Re-inventing Mathematics Intervention: Focus on Conceptual Development by Juli Dixon [Part 2 of Re-inventing Intervention]

The first of Six Features for Re-inventing Intervention is to focus on conceptual development. The argument for the six features is laid out in the introduction to this series. I encourage you to start with the first blog in the sequence if you haven’t read it already. In that blog, I shared an example of a 4th grade intervention group where the teacher presents students with the following multiplication example and talks through it with language typically used when multiplying multidigit numbers.

Prior to reading on, think about what language you would use as you solved this problem by applying the standard algorithm if nobody was listening. What follows will be much more impactful if you pause and follow through with this request.

Here is what I would likely say:

Two times 6 is 12, put down the 2 and carry the 1. Two times 8 is 16, add the 1 to get 17. Put down the 7 and carry the 1. Two times 4 is 8, plus 1 is 9. Put down a 0. One times 6 is 6. One times 8 is 8. One times 4 is 4. Two plus 0 is 2 and 7 plus 6 is 13. Put down the 3 and carry the 1. One plus 9 is 1, plus 8 is 18. Put down the 8 carry the 1. One plus 4 is 5.

Are you surprised? I spend my academic life focused on sense making in mathematics, and then when I perform the standard algorithm for multidigit multiplication when nobody is listening, the language I use for myself is basically nonsense. I start out fine when I multiply 2 times 6 to get 12. However, what does it mean to “put down the 2 and carry the 1”? Is it a 1? How do you “carry” a number? This language is confusing at best and inaccurate at worst. If the goal is to focus on sense making and I use this language with students, I am not supporting them to understand the process of multidigit multiplication1.

Should I revise the self-talk I am using? My response might seem to contradict my position, but I don’t think I need to change it. The reason I think my self-talk is okay is because I already understand the algorithm. I know that when I say “carry” I really mean that I am regrouping ten ones to make one ten. I am recording the remaining ones in the ones column and the one ten in the tens column to add to the tens. It is more efficient for me to say “carry the one” than all of that.

Efficiency is the issue though. In our rush to “catch our students up” we are jumping to the efficient language too early. This language should evolve from an understanding of the standard algorithm. When students are in an intervention setting and focused on multidigit multiplication, it is likely that they have already demonstrated struggles with this algorithm. Saying it more often and more slowly in the same way students sat through in whole-class instruction is not the answer. However, we also can’t use our own, adult self-talk as teacher talk. Our teacher talk, at least initially, must be grounded in conceptual understanding. Consider the difference as I talk through the algorithm using teacher talk.

I am multiplying 486 by 12. I think of 12 as 2 ones and 1 ten. I start by multiplying 486 by 2. Two groups of 6 ones is 12 ones. That can be regrouped as 1 ten and 2 ones. I record the 2 in the ones place and I record the 1 ten in the tens column to combine with the tens after I multiply them. Two groups of 8 tens is 16 tens. I add the recorded ten from earlier to get 17 tens in all. Seventeen tens can be regrouped as 1 hundred and 7 tens. I record the 7 tens in the tens place, and I record the 1 hundred in the hundreds column to combine with the hundreds after I multiply them. Two groups of 4 hundreds is 8 hundreds. I add the recorded hundred from earlier to get 9 hundreds in all. I record 9 hundreds in the hundreds place.

Now I multiply 486 by ten. Ten times 6 ones is 60 or 6 tens and 0 ones. I record the 6 in the tens place and the 0 in the ones place. Ten times 8 tens is 8 hundreds. I record the 8 in the hundreds place. Ten times 4 hundreds is 4 thousands. I record the 4 in the thousands place. Now I add 971 + 4860 and regroup as necessary. My finally product is 5832.

Using conceptually based language to describe the multidigit multiplication algorithm is a lot! It probably took twice as long, and that’s without describing the process of adding the partial products 971 and 4860. No wonder so many teachers skip this stage of the process. This leads us to reconsider the point of intervention. Is the goal to promote efficiency or understanding? The answer should be both. What is important to consider is the order with which they are addressed. We need to begin with understanding. Focusing on understanding is typically inconsistent with efficiency of language. It is often cumbersome and slow. But it is necessary. Returning to our multiplication example, to support students to make sense of multidigit multiplication, it is often helpful to back up and focus on sense making with single-digit multiplication and use that to lead into multidigit multiplication. For example, we can make sense of 2 x 6 as two groups of six. What about 2 x 86? This can be thought of as two groups of 86. We can break apart the 86 to find two groups of 6 and two groups of 80 then combine them. This extends nicely to 2 x 486. But what about 12 x 486. This is more complicated. Can we break apart both the 12 and the 486? Absolutely. That is what happens with the multidigit multiplication algorithm when we make sense of it. This is supported by using base ten blocks and the area model for multiplication.

Once students understand what is happening when we multiply the partial products (the 486 by 2 ones then the 1 ten in the task included here), the focus can shift to connecting the process to a more efficient solution path. The next blog in this series will address the need to be explicit when connecting conceptual development to procedural fluency. Stay tuned! In the meantime, please share your feedback on twitter @thestrokeofluck!

1 My colleagues and I discuss the conceptual underpinnings of multidigit multiplication in depth in chapter 2 of Making Sense of Mathematics for Teaching Grades 3-5 (Dixon, Nolan, Adams, Tobias, & Barmoha, 2016).