Don’t get me wrong, procedural fluency is very necessary. However, if procedural fluency is your only means of understanding you have a huge problem. This problem extends pass performing well on a state assessment. When procedures fail you, what do you have to rely on but conceptual understanding. An Englishman put it this way, “The only place procedural understanding brings success with problem solving is in American classrooms. In all other cases, we must rely on our conceptual understanding.” (I hope I quoted you accurately Phil.).
So when I discovered that majority of my students were procedurally fluent with converting fractions to decimals and didn’t understand at all why the rules they applied so successfully even worked, I couldn’t turn a blind eye. In the midst of feeling like I was ready to give into the ways of rules and tricks without understanding, I took a stand. I would not give up on them and I would not give up on my beliefs.
This lesson was implemented.
I thought it would be the bridge to understanding for my kids. I figured if I started from the bottom, the basics, with modeling things would click. What I discovered was they had minimal experience with modeling. Most didn’t know where to begin with setting up a model to demonstrate division. They didn’t get the idea of the physical manipulation of the blocks showed the division. Instead they thought they needed to represent the dividend, divisor as well as the division symbol. I also discovered they were limited in their reasoning abilities due to desire, experience or cognition. Needless to say, of the papers I’ve reviewed, no one was able to make the connection between why the rule in which they are proficient works.
I’m not giving up though. My next steps will be more explicit. Not explicit in telling them why it works. But explicit in my questioning as we model the division of 1 divided by 4. “What can we do to physical put 1 whole into 4 groups?” Students should conclude the whole can be exchanged for ten tenths. Draw the model. “What number is represented by the model?” My goal is for students to see adding the decimal point and zero is the ten tenths needed to physical put 1 whole into 4 groups. So the problem doesn’t become ten divided by 4 but actually 10 tenths divided by 4. Which is still 1 divided by 4.
I know moving forward it will be important to spend time working with modeling with manipulatives . I’ll probably end up using some fifth grade tasks from the GA math frameworks to help activate prior knowledge through the use of modeling with manipulatives.