|INFINITY - Very Large Numbers - and very small numbers
Fractions and Infinity
It is interesting that we do not study infinity in early mathematics programs. In fact, in Ontario, Canada, where we teach, it isn’t even mentioned in the curriculum until grade twelve polynomials. Maybe the designers of the curriculum thought it too difficult a concept for young mathematicians to understand. Why is it that we think that infinity is difficult to understand? It is certainly easier for us to understand than some very large numbers. Try to imagine the 600 billion dollar coins of the Canadian government’s total debt. It is difficult to envision this many tangible things. Infinity however is a relatively easy thing to understand, it is so many that it is undefinable. Children do not have difficulty with the undefinable in their imaginations, just like they have no problem understanding poetry, beautiful artwork, and music, or ideas like truth, justice, and honesty.
Up until grade three, fractions for students are the simple halves, fourths, fifths and so on. It is specifically written that students are to study fractions “without using numbers in standard fractional notation.” They are to understand all about the even parts that divide up a whole or a set, and the parts of this division that are fractions of a whole or a set, but not with numbers. Or with ideas like numerators or denominators. We wondered if perhaps the students might have a more sophisticated understanding of fractions than the curriculum gives them credit for. Might it be possible for children to learn about fractions without an endless series of colour-the-fraction worksheets?
We also wondered if we could wrap all of the fractions up in a problem that lets young students appreciate the beauty and surprise of an infinity of fractions being able to be held right in the palm of their hands.
The idea of fractions was wrapped up in a paradox not unlike the famous paradoxes of the ancient Greek philosopher, Zeno. In order to walk out the classroom door, we first have to get to a point halfway to the door. Once there we still need to get to a point halfway of the remaining distance. And so on and so on. The students quickly catch on to the idea that there is always something left when you cut something in half, i.e. half. They also see that the remaining half is getting smaller and smaller, and we might never get out the door when we think this way. We were ready to model this paradox.
The students were given a series of squares divided up into an 8 by 8 grid. In the first square, they were asked to colour half of the whole square. In the next square, they were asked to colour half of the previous amount or one fourth. This kept repeating until they got down to one square remaining in the grid. They were asked if it was possible to keep going. “Sure,” was the reply. They just needed sharp pencils to draw the fractions inside a really small square. Prompted by watching a video of mathematician Graham Denham from the University of Western Ontario doing this same activity, the students were able to imagine that their fraction drawing might never end. “There’s no reason to stop just because your pen’s not very sharp, in your mind you can see how this pattern continues,” he prompted. At this point the students had drawn and easily named one half, one quarter, one eighth, one sixteenth, and all the way until at least one sixty-fourth. Many students just continued on until they ran out of printed squares on the page which brought them to 1/4096th. Interestingly, these particular students all figured out methods to define these fractions numerically. They either doubled the numbers in their heads, added or multiplied them on paper, or picked up a calculator.
Next, the students were asked to imagine that we have kept going and that we have an infinite number of fractions drawn, what shape do they think would be made if they could cut them all out and put them together? How big would it be? Answers of “huge, bigger than the school, bigger than the universe” were given. They were then asked to cut out the fractions they had and to start putting them together in a shape. Around the classroom, as they were assembling fractions, came startled exclamations of, “Hey! It’s just the square! It all fits into the original square!” They had just discovered that the sum of ½ + ¼ + ⅛ + . . . = 1. They could pick up their newly assembled square filled with fractions and know that they had infinity in the palm of their hands.
You might remember this from your time in high school math class.
Instead of heading into a series of worksheets with the idea of practicing their fractional knowledge, we made “stained glass” squares filed with the series 1/2n were shown a different way to colour half of the 8 by 8 grid and asked if they could think of other ways to colour half that might be even more beautiful. There was a variety of different ways the students used to fill out ½, ¼, ⅛, and so on.
During the fraction and infinity portions of the lesson, the students were positioned by the teachers as meaning makers and discoverers of mathematics. The activities were presented in a puzzling way that had the students trying to find and understand the math that they were playing with. The students eagerly shared their discoveries with the teachers and were encouraged to share with their peers. By deliberately putting the students into the role of meaning makers who were responsible for sharing their discoveries, the students took an active role in their learning. They were an integral part of the learning process, they were not just the repeaters of attained knowledge they would have been had they been given fill-in-the-fraction worksheets. As such, all of the students understand their grade level fraction expectations. More importantly, they have played with much more complex ideas about fractions and numbers that they will encounter formally as they move through the grades.
After the learning activities, the students were asked to think about how they might take on a teaching role with their parents as the students. A script (lesson plan) was co-created so that we could all ensure that our parents learned about fractions, but most importantly they would also get to experience the surprise at being able to hold infinity in their hand.
Initially, the teachers are positioned as a keeper of the secrets. This is an unusual role to take on as a teacher, but it is exactly the role you would take on if you were a storyteller, a movie maker, a comedian, a poet to name but a few. The important part of being a keeper of secrets is not that you keep everything to yourself, but that you keep all the surprising bits under wraps until they are uncovered by the audience - in this case, the students. The ultimate surprise was that infinity could be held in the palm of your hand. This fact was left out until the students were absolutely Along with being a keeper of secrets, the teachers were guides. They guided the students through the activities, keeping them roughly on the path of the mathematical narrative, leading them towards the surprise. As guides, the teachers must listen carefully to what the students are telling them about the math they are experiencing, helping the students to make meaning out of the puzzling activities. While the students get to experience surprise and pleasure at the mathematics revealed to them, the teachers also get to experience the pleasure of being the storyteller of the mathematical story of infinity and fractions.
The parents were able to experience the same mathematics activities as their children with the students acting as the teachers. They were asked for focussed feedback - What did your child share with you about their math learning? What did you learn from this math-at-home activity?
- He told me he learned how to hold infinity. Even if things are not visible to your eye it doesn’t mean they don’t exist.
- Infinity is anything. Infinity has no limits.
- The children’s imagination can figure out and answer a lot of everyday questions.
- It is important to know what fractions are and what they represent.
- It increased the awareness of how maths can be fun and practical.
- He taught us that we are only limited on paper, not our minds.
There is a twofold reason to share these mathematics activities with the parents. Besides letting the students take on the role of teacher, they get to be the keeper of secrets and guide their parents through the same narrative they have experienced. By sharing these surprising and pleasurable mathematics activities, the parents themselves become more open to this type of activity, they become aware of another way to learn, and in fact they begin to expect it.
Implications for Practice
This activity perfectly illustrates that we can wrap important, but unsurprising concepts like fractions up in a different context than what is normal. Fractions are very important, rational thought becomes key to math success later in the students’ learning. It is just that at the primary grades’ level, these numbers are not always taught in a surprising or pleasing manner. By wrapping fractions up in a paradox that tends to infinity, the students have been given the chance to open their minds to this important concept, and to play around with a math idea that they will not see again until they take calculus and learn about limits.
The students had no difficulty making half, then half of half, and half of half of half and so on. In fact, they were even able to calculate the names for these very small fractions. No worksheets were done by students, only fraction art and problem solving questions. The students were able to understand a great deal about fractions like how to gather fractions together into a whole, and even to relate fractions to division. They learned this through working out a paradox that plays with infinity. This was as surprising and pleasurable to teach as it was to learn.
The Math Performance Festival is funded by the Imperial Oil Foundation, the Fields Institute, Research Western, the Faculty of Education at UWO, and the Canadian Mathematical Society. A project by George Gadanidis (UWO), Marcelo Borba (UNESP, Brazil), Susan Gerofsky (UBC), and Rick Jardine (UWO).