“I hate math!” “Math is too hard!” “Why do I have to explain my thinking?” At every turn, students and now parents are asking this same question.
Let’s step back a moment and remember our own frustration when we were in school and our teachers made us “show our work.” In our day, “showing your work” meant writing out all of the steps when solving an equation or geometric proof. Or it meant writing out the steps to a long division problem. For those of us who could “do it in our heads” and, therefore, pushed back, this caused friction–both in school, with the teacher, and at home, with our parents, as they pointed to the paper and told us we couldn’t get up from the table until we wrote out all the steps.
Today, with Common Core, phrases such as “explain your thinking” and “justify your answer” take on a new meaning–and some students are pushing back. As we collectively emerge from over a decade of math teaching and assessment that, overall, emphasized more computation than problem solving, it is not surprising that we need to teach students and their parents what it means to explain our reasoning.
Take this example of a math problem about the water level in a rectangular prism.
A sealed rectangular container 10 inches by 12 inches by 6 inches is sitting on its largest face. If it is filled with water up to a level 2 inches from the top, how many inches from the bottom will the water level reach if the container is placed on its smallest face?
Answer: 8 inches from the bottom
Here is one explanation of the mathematical reasoning behind this answer.
A rectangular container is the 3-dimensional solid called a rectangular prism. To find the volume of a rectangular prism, or the amount of water inside the container, you multiply length times width times height. The largest possible face of this rectangular prism is the rectangle that is 10 inches by 12 inches or 120 square inches. The height of the container is 6 inches, so the water rises 4 inches from the bottom. Therefore, to find the amount of water with the container on its largest face (10 x 12), I multiplied the three numbers 10, 12, and 4, which equals 480. There are 480 cubic inches of water in the container. Using this same thinking, the smallest face is the rectangle with the smallest area, which is 6 inches by 10 inches, or 60 inches squared. This means the height is now 12 inches high. I know that no water left the container, so I need to find the dimension of the height of the container with the same volume of 480 cubic inches. Given the formula, length times width times height, or 6 x 10 x ?, the height of the water must be 8 inches because 6 x 10 x 8 equals 480 cubic inches.
Now that you have seen one possible explanation, how ready and willing do you think your child is to explain his thinking? The next time you hear your daughter or son (it is more likely that it will be our boys who say this because more of them are reluctant to want to write), say, “I hate math!”, ask him to explain why. It will be good practice.