Teaching Strategies
One of the strongest features of the CCSS-M is the Standards for Mathematical Practice (See Appendix). Addressing a clear need for students to develop problem solving skills, these practice standards are identical from kindergarten through twelfth grades and describe the habits of strong mathematicians. According to the TIMSS Videotape Classroom Study (Stigler, Gonzales, Kawanka, Knoll, & Serrano, 1999), teachers often "design lessons that remove obstacles and minimize confusion [where] procedures for solving problems would be clearly demonstrated so students would not flounder or struggle." Lessons that are planned from this perspective limit students' abilities to deeply understand content. I want to do the opposite, but not with abandon. As explained below, I have come to believe through my research that successful problem solving experiences depend not only on rich tasks, but also on the careful planning and execution of lessons that include time for reflection and consolidation. In order to promote rich mathematical exploration and discussion, I will use the Five Practices Model (Smith and Stein, 2011) to design and execute thoughtful lessons.
The planning process requires first, specifically setting a goal and selecting a set of high quality problems that will help students reach it. While this seems like an obvious and perhaps simple step in preparing a lesson, much thought is required to align the task with the mathematical and problem solving goals of the lesson, as it serves as the foundation for all other parts of the work. Consideration of the difficulty and interest levels it involves will be of primary concern in this unit, as well as making relevant connections to students' lives.
Anticipating, the first of the Five Practices, requires me to engage with the task prior to administering it with students to anticipate possible solution strategies students may come up with during the lesson. At this point I also try to anticipate misconceptions that may surface during their work so that I am prepared with questions to redirect them. While it will be unlikely that I will anticipate all of the ways students will solve the problem, taking the time to think it through in different ways will better prepare me to quickly analyze them during class while guiding student groups.
As students work, I will be monitoring and selecting their responses, paying attention to their strategies and asking individuals and groups about their thinking while they progress collaboratively through the task. During this time, I will be able to identify students whose thinking should be shared with the class during our whole class discussion and summary. Having a monitoring tool with which to keep track of students' strategies is a quick way to stay organized and remember what I see at each group. I will take note of general approaches such as pictorial, verbal, algebraic, etc., so that each type of strategy is represented in the whole class culminating discussion of the day's work. I will also make a checklist of strategies I anticipate, with room for unanticipated strategies that may come up, so that it is easy to organize their sharing out in a way that logically develops the key mathematics from the task. I plan to be careful to share many strategies so that every learner has something to connect to.
Sequencing and connecting occur as part of the whole class discussion after student groups have had time to work together on the task. Sequencing student responses is directly related to achieving the goal of the lesson, as students who share their strategies help inform their peers' understanding as well as their own. A typical sequence might begin with a concrete strategy such as a drawing and move toward more abstract, algebraic strategies. This approach allows everyone an entry point into the discussion, but leaves room for extension as appropriate. While students present their strategies, the teacher helps connect the responses and tie them back to the goal of the lesson through highlighting patterns among and the level of efficiency of each shared idea. In recent years there has been a paradigm shift toward conceptual teaching of mathematics. I favor this, but also believe that, unless carefully executed, there is a danger of leaving too many loose ends for students to truly synthesize their learning. This step of connecting offers an opportunity for everyone to get on the same page with what was learned that day and to walk away with common vocabulary and knowledge.
The CCSS-M Standards for Mathematical Practice cannot be taught in isolation; they occur spontaneously during effective problem solving activities. In my research a description that resonated with me was, "These eight Practices call attention to what it means to be mathematically proficient, to go beyond simply memorizing facts and formulas. When our students experience mathematics through these Practices, they have repeated opportunities to make sense of the ideas and to build a deeper understanding of skills and concepts," (Connell, 2013).
Unfortunately, most students do not intrinsically value the struggle that understanding mathematics presents. They often have preconceived notions about who is "good" or "bad" at math, based largely on answer-getting and not strategic thinking, which prevent them from persevering in problem solving (Mathematical Practice Standard 1). In my students' cases, they have not had much, if any, experience grappling with problems they did not already know the algorithm for solving. Therefore, I plan to draw attention to evidence of these practices as they surface during student work time in class through verbal praise, and through student self-reflection in writing on a weekly basis where they will assess themselves on the practices they exemplified that week, providing evidence to support their claims. These will be featured on a classroom display.
Focusing on problem solving has proven benefits. However, many of my students lack a lot of basic number sense, which hinders their progress and can lead to frustration and abandonment of rich problems. To address this deficit in a meaningful way, I will create strings of number talk prompts with interesting patterns that illustrate the properties of arithmetic and elicit a variety of strategies from students. This strategy is based on the work of Sherry Parrish who explains, "Classroom number talks, five- to fifteen-minute conversations around purposefully crafted computation problems, are a productive tool that can be incorporated into classroom instruction to combine the essential processes and habits of mind of doing math" (Parrish, 2010).
Establishing a clear protocol for number talks is important, and I plan to follow Parrish's fairly closely as we put this routine into practice 2-3 times per week. Students will gather on carpet in a corner of my room for the talks so that they use mental math to solve rather than paper, pencil, or calculator, and to make this routine a clear part of class with a designated space. The problem will be presented and written on the board. Students solve mentally and indicate by placing their thumbs on their chests when they have arrived at a solution using one strategy. This personal signal to the teacher eliminates the raising of hands, which often discourages others from continuing on their solution paths because they feel that someone already knows the answer and therefore the process is over. Students are encouraged to find multiple ways to arrive at their solutions, and they add fingers up to indicate each new strategy they find. This keeps everyone engaged during the wait time needed to give everyone an opportunity to think through their processes.
Once nearly all or all students have at least a thumb up, I ask for answers to get them out of the way. Often there will be more than one answer, especially at the beginning of a new type of problem string. I list all of them at the top of the paper and circle the one we are able to prove at the end. Next, I ask for volunteers to defend one of the answers on the list. If I know a student had a misconception and she or he volunteers to defend the answer I tend to call on that student first to illuminate the misguided thinking and diversify the thinking that is shared; we often learn more from incorrect answers than from correct. This, of course, depends on a strong classroom culture where mistakes are valued and safety has been established. This process continues until we have exhausted the contributions. Many misconceptions flesh themselves out along the way, but the process certainly takes longer when we first begin and students are learning the nuances of number talks. Over time, the five- to fifteen-minute range becomes much more realistic.
All of these strategies share the common thread of student understanding guiding lessons and determining the direction of the unit. An essential component of this unit, and arguably of effective teaching, is one that is counterintuitive to the traditional view of mathematics teaching and learning: flexibility. We want our students to become flexible in their approach to and understanding of mathematics, and are now realizing the importance of modeling this in our teaching. This requires depth and breadth in our own competency of what we teach, which is something that was significantly lacking in my own experiences as a student and something I have worked hard to develop throughout my post-high school education that continues today.
The process of filling our own content gaps as teachers is easier stated as a need than accomplished. My certification was for kindergarten through grade 9 all content areas, but after two years of teaching sixth grade all subjects, I knew that mathematics was my passion. I participated in a teacher education program at the University of Chicago called SESAME where I took classes from university professors to earn my endorsement in middle school mathematics. Since then, I have attended hundreds of professional development events, but had never encountered the depth of content learned in these courses until engaging in this summer's seminar. While the level of content I learned was beyond what I teach, my students have benefited greatly from my greater depth of knowledge that allows me to connect what I am teaching to their future learning; this perspective enables me to understand the importance of the content I teach, and to communicate it better to my students. My participation in this summer's seminar reinforced the benefits of content-based learning.
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