本研究目的，主要在探討開放性試題於學生數感概念的可行性，以四等分制來進行分析，透過多重Rasch 模式的分析方式，是否能發現不同年級學生對不同題目間是否有認知層次的不同。由於本文為一探索性的研究，故在題目的選擇上，先使用數感的實作評量試題，以找出現階段學生是否對數感概念在學習上是否產生落差。

　　有鑑於古典測驗理論下的分析所採用的指標，像是難度、鑑別度、信度，都會受樣本的影響(sample dependent)。也就是說，所得到的這些指標，會依據受測樣本的不同而有所差異，同一份試題施測於兩個學校，在A 校所得到的試題參數，和在B 校所得到參數有可能會截然不同。另一方面，古典測驗理論下來估計受試者的能力值，容易受到試題的影響 (item dependent)，同一位學生在一份簡單的考試卷可得高分，而在一份難的考卷卻會得低分，只要測驗的題目不同，受試者所得到的能力估計值就會有很大的差異，在不同試卷的題目，即使內容相近，不同受試者之間的成績也無法做直接的比較與對照。然而，試題反應理論能補足古典測驗理論分析的缺點，進行較嚴謹的分析。

　　多層面的Rasch 模型，是延續Rasch 模型發展而來，由於可以同時考量試題難度和評分者嚴厲度之間的關係，被廣泛的使用在許多不同的領域。

　　實作評量與另類評量(alternative assessment)及真實評量(authentic assessment)十分類似，但另類評量為廣泛的指有別於傳統的紙筆測驗，真實評量則著重於評量內容於現實生活的結合，實作評量則是重視學生參與建構、並進行實作的過程與結果。

　　數感的重要，可從Yang 與Wu (2010)提出四個理由。第一，數感可以表現出彈性、創造、合理與效率的思考模式 (Dunphy, 2007)。第二，對於數量、數字、運算的主要概念，能有效率及有彈性的應用至生活中。第三，成年人在的數學思考的發展也與數感有直接的關係，Dehaene (1997)與Berch (2005) 提及成人應要進行直覺的數感來使未來的數學思考與應用更加豐富。第四，過度的計算不僅會限制孩子在數學的思考與理解，也會阻礙孩子在數感上的發展 (Burns, 1994；Kiplatrick et al., 2001；Reys & Yang, 1998；Yang & Li, 2008)。

　　本研究從桃園縣取樣四、五與六年級，四年級有四個班級，其他年級各三個班級，每年級約一百位學生，共三百多位，以發現實作評量試題在估測概念的應用。本研究設計實作評量試題，試題性質為應用、分析、評鑑與創造等向度。每題均有三到四子題，每一題組以3 分制進行評分，最高為3 分，最低為0 分，以進行設計。得3 分者完全正確，2 分者為少部份錯誤，1 分等級者為部份正確，0分者為完全錯誤。

　　研究結果發現，有關電器組的評量目的為要求孩子用他自己的策略選擇後，2再進行計算，在此部分的成就表現，幾乎全班都有25 人答對。雖然子題二，進行了減法運算，學生也都能回答正確。於第三子題，雖出現加減法的混合運算，每班也約有20 位學生都能列出正確的算式，進行解題。由此可看出，四至六年級學生在基本計算的數概念部份有一定的學習水準，學生亦能接受此種題型的出題方式。

　　有關聰明的科南，目的要學生能覺得問題是否適當，能提出理由、判斷圖表、製作長條圖與設計問題來問同學。學生在第一子題提出很多不同的理由來贊同或反對這一問題，不論學生的理由如何，只要合理就視為正確的答案。於此，也對課本將長條圖延後至六年級的適當性，提出證據，學生可於四年級學習，而不需延到六年級。最後，要求學生由圖表來想出一個問題，有一班約半數同學都能提出，學生表現均相當優異。如學生用折線圖，用正字標記標示等，這都是相當合理的表達方式。

　　於哈利波特題組的目的在於測量孩子在時間的量感。在第一子題中發現，學生在時鐘的標示部份，出其意料的相當簿弱，一班少於十人答對。可能與學生沒有進行撥鐘的實作有關。如果學生沒有進行實作，即使學生到高年級，觀念還是不清楚。最後請學生自行設計一個上午的時間表，學生答對的表現也的確不多，每班約5 人。因為本題有很多學生產生誤解，誤認為自己可以安排行程，也就是未來在設計題目時要更加明確。但若日後學校考試能針對此一方向出題，學習表現應會更佳。

