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# 國小二年級學生在古氏積木、錢幣、櫻桃表徵物問題下的位值概念研究

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doi: 10.6278/tjme.202010_7(2).002

## 國小二年級學生在古氏積木、錢幣、櫻桃表徵物問題下的 位值概念研究

1桃園市中壢區中平國民小學

2國立臺中教育大學數學教育系

（個位問題18 題，十位問題 18 題）為研究工具，探討二年級學生在三種表徵物（古氏積木、

（一）渾沌期；（二）建構期；（三）理解期，接著以變異數分析考驗學生在不同表徵物問題表現 的差異。本研究的主要發現為：（一）63.6%的國小二年級學生已建構二位數位值概念，達層次 三「理解期」，24.6%的學生在「建構期」，而 11.8%的學生還在層次一「渾沌期」；（二）學生在 三種表徵物的個位問題並未出現表現差異，但在十位問題上，錢幣表徵問題的表現優於古氏積 木表徵，且古氏積木表徵優於櫻桃表徵；（三）學生在非例行性十進位及一個一個數的十位問題 表現不如例行性十進位問題。

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Corresponding author：Yuan Yuan，e-mail：yuanyuan@mail.ntcu.edu.tw Received：14 September 2020;

Accepted：23 October 2020.

Tsai, H. H., & Yuan, Y. (2020).

Second graders’ concepts of place value represented by problems involving cuisenaire rods, coins, and cherries.

Taiwan Journal of Mathematics Education, 7(2), 25-44.

doi: 10.6278/tjme.202010_7(2).002

## Second Graders’ Concepts of Place Value Represented by Problems Involving Cuisenaire Rods, Coins, and Cherries

Hsiao-Hui Tsai 1 Yuan Yuan 2

1 Chung-Ping Elementary School, Taoyuan

2 Department of Mathematics Education, National Taichung University of Education

The objective of this study was to examine students’ developmental levels and performance in relation to the topic of place value. Accordingly, questions featuring three mathematical representations were used: cuisenaire rods, coins, and cherries. A total of 431 second-grade students were enrolled from four elementary schools in Taoyuan and New Taipei City. The research tool was a self-developed test with 18 questions on place value for units and tens. On the basis of the students’ test results, questions were sorted according to difficulty, and four-fifths of the questions on ones and tens were used as the criteria to create three levels for classifying students’ development in relation to place value: chaos level, construction level, and understanding level. The main findings of this study are outlined as follows: (1) 64.6% of the second-year elementary school students constructed a two-digit place value concept and reached the “understanding level,” 24.6% were at the “construction level,” and 11.8% remained at the

“chaotic level.” (2) No differences existed in students’ performance in the three representations of the problems involving units. However, on the problems involving tens, students performed better in the coin representation problem than they did in the cuisenaire rods representation and better in the cuisenaire rods representation than they did in the cherry representation. (3) For problems involving tens, students performed better in problems represented in routine manners than in non-routine and one- by-one representations.

Keyword: Cuisenaire rods, Place value, Representation, Canonical base 10

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### 壹、緒論

（Chan et al., 2014; Miura & Okamoto, 1989），Wong（2018）也研究發現一年級兒童對整數的數 量和位值的理解，會影響四年級理解分數和小數的知識。由此可知，位值的理解對於國小兒童 學習數學乃是重要的基石。

（2016）發現 7 歲兒童在學習數的位值概念時，符號式的教學方式適合高能力學生，而積木呈 現的教學方式則適合低能力學生；許舒淳（2013）研究一年級低成就學生在不同虛擬教具教學 下的進步情形，結果顯示虛擬積木組的學生進步效果比虛擬錢幣組好。教具、圖片是作為連結 抽象概念的表徵，可以幫助學生建立數學符號和意義間的連結（Uttal, O’Doherty, Newland, Hand,

& DeLoache, 2009）。由上述研究結果可知，使用不同的教具表徵進行教學，對學生的學習表現 有不同的影響。

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（一）國小二年級學生的位值概念層次發展為何？

（二）國小二年級學生在古氏積木、錢幣與櫻桃表徵物問題的位值概念表現差異為何？

### 貳、文獻探討

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1970 和 1980 年代，許多探究兒童如何發展數字系統與二位數位值概念的文獻紛紛出爐。

