What are the chances of completing a Disney-Pixar Domino Stars collection from a fixed number of packets?

It’s a strange thing for a mathematical person to say, but some people are luckier than others. Luckily for our son (or his parents), we shop at Countdown, which means he received a regular supply of Disney-Pixar Domino Stars throughout the 6.5 week promotion.

It was all a bit of a yawn to me, this time round. Having calculated the chances of collecting a full set of Dreamworks Heroes Action Cards, I was neither surprised nor frustrated by the increasing number of duplicates as his collection grew. The theoretical hyperbolic growth of the number of packets he would have to open to find a new domino implied a spend of $4480 to complete a set of 50. Almost $700 per week?? Not very likely.

To cut a long story short, on the last day of the promotion, we trudged home from a swap meet needing just two more to complete the set. Our chances of finding those last two looked bleak…except that a neighbour had just dropped off a big bag of 22 packets in our letterbox.

As I said, some people are luckier than others, and to our amazement, our son actually found the very two that he needed! Now how lucky is that?! I mean, what are the chances…?

Without going into too much detail, you can either focus on what you don’t want to happen, i.e. you get none of one of the coveted dominoes and any number of the other:


or you can focus on what you do want to happen, i.e. you eventually find one of the two that you’re looking for, and then hope to find the other one amongst the packets remaining:


But a friend of ours came up with an expression that best reflects what really happened, i.e. our son stopped opening packets as soon as he found the two he needed:


Reassuringly, all three expressions evaluate to this rather impressive looking fraction:


So it was very close to a 1 in 8 chance. Our son doesn’t realise just how lucky he is.

Dr Audrey Tan, Mathmo Consulting
September 2015

What about the children?

Over the past two weeks, there has been a flurry of public discussion about primary maths education in New Zealand.  Two reports, released within a week of each other, came out with the same message: our primary school children are performing very badly in maths.  Not exactly news, but it’s good to see the nation talking about it.

Releasing the Crown’s National Monitoring Study of Student Assessment on Mathematics and Statistics (2013?!) late on a Friday before the long Queen’s Birthday weekend was a possible reason for the muted response to the latest depressing statistic: only around 41% of approximately 2000 Year 8 students in 2013 met the expected level of achievement.  Our primary school leavers are struggling with decimals, fractions and percentages, and don’t I know it…

But what really got people riled up was The New Zealand Initiative’s report, “Un(ac)countable: Why millions on maths returned little”, written by Rose Patterson.  As the Minister of Education said herself, it provided a “fresh perspective” on New Zealand’s maths learning woes.

Now that the media hullabaloo is settling down, let’s try to set the record straight.  The report investigated and established the failure of the Numeracy Project by:

  • providing evidence of a decline in student maths performance that aligns with the roll-out of the Project;
  • debunking the myth that it’s not such a bad thing that Kiwi kids don’t “know” anything anymore because their strength lies in the higher-order areas of “applying” and “reasoning”, unlike their east-Asian rote-learning counterparts. Sorry, but it turns out those east-Asian kids are not only better at “knowing”, they’re also better at “applying” and “reasoning” because they’ve actually got some knowledge to work with.

The report also investigated maths teaching quality in our primary schools, citing a study in which a significant proportion of 125 student teachers were unable to answer some basic primary-level maths questions.  Personally, I feel this raises some serious questions about the quality of the Bachelor of Education degree.  Is it not reasonable to expect that all graduating primary school teachers should be able to do primary school maths?

Critics called it unfair, but Rose Patterson was professionally compelled to examine teacher quality after interviewing curriculum writer Vince Wright and maths education researcher Jenny Young-Loveridge.  Both interviewees prefer to blame the failure of the Numeracy Project on its poor implementation by teachers rather than its flawed ideology.  It really wasn’t the Herald’s finest moment when it accused the report, and by association the Minister of Education because she agreed to launch the report, of criticising teachers.  Not one single journalist mentioned the report’s actual conclusion: that teacher quality was unlikely to have changed over the past 15 years, and that the decline in student maths performance was due to the Numeracy Project’s multiple strategy approach to numeracy and the loss of emphasis on the basics.

