Restoring confidence in mathematics education in New Zealand

Timely comment on the Maths debate in NZ from Dr Audrey Tam

Mathmo Consulting

In my previous post, I provided an overview of the past 20 years of mathematics education and declining student achievement in mathematics. With the announcement of a refresh of the New Zealand Curriculum (NZC), this seems like a good time to discuss the first steps towards restoring confidence in mathematics education in Aotearoa New Zealand.

Explicitly value the culture of mathematics and mathematicians

Earlier this year, I attended a Ministry of Education (MoE)-facilitated hui for Mathematics and Statistics. It was explained that the Ministry desires a bicultural curriculum that explicitly values all cultures, and that we need to shift to a decolonised, anti-racist curriculum.

Immediately, we are faced with a paradox. The word bicultural suggests that we should explicitly value only two cultures, or at least value two cultures more highly than other cultures. Does that not sound a little bit racist?

I do not wish to discuss…

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Replanting Creativity during post-normal times: a critical response.

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In October last year a report by Peter O’Connor et al., entitled “Replanting Creativity during post-normal times” was released by the Centre for Arts and Social Transformation at the University of Auckland. It has been in the press (here) and lectured on this year and seems to have been well received and viewed as general advocacy for the arts. The research project involved the creation of a student survey instrument that was used to measure how schools fair in fostering creativity and even came up with ‘total creativity score’ for each school. In this blog I critique two aspects of this report – the research methodology (and therefore the findings) and the authors’ contention that in NZ schools the arts are dying and do not foster creativity.

The research centers around the creation of a Creative Schools Index comprising 11 dimensions that are hypothesized to foster creativity – collaboration, problem solving, critical thinking, playfulness, environments, divergent thinking, innovation, discipline knowledge, risk taking, synthesis and curiosity. With some trialing, a 56 question survey was developed to give to students that could serve as a measure of how well individual teachers and schools provided for the development of creativity. Data was collected in 17 schools and from 1973 students.

Despite the authors claim that their Creative Schools Index has provided schools with ‘valid and reliable data’ it is by no means clear that this is the case. To establish reliability it would be necessary to resurvey the same set of students at a later time period, perhaps a week or a month or a term later to see by how much individual student answers vary. Despite extensive statistical treatment of the data, this does not appear to have been undertaken. To determine validity, some independent measuring of the 11 creative dimensions should have been carried out. This could have been survey of teachers or researcher in class observation. This would have provided some idea of whether student’s perceptions represent reality.

The authors find that across all 11 dimensions, the frequency of creative opportunities diminish across the school system (presumably primary to secondary) and that NZ schools are not valorising or encouraging creative dispositions nor building the skills or knowledge in the creative process. These conclusions in particular are hard to reconcile with the strength of art, drama and music in NZ secondary schools. In almost every secondary school in NZ students receive compulsory music, art and drama lessons at the junior level and at the senior level they have the opportunity to develop creative skills and knowledge in each of these areas as self-selected subjects. Many schools have drama productions and musical groups students can participate in. And this doesn’t touch on English which teaches creative writing, or Graphic Design, Photography, Media Studies, all of which foster and give opportunities for creativity. So it seems, as a NZ teacher said on twitter:

“I always struggle to name any other time in a person’s life when they have more access to music, theatre, arts, and so on than when they’re at school.” Ben Duckett @bjd8747

It seems in framing creativity primarily as a set of transferable skills and dispositions the reports authors underplay the powerful role that schools play in developing the declarative and procedural knowledge that creativity in different fields ultimately depends on. Could schools do better? Probably. But to argue that schools are where ‘creativity goes to die’ (headline of newsroom article by lead author Peter O’Connor) based on student surveys and in the face of contrary evidence observable in many NZ schools seems unreasonable.

Running through the Minefield

From Zoe Enser, thoughts on research and evidence in teaching.

