The Power of Routines to Develop Computational Fluency: “Rename The Number” In classrooms in which mathematics is centered on sense making, children can be observed talking to each other about their thinking, debating and analyzing each other's ideas, questioning and challenging each other in positive thought provoking ways. In this way they begin to make sense of numbers and operations and how they relate to one another (number sense). Students are not required to use a particular algorithm when adding or multiplying, but encouraged to explore various effective strategies for operating with numbers. When doing double-­‐digit addition, for example, when adding 49 + 68 most of us were told to start in the one's column, write down the seven and “carry the one,” etc. Many of us believe that algorithms like this one are not only efficient ways of operating on numbers, but that they are the only efficient ways. These beliefs are myths. Algorithms are culturally determined and no one algorithm is used universally. In Common Core State Standards influenced math classes, teachers emphasize sense-­‐making and reasoning and highlight strategies that bring out big ideas and underlying structures in mathematics. instead of memorization and rote practice of procedures. Carefully constructed mathematics routines can be used to develop students’ capacity to reason mathematically and compute fluently. Starting the day with a carefully selected math routine is one way to help students cultivate mathematical fluency. Here are some criteria for selecting activities worth revisiting. The activity: • centers on “big ideas.” • can be instrumental in developing a repertoire of mathematical strategies. • should have multiple access points. • needs to be open-­‐ended so that children can extend the activity as far as they are able. • will sustain interest for l0 to 20 minutes. • can be easily accessed and completed to some meaningful degree when a child arrives late. • can be used as a formative assessment tool. Rename the Number Rename the Number is one activity that meets all the criteria listed above. Select a target number, for example, 30. Ask students to write as many expressions or equations that rename the target number as they can for a short, specified period of time (e.g., 3-­‐5 minutes). Some ways to rename this number might be: 31 – 1, 6 x 5, 100 – 70, and (4+9+3+20) – 6. While the students are working, circulate and observe the various approaches they are taking. Then consider which of the students is doing something worth highlighting in a whole-­‐class discussion. Select one to three student expressions/equations to present to the class. Use these to highlight a mathematical big idea or strategy. For example, if someone wrote, 5 x 6 = 30 and then wrote (5 x 5) + (5 x 1) = 30, you could use this example to discuss the distributive property of multiplication over addition. Or if someone wrote 10 + 10 + 10 = 30 and 3 x 10 = 30, use these equations to discuss the connection between repeated addition and multiplication. 1 The Power of Routines to Develop Computational Fluency: “Rename The Number” When selecting a target number consider the following: Which number to assign and why? In picking a number, its magnitude is one characteristic to consider (e.g., is the target number a 1, 2 or 3-­‐digit number or a fraction?). Another thing to consider is how to use constraints (e.g., you cannot use addition to make the target number). When constraints are imposed any number might be appropriate in upper grades. For example, if you stipulate that students must use division or fractions in their examples, then you have added a constraint that has increased the cognitive demand of the task. When selecting the target number and constraints, be sure to try creating equations/expressions that meet the constraints before offering the task to students. Then predict what expressions students may come up with and which ones might be worth highlighting in order to bring out big ideas and mathematical properties. Access and Challenge Differentiation occurs naturally in Rename the Number. Given the target number 30, a fourth grader might begin with the equation, 29 + l = 30; another might write (l00/l0) + 20 = 30. This means that students’ responses will vary from the basic to the sophisticated. Some will make one or two complex sentences; some will find and record math equations that follow a pattern. Still others will grab a calculator and work with very large numbers or experiment with operations. Thus the activity provides multiple entry points and a high ceiling which are essential when differentiation is a goal. The open-­‐ended format of Rename the Number has advantages over the more traditional forms of putting a few arithmetic problems on the board for students to compute. Because it invites children to play with numbers and relationships, they develop flexibility in how they think about numbers and how they organize their ideas. Rename the Number also allows children to experiment and construct meaning of big mathematical ideas. One that easily arises is equivalence (e.g., 16 x 4 = 8 x 8). Equivalence is the idea that “any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value.” (Charles, 2005 NCSM) Other big ideas that might arise as students play with numbers are: (1) the inverse relationship between addition and subtraction or that between multiplication and division; (2) part-­‐whole relations (e.g., 18 = 17 + 1 = 15 + 3); (3) the commutative property of addition or multiplication (15 + 3 = 3 + 15; 9 x 2 = 2 x 9); and (4) the connection between repeated addition and multiplication (e.g., 30 = 5 + 5 + 5 + 5 + 5 + 5 = 6 x 5). These big ideas sometimes emerge as students find and explore patterns as in the list of equations given below: Examples:
25 + 5 = 30
26 + 4 = 30
27 + 3 = 30
28 + 2 = 30
2 The Power of Routines to Develop Computational Fluency: “Rename The Number” 29 + 1 = 30
30 + 0 = 30
What’s next?
