Last year I volunteered in a third grade class. The school mainly uses Investigations 3. During a visit last spring, the class was working on the second session of the fractions unit. I was excited to find out about the students’ beginning understandings of fractions.
I sat with a group of four students who were working on making fraction sets. Their task was to fold each of 5 sheets of paper into two, three, four, five, and six equal parts and then to label each piece.
The students made 4 equal pieces by first folding the paper in half and then in half again. They made eight equal pieces by making 4 equal pieces and then folding those in half.
Making three equal pieces was more challenging. They tried, and made unequal pieces. When I asked, “Would it be fair if I had this piece and you got this piece?” they shook their heads no. Jerome figured out how to make three equal pieces and proudly shared how he did it.
When it came time to label each piece with a fraction, Samantha correctly labeled the halves, but announced “The 1 on top is 1 whole and the number on the bottom is the number of pieces.” We reviewed what they had discussed in class about what the numerator and denominator stood for and then she and Jerome tentatively labeled the pieces in the rest of their fraction sets correctly.
Noor labeled her sixths: 1/6, 2/6, 3/6, etc. I reminded her that the number on top told the number of pieces. I pointed to the piece she labeled 2/6 and asked, “How many pieces is this?” She said one and changed each of her labels to 1/6.
Jung labeled one of her eighths 2/4. I asked her what the 2/4 stood for and, as she talked, I realized that it stood for the array that the eighths made — 2 rows with 4 rectangles in each row! I acknowledged that it did in fact make a 2 by 4 array, but reminded her that we were thinking about the piece of paper as one whole and were writing fractions to show how much of the whole each piece is.
Watching these students work made me think about the complexity of understanding fractions and fraction notation. Fractions require thinking about the relationship between parts and wholes. To understand fraction notation, you have to understand that the numerator and denominator are not just two separate numbers – the relationship between them is important.
The students’ struggles with making thirds and labeling the fraction pieces made me think about what the activity of making fractions sets provides. Some might wonder: why ask them to make fraction sets? We could give them a pre-made set that shows them what halves, fourths, eighths, thirds and sixths look like and how they should be labeled. Pre-made fraction sets can be a useful reference, but I would argue that, as students are just beginning to make sense of equal parts of a whole, making the sets is critical. It was working to make them that uncovered the students’ thinking and gave them an opportunity to puzzle through important ideas.
When students create fraction sets they make equal pieces themselves. They figure out how to label the fraction pieces and notice relationships among the different-size pieces. Working with these students I learned a lot about what the students were thinking, and was reminded of the many ideas that contribute to the understanding of fractions. I wondered what confusions or revelations would come up as they worked with non-unit fractions?
Making fraction sets takes time and the learning may seem messy. However, taking the time to puzzle through important ideas like these is the way I believe that students really make sense of mathematics– it should not be a clean easy path!