Teaching it Real

One of the phrases I’ve seen used over the past few years has been ‘the evidence says’. Well, to be more precise, ‘the research says’. It is a phrase which can either provoke people into action, grabbing onto the newest book or document that comes out ready to change the world with it, or send them into spasms of eyerolling, dismissing it with a well-placed ‘research can prove anything’ or ‘it could never happen here.’ These reactions are understandable from both positions. On the one hand teachers and leaders would love there to be a silver bullet which will ‘fix’ all of the issues we face day to day. On the other there has been a lot of guff which floated around either under the label of research or having mutated into something which bears no relation to what was originally said. With all that in mind then, is there…

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On learning something new…

person playing wind instrument
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New learning is a fragile thing. I am reminded of this as I try to improve my skill on the flute, not my main instrument. I have the advantages of knowing multiple scales as a mental map in my head, of already being a fluent reader of notation, and I even have the correct flute fingering for different notes memorised. But I am humbled by the lack of a consistent embouchure and the fact that the knowledge of which fingers to use for each note is not sufficiently automatic for me to play well or reliably. I experience what I see my beginner trumpet students experiencing – knowing the correct valve to push down but their fingers do something else altogether, seemingly at random. Also the ability to play something correctly once only to subsequently fail to do so again and again.

Apart from giving me new empathy for my beginning trumpeters, this has caused me to reflect on the learning process more generally in terms of how fragile new learning is and how much consolidation is required for it to stick. I believe that teachers need to make deliberate efforts to teach for memory whether they are teaching knowledge, processes or skills. Simply giving students an ‘experience’ of a topic or skill  is not enough to result in long term learning. Even explaining a concept so that they have a good understanding of it is not enough if that understanding is soon forgotten. That ‘aha’ moment is necessary but not indication of permanent learning.

Do your students get enough practice using the knowledge or skills you are teaching them for it to become embedded in their long term memory? Two methods with strong evidence from cognitive science/educational psychology that can help are low stakes retrieval practice (quizzing) and elaborative interrogation (getting students to demonstrate and expand on concepts via questioning). If you are a learning a new skill then ample rehearsal of that skill is also an obvious step. Another technique teachers can use is cumulative assessment, in which material from previous topics is incorporated into end of topic tests, to help ensure knowledge is retained.

Simply put it takes a hang of a lot of going over for most students (human beings generally in fact) to ‘get it’ and ‘remember it.’ An interesting piece of research is ‘Project Follow Through’ and more specifically the success of a form of classroom teaching called Direct Instruction, developed by a Siegfried Englemann. His system, which enjoyed unparalleled success raising the achievement of underprivileged students in the US involved lessons in which 80% of each lesson was repetition of previous content. This feature was likely a big factor contributing to the success of the method. Of course the challenge is to keep the processes of embedding of knowledge/skills varied and interesting enough so that our classes don’t become too Gradgrindian.


How Learning Happens – Seminal Works in Educational Psychology and What they Mean in Practice: A review

book cover

This is a satisfyingly thick book with short chapters allowing one to dip in and out without losing track. The title is rather cool, and I can imagine producing it with a flourish in the staff room to impress colleagues. Sadly though, in my experience most will not be particularly interested. Teachers seem to be most excited by professional discourse that addresses the affective side of education – anything that speaks of making students feel safer, more connected and more loved will create a buzz of interest – and it goes without saying that this will raise student achievement. This predominance of heart over mind ignores more intellectual aspects, such as the applications of cognitive science or educational psychology, which only rarely find their way into consideration, despite their significant effect on student progress and well-being. It is these later features that Paul Kirschner and Carl Henrick address in their book.

How Learning Happens selects 28 key works of research in educational psychology from 1960 to 2013. Each work gets a chapter in which the reader is informed why they should read the article, a description of the research, the implications for teaching practice plus a useful bullet pointed ‘takeaway’ summary at the end. There are nice introductions to Sweller’s cognitive load theory, Geary’s biologically primary and secondary knowledge, Pavio’s dual coding, Rosenshine’s principals of instruction and Black and Wilam’s assessment for learning, along with a range of perhaps less commonly known works such as Rothkopf’s Concept of Mathemagenic Activities (no, not about maths, but activities that promote learning!) and Bandura’s work on self-efficacy. Each chapter is supported by key references with QR codes which provide a convenient way of accessing supporting material. The book is rounded out with a final chapter entitled ‘The Ten Deadly Sins of Education’ which addresses some of the common myths in education such as Ken Robinson’s ‘schools kill creativity.’