I am pleased that I had the opportunity to speak publicly in support of our teaching workforce.  Frankly, it is shameful that the people who are supposed to be looking after our teachers chose to not defend them.  Instead, the NZEI Te Riu Roa president said underfunding for teacher professional development (PD) was to blame for the poor results, even though the report pointed out that New Zealand spends a lot of money on maths PD – more than most other countries, in fact.  Even after the Minister of Education responded to the “maths problem” by promising to raise the quality of maths teaching through more PD, the NZEI president still wasn’t happy, saying things would only improve if the training was better than what’s currently available.

Sigh. Result. By the way, has anybody thought about the children lately? When political and professional pride get in the way of helping our kids, it is really sad.

Had certain things happened or not happened, the political response to the Un(ac)countable report might have panned out quite differently.  But, having been to Wellington and heard with my own ears the Minister of Education’s response, it is clear there is more work to be done.  Nobody could have held the Minister accountable for the mistakes of past governments.  In her own words, the Numeracy Project “was in line with international thinking at the time”, so she missed a great opportunity to renounce it and become the public hero.  Instead, everything’s gonna be alright now that we’ve got National Standards.  Erm, would these be the same National Standards that her Ministry just deemed as lacking dependability?

What has been truly heartening, however, is the public’s response.  The Herald’s suspiciously unoriginal editorial and the unrepentant curriculum writers’ opinion editorial have been met with mockery akin to the Emperor’s New Clothes.  It’s good to see the public ain’t buying it any more.

I am proud of this campaign’s role in bringing the debate to this point, but merely talking about it won’t help our children.  A shocking amount of taxpayers’ money has been spent on what can only be described as a failed experiment.  By allowing it to continue, we are failing our children.  From here on, we are all accountable.

Dr Audrey Tan, Mathmo Consulting
June 2015

Open letter to Hon Hekia Parata MP, Minister of Education

Dear Minister Parata,

I write in response to your speech at the launch of the New Zealand Initiative’s report “Un(ac)countable: Why millions on maths returned little” on 4 June 2015.

With respect, the Crown’s recently released National Monitoring Study of Student Achievement has NOT shown that the system is failing a minority of students. To quote: “The curriculum expectation at Year 8 is that students will be working solidly at Level 4. About 41 percent of [approximately 2000] Year 8 students achieved at Level 4 or higher on the KAMSI assessment.” In actual fact, the system is failing the majority of primary school students in New Zealand, not a minority.

It is incorrect to suggest that the Un(ac)countable report continues the age-old debate in education between those who believe in rote learning and those who place a higher value on critical thinking. By doing so, you have precisely proved the report correct: “Rather than striking a good balance between instrumental learning and relational learning, and enabling the two to build on each other, they tend to be falsely dichotomised. They should work in tandem.

Looking at the graphs, we can see that Year 5 student performance in TIMSS has been on the decline since 2002, Year 9 student performance in TIMSS has also been on the decline since 2002, and PISA 2012 showed a sharp decline in the performance of 15-year-old students in mathematical literacy, in stark contrast to the OECD average. To say that New Zealand student performance in these international assessments has “declined slightly in recent years” is something of an understatement.

Striking the right balance between the practice and mastery of basic skills and developing higher-order thinking is much easier than you claim, and should be guided by evidence. A report released last month, written by mathematician Assoc. Prof. Anna Stokke for the C.D. Howe Institute in Canada, explains that “studies consistently show direct instruction is much more effective than discovery-based instruction, which leads to straightforward recommendations on how to tilt the balance toward best instructional techniques.

