Teaching Strategies
Thinking Thematically: Everyday Classroom Scenarios
Throughout this period of research and planning, I considered not only the content to be taught but also the structure of the unit itself. My first decision was to organize my unit around a topic that my students would be both familiar with and excited by. My initial reasoning had to do with knowledge of my student population. 22 of my incoming 27 students are English Language Learners. Research supports the idea that these students learn better when they are able to make connections from day to day within a unit. As lessons proceed, students can draw upon what they learned earlier to steadily build their schema around a topic. With this in mind, I chose the theme of using addition and subtraction to address everyday classroom and school-based scenarios.
Why this theme? For one, I will be teaching this unit at the start of the school year, an exciting time for third grade students. Each day they reacquaint themselves with the structures and systems of the school and their classroom. At my school, early September finds students renewing their excitement over rituals such as tracking their classroom’s days of perfect attendance or counting their number of personal “Caught Being Good” tickets to earn a small prize. Meanwhile, students receive new classroom jobs such as distributing laptops from the computer cart to their classmates or filling out the daily lunch count.
In all of these scenarios (and many more), students can develop and put to use their understanding of the relationship between addition and subtraction. As such, my choice to build a unit around these everyday classroom scenarios takes advantage of my students’ natural excitement for the topic. My choice to make use of these scenarios also makes sense from a mathematical perspective. These activities make up the daily experience of my students at school, and, as a result, my students already have their own strategies and ideas around how to count or track the quantities involved in these scenarios. These personal strategies will be great starting points for building deeper understanding and will provide great fodder for student discussion. Finally, these problem scenarios will be useful because they are relevant throughout the entire school year and so can afford my students many opportunities for revisiting and practicing the mathematical concepts addressed in the unit.
Thinking Structurally: A Progression of Learning
Having a theme is not enough, of course. Throughout my planning, I also considered a number of possible progressions for how I would introduce and develop both the theme and the content. Given past experience, I was pretty sure I would enter my classroom and encounter a number of students who struggled to solve first-grade level comparison problems. I wanted to structure my unit to ensure that, after a few weeks of learning, these same students would be quite adept at solving two-step problems of various types and combinations.
My unit will begin with an introduction to the theme. I will present my students with vignettes of a few classroom scenarios and ask for ideas about what the student in the story should do. An example might be: “Michael is taking the lunch count and needs to know how many students are in class. He looks around the room and sees three empty desks.” This simple, broad description of a scenario can be the basis for an interesting discussion about quick and effective approaches. Michael can count the students he sees one by one. He can count the boys and girls separately and find their sum. Or, more efficiently, he can take the total number of desks (given there is one for each student) and subtract the number of empty desks. Together the students will brainstorm a list of scenarios in which they can use addition and subtraction and at school. This ongoing list will become an anchor in my classroom, a tool to continually reinforce the real-world context of the unit’s content.
After this introduction, the unit will progress according to two main phases, each with its own type of assessment (Figure 1). In the first phase, I will deliver a series of lessons on one-step problems, with more time and attention paid to comparison problems. When teaching these problems, I will scaffold rigor by first beginning with “equalizing” language and moving on to static wording, as described above. It will be during these lessons that I introduce the bar model and help my students use it to reason through quantities and their relationships. Modeling will be especially helpful for students to reason through the more challenging “inconsistent” problems. Knowing that my students will benefit from continued exposure to a wide variety of these one-step problems, I will formally assess them at this point in the unit by asking them to create a “workbook”; each student will develop a problem about a school scenario as well as create an answer key that shows a clear method for solving. I will compile these examples so that students can get further practice by solving the problems of their peers.
In the second phase, students will learn to solve two-step problems, which I will scaffold according to difficulty. This means I will begin with easier one-step component problems, such as change and part-part-whole types, and later combine the more complex “inconsistent” comparison types. As they continue to develop what they have learned about bar models, my students will be able to visualize the multiple quantities and relationships within these increasingly complex scenarios. In this phase, I expect students to discuss multiple solution methods and justify their strategies by referring to their models. It is my goal for students to be able to decompose these two-step problems as a chain of two simpler problems linked by an unknown. Only after they solve for this first unknown will they have enough information to solve the problem in its entirety.
