Thinking about Infinity with Elementary School Students
By Jana Mohr Lone
In the beautiful picture book Infinity and Me, written by Kate Hosford and illustrated by Gabi Swiatkowska, a young girl, Uma, looks up into the night sky and starts to feel “very, very small.” She begins to wonder about the nature of infinity.
Uma then asks various people how they imagine infinity. She receives a wide range of responses, such as “a giant number that keeps growing bigger and bigger forever” and a family, which could “go on forever.” She notices that it’s hard to talk about infinity without mentioning “forever,” and considers whether there is anything she would like to do forever.
Uma considers having recess forever, but then questions whether, if there’s no school before or after recess, it would still be recess. In the end, she stares at the night sky while snuggled next to her grandmother, and she observes that the sky doesn’t seem as “huge and cold” anymore.
I read this book to a group of fourth-grade students after one of the students had suggested that we have a conversation about infinity, saying that she often wonders about it. The students had the following questions about infinity:
If infinity can be a family, how would it be infinite?
Why does infinity go on? Why is it called infinity? What is infinity?
Do we live in infinity?
If infinity never ends, how did it begin?
Can infinity be different sizes? Can infinity overlap? Infinity goes on forever—can it be destroyed?
Is your mind able to imagine infinity?
How was infinity made? Why was infinity made?
Does anything go on for infinity? Is infinity just when all the numbers run out?
Can the word ‘infinity’ hold what infinity is?
The students were especially curious about the idea suggested in the book that infinity could be imagined as a family.
“Is family really infinite?” one student asked. “What if you have no siblings and then you have no children? Wouldn’t that be the end of your family?”
“But you would still have family in the world, even if you didn’t know about them,” another student responded. “There would be cousins and second cousins and so on. I don’t think families ever end.”
“We once had a 100-year-old neighbor, and he said he was the last of his family. He had no children and no brothers or sisters or nieces or nephews. He said that when he died, his family would come to an end.”
“I guess it matters what you mean by family,” said another student. “If you think about it, we all started with the same bacteria, before the first human being. So we are all family, in a way.”
“But then would talking about your family really mean anything?”
“Also would that mean that when we say the word ‘family,’ we might not even be talking about the same thing? Because when we talk about family, we don’t usually mean everyone in the world.”
“There are lots of definitions of family,” pointed out a student.
“Are the possible definitions of family themselves infinite?” I asked.
“Maybe that’s true of almost every word we use,” a student replied.
This led us into an exchange about whether we can ever really imagine infinity. One student said that she didn’t think so, the concept was “too big for humans to be able to understand.” Another student commented, “I feel like infinity is un-sayable and also un-countable, so I feel like the word infinity is just there to help you wrap your mind around it.”
Other students thought that we could imagine it, if we think about it as something that just goes on and on, and never stops. Like the infinity symbol, one student noted.
The students questioned whether something can be infinite if it had a beginning but has no end. Does it have to be endless in both directions? A family, for instance, might go on forever, but it might have had a beginning. Some students noted that eventually the earth will no longer exist, and so human beings (and families) won’t actually go on forever.
How big is infinity? Are there are different sizes of infinity? We talked about whether the infinite set of real numbers, for example, is larger than the infinite set of whole numbers. Some students said they could see why one infinity would be larger than another, and others insisted that if an infinite set goes on forever, it makes no sense to talk about one infinite set being larger than another one.
“Forever means forever! If two things go on forever, how can one be bigger than the other?”
I think this is one of those conversations that could have gone on forever, but it was time for recess.
