 Hello, my name is Kerry Erdman and I am a member of the Edmonton Regional Learning Consortium Mathematics Team. Thank you for joining me today to explore division personal strategies in the Division II Mathematics Classroom. Before we begin our podcast, I would like to take a moment to thank ERLC for this learning opportunity as well as Alberta Education for the grant that made this professional learning opportunity possible. One of the main changes in the new revised Alberta Mathematics Curriculum in the area of division is the development of students' personal strategies. This is the area we're going to delve further into in this podcast. So 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 that make sense to them. Memorized procedure is how many of us were taught. When we learned about additions, subtraction, multiplication, and division, most of us learned these through formal algorithms or step-by-step procedure. 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 nine 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 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 is 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 memorize steps and rules for getting the right answers. 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 begin to focus only on procedures. Once students understand the underlying mathematics, it becomes easier for them to remember procedures because they actually 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. This is still very important. However, the kind of practice students engage in may be different, may look different. Merlin Burns states in 1994 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 and practiced. 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're going to look at ways we can help our students develop division personal strategies, and what some of these personal strategies may look like in the classroom setting. So let's begin by having a look at some of the variety of ways students are approaching division 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 division, there are four main parts. First, the dividend is the number of items you start with that are being divided up into groups. In this example, the dividend would be 289. The divisor is the number of groups we are breaking the dividend into. Here, the divisor is 4. The quotient often misnamed, or the answer, is the number of items in each group. Here, 72 R1. R1 refers to the fourth part of division, or the leftovers which we call the remainder. The remainder must always be smaller than the divisor. So let's begin by looking at a division question that I asked in my grade 5 classroom when we were working on division. I presented my students with the equation 56 divided by 8, and had them apply personal strategies to solve the question. 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 the problem by using related multiplication facts. The student recognized that 56 divided by 8 had a related multiplication fact. They thought 8 times some number would give them a product of 56. The student knew that 8 times 7 gave a product of 56, and then they used this knowledge to revisit the division question and plug in the missing answer of 7. A second student approached the same question by using a combination of methods. This student started with a related fact that was familiar, in this case, 40 divided by 8, being 5. The student then used skip counting to get to their destination. The student then added the groups from skip counting to the related fact to yield an answer of 7. Students 3 and 4 both applied what we refer to as a having strategy. Student 3 began by dividing 56 by 2 to get 28. They then took this partial answer, 28, and divided it by half of 8, or they divided it by 4, to come to the answer of 7. Student 4 began by dividing by half of 8, and then the remaining factor of 2 to come to the final answer of 7. Next, let's have a quick look at five strategies that your students might use in your classroom and what some of these might look like in your classroom. The first strategy here is expanding the dividend. In example A, we look at the equation 2718 divided by 3. If we approach this question by expanding the dividend using place value, we would look for multiples of 3 in the dividend that we are familiar with. In this example, I have expanded the dividend to 2718. Recognizing that both of these numbers are multiples of 3, I then just need to figure out the answers for these two equations separately. So 2718 divided by 3 equals 900, and 18 divided by 3 equals 6. I then add the partial quotients together to get my final answer, or 900 plus 6, which is 906. The same concepts can be applied to decimal numbers. In example B, 20 dollars and 32 cents divided by 4, I have decided to break down 20 dollars and 32 cents into 20 dollars and into 32 cents respectively. Recognizing that both of these numbers are multiples of 4, I then divide these simpler equations to get partial quotients. In other words, 20 dollars divided by 4 is 5 dollars, and then 32 cents divided by 4 is 8 cents. When I add these partial quotients, I get my final answer of 5 dollars and 8 cents. A second strategy included here is breaking up the dividend. In this example, we begin by recognizing that 192 is composed of multiples of 6. Here, I recognize 192 as 180 plus 12. We then take our parts and divide by our divisor of 6. 180 divided by 6 equals 30. 12 divided by 6 equals 2. Adding these two partial answers together, we find our final answer of 32. A third strategy is adjusting the divisor and dividend. In example A, 620 divided by 5, we could double both divisor and dividend to make for an easier calculation. Yielding 1240 divided by 10, which would equal 124. In example B, 850 divided by 50, doubling would give us 1700 divided by 100 equals 17, an easier calculation. In example C, if we multiply our divisor and dividend by 4, we would get 1400 divided by 100 equals 14, a much easier equation for some students. Our strategy I've included is to divide tens and hundreds by one digit. Here, in this example, 180 divided by 3, we would think of 180 as 18 sets of 10. Using a related fact, 18 divided by 3 would equal 6. So 18 sets of 10 divided by 3 sets of 10 would equal 6 sets of 10. We would then recognize that our answer is actually 6 tens or 60. The last strategy I've included today is looking at inverse operations. Multiplication and division are inverse operations. In fact, one of the most common definitions for division is that it is the opposite of multiplication. The multiplication division sentences of a set of particular numbers is sometimes referred to as a fact family. On the screen right now, I have what could be called the fact family of 4, 6 and 24. Students can often use these fact families to help them solve division questions. So how can we help students develop personal strategies in our classroom? Use story problems frequently or provide a context for problems within your classroom. This helps the students to actually connect to their learning, to personally connect to their learning. Make sure you're allowing for a range of methods and allowing students to use the methods that they are most comfortable using. Be sure to encourage discussion about the thinking that students were using when working through their problems. Use the strategies that students have presented in their work. Help students practice with different strategies so that they can find methods that are meaningful for them. Be sure to support student thinking with written recordings of their thinking. Make sure to model this so students learn quickly how they can record their thinking. As students develop their own skill, insist that they record the steps in their thinking too. And make sure you know your staff so that you can start to label the strategies for students, such as that's an adding up strategy. Soon they will develop the language of mathematics and also a repertoire of strategies that they can draw from for future questions. Some teachers ask, what do I do with the problem solving strategies that I used to teach? 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 become background information for teachers to understand strategies that they see students using in their classroom. Thank you for joining me in this podcast. For more information on personal strategies in the division area, please take some time to check out the noted web link here from Albert Education, as well as the rich book resources listed here.