The Numeracy Development Project has influenced the teaching of mathematics in New Zealand. It has changed the language people use to talk about mathematical understanding, introducing the terms “multiplicative thinking”, “part-whole” and “proportional reasoning” to the teacher toolkit. It has empowered some teachers to think differently about the teaching of mathematics. It has brought “number” front and centre, often crowding out algebra, geometry, measurement and statistics, which are now commonly called the strands. It has baffled a large number of parents. Has the Numeracy Development Project been a success? If not, how can we fix it?

I have been pondering about the efficacy and side-effects of the Numeracy Project in New Zealand. I have heard criticisms from Primary and Secondary teachers, and defense and explanation from advisors. I have listened to a very illuminating podcast from one of the originators of the Numeracy Project, Ian Stevens, I have had discussions with another educational developer who was there at the beginning. I even downloaded some of the “pink booklets” and began reading them, in order understand the Numeracy Project.

Then I read this article from the US organisation, National Council of Teachers of Mathematics, Strategies are not Algorithms, and it all started to fall into place.

The authors explain that researchers analysed the way that children learn about mathematics, and the stages they generally go through. It was found that “Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.” They claim that in the US “(t)he idea of “invented strategies” has been distorted to such a degree that strategies are being treated like algorithms in many textbooks and classrooms across the country.” I suspect this statement also applies in New Zealand.

## Strategies taught as algorithms

Whitacre and Wessenberg refer to a paper by Carpenter et al, A Longitudinal Study of Invention and Understanding in Children’s Multidigit Addition and Subtraction. I was able to get access to read it, and found the following:

“Although we have no data regarding explicit instruction on specific invented strategies, we hypothesize that direct instruction could change the quality of children’s understanding and use of invented strategies.** If these strategies were the object of direct instruction, there would be a danger that children would learn them as rote procedures in much the way that they learn standard algorithms today**.” (Emphasis added)

Were they right? Are the strategies being taught as rote procedures in some New Zealand classrooms? Do we need to change the way we talk about them?

## How I see the Numeracy Development Project (NDP)

The NDP started as a way to improve teacher pedagogical content knowledge to improve outcomes for students. It was intended to cover all aspects of the New Zealand Mathematics and Statistics curriculum, not just number. Ian Stevens explained: “Numeracy was never just Number. We decided that in New Zealand numeracy meant mathematics and mathematics meant numeracy.”

The Numeracy Development Project provided a model to understand progression of understanding in learning mathematics. George Box once said “All models are wrong and some models are useful.” A model of progression of understanding is useful for identifying where we are, and how to progress to where we would like to be, rather like a map. But a map is not the landscape, and children differ, circumstances change, and models in education change faster than most. I recently attended a talk by Shelley Dole, who (I think) suggested that by emphasising additive thinking in the early school years, we may undo the multiplicative and proportional thinking the students had already. If all they see is adding and subtracting, any implication towards multiplicative and proportional thinking is stifled. It is an interesting premise.

The Numeracy Project (as it is now commonly called) suggested teaching methods, strongly based around group-work and minimising the use of worksheets. Popular invented strategies for arithmetic operations were described, and the teaching of standard algorithms such as vertical alignment of numbers when adding and subtracting was de-emphasised.

An unintended outcome is that the Numeracy Project has replaced the NZ curriculum in some schools, with “Number” taking centre stage for many years. Teachers are teaching invented strategies as algorithms rather than letting students work them out for themselves. At times students are required to know all the strategies before moving on. Textbooks, worksheets and even videos based around the strategies abound, which seems anathema to the original idea.

## Where now?

So where do we go from here?

To me empowerment of teachers is pivotal. Teachers need to understand and embrace the beauty of number theory, the practicality of measurement, the art and challenge of geometry, the detective possibilities in data and the power of algebra to model our world. When mathematics is seen as a way to view the world, and embedded in all our teaching, in the way literacy is, maybe then, we will see the changes we seek.

