根據本研究的結論，提出以下三點建議供後續相關研究者或執行單位 參考:
(一)數學補救教學的實施應兼顧教師與學生中 /6 '並提供學生表達想 法的機會
文獻指出，對於數學學習困難的學生，除了提供明確且有系統的教學 示範外，也應提供學生視覺化表徵以及表達自己想法的機會( Doabler et al., 2012)。意即，兼顧教師的示範與學生的探索，才能培養學生完整的數學 知識與能力。但本研究發現，現場教師實施攜手計畫數學補救教學時，大 都以教師的示範、講解、練習與回饋為主，較少採以學生為中心的方式進 行，這將使低成就學生的學習緊焦在技巧的熟練，而非在概念的理解與探 索，這恐將影響學生未來高層次數學的學習。因此未來在進行相關師資培 訓時，應針對有效數學補救教學的策略進行探究，並提供教師相關的協 助，以利數學補救教學的實施與成效。
當代教育研究季刊第二十三卷第一期， 2015 年 3 月，頁 113-147 137 Contemporary Educational Research Quarterly Vol.23,No. I
(二)規劃與發展合適的數學補救課程
本研究發現，教師在補救教學時並未精確地根據學生的起點行為，來 進行課程的規劃與決定，且教師表示執行數學補救教學最大的困難之一，
便是缺乏適合學生程度的數學補救教材。因此，發展適合不同學習程度學 生、且符合數學概念階層的國小數學補救課程，是目前數學補救教學實施 的當務之急。數學補救課程的發展，或許可以採用徐偉民 (2011e) 發展 的模式，以能力指標的分年細目為基礎，根據分年細目中呈現的數學概念 結構，由基礎概念到進階應用的歸類與重組，形成不同回合的學習主題，
以適合不同程度的學生;或根據目前教育部已經公告的數學補救課程
(http://priori.moe.gov.twl) ，結合診斷評量的結果，在未來師資培訓上，
協助教師根據診斷結果與教育部公告的補救課程，來進行不同單元或概念 的編排，在符合學生起點行為與數學概念邏輯結構的前提下，發展適合低 成就學生的數學補救課程。因為課程的合適程度，將是影響數學補救教學 成效的關鍵。
(三)結合校內外資源，採取行動研究解決教師在數學補救教學中的困難 大多數教師表示，補救教學實施有困難時，會尋求學校行政與其他教 師的協助，很少尋求校外專家的協助。因此建議學校可以結合校內外的資 源，與校外學者專家(如大學教授)共同合作，透過行動研究，針對數學 課程決定與教學實施兩個面向，來構思合適的行動方案，並在嘗試與反省 中持續修正，以使教育公平性的理想可以真正落實。
誌謝
本文感謝科技部提供經費協助(計畫編號:NSC 99-2511-S-153-013)
與參與研究的教師。本文的論點為作者所有，不代表科技部。
138 徐偉民、 ill曼麗:國小攜手計查數學補救教學課程決定與教學實施之探究
[ Summary ]
INVESTIGATION ON CURRICULUM DECISION AND INSTRUCTION IMPLEMENTATION OF MATHEMATICS REMEDIAL INSTRUCTION AT AFTER-SCHOOL ALTERNATIVE PROGRAM IN
ELEMENTARYSCHOOLS
INTRODUCTION
An after-school remedial instruction program has been implemented in Taiwan since 2006 to give remedial instruction to students who are behind in school learning,provide all students with an opportunity for fruitful learning, and fulfill the goal of education equity. Mathematics, in particular, is one main subject of the remedial instruction program. However,previous research in Taiwan rarely probed into how the mathematics remedial curriculum was determined by a teacher and how a teacher implemented mathematics remedial instruction. Considering that curriculum decisions and the implementation of ins甘uction are key influencing factors of whether the instruction of remedial mathematics is effective, we propose to center on curriculum decisions and the implementation of instruction in this study, understand how teachers give mathematics remedial instruction and teachers' concerns in the instruction by means of observation and interview,as well as find out how teachers determine the content of mathematics remedial curriculum by means of questionnaire surveys, how teachers caπY out remedial instruction, and difficulties encountered in the two dimensions of remedial instruction in order to have a better understanding of the current status of mathematics remedial instruction in Taiwan.
METHODS
Data for this study was collected in the 2010 academic year. A total of eight teachers participated in the observation and interview while 300 teachers participated in the questionnaire survey. With reference to a framework for observing mathematics instruction proposed by Artzt and Armour-Thomas (2002) and Stein, Remillard and Smith (2007) emphasis that mathematical problems should be the core of mathematics instruction,we formulated a scale
當代教育研究季刊第二十三卷第一期· 2015 年 3 月，頁 113-147 139
Contemporary Educational Research Quarterly Vol.23,No.1
for observing mathematics remedial instruction and proposed to understand the current status of mathematics remedial instruction from the three perspectives: Mathematical problem,learning environment, and discourse. A semi-structured interview was employed, and the outline of the interview encompassed three aspects: Teachers' considerations in deciding the content of mathematics remedial curriculum, the implementation of mathematics remedial instruction and assessments,and difficulties that teachers perceived in course of instruction. In addition to teachers' basic information (gender, teaching experiences,educational degree,etc.),the questionnaire covered 11 question items in the three aspects: Curriculum decision, instruction implementation, and difficulties encountered in curriculum and instruction.
