Students’ academic work in school is defined by the academic tasks that are embedded
in the content they encounter on a daily basis. Tasks regulate the selection of information
and the choice of strategies for processing that information….Students will learn what a
task leads them to do, that is, they will acquire information and operations that are
necessary to accomplish the tasks they encounter.
—Doyle (1983, p. 162)
Theories of mathematical learning suggest that the nature of classroom activities, or “tasks,” affects students’ abilities to learn mathematics with understanding (Doyle, 1983; Hiebert & Wearne, 1993). Hiebert and Carpenter (1992) contend that the structure and depth of knowledge depends on learners’ prior knowledge and their ability to access it and connect complex mathematical ideas to one another within a broader network of understanding. Students with connected knowledge structures are better equipped to engage in reasoning and problem solving and better primed to transfer learning and adapt understandings to new contexts (Bransford, Brown, & Cocking, 1999). Therefore, the mathematical tasks in which students engage should facilitate and support students’ conceptual understanding of mathematics, fostering deep connections among mathematical ideas (Hiebert & Carpenter, 1992).
The majority of time students spend in the classroom is on tasks (Hiebert & Wearme, 1993), and thus the quality thereof and their potential to advance students’ thinking is paramount. Instructional planning is an important precursor to effective teaching, and when designing instructional tasks, teachers should identify the following:
1. What are the important mathematics procedures and concepts for students to understand and why?
2.At what level and in what ways do students understand the mathematics prior to the task? In other words, what is students’ prior knowledge (including both their knowledge of the topic and at what cognitive level)?
3. Is the content of the task identified in standards for teaching mathematics? What is the relationship of this content to the critical foundations of algebra and major topics of school algebra ?
4. How does the task move students’ mathematical understanding forward? How does the task foster students’ construction of new understandings of mathematics?
5. How does the task provide all students access to mathematics by building on their prior knowledge? How does the task offer multiple entry points to engage in higher-order mathematical thinking?
6. How does the task offer students opportunities to experience multiple representations of concepts? In what ways does the task allow students to employ multiple problem-solving strategies in varied contexts?
How teachers design tasks for students has many implications for if and how students learn mathematics (Boston & Wolf, 2006; Stein, Grover, & Henningsen, 1996). High-quality tasks are those that are well matched to a student’s level of understanding, providing a balance between challenging the student and limiting frustration and inadvertent misunderstandings and errors.
High-quality tasks foster students’ abilities to reason, solve problems, and conjecture (Matsumura, Slater, Junker, Peterson, Boston, et al., 2006). When tasks are too narrowly conceived, for example, requiring a student to memorize a formula or procedure without reference to prior knowledge and potential applications or variations of that procedure, students may learn a disconnected skill that does not contribute to the student’s broader mathematical competence. Developing fluency with operations and computations is critically important to students’ mastery of mathematics ; tasks should ensure that students master standard algorithms and procedures, but at the same time they also should come to understand when, how, and in what ways to use those procedures to solve problems of different kinds. In sum, teachers must know how to design meaningful instructional tasks that are aligned to their students’ prior knowledge and hold the potential to deepen students’ conceptual understanding of mathematics. For example, when young children learn to count fluently forward and backward, this skill should be connected to the concept of cardinality or quantity
comparisons and teachers may use measurement tasks to illustrate differences between numerical rankings. As appropriate to the activities used, connections should be made to counting, combining and recombining sets, and early understanding of the principles of addition and subtraction.
Wiggins and McTighe (2001) recommend that teachers establish clear learning goals and clear criteria by which they are assessed. Learning goals can be used to construct a progression of learning activities. In mathematics, learning progressions should align with the major topics of school algebra, but they need to be more detailed so as to delineate skills in a coherent, sequenced manner and so as to establish transparent
targets for assessment (Heritage, 2008). Brief assessment of students’ mastery of key concepts and skills in a learning progression increases teachers (and learners’) awareness of any gaps between the desired learning goal and the students’ current knowledge and skill (Ramaprasad, 1983; Sadler, 1989). When teachers know where the students are experiencing difficulty, they can use that information to make the necessary instructional adjustments, such as reteaching, allowing extra opportunities for practice, providing instruction in small groups, or changing the method or type of instruction. More frequent assessment provides teachers and students with more immediate feedback on performance and allows self-evaluation on teaching practices and
student performance.
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