High School Students’ Understanding Of The Exponent And Exponential Function

The exponent and exponential function are important concepts that play important roles in high school mathematical curriculum. They are closely connected with the logarithm and logarithmic function and also of great help to the learning of advanced mathematics. The curriculum standards of high school mathematics both abroad and at home have clear regulations and requirements on the learning of the exponent and exponential function. As one of the indispensable part in the College Entrance Examination, the exponent and exponential function are also widely applied in our daily practical life.There are few studies on students’understanding of the exponent and exponential function in our country.The aim of this thesis is to describe and analyze students’levels of understanding of exponent and exponential function within the context of APOS theory and to explore how familiar students are with the magnitude of exponentiation intuitively. The major research questions are specified as below:(1)In what aspects do the high school students have obstacles of understanding exponent? How can their levels of understanding of exponent be categorized? What are the correspondent manifestations? Are there any differences between the research findings of this thesis and the foreign studies of the same kind?(2) What are the high school students’ levels of understanding of exponential function? What are the correspondent manifestations? Are there any differences between the research findings of the thesis and the foreign studies of the same kind?(3) How is students’ the estimations of exponentiation intuitively at different ages? Are there any differences between the research findings of the thesis and the foreign studies of the same kind? Or any historic similarities?In this thesis, a questionnaire survey was conducted among307high school students at three different grades in a national key high school in Yancheng, Jiangsu Province. The data are handled through LatentGold, the software of latent class analysis (LCA). From both quantitative and qualitative analysis, the following conclusions can be drawn:(1) High school students have a good command of the concept of exponent. For them, the most difficult part is the power of negative integer exponent. With the help of the LCA software, the results suggest that two levels of understanding of exponent can be identified. The two clusters are action level and object level. At the former level, students feel it more difficult in the extension of the prototype, especially the exponent with the form ax when x is a negative integer or a fraction. Compared with the findings of the foreign studies of the same kind, the Chinese students have deeper understanding of exponent, especially the fractional exponent.(2) High school students’representation of the exponential function lacks diversity, most of which focus on some principles and properties. The analysts of LCA software reveals that students are classified into two groups when understanding exponential function, which are action level and process level. At the latter level, students can understand the exponential function as a process of exponentiation and can derive the properties of exponential function. Compared with the findings of the foreign studies of the same kind, foreign students have more limited understandings of exponential function, only reaching the action level.(3) Students have an underestimation of the exponential process. Students at the three different age brackets have different patterns of estimation of exponential expression and have different characteristics. The result is in accordance with the findings of foreign studies. Meanwhile, historical parallelism exists in the students’patterns of estimation.Based on the conclusions above, some teaching suggestions are put forward as below:(1)Over emphasis of the prototype of exponent "positive repeated multiplication" should be avoided in teaching which the prototype of exponent can be extended.(2) Representation of understanding exponential function should be enriched so as to help students internalize the properties of function rather than to memorize them inflexibly.(3) Students’ estimation ability should be improved as much as possible in students’ learning of exponential process.