Last week a few upper elementary classes started to explore different methods to divide multi-digit numbers. Many were familiar with repeated subtraction for numbers more than one and some even had experience with the long division algorithm. I asked students to explain their reasoning for the steps needed to divide numbers using repeated subtraction and long division. All of the students were able to explain the reasons for using repeated subtraction. Students gave quality answers and were able to communicate why each step was performed. Some students even related repeated subtraction to a number line. I then asked the students the reasoning for using the long division algorithm. I heard responses like “it’s quicker” or “that’s what I was told to use” or “you just do this and this.” I could tell that there was a disconnect between the shortcut and having a conceptual understanding. Students understood the steps but couldn’t provide solid reasoning to why you would bring down the next number.
The class then had a conversation about the importance of being able to clearly explain their mathematical thinking. The students that knew how to use the long division algorithm were getting correct answers, but couldn’t tell me why. Blindly following procedures can lead to holes in understanding. Explaining the reasoning behind completing a problem is important. Honestly, I don’t mind if the students use algorithms like the above picture if they already have a descent understanding. The problem I have is that sometimes this is the only way students are taught how to divide large numbers. The problem becomes steps –> answer without understanding.
After the class had a discussion about long division we explored the partial quotients algorithm. I explained to the students that this was another method to divide larger numbers. As we progressed through and explained the steps, students became more aware of how partial quotients seemed to make sense to them. For many, this was the first time exploring the partial quotients algorithm. The students were able to explain why each step was taken in the process.
I believe some of the students were also relieved that they didn’t have to get the partial quotient “correct” the first time. By breaking apart the problem students were starting to see the correlation between repeated subtraction and the partial quotients algorithm. More importantly, students were able to explain their reasoning for completing the problem with this method.
As we move forward, I feel as though students are becoming better at explaining their mathematical thinking. It doesn’t matter to me what method the student decides to use in the future as long as they are able to justify their reasoning. This thinking could also lend itself to just about any type of computation skill. Last week reminded me of the need to expose students to multiple strategies to complete problems. Providing these strategies can assist students in becoming better at explaining their mathematical thinking.
As my students progress through their fraction multiplication unit I came across another example of why using multiple strategies matters. In the past students learned how to multiply mixed numbers by 1) Convert the the mixed numbers into improper fractions 2) Multiply the numerators and denominators 3) Covert the improper fraction back into a mixed number. This is how I was taught to multiply mixed numbers. Although this method seems to work, students had trouble explaining why they completed each step. So early this week I decided to use a different strategy in class.
Students were able to visually represent this multiplication problem and the steps to solve seemed more logical.