Subsequently, statistical analysis was performed based on the number and percentage of teachers' answers to the questionnaire.
RESULTS
1. Implementation of Mathematics Remedial Instruction and the Teachers' Concerns
Mathematics remedial instruction of the eight teachers who were selected as subjects in this study is presented in Table 1. Judging from the data in Table 1, mathematical symbols were heavily used to present mathematical problems by the eight teachers when giving mathematics remedial instruction, mathematical problems involving mathematical calculations accounted for the majority of all mathematical problems,and the instruction was largely given straightforwardly. With respect to instruction methods and teacher-student interaction, most teachers explained and demons仕的ed to the entire class before letting students practice solving mathematical problems individually and giving students individualized instruction, and there was no interaction among students. Factors in relation to instruction and curriculum were teachers' most frequent concerns when giving mathematics instruction, respectively accounting for 22.7% and 18.5% of all teachers' concerns.
Teachers' third concern was students' learning performance and attitudes, which accounted for 16.6% of all teachers' concerns. The three concerns all together accounted for 57.8% of the interviewed teachers' responses, indicating that the three factors were teachers' foremost concerns when giving mathematics remedial instruction.
140 徐偉民、劉曼麗:國小攜手計查數學補救教學課程決定與教學實施之探究
Table 1
The statistical table ofteachers' mathematics remedial instruction
Dimension / Teacher ObTl的ObT2的ObT3叫ObT4的ObT5
m
ObT6的ObT7(2)ObT8州The numberof 角
questIOns
Problem Representation S NJ P
Motive D D D
Note1. ObTl(6)stands for the observation ofTlteacher's instruction
of6月 grade ，
and so forth.L-G L-G L-G
Note2. NJ stands for“unable to determine"; S stands for "symbolic representation",P stands for“iconic representation"; D stands for“starting directly"; B-D stands for
“reviewing prerequisite knowledge prior to starting directly"; E stands for“life experience"; C stands for
“
conceptual problems", A stands for "calculation problems",R stands for“application problems"; L stands for“giving a lecture to the entire class",Gstands for“individualizedins甘uction弋 L司Gstands for“giving a lecture to the entire class prior to giving individualized instruction"; Wstands for“interacting with the entire class",Istands for "interacting with individuals", W-I stands for “interacting with the entire class prior to interacting with individuals".2. Teachers' Curriculum Decision and Instruction Implementation of Mathematics Remedial Instruction
Given that only 277 out the 300 teachers who participated in the questionnaire survey had perfo口ned mathematics remedial instruction, we only analyzed the 277 teachers' answers to the questionnaire. Teachers' thoughts on curriculum decisions of mathematics remedial instruction and results of teachers' contemplation are shown in Table 2,which revealed that teachers set the objectives of a remedial curriculum prim訂ily based on students' existing mathematical knowledge and skills and secondarily on teachers' previous teaching experiences. With respect to teachers' decisions on the curriculum content, most teachers used textbooks for students in that grade as the core content of remedial instruction and added supplementary teaching materials now and then. Meanwhile, 25.3% of teachers used textbooks for students in other grades as materials for remedial instruction to better suit students' existing mathematical knowledge and skills.
當代教育研究季刊第二十三卷第-期， 2015 年 3 月，頁 113-147 141 Contemporary Educational Research Quarterly Vol.23,No. I
Table2
The statistical table ofteachers' curriculum decisions ofmathematics remedial instruction Dimension Decisions on curriculum0峙的卸的 Decisions on curriculum content
Guidelines _.. _. . _ Textbooks~ Textbooks and
Objectives Objectives Personal C"''''__'n' .~~.:;~..~Textbooks
d J L - o f of teaching p:122ce stuL ofother d ; L s u t Z : r rnaKIng curriculum teachlng expenences grade grades matEnals
Fr官quency 82 88 145 245 176 70 88 172 Percentage 29.6 31.8 52.3 88.4 63.5 25.3 31.8 62.1
The statistical results of the implementation of mathematics remedial instruction are revealed teachers' remedial instruction centered on enabling students to understand mathematical concepts and performing basic mathematical calculations, emphasized on association with students' life experiences, and adopted individualized instruction as the principal mode of instruction. Except for differences in the presentation of instruction and results at the stage of observation, both the focuses of instruction and the implementation of instruction were consistent with results of observation.
Also, 85.6% and 77.6% of teachers expressed encountering difficulties in remedial curriculum decisions and the implementation of remedial instruction, the primary reason being a great divergence in individual students' mathematical knowledge and skills.