mathematics

Proactive and preventative: Why this new fix could save reading (and more)

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When our research on supporting reading, writing, and mathematics for older – struggling  – students was published last week, most of the responses missed the heart of the matter.

In Australia, we have always used “categories of disadvantage” to identify students who may need additional support at school and provide funding for that support. Yet those students who do not fit neatly into those categories slipped through the gaps, and for many, the assistance came far too late, or achieved far too little. Despite an investment of $319 billion, little has changed with inequity still baked into our schooling system. 

Our systematic review, commissioned by the Australian Education Research Organisation, set out to identify the best contemporary model to identify underachievement and provide additional support – a multi-tiered approach containing three levels, or “tiers” that increase in intensity.

(de Bruin & Stocker, 2021)

We found that if schools get Tier 1 classroom teaching right – using the highest possible quality instruction that is accessible and explicit – the largest number of students make satisfactory academic progress. When that happens, resource-intensive support at Tier 2 or Tier 3 is reserved for those who really need it. We also found that if additional layers of targeted support are provided rapidly, schools can get approximately 95% of students meeting academic standards before gaps become entrenched.

The media discussion of our research focused on addressing disadvantages such as intergenerational poverty, unstable housing, and “levelling the playing field from day one” for students starting primary school through early childhood education. 

These are worthy and important initiatives to improve equality in our society, but they are not the most direct actions that need to be taken to address student underachievement. Yes, we need to address both, BUT the most direct and high-leverage approach for reducing underachievement in schools is by improving the quality of instruction and the timeliness of intervention in reading, writing, and mathematics.

Ensuring that Tier 1 instruction is explicit and accessible for all students is both proactive and preventative. It means that the largest number of students acquire foundational skills in reading, writing, and mathematics in the early years of primary school. This greatly reduces the proportion of students with achievement gaps from the outset. 

This is an area that needs urgent attention. The current rate of underachievement in these foundational skills is unacceptable, with approximately 90,000 students failing to meet national minimum standards. These students do not “catch up” on their own. Rather, achievement gaps widen as students progress through their education. Current data show that, on average, one in every five students starting secondary school are significantly behind their peers and have the skills expected of a student in Year 4:

Source

For students in secondary school, aside from the immediate issues of weak skills in reading, writing and mathematics, underachievement can lead to early leaving as well as school failure. Low achievement in reading, writing, and mathematics also means that individuals are more likely to experience negative long-term impacts post-school including aspects of employment and health, resulting in lifelong disadvantage. As achievement gaps disproportionately affect disadvantaged students, this perpetuates and reinforces disadvantage across generations. Our research found that it’s never too late to intervene and support these students. We also highlighted particular practices that are the most effective, such as explicit instruction and strategy instruction.

For too long, persistent underachievement has been disproportionately experienced by disadvantaged students, and efforts to achieve reform have failed. If we are to address this entrenched inequity, we need large-scale systemic improvement as well as improvement within individual schools. Tiered approaches, such as the Multi-Tiered System of Supports (MTSS), build on decades of research and policy reform in the US for just this purpose. These have documented success in helping schools and systems identify and provide targeted intervention to students requiring academic support. 

In general, MTSS is characterised by:

  • the use of evidence-based practices for teaching and assessment
  • a continuum of increasingly intensive instruction and intervention across multiple tiers
  • the collection of universal screening and progress monitoring data from students
  • the use of this data for making educational decisions
  • a clear whole-school and whole-of-system vision for equity

What is important and different about this approach is that support is available to any student who needs it. This contrasts with the traditional approach, where support is too often reserved for students identified as being in particular categories of disadvantage, for example, students with disabilities who receive targeted funding. When MTSS is correctly implemented, students who are identified as requiring support receive it as quickly as possible. 

What is also different is that the MTSS framework is based on the assumption that all students can succeed with the right amount of support. Students who need targeted Tier 2 support receive that in addition to Tier 1. This means that Tier 2 students work in smaller groups and receive more frequent instruction to acquire skills and become fluent until they meet benchmarks. The studies we reviewed showed that when Tiers 1 and 2 were implemented within the MTSS framework, only 5% of students required further individualised and sustained support at Tier 3. Not only did our review show that this was an effective use of resources, but it also resulted in a 70% reduction in special education referrals. This makes MTSS ideal for achieving system-wide improvement in both equity, achievement, and inclusion.

Our research could not be better timed. The National School Reform Agreement (NSRA) is currently being reviewed to make the system “better and fairer”. Clearly, what is needed is a coherent approach for improving equity and school improvement that can be implemented across systems and schools and across states and territories. To this end, MTSS offers a roadmap to achieve these targets, along with some lessons learned from two decades of “getting it right” in the US. One lesson is the importance of using implementation science to ensure MTSS is adopted and sustained at scale and with consistency across states. Another is the creation of national centres for excellence (e.g., for literacy: https://improvingliteracy.org), and technical assistance centres (e.g., for working with data: https://intensiveintervention.org) that can support school and system improvement.

