Today's schools are barely recognisable to adults who went to school in a different era. Grandmother Denise Mills and education reporter Simon Collins spent two days in a Year 4 "modern learning space" to see just how different things really are. Today: maths.
Getting the right answer is no longer good enough in our primary school maths classes - you have to be able to explain the "strategy" that got you there.
About eight Year 4 students sat on the floor at The Gardens School in Manurewa as teacher Elizabeth Barrowman gave them maths problems to solve.
"52 + 12," she wrote on the whiteboard.
The children wrote in their exercise books on the floor: "52 + 12 = 64" (if they got it right). But then they had to explain how they got that answer.
Many struggled to explain. Barrowman wrote on the whiteboard to summarise one student's strategy:
"50 + 10 = 60.
"2 + 2 = 4.
"52 + 12 = 64."
But other strategies were also possible and equally acceptable, such as:
"52 + 8 = 60.
"60 + 4 = 64."
Barrowman gave the children small wooden blocks to make the exercise concrete - little square blocks a bit bigger than a pea to represent single numbers, and long skinny blocks to represent tens, so the children could see visually that 10 ones were the same as one ten.
Five tens and two ones, plus 12 ones, added up to five tens plus 14 ones. And that was the same as six tens and four ones, 64.
Then she gave them plastic cards divided into 10 squares on each card. The students marked off crosses on five cards and on two squares of a sixth card, then filled in the rest of that card with another eight crosses and added four more crosses on a seventh card to get the same result: 64.
Eight children all trying to do these exercises at different paces on the floor looked very confusing to an onlooker, but it was aimed at making sure the children actually understood what they were doing when they added 12 to 52. They were not just following an abstract formula.
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In a small breakout room, another teacher was doing a similar exercise with another half-dozen children using coins on cards, showing that 10 10c coins filled up a whole card of 10 squares and was equivalent to $1.
In the cooking room downstairs, the teacher aide was helping another group to measure out the ingredients in a recipe, weigh them on scales, use their iPads to photograph the weights and record them in their exercise books.
On another day, students were given maths problems and had to think of real-life situations in which the problems might arise such as, "If you were planning to cook for 52 people, and then another 12 people arrived unexpectedly, how many people would you have to cook for now?"
"We want them to have an understanding of number," Barrowman explains.
"We have calculators, but we need our children to have the critical thinking to recognise whether an answer they have on the calculator actually makes sense.
"I know when I was a child at school, the number was the number. We want our kids to know that numbers can be broken up, you can take them in chunks but, to know that, you have to know about place value [ones, tens and so on].
"There is more than one way of doing things. Children will gravitate to the way that they are most comfortable with."
Yolande Mathlay, maths leader for The Gardens School's junior years, says the children still learn their times tables - although not in one big hit.
"In Years 2 and 3 we just focus on letting them have their two times tables, their five times tables and their tens. Those are the easiest ones. And as they move higher up we introduce the three times tables and the fours and so on," she says.
"But we have something just prior to that, and that is where they need to understand what multiplication actually is. So we don't just rote-learn our times tables, we need to understand three groups of two and what does it mean.
"We use the materials to show them: 'This is three groups of two, so that means 3 times 2.'
"We still teach times tables, but not in the same way that we learnt it when we were at school. We use real-world stuff, such as, 'Three children ate two bananas each.' And we really draw the pictures and have little objects and materials to show them three children each eating two bananas."
"We work collaboratively," Mathlay adds.
"We allow or give the children an opportunity to work independently or collectively with each other on how they solve problems and how they manipulate materials.
"So that's the main difference, it's a lot of collaboration. We have found that it has really enhanced their learning. It's so much better to bounce an idea off somebody, you learn so much better that way."
This way of teaching maths was introduced across all NZ primary schools by the Ministry of Education in the year 2000 and was known as the "Numeracy Project" - a term still used at The Gardens School, even though the project has been blamed for New Zealand's declining maths scores ever since 2000 in the Programme for International Students Assessment (Pisa).
As Mathlay describes it, the children work through three phases: materials, imaging and number properties.
"We have a little bit of a process in terms of giving them an opportunity to manipulate materials with their hands, but then follow on with imaging. That is masking the materials so that they try and use their brain to image, or imagine, the answer," she says.