For a beginning teacher just starting out or experienced teachers wanting to broaden their knowledge by absorbing some of the science around learning, this is an excellent resource. Not every work discussed in the book appealed to me – a few me struck me more as arguments from imagination in which the researchers had perhaps mapped out an educational thought space without much evidence, but that’s psychology for you – it wasn’t so long ago that major theories in psychology were built on studies where n=5.

A minor quibble for me came in a grey breakout box (p 128) explaining standard deviation and effect size. The authors present effect size (d) as way of comparing the relative impact of different teaching approaches determined through research, much as Hattie does in his rankings of educational interventions link . In fact they make a point of mentioning Hattie’s hinge point of 0.4 as a cut-off below which ‘you could just as well done nothing.’ This is quite erroneous. For one, effect sizes resulting from research into different types of educational interventions cannot be meaningfully combined (as Hattie does in meta meta analyses) or easily compared. Many factors such as the quality of experimental design, sample size and even the age of the students (given spread of academic achievement data tends to increase with age) will affect the effect size (more info here).  And teachers need to know this if they are to grapple with the findings of research.

While reading the book I excitedly tweeted a few paragraphs and although there were likes and retweets, some interesting cautionary comments were made by Christian Bokhove an Ed Prof at the University of Southampton. The first related to a Danish study (link) referenced in the chapter on direct instruction (p182). The study identified a range of features typical of ‘learner-centered education’ and looked at data on academic achievement and concluded that it had a negative impact, particularly more so on low socio-economic students. To a fan of direct instruction like me this was gold but Christian pointed out the limitations of the study link  – a reminder of the dangers of taking studies cited in support of claims at face value.

A second paragraph I tweeted concerned the authors stating ‘motivation leads to learning’ as the 8th of their 10 Deadly Sins Of Education (p302). They observe how often one of the keys to improving outcomes for students is seen as improving their motivation and engagement, and claim that research shows that there is neither a causal nor reciprocal relationship between motivation and learning – it is always the one way – from learning to motivation. This clearly resonated with many on edutwitter, as it did with me. So often we hear arguments along the lines of ‘if only we make it more interesting they will engage and learn better’ or ‘we just have to get the students more motivated.’ The focus is rarely on how can we teach more effectively, give the student a taste of success, and then reap the natural flow of motivation that results. Nevertheless, it was pointed out by Christian and others that Kirschner and Hendrick have taken quite a narrow slice of the research on student motivation to support their point, and that there is a reasonable amount of evidence to support a reciprocal relationship between motivation and learning.

I recommend this book highly, it really is a ‘one shop stop’ for someone wanting to dive into evidence-informed education for the first time and provides many starting points for digging deeper into the science of learning.
















Play: nature’s default mechanism for learning and it’s role in schooling

photo of child s hand playing clay

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If you work in a primary school or have children attending one you may be aware of a growth in popularity in recent years of play-based approaches. It is not just a grass-roots teacher-led movement, teachers can undertake professional development courses concerning play (link) and Cambridge University even has a research unit devoted to it and a professorship sponsored by Lego (link).  This blog is my attempt to examine the justifications for these approaches and the nature of the evidence supporting them.

There are three reasons I see that might be used for including play in the primary curriculum, above and beyond what students experience at interval and lunchtime. The first stems from recognition of the important role that play has in neurological and social development of children and the fear that perhaps they are not receiving enough play to adequately facilitate this. Both these are supported by evidence. Research documents the importance of play in the development of mammals and children eg. Burghardt Bekoff Ginsburg. and also evidence that the amount of outdoor play children experience has decreased significantly over the last decades in favour of indoor activities such as screen-time that often do not involve other children Twenge.

