# The Arrow Way

On the way home tonight, my second-grade son Aaron starting talking about something he’s learning called The Arrow Way. It’s part of the EngageNY curriculum.

There’s a lot of positives in this representation, especially when it’s also coupled with number lines. But why couldn’t the problem above just skip the words “using the arrow way”? Or in problems like this, why require students to use this method?

The result is that some kids think this is a “rule” or “law” instead of a representation/aid/solving method. This can cause misconceptions as kids focus on weird details like what is and isn’t allowed in the representation, rather than on tying the representation to existing knowledge and problems. The visual is nice, but once it’s named, it becomes as important as the other named things kids know in math, and at this point there aren’t many.

Here’s the conversation with Aaron (age 7).

“We have this new rule, it’s called The Arrow Way. The arrow way isn’t a story, it’s a math law.”

“What is it?”

“I don’t know.” (laughing)

“Come on, tell me for real.”

“The rule is +1, -1, +10, -10, and you gotta put one of these above the arrow.”

“So if I started with 50, what could I do?”

“Add 10 to make 60!” (He asked me to make the 60 bigger, “like I’m shouting.”)

“Okay. What else could you do, starting from 50?”

“Minus 10 to make 40! Or you could minus 1 to make 49! And there’s nothing else.”

“Ok. Could you start with 50 and make 70?”

“No. You can’t add 20, because it’s not in the rule.”

“Could you add 100?”

“No! That’s not in the rule!”

“Could you use more than one arrow to make 70?”

“No! Not at all.” (laughing)

“Can you use more than one arrow together?”

“I don’t know, but you can only start from zero, in the rule.”

“Ok. Is that true? Why can you only start from zero?”

“Because that’s a math law, also.”

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About Bowen Kerins
Bowen is a mathematics curriculum writer. He is a lead author of CME Project, a high school curriculum focused on mathematical habits of mind. Bowen leads professional development nationally, primarily on how math content can be taught with a focus on higher-level goals. Bowen is also a champion pinball player and once won \$1,000 for knowing the number of degrees in a right angle.

### 5 Responses to The Arrow Way

1. I see both sides of this argument. If providing multiple strategies, then isn’t there value in requiring some limited forced practice of the strategy to confirm that students have an opportunity to use the strategy enough times to begin to understand why it is useful? It is clear he doesn’t understand it now but that means that if he wasn’t asked to use the strategy explicitly he wouldn’t, so he would never have the opportunity to figure out better how it works.

I’m actually thinking of the proof analysis strategies provided in CME geometry. There are definitely problems in there that say “use a reverse list to plan this proof”. I actually did not force students to use proof blocks and flowcharts this year as many times as I should have and now I think some students who could really benefit from that organization system have missed the opportunity to really see how powerful the strategy can be.

At the same time you are absolutely right that we don’t want students losing the forest for the trees. I definitely had some students trying to use proof blocks and flow charts far longer than they should have despite the strategy not really working for their thought process. Those I had to assure that the strategy itself didn’t matter. It is a tricky balance.

• Bowen Kerins says:

One aspect of this I haven’t explored yet is why he thinks of this as a “rule” and a “law” instead of a “method” (EngageNY later calls it a “simplifying strategy”). I would have expected him to know the difference, maybe he doesn’t. I agree with you that some practice with a new method is needed, but how long is a very difficult question.

Interestingly he’ll solve 49 + 23 as 49 + 1 + 22, using a strategy to make a ten, but isn’t tying that or previous number-line work to The Arrow Way.

2. Maya Quinn says:

How does he compute 15 + 15?

How does he compute 18 + 12?

Suppose you suggest a new method, called the Squiggly Way.

In the arrow way, you can add or subtract:
1, 10 (which is 10*1), or 100 (which is 10*10).

In the Squiggly Way, you can add or subtract:
1, 6 (which is 6*1), or 36 (which is 6*6).

Is the Squiggly Way better than the Arrow Way?
Is the Arrow Way better than the Squiggly Way?
Are they both equally good?
How can you tell?
Why x4?

• Bowen Kerins says:

I’m pretty sure the Arrow Way is better than the Squiggly Way simply due to the correspondence with base-ten place value. I would not consider these to be equally good unless I had three fingers on each hand.

In computing 18 + 12, my son will usually do 8 + 2 = 10 then add that ten to the total of the other tens to get 30. Other times he will break the 12 and do 18 + 2 = 20 then 20 + 10 = 30.

• Maya Quinn says:

Sorry, I should have provided more context: The point of asking the Squiggly Way question(s) was precisely an attempt to emphasize the importance of base-10 properties in using the Arrow Way (whereas a Way based on powers of 6 is not as useful for base-10 arithmetic).

I was not directing the questions at you, but rather suggesting them as possible questions for a student who is learning about the importance of decomposing numbers by place value in the context of using the Arrow Way.