"Then we move on to what is called number properties. That is when we want to understand how they got to the answer."
At primary school level and in fact up to Year 10, the ministry says: "Children should do most calculations in their heads. They should only use pencil and paper or a calculator when the numbers are hard."
At The Gardens School, this basic learning is backed up by computer programs such as Kahoot and, especially, Mathletics. In Year 4, every student must earn at least 1000 points in Mathletics every week - effectively a form of maths practice.
They are expected to do their Mathletics mainly at home, but they can also work towards their 1000 points in class if they complete their other work quickly. Parents were told in a February newsletter: "Students that do not complete this will spend some of their Friday lunchtime doing so."
At the whole school assembly on the Monday morning before the Easter holidays, more than 30 students were called up to the front to receive various awards, including several for the most Mathletics points earned in the week.
Denise Mills, a 67-year-old grandmother of two girls in the school, says there are far more awards than she remembers in her school days.
"The reward system with the certificates, I think it's great. With five certificates they get to high-five with Mrs Fowler [the principal]," she says.
"They learn better because they are self-motivated. I know from my grandchildren that they are keen to go up to the next level of learning books, they are keen to get the numbers in Mathletics."
Professor John Hattie has said the Numeracy Project reduced the emphasis on basic knowledge such as the times tables.
"Teachers loved it because they didn't have to know the maths, they only had to know the strategies," he said in 2017.
"It led to 10 years of the greatest decline in maths. People have woken up to the fact that that was one of our biggest failures."
New Zealand's Pisa scores for maths, measuring maths achievement of 15-year-olds, declined steadily from 523 in 2003 to 495 in 2015 on a scale set to an international average of 500.
However maths achievement by Year 5 students, perhaps a better measure of the impact of the Numeracy Project, has flatlined in the Trends in International Maths and Science Study (TIMSS), fluctuating insignificantly between 486 and 493 over the same period after improving significantly between 1995 and 2003.
Whatever the cause, our Year 5 students have scored consistently below the international average since the TIMSS survey started and well below all other Anglophone nations where the test results are not distorted by comparing across different languages.
Briar Lipson of the NZ Initiative thinktank says children need to get basic facts such as their times tables into their long-term memory before they can handle any maths problem in their heads.
"Our [short-term] working memory can store perhaps five to seven items at any one time. No matter how brilliant you are, you can only store that many items in working memory," she says.
"The only way to cheat and move beyond that is to have knowledge in long-term memory. When we add GST, we know it's 15 per cent and have good stores of knowledge to work it out such as taking 10 per cent of the number and then half of that again for the other 5 per cent.
"But if you are someone who doesn't have that in long-term memory, and are provided with a problem, your working memory is overloaded with all the things you have to work out because you don't know your times tables."
Mathletics also has its critics. Mark Bracey, a Howick Primary School teacher who blogs at Ease Education, says programs like Mathletics involve very little thought - and therefore very little learning.
"I'm running the Mathletics program at our school and when I was asked to provide certificates for the top learners I thought, 'Oh my goodness, actually this is just participation - the longer you sit at the computer, the more points you get,'" he says.
"Is that a good use of your time? I'd argue not, because I know how I got those children learning maths. It's actually simple: it's good teaching, it's getting the kids to think.
"Those Mathletics programs are very passive, I'd rather have children doing computational puzzle cards on the floor. A lot of it [use of Mathletics] is keeping the kids quiet while you get on with your group.
"The kids love the Mathletics quiz, but I'm challenging them to be creators.
"To be creative you have to have a good knowledge base. The problem is that we sometimes get knowledge, knowledge, knowledge, there's no time to consolidate that knowledge and be creative, and then go to the next level of knowledge."
Barrowman says The Gardens School, like many others, is constantly refining the ways it teaches maths.
"When the Numeracy Project initially came out, we became too focused on number. That meant that the geometry, the measurement, the statistics and all the other strands of the curriculum got neglected," she says.
"So now in the modern interpretation of the Numeracy Project, we have been more careful about making sure we are giving the whole curriculum coverage."
Monday: No more rows of desks.
Tuesday: Learning on devices.
Wednesday: Literacy - reading to understand.
Today: Maths - the right answer is not enough.
Tomorrow: Self-directed learners.
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