I have not taught in a NZ school for around 10 years now but I am surprised with the direction the Numeracy Project appears to have taken. I was there for its implementation under the watchful eye of Len Cooper, who once told me to never blindly accept new theories as the saviour of education- to question and challenge when necessary. I am wondering if his guidance has made me the Maths teacher I am today, one who encourages discussion, collaborative learning and a sharing of strategies. I have always used the NP as a tool kit, not as a stand alone programme and I have always dipped into it as needed, using it alongside other resources as this was the way I believed it had been developed. On a personal note it allowed me to “understand” Mathematics in a way I had never been taught in school and this was necessary for my personality, I have never been a great “rule follower”.

I am a strong supporter of the NP as it changed the way I approached my teaching. Number in my opinion is important because unless you understand place value and how it works you will struggle understanding fractions, percentages and ultimately the strands such as statistics and geometry as they are all linked to number and vice versa. I don’t think that the NP is the issue, it is the way it has been used. As the saying goes, “A bad tradesman always blames his tools”. Whoever took the NP and twisted it to the way I have been hearing some teachers are using it, they have a burden to bear.

Hi Kat

Thank you for your comment. I totally agree that there are good aspects about the NP. It is very good to hear from someone it helped. I suspect some of the originators of the NP are somewhat alarmed at what their baby has grown up to look like.

Thank you so much for posting this useful summary. After some baffling discussions with my child’s maths teacher I had done some reading up but hadn’t yet worked through the information trail you’ve provided above.

To give you some background in our (primary 4) classroom, yes strategies were generally being taught as algorithms. As it was explained to us, by a frontline teacher, this message has been distilled into “If children learn multiple ways of doing the same thing they are supposed to understand the maths at a deeper level” and subsequently be better off down the track.

What this looked like to most of our peer parents was children that had already internalised a concept were being held back to learn 5 extra ways of doing the same thing. Even stranger, at least to the parents, was the absence of any rote learning – resulting in, a reasonable number of, children who were unsure which technique to apply in which scenario and a lack of practice to actually apply any of them with an acceptable level of efficacy.

My initial reaction had been (after discussions with the teacher) to help my child with practice to increase their ability to actually apply their favourite technique in practice. Thanks to your article I will now be engaging with my child more to guide them and help further their ability to work out how/why some of these techniques work for themselves.

Thanks Nik – so glad I could be of help. The holding-back to get more strategies is definitely a worrying feature.

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Thank you for articulating what I’ve been thinking since listening to the podcast you mentioned above earlier in the year. We use NDP to direct our judgement on National Standards. To not use it as such would be a challenge that would need addressing as it currently gives us clear criteria – however misguided it may be.

I find National Standards even more problematic than the Numeracy project. At best a dubious impersonation of US and UK schemes that have been shown to lower educational achievement, discourage collaboration and narrow curriculum.

Thanks Nic, an interesting article.

I feel the Numeracy project was flawed from the outset with the teaching of the language of strategies and the need to remember the names and the strategies.

This created confusion and doubt in kids. The best maths teaching I have done is in recent years, focusing on kids to use their heads, and when this doesn’t work, to use materials of some sort. No names given to any strategies- just building and using knowledge of numbers.

Hi Nic,

Great analysis! There are some useful things but I’m afraid it missed the mark and we need this conversation urgently. I despair at the thought of how many kids I’ve switched off from enjoying maths. Clearly, we can do better.

I graduated just as NPD hit schools and over the years I’ve watched it become a prescriptive tick the boxes (or the I Can Sheet) exercise. Implementing it was so daunting and so over complex it became an overwhelming content delivery system. The endless PD was like we were wrestling a mammoth and to be honest, it compartmentalised maths and number beyond any meaningful application. There were plans to be followed, ability groupings to be maintained (ugh!), books to practice from and follow-up assessment, all supported by enough resources (paper and material) to bury the pyramid of Giza. There was only ‘this’ way to do it. Worst still (for me) was the belief of keeping kids back a Stage because they were not following the learning plan or couldn’t show all the strategies. At no stage was I ever told the strategies could be imagined!!

And it was very boring for kids, disadvantaging those who a) developed a fixed mindset of their maths ability thanks to grouping, b) needed a real life context to make connections (Pirate Boxes???). I began to adapt two years ago by teaching more of it through the strands and brought it back into real life contexts.

I welcome any step forward from here.