Your commitment to raise the quality of maths teaching in New Zealand is welcome. However, the Un(ac)countable report shows that teacher quality is unlikely to have changed over the last 15 years, and the true reason for the decline in New Zealand student performance in mathematics is the loss of emphasis on the basics. Until the Ministry acknowledges the overwhelming evidence and addresses the deficiencies in curriculum content and delivery, throwing more money at professional development for teachers will, sadly, have little effect. That is why I have decided to share this letter, so that parents, teachers and principals can also examine the evidence and make appropriate choices for the children in front of them. By working together, I am confident that we will bring back column addition to New Zealand’s early primary maths curriculum.

Yours sincerely,

Dr Audrey M. Tan
Mathmo Consulting

Live interview on TVNZ’s Breakfast

“As an (old) ex teacher I absolutely applaud your appearance and comments on Breakfast TV this morning. You were better than excellent and I couldn’t agree more with EVERYTHING you said.” – G Blackwell

“For me, the most encouraging aspect you shared was that it is possible to turn things around quite quickly, even after all the years that have been wasted trying to teach maths using a method that clearly doesn’t work and also disadvantaged the students. In other words, because you were so very specific about what needed to change and how, you brought a sense of hope, not only to the students who may have been listening to the interview, but to the parents who have felt anguish seeing their children robbed in their mathematical education and knowing this would negatively impact their futures. Thank you Audrey.” – S Maxwell

Numeracy Project was a failed experiment
A report shows Kiwi kids are failing maths in higher numbers than ever before. Dr Audrey Tan says the $70M Numeracy Project was a failed experiment, and gives her recommendations on how to fix the current New Zealand primary school maths curriculum.

Kiwi kids are failing to grasp maths fundamentals

Pardon the cynicism, but why on earth would the Ministry of Education release a report late on a Friday before a long weekend, if not to bury bad news?

The National Monitoring Study of Student Achievement 2013 found that only 41 percent of [approximately 2000] Year 8 students achieved at Level 4 or higher on the KAMSI assessment. The curriculum expectation at Year 8 is that students will be working solidly at Level 4.

It is hardly any wonder that older primary school children are failing to grasp maths fundamentals such as fractions and decimals. Too many of them are still muddling their way through the list of whole number strategies, such as “doubling and having”, “tripling and thirding”. Assuming they have already learned how to perform column-based multiplication (which many of them haven’t), their time would be better spent experiencing decimals, fractions and percentages in a variety of ways. And I don’t mean colouring in pictures of fractions; I mean learning how to actually do maths with these numbers.

A curriculum that emphasises practice and mastery of the basics would give all children, regardless of their ethnicity or socio-economic background, the best chances of success with maths. Just look at what one Decile 1 school teacher and her students, mostly Pasifika and Maori, achieved in just five months.

We don’t have to wait for the Ministry to tell our teachers what to do; we can’t afford to. The path ahead is obvious. Let’s just get on with it.

Dr Audrey Tan, Mathmo Consulting
May 2015

Report finds Kiwi kids are failing to grasp maths fundamentals – National – NZ Herald News
Maths standards achieved by New Zealand children are sliding dramatically between their fourth and eighth years at school. – New Zealand Herald

Ministry’s concern over gaps in NZ maths teaching

Readers will be pleased to know that we already have a name for “space and shape” mathematics. It’s called geometry.

I put it to the Ministry of Education that students’ lack of exposure to “formal maths” is a direct consequence of students’ over-exposure to numeracy strategies. There are only so many hours in a school day, after all.

But let’s remain optimistic. The Ministry may never admit that the Numeracy Project was an abject failure, but the report paves the way for them to quietly sweep it under the carpet. The term “formal maths” suggests more direct teaching and less discovery-based learning.

Concern over gaps in NZ maths teaching – National – NZ Herald News
Students are not being taught enough “space and shape” mathematics and the “huge” learning gap is hurting achievement, the Ministry of Education says. – New Zealand Herald

School reforms ‘need to be tested’

“Too many education reforms are failing to measure success or failure in the classroom,” said Dr Andreas Schleicher. He is right, but in 2009, New Zealand’s National Education Monitoring Project (NEMP) concluded that “…over the 12 years from 1997 to 2009 there has been 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.”

