第四章 研究結果
第二節 建議
根據本研究結果，工作記憶作業與概數感作業在國中階段仍有發展的空間，
部分研究也顯示，工作記憶能力和概數感能力能夠透過訓練而提升（Klein &
Boals, 2001; Wilson 等, 2006），但其研究對象為成人或國小學童，是否適用於國 中階段學生，還有待進一步研究。目前能夠確定的是，工作記憶能力和概數感能 力均能有效預測學生的數學成就，本研究結果可做為教師教學的參考依據，學生 的工作記憶容量不夠時，教師能夠提供類似「做筆記」或「列出計算過程」的策 略，藉以提昇學生在數學學習上的成就。以三年級第三次學習評量中關於機率的 題目為例，說明何謂列出計算過程極其重要性。題目如下：
平面坐標上，在－
1
＜x
＜4
、0
＜y
＜6
的範圍內，任取一點 （a , b
），且每一點被取到的機會相同。若a
、b
均為整數，則 （a , b
） 在直線方程式3x
＋y
＝6
上的機率為何？這個題目應該分為三部分，第一部分是 a、b 的值，第二部份是（a , b）為方程 式的解，第三部分為機率是多少。如果要同時處理三個部分的運算，會造成工作 記憶的負荷過大，教師可建議學生一一處理，首先第一部份根據 a、b 均為整數 的條件，可以得知 a = 0, 1, 2, 3，b = 1, 2, 3, 4, 5，要求學生先兩兩成一數對寫下 來：（0 , 1）、（0 , 2）、（0 , 3）、（0 , 4）、（0 , 5）、（1 , 1）…以此類推，共 20 組數 對。然後進行第二部分，分別代入方程式中，看看是否為方程式的解，並將結果 作記號。最後依機率的定義，計算出是方程式解的數對數量佔全部 20 組數對的 數量的幾分之幾，即為答案。這裡要強調的是，「將 20 組數對一一列出」，因為 對大部分工作記憶能力較好的學生而言，因為維持和運作訊息的能力較好，他們 不需要「將 20 組數對一一列出」，可以直接判斷數對是否為方程式的解然後算出 機率；但對工作記憶能力較弱的學生而言，因為能維持的訊息有限，使得在判斷 是否為方程式解的時候，總是會漏掉幾個數對，以致於最後計算機率時發生錯誤，
74
若教師在此時提供「將 20 組數對一一列出」的策略，即可降低工作記憶的負荷 量，使學生能順利解決問題。教學上除了提供幫助降低工作記憶負荷量的策略外，
可能還需要在教學上幫助學生熟悉降低工作記憶的策略，像是如何依序寫出適當 的算式，甚至如何排列算式等，可能都需要列入教學當中。
本研究對象為南部某市立完全中學國中部一、二、三年級學生，以校內三次 數學科學習成就評量做為學生數學成就的依據，因此本研究結論不宜過度推論至 其他年級或其他學科之學習成就。建議未來可嘗試選用其他不同的作業進行測驗，
並且持續追蹤這群受試者的數學成就，作為縱向研究的樣本，能夠更進一步確認 工作記憶、概數感與學生數學成就的關係。
關於作業設計的部分，本研究中的記憶更新作業包含了個位數的加減法，
此運算雖然非常基本，但也可能因此造成記憶更新作業的結果與數學成就有如此 高的相關，建議未來可以修改記憶更新作業，使用其他非數學甚至沒有數字的運 作，來測試一般的記憶更新能力與數學成就之間的關係，若能得出類似的效果就 可以說明在工作記憶中的記憶更新能力，是達到數學成就的重要因素。
根據本研究結果，工作記憶能力及概數感能力均能有效預測學生的數學成就，
若能透過相關作業的測驗發覺學童在數學學習上的困難，便可介入補救，然本研 究所使用的工作記憶測驗組並不適合年紀太小的孩童，因此建議未來需設計出適 合測量國小學童中央執行系統能力的工作記憶作業，將研究對象向下延伸至國小 學童甚至學齡前幼童，若能及早發覺學童在數學學習上的困難，便可及時介入、
提早實施補救教學。
75
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trial 15 slide 1 slide 2 slide 3 slide 4 slide 5 slide 6 slide 7 slide 8 slide 9 slide 10 answer
position 1 4 -1 +5 +1 9
position 2 6 +3 9
position 3 1 +2 +4 7
position 4 8 8
trial 16 slide 1 slide 2 slide 3 slide 4 slide 5 slide 6 answer
position 1 6 6
position 2 9 -5 +4 8
position 3 8 8
position 4 7 7
trial 17 slide 1 slide 2 slide 3 slide 4 slide 5 slide 6 slide 7 slide 8 slide 9 slide 10 slide 11 answer
position 1 6 -4 +6 8
position 2 4 -1 +5 8
position 3 8 -2 -4 2
position 4 1 1
position 5 9 9
82 附錄 2 句子廣度作業材料
yes response trial no response trial
外頭風聲很緊，這件東西根本脫不了手 我沒帶錢出門，不幸遇到老友才沒丟臉
83
84
85 附錄 4 數學學習評量試題
86
87
88
89
90
91 便會響起。」這些訊息。在當天基德先後輸入了 4189、1603、1881 三組號碼，
分別得到錯誤代碼 13、37 與 25。聰明的你，請幫助基德找出這兩位數字究竟 是什麼？
92
93
94
7.
3 4 2
3 4 2
(369) (468) (2280) (123) − (234) − (456) = ?
8.化簡
2(2 7) 2 1
3 6 4
x + − x + −
9. 解
2(2 7) 2 1
3 6 4
x + − x + =
，得 x＝______。10.
199 571 3999
−
的倒數與其相反數的乘積為何？95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
E 兩點。已知∠C＝90°，∠B＝35°，則 ＝________度
9. 如上圖(八)，△ABC內接於一圓，且過 B 點的切線與
AC
的延長線交於 D。若∠BAC＝30°，∠D＝35°，求 ＝_______度
10. 如上圖(九)， AB 、
CD
為圓內的兩弦，且兩弦相交於 P 點。若 AP ＝x
，CP
＝
x + 3
， BP ＝3 x − 1
， DP ＝x + 1
，則 x＝________11. 如右圖，長方形紙條中，已知 AB ＝2， AD ＝3。將此紙條沿著對角線
AC
對 摺，如圖所示。設 P 為 AD 與BC
交點，則 BP ＝________C D
B A
P
D B C
A
110
111
112
113
114
115
116
117
118
119
5.5.
5.5. 如圖7，在直角三角形ABC中，∠C＝90°，AC ＝8，BC ＝6。若I是∠CAB、∠CBA 之角平分線的交點， IE ⊥ AB ，則∠AIB的度數為 (6) 。
6.6.6.
6. 如圖8，△ABC是圓O的內接等腰三角形， AB ＝
AC
，延長CO
交 AB 於D。若∠BDC＝84°，則∠B的度數為 (7) 。
圖7 圖8
7.
7.7.
7. 如圖9，在△ABC中，已知∠C＝90°，G為△ABC的重心，
GH
⊥ AB ，若AC
＝ 32，BC
＝24，則GH
＝ (8) 。8.
8.8.
8. 如圖10，D為△ABC中¯BC 上的任一點，且G1、G2分別為△ABD、△ACD的重心，
若△ABC的面積為36平方公分，則△AG1G2的面積為 (9) 平方公分。
圖9 圖10
C
A B
E
I
120
12.12. 某班有33位同學(座號01～33)，導師利用下面的亂數表，自第三列第四行開 始，由左向右每兩位一數，抽出3位同學負責抬餐桶，則這3位同學的號碼分