Mental mathematics has value beyond middle school. It builds students' confidence, promotes flexible thinking, and supports learning other mathematics. The article shares sample objectives and ideas for teaching and assessing mental mathematics objectives.

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- Author or Editor: Rheta N. Rubenstein x

### Rheta N. Rubenstein

In the late 1980s we challenged ourselves as a profession to meet the goals described in the *Curriculum and Evaluation Standards for School Mathematics* (NCTM 1989). We envisioned programs that would “encourage and enable students to value mathematics, gain confidence in their own mathematical ability, become mathematical problem solvers, communicate mathematically, and reason mathematically” (NCTM 1989, 123). In particular, for secondary school students we sought to present a core curriculum consisting of a common body of mathematical ideas accessible to all students. These visions were further detailed in the specific content standards, in the *Professional Standards for Teaching Mathematics* (NCTM 1991), in an assortment of *Addenda books*, and, this past spring, in the *Assessment Standards for School Mathematics* (NCTM 1995). For many people, however, among them teachers, parents, students, and school administrators, these *Standards* documents were merely visions, perhaps even pipe dreams.

### Rheta N. Rubenstein

When I began teaching in the late 1960s, we had no videotapes, commercial manipulatives, or calculators to create engaging learning activities for students. I recall spending a good part of my first year searching desperately for ways to motivate my seventh and eighth graders and help them learn mathematics. One activity that I stumbled across worked magically. I thought, “This is a gem,” and I continue to cherish it today. I call it the “function game,” but my students and others have called it the “input-output game,” “guess my rule,” or the “computer game.” I understand that the game first gained prominence in the “new math” era, but it must have been around in some form much earlier. Today, the game is available on many computer systems and is popular with students and teachers. The computer versions, however, lack many dimensions of the live version.

### Rheta N. Rubenstein

*Principles and Standards for School Mathematics* reminds us that communication is central to a broad range of goals in mathematics education (NCTM 2000). These goals include students' being able to (1) organize and consolidate mathematical thinking; (2) communicate coherently with teachers, peers, and others; (3) analyze and evaluate others' strategies; and (4) use language to express mathematics precisely. One part of communication is acquiring mathematical language and using it fluently. This article addresses learning vocabulary as one dimension of mathematics communication.

### Rheta N. Rubenstein

If students are going to become independent learners, they need to read mathematics textbooks successfully, as well as solve problems. However, motivating students to read lessons ahead of class can be a challenge. One method that helps is giving brief, frequent, low-stress quizzes on the basic ideas in the reading. They help students focus on critical ideas and give them regular feedback on their learning. These quizzes help the teacher gauge students' grasp of introductory ideas and assess their readiness for more new material.

### Rheta N. Rubenstein

Varieties of computational estimation were explored in relation to other mathematical skills and sex differences. The subjects were eighth graders (144 girls and 165 boys) from seven schools. Open-ended estimation was the most difficult, followed by estimation relative to a reference number and estimation within an order of magnitude. Verbal tasks were not more difficult than numerical tasks, but decimals were more difficult than whole numbers, and quotients were more difficult than products, which in turn were more difficult than sums and differences. Boys scored higher than girls on the total estimation test and on the order of magnitude scale. Stepwise regression analyses indicated that estimation performance was best predicted by skill in operating with tens.

### Rheta N. Rubenstein

The function game is a powerful and motivating tool for engaging middlegrades students in mental mathematics, problem solving, communication, and inductive reasoning (Rubenstein 1996). The game can also be used to help students achieve the goals of NCTM's Algebra Standard for grades 6–8; that is, to “represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules” (NCTM 2000, p. 222). (For a simple electronic version of the game, use the applet on the CD-ROM in Cuevas and Yeatts [2001].) This article will show how the function game format serves as a launchpad to help students build, distinguish, and translate between two basic forms of patterns.

### Rheta N. Rubenstein and Denisse R. Thompson

A tool used in reading theory is adapted to help mathematics teachers ask good questions that help students interpret displays of information.

### Judith M. Flowers and Rheta N. Rubenstein

A sequence of mental math problems using reasoning can boost students' understanding and confidence in performing multiplication.