Monthly Archives: November 2021

Turtle Heads All the Way Down

Yesterday I shared the following thread on Twitter about an exchange between me and my daughter:

The tl;dr version is that my daughter was introduced to something called “turtle head multiplication” – something I am well aware of and that makes me cringe – and I explained to her why I want her to focus her time and energy thinking about *math* while she’s doing math rather than getting distracted by stories of turtles, butterflies, or cowboys on horses eating donuts.

So what is turtle head multiplication?

I took a screenshot from this YouTube video

Essentially, it’s the standard US algorithm for multiplying multi-digit numbers. I have nothing against the standard algorithm. What I dislike is dressing it up in a story that distracts from learning actual mathematics, such as why the 0 (not an egg) is needed. It’s not an egg, nor is it a “magic 0” (another name I’ve heard it called). Rather, it’s because when you multiply 94 × 3, you’re really multiplying 94 × *30*. That product is going to be 10 times greater than the product of 94 × 3, so all of the digits in 242 shift one place to the left:

  • 2 hundreds become 2 thousands (200 × 10 = 2,000)
  • 8 tens become 8 hundreds (80 × 10 = 80)
  • 2 ones become 2 tens (2 × 10 = 20)

The digit in the ones place was shifted into the tens place, so now there are 0 ones left, which is why the product 2,820 ends in a 0. This is also referred to as a placeholder 0 because it helps us accurately recognize the values of the other digits. We use placeholder 0s in all sorts of numbers, such as:

  • 302
  • 320
  • 3,000,200

Why do some teachers teach turtle head multiplication?

I believe teachers introduce it with the best of intentions because teaching the standard US algorithm for multiplication is hard.* They have likely tried teaching the standard US algorithm in the past, noticed common mistakes students make, and eventually looked to the internet or a fellow teacher for advice about making it easier for students to remember and follow the steps of the algorithm.

* If it’s taught in isolation from other ways of thinking or working with numbers. If it’s taught as a series of steps without connection to other concepts such as place value or other strategies such as partial products.

I appreciate these teachers being reflective and looking for ways to support their students. That I love. However, the small change I want to suggest is bringing these questions along whenever a teacher is looking for tips or advice about how to support students:

  • Is this idea I’m hearing about, and potentially going to implement, focusing on mathematics, or is it distracting from mathematics with a cute picture/rhyme/story?
  • If my students are struggling with remembering steps, why might that be? What mathematical ideas might they not understand that is making these steps hard to remember?

In the case of turtle head multiplication, I would say it distracts from the mathematics. If students are telling a story about a turtle head (not even a body!) wearing a collar and laying an egg, then we’re not focusing our limited and precious time with our students on actual mathematics. We’re not stopping and identifying the important mathematical ideas our students need to go deeper with that will help them make more sense of the algorithm and remember the steps they need to take (and why they’re taking them!).

If not turtle head multiplication, then what?

I’m going to pause here, but I plan to write a follow up post shortly providing one answer to this question. I’ll give you a hint that it has to do with having a clear progression of understandings and experiences that build students’ knowledge of multiplying with multi-digit numbers.