Restoring confidence in mathematics education in New Zealand

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 racism here. In the context of mathematics, I don’t have to. When writing a good maths curriculum, it is the culture of mathematics and mathematicians that needs to be explicitly valued. Studying the common language of mathematics is naturally inclusive and promotes equity in the system. The beauty of mathematics is revealed to people when they realise, not only can they apply mathematics within their own cultural context, it is the same mathematics that can be applied in other cultural contexts. Mathematics is a unifying approach that fosters understanding between communities.

The social landscape of Aoteroa New Zealand is changing rapidly. Of course, local schools and kura must respond to the cultural needs of its learners and whānau, but continually writing new maths problems is exhausting for teachers and hardly a sustainable approach. It’s not mathematical thinking, either. Teachers should contribute the rich materials they write to a central bank of resources, for the benefit of other teachers and their students. As the database grows, we will spot and appreciate patterns, the same mathematics being applied in different contexts and cultures. This must surely be the ultimate in cultural sustainability.

Celebrate column addition (and other column-based methods)

“Education in New Zealand is a student-centred pathway providing continuous progression and choice.” – Ministry of Education website

It’s hard to imagine any education system in the past that wasn’t centred around students. I cannot think of anything more student-centred than giving children the wisdom of our experience and empowering them with the best mathematical tools available, so that they can make continuous progress with their maths and have a full choice of subject options at secondary school and beyond.

Our decimal number system is a place-value system that was designed to be used in columns. The decision to actively discourage our youngest children from lining up the columns not only disrespects one of the earliest key breakthroughs in mathematics, it disadvantages the children who need it the most. I don’t have to explain the benefits of column addition to any parent or caregiver who has seen their child struggle with the cognitive load of a mental strategy.

If this idea still rankles anyone in the education sphere, please read this discussion and note England’s progress since they brought back column addition eight years ago. Or, more locally, look at the progress achieved in a Decile 1 class of Year 7/8 students who caught up on three years of knowledge in five months.

Respect the cognitive science of learning and support children to learn their times tables

Despite this being an achievement objective in the NZC, it does not seem to be happening.

I do not support rote learning times tables if rote means “mindless recall without understanding the concept of multiplication”. But the messaging is that children should be using “number properties” to work out multiplication facts before committing them to memory. This is a false prerequisite.

This topic was discussed years ago, but more recently, I have come across some alarming research, detailing observations of the concept of multiplication being introduced in New Zealand classrooms. It was mathematically unsound, and the poor children were so confused, they ended up regressing. We’ve got serious problems.

Cognitive neuroscience informs us that fluency with basic facts really does matter. See discussion here.

More whole-class, explicit teaching

We already know a lot about what makes great teaching, including Rosenshine’s Principles of Instruction. We don’t need to reinvent teaching, just like we don’t need students to reinvent mathematics.

New Zealand’s own John Hattie has synthesised the results of tens of thousands of studies to measure the effectiveness of different teaching practices. The average effect size is 0.4, meaning that teachers should be aiming higher than that.

Direct instruction (0.60) and explicit teaching (0.57) are more effective than inquiry-based learning (0.4) and discovery-based teaching (0.21).

Scaffolding (0.82), deliberate practice (0.79) and linking new learning to prior knowledge (0.93) all have very high effect sizes.

Within-class ability grouping has a very low effect size (0.18). There has been plenty of discussion about “labelling” children, but there are far more pragmatic reasons for abandoning it. Preparing multiple lessons creates more work for the teacher and significantly reduces the contact time that any one student has with their teacher, resulting in low engagement levels and poor learning.

When done well, whole-class teaching means that students remain engaged for longer and everyone moves forward together, generating effective classroom discussion (0.82). This does not necessarily exclude group work, but the objectives of social, co-operative learning are quite different and should not replace whole-class teaching and individual practice for the core learning of mathematics.

