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Background: Number sense is a key topic in mathematics education, and the identification of children's misconceptions about number is, therefore, important. Information about students' serious misconceptions can be quite significant for teachers, allowing them to change their teaching plans to help children overcome these misconceptions. In science education, interest in children's alternative conceptions has led to the development of three-and four-tier tests that not only assess children's understandings and misconceptions, but also examine children's confidence in their responses. However, there are few such tests related to mathematical content, especially in studies of number sense. Purpose: The purpose of this study was to investigate children's performance and misconceptions with respect to number sense via a four-tier diagnostic test (Answer Tier → Confidence rating for Answer Tier → Reason Tier → Confidence rating for Reason Tier). Design and method: A total of 195 fifth graders (10-11 years old) from Taiwan participated in this study. The four-tier test was web-based and contained 40 items across five components of number sense. Findings: The results show that (1) students' mean confidence rating for the answer tier was significantly higher than for the reason tier; (2) an average of 68% of students tended to have equal confidence ratings in both answer and reason tiers; (3) students who chose correct answers or reasons had higher mean confidence ratings in most items (36 out of 40) than those who did not; and (4) 16 misconceptions were identified and most of them were at a strong level. Conclusion: The four-tier test was able to identify several misconceptions in both the answer and reason tier and provide information about the confidence levels. By using such information, teachers may be better positioned to understand the nature of learners' misconceptions about number sense and therefore support their pupils' progress in mathematics.

An experimental study concerning the performance of a rectangular fin in both dry and wet condition was carried out. The visual observation of the dehumidifying phenomenon identified four regions, including the fully dry, very fine droplet, larger droplet, and film-like region. Experimental results showed that the effect of dry bulb temperature on the wet fin efficiency is very small. The dry fin efficiency is about 15–25 percent higher than that of the corresponding wet fin efficiency. For fully wet condition, the effect of relative humidity on the fully wet fin efficiency is also small. For partially wet surface, a considerable influence of the relative humidity on the fin efficiency is encountered. Test results for the fully wet fin efficiency agree well with some of the previous studies but disagree with some of the previous investigations. The main cause to this controversy may be attributed to the formulation of the relation between the humidity ratio and the fin temperature. The fully wet fin efficiency decrease slightly with increase of fin base temperature. However, the effect of fin base temperature on the dry fin efficiency is relatively small.

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