A second reason, related to the first, is that academic learning is best delayed  till the age of 7, based on the idea that the child’s brain is not developmentally ready for such learning till then. This seems to be based in part on Piaget’s stages of development in which children do not develop the ability to think logically until they enter the ‘concrete operational stage.’ However, despite the dogged persistence in educational circles of Piaget’s theories, significant and valid criticism casting doubt on them has been made in the light of more modern research (link).

Arguments about school starting age are also made based on the success of the Finnish education system in which children don’t start school till age 7. However, in our more diverse society here in NZ, delaying the acquisition of vocabulary, decoding text and maths skills till age 7 may miss an important opportunity to close the gap between the children that gain this knowledge from their parents before they even start school and those from less privileged backgrounds who do not.

The third reason play-based approaches are advocated relates to the romantic meta-belief underpinning progressive educational thought – that children learn better through play or the most ‘natural’ or ‘spontaneous’ method possible. In essence, play has become the ‘new’ discovery learning which seems to find adherents in every generation of teachers despite overwhelming evidence showing explicit instruction of a structured curriculum by a teacher is more effective Rosenshine Mayer Kirschner. Let’s not forget also that the very successful Direct Instruction model of Englemann in Project Follow  Through involved teaching primary children.

Some might argue that surely a play-based approach would still be great for students to learn oracy, motor-coordination and important social skills like self-regulation. However, I would argue that in the environment of a school we have the opportunity to deliver structured programs such as music, drama, and PE which would deliver these outcomes much more effectively. There are still arguments to be made for free-play as an intrinsically unique activity of benefit to children eg Gray. but perhaps lunchtime and interval are enough?

So there are many questions to be answered around the inclusion of play-based learning in primary schools. Is free-play or structured play valuable in the first years but not in later years? Is there an opportunity cost relating to the closing of knowledge gaps between more and less privileged children? How sound is the underpinning reasoning for including play-based approaches? And lastly, but most importantly, where are the quantitative longitudinal studies demonstrating its efficacy over other approaches?

Footnote: If you thought the science of why mammals play was settled read this

A large scale multi-school, multi-nation well-being intervention: Convincing evidence or not?

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A few weeks ago, 600 teachers in my district ranging from early childhood to secondary, attended a professional development event to launch a new district wide well-being initiative. The speaker was Dr Denise Quinlan, director of the NZ Institute of Well-Being and Resilience and graduate of the University of Pennsylvania Center for Positive Psychology.  Denise made the case for the introduction of well-being interventions in NZ schools and cited a key piece of research in support of her arguments. The research is a PhD thesis by Adler Alejandro, entitled “Teaching Well-Being Increases Academic Performance: Evidence from Bhutan, Mexico and Peru”  and carried out at the Center for Positive Psychology at Pennsylvania University. The thesis can be downloaded here.

Being interested in assessing the quality of the evidence behind educational interventions, especially one that brought 600 teachers together, I read the thesis. For a PhD study it is a spectacular piece of work spanning 3 nations and a huge number of schools and students. The first study was in Bhutan, a 2nd in Mexico and a 3rd in Peru. In all cases the well-being intervention caused significant increases in student well-being and academic performance. Furthermore, the research used randomisation, an active control group, single blind experimental techniques along with large samples sizes and international replication which surely means it is a convincing piece of evidence. Or is it?

In each country the intervention consisted of roughly the same ten curriculum elements, customized somewhat to the local culture. Here are the elements for Bhutan:


The number of different elements strikes me as problematic: Which of them, or which combination of them, caused the increase in student academic performance? It doesn’t matter, you might conclude, what matters is that it worked. But what if most of the benefits where due to one or two elements and the rest had no effect, surely we would want to know if we were trying to design a similar intervention?

But the complications don’t end there. In addition to the ten elements outlined above, teachers in the intervention group were also taught how to give written and verbal feedback to students with a positive rather than negative or corrective focus from the teacher – something Alejandro claims (without evidence) is typical of teachers. Also the teacher training was designed to be a significant personally transformative experience, presumably from a positive psychological perspective. So the possible causal effects for changes in student performance start to compound. Did positively transformed teachers teach their curriculum with more vigour than usual resulting in higher academic performance?

