The topic of similarity plays an essential role in developing students’ deductive reasoning. However, knowing how to teach similarity and understanding how to incorporate deductive reasoning and proof along with plane geometry remain a challenge to both school curriculum creators and teachers. This study identified the problems and characteristics regarding how similarity is treated in secondary mathematics textbooks in Hong Kong in the past half century. The content analysis method was used to analyze six secondary mathematics textbook series published in different periods. From the epistemological perspective of the textbook contents, our analysis shows the historical context and learning trajectories of how similarity was treated in school curriculum. The natural axiomatic geometry paradigm is not emphasized too much at different stages and most of the textbooks did not provide formal proofs of similarity. The intuitive idea was gradually con-solidated into a formal definition of similarity. Furthermore, the way that rigorous geometric de-duction can be performed from intuitive concepts and experimental geometry to the idea of proofs and formal proofs is also discussed. Copyright © 2021 by the authors.

Textbooks play an important part in teaching and learning mathematics. However, textbook research on spatial ability has so far received little attention. In this study, we investigated whether school mathematics textbooks provide enough opportunities for primary students to learn three-dimensional shapes and help them acquire spatial ability. Three mathematics textbook series used in Hong Kong were analysed through content analysis. Findings showed each series of textbooks contained five domains for introducing solid figures, which included real-life examples, understanding concepts, measurement, finding laws, and developing spatial ability. The development of spatial abilities was mainly shown in the production of three-dimensional graphics. What students experienced in the learning of 3-D shapes was relatively narrow. Geometric graphics were mainly limited to the conventional shapes at the early stage. Activities to explore three-dimensional graphics were insufficient. It was suggested that combining the surrounding world of children with geometry in the classroom was not just a criterion for textbooks but also the focus of classroom teaching. The flexibility of teachers’ adoptions of textbooks is also worthy of further investigation. Copyright © 2021 by the authors; licensee Modestum.

數學教科書的設計和內容鋪排對教師的教和學生的學均十分重要。除了學習目標、學習內容和相關的學習活動外，教科書所提供的可能的學習軌跡亦指出了如何帶領學生從一個知識點過渡到下一個知識點，這對於教師實施教學至關重要。這種學習軌跡的佈置並非一成不變的。本文以中學數學相似圖形這一課題為例，分析香港在不同時期的數學教科書中的內容編排，從中尋找該課題在教科書中佈置的歷史脈絡和變遷，並以此討論對於該課題合理的教學鋪排方式，也希望對其他數學課題的教學有所啟示。 Copyright © 2020 國立臺灣師範大學數學系、台灣數學教育學會.

數學是一門具有抽象性的學科,數學知識表徵是教材編寫的關鍵,教材對知識進行合理表徵,有利於抽象的數學知識直觀形象化。香港地區在教材編寫方面吸收了英國教材編寫的經驗,但我國對香港地區的教材研究相對較弱。因此,本研究選取西師版教材與香港版教材,比較分析兩版教材在數學知識表徵上的異同,汲取香港教材在教材知識表徵方面編寫的經驗與長處,為修訂與改進本國教材以及教師對教材知識表徵的處理提供有價值的參考意義,優化教師的教學方式,促進數學教師的發展。本文採用了文獻研究法、文本分析法、案例分析法和比較研究法等研究方法。基於數學表徵的含義,建立教材知識表徵的比較框架,對西師版教材與香港版教材的數學知識表徵形式、多元性以及關聯性進行靜態比較,再以大陸教師與香港教師的課堂錄像轉譯成文本形式,統計分析兩位教師在代數課和幾何課中對教材知識表徵的使用情況,對教材知識表徵進行了動態比較,最後根據研究結果對我國教材的編寫提出了參考建議。本研究得到了如下的結論:(1)教材知識表徵形式方面,兩版教材都是以文字表徵為主,但香港版教材比西師版教材更加注重圖像表徵。(2)教材知識表徵多元性方面,香港版教材比西師版教材更加注重多元性。(3)教材知識表徵關聯性方面,兩版教材都比較注重知識表徵形式的關聯性,並都以並列關係和承接關係為主,但香港版教材並列關係所佔比例大,而西師版教材承接所佔關係比例大。(4)教師對教材知識表徵使用情況方面,大陸教師和香港教師對教材知識表徵進行完全使用的情況較多,香港教師在對教材知識表徵進行改編使用的部分比大陸教師多,在教材知識表徵的創新使用相對比大陸教師少。根據上述研究結論,為我國教材的編寫提出了以下參考建議:第一,注重加強教材的多元表徵,促進學生對知識的內化;第二,注重教材知識表徵形式間的聯繫,促進多元表徵形式的轉換;第三,注重教材知識表徵形式的課堂處理,促進數學知識的建構。

