As I was walking down the hallway in one our our elementary schools the other day, one of the teachers rushed out of her room to catch me. “I love teaching this way!” she said with shining eyes. “The activities are exciting and challenging. The kids can't wait to do them. They are learning so much! I have really seen changes in how they attack problems. The only thing that bothers me is that I feel that all my students—not just the ones who have been identified as gifted— could benefit from these activities.

### Mary L. Giannetto and Lynda Vincent

Many high school teachers have concerns about their students' ability to apply mathematical skills to other disciplines and situations. Some teachers believe that students should not “learn math in a vacuum.” One way to enhance students' understanding of the concepts learned in mathematics class is to apply mathematical skills to other subject areas, especially science and technology.

### Drew Polly

Try these approaches to boost your students' understanding and higher-order thinking skills.

### Wendy B. Sanchez

Educating students—for life, not for tests—implies incorporating open-ended questions in your teaching to develop higher-order thinking.

### Keith Weber

The purpose of this article is to investigate the mathematical practice of proof validation—that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and examplebased reasoning. Conceptual knowledge plays an important role in the validation of proofs. The practice of validating a proof depends upon whether a student or mathematician wrote the proof and in what mathematical domain the proof was situated. Pedagogical and epistemological consequences of these results are discussed.

### Alfinio Flores

The activities described here are creative, informal, intuitive geometry activities that stress higher-order thinking. The activities were completed by students in twenty-one bilingual classrooms in grades K-3 in a laboratory, hands-on setting.

### Virginia E. Usnick and Patricia M. Lamphere

### Edited by George W. Bright

Calculators help teachers teach in ways thot encourage students to become actively involved in their learning. This personal involvement is certainly consistent with the goals set forth in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). Calculators also allow the exploration of exercises that, although beyond the computational expertise of the students, may furnish opportunities for development of higher-order thinking skills

### Michael Wong

By using body parts and sightlines to find the distance to an object, students can reinforce the study and application of similar triangles. This activity is a good combination of tactile learning, practical application, careful observation and modeling, and higher-order thinking. The teacher may subsequently pursue the material in greater depth or use it as a starting point for excursions into other fields of study.

### Joel A. Bryan

During the thirteen years that I taught high school physics and mathematics, I found that my physics students typically came to class excited to learn. As in all science classes, they interacted with fellow classmates while performing laboratory investigations and other group activities requiring higher-order thinking skills. To create a similar experience for my mathematics students, I developed a laboratory investigation for my precalculus class. These students responded just as favorably as my physics students to hands-on data collection activities.

### Wei Sun and Joanne Y. Zhang

In its Principles and Standards for School Mathematics, the NCTM suggests that fluency with basic addition and subtraction number combinations is a goal in teaching whole-number computation (NCTM 2000, p. 84). A mastery of lower-order skills instills confidence in students and facilitates higher-order thinking. The ability to automatically recall facts strengthens mathematical ability, mental mathematics, and higher-order mathematical learning. Without this automation, students have difficulty performing advanced operations.