Pertinent to discuss also is the nature of the control group. Teachers in the control group were trained to teach human anatomy, psychology and nutrition to students, and did so for the same amount of time as the intervention group. Perhaps I am being unreasonable, but the difference between the intervention and control group programs seems too big to adequately compensate for intervention effects (eg. the power of enthused teachers doing something new with students).

The crunch question though, is does this research provide enough evidence to underpin the design and implementation of similar well-being interventions in NZ schools? Adhering closely to the design of the intervention used may give the best chance of success. But we don’t know enough about what caused the positive effects to adjust the program to our NZ context. For example most NZ secondary schools already teach a Health curriculum based around well-being. Does that mean many of our students already have good well-being skills, more so perhaps than students in Bhutan, Mexico or Peru? Is the opportunity cost of implementing this particular program with high fidelity too high compared to other possible initiatives such as improving the quality of literacy and numeracy instruction or creating knowledge-rich curricula, all of which would improve educational achievement and the life-long well-being of students?


Fueling teacher practice: Av-gas or Liquid Hydrogen?

selective focus photography of gasoline nozzle
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Aviation gas has an energy density of 33.5 MJ/L whereas liquid hydrogen only has an energy density of 9.3 MJ/L. If ideas are the fuel that drives teachers then too many are running on liquid hydrogen.  Teacher practice is based around a number of key ideas and too many smart, dedicated, caring teachers are being given the wrong ones.

A popular one is ‘it’s all about relationships.’ The implications are students will learn best when the teacher knows their background allowing a positive relationship to be formed and thus the teacher will be able to optimise their teaching and behaviour management of said student. Trouble arises because, although there is a kernel of truth in this, it is flawed as a starting point.

To begin, most teachers have between 20-30 students in their class that can only be successfully managed as a group, not as individuals. If the teacher is in a high school, that number increases to between 100-120 students, often seen all in one day. In these circumstances, the best chance of a positive relationship between a teacher and their students depends on the teacher establishing firm control of student behaviour dynamics so that effective teaching and learning can take place. The best teachers are assertively in charge, approachable yes, but able to command student attention and co-operation to foster learning. They are not setting out to befriend students as a strategy.

Knowing more about the background of students – and yes, some have very difficult backgrounds – can cause a teacher to lower expectations. This might involve going easier on them over behaviour issues or work completion, both of which are unhelpful for the student concerned. For example, a student who is weak at maths, needs more practice at maths, not less. A maths teacher knowing a student is weak at maths but doesn’t do homework because his parents fight every night is not helping them by letting them off work.

The principal of the high school I attended as a student insisted that his teachers be firm, fair and friendly – in that order. Being firm, fair and friendly is not the same as being friendly, fair and firm, which is can be the case with more student-centered approaches. The later approach can place the teacher in the position of soliciting the friendship and good-will of students.



Teaching ‘Sticky’ and ‘Non-sticky’ Kids

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Whether we like it or not, some kids do better at school than others. A good part of this is beyond our control, determined by their genetic inheritance. In his book ‘Blueprint: How DNA Makes Us Who We Are’  Robert Plomin finds that on average, between 60 and 70% of educational achievement is down to genetics. This still leaves a significant 30-40% determined by the environment which schools, parents and peers can affect. Obviously educational achievement is strongly informed by intelligence and a useful way to look at it, conceived and researched by the psychologist Raymond Cartell, describes it as consisting of two parts – fluid intelligence and crystallised intelligence.

Fluid intelligence is the genetically determined part and consists of our working memory capacity and ability to use logic and reason to solve new unfamiliar problems without depending on prior knowledge. It is worth noting that nobody has yet found a way to improve fluid intelligence, despite the intensive effort to develop cognitive training programs. Crystallised intelligence consists of the knowledge you have in long term memory – what you know. It is this second part that parents and schools can affect, and as David Didau argues so persuasively in his book ‘Making Kids Cleverer – A manifesto for closing the advantage gap,’ increasing crystallised intelligence should be the principle aim of schooling.