Kamii（1986）以皮亞傑的理論來檢視兒童位值概念的發展，皮亞傑認為數概念屬於邏輯數學知 識（logico-mathematical knowledge），此知識為兒童在心中建立對概念的理解，非直觀而得。所 以兒童之所以有學習位值概念的困難，肇因於反思抽象（reflective abstraction）的能力不同，也 就是人們在心中對數學概念所建構的意義各有不同。於是 Kamii 根據此理論對兒童進行研究，

Ross（1986）認為數字的部分與全體關係（part-whole relationship）是理解位值的先決條件，他 對60 位國小二到五年級的學生（每個年級 15 位學生）做了一系列測驗，並提出兒童位值概念 發展的五個階段：階段一，學生只認識數字的整體；階段二，學生可以分別十位與個位的位置；

「個位」數值已十分熟悉，但只有 60%的學生成功指認「十位」數值。袁媛等人（2017）以自 編的位值測驗，檢測桃園市國小一、二年級學生之二位數位值概念發展，其中只有 50%的二年 級學生已達理解個位、十位的階段。由於低年級為建立基礎數概念的關鍵期，課程內容也多著 墨於位值概念的建立，但學生十位位值概念的建立並不理想。

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1 長度線段圖與成比例線段圖呈現問題的方式

2 兩種不同的表徵(a)不規則排列的表徵(b)線性排列的表徵。引自”Materials count: Linear- spatial materials improve young children's addition strategies and accuracy, irregular arrays don't,” by J. Schiffman and E. V. Laski, 2018, Plos One, 13(12), p.4.

「十」。透過古氏積木所呈現的位值概念，可以讓兒童明白數系統，古氏積木成比例的特性也能 讓兒童明白數系統中的十進位性質（Chan et al., 2017; Hiebert & Carpenter, 1992）。然而，已具備

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### 參、研究方法

（一）位值概念測驗

(one-to-one collection)

(one-to-one collection)

(canonical base 10)

(noncanonical base 10)

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1. 古氏積木表徵物題目（測驗第 1 題）

2. 錢幣表徵物題目（測驗第 13 題）

3. 櫻桃表徵物題目（測驗第 25 題）

1. 一個一個數：如第 4 題的第 4 個選項所示，20 個積木是一個一個排列。

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2. 例行性十進位表徵：如第 2 題的第 4 個選項所示，30 個積木是以 10 個積木為一條，共 3 條 積木來排列的，以此稱為例行性十進位表徵，因其呈現方式和十進位位值相同。

3. 非例行性十進位表徵：如第 6 題的第 2 個選項所示，10 個積木是以兩條 5 個積木來排列，此 排列方式迥異於十進位呈現方式，故稱為非例行性十進位表徵。

（二）計分方式及發展層次判定

（三）位值概念測驗的預試與修正

「十位」題目的信度值為 .97。整份測驗的信度係數值達 .95。吳明隆（2013）指出研究工具的 內部一致性估計值須達 .80 以上，才普遍被接受，因此本測驗工具預試結果的信度值佳。

1. A 卷順序（黃卷）：「古氏積木」—「錢幣」—「櫻桃」

2. B 卷順序（藍卷）：「錢幣」—「櫻桃」—「古氏積木」

3. C 卷順序（粉卷）：「櫻桃」—「古氏積木」—「錢幣」

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C 卷有 139 份。研究者回收位值概念測驗後，將學生的答案鍵入 Excel 檔案中，並以資料轉換的 方式算出學生的得分，先以描述性統計呈現學生在各試題的答題表現，據以判定學生的二位數 位值概念發展層次。再依學生在三種不同表徵物問題的得分表現進行相依樣本變異數分析，以 考驗學生在三種問題表徵物問題的表現是否有顯著的差異。

### 肆、研究結果與討論

（一）二年級學生的測驗結果

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2

（二）二年級學生的位值概念發展層次

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3

Ross（1989）發現二年級學生答對6題數字對應測驗題目的人數為0，沒有答對任何一題的人 數為8人（總人數15人），占53.33%。與本研究相比，我國二年級學生的位值概念發展比該研究 的學生（美國加州）好。這有可能如Miura與Okamoto（1989）所述，亞洲語系兒童因數字語言 與十進位結構相仿，所以學生在學習位值時較能得心應手。袁媛等人（2017）發現二年級學生 約半數達到層次三，本研究一開始也是以答對題目的80%作為通過標準，而二年級學生達到層次 三的有63.6%，略高於袁媛等人的研究結果。由於後者的研究對象有較多的新住民與原住民學生