While past national agreements in Australia have emphasised local variation across the states and territories, our research findings highlight that systemic equity-based reform through MTSS requires a consistent approach across states, districts, and schools. Implemented consistently and at scale, MTSS is not just another thing. It has the potential to be the thing that may just change the game for Australia’s most disadvantaged students at last.

From left to right: Dr Kate de Bruin is a senior lecturer in inclusion and disability at Monash University. She has taught in secondary school and higher education for over two decades. Her research examines evidence-based practices for systems, schools and classrooms with a particular focus on students with disability. Her current projects focus on Multi-Tiered Systems of Support with particular focus on academic instruction and intervention. Dr Eugénie Kestel has taught in both school and higher education. She taught secondary school mathematics, science and English and currently teaches mathematics units in the MTeach program at Edith Cowan University. She conducts professional development sessions and offers MTSS mathematics coaching to specialist support staff in primary and secondary schools in WA. Dr Mariko Francis is a researcher and teaching associate at Monash University. She researches and instructs across tertiary, corporate, and community settings, specializing in systems approaches to collaborative family-school partnerships, best practices in program evaluation, and diversity and inclusive education. Professor Helen Forgasz is a Professor Emerita (Education) in the Faculty of Education, Monash University (Australia). Her research includes mathematics education, gender and other equity issues in mathematics and STEM education, attitudes and beliefs, learning settings, as well numeracy, technology for mathematics learning, and the monitoring of achievement and participation rates in STEM subjects. Ms Rachelle Fries is a PhD candidate at Monash University. She is a registered psychologist and an Educational & Developmental registrar with an interest in working to support diverse adolescents and young people. Her PhD focuses on applied ethics in psychology.  

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

mathematics, Maths curriculum, STEM education, , ,

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.

Why kids under five must start learning to code

digital technology in schools, Early childhood education and care, mathematics, Maths curriculum, teaching coding, Teaching coding in schools, , , , ,


There’s a lot of pressure to learn coding in primary school to develop 21st century computational skills. But I think we should start in preschool.

Schools and governments recognise the need for teaching 21st century skills. We can see the evidence for that in the Australian Curriculum: Mathematics. But just as we teach preschool children the fundamentals of reading, we must now include computational thinking and coding.

While many early childhood services may have digital technologies, how they are presented and taught to young children, requires more focus and perhaps, upskilling of educators.

In my research we use cubettos. These are simple wooden square box robots with a smiley face that come with coding boards with colourful plastic pieces which make the robots move. We also uses bee-bots, small plastic bee-shaped robots with simple programming buttons on their backs, and blue-bots, clear cased bee-shaped robots that respond to commands sent from a computer or iPad.

I have been exploring how children learn to use them, and how educators support their use, as an extension to my PhD research. With 3-5 year olds, they realise that the pieces you put in the coding board, or the buttons you push, make the robots move in particular ways, and you can start explaining how they move and why they follow our commands.

Learning the basics of robotics at this age will set the foundation for primary school learning. It’s a great introduction to pre-maths, algorithms, counting and problem-solving: they learn this is the ‘recipe’ that moves the robots along.

While some early childhood education services have some robots to play with, what’s missing is the opportunity for richer learning that these devices offer.

Some educators will have a blue-bot or bee-bot and they might push the buttons to create a code for the children, but they’re not necessarily taking the next step to explaining the concept of coding.

The aim for my research is to think about the best way to equip educators to teach coding to pre-schoolers through play, for example whether it will be creating an instruction manual or workshop or something else.

Screens and play

Another aspect of technology is the use of screen technologies among the 3-5 year old age group, both real and replica or broken, which I call ‘imaginative’ technologies. Children want to use real technologies in play, but imaginative technologies are the next best thing. Children today live in a digital world and, given the opportunity, will readily use technology to meet their play needs.

Through my research, which has included interviewing 84 educators in the New England region alongside my colleague Dr Marg Rogers, I have found some educators are reluctant to incorporate real technologies into their classrooms for use by very young children.

Some educators and parents believe early use of technology will reduce their child’s creativity and imagination. Others encourage it. I have found with people holding such strong views, discussions on the subject can be a minefield.

Many services don’t provide real technology for children. They might use an iPad or camera, but it’s very controlled and directed, and the children are not given enough time with the technologies to develop skills and to learn.

The anxiety around using screens could also be further compounded and confused by current national guidelines on screen use.

National guidelines out of step

The current national guidelines recommend children under two are not exposed to any screen time. But this is really out of sync with home or modern life. So, it’s an interesting question for technology researchers like myself – do we follow the guidelines and not give technology to children? But children see technology and in their imaginative play, they want to copy what adults do.

They see people on phones, taking photos and typing on computers from an early age. How can you then have a ‘home corner’ in early childhood education centres that don’t have any of that? How many restaurants take orders on a phone or iPad? How can children re-enact what goes on in a restaurant without technology? Same with a doctor’s surgery or a supermarket. Technology is everywhere.