Dr. Where is your evidence? What analysis have you also done of Poutama Tau? What are your thoughts on integrated curriculum?

Tell me more. What would you like evidence of? Tell me about Poutama Tau and integrated curriculum.

Talk more if you can, about ‘invented strategies’.

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Poutama Tau developed by the likes of Tony Trinnick and others is the Maori-medium equivalent of the Numeracy Project.

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When I talk about integrated curriculum, I talk about misxing strands together and teaching different subjects together.

As an example: We recently taught ‘patterns and relationships’ by teaching line dancing, we also taught ‘patterns and relationships’ through using tititorea sticks.

http://www.teara.govt.nz/en/traditional-maori-games-nga-takaro/page-5

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i put forward that if we used more integrated curriculum than we can teach strand more effectively.

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I must also add, that we make sure that children are generally numerate up to Stage 4, before introducing strand… and until they get to stage 4 we focus more on Number, to make sure they have a firm grounding.

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Just my thoughts.

Totally agree Nicola. It was me who chatted with Ian Stevens in that podcast. I truly wish I had known the true intention of the NDP years earlier. Are you aware of the PD teachers had at the time it was introduced? Unbelievable…we were told that we MUST use the pink books almost as scripts and the examples etc in it. Anyway, I work with many different curricula globally and in the most recent one I’ve been working with they emphasise even at grade 3 (year 4) students don’t need to name their strategy or choose most efficient etc…they need the opportunity to solve problems and explore using manipulatives/diagrams etc.

Thanks Mandy. Your podcast really got me thinking and has done the same for many others also.

Hi Dr NIc

All the above very very interesting as is your Jo Boaler article. Jo is

I was involved in the Secondary Numeracy Project implementation 2002 thru to 2010. I witnessed many workshops by primary and secondary facilitators and teachers and saw quite a bit of everything your bloggers have mentioned. The many ways to add being seen as a necessary step before proceeding for example. I must have run a hundred workshops myself on different topics.

I want to add that there was an “Addition Addiction” infesting the primary mathematical world and it continues to exist today. I identified this and reported on it in one of my newsletters as “Hogan Declares War on Addition!” 2007. It caused a ripple of comments. This fascination with addition has a cause.

I tested teachers and students alike to see how complex their thinking had become in solving problems using mathematics they knew. I found out that the majority (80%+) of primary and intermediate teachers are additive thinkers. This means that the first approach to any problem is to look for some additive solution. Additive thinking is OK in the right circumstance and on spreadsheets adding is commonly used. It is more complex than counting which is all you need to solve every problem in the world if you have enough time. Additive thinking is to me a one dimensional approach, counting is zero dimensional. Both are useful.

Efficiency of thought and linking the ideas beyond simple added connections, invoking factors, multiples and primes is where thinking becomes multiplicative. Being multiplicative is connecting ideas in a 2 dimensional way. Here is a problem to test yourself.

I buy 36 bales of hay for $4.50 each. How much do I pay altogether?

Have a go at solving this in a counted way, additive way, multiplicative way, proportional way and share your answers with someone else. Discuss and ponder the efficiency and what the different solutions show. I hope you confirm you are actually discussing complexity of thinking.

One teacher in charge of a Year 9 programme told me the answer was 45 + 45 +45 +9 + 9 + 9 and that the answer was therefore 135 + 30 take away 3 = $162. What was this person thinking to solve this problem?

The scary thing for me was that this particular teacher did not have any other strategy than a calculator to check the answer.

So if teachers are mostly additive thinkers perhaps they are only able to teach additive thinking. We are all limited by the complexity of our own thinking. Maybe the Numeracy Project did not produce the expected leaps for the reason that the teachers had not learned that more complex demands of multiplicative and proportional thinking in the first place. Not yet!

I will keep an eye on this blog Dr Nic. Thanks for your work and promotion of mathematics and statistics. The Dragons are fun by the way. HG

“So if teachers are mostly additive thinkers perhaps they are only able to teach additive thinking.We are all limited by the complexity of our own thinking.” This is so important to understand. Mathematics confidence plays a big part here too. Teachers will often default to teach as they were taught if unsure of new methodology.