The OECD’s PISA study in 2012 found that student-oriented learning was negatively related to mathematics performance in every education system – that evidence is very hard to ignore. If it doesn’t work for 15-year-olds, for whom it might be plausible because they should have the knowledge and maturity by then to be more self-directed in their learning, then it’s not going to work for younger children.

The same research found that teacher-directed instruction was positively related to mathematics performance, but high levels did become negatively related.

Thus, we have a strong case for mostly whole-class explicit teaching, with student-oriented approaches on the fringe. Perhaps the progressive educationists who support constructivism (the theory of learning in which children construct their own knowledge through their own experiences) forgot that the school day hasn’t gotten any longer. If students are to keep up with the rate of learning required to meet curriculum expectations, we simply cannot wait for them to discover maths by themselves, or risk them giving up when it starts to feels too hard.

There may be no silver bullet to fix New Zealand’s big maths problem, but bringing back column addition is probably as close as we will get. Let our young children line up the columns again and see their confidence restored, and likewise their teachers’ too. Then we can look forward to building a rich and flexible maths curriculum that will meet the needs of all learners.

Dr Audrey M. Tan
May 2021

20 years wasted – enough is enough

In December 2020, RNZ reported that New Zealand’s Year 9 students recorded the worst-ever results in maths and science.  Four years earlier, they reported that the same generation of students, New Zealand’s Year 5 students, were the worst at maths in the English-speaking world.

We have far too many students struggling with basic numeracy tasks. Looking at the TIMSS 2019 results for Year 5 students,

NB: In 2019, TIMSS conducted their survey on paper in some countries (including New Zealand and Australia), and electronically in other countries (including England and Singapore).  Relative placings and international averages are for the paper survey only.  Combined relative placings on the questions above differ by no more than one place.  Combined international averages on the questions above differ by no more than four percentage points.

In case you hadn’t noticed, that last question was multiple choice. New Zealand’s success rate is exactly the same as in 2015, and worse than what we would expect from random guessing (25%).  An earlier cycle of TIMSS suggests a constructed response success rate would have been much lower.  

This is not exactly news. Looking at the TIMSS 2015 results for Year 5 students,

In TIMSS 2011, New Zealand’s Year 5 students finished last-equal among peers in participating developed countries:

If you think it doesn’t matter that children can’t perform these basic numeracy tasks (e.g. “they don’t need to calculate any more because we have calculators” or “it’s more important to develop their reasoning and problem solving skills”), then think again.  When examining the performance of all countries participating in TIMSS 2019, there is a strong positive correlation between performance in the Number content domain and performance in the other two content domains, Measurement and Geometry, and Data. Similarly, there is a strong positive correlation between performance in the Knowledge cognitive domain and performance in the other cognitive domains, Applying and Reasoning. In other words, strong numerical proficiency paves the way for success in all aspects of Mathematics and Statistics, and you can’t solve higher-order problems or perform complex reasoning unless you have a good bed of knowledge to begin with.

What on earth has gone wrong?  

Twenty years ago, a radical new approach to teaching mathematics in New Zealand, known as the Numeracy Development Projects or just the Numeracy Project, was rolled out across the nation before there was any robust proof of its effectiveness.  In fact, the decade-long roll-out was the research experiment. 

The goal was to raise student achievement by strengthening the capability of teachers through professional development.  The philosophy was to prioritise conceptual understanding over procedural knowledge and skills (frequently called “rules”, with a negative inference). Despite acknowledging the interdependence of “knowledge” and “strategy”, the teaching of knowledge was relegated to a mere “ten-minute whole class warm-up at the beginning of lessons“. Written methods such as column addition and subtraction, divisively labelled as “algorithms”, were not to be taught to students, if at all, until students had jumped through a series of hoops for five or six years, calculating in their heads using increasingly complex mental strategies.  The easiest methods for adding and subtracting numbers literally became the last lessons on addition and subtraction.

The Numeracy Project researchers created their own stages of progression called the Number Framework, and so the research conveniently showed that students were, um, progressing.  