In a new research project, I will ask children aged 3-5 what they want in their imaginative play spaces and if they can make (out of recycled materials), what they need to in order to have an imaginative play space reflective of the real thing. For example, having iPads in a restaurant to take the orders or look up recipes. I believe real technologies also need to play a larger role in early childhood education.

And I don’t believe we can stop children accessing technologies.

I think it’s a bit disrespectful to not let children use technology in their play. How does it compute in their brains that technology is everywhere in their world, but they are not allowed to use or understand it? It must be confusing for them. I have found, educators need to support children’s technology use in positive ways. I’d like to see non screen-based coding, and iPads with select apps chosen for the learning that’s possible, including to document their own learning, in the preschool years.

Children need to learn how to use technologies and when it is appropriate to use them. Hopefully further research will help to guide the provision of technologies and guidelines for its use, support children’s ethical behaviour and reduce some of the discomfort educators and parents feel around the inclusion of working technologies in children’s lives.

Jo Bird is a senior lecturer at the University of New England, Armidale. Her PhD explored children’s use of digital technologies in imaginative play and the educators’ provision of the various devices, both working and imaginative. Her research interests include children’s play, the use of technologies by both children and educators and early childhood leadership. She loves presenting, both her research and inspiring others to use technologies in creative ways with children and to recognise their leadership worth.

Lazy, crazy mathematicians and other myths we need to bust

mathematics, Maths curriculum, STEM education, , , ,

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.

Five questions to ask if you think teaching problem-solving works

Curriculum review, Maths curriculum, teaching numeracy

Every few decades there is a campaign to include general problem-solving and thinking skills in school curricula. The motivation is understandable. Everyone would like our schools to enhance students’ critical thinking and problem-solving skills. Because it is so obviously important for students to have such skills, these campaigns are frequently successful in including thinking and problem-solving skill modules in the curriculum. Unfortunately, success in introducing thinking and problem-solving into curricula has not been matched by successful educational outcomes. Across the world, there is a consistent failure to actually improve problem-solving performance. We now know enough about human cognition to know why attempts to teach general cognitive skills such as problem-solving will always fail. Here are a few questions that those advocating for the curriculum to include problem solving in areas such as mathematics might want to consider.

  1. What sophisticated, learnable and teachable problem-solving strategies do you personally use when solving a novel problem? If you cannot describe the strategies you use, what hope do you have of teaching them? At least consider the possibility that there are no learnable and teachable general problem-solving strategies.
  2. Ignoring the problem that no novel, general problem-solving strategies have ever been devised, what evidence is there from randomised, controlled trials that teaching general problem-solving strategies improves problem-solving performance? If, after dozens of years attempting to find a body of evidence for the efficacy of teaching general problem-solving strategies, no such bodies of evidence exist, we must at least consider the possibility that they will never exist.
  3. The relevant randomised controlled trials have been run. Within a cognitive load theory context (Lovell, 2020; Garnett, 2020; Sweller, Ayres and Kalyuga, 2011) dozens of experiments from around the globe have compared learners solving classroom problems as opposed to studying a worked example demonstrating the solution. For novice learners in an area, the results overwhelmingly indicate improved performance by the worked-example group over the problem-solving group. Why? Humans are amongst the very few species that have evolved to obtain information from other members of the species. We are very good at it. We can obtain information by problem solving but it is a slow, inefficient technique. If available, novel, complex information always should be obtained from others during instruction rather than attempting to generate it ourselves.
  4. There is evidence that problem solving can be superior to studying solutions but it only occurs when students are already knowledgeable in the area. They need to practice problem solving. There is no evidence that knowledgeable students are better at solving novel problems outside of their areas of knowledge. Why does practice at solving problems only become effective once we become reasonably knowledgeable in the relevant curriculum area? Cognitive load theory provides an answer that is beyond the scope of this statement (see also Martin and Evans, 2018 ).
  5. Ignoring the lack of evidence from randomised, controlled trials, why do correlational studies on data from international tests consistently demonstrate that the less guidance learners are given when learning, the less they learn? (Note, problem solving is associated with minimal guidance.) (Oliver, McConney and Woods-McConney, 2019; Jerrim, Oliver and Sims, 2019)

As indicated above, cognitive load theory provides one answer to this set of questions. The theory uses our knowledge of human cognition and evolutionary psychology to devise novel instructional procedures. It explains why (a) when dealing with novel, complex problems, studying worked examples is superior to solving the equivalent problems, (b) why solving problems is superior to studying worked examples when levels of expertise have increased in a particular domain, and (c) why attempting to teach non-existent problem-solving strategies, by taking time away from teaching subject matter, reduces students’ performance on international tests.

So, how can we increase problem-solving skill? By increasing domain-specific knowledge. Expecting anyone to engage in sophisticated problem solving and critical thinking in areas where they have minimal knowledge is futile. Lots of domain knowledge allows critical thinking and effective problem solving to occur naturally and automatically. Attempting to teach general problem-solving skills rather than knowledge, does not.

John Sweller is the emeritus professor of educational psychology in the School of Education at UNSW.

How to do the sums for an excellent maths curriculum

Curriculum review, Maths curriculum, teaching numeracy, ,

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.