MathEd, Section 4
October 11, 2010
According to Van de Walle, Karp, and Bay-Williams, "Students should not just think of remainders as 'R 3' or 'left over.' Remainders should be put in context and dealt with accordingly" (p. 158). I find this statement particularly interesting because the mathematics lesson that I observed in my 5th grade classroom focused on long division with remainders. This concept was taught to the students utilizing the "R 3" or "left over" method. Despite the fact that this strategy may be in disagreement with Van de Walle and his colleagues, I feel that the lesson was effectively taught and fulfilled many categories of a productive mathematics lesson. In particular, my mentor teacher did a fine job of using a variety of grouping structures throughout the lesson along with using beneficial models to support learning.
Most educators are aware of the benefits of using a variety of grouping structures such as individual, pairs, small groups, and whole group instruction. When a teacher effectively employs flexible groupings, he/she is "allowing students to collaborate on tasks [and] provides support and challenge for students, increasing their chance to communicate about mathematics and build understanding" (Van de Walle, Karp, & Bay-Williams, p. 67). In this particular lesson, the students were paired up with their seat partner and were then given a blue bag with manipulative blocks and other materials in it. For each aspect of the lesson, the students were instructed to work together with their partner and discuss methods. Once the students began working on the set of four practice problems, they were reminded each time to "show work and work together!" Van de Walle and his colleagues imply that sometimes students work in partner groups because "the nature of the task best suits only two people working together..." (p. 67). I feel that the activity used to learn this lesson is best structured for individual or partner work due to the limited amount of manipulative blocks in each bag. The partners were able to verbally discuss with each other the methods, issues, and discoveries that they experienced throughout the lesson. At one point, however, an issue arose where there were not enough "1 block" manipulatives for a given problem. The solution was for two partner groups to come together to form a small group of four. I was unsure about how well this type of grouping would function given the character of the assignment but it turned out to work just fine. In this type of group structure, more students were able to share their thoughts with one another and clarify problems that one partner may not have had the ability to do. Groups are normally selected based on students' varying academic abilities (Van de Walle et. al), and although this was not specifically the case, it inevitably worked out that in a group of four students, not all were at the same academic level. Through experiencing our own MathEd activities during class time, it is clear that using a variety of grouping structures is beneficial. My peers and I have been given the opportunity to work independently, work with partners, and work with our small group tables on a variety of tasks, sometimes even circulating the room while completing them. I have been able to develop an idea of advantages and disadvantages to each type of grouping structures by analyzing the productivity and accuracy displayed by each group.
Mid-way through the lesson on long division my mentor teacher stopped the students and explained that the reason for using manipulative blocks was for them to "think about division in a different way." These 5th graders had difficulty demonstrating the ability to do long division during the previous math lesson, and so, the teacher decided to provide an alternative method. Using models such as manipulatives, calculators, or visuals support objectives. In support of my mentor teacher's train of thought, Van de Walle and his colleagues explain that "The more ways that children are given to think about and test an emerging idea, the better chance they will correctly form and integrate it into a rich web of concepts and therefore develop a relational understanding" (p. 27). The students were challenged to first model "16" with the manipulative blocks and then to divide that "16" into three equal groups. After allowing the students a few minutes to attempt this task, my mentor teacher asked the students, "Can you get three equal groups with all of the blocks into it?...No!" This is how the students realized that they could get most of the manipulative blocks into three equal groups, however, there would be one remaining block, which the teacher exclaimed was "Ok!" This task was then translated into a division problem numerically represented on the chalkboard as 3√16, which produces a solution of 5 R1. The students were able to compare their three equal groups of five "1 blocks" with this numerical solution. Supporting the use of these manipulative blocks strengthens "the ability to move between and among these representations improving student understanding and retention" (Van de Walle et. al, p. 27). In addition to the physical manipulatives provided to the students, each pair was given a small chalkboard to display their numerical answers on for the remainder of the practice problems. This is essential to note because it offers the students who learn best by writing and visually seeing the numbers an opportunity to comprehend the material, whereas the hands-on visual learners had their chance when using the manipulative blocks. In class, we have had the pleasure to complete many hands-on activities/tasks using various types of physical materials or manipulatives. The use of cards, beans, blocks, etc. has proven to be beneficial in learning concepts that we may not have been all too familiar with prior to class instruction, our recent unit on alphabitia being a perfect example.
Not all students learn at the same rate and not all students learn in quite the same way. As a future educator, it is important to understand this and to do the best I can to use a full range of instruction methods to increase instructional effectiveness. Two ways in which I can accomplish this is by using a variety of grouping structures and by using an assortment of models. The lesson that I observed in my 5th grade classroom on long division with remainders productively demonstrated many of the categories that would classify an effective lesson.
Van de Walle, John A., Karp, Karen S., and Bay-Williams, Jennifer M. Elementary and Middle School Mathematics, Teaching Developmentally. Seventh Edition. Pearson Education, Inc. 2010.