 Hello, my name is Kerri Erdman and I am a member of the Edmonton Regional Learning Consortium Mathematics Team. I'd like to thank you today for joining me to explore multiplication personal strategies in the Division II Mathematics Classroom. I would like to take a brief moment of your time before we begin to thank Alberta Education for the grant that made this professional learning opportunity possible. One of the main changes in the new revised Alberta Mathematics curriculum is the development of students' personal strategies in the area of multiplication. This is the area we're going to delve further into in this podcast. Let's begin by looking at what personal strategies are. Personal strategies can be defined as meaningful steps students take to solve a problem when using addition, subtraction, multiplication or division. The revised program of studies emphasizes students' understanding concepts they learn, not simply memorizing procedures or facts. Our task is to help students develop methods that are mathematically sound and make sense to them. Memorized procedure is how many of us were taught. When we learned about addition, subtraction, multiplication and division, most of us learned these through formal algorithms or step-by-step procedures. We did not always understand why we did each step or why we did the steps in a specific order. The revised program of studies emphasizes students' understanding concepts they learn, not simply memorizing procedures or facts. The revised kindergarten to grade 9 Alberta Mathematics program states, students think about numbers and operations with numbers in a variety of ways. Students also problem-solve using different strategies. We must honor these different ways of thinking in our teaching of mathematics. This means we must provide opportunities for students to represent their thinking in a variety of ways, rather than prescribing how students will record mathematics symbolically. It's important for teachers to realize that no matter what strategy they may teach, students will process it in many different ways. Think about this statement. If we accept the theory of constructivism, that knowledge is constructed by the learner, having students develop their own personal methods and strategies when approaching math problems is a natural conclusion. This isn't really a new idea. The learner should never be told directly how to perform any operation in arithmetic. Nothing gives scholars so much confidence in their own powers and stimulates them so much to use their own efforts as to allow them to pursue their own methods and to encourage them in them. This quote was taken from Colburn back in 1970. When children receive instruction before they have the foundational ideas necessary to understand the mathematics presented in the problems, they tend to just memorize steps, steps for getting the right answer. This breaks down at the point when true understanding becomes necessary for further growth. In time, the students actually stop looking for meaning and they tend to focus only on procedures, thinking that procedures will get them the right answers. Once students understand the underlying mathematics, it becomes easier for them to remember procedures because they make sense. Conceptual understanding leads to procedural fluency. They can also reconstruct a procedure if they forget a step. Conceptual understanding and procedural fluency are intertwined. We do not teach conceptual understanding at the expense of procedural fluency. Practice is still important, however. The kind of practice students engage in may look different in your classroom. Marilyn Burns in 1994 states that imposing the standard arithmetic algorithms on children is pedagogically risky. It interferes with their learning and it can give students the idea that mathematics is a collection of mysterious and magical rules and procedures that need to be memorized in practice. Teaching children sequences of prescribed steps for computing focuses their attention on following the steps rather than on making sense of numerical situations. So today in our podcast we are going to look at ways we can help our students develop personal strategies and multiplication and what some of these personal strategies may look like in the classroom setting. So let's begin by taking a closer look at some of the variety of ways students are approaching multiplication problems in the classroom. Often before we begin in the classroom we need to make sure that everyone is working with a common vocabulary. In terms of multiplication there are three parts. The numbers being multiplied together are called factors or multiplicands. The results or answers called the product. Two factors yield a product or more specific vocabulary would be the multiplicand by the multiplier yields the product. Now that we have a common vocabulary let's begin by looking at one multiplication question that I have asked in my Grade 5 classroom when my class was working on multiplication. I presented my students with the equation 34 times 23 and had them apply personal strategies to solve this equation. Let's look at how my students approached this problem. There were four main strategies presented by my students on this question. The following strategies have all been taken from student papers. Student 1 approached this question by recognizing place value and then multiplying 4 by 3, 30 by 3, 20 by 4 and 20 by 30 recording their multiplication as partial products. They then added these partial products together to arrive at their answer. This method is quite common in my classroom setting. Student 2 chose to apply lattice multiplication to this question. To use this strategy on the question students begin by drawing a 2 by 2 grid. Students draw diagonal lines from the right top corner of each square on their grid. They then place the digits of one multiple across the top of the grid and the digits of the second multiple down the side of the grid. The students then multiply the two values that correspond on each of the grid squares recording the product in each of the squares. The students then add up these diagonal values and the values down the left and across the bottom are the final product in place value format. Many of my students were very drawn to this method for double-digit multiplication. Students 3's strategy was to model using base 10 blocks and then to sketch an array. The student then calculated the partial products of the four parts and added these products together to arrive at their answer. Some of my students do not use the actual manipulatives but skip to the sketching part of this method giving them a visualizing technique to help them figure out their answers. Student 4 chose the standard algorithm for multiplication. This is the memorized procedure that many of us are familiar with. All students, regardless of the strategy they chose, were successful at arriving at the correct answer. Now let's have a more in-depth look at some other strategies that you may come across in today's Mathematics Classroom. I have included eight strategies that have been introduced to me through ATA sessions, Macata sessions, ERLC sessions, and through resources by Mathematic Gurus such as Catherine Twomey-Thosnell, Mary Ann Small, Marilyn Burns, and John Vandewal. The first strategy we will look at is distribution, front-to-end multiplication. In example A, we would begin front-to-end multiplication by using place value. In this example, we would think of multiplying 3 by 700, then 3 by 40, and lastly 3 by 2. We then would take these partial products and add them all