Yes, it’s how we learned it. Yes, most people will say, “It worked for me, so it will work for them!” The standard algorithm has definitely been a bone of contention in many conversations. On another, but connected note, while watching Spiderman into the Spider Verse, my youngest was singing along with one of the songs. A friend of mine made the comment, “Kids can hear a song once and be able to sing it, but yet they can’t remember things in school.” (With the voice of Carrie) I couldn’t help but wonder, “If concepts had more structure that is easily identified by students, would learning be more fluid”. This reminded me of a talk I watched of William McCallum on The Story of Algebra. In it he discussed finding structure within a poem to help memorize the poem. He then connected it to the structure found within the Common Core Standards.

In 5th grade, students solve division of whole numbers using partial quotients. A huge component of partial quotients is considering the entire quantity of the dividend and the application of place value understanding. For example:

When dividing 691, students should think how many 8s are in all of 691. After working with multiple of 10s in prior opportunities, students will conclude there are about 80 eights in all of 691. Fast forward to 6th grade when students are first introduced to the standard algorithm for division, it’s important to carry that same thinking with a heavy undertone of place value understanding.

In our first session as Great Girl Mathematicians, we reviewed division using partial quotients. By the second session, a young rising 6th grade male joined the group, thus removing Girl from our group name. The second session was spent connecting partial quotients to standard algorithm.

We began our session with the partial quotients strategy, but instead of writing the quotients to the side, we staked them on top as they would see them within the standard algorithm. They could also see the place value connection of the why behind the standard algorithm. I paralleled this approach with the standard algorithm. I explicitly modeled my thinking:

“I notice there are 69 tens in 691. I know 8 x 8 is 64, but I need 64 tens. Therefore, I can multiply 8 by 8 tens which is 64 tens. Because it is 8 tens, I will record the 8 in the tens place of the quotient.”

As I did the subtract, I attended to precision by stating, “69 tens minus 64 tens is 5 tens”. Speaking of it in this manner gives meaning to why the next number is ‘brought down’. In this problem, the next number is in the ones place, resulting in 51. Because 8 x 6 is 48, the 6 is placed in the ones place of the quotient.

As the kids worked through their practice problems, some needed to use the staked partial quotients before rewriting it as the standard algorithm. Others caught on quickly. While others grappled with where to place the numbers in the quotients. Using place value as our guiding thought helped them to make sense of why to record numbers in certain places.

This approach debukes the cheeseburger mnemonic device which causes students to not consider the magnitude of the dividend making the thought “does my answer make sense” seem like a distant after thought. Being explicit about the place value understanding creates the structure to help students make connections between and among concepts.

Once you invent or become open to this method you can see so many paths. 100 8s is too much. If 100×8=800 how much is 50×8? 400. That leaves 291. 25×8 must be 200, leaving 91. 11×8 is close, so 50+25+11…86 and 3/8. And I love how it extends to decimals and fractions, since it’s actual division.

It’s this flexibility in thinking that should be the goal. Not that we teach them as procedures, but that we as teachers create opportunities for students to invent and become open to their own mathematical thinking.