So we did have an effective evaluation in place. What happened next? NEMP got shut down and the new curriculum remains intact. It really is time for parents and teachers to demand something better from the Ministry of Education.

School reforms ‘need to be tested’
Trillions of dollars are spent on education reforms around the world without any effective evaluation to see if changes have worked, says the OECD.

The ineffectiveness of modern teaching methods

“A review of more than 200 research studies to identify teaching elements with the strongest evidence of improving attainment was published by the UK’s Sutton Trust in October 2014. It identified common practices that have no grounding in research but can be harmful, including using praise lavishly, allowing learners to discover key ideas by themselves, grouping students by ability and presenting information to students based on their preferred learning style.” – Emeritus Prof. William Reville

The reason why modern teaching methods don’t work
Whole-class teaching, in which children learn to use their long-term memory, has been abandoned for a more personalised, naturalistic approach, and it’s been a disaster.


Podcast: When Math Doesn’t Make Sense

This is an amazing interview with Assoc. Prof. Anna Stokke, co-founder of WISE Math in Canada. If you can, please find a quiet half-hour to listen to it. It’s long, but it covers everything you need to know about the current state of primary maths education in western countries.

Frankly, there is nothing more that needs to be said. After listening to this podcast, you’ll be thinking…’this makes perfect sense, so WHY are we still fighting the fight??’


Interview with TVNZ’s ONE News

At the end of 2014, TVNZ’s ONE News put together a story on the absence of column addition in the early part of New Zealand’s primary maths curriculum and the difficulties that parents face when trying to help their kids with their maths homework. I offered to show them a school that had embraced column addition and my approach to teaching maths, to see the amazing difference it had made to their students.

Naturally, the two-minute piece did not go into much depth – they failed to mention the serious consequences of not allowing children to practise column addition, e.g. half of New Zealand’s 9-year-olds cannot add two three-digit numbers – so, for those who are interested, here are the questions they had prepared, together with my answers.

Q: What’s the difference between what the kids here are learning and the current curriculum?
A: In fact, they’re still learning the same stuff, but the main difference is that students here are given the freedom to choose the methods that work best for them, and that includes the vertical methods. They also appreciate the importance of knowing their times tables off by heart, and they spend time working on that.

Q: Why should the vertical methods be taught to kids earlier?
A: The current curriculum asks children to do too much in their heads before they have the mental capacity to cope with it. The vertical methods enable children to work efficiently with larger numbers, without the cognitive overload. So really, something like column addition should be the first lesson in the book, not the last.

Q: Proponents of the current curriculum say the vertical methods are too confusing until children have a good understanding of place value. What’s your response?
A: That makes no sense to me. Understanding place value is important, and if taught well, lining up the columns really ought to help with that – that is precisely what our decimal number system was designed for! But I’m not talking about going back to the old days when children learned by rote without understanding. It does require good teaching to make sure that they’re developing their understanding as they practise these methods. And once they’re older, when they have that understanding and can hold more in their heads, then they’ll be able to migrate onto more sophisticated strategies, and quite easily too.

Q: They also say that it’s not just about finding the right answer, it’s about teaching kids to think mathematically. What’s your response?
A: Absolutely, I couldn’t agree more! And that’s the fallacy of the current curriculum. Spending lots of time discussing various strategies for adding or multiplying two numbers is not a good use of children’s time. We need kids to be thinking at a higher level, solving practical problems, and not be held back by the numbers. The emphasis should be on knowing WHEN to add, subtract, multiply and divide, and we need to equip children with the skills they need to just get on with it.

Q: What results has the school had from changing the way these kids are taught?
A: The transformation in these kids has been astonishing! They used to hate maths, but now they love maths. In just five months, one class went up by three IKAN stages, and now they’re working at the expected level for their age.

The story screened on Monday 26 January 2015, just as children are getting ready to go back to school.

Special thanks to the school for allowing us to film on site.

Dr Audrey Tan, Mathmo Consulting
January 2015

Primary school maths – Old column method better?
News article Jan 2015, TVNZ.