本研究主要是嘗試從基模的層面來評估國中小數學資優生的數學解題能力，為此，研究者自行研發三份評量工具，分別是耕作問題評量工具、活動範圍評量工具和倒著走評量工具，當中牽涉到比例、面積、周長、耕作問題、時間等數學概念。本研究的研究對象為24名國中小數學資優生，當中大部分曾代表台灣參加國際性的小學數學競賽。耕作問題和活動範圍兩份評量工具是以全體24名學生為受試對象，此二次施測各歷時一小時。這群受試者中有五位更接受倒著走評量工具的評估，並於其解題後進行半結構性的訪談，藉以了解其解題思路，全程歷時一個半小時，並有錄音、錄影。
至於，耕作問題和活動範圍這兩份工具的評估重點是要評量學生在相關基模知識、瞭解題目資料的程度、猜題、分解題目以及解題等層面的表現，學生須獨自作答，並紀錄在各項目中作答所耗費的時間；而倒著走評量工具的評估重點是想了解相關基模知識在解題過程中所產生的干擾情形，並以個別施測與訪談的方式進行，且在訪談過程中鼓勵學生盡可能說出自己的想法。
在耕作問題與活動範圍評量工具的資料分析方面，除了採用量的方法分析學生在各項目的表現是否與解題表現有關之外，並以質的方法分析三組具有強、中、弱基模知識的學生，比較他們解題的表現有什麼實質的不同，最後再從強和弱基模兩組中各挑選一名作深入的分析。在倒著走評量工具的資料分析方面，則先分析五名學生對於數字、文字、圖形和符號在心智中逆向思考的能力，然後分別從時間對應流程圖及訪談原案的角度來比較學生解倒著走問題的表現。
從整體的印象來說，受試者在耕作問題上的解題表現較優於活動範圍。從量方面的分析得知，在耕作問題評量工具上，受試者在瞭解題目資料的程度和解題表現之間有顯著相關；在活動範圍評量工具上，則是受試者所具備相關基模知識和分解題目與解題表現之間有顯著相關。
從耕作問題及活動範圍之質的分析結果發現，基模知識較薄弱的學生，其基模知識似乎存有較多的迷思概念，且概念與概念彼此之間缺乏連結；而基模知識連結較豐但並不精緻的學生，其基模知識中的迷思概念雖似乎少於基模知識較薄弱的學生，但其精緻的知識卻遜於基模連結十分豐富的學生。至於基模知識十分豐富的學生，其相關概念較精緻，概念之間的連結較豐。
至於，倒著走評量工具之分析結果，有資料顯示若受試者相關的基模知識太強的話，有可能反而成為解題過程中的絆腳石。
根據以上的研究發現與心得，本研究建議教師宜在教學過程中將數學概念做更精緻解釋，且提供多元的方式來呈現教材內容，並考慮採用類似本研究的評量工具，以作為認知診斷之用。至於日後的研究可加以考慮學習風格、後設認知的能力、家庭背景等因素作更全面的探討。

The purpose of this study is to explore the possibility of a new approach to assess mathematical problem solving ability from a schema perspective on a group of sixth and seventh grade mathematically gifted students. This is done by developing three specially designed problems. The basic concepts behind these problems included ratio, area, length, time, etc., that are all covered in the mathematics curriculum of the elementary school. A total of twenty-four subjects were involved in this study, and most of them had represented Taiwan to participate in international mathematical competitions. All subjects were required to solve the first two problem sets within an hour. As for the last problem, it was solved by five specially chosen students, which was then followed by a semi-structured interview. The whole process was videotaped and last for one and a half hours in order to further understand their reasoning process.
The first two problem sets were so designed as to evaluate the extent of related schema knowledge of the subjects, as well as their ability to identify the given data in the problems, ability to guess what question will be asked based on part of the information of the problem, ability to decompose the problems into a set of small problems, as well as their problem solving ability. These two problem sets were required to be solved individually and their solution times were recorded for each subproblem. The purpose of the last problem is to understand the role of related schema knowledge regarding whether it will enhance or disturb students’ problem solving performance.
Both quantitative and qualitative procedures were adopted for analysis of the first two problem sets. In particular, students were classified into three levels of schema knowledge as well as three levels of problem solving performance. A detailed analysis was then performed to compare if there was qualitative differences among students of difference schema levels with respect to the quality of their solutions. As for the last problem, after assessing the abilities of reverse thinking of the five students, a detailed analysis was then performed to compare their problem solving strategies by visualizing their solution process.
For the first problem sets, it was found that the students’ ability to identify the given data in the problems was highly correlated to their problem solving performance. As for the second problem sets, the extent of their related schema knowledge and their ability to decompose the problem into smaller problems were found to be highly correlated to their problem solving performance. In general, it was found that for those students with weaker connections among concepts, their schema knowledge were also more choppy, and that they were weaker in terms of deriving extra information from the given data in the problems. Meanwhile, they tended to be less careful with respect to breaking down the problems into smaller problems, as well as less able to implement their plans nor to solve the problems. As for those with medium schema knowledge, they were more able in extracting extra information from the given information, as well as representing the problems in another formats. Yet, such treatments of the problems were usually not very much in depth, and the success rate was not very high. Finally, for those students with high schema knowledge, they exhibited very good connections among concepts, and could utilize, to a better extent, their schema knowledge together with the given information of the problem to generate useful information to solve the problems. Further more, they could break down the problems into subproblems in a more extensive way. Finally, they could implement their plan better and achieved better rate of success.
The major finding with respect to the last problem was that strong schema knowledge could sometimes hinder the process of problem solving, by channeling the solution into a wrong direction.
The major suggestions of this study were that teachers should consider providing more accurate explanations of mathematical concepts, provide multiple representations of the problem, and consider using a comprehensive way to assess students’ mathematical cognitive knowledge, parallel to the examples used in this study. Topics for future research were also suggested at the end of this report.

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