns.

In school curricula worldwide, a function is defined as a special kind of mathematical relation that has two features: *arbitrariness* and *uniqueness*. According to the *arbitrariness* feature, a function does not need to be defined *only* on number sets, neither does it need to demonstrate regularity (a particular graph or expression)

As noted, sometimes functions are described as ‘machines’ or ‘input-output relations’. For example, if *x* =1 (an input), *y* = 2 (its output) for the function *y* = 2*x*. If *x* = 1, *y* cannot be both 1 and 2. That is, for every input there is *only* one output, and this is exactly the *uniqueness* condition of functions.

Research shows that these features of functions can be quite abstract and difficult to understand for school students. In Clement’s (2001) study, for instance, only four out of 35 high school students were able to give a definition of function incorporating these features.

I was curious about what analogies teachers would use to describe the concept of function.

**Teachers’ analogies to the concept of function **

Among the total of 224 responses to my open-ended questionnaire, there were 61 instances (representing 26 teachers) where the teachers made an analogy between a function and real-life situation.

The most popular analogies were the conventional ‘machine’ (18 occurrences) and ‘factory’ (4 occurrences) analogies, followed by a novel ‘child-mother linkage’ analogy (12 occurrences).

One of the teachers, for example, wrote:

[A function is like] Mother-child relation (every child surely has a (birth) mother).

According to another teacher:

If Set A is the set of children, Set B is the set of mothers: (i) no element will be left out in Set A as every child has a (birth) mother; (ii) a child cannot have more than one (birth) mother and also not all mothers have children.

In this ‘child-mum-linkage’ analogy, by comparing a function with the biological linkage between a child and mum, the teachers infer that:

- The domain and codomain of a function can be anything – such as children and birth mums – and this promotes understanding of the
*arbitrariness*condition of functions. - Every domain member is mapped to codomain members – like every child has a birth mum, and each domain member is mapped to
*exactly one*codomain member – just like a child cannot have two birth mums. These two similarities between a function and child-mum linkage promote understanding of the*uniqueness*requirement of functions.

Also, some concepts related to function, and some specific types of functions, can be illustrated through this analogy:

- Some codomain members may remain unmapped – just as some mums may have adopted – but not biological – children (the concept of upper bound for the range).
- All domain members can be mapped to a single codomain member – similar to all children in the domain may share the same mum (a constant function).
- Each range member is mapped to no more than one domain member – just like no child in the domain shares the same mum (one-to-one function).
- No codomain member is unmapped – like every mum in the codomain is a birth mum (onto function).

**Final words **

My study shows that analogies can be useful cognitive tools for making abstract and complex ideas accessible for students including the formal definition of function. Supporting students in learning the formal definition of function, that contains the *uniqueness* and *arbitrariness* features, may support their understanding of the *covariation* idea, and that can be a focus for future research.

*Vesife Hatisaru is a Lecturer in Secondary Mathematics Education in the School of Education at Edith Cowan University and an Adjunct Senior Researcher in the School of Education at University of Tasmania. Her research interests include associations between teacher knowledge and student learning outcomes and teaching and learning practices in maths classrooms.*