Or were they? In 2005, the National Education Monitoring Project reported that students were “improving in tasks that require quantitative reasoning skills, but declining in basic mathematics facts and solving simple number problems.”

In 2009, “there was no meaningful change in number task performance between 2005 and 2009, for either year 4 or year 8 students.  The most notable change in performance was a decline for year 8 students on multiplication problems, where changes in computation strategy were clearly evident.”

The long-term trend from 1997 to 2009 was “a small net improvement in mathematics performance at year 4 level (held back from a larger improvement by the decline between 2001 and 2005 in basic fact knowledge), and essentially no net change in mathematics performance at year 8 level.”

It was clear the researchers needed to do something to address the deficiencies shown up by NEMP.  They did do something.  They shut down NEMP.

And, presumably, knowing their jobs and their Numeracy Project facilitators’ jobs were on the line, they secured their future by having the Number Framework embedded into the revised New Zealand Curriculum and writing the Mathematics National Standards to align with their aspirations, just as the experiment was coming to an end.

In 2010, the awful truth could not be hidden any longer.  The final, longitudinal Numeracy Project study concluded that “the absolute levels on the Framework attained by students were in many cases well short of the numeracy expectations for students at particular year levels stated in the New Zealand Curriculum and in the Mathematics [National] Standards.”

An estimated $100M (based on $70M spent in the first seven years) of taxpayers’ money had been spent on a revolutionary approach to teaching maths, and it didn’t work.  The experiment had failed.

But it was too late and presumably too embarrassing for the Ministry of Education to pull the plug, and much easier to go along with the idea that it was a “temporary situation while teachers are continuing to upskill themselves”.  Apparently, “many teachers stuck very closely to the printed NDP resources (the “pink” books)”, which “could reflect the low levels of confidence that many teachers still have [despite two years of professional development]…In retrospect, it may be that teachers needed to receive support by facilitators for considerably longer…”.

It’s remarkable how the teachers could be criticised for doing exactly as they were told. Did the Ministry ever wonder whether the pink books were the problem? And what does “considerably longer” than two years of professional development look like? It suggests that not even our student teachers would be adequately prepared to teach mathematics by the time they complete a three year Bachelor of Teaching degree.

In New Zealand, a distinction is made between mathematical (content) knowledge and pedagogical (content) knowledge (i.e. how to teach maths). The parallels with the Numeracy Project are uncanny: initial teacher education (ITE) focusses almost entirely on pedagogical knowledge, with little regard for mathematical knowledge. A 2012 survey of first-year student teachers showed that “students enter ITE with minimal levels of mathematical content knowledge…It is questionable whether their performance can be brought to an acceptable level.  Currently students are not assessed before graduation to ensure they meet numeracy competency requirements.”

If publishing the results of this survey was intended to raise the flag on the universities’ inadequate preparation of their student teachers, it fell on deaf ears. To this day, still only one university in New Zealand bothers to assess the numeracy of all their student teachers before graduation.

What can be made of such a muted response by the universities? There is little incentive to improve the training of pre-service teachers if there is an opportunity to sell Ministry-funded professional development to in-service teachers. In the decade following the Numeracy Project, primary school teachers in New Zealand engaged in higher levels of mathematics professional development than the international average. Yet, the “temporary situation” of 2010 has not improved. Page 53 of this report provides insight into what some Ministry-funded professional development programmes look like.

If, on the other hand, publishing the results of this survey was intended to provide a reason for the failure of the Numeracy Project, the researchers had left it too late. They weren’t assessing students taught in the days of old; they were assessing the earliest victims of their own experiment. These kids were now feeding back into the system as teachers.

Enough is enough. The results speak for themselves. Our primary school teachers are not at fault for a flawed curriculum based on an academic theory of learning. Our children deserve a quality mathematics education; right now, some of them are not even getting a basic one.

What should happen next? That will be addressed in my next post.

Dr Audrey Tan, Mathmo Consulting
April 2021