The unit will culminate with a summative assessment of questions for both one- and two-step problems. I will conduct an item analysis to determine which problem types, or combinations of problem types, were easiest for my students and which were most difficult. This information will help me decide which problems my students still need to practice, as well as determine which students are in need of continued support. Later in the year, I will teach a similar unit in which I help students use modeling to reason through various one- and two-step multiplication and division problems. The results of this assessment will help me make decisions about how I might bridge my students’ understanding of addition and subtraction toward a full understanding of multiplication and division.
Structured Problem Solving
So far I have described the theme of the unit and the progression of its lessons. Yet there are more variations in types of problems than I could possibly show my students during this time. How, then, would it be possible to achieve the goal for students to be able to solve any individual addition and subtraction problem they come across? The purpose, ultimately, is for students to develop the habits of mind of analyzing and reasoning through all the types of problems so they can approach any addition or subtraction problem successfully. To develop and transfer this level of higher-order thinking, students must experience regular problem-solving-based lessons and routinely compare and discuss the reasoning behind various approaches to problems. To achieve this end, I use a routine called “structured problem solving” as the daily approach to mathematics instruction in my classroom.
Common in Japan, the purpose of structured problem solving is to both generate interest and excitement in math and to help students develop and discuss ideas that yield robust understanding of math content. Lessons begin with the introduction of a carefully worded problem. Students work alone, or sometimes collaboratively, to solve the problem using what they already know. Next, the teacher selects certain students to present their solution methods to the class. These methods provide the basis for a whole-class discussion in which students compare their mathematical ideas. The teacher facilitates this conversation to help students pull their ideas together and arrive at some new learning. Near the end of the lesson, the teacher helps the students summarize their learning. Finally, students often reflect on what they have learned in a math journal.11
There are three important features of a typical structured problem solving lesson: careful creation and delivery of a problem, facilitation of classroom discussion, and strategic display of student ideas. Below I will describe each of these features and how I use them in my classroom.
To create a good problem, a teacher should think about what would motivate the students to want to solve it. It is not enough for students to want to find a solution only because their teacher expects them to. A good problem should also present some exciting tension that the students are eager to approach and ultimately resolve. This tension should arise from the context of the problem itself and from how the teacher chooses to present the problem. For example, my class gets an extra recess block as a reward after they accumulate 25 days of perfect attendance. Before presenting my students with a problem about how many days remain before they will get their incentive, I might first ask them what games they would play during an extra recess period or to estimate how soon they think they can accumulate their 25 days. I have found that the excitement the teacher builds when leading up to the question in this way can often propel students into an eager mathematical investigation of the problem itself.
In Japan, the facilitation of classroom discussion is referred to as neriage, literally “kneading” or “polishing.” In other words, the artful teacher uses discussion as a way of “polishing” student ideas so that what is produced is a refined understanding of the concept at hand. In my opinion, quality neriage is one of the hardest things I do as a teacher. It begins before the lesson, when I decide what exactly I want my students to learn and then think about all the ways they might try to solve the problem. While students solve, I observe the students’ solution methods and choose a handful of different ideas that I think will help move us toward the new learning I have planned. I then select certain students to present these ideas to the class, using a sequence that I think will help students move from a simple to a more sophisticated understanding of the concept. Then the real work begins. I ask questions that get students to compare the mathematics within each of the solution methods, so that the class arrives not only at a correct solution, but a robust understanding of the mathematics that got them there. Of course, given the complex and at times unpredictable nature of student thinking, this process is a mixture of planning and improvisation, anticipation and reaction. This is what makes neriage so challenging, but also so thrilling.
The third feature of structured problem solving involves how the teacher displays student ideas on the board. Ideally, the board work should develop strategically to “tell the story” of how the mathematical understanding developed throughout the lesson. Ideally, everything written and posted on the board stays there. Nothing is erased. Typically, I begin with a captivating image that has to do with the context at hand, in this case a classroom scenario. Once I build excitement about this topic, I introduce the problem and write it alongside the image. Before releasing the students to solve, I might survey the room for some initial solution ideas and make a list of these possible methods under the problem. Later, as students present their ideas, I carefully represent each student’s method on the board in a way that makes their thinking visible. Once all strategies are up, these became the central reference points for class discussion. I continue to add important student ideas and key points as they arise during discussion. At the lesson’s end, I synthesize the new learning in the form of a summary on the right side of the board, often posing a reflection question alongside the summary (see Figure 2).
Figure 2: Board display for a lesson on finding the area of L-shaped figures.
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