Читать книгу Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong - Страница 23

To guide your instruction and even enhance your own capacities for problem solving, we have developed a model for solving word problems that puts the emphasis squarely on learning to mathematize (Figure 1.4). The centerpiece of this model is what we call the “mathematizing sandbox,” and we call it this for a reason. The sandbox is where children explore and learn through play. Exploring, experiencing, and experimenting by using different representations is vital not only to developing a strong operation sense but also to building comfort with the problem-solving process. Sometimes it is messy and slow, and we as teachers need to make room for it. We hope that this model will be your guide.

Description

FIGURE 1.4 A MODEL FOR MATHEMATIZING WORD PROBLEMS

The mathematizing sandbox involves three steps and two pauses:

Step 1 (Enter): Students’ first step is one of reading comprehension. Students must understand the words and context involved in the problem before they can really dive into mathematical understanding of the situation, context, quantities, or relationships between quantities in the problem.

Pause 1: This is a crucial moment when, rather than diving into an approach strategy, students make a conscious choice to look at the problem a different way, with a mind toward reasoning and sense-making about the mathematical story told by the problem or context. You will notice that we often suggest putting the problem in your own words as a way of making sense. This stage is critical for moving away from the “plucking and plugging” of numbers with no attention to meaning that we so often see (SanGiovanni & Milou, 2018).

Step 2 (Explore): We call this phase of problem solving “stepping into the mathematizing sandbox.” This is the space in which students engage their operation sense and play with some of the different representations mentioned earlier, making translations between them to truly understand what is going on in the problem situation. What story is being told? What are we comparing, or what action is happening? What information do we have, and what are we trying to find out? This step is sometimes reflected in mnemonics-based strategies such as STAR (stop, think, act, review) or KWS (What do you know? What do you want to know? Solve it.) or Pólya’s (1945) four steps to problem solving (understand, devise a plan, carry out a plan, look back) or even CUBES. But it can’t be rushed or treated superficially. Giving adequate space to the Explore phase is essential to the understanding part of any strategic approach. This is where the cognitive sweet spot can be found, and this step is what the bulk of this book is about.

Pause 2: The exploration done in the mathematizing sandbox leads students to the “a-ha moment” when they can match what they see happening in the problem to a known problem situation (Figures 1.2 and 1.3). Understanding the most appropriate problem situation informs which operation(s) to use, but it also does so much more. It builds a solid foundation of operation sense.

Step 3 (Express): Here students leave the sandbox and are ready to express the story either symbolically or even in words, graphs, or pictures, having found a solution they are prepared to discuss and justify.