（約十分之一），而本研究對象雖為同地區學校之學生，然新住民學生身份比例較少（每班1～

2人），因此可能造成表現上的差異。另一方面，兩個研究的施測時間點也有不同，袁媛等人的 研究收集資料在4月初，而本研究在4月底至5月初，因二年級學生在課程安排上也正加強學習數 的位值概念建立，一個月左右的時間差也可能造成兩次測驗結果略有不同。

（一）學生在三種表徵物問題上的表現差異

2 = .03，代表三種表徵物問題的平均數之間有顯著差異存在。以Scheffe法進行事後比較，發現 國小二年級學生在古氏積木問題的平均得分與櫻桃問題無顯著差異，而兩者得分均顯著低於錢 幣問題。

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4

1.古氏積木問題 9.69 3.05

11.53*** 2 > 1 = 3

2.錢幣問題 9.97 2.83

3.櫻桃問題 9.65 2.88

*** p < .001

（二）十位題目中三種表徵物問題之表現差異

（SD = 2.42）。以重複量數變異數分析，考驗三種表徵物問題平均數的差異顯著性，結果F(2, 860)

= 21.55，p < .001，2＝.05，顯示在十位題目中三種表徵物問題的平均數之間有顯著差異存在。

5

1.古氏積木問題 4.16 2.46

21.55*** 2 > 1 > 3

2.錢幣問題 4.41 2.28

3.櫻桃問題 4.03 2.42

*** p < .001

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= .10。學生在十位題中的一個一個數題目上，則是錢幣問題表現最優異，古氏積木與櫻桃問題 沒有表現上的差異，F(2, 860) = 6.12，p = .002，2 = .01。而學生在十位問題中的非例行性題目 上，則是三種表徵物均沒有出現差異。另外，學生在例行性十進位的題目表現上比一個一個數 以及非例行性十進位的題目好，然而一個一個數與非例行性十進位題目兩者之間沒有差異，F(2, 860) = 13.67，p < .001，2 = .03。

### 伍、結論與建議

（一）六成以上的國小二年級學生已能掌握十位數字的意義，並能正確選出代表其數值的不同 表徵形式

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（二）二年級學生在錢幣表徵物問題上有最佳的表現

（三）學生在例行性十進位櫻桃表徵物問題表現較差

（四）學生在非例行性十進位及一個一個數的十位問題表現不如例行性十進位問題

（一）位值概念教學需善用及考量表徵物的特性引導學生學習，並強調數值的意涵

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（二）教師在做位值教學時可補強非例行性的表徵物排列方式

（2001）指出能以非例行性方式排列數字的學生，其位值概念是良好的。但學校教學所使用的 古氏積木或實物教具，通常為了教學生數字的十進位系統，會強調例行性排列的表徵，練習題 也多以此種型式出現，例如：問 78 有幾個十，幾個一。教科書和教師會傾向強調學生要以例行 性十進位方式呈現，如：7 個十，8 個一，這會無意間讓學生產生迷思，以為數字只有例行性的 表示方式。另外，在一年級數學教材中有「分與合單元」，它是介紹加減法的前置經驗。但它隱 含的概念是數字有多元的組合和分解方式，可惜教材中較強調 10 以內的分與合。所以學生在之 後學到加減法進退位問題等需要以非例行性位值概念思考數字時，就會流於機械式運算。只知 操作卻不知其運算模式是因數字的組合方式多元。所以建議教科書編輯者在編審時，可設計將

「分與合單元」與「位值」教學做融合，讓位值教學不再只強調例行性的表示方式，也讓學生在 學習以分與合為基礎的加減法進退位單元時，能融會貫通數字的多元呈現方式。除此之外，教 師在以多元表徵，如古氏積木、錢幣、花片……等呈現數字時，應注意不要只是強調例行性的 表示方式，也應讓學生理解非例行性的表示方式亦是正確的，避免讓學生產生迷思概念。

（三）未來可進一步探究學生在錢幣表徵物表現較佳的可能原因

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「世俗化」（ secularization）一詞是當下宗教社會學研究中使用

Based on Cabri 3D and physical manipulatives to study the effect of learning on the spatial rotation concept for second graders..

A study on the spatial orientation ability for sixth grader students of elementary school― using three-dimensional views (Unpublished master’s thesis). National

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