One reason for the achievement or advantage gap is the amount of knowledge and vocabulary students enter school with. Students that have been read to extensively, been part of many dinner time conversations, talked at more by adults, taken on trips to libraries, museums etc., have a significant advantage. These students have more crystallised intelligence. This prior knowledge permits them to understand and retain a greater percentage of everything the teacher says, both instructions for activities and explanations of concepts. It is as if their minds possess intellectual velcro or stickiness.

Because significant numbers of children arrive at school without this useful prior knowledge or mental stickiness it is important for teachers to do two things:

  1. teach a knowledge-rich curriculum explicitly
  2. employ specific strategies to help students remember what they learn

Teaching a knowledge-rich curriculum ensures any cultural knowledge deficits are ameliorated as much as possible. Explicit teaching means not setting up learning activities which privilege students with more prior knowledge, such as inquiry based learning or semi-discovery approaches, but carefully breaking down topics and explaining them directly to students, questioning to check understanding and giving lots of guided practice using the knowledge.

Whether they admit it or not, every teacher wants, and aims for students to remember lesson content, however few take deliberate steps to help students remember what they learn. An exception would be teachers who believe they are teaching transferable general skills and that detailed content can be looked up on google as required so there is no point in remembering it, but cognitive science is very clear that we can only think with the information we have in our long term memory, making this approach more than a little flawed.

Strategies to help students remember learning such as key vocab, definitions and procedures can include frequent low-stakes tests or student self-quizzing using knowledge organisers, chorusing responses and online games such as Kahoots. The aim is for the knowledge to be transferred into student’s long term memories permanently, not just for an exam or end of topic test. To this end cumulative rather than modular end of topic tests which include content from previous topics helps cement learning.




Starting points: teaching other people’s kids

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In my last year of high school my favourite subject was physics. I enjoyed mechanics the most, calculating velocity, acceleration, angular momentum etc., and the steps in logic in working them out. I remember one key strategy for solving exam style problems imparted to us, which was to always start with what you know. This meant something very specific, which was writing down in a list all the parameters for which values were given in the question and then reading carefully to figure out what parameter you were being asked to calculate.

Today starting with ‘what you know’ seems to me to be the most logical way to address the enterprise of teaching other people’s kids. Except instead of a few values given in an exam question, ‘what you know’ becomes the ‘what we know’ in a broader scientific sense. It means considering what can be reasonably concluded from empirical investigations in relevant fields – which conclusions are supported by evidence and better still supported by a plausible theory?

I should explain I use the term ‘teaching other people’s kids’ because I think if you were, for example, teaching your own kids, you’d be entitled to teach them any which way you choose, whereas when you are employed to teach other people’s kids, you and the school should be using proven methods, as befitting any professional providing a service to the public. This is where starting from ‘what we know’ in a scientific sense comes in.

In my experience most parents and many teachers are not aware that much educational practice, both historically and currently is not developed with reference to empirical evidence, but is based instead on beliefs about how students learn. For example – students learn best when they work independently, when they choose their own topics, when they learn in an authentic context or when the teacher’s lesson delivery matches their preferred learning style. Teachers and principals, must ask ‘what is the evidence for this?’

It is astounding for me to reflect, 16 years into my career, that my teacher training did not take account of nor reference knowledge from the scientific fields relevant to learning. Even by the start of this century the fields of cognitive science and educational psychology had established reliable insights to guide teachers in their practice. There were also considerable bodies of research directly speaking to what works and what doesn’t in schools and in the classroom.

Why should we be privileging science over belief as the basis for teaching? This is because, although far from perfect, it is the best hope we have of figuring out what works and why and distinguishing that from our own folk theories, anecdotes and biases. And our children deserve this. It doesn’t mean that there is a piece of research to back every decision a teacher or school makes, but that their decision making should be informed where possible by research.