# Dedicated to Teachers

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.

As I sat down with a pair, I asked them to explain their thinking to me. They did so, using the algorithm for fraction multiplication and the formula for area. Then, they told me how they thought their answer was unreasonable because the area of a rectangle should be larger than the dimensions. As I probed further, inquiring as to how they knew this, they gave whole number examples to defend their assertion. So, I asked if we could look at a different problem to see if their claim was always true. Our floor tiles happen to be 1 square foot. I asked the kids to mark 1/2 foot by 1/2 foot on a tile. Immediately, they saw that the area was 1/4 of the whole tile. They now could see that their original answer of 3/8 of a square foot was reasonable because they were able to blend their more recent knowledge of multiplying fractions with that previous knowledge of area when dealing with whole numbers. They had realized a deeper meaning of multiplication. When we help students build those bridges between previously learned and new concepts, the light bulbs start to go on and math makes sense.