Learning Basic Single-Digit Combinations (Facts): Developing Important Ideas in Mathematics; Part I – The Case

Posted by on Sep 1, 2017 in Elementary | One Comment

Submitted by:
James Brickwedde, MCTM VP Elementary
Project for Elementary Mathematics, www.projectmath.net

This is the first in a series of articles that proposes a different instructional approach and purpose for becoming fluent in recalling basic single-digit combinations in all four operations. Becoming fast and accurate in recalling these combinations by rote has been a long time scared cow once students enter second grade. Many times, young children have been excluded from working in more enriched mathematical settings based on slow performance on timed tests assessing the capacity to recall these combinations. What I hope to persuade you of over the course of these articles this year is, by taking an algebraic approach towards the rich mathematical ideas that underlay single-digit combinations, you are helping students develop effective strategies in deriving which in turn are the exact strategies needed when operating efficiently with multidigit numbers.

Take the following mathematical expression:

7 + 9

This is a combination many children take a long time committing to memory. Consider how posing the following number string to students might help them develop a strategy to effectively solve for the sum of 7 + 9 en route to fluency recall.

7 + 9 = 7 + 10 – 1     True or False? Why true or why false?

7 + 9 = 10 + 9 – ☐   What needs to go in the box to make the number sentence true?

7 + 9 = 7 + 7 + ☐

7 + 9 = 6 + 10           True or false? Why true or why false?

7 + 9 = ☐                 Which strategy from above can you use to figure out 7 + 9?

Historically, teachers in the United States have sent the basic single-digit combinations home with students with the directive to memorize followed with timed tests to assess the students’ capacity to simply recall them. Teachers frequently complain that students who don’t know their combinations fall back to counting on their fingers to determine the result. I have been in and around elementary classrooms for 28 years. This is an on-going complaint that precedes my time as a teacher. I fully agree that the more fluent students are in recalling the single-digit combinations the easier time they have when operating with multidigit numbers. The professional question is, how do we get children to attain this level of recall?

A Learning Theory Perspective

A little visit with learning theory and what research has revealed is worth a moment’s time. Researchers have demonstrated that children progress through developmental stages of understanding and strategy use (Carpenter et al. 2015). Children progress through the stages of direct modeling (a concrete level where two blocks and five blocks are laid out then all the blocks are counted starting from one) and counting on (initially starting from 2 then counting on five more numbers; later on being able to count on from the larger starting with five then counting two more numbers). The question is, what is the next stage? To understand next progressions, let’s revisit the notion of ‘the zone of proximal development.’

Think of what many elementary teachers have come to view regarding children’s developmental progressions in literacy development. The books children read independently are at their comfort level. That is different from the texts used during instruction that are best set at the student’s next developmental level. Read aloud literature is often a level ahead of the instructional level with the intent to develop children’s oral comprehension. This ‘zone of proximal development’ that we use in literacy holds true in strategy development in mathematics. If I don’t know 7 + 9 at a recall level yet, I will revert to my comfort level to solve the problem. If counting on is the only other strategy that I know, then it is that level to which I will revert. So what is in between counting on and fact recall, or in the case of multiplication, skip counting and fact recall?

Carpenter & Moser (1983) in their work with children articulated what some children start to do on their own: use deriving strategies. Derived strategies are based on the mathematical principles of decomposition of number and use of relational thinking to use a known single-digit combination to figure out the unknown combination. The combinations posed earlier are examples of derived strategies. View the following video of a group of first graders working with a number string that has as its initial focus using the doubles plus and minus one strategy.

There are several things to notice in this episode. Jamie clearly knows intuitively that 6 + 7 is thirteen and intuitively did so knowing that 6 + 6 + 1 is the same as 6 + 7. However, when asked explicitly to state where the other six was coming from, his ability to place into language his intuitive thoughts reached a limit. It was through the voice of Sophie making the decomposition of seven into 6 + 1 that Jamie was able to continue explaining his thinking. The teacher’s role of capturing the student’s thinking in a combination of informal and formal notation allows the students to consider the trace of their strategies, giving a visual form to wrap their voice around.

Further on in the episode, even though the instructional intent was to foster a conversation around the doubles plus or minus one strategy, Grace articulates a Make a Ten strategy. She decomposes one of the sixes into three plus three in order to combine 3 + 7 to create a ten, then follows adding the other 3 to 10 to find the answer 13. What is intentional about the number string presented to the students is the goal of explicitly exploring and nurturing derived strategies en route to being able to recall the fact combinations. Therefore, if I don’t automatically know the sum of 7 + 9 yet, and if I have been encouraged to make derived strategies my comfort zone, instead of falling back on counting strategies, deriving becomes the my fall back strategy. The mathematical processes used in deriving are far more advanced than those used in counting on.

What are the benefits?

So why not continue to just have students go home and memorize these combinations? Isn’t spending time on derived fact strategies just a waste of class time? As to the former, America has been asking students to memorize for rote recall for generations and it only works for the few. Too many students, who otherwise have decent math skills, do not succeed with this approach and end up with extreme anxiety issues around math in their daily lives. As to the latter response, becoming skilled in using derived strategies with single-digit combinations are the very strategies that students need to use to be fluent with operating on multidigit numbers. The strategies also draw out the algebraic properties of operations. Therefore, deriving strategies develop build transferable math skills useful for more sophisticated mathematics encountered later in their learning.

Consider this number string designed around the core strategy of Make a Ten.

7 + 3

7 + 5 = 7 + 3 + ☐      How can you use what you know about 7 + 3 to help you figure out 7 + 5?

17 + 5                 How can the same strategy used for 7 + 5 be used to solve 17 + 5?

47 + 5

7 + 5

Beyond developing a specific strategy to learn 7 + 5, students are also developing fluency around decomposing numbers into relevant subparts, and they are using the associative property of addition. They are also seeing how the relationships of strategies used for single digit combinations can be used with multidigit combinations. The ideas and skills are both connected and generative.

Fluency Standard

According to the research by Kamii (2000), a student who gives the total to a single-digit combination in two seconds or less is doing so at the recall level. If more than two seconds, the student is using some level of calculation. The question to be investigated with the student is what strategy is being used to calculate. Carpenter et al. (2015) in their work have found that students who are fluent in recalling single-digit combinations relied on derived strategies to get to the recall level. Those derived strategies are so well practiced that the combinations are answered within two seconds or less.

Coming Up in Math Bits

Next month, multiplication strategies will be explored. In future months, I will return to subtraction, division, more specific connections to multidigit operations, and homework support. A focus will be on using number strings, not only in number talk situations, but also integrated into instruction, seatwork, homework, games, etc. What is the role of time tests, and at what point in time are they useful? Where I can, I will provide some video clips that capture students thinking this way.


Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L. & Empson, S.B. (2015). Children’s Mathematics: Cognitively Guided Instruction, 2nd Edition. Portsmouth, NH: Heinemann.

Carpenter, T.P., Franke, M.L. & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School. Portsmouth, NH: Heinemann.

Kamii, C. (2000). Young Children Reinvent Arithmetic: Implication of Piaget’s Theory, 2nd Edition. New York: Teacher’s College Press.

1 Comment

  1. Sarah Johnston
    September 4, 2017

    Hi Jim,

    I enjoyed your succinct article about number strings and derived facts. With school starting tomorrow, it is a nice reminder for me about best practice. It was fun to see you teaching your first graders so long ago. I remember Jamie from seeing him before in your videos.

    I will look forward to the next article, and I’ll see you on Sept. 27.



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