Learning Basic Single-Digit Combinations (Facts): Developing Important Ideas in Mathematics; Part II – Developing Number Strings to Build Derived Strategies

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

Last month, I laid out the reasoning behind a different approach towards learning the single-digit combinations with sums between 1 and 20. This position is based upon three core ideas from research. The first is that if an individual does is not able to recall instantly a particular combination, he or she will revert to a comfort level strategy to calculate the answer. Due to most generational practices around learning these single-digit combinations here in the United States, that comfort level strategy is counting on or back on one’s fingers. Related to this reference to learning theory is the second core idea. There is a developmental strategy level between counting on/back and being able to recall combinations automatically. That stage of development is derived strategies. This means that significant instructional time needs to be spent nurturing derived strategies with students at an instructional level so that these new strategies become the students new comfort level. Thirdly, these derived strategies are the very strategies students need to develop to be fluent decomposing, reconfiguring, and operating on multi-digit combinations and key to working with fractions. Derived strategies draw out the algebraic properties and relational thinking around equivalencies that underlay the four operations. An instructional focus on derived strategies, therefore, develop multiple concepts and skills that students need as they progress through future levels of mathematics.

Figure 1: Developmental Levels of Children’s Solution Strategies (Carpenter, et al. 2015)

This month, I would like to concentrate on ways to develop number strings with students. Multiplication will be featured as the operation around which these number strings can be formed. The core ideas expressed here are readily extended to the other three operations. The ideas to ruminate on are, a) how to initiate a string to gain maximum participation, b) how to foster the relational thinking from one element of the string to the next, c) how to capture student thinking through various forms of representation, and d) how to use questions and prompts that elicits and clarifies student thinking.

To begin, view the video by clicking on the link.

As I have come to develop these sequences for my students (who now include adults), I begin with a combination that will gain maximum participation among the students. In this instance, the string began with the square 7 x 7. The products of square combinations such as & x 7 are often known before some of the neighboring combination. Beginning with five combinations (x 5), doubles (x 2), and tens (x 10) are key examples of where to start in order to gain maximum participation and to begin to stretch relationally to harder neighboring combinations. The next in this video string was 7 x 9 as this is a combination that students take a long time in gaining automaticity. The initial response by Milda (64) is an example. Think about the products of 48, 49, 54, 56, 63, and 64. Which combination(s) goes with which product is difficult for students to sort through if a strict rote memorization process is relied upon to recall each individual combination. Some teachers, as a result of well intentioned reasons, teach child mnemonic devices and chants to recall specific combinations. Erika is an example of a student having been taught the nine fingers trick to recall 7 x 9. While she got a correct answer, she was not engaged in mathematics. That memory device is limited in scope and is not foundational for any future mathematics.

The prompt specifically posed to students was If you know what 7 x 7 is, what would be 7 x 9? This prompt is mathematically aimed at strategy development whether or not you already know the combination at an automatic level. Such prompts foster relational thinking such that a network of related strategies combination to allow automaticity in recalling a product. As the conversation in the video unfolded, more than one strategy emerges, not just those based on 7 x 7. Nevertheless, 7 x 7 serves as a catalyst for those students who need to move off of skip counting beginning from zero.

Raleigh is able to articulate that seven nines is the same as having seven sevens plus two more sevens. Captured on the board by the teacher, this equation traces Raleigh’s thinking: 7 x 9 = 7 x 7 + 7 x 2. One could further clarify his thinking by inserting the unspoken decomposition of nine being the same as seven plus two as in 7 x 9 = 7 x (7 + 2) = 7 x 7 + 7 x 2. That is an in-the-moment decision that you as the teacher need to make. What Raleigh demonstrated confidently, whether he is explicitly aware of it yet or not, is the distributive property of multiplication over addition. The representation serves not only for Raleigh as a means to clarify his thinking but serves as a means for others in the classroom to reflect visually upon his verbal descriptions and make mathematical sense of his work. Adding to the representation is a decision based on your reading the level of understanding that appears evident among those listeners.

Ed follows Raleigh with a compensation strategy relating 7 x 9 to 7 x 10. Ed uses the articulates that 7 x 9 = 7 x 10 – 7 x 1 as he describes that he needed to subtract “one seven times” to end up with seven nines. The representation placed on the board in this episode shows 10 x 7 – 7. Whether or not to use “- 7” or “1 x 7” again depends upon the novelty of the strategy among the class members, in which case 1 x 7 adds clarity, or if that has become collectively understood, in which
Nevertheless, compare the foundational use of the distributive property of multiplication over addition, and compensation strategy used by these two students with the limited capacity of teaching children mnemonic devices. To use an analogy based on the Three Little Pigs, Raleigh’s and Ed’s thinking allows each of them to build a house of brick, where the past teacher’s good intentions to teach mnemonic devices leads to a house of straw.

The third element in this string was 7 x 16. This is not a single-digit combination. However, extending a string to just outside the range of single-digit recall fosters the understanding that the strategies used to derived the single-digit combinations are the very strategies needed to work with multi-digit combinations. This nurtures the relational thinking that underpins the associations among a network of mathematical ideas.

Devan responds with the strategy of 7 x 10 + 7 X 6, a use of the distributive property of multiplication over addition by decomposing the sixteen into its place value components (16 = 10 + 6). Celeste uses the information from the two previous expressions that combine to make sixteen; 7 x 16 = 7 x (7 + 9) = 7 x 7 + 7 x 9. Emma uses the first element in the string, 7 x 7, uses that twice, then understands that she still needs two more sevens to combine for a total of 16 groups (16 = 7 + 7 + 2). Celeste’s and Emma’s strategies both represent the use of the distributive property of multiplication over addition, just different decompositions than Devan’s. Both, however, were built relationally from previous pieces of information. Sachin uses the compensation strategy that is built around 7 x 20 – 7 x 4. He builds up to seven twenties via knowing seven tens and doing that twice.

The richness of strategies and the evidence of algebraic and relational thinking is strong among those students that shared in this video clip. The number string could have taken other directions. With the formation of any future string that you design on your own, the main element is what is the mathematical goal, how to start to gain maximum participation, and how do to begin to stretch the next items in the string to foster relational thinking. While your string may be designed to foster a conversation around a particular strategy, being open to any and all strategies is essential. What relationship is evident to one student is not necessarily the relationship seen by another. The diversity of thinking is to be celebrated. The ultimate goal is to scaffold students off of counting by ones or skip counting from zero to become comfortable with organizing around key landmarks. These derived strategies build the very foundation to then begin operating with multi-digit numbers. In essence, use what you know to figure out what you don’t know. That is a key element for progressing further in mathematics!