# Teachable Moments in Math

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## The Equal Sign

*Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false*. (1.OA.D.7)

*x*+ 5 = 11 −

*x*make sense and open the door to new strategies for solving complex problems.

*Why this concept is important*. Understanding that the equal sign signals a relationship between quantities, that it isn't just a prompt to "give the answer," has been shown to offer lasting benefits as students move into the more abstract areas of mathematics in algebra and beyond (Knuth, Stephens, McNeil, & Alibali, 2006). Teachers can use discussions like the one described in the opening vignette to uncover misconceptions about this symbol and address them before they become an impediment to future learning. Engaging in frequent discussions about the symbol and its meaning in the early grades is crucial because developing a comprehensive understanding of the equal sign is a complex process that happens over time, not in a single lesson (Carpenter, Franke, & Levi, 2003).

*How you can enhance it*. Primary grade teachers can help pave the way for a smooth transition into algebra by making two shifts in their day-to-day practice.

*equals*with synonyms such as

*is the same as, has the same value as, balances*, or

*is worth the same as*. Make a poster with the equal sign in the center and a web of synonyms around it, and include a drawing of a balance scale or teeter-totter as a visual reminder. Keep this poster in a prominent place as a reminder for students to use a wide range of terms as they build a deep understanding of the symbol.

## Cardinality

*Understand the relationship between numbers and quantities; connect counting to cardinality*. (K.CC.B.4)

*Why this concept is important*. Cardinality is the ability to bring meaning to the counting process. It opens the door to using numbers for describing and comparing and lays the foundation for combining (adding) and separating (subtracting). Many of the early strategies children will develop for solving addition and subtraction problems rely on a meaningful understanding of counting (Clements & Sarama, 2009; Cross, Woods, & Schweingruber, 2009).

*How you can enhance it*. Two instructional shifts are helpful here. First, don't assume that a child's ability to recite numbers in order means that he or she comprehends counting and quantity. To find out, ask, "How many?" after each counting task, even when modeling those tasks. When taking the lunch count in the morning, count the raised hands—and then be sure to ask, "How many children ordered hot lunch today?"

## Properties

*Apply properties of operations as strategies to add and subtract*. (This skill appears in several 1st and 2nd grade standards.)

*Why this concept is important*. Simplifying calculations isn't the only benefit of knowing and using properties. Mathematical Practice 7 states that "mathematically proficient students look closely to discern a pattern or structure." Students who develop a habit of mind for problem solving that includes reflection and planning ahead will be able to use this skill to great advantage throughout their mathematical careers. Students without this capacity have a tendency to plunge headlong into every problem without first taking a step back to identify the goal and consider multiple solution paths. Whether the context is single-digit addition or more advanced topics, making and using generalizations in clever ways to simplify seemingly complicated problems are essential problem-solving skills.

*How you can enhance it*. The primary instructional shift here is to create regular opportunities for students to make and test generalizations about numbers and operations. For example, pose a set of problems containing number pairs like 5 + 2 = ? and 2 + 5 = ? Observe students as they solve them, and watch for a student who immediately knows the answer to the second problem after solving the first. Ask this student to share his or her strategy with the class, and see who agrees or disagrees. Follow up with questions like, "Will this always be true, even for large numbers? How could we know for sure? Is this true for subtraction, too?"

## Composing and Decomposing

*six*Common Core standards and across

*three*mathematical domains in the K–2 standards is a strong indication of their importance.

#### Figure 1. Decomposing and Recomposing Geometric Shapes

*compose*and

*decompose*to describe similar thinking applied to quantities. Consider the problem 27 + 19 = ___. A student who can compose and decompose numbers could solve this problem by breaking the numbers apart and putting them back together in convenient and clever ways (see fig. 2).

#### Figure 2. Decomposing and Recomposing Numbers

*Why this concept is important*. Students who develop flexible thinking about numbers early in their schooling are poised to develop complex mathematical thinking as they progress through the grades. Students who can decompose and recompose numbers see many options when presented with a challenging computational problem. Students without this ability typically have only one way to approach it: They line up the numbers vertically and follow a memorized procedure. The ability to compose and decompose numbers also enhances students' knowledge of place value and landmark numbers, such as 25 and 75, and potentially even their reasoning with negative numbers.

*How you can enhance it*. Use every opportunity to encourage flexible thinking about numbers. Ask, "What's another way to think of that number? How could we break that number into simpler pieces?" Pose number riddles with clues like, "I'm thinking of a number that's made of 4 tens and 7 ones. What's my number?" Or, "If you take my number apart one way, you can see 25 and 25 and 5. If you take it apart another way, you can see 40 and 15. What's my number?" Emphasize finding combinations that use "friendly numbers," like decade numbers (multiples of 10), and easy referents, like 25 or 75.

## Unknowns

*Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions*. … (1.OA.A1)

*unknown*is associated with variables in algebra, but that's not the intent in this standard. The point here is to ensure that students go beyond solving the traditional, straightforward word problem formats so often found in textbooks.

*Result unknown*: "Dina had 12 marbles. She gave her cousin 7 marbles. How many marbles does Dina have left?"*Change unknown*: "Dina had 12 marbles. She gave her cousin some marbles. Now Dina has 5 marbles. How many marbles did Dina give her cousin?"*Start unknown*: "Dina had some marbles. She gave her cousin 7 marbles. Now Dina has 5 marbles left in her bag. How many marbles did Dina have at the start?"

*Why this concept is important*. Problems with unknowns in different positions promote multifaceted understandings of relationships among quantities and encourage the development of robust problem-solving skills. When children begin solving word problems, they typically use cubes, drawings, or fingers to represent and act out the situation. A "result unknown" problem (like the original marbles problem) lends itself to this strategy.

*How you can enhance it*. Shift the emphasis for solving word problems from a routine approach in which most of the problems follow a formula or pattern to one in which there's plenty of variation in the types of problems posed. Students will develop new solution strategies when they can't solve the problems using their original methods (Carpenter, Fennema, Franke, Levi, & Empson, 1999).

## First, Understand

*Authors' note*: The standards quoted in this article are from National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010).

*Common Core Standard Standards for Mathematics*. Washington, DC: Authors. Retrieved from www.corestandards.org/Math </ATTRIB>

##### References

ASCD. (2012). *Fulfilling the promise of the Common Core State Standards: Moving from adoption to implementation to sustainability*. Alexandria, VA: Author. Retrieved from http://educore.ascd.org/resource/Download/1d60f46d-b786-41d1-b059-95a7c4eda420

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. (1999). *Children's mathematics: Cognitively guided instruction*. Portsmouth, NH: Heinemann.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). *Thinking mathematically: Integrating arithmetic and algebra in elementary school*: Portsmouth, NH: Heinemann.

Clements, D. H., & Sarama, J. A. (2009). *Learning and teaching early math: The learning trajectories approach*: New York: Routledge.

Cross, C. T., Woods, T. A., & Schweingruber, H. (2009). *Mathematics learning in early childhood: Paths toward excellence and equity*. Washington, DC: National Research Council.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. *Journal for Research in Mathematics Education, 36*(4), 297–312.