本研究的目的是希望探討國小學生對於因數與倍數之學習進程，並初步發展因數與倍數學習進程評量試題。基於本研究之研究目的，研究者進一步提出兩點研究問題，陳述如下：
1.國小學生對於因數與倍數之學習進程的內容為何?
2.本研究所初步開發之因數與倍數學習進程評量試題的可行性為何?
在研究流程方面，研究者首先對國內外學習進程的相關文獻進行討論，以釐清學習進程的意涵、特徵與研究方法；再根據所討論出的學習進程研究方法發展國小因數與倍數學習進程架構，並透過評量施測的方式檢驗此架構的內容。
本研究所採用的研究方法是文獻分析法與調查研究法。研究者係以過去對於因數與倍數的相關研究文獻為基礎，同時為兼顧過去的研究結果與現階段學生的實際表現情形，研究者針對高雄市某國小6位六年級學生以及二、三、四、五年級各2位學生，共14位學生進行訪談，以瞭解不同年級學生在因數與倍數相關概念上的表現，並協助發展初步的學習進程架構。
擬定初步學習進程的內容後，研究者將以此學習進程的內容進行試題的開發，對用以測量學生在學習進程中所屬階層的評量試題進行初步的發展。在評量施測階段，研究者蒐集高雄市某五所國小三、四、五、六年級共619位學生做為研究對象進行施測，以檢驗研究者所初步發展之學習進程與學生的學習表現是否相符。
本研究一共發展出三套學習進程，包含整除概念學習進程、因數概念學習進程以及倍數概念學習進程，研究的結果發現：（1）在整除概念學習進程中學生必須先初步理解整除的概念，才能進一步掌握「a被b整除」與「a整除b」的語言使用；（2）在因數概念學習進程中，學生必須先從乘、除法算則理解因數關係，才能進而掌握因數的概念，之後開始留意到兩數共同的因數，而發展出公因數的概念，最後能夠從公因數的概念理解中，精緻出最大公因數的意涵；（3）在倍數概念學習進程中，學生必須先從乘、除法算則理解倍數關係，才能進而掌握倍數的概念，之後開始留意到兩數共同的倍數，而發展出公倍數的概念，最後能夠從公倍數的概念理解中，精緻出最小公倍數的意涵；（4）學生在初步掌握因、倍數概念的意涵時，即能夠開始留意到「因、倍數互逆」的關係，並且隨著學生對於因、倍數概念的掌握程度，學生能夠進而對「因、倍數互逆」的關係進行瞭解與交叉應用；（5）學生能夠解決高層次的例行性試題，但卻無法解決較低層次的非例行性試題。
本研究所初步開發的學習進程評量，在整除概念試題的部分，能夠將498位學生進行學習進程階層的歸類，占總樣本數的80.4%；在因數概念試題的部分，能夠將348位學生進行學習進程階層的歸類，占總樣本數的76.7%；在倍數概念試題的部分，能夠將453位學生進行學習進程階層的歸類，占總樣本數的73.2%；研究者認為本評量工具在測量學生學習進程階層的功能上，具有一定程度的可行性，並且在進行調整與修正後，將能夠對學生在學習進程中的階層表現有更高的解釋能力。
根據上述的研究發現，本研究建議，未來的研究者可以針對學生的學習採用追蹤性的研究方式，以本研究所發展的學習進程內容，針對學生在整除、因數或倍數概念的學習過程進行更深入的探討，在評量工具的設計與使用方面，試題設計應兼顧例行性試題與非例行性試題，並建議使用更多的評量試題以及多元的判準原則來判斷學生的所屬階層。

The purpose of my dissertation is to look for the elementary school students’ learning progressions on the divisor and multiple and to develop initially the assessments about the learning progressions on the divisor and multiple.
On the base of my research purposes, I have two research questions as follows.
1.What are the contents of the elementary school students’ learning progressions on the divisor and multiple?
2.How well are my initial assessments concerning the learning progressions on the divisor and multiple?
When it comes to my research procedures, I discussed the relative paper, native or foreign, at first so as to clarify the meaning, the character, and the research mode of the learning progressions. Then I based on the discussed research modes to develop my learning progression instrument for elementary school students on the divisor and multiple, and I tested my instrument by testing some elementary school students.
My research methods are survey research and documentary analysis. Based on the relative research paper about divisor and multiple and based on the past research results and based on the nowadays students’ performance, I asked fourteen students, including six 6-grade, two 2- grade, two 3- grade, two 4-grade, two 5-grade students in Kaohsiung, so that I may understand those different grades students’ performance on the concept of divisor and multiple, and that would help me develop my learning progression instrument.
After my initial instrument, I built my assessment to test the attributive level of the students. In this stage, there were 619 students, ranging from 3-grade to 6-grade of five elementary schools in Kaohsiung, participating my assessment and finally I tested their scores and their performance to see whether they agreed with.
There are three learning progressions in my research, including the learning progression of concept of being divided with no remainder, the learning progression of concept of divisor, and the learning progression of concept of multiple. Based on my result, I found that (1) on the learning progression of concept of being divided with no remainder, the students had to clear the concept first, then they would differ the sentence “A is divisible by B” from the sentence “A divides B”; (2) on the learning progression of concept of divisor, students have to understand the relation among every element from multiplication and division first, then they will know the concept of divisor; after they master the concept of divisor, they look out that a number may, at the same time, be a divisor of two different numbers, and they will develop the concept of common divisor; at last, they extract the meaning of great common divisor from understanding the concept of common divisor; (3) on the learning progression of concept of multiple, students have to understand the relation among every element from multiplication and division first, then they will know the concept of multiple; after they master the concept of multiple, they look out that a number may, at the same time, be a multiple of two different numbers, and they will develop the concept of common multiple; at last, they extract the meaning of least common multiple from understanding the concept of common multiple; (4) when they first understood the meaning of divisor and multiple, they knew the inverse relation of divisor and multiple; also, with their extent about those concepts, they may apply the concept of inverse relation of divisor and multiple; (5) students could solve the highest level routine examinations, yet could not solve lower level non-routine examinations.
Based on the assessment, I distributed 498 students (80.4%, accounting for all 619 students) to my learning progression of concept of being divided with no remainder, 348 students (76.7%, accounting for 454 students) to my learning progression of concept of divisor, and 453 students (73.2%, accounting for 619 students) to my learning progression of concept of multiple. I thought that my assessment did work for some extent, and after some adjusting and revising, it may explain much more about students’ levels of learning progressions.
From the above result, I suggest that the future researcher use my learning progression instrument to track students’ learning circumstance so that they may do some more conferring. As for the assessment, I suggest that the future researcher give consideration on routine tests and non-routine tests, as well as use more quantity of tests and multi-principles to judge which level the students have.

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