In his new book, Comprehending Problem Solving, Arthur Hyde discusses how to help students develop a deeper, richer understanding of mathematical concepts. He argues that although helping students learn the concepts behind procedures requires more effort up front, it will benefit you and your students in the long run because they will have already laid a solid foundation for subsequent learning.
In today’s blog, which is one of the classroom stories included in Comprehending Problem Solving, elementary math specialist Sara Garner describes an encounter with a pair of puzzled students. Notice how her thoughtful questioning and feedback helps them see multiplication in a different way.
A Deeper Meaning of Multiplication
by Sara Garner
I view the process of helping students assimilate new knowledge into prior knowledge as one of the most exciting parts of my job. I was able to build that bridge for students recently as they were working on C.C.5.NF.4 (apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction) with a problem involving finding the area of a small plot in a garden. The dimensions for the garden were both fractional parts of a foot. The students had recently learned how to multiply fractions. They had built hands-on experiences finding actual areas. However, as they encountered this particular problem while working in partners, several students had the same response. When they multiplied the two fractions (1/2 ft. x 3/4 ft.) and got 3/8 of a square foot, they said, “That’s not reasonable. The area can’t be smaller than the dimensions were in the first place.” Normally, hearing kids say an answer is not reasonable is music to my ears. It means my students are really thinking about what makes sense. In this case, though, it was also an opportunity for me to see that the kids weren’t putting together their knowledge of fraction multiplication and area quite yet.