Lower secondary school students’ understanding of algebraic proof

Secondary school students are known to face a range of difficulties in learning about proof and proving in mathematics. This paper reports on a study designed to address the issue of students’ cognitive needs for conviction and verification in algebraic statements. Through an analysis of data from 418 students (206 from Grade 8, and 212 from Grade 9), we report on how students might be able to ‘construct’ a formal proof, yet they may not fully appreciate the significance of such formal proof. The students may believe that formal proof is a valid argument, while, at the same time, they also resort to experimental verification as an acceptable way of ‘ensuring’ universality and generality of algebraic statement

High-Performance Shuffle Motor Fabricated by Vertical Trench Isolation Technology

Shuffle motors are electrostatic stepper micromotors that employ a built-in mechanical leverage to produce large output forces as well as high resolution displacements. These motors can generally move only over predefined paths that served as driving electrodes. Here, we present the design, modeling and experimental characterization of a novel shuffle motor that moves over an unpatterned, electrically grounded surface. By combining the novel design with an innovative micromachining method based on vertical trench isolation, we have greatly simplified the fabrication of the shuffle motors and significantly improved their overall performance characteristics and reliability. Depending on the propulsion voltage, our motor with external dimensions of 290 μm × 410 mm displays two distinct operational modes with adjustable step sizes varying respectively from 0.6 to 7 nm and from 49 to 62 nm. The prototype was driven up to a cycling frequency of 80 kHz, showing nearly linear dependence of its velocity with frequency and a maximum velocity of 3.6 mm/s. For driving voltages of 55 V, the device had a maximum travel range of ±70 μm and exhibited an output force of 1.7 mN, resulting in the highest force and power densities reported so far for an electrostatic micromotor. After five days of operation, it had traveled a cumulative distance of more than 1.5 km in 34 billion steps without noticeable deterioration in performance.\u