together to get our product. In example B, we break the second factor up. We would multiply the factor 4 by $5, then 4 by 30 cents, and finally 4 by 2 cents. Again, we would get partial products, and again, you would add these partial products together to get your final answer. A second strategy to consider is extending using the distributive property. In example A here, we think of 49 is 50 minus 1. So 49 times 7 would be the same as 50 times 7, subtract 1 times 7. Our thinking would be 50 times 7 is 350, taking away 1 times 7, which is 7, or 350 take away 7 equals 343. In example B, we could think of 11 as 10 plus 1. So 23 times 10 plus 23 times 1 would equal 230 plus 23, or our final answer of 253. Another strategy is to apply the commutative property and to recognize that when you are multiplying, you can multiply numbers in any order. In this example of 4 times 70, we can recognize that 4 times 70 can be broken further down to 4 times 7 times 10. We now can rearrange our order and look for a more simpler multiplication. In this example, I have rearranged it to group 4 by 7 first and then multiply by 10, or 4 times 7, which is equal to 28, and then multiply by 10 to reach my final answer of 280. The strategy of rounding up and adjusting can come in very handy in some problems. If we look at our first example here, example A, $599 times 3, we can choose to round $599 up to an even $600. We then can decide to multiply that $600 by 3 to get the product of $1,800. Because we rounded up by 1, we then subtract one group of 3 to get our final product of $1,797. In example B, we recognize that $9.97 is very close to $10. $10 times 3 equals $30. We then recognize that we have now three extra sets of $0.03 or $0.09. We then must subtract this amount from our $30 to reach our final product of $29.91. In our next strategy, students recognize that when multiplying two numbers, they can divide one factor and multiply the other by the same amount and that this does not change the product. This is called using the associative property. If we decide the amount we want to change by is 2, we refer to this strategy as the doubling and having strategy. Using the doubling and having strategy to calculate a product, you can divide one number by two to get half of that number and then double the other number. This can help result in an easier calculation while keeping the equation balanced. In the example A above, 16 times 5, you can divide the 16 by 2 and choose to multiply the 5 by 2 without changing the balance of your equation, ultimately creating a simpler multiplication equation. In example B, you can divide 250 by 2 and multiply 5 by 2 to create a simpler multiplication equation. Another strategy is to try to make one of the factors a multiple of 10, if possible. Let's look at how you could multiply when one number is close to 10. I've chosen the example 8 times 67 for us to look at. One method a student might choose to use could be to look at the numbers and decide to round the 67 to 70. The students then think 8 times 70 equals 560. The student then recognized that they have three extra groups of 8, so multiply 3 by 8 to equal 24. They then subtract 24 from 560 to get the product of 8 times 67, or in this case, 560 minus 24 equals 536. A different student might look at the same example and solve it in a similar manner. They may wish to round the 8 up to 10 and think 10 times 67 equals 670. Here they recognize that they now have two extra groups of 67. So they would multiply 2 times 67 to equal 134. They then would subtract the 134 from 670, giving them the final product of 536. Both of these strategies are acceptable and yield the correct response. Another strategy that students may use is the partitioning strategy. Students are given a question. Here the question is, in a building, there are 53 floors with 41 offices on each floor. How many offices are in the building? A student might begin by finding the sum of 10 sets of 41 to be 410. They then might add on four more sets of 410 or 1,640, and then finally add on three sets of 41 or 123. Adding all of their calculations together, they would see that there are 2,173 offices in this building. In this example, can you see how this student is thinking? Why is the student dividing by 2? Alberto recognized that in order to find out the number of books, he would have to multiply 177 by 17. He began by breaking 177 into place value parts, starting with the one values and multiplying these separate parts by the 10 value of 17. Once he completed these partial products, he added them together. So essentially he has multiplied 10 times 177 at this point in his thinking. He then recognizes that 17 is equal to 10 plus 7 or 10 plus 5 plus 2. And to figure out the value of 5 sets of 177, he takes the product of 10 and divides it by 2. He then adds this onto his original answer, giving him 2,655 or the value of 15 times 177. He recognizes that he still needs 2 more sets of 177, so adds these together, yielding 354. His final step then is to take this amount of 354 and add to his previous value of 2,655. His new answer is 3,009 or 17 sets of 177. For some students partitioning or breaking the equation into smaller parts like this makes larger equations more manageable for them. Using the compensation strategy, a student might approach this problem by multiplying 10 by 250 and then having the answer. Here the student might multiply 20 by 70 to get 1,400 and then subtract 210, the three extra jars of 7 ladybugs to get 1,190 ladybugs all together. These are just some of the personal strategies that you may come across in your division two mathematics classroom. So how can you help students develop personal strategies within your classroom? Some suggestions include, use story problems frequently or provide a context for problems. Students learn better when they can connect personally to concepts that are interested. Allow for a full range of methods, remembering that no matter what strategy we may teach, students will process it in many different ways. Challenge students to solve a problem in more than one way. This forces students to think of problems in more than one way and to explore new strategies to show you what they know. Encourage discussion about the thinking the students were using when working through their problems. This forces the student to actually reflect upon their learning. Use a strategy that a student has presented in a new problem. Label the strategy using the student's name. For example, Debbie's strategy. This helps students practice with different strategies so that they can find methods that are meaningful to them in a non-threatening atmosphere. Support student thinking with written recordings of their thinking. As students develop their own skill, insist that they record the steps in their thinking too. And know your stuff so that you can start to label the strategies for students. They'll develop the language of mathematics and a repertoire of strategies to draw from. Remember these? Some teachers ask, what do I do with the problem-solving strategies I use to teach in my classroom? Strategies like act it out, draw a diagram, look for a pattern, use a table, guess and check, work backward, solve a simpler problem. These strategies can now be used as background information for teachers to understand strategies that their students are using in the classroom. Thank you for joining me in this podcast. For more information on personal strategies in the multiplication area, please check out the above noted web link from Albert Education, as well as the rich book resources listed here.