maths

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

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.

How to do the sums for an excellent maths curriculum

As we await the release of a new Australian curriculum for mathematics, debates about its contents are developing. As is typical with educational debates, the issues are often painted in binary terms: traditional vs progressive, explicit teaching vs problem solving, content vs skills, procedural vs conceptual knowledge. In mathematics education, these debates have existed for some time, pitting supporters of explicit teaching of clearly defined content against those who advocate for more opportunities for mathematical problem solving and reasoning. However, in order to produce mathematically-able citizens at one end of the spectrum, and at the other, the mathematicians of the future, we need both. We need students with extensive mathematics knowledge and skills, who can also think flexibly and creatively with that knowledge to be confident problem solvers. 

So what do teaching approaches have to do with the new Australian curriculum? A curriculum should outline what is to be taught alongside its purpose and intent. In addition to outlining the mathematics content for each year level, the current Australian curriculum for mathematics includes four proficiency strand: understanding, fluency, problem-solving and reasoning. The strands describe “how content is explored or developed; that is, the thinking and doing of mathematics.” These proficiencies are integrated with the content to ensure that students can work adaptively with mathematics content rather than just be competent users of set procedures. The world that students will inhabit in the future will require them to have strategies to solve complex problems and apply their mathematical content knowledge to unfamiliar situations. Taking on these challenges will require them to be confident and experienced problem solvers. Knowing how to apply mathematical procedures and algorithms only to familiar situations is not sufficient. 

The world that students will inhabit in the future will require them to have strategies to solve complex problems and apply their mathematical content knowledge to unfamiliar situations.

Criticisms of learning approaches that enable students to problem solve or to reason mathematically often assume that teachers are trying to teach generic skills in the absence of content. This is not the intent of the Australian mathematics curriculum, where it is expected that the proficiencies are integrated with the mathematics content. Students need sound content knowledge in order to draw on that knowledge to solve problems. They cannot think critically without sound underlying knowledge to think critically about. In mathematics they need both procedural and conceptual knowledge, which develop iteratively alongside one another. Teachers have a role to play in providing quality explicit instruction which takes into account cognitive load theory but students also need time to explore and compare alternative problem solutions and to communicate their mathematical reasoning.  

We know that mathematics is a polarising subject. Most people have strong views about their attitudes towards the subject and their capacity to do mathematics. Such views were generally formed as a result of their schooling. Unlike other school subjects, mathematics induces widespread anxiety and negative feelings, often attributed to traditional teaching and assessment practices. These negative attitudes cause many learners to opt out of mathematics in the senior years of schooling and to disengage with the subject from an even earlier age. The TIMSS data tells us that more positive attitudes, in terms of valuing, liking and confidence with mathematics are related to higher achievement. So teaching approaches that support the development of positive attitudes are vitally important so that students don’t start to consider themselves as a ‘non-maths person’, losing interest in mathematics early in secondary school. Traditional explicit teaching, while appropriate some of the time during a teaching sequence, if overused, can alienate some learners, leading to disengagement and a reduction in student confidence leading to a decrease in enrolments in mathematics in senior years of schooling and beyond. 

The new Australian Curriculum is an opportunity to create a curriculum that will enable students to develop a deep understanding of and appreciation for mathematics. The curriculum should provide guidance for teachers so that they can teach students in ways that develop understanding and competence but also develop an interest in the subject and a desire to learn more. We want our students to finish school with positive attitudes to mathematics and confidence to use mathematics within their personal and working lives. This will only happen if teachers provide a balanced approach to learning mathematics, with time for students to learn from explicit instruction, but also to apply their creativity and knowledge to unfamiliar situations, building confidence in themselves as learners, preparing them for whatever the future holds.

Professor Kathryn Holmes is the director of the Centre for Educational Research in the School of Education at Western Sydney University. She is also the asssociate dean, research, School of Education

Associate Professor Catherine Attard is the deputy director of the Centre for Educational Research in the School of Education, Western Sydney University. She is also president, Mathematics Education Research Group of Australasia

 Tomorrow Emeritus Professor of Educational Psychology at UNSW John Sweller adds to the debate.