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CONNECTING PROCEDURAL FLUENCY AND CONCEPTUAL UNDERSTANDINGWith computation, traditionally, and in today’s mathematics classrooms, there seems to be a lot of dependence on memorization that has shortcircuited the learning of fundamental arithmetic operations and the development of analytical skills and the use of properties. Facility in multiplication involves both an understanding of the concepts and memorization of the facts. While some experts agree that students must quickly retrieve multiplication products from memory, current research draws varied conclusions concerning the effectiveness of various approaches to helping students memorize multiplication facts. It is important and helpful for students to learn the basic mathematical facts including multiplication, but they should also understand the meaning of the basic operations involved in the process before being expected to have fact fluency (Van de Walle, Karp, & BayWilliams, 2014). NCTM affirms that developing number sense, gaining fact fluency and understanding arithmetic operations through conceptual understanding is a critical aspect of the elementary mathematics curriculum. Here are a few examples: “The curriculum should focus on the development of understanding, not on the rote memorization of formula.” “The 912 standards call for a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures and by proficiency with paperandpencil skills to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving.” “Classroom observations should gather information about whether mathematics is portrayed as an integrated body of logically related topics as opposed to a collection of arbitrary rules that students must memorize.” Employing such alternate ways to teach multiplication in the elementary grades can help reinforce conceptual understanding. For example, one may employ pictorial representations to review, demonstrate, and teach multiplication in elementary classrooms. This can help facilitate discussions about strategies for finding an answer prior to asking them to memorize. Such an approach can not only help to strengthen the skills of weaker students who often struggle to do multiplication through memorization but also help the students who often just memorize to visualize and verify their answer. Representations that illustrate multiplication and division. A representation that has generality is one that has a lot of general use (has versatility) to represent different mathematics contexts. A number line is a representation that allows one to use it for counting and cardinality, to represent addition and subtraction, and for many other purposes. In multiplication, we can illustrate the interpretation that multiplication is repeated addition. This number line representation can help illustrate the commutative property with jumps as shown in the figure: Four jumps of 3 is 12, and three jumps of 4 is also 12 (see Figure 4.5). Figure 4.5 Using a number line to represent multiplication. Source: Authors. MultiplicationArea Model. The array and area model can be a great way to teach multiplication and has great connections to measurement and geometry concepts of considering the length and width of rectangular shapes. Using the area model students can look for patterns in multiplication and the commutative property. Using a giant multiplication chart, students can build the arrays for the facts and look for patterns in numbers. Think about it! What opportunities for SenseMaking and Pattern Seeking are afforded with this visual below? Figure 4.6 Using the giant multiplication to look for visual and numeric patterns. Source: Authors. One of the activities that many teachers engage in is creating a giant multiplication chart to actually show students that they do not have to memorize all 100 facts (up to 10) or 144 facts (up to 12). In fact, the zero property and the identity property can easily knock out many. Then, we start with familiar facts like times 2 and times 5. The clock is a friendly reminder to count by 5s and students love the singsong nature of skip counting by 5s. Times 3 is often thought of “times 2 plus 1 more set” for example 3x6 is 2x6 = 12 plus 6 more to get 18. Rehearsal may be necessary for students to build up quick recall and develop automaticity. Times 4 is often thought of as “double doubles” so if I know double 7 is 14 which is 2 x 7 then I can double that and get 4x7 is 28. That same idea can be used for times 6, double my times 3, and I know my times 6. Times 9 is always popular because students take pride in learning what they believe are the toughest facts, and they seem to be very proud when they learn 9 x 9 = 81. But times 9 has lots of wonderful patterns like the digital sums for all the products being always equal to 9, like 18 is 1 + 8 = 9 and 27 is 2 + 7 = 9 and so on. But more importantly 9 is 1 set away from 10 so they can use the multiples of 10 to get to a familiar fact they know, for example, I know 8 x 10 is 80 and 8 x 9 is just one set of 8 less than that so it will be 80  8 = 72. For some abstract thinkers, they may think (n x 10)  n = n x 9 Text Box 4.7 Understand Properties of Multiplication and the Relationship between Multiplication and Division CCSS.Math.Content.3.OA.B.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.) CCSS.Math.Content.3.OA.B.6 Understand division as an unknownfactor problem. For example, find 32 + 8 by finding the number that makes 32 when multiplied by 8. The giant multiplication chart above can show how the commutative property allows for the beautiful symmetry of facts. Knowing that 3 x 8 is the same as 8 x 3 can help students feeling overwhelmed at first learning all the facts by cutting the number of facts in half. These compositions and decompositions may help students use facts that they know to find easier ways to learn the facts. In addition, it provides students an opportunity to be pattern seekers and discover patterns and strategies; they can use to make sense of the facts they are learning. For example, using this visual, one can see the square facts prominently appearing diagonally across the chart. One might also notice prime numbers only have long boring rectangles going along the edges x 1. This is a perfect connection to prime and composite numbers and discussion about factors. In addition, keen pattern seekers might love to analyze the pattern of what happens when one multiplies even times even, even times odd, and odd times odd. Properties can help students build a deeper understanding of operations and provide support for learning mathematics facts. One of the ways to build students’ mental mathematics flexibility is to use the associative property or the distributive property to compose and decompose facts. Using the area model, students can use the distributive property to used derived facts to learn trickier facts (see Figure 4.7). Figure 4.7 Using distributive property to decompose multiplication facts. Source: Authors. Figure 4.8 Multidigit multiplication and division using the area model. Source: Authors. The area model is a representational model that has generality because this visual can be used in later grades to represent multidigit multiplication along with multidigit division (see figure 4.8). The idea of distributive property comes into play as we consider the partial products and other methods of multiplication below. Multiple Representations for Multidigit Multiplication Let us now consider multiplying 23 x 12. The traditional method is as follows:
We can now look at multiple pictorial representations of the same problem that are each motivated by several important mathematical concepts. First, the distributive property allows us to consider the following representation:
One can alternatively consider:
If we were to use expand both 23 and 12 using respective landmark numbers 20 and 10 we can again use a representation that yields the famous FOIL (FirstOutside InsideLast) which is connected to the partial products and can be seen as the areas of the respective rectangles in the figure.
Other pictorial representations, with a historic perspective, include the application of the vertically and crosswise multiplication introduced in several cultures (including Indian and Chinese) as well as other procedures such as the Jalousie multiplication (also referred to as Lattice multiplication) introduced by the Persian mathematician al Karaji in his book Kafi fil Hisab. While the latter technique is not intuitively obvious, the former technique brings out the importance of place value. These are demonstrated next (see Figure 4.9). Figure 4.9 Other pictorial models for multiplication. The above figures show vertically and crosswise multiplication on the left and Jalousie multiplication on the right. Supplementing current teaching methods with such pictorial models help the students to visualize the answer and make sense of the traditional algorithm. Think about it! How does providing multiple visuals and approaches help students detect patterns in multiplication and properties of multiplication and become efficient in learning the facts and building conceptual understanding and procedural knowledge? 
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