If you follow the pattern 31 + (– 1) = 30 would be next. This might initiate a student wondering about subtraction or negative integers. Assessment Rename the Number, when done in the manner described—namely giving students the opportunity to come up with as many equations as possible within a short period of time, rather than having each class member come up with one way to rename the number, is a useful formative assessment tool. It gives teachers a window into students' thinking and strategies. This in turn provides the teacher with an opportunity to differentiate instruction and/or to use the information for guiding whole-­‐group discussions. If we ask children to keep a math log or journal, each time they engage in Rename the Number, they can date and record their expressions/equations in their notebooks. Over a long period of time, the journal entries can serve as a barometer of a student's progress. Having this kind of long-­‐term assessment can help a teacher maintain a positive perspective on a student’s development, which may be especially helpful when a child is not meeting grade-­‐level expectations. Examine the work of each individual child over time and ask, “What is revealed and when do new strategies begin to emerge?” For example, notice which children think in terms of patterns, which ones work quickly in an organized fashion, which children like to play with numbers and challenge themselves and which students are sticking to simple, straightforward equations, etc. As we learn to examine student work in light of strategies and/or big ideas, we can use this information to assign specific games or tasks to support a child’s development of more complex ideas or to increase fluency (employing differentiation of another sort). Rename the Number used in this manner transforms the role of the teacher into the role of a researcher. Through observing and conferring with students and examination of student work, we learn to assess what each student understands, where the gaps are, and what opportunity we might provide for that student to deepen understanding or bridge a gap in knowledge. This activity can also give a teacher a big picture view of the class as a whole. Examine a class set of papers to get a very clear idea about the range of understanding in the class. Sort the papers into piles based on student thinking to determine the various approaches and strategies presently in use in the class. Then consider which new strategies could be introduced or highlighted or connected and use students’ examples to do this. The Role of Discussion Taking the time to carefully select and discuss children’s equations or expressions is an important component of the activity. Through this interchange of ideas children learn how to articulate their understanding of various math big ideas and strategies. By discussing examples of student-­‐
3 The Power of Routines to Develop Computational Fluency: “Rename The Number” constructed equations that are chosen because they can bring out an important big idea or highlight a more effective strategy, students develop “number sense” and computational fluency. One way to engage in robust discussion is to gather students in a circle after everyone has had a few minutes to construct equations and the teacher has written a pre-­‐selected student equation on the board. The class is asked to determine whether the equation is correct and to convince each other of their findings. If it is not correct, students try to figure out what needs to be done in order to get to the target number. For example, if there was a miscalculation and the answer when checked by the group was really 35 and not 30, students then figure out how to get back to 30 (e.g., subtract 5). The focus is not whether the answer is right or wrong, but on the reasoning and whether or not the strategy is useful mathematically. Another option for this activity is to choose a series of equations to examine. For example, if a student explored a pattern, the teacher might put up a few of the equations and ask the rest of the students to describe the pattern and figure out why the pattern works. Example:
25 + 5 = 30
26 + 4 = 30
27 + 3 = 30
A conversation about the pattern might reveal that the sum stays the same because one addend increases by one while the other decreases by one. This strategy, compensation, is based on two mathematical big ideas: the associative property of addition and equivalence (25 + 5 = 25 + (1 + 4) = (25 + 1) + 4 = 26 + 4) and can be used for making addition problems easier to compute (e.g., when adding 499 + 387, change the 499 to 500 and the 387 to 386!). Focusing discussions on specific strategies for mental computation a teacher can help students understand how to look to the numbers first before picking a computation strategy. By carefully highlighting specific strategies, a teacher develops the class’ collective computational repertoire. Another important strategy to develop is that of constant difference. By increasing or decreasing the minuend and subtrahend by the same amount an infinite number of equations are created with the same difference. This doesn’t seem to be especially significant in the following pattern: 8 – 5 = 3 7 – 4 = 3 6 – 3 = 3 However, continuing the pattern will result in some curious number sentences such as: 2 − −1 = 3 −1− −4 = 3 4 The Power of Routines to Develop Computational Fluency: “Rename The Number” The constant difference strategy is very effective in the following example: 72 – 36 = 69 – 33 Notice how reducing both the minuend and the subtrahend by 3 maintains the distance between these two numbers. The second expression is easier to compute mentally. You can also add 4 to both the minuend and the subtrahend: 72 – 36 = 76 – 40 again resulting in a problem that can be easily solved mentally. On occasion some children will come up with things that are not generally taught at the grade level (e.g., the concept of square roots in fourth grade). Through the activity of Rename the Number, a student one day gave an example in which he used a square root symbol. Apparently, his older sister had showed him the symbol and explained the idea. He then applied the concept correctly in an expression he made up about the target number. When he shared his expression, other children became interested in the idea of square root. Some of them figured out how to appropriately use the symbol and began to incorporate it into their own expressions. Of course, when you start to work with square roots, exponents come into play. That prompted a conversation about multiplying a number times itself, and the various ways one could notate that idea, e.g. 4 x 4 = 4². Additional Ideas for Math Routines Rename the Number does not need to take more than l5 minutes once the routine has been established. If only two or three children share an example each time, that is enough as long as you vary who the children are and focus on developing a repertoire of mathematical strategies. This is especially true if the children are recording their math sentences in their math journals because you can look over their work whenever you choose. There is no need to correct each sentence or even look at every math log entry. The logs can be thought of as “scrap paper” and the time as thinking time rather than as finished products that demonstrate performance. Select entries to focus on and make sure that you have looked at each journal at least once every two weeks to see how students are progressing and what new strategies and understandings are showing up in their work. Conclusion Math routines can become an enriching part of your mathematics program when they are carefully crafted, open-­‐ended, have multiple entry points, revolve around big ideas and the development of mathematical strategies. Rename the number is one example of a routine that meets these criteria. Lucy West Antonia Cameron Anne Burgunder 5