Vesife Hatisaru Archives

It’s all about balance: why analogies might make maths easier to understand

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 2x + 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 = 2x + 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 = 2x. 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.

Lazy, crazy mathematicians and other myths we need to bust

Creating opportunities for students to develop ‘healthy’ images of mathematicians and mathematics is paramount. The images of mathematics or mathematicians that students hold have a huge impact on their learning outcomes. For example, the perceived negative image of mathematicians by students could result in unhappiness in mathematics classrooms or a loathing of mathematics (Hatisaru & Murphy, 2019).

Why is it important? Maths matters because it impacts life quality, income and national development.

Since 2009, I have aimed to understand school students’ images of mathematics and mathematicians (Hatisaru, 2020). What views do they have about mathematicians and their work? What are the connections between students’ views about mathematicians and their attitudes towards mathematics? What views do they have about the needs for mathematics? How do they perceive their mathematics classroom?

Students’ images of mathematicians and mathematics are developed throughout years and impacted by several different factors. From the investigations of myself and others it is clear that experiences in mathematics classrooms contribute to students’ perceptions. Other factors include representations in media and popular culture, and family or society related factors.

For example, Wilson and Latterell (2001) found that in movies, literature, comics, and music mathematicians are portrayed as insane, socially inept. Darragh (2018) too.

Ucar et al. (2010) examined the image of mathematicians held by a group of 19 elementary school students and observed that the students described mathematicians as ‘unsocial, lonely, angry, quiet who always work with numbers’. (p. 131).

Picker and Berry (2000) introduce a cycle of the perpetuation of stereotypical images of mathematics and mathematicians (for example ‘mathematicians are weird’ or ‘mathematicians are asocial people’). According to them, this cycle begins with exposition of different cultural and societal stereotypes via TV, cartoons, books, other media, also via peers and adults through negative repeating phrases.

Among students there is a dominant male perception of mathematicians (e.g., Aguilar et al. 2016; Picker & Berry 2000). In Picker and Berry’s study, which included participants from 5 different countries, students sometimes associated negative or aggressive behaviours to mathematicians such as being large authority ﬁgures, crazy men, or having some special power.

Darragh (2018) examined 59 young adult fiction books to identify the depiction of school mathematics in them. Mathematics was more commonly portrayedto be “nightmarish; inherently difficult; something to be avoided: …”

“Mathematics teachers in particular bore the brunt of negative portrayals and were depicted as ridiculous, sinister, insane, and even dispensable; in short, they were positioned as villains.”

Students then meet teachers who lack awareness of stereotypes of mathematics and mathematicians, and sometimes they themselves hold certain stereotypes. Through teachers and the media, students are affected by certain attitudes such as ‘they must be quick at mathematics to be good at it’, or ‘mathematicians are a privileged group who have the special ability to do mathematics’. These messages and others, according to Picker and Berry, contribute to the formulation of the perceptions of mathematics and mathematicians in students’ minds. Jo Boaler, too, indicates that in her writings on the (important) role of holding a Growth Mindset in mathematics.

Over time, Picker and Berry continue, students develop attitudes and belief systems towards mathematics and mathematicians that may lead to generalisations or stereotypes. The cycle completes with the exchanging of students’ views with others. As a part of society, each student now contributes to others’ images of mathematicians and mathematics.

Given that some students hold negative images of mathematicians and mathematics, and their images are impacted by school-related factors, it is important that school educators are aware of student images.

For about four years now, in my interactions with schoolteachers in several different conferences, workshops, and professional learning events, I have noticed that some teachers use the phrase ‘Since mathematicians are lazy …’ often when they introduce some ‘short-cut’ methods or procedures to their students. Once, for example, the context was solving the problem: 27 + 28 + 13 = ? The teacher’s language practice was: ‘Since mathematicians are lazy, they add 27 to 13 first, which is 40, and then add 28 which gives 68’.

In fact, the mathematical behaviour behind this solution is ‘efficiency’ (Cirillon & Eisenmann, 2011) rather than ‘laziness’. The mentioned ‘lazy mathematicians’ know that, according to the associative property, 27 + 28 + 13 = (27 + 28) + 13 = (27 + 13) + 28. In this case, adding 27 to 13 first is a lot easier than adding 27 to 28 as 7 and 3 makes 10.

Using this property for solving a problem such as 138 + 44 + 12 + 6 = ? makes the calculations even easier: adding 138 to 12 gives 150, and adding 44 to 6 gives 50. The sum of 150 and 50 is 200. Once again, the reason for mathematicians’ desire to use these approaches is not ‘laziness’ but their desire for ‘efficiency’. They also see mathematics as a connected body of knowledge. That is, they use the same property in solving algebra problems (e.g., 17x + 21y + 43x + 19y = (17x + 43x) + (21y + 19y) = 60x + 40y).

In the short term, ‘mathematicians are lazy’ types of messages may appeal to students, but in the long term, they may contribute to the development of (negative) stereotypical images of mathematicians in students. It may prevent students from ‘seeing’ the reasons behind mathematical procedures, and the beauty and connectedness in mathematical ideas. Furthermore, they are morally wrong: Are mathematicians really ‘lazy’? Have we met all mathematicians? Have we measured their relevant attitudes? Were they found to be ‘lazy’ based on those measurements?

While we cannot control messages in the media or popular culture, as also Cirillon and Eisenmann tell us, we could carry and share best messages with our students. My suggestion to schoolteachers, and all other actors in mathematics education including parents and family members, is that we use alternative phrases. Why not use: ‘Since mathematicians are creative …’, ‘Since mathematicians seek to find alternative approaches …’, or ‘Since mathematicians desire to use more efficient ways …’.

These messages are not only more representative and morally more appropriate, but they also have more value in developing images of mathematicians and mathematics in students that are closer to the reality. Moreover, they could contribute to establishing ‘healthier’ relationships between students and mathematicians and mathematics.

Vesife Hatisaru MEdB, MEdM, PhD, MEdD is a lecturer in Mathematics Education (Secondary) in the School of Education, Edith Cowan University Joondalup, and an adjunct senior reseacher in the School of Education, University of Tasmania. She had a long career as a secondary school mathematics teacher before entering academia.