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Procedural fluency is the ability to accurately and efficiently conduct operations and mathematics practices.

Correction and Feedback

Receiving immediate feedback about performance is particularly important in mathematics. If students are performing an operation incorrectly, they should be told which parts correct and which parts are incorrect. Showing students patterns in their errors is an important source of feedback. Students also need to learn to check their own work and monitor their errors. Working in pairs can help in this process of checking and monitoring because students benefit from a peer’s help when they may not from the teacher. Remember, feedback includes pointing out improvements as well as needed changes. Students in Ms. Wong’s math class were given a worksheet to practice their new skill of using dollar signs and decimal points in their subtraction problems. Ms. Wong told the students to do only the first problem. After they completed the problem, they were to check it and make any necessary changes. If they thought that their answer to the problem was correct, they were to write a small c next to their answer; if not, they were to write a small i for incor-rect next to the answer. They were also to indicate with a check mark where they thought they had made a mistake. Ms. Wong moved quickly from student to student, check-ing their work. Students who had the first problem correct were given task-specific praise and directions for the rest of the problems: “Good for you. You got the first problem correct, and you had the confidence, after checking it, to call it correct. I see you remembered to use the dollar sign and decimal points where they were needed. After you fin-ish the first row, including checking your problems, meet with another student to see how your answers compare. Do you know what to do if there is a discrepancy in your answers? That’s right. You’ll need to check each other’s problem to locate the error.” For the students who had solved the problem incorrectly, Ms. Wong stopped by each student’s desk and said, “Tell aloud how you did this. Start from the beginning, and as you think of what you’re doing, say it aloud so I can follow.” Ms. Wong finds that students often notice their own errors, or she will identify some faulty thinking by the students that keeps them from cor-rectly solving the problem. Once the error has been found and corrected, Ms. Wong asks the student to do the next problem, again saying what is being done aloud. If correct, Ms. Wong gives task-specific praise and directions for the rest of the problems.

Alternative Approaches to Instruction If a student is not succeeding with one instructional approach or program, the teacher should not hesitate to make a change. Most students learn best when they are provided prerequisite skills to sup-port the math processes and opportunities to practice with feedback. Consider changing resources if students are hav-ing difficulty, including adjusting textbooks, workbooks, math stations, and manipulatives.

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Applied Mathematics Concrete and representational materials and real-life applications of math problems make math relevant and increase the likelihood that students will transfer skills to applied settings such as home and work. Students can continue to make progress in mathematics throughout their school years when they have the underly-ing foundational scaffold from which to build their skills and problem solving. The emphasis needs to be on problem solving rather than on rote drill and practice activities. The term situated cognition refers to the principle that students will learn complex ideas and concepts in the contexts in which they occur in day-to-day life (real-world application). Students need many opportunities to practice what they learn in the ways in which they will eventually use what they learn. This is a critical way to promote the generalization of mathematical skills. For example, when teaching measurement, a teacher can give students real-world application opportunities to use the mathematics they are learning, such as measuring rooms for carpet, determining the mileage to specific locations, and so on. When Ms. Wong’s students successfully used dollar signs and decimals in subtraction, she gave each of them a mock checkbook, which included checks and a ledger for keeping the balance. In each of their checkbooks, she wrote the amount of $100.00. During math class for the rest of the month, she gave students “money” for their checkbook when their assignments were completed, and their behavior was appropriate. She asked them to write her checks when they wanted supplies (pencils, erasers, chalk) or privileges (going to the bathroom, free time, meeting briefly with a friend). Students were asked to maintain the balances in their checkbooks. Students were penalized $5.00 for each mistake the “bank” located in the checkbook ledgers at the end of the week, much like a charge a real bank would make for an overdrawn account.

 

Generalization

Generalization, or transfer of learning, needs to be taught. As most experienced teachers know, students often can perform skills in the special education room but cannot perform them in a regular classroom. To facilitate the transfer of learning between settings, teachers must pro-vide opportunities to practice skills by using a wide range of materials, such as textbooks, workbooks, manipulatives (e.g., blocks, rods, tokens, real money), and word problems. For example, the teacher could have students measure different objects with things (unsharpened pencils, sheets of construction paper, or newspaper pages) rather than rulers or yardsticks. Teachers also need to systematically reduce the amount of help they provide students in solving problems. When students are first learning a math concept or operation, teachers provide a lot of assistance in performing it correctly. As students become more skillful, they need less assistance. Teachers must remember that generalization or transfer of learning must be planned for rather than “teach and hope” that it will occur. When Ms. Wong’s students correctly applied subtraction with dollars and decimals in their checkbooks, she asked them to perform similar problems for homework. Ms. Wong realized that before she could be satisfied that the students had mastered the skill, they needed to perform it outside her classroom and without her assistance.

Participation in Goal Selection Allowing students to participate in setting their own goals for mathematics is likely to increase their commitment to achieving goals. Students who selected their own mathematics goals improved their performance on math tasks over time more than did those students whose mathematics goals were assigned to them by a teacher (L. S. Fuchs, Bahr, & Rieth, 1989). Even very young children can participate in selecting their overall mathematics goals and can keep progress charts on how well they are performing.

 

Instructional Approaches Students in the United States have scored well below others (Taipei, South Korea, Sin-gapore, Hong Kong, Japan) in mathematics proficiency in grades 4 and 8 on the international assessment of mathematics, Trends in International Mathematics and Science Study (Provasnik et al., 2016). We need to examine how we teach. It may be advantageous for math teachers to con-sider focusing instruction on the development of fewer mathematical topics that are the more important ones so that students become truly proficient. This approach is used in many other countries that have demonstrated successful outcomes in mathematics (Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005). The National Research Council (NRC) (National Research Council, 2001) indicates that “mathematical proficiency” is the essential goal of instruction. What is mathematical proficiency? It is what any student needs to acquire mathematical understanding. The NRC describes five inter-woven strands that compose proficiency. Consider how you are integrating these strands into your instruction. Also, consider how you might determine whether the students you teach are making progress along each of these strands.

1. Conceptual understanding refers to understanding mathematic concepts and operations.

2. Procedural fluency is the ability to accurately and efficiently conduct operations and mathematics practices.

3. Strategic competence is the ability to formulate and con-duct mathematical problems.

4. Adaptive reasoning refers to thinking about, explaining, and justifying mathematical work.

5. Productive disposition is appreciating the useful and positive influences of understanding mathematics and how one’s disposition toward mathematics influences success.

See Apply the Concept 11.2 for suggested instructional practices.

It is particularly important for teachers to design mathematics programs that enhance learning for all students, especially those with diverse cultural or linguistic back-grounds. See the next section for suggestions on how to do this.

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