Child Turns Math on its Head
This is a fascinating story and an eye-opener on the flexibility of the learning process.
When we think of the first math lessons that toddlers learn in life, we immediately think of counting. This includes being able to recite numbers in order and being able to point to a series of objects and correctly count each one once (one-to-one correspondence). It is often true in preschools, particularly Montessori preschools, that children are not allowed to learn a more complex activity (such as multiplication) until they have mastered its pre-requisite (in this case, counting).
However, the learning process is not always linear, as you shall see:
A close family friend took a road trip to Yosemite National Park. Seeing the trees whizz by, their 26-month-old daughter, Ariana, who was in the back seat, expressed a desire to count them. At that age, she knew how to count a few numbers but got mixed up by the order. So, she counted something like, 1, 2, 3, 8, 7, 9. Her uncle who was with them immediately translated the miscounted numbers into a song with a refrain, “crazy numbers!” Of course, the catchy song cemented Ariana’s number order confusion in her mind for a long time.
Soon after that, feeling rather perplexed about counting, Ariana formally announced that she only needed to know how to count up to her age. She felt this was perfectly rational, and no one should expect her to know how to count beyond her age.
Sure enough, 10 months later when she turned 3, she kept her self-imposed rule and added just one more number to her repertoire, counting to 3… and no more.
While most parents would react by cajoling their child to learn their numbers, Ariana’s mother and father let their daughter take the lead.
To provide a little background, Ariana did not attend preschool. Her parents were planning to homeschool her beginning at the age of 5. In line with their philosophy, they did not work with her on number concepts: They did not do counting games; they did not challenge her with math problems, etc. They were perfectly fine allowing her to learn how to count at her own pace.
Because Ariana convinced herself that she could only count to 3, she was faced with a problem every time she needed to count the number of stones she gathered, or describe the number of tangrams that she used to make her design, or the number of blueberries she wanted to eat, or the number of eggs she found on Easter.
All by herself, without any suggestions from an adult, she figured out how to count in groups of 3. So, she would say, “I have two 3s and two extra.”
Astonishingly, she kept this self-imposed rule in place that entire year – never counting above 3, and when she turned 4, she dutifully added one more number to her repertoire.
So, naturally, in her entire fourth year of life, she was counting everything by groups of 4, and remarkably, did not add the next number until she turned 5! And when she was 5 – a time when children her age were counting up and down to and from 100 – she would religiously only count to 5, and so counted all her objects in groups of 5.
It was only when she neared 6 that she threw aside these restrictions and learned to count up to 100 and then 1000 in a matter of a few months.
By the time Ariana was 6, the home schooling curriculum introduced counting by 1s, 2s, 3s, 4s, 5s, 6s, and some 10s through a story and rhythm song. By then she also learned about addition, subtraction, multiplication, and division – also through stories and also with a very light touch (no worksheets, no drilling).
Despite this lack of formal math training, Ariana seems to have a heightened number visualization ability. For example, when Ariana’s mother was having a casual conversation with her about the date of her friend’s birthday party, she mentioned the party was 2 months prior, or 60 days. Ariana’s mom asked her if she knew how many days were in a month. At first Ariana said 29, then she re-evaluated while talking out loud, “50-10-40-20-30! 30 days,” she said. She came up with the solution herself by portioning out the number of 10s in 60 into two groups – subtracting from one group, and adding to the other until she got to the mid-point, 30!
As a former first grade teacher, I tried to remember if any of my previous students could have done the mathematical gymnastics that Ariana had done in her head. Perhaps, one or two of my very brightest just might have succeeded. But Ariana had done this with very little formal math practice.
Another example of Ariana’s visualization skills was demonstrated just after turning 7. Ariana’s mom was prepping for the upcoming school year and found that the home-schooling curriculum was adding story problems to the curriculum. The challenge question for the year, which was supposed to be asked after actively practicing similar but simpler problems over ten months of home-schooling, seemed easy enough to Ariana’s mom. So without giving advanced preparation, she read the problem out of the book to her.
“We have four children and each of them wants an apple. There are ten apples in the fridge. How many will be left if each child takes one?”
Ariana may have misheard the problem because she first said, “two apples.” Her mom replied, “really? There’d be two apples left if each one takes one?” To that Ariana quickly explained, “No, there’d be six. But they want as many as they can have, so they’d each get two and there’d be two left.” Obviously Ariana could not be constrained by someone else’s story problem. Ariana went on to say, “If the remaining two apples were cut in half, each kid would get 2½ apples.”
Pretty good number juggling for someone who has never attended a regular school!
At this point, Ariana’s little 3-year old brother chimed in that he wanted three apples because he’s 3 and Ariana should have seven because she’s 7. Apparently this age-number obsession runs in their family!
Did Ariana’s self-imposed counting restrictions over three years actually wire her brain to develop higher-level math skills at an earlier age? I wonder.
Was she able to visualize numbers in various groupings better because of her unusual trajectory of learning math? Most probably yes!
Naturally, every child learns differently and very few children will come up with a self-imposed rule on counting as Ariana did – especially for such a long period of time. And of course, constraining a child to limit his or her counting would be ludicrous. But what about encouraging grouping activities beginning at age 3? Would that help children visualize numbers more clearly in their heads and allow them to do number manipulations easily and effortlessly by age 6?
If nothing else, Ariana’s path of learning challenges the methodical step-by-step teaching that is common in preschools today.