Grade ones multiplying, dividing, sharing quantities, or making arrays. It sounds unlikely, but is it? Given a really good mathematical narrative to follow, would it be possible for some of the youngest of our learners to step well beyond what is generally expected of them? This is the question we asked ourselves as we stepped into a world of mathematics pedagogy where more things are possible than you might imagine.
Two things were in our heads as we planned to see if grade ones could understand the concepts of multiplication and division. We needed a really good mathematical narrative for the students to follow and we needed to use manipulatives and representations to strategically bring out the mathematics. The narrative we chose was perfect. And the Doorbell Rang follows the mathematical adventures of two children sharing twelve cookies as more and more people show up to their home. How many cookies does each person get now? A perfect model of division as sharing. The manipulatives we chose were linking cubes as these can be quickly arranged and re-arranged as the situation in the story changed. To make representations of the arrays that the students would be making with the cubes, we used bingo dabbers and grid chart paper as these result in colourful displays that are quick and easy to make.
In grade one, students need to be able to show conservation of number, i.e., 12 is 12 even if it is spread out, to compose and decompose numbers, and count by 1s, 2s, 5s, and 10s. Conservation of number is a slippery concept. 12 is 12 even if you spread out your counters, but 12 cookies shared with 2 people is different than 12 cookies shared by 6 people even though the total number of cookies has not changed. Decomposing 12 into different equal shares is a way to link the specific expectations for grade one with later learning in grades two and three where sharing, division and fractions are introduced. Arrays lend themselves to be counted using skip counting. A 3 x 4 array can be counted by 3s or 4s in a concrete or representational manner rather than the rote memorization of counting by 2s, 5s or 10s.
The students were mathematical explorers and storytellers. As we shared the story with them, they modeled different ways to show how the cookies could be shared. Working in pairs, they represented each problem by re-arranging their linking cubes to match the dilemma the children in the story faced. After having gone through the entire story and having modeled each case of cookie sharing, the students were given bingo dabbers and grid chart paper to explore different ways to make twelve. They had no difficulty making different arrays of twelve.
The teachers role was primarily that of a story teller, a problem poser and a summarizer. As the story was being told, the teacher would pose the problem of how to represent the cookie sharing with linking cubes, encouraging pairs to share and to verify if the arrangements were valid. After the story telling, the students were invited to help each other to summarize the mathematics, to make a display of common understanding for the class. It was notable that students were able to articulate not only that there are different ways to compose 12, but that it was the information in the story that had taught them. They were also able to make generalizations about how the more people they had, the less cookies they would have. One intrepid mathematical explorer even noted that 12 cookies shared with 24 kids is one half a cookie each.
As a consolidation to help the students remember what they had learned, they were asked to make their arrays into pictures of other things. From the imagination of the students came arrays ‘dressed up’ as rocket ships, caterpillars, and houses. This served the students as a focal point for the memorable mathematical learning experience they had had. To make the experience even more memorable, comments the students had made in class during their learning as well as comments they had made during their class summaries were put together to make lyrics for a song. The strength of making lyrics up from the words of the students is that they can now all hear their own words - it is themselves singing about mathematics. The song was recorded and made into a video with pictures of them during the learning time. This video was posted to the Mathfest.ca website so that it might reach a larger audience and so that students could share their learning at home with their parents.
As elementary teachers, we are very good at modeling numbers and mathematical situations with manipulatives before having students work in the abstract with numbers. In this instance we added a step in between the concrete and the abstract: the representation. Being able to represent numbers and concepts in pictorial form is a powerful bridge between the concrete and the abstract that we will have to try to implement more often. The student who very easily showed how 12 ÷ 24 = ½ was able to simply prove her thinking using a bingo dabber array and a marker to divide each ‘cookie’ in half. There would have been no elegant way for her to have accomplished this with linking cubes.
Not only were the students able to demonstrate and understand conservation of number, they were able to easily and elegantly show how within this number are other numbers that compose the number 12. Making arrays to show sharing afforded the students the ability to understand a concrete model for skip counting, division, and multiplication which should serve them very well as they continue on their mathematical journeys in following years. The learning of these concepts was made richer by having them revealed through a mathematical narrative. The students all had fun, and enjoyed the puzzles of making different arrays to help the children in the story share. This mathematical learning was delivered through narrative, but was also thoroughly embedded in the natural setting of children sharing and eating yummy cookies. They had not only thoroughly met the specific expectations of the grade one program, they had successfully set up their learning for years to come.