Duncan says that the better children are at learning maths as preschoolers, the more likely they are to succeed academically. Early maths skills are more consistently predictive of later achievement than early reading skills, he adds.
At Lakeview in Takapuna, children are exposed to both literacy and numeracy in their daily activities. It's a programme that has been recognised by the Education Review Office in its latest report.
Tod-Bosch says although they don't have a dedicated maths lesson, "we do very consciously make sure we have numeracy-strong activities available at all times". In addition to standard games and puzzle activities, they include practices such as growing and measuring plants and monitoring the hatching of eggs on a graph. She does not, however, give either maths or literacy dominance in the preschool set-up. "Even our NZ curriculum doesn't advocate one subject over the other."
Getting the best out of your preschooler doesn't mean you have to enrol your 4-year-old in extra tuition, says Duncan. Rather, he suggests a return to board and card games. "Maths learning is about being able to do things like count squares on a board and play board games - coupled with parenting, that builds maths into everyday learning experiences."
Duncan - whose recent work includes estimating the role of school-entry skills and behaviours on later school achievements - was in NZ last month as the Sir Frank Holmes Visiting Fellow in policy studies at Victoria University.
Children used to play a lot of board games, he notes, and that's still the case for white, middle-class families. Board games like Snakes and Ladders have been shown to build children's understanding of the number line, which is key for mastering addition and subtraction, he says. "But low-income children play a lot of video games."
That doesn't mean you have to throw away the PlayStation or iPad; just use them sparingly. A PlayStation may be good for hand-eye co-ordination but not for basic academic skills.
He adds that bringing out the best in children involves parents interacting with them. "Parents should be aware of these simple lessons: that building these early maths and literacy skills into their daily interactions is fun to do and helps children be better ready for school. Be sure that video games are off-set with board games."
The studies which back up his findings were conducted in the US, Canada and the United Kingdom.
NZ's early-childhood teaching levels are of a very high quality, Duncan says, "but we could reasonably predict the results would be similar here".
At the University of Auckland's faculty of education, lecturer Shiree Lee says toddlers' mathematical development begins much earlier than originally thought. Writing in the New Zealand Educational Review, Lee describes her observations involving children aged between 12 months and 3 years, who demonstrated highly developed mathematical skills.
"The toddlers in my study used abstract ideas, prior experiences and other people for purposes that may not generally be recognised as being mathematical. This exciting learning can further develop through play experiences in everyday settings," wrote Lee, who exhorts adults to pay closer attention to those developing skills.
"Next time you have the opportunity to watch toddlers play, ask yourself if you can see or hear whether they are engaging in mathematical thinking."
Lee's colleague, Peter Hughes, principal lecturer at the University of Auckland's faculty of education, previously taught maths at high schools in England and NZ. He cautions parents not to mistake counting for knowing how numbers work.
"If you can count to 16, does that mean you know what 15 plus one is?"
He adds that the aspect that tends to be ignored by parents is spatial visualisation - something he believes to be as important, if not more important, for children to learn. Ten and 11-year-olds struggling with maths lessons may have poor spatial awareness, says Hughes.
Back at Lakeview, the party proved a huge hit. The children, who had been practising since October for their end-of-year play, had to know how to time their stage arrival: first was flowers, second was bees, third was the pukeko and so on.
Even at Christmas, there's still room for maths.
Babies' brains as absorbent as sponges
Tina liu is performing a routine task: changing a nappy. At the same time she's introducing her 18-month old charge to mathematics. "I tell her 'I'm going to count to 10 and then we'll be done'," she explains. It's the same when the babies are rocked to sleep: the song sung to them has the rhythm of numeracy to it.
Liu, who has a Bachelor's degree in early childhood education, works at Auckland Girls' Grammar Early Childcare Centre, which caters for 32 children, aged from 3 months to 5 years.
Head teacher Jane Archer has been here nine years.
"Maths doesn't have a particular time of day here," she says. "It starts from the moment the children can say the [security] code on the gate - and then they're lifted up to identify those numbers. They learn very quickly."
Board games, cooking, singing, puzzles, building: numeracy is enmeshed in everything the children do, says Archer.
The centre has no structured timetable - other than for meals, nor electronic games. But there is a piano and other musical instruments.
Archer believes Duncan has been too prescriptive in his observations. "He sounds like he's isolated maths into a package, whereas here it's woven throughout the day."
She also believes literacy and numeracy are equally important in a child's early years.