　　慶生會的目的在於測量學生對於因倍數概念與發現乘除法的關係。由於因倍數是在高年級的教學，所以預期學生在四年級時，應尚未教到本單元。然而，在學生的表現中，學生未回答的比率不高，只有四年級一班的學生有25 人，其它班級的表現都相當優質，如有提出一種的分法，二種以上分法者。在第三子題的難度雖然增加，學生仍可以用分法的方式進行解題，雖然答對人數不多，但可鑑別出較高層次的學生。

　　本實作評量的結果發現，學生在開放性問題的表現相當多元，可以測量出學生在數學方面的能力，期望在未來的大型考試中，看到這樣有深度的題目在國內中出現。
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　　The purpose of the current study is to propose the possibility to design the?open-end question to examine students’ number sense. Four ranking level were design to analyze students’ achievement. Via the muti-facet Rasch Model approach, researcher aims to examine whether different graders’ on different questions perform differently. Because this is an exploratory research, the questions were focused on the number-sense field to find whether students have difficulty in fourth to sixth grade levels.

　　In terms of the parameters of classical test theory were influenced by the sample size, such as difficulty and discriminatory. That is, the parameters will perform differently because students are in different schools. The results of students in A school are different from those in B school. Besides, the estimates of students ability are depend on items. The same student score higher on one exam, but lower on another. The parameter of ability estimator varied dependent on items. However, the item response theory can be used to avoid the problems as shown above.

　　The assumption of multi-facet Rasch model is based upon Rasch model. The model is able to take into account the item difficulty and the grader’s aspects, which can be applied to different fields.

　　The performance based assessment, alternative assessment and the authentic assessment are similar. The alternative assessment is extensively used differently with respect to the paper-pencil test. The authentic assessment stressed on the contents related to daily life, but the performance assessment emphasize on that students need to construct the knowledge by themselves.

　　The importance of number sense can be driven from Yang and Wu (2010) with four reasons. First, number sense can help students to show flexible, creative, reasonable and efficient thinking model (Dunphy, 2007). Secondly, with respect to the concept of quantity, number and operation, students can apply to their daily life.
Thirdly, adults thinking development is directly related to number sense. Dehaene (1997) and Berch (2005) proposed that adults should utilize intuition to develop their mathematics thinking ability. Fourthly, over practice not only limits students’ thinking and understanding ability, but restricts students’ development in number sense (Burns, 1994；Kiplatrick et al., 2001；Reys & Yang, 1998；Yang & Li, 2008).

　　Total were 300 participants were sampled from Chungli, Taoyuan County. Four classes students were from fourth grade, and three classes were from fifth and sixth 4 grades, respectively. The aspects of items were designed for students to analyze, apply, evaluate and create questions. The score ranged from 3 to 0. Students graded 3 were totally correct. They scored 2 were mostly correct. Students graded 1 were partially correct and 0 were totally incorrect.?

　　The results found that about the items for electricity aims to ask students to take their own strategy to solve the problems. For each class, over 25 students were correct.

　　In the second question, most students answered correctly. For the third question, although students need to add and subtract, more than 20 students’answers were correct. Most students from fourth to sixth grade levels performed correctly in the first item.

　　The second item is related to statistics data. Students need to propose the reason, evaluate the graph, to draw a bar chart and to design a question from the graph. Students gave many different reasons to approve and against the question. As their answers are reasonable, they were graded correct. The results provided the evidence that statistics chart can be taught at fourth grade level instead of sixth grade. Finally, students were asked to propose a question based on the chart. About one half of students perform well.

　　The third item aims to measure students’concepts of time. Most students calculate the time correctly, but few draw the clock time precisely. They may lack of experiences to set a clock. If students in low grade level do not have the experiences, they will make the same mistake in high grade levels. The question also asked students to design a schedule. Few than 5 students made the answers correct. It may results from that students misunderstood the problems, so that most students arranged the schedule by themselves regardless of the questions suggested.?

　　The item of birthday party is to evaluate the concept of factors and the relationship of multiplication and division. Due to related contents are taught in the high grade levels, fourth graders may have difficulty to deal with the problem. However, students in fifth and sixth grade levels did not perform well.?

　　The results found that students have shown different ideas in these four items. The design of the open-end questions can help us to diagnosis students’mathematics ability in multiple aspects. We expect to have these open-end questions applied in Taiwan Assessment of Student Achievement (TASA) in the near future.
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