As math instructors aren’t we always just walking that fine line between fluency and conceptual understanding? We know that the answer lies in both. Our students need to have fact fluency, but that fluency is only meaningful if they can problem solve. We know this, we teach this, we preach this – but you won’t believe how I saw this play out with a new student in my class.

During the first couple weeks of school I like to work problems out with each of my students during independent work time. I like to talk to them and watch them work so I can hopefully get a better understanding of how they process math. And I have to say that I saw something this year that I have never seen before from a sixth grade student. We are studying decimal operations right now and the problem I’m about to show you was a decimal division problem.

Here’s the problem. (See Step 1)

So he realized that the first step was the move the decimal in the divisor and then dividend. Then he knew to see how many times 4 goes into 11. Here is where it gets so interesting! He applied a division strategy of drawing a picture and representing the division with circles. He drew 11 circles and then grouped them into groups of 4. He realized that he could only make 2 complete groups of 4 in 11, so he put a 2 above the 11. Here’s a picture of his work. (See Step 2)

Next he knew that he needed to multiply 4 times 2. Let’s stop for a quick second, this should be a fluent memorized fact, right? Not here! Instead the student drew 2 sets of 4 tallies and counted that there were 8 tallies together. Here’s another picture of this step. (See Step 3)

By this point I was completely mesmerized. He has a great conceptual knowledge of the meaning of multiplication and division, but this process is so slow! He continued working the problem this way and got the right answer, but I swear it took close to twenty minutes! This was such an eye opening experience and further cemented the fact that we have to prioritize our students building fluency for their own efficiency. Have you ever seen anything like this? I was amazed, impressed, and absolutely shocked – and I sent him home with some flashcards.