Single Digit Addition Strategies

Learning the strategy types and trajectories can help you:

Look for and select student-generated strategies to discuss in class.
Know when to (ever so gently) nudge students along in their journey.

Before we dig in, a few important notes:

Each strategy represents some form of sophistication and efficiency, even if some strategies are more abstract than others.
Strategies are different than models. One model (a number line, for example) can show many different strategies. Knowing this distinction can help us nudge kids to use more abstract strategies instead of just switching models (unless the change in model reveals an important relationship).
“It’s a strategy if students came up with it but still a procedure if directly taught” – @mathteacherlove. Use your position of authority to orient students towards each other, since learning a strategy from a peer is very different than learning it from the teacher or textbook. (More on that here).
Strategy development is not linear. Students move from one strategy type to another fluidly, sometimes within the same problem. Also, students use different strategy types depending on the numbers (3+5 vs. 3+9), even within the category of “single digit addition.” It is important, therefor, not to label the student (“Evan is a counter”). Instead, label the strategy and consider the context (“Evan is using a counting strategy for 5+3, maybe because counting on 3 times is relatively easy? Let me see what he does with 5+8”).

That said, let’s look at what might dismissively be called “drawing a picture” but is the act of problem solving itself: Direct Modeling. This strategy is “distinguished by the child’s explicit physical representation of each quantity in a problem and the action or relationship involving those quantities before counting the resulting set” (Children’s Mathematics, 2nd Edition p. 29). Let’s use this context to make sense of the strategy types:

You have 8 library books. You check out 7 more books from the library (generous lending policy, no?). How many books do you have now?

If you Directly Model, you might draw 8 circles for the library books you have, 7 circles for the library books you checked out, and count all the circles to find the total. The quantities could also be represented by physical objects, like cubes.

Direct Modeling is sophisticated because:

It represents sense-making. Rather than waiting to be shown how to solve, students are doing what is intuitive and natural to them by drawing or physically modeling the problem.
It reveals important relationships that help us make sense of Derived Facts (more on that in a minute).
Although this strategy is less abstract than others, the act of abstraction is still present – the circles/cubes are a metaphor for the library books.

Next comes Counting Strategies, where students begin to abstract some of the quantities in the problem. For example, they might start at 8 and count on 7 times (“eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen”). Notice that Counting Strategies are built on the foundation of Direct Modeling.

Counting Strategies are sophisticated because:

Students take a step towards abstracting the quantities in the problem. Here, only the 7 is represented fully (in this case on the fingers), the 8 and 15 are abstracted.

Over time, Derived Facts emerge. These strategies are what we think of when we describe “number sense” – a playful flexibility with number relationships that allows students to reconstruct problems to their liking.

For this problem, a student might think “8 is close to 10, it just needs 2 more. I can get 2 from the 7. Now I have 10 and the 5 left over from the 7, so 15!”

Derived Facts are sophisticated because:

Like Direct Modeling, they represent the act of sense-making. “What do I already know that can help me solve this problem?”
Without being shown, students use relationships based on the informal use of the properties of operations. Here, solving 8+7 by thinking of 8+(2+5)=(8+2)+5 uses the associative property of addition. Students don’t need to know the definition or notation to use it (those conventions get in the way of informal intuition in the beginning).
They reinforce flexibility and the understanding of equality. If you know you can change a problem to an equivalent but easier problem (for you), you’re empowered by relational thinking (which leads directly to algebraic reasoning).

After lots of work with number relationships and various levels of abstraction, students will start to Recall some facts. Then more facts. Then all of them. This takes time and (to be blunt) you don’t have much control over it. You can give them lots of opportunities to Directly Model and use Derived Facts, since both support recall.

Eventually, students’ brains will say “Hmm, we seem to be coming across 8+7=15 a lot, time to store it.” When asked, students will have a hard time describing how they arrived at the answer. This is why I stopped asking students to detail a strategy if they say “I just know.” Instead, I give them a related problem to test the flexibility of their new knowledge: “If 8+7 is 15, what is 7+8? What about 8+8? What about 8+9?”

Recall is not a strategy – it is a byproduct of exposure and flexibility. The sophistication of Recall exists in its connection to Derived Facts and Direct Modeling.

Students who forget a previously known fact can dip back into Derived Facts or Direct Modeling (or a quick mental visualization of a Direct Model) to arrive at the answer. This has the added benefit of reinforcing student agency (“If I don’t know something I can figure it out”) and the properties of operations (“I’ll change this into an equivalent problem I can solve”). This eventually happens within 3-5 seconds, meaning that the forgotten fact doesn’t slow the student down. I still regularly dip back into my Derived Facts for multiplication to re-remember 7×8.

Derived Facts usually involve only a few steps and little or no counting. This means lower cognitive load and increased accuracy compared to Counting Strategies and Direct Modeling.

The relationship between Direct Models and Derived Facts is powerful, and is stronger than the connection between Counting Strategies and Derived Facts. In the picture below, it is easier to see (8+2)+5=10+5 in the Direct Model than in the Counting Strategy.

Although Derived Facts and Recall are the ultimate goal, deep understanding can’t be rushed. Use the sophistication of each strategy type to appreciate and leverage students’ strengths. Then, carefully, nudge them to the next level of abstraction.

Questions to consider:

What activities invite students to use strategies that makes sense to them?
What classroom structures give students opportunities to make connections between strategies?
How do you nudge students to try a more abstract strategy without taking over their thinking/agency?
What other Derived Facts might students use for 8+7? Would students use the same derived facts for 5+7, or does that slight number change affect what is “efficient”?