我國香港地區和大陸有著相同的文化底蘊,在數學教育理念方面有很多相似之處。21世紀以來,香港在借鑒國外教育經驗和本地教育實際的情況基礎上進行了多次課程改革。在最新公布的TIMSS排名中,香港的學生在數學上取得了不錯的成績。函數是高中數學課程中重點內容,函數思想貫穿著整個高中數學的學習。三角函數是高中數學初等函數中的一個重要部分,學習三角函數可以加深學生對函數內容的進一步理解,同時也可以培養學生的邏輯思維能力和數學建模的思想,提高運用所學知識解決實際問題的能力。本文將運用課程難度模型對大陸普通高中人教A版教材與香港高中數學牛津版教材的三角函數部分的課程難度進行比較研究。研究的主要內容和結果:1.運用史寧中教授的課程內容難度模型對三角函數部分的課程難度進行了比較研究,得出:在課程時間上,人教A版的三角函數課程學習時間是32課時,牛津版的三角函數課程時間是25課時;在課程廣度上,人教A版的三角函數課程廣度比牛津版的三角函數課程廣度要更大;在課程深度上,人教A版的三角函數課程深度稍大於牛津版的三角函數課程深度。2.運用了鮑建生教授建立的數學題綜合難度模型對人教A版和牛津版教材三角函數部分的習題進行了比較研究,從探究、背景、運算、推理和知識含量五個因素上對習題進行了分析。人教A版教材在“探究”“背景”“運算”三個因素上高於牛津版教材,在“推理”“知識含量”因素上牛津版教材略高於人教A版。兩個版本的綜合難度模型的雷達圖大致的形狀基本上相似,在“運算”水平上人教A版的加權平均明顯要高於牛津版的,說明人教A版的教材習題計算量相對而言比較大;在“推理”水平上,牛津版教材相比人教A版更高,說明牛津版教材在習題設置上會更注重學生的推理過程。

中國內地自2001年推出基礎教育新數學課程後，於2012年頒佈了新的修訂版。數學教育的目標從“雙基”擴展至“四基”，亦即由“基礎知識”“基本技能”增加了“基本活動經驗”及“基本思想”。這一改變符合國際數學課程的大趨勢，也是課程研究中“兩個p”的兼容性的問題。其中一個p表示學習結果（product），包括知識內容的學習；而另一個p表示學習過程（process），包括內在能力的培養。故此，探討兩個p之間的關係是值得我們深入探究的。這個課題並不局限中國內地，也不局限於數學科。本文以定性的內容分析法分析了香港三套常用的小學數學教材，以立體圖形這一課為例，解釋了它們在各環節的側重點和教學脈絡，最後給出了教學上的啟示。 Copyright © 2014 首都師範大學出版社.

What is learning and how could the learning in mathematics be enhanced has fallen into debate for many years. Singapore rank number one in both TIMSS and PISA in 2015. What are the insights we could earned from such competition? Regarding to the facts that Singapore is one of the first countries in implementing the metacognition into their mathematics curriculum in 1990s (Chan & Leung, 2013), metacognition may possible be the answer we are searching for. There is no doubt that textbooks take a very important role in this. White & Mesa (2014) indicated that lecturer rely highly on textbook in their teaching especially the tasks assigned. However, researches with textbooks related to mathematical content and researches on their potential in developing metacognitive knowledge are limited (Mesa, 2004; Mesa, 2010). Especially, comparisons between metacognitive strategies among textbooks in different countries are very limited. Thus the following questions are proposed to be investigated. 1. Does mathematics textbook emphasis enough metacognitive strategies in enhancing students' metacognition? 2. Does the pattern of using metacognitive strategies in mathematics textbooks different among countries/cities? The assessment is divided as seven categories: think aloud, questioning, self-questioning, error detection, group activity, alternative strategies and reflection. It is identical to those suggested by Chan and Leung (2013). Result shows that think aloud is the most popular strategy used in textbooks among Singapore, British and Hong Kong. However, other strategies weight about only 20% in total. It is suggested that more metacognitive ingredients could be added in textbooks in enhancing students' metacognition.

以古德萊德對課程理解框架為參考，此研究從理解和實施的課程層面，調查和比較了北京、臺灣、香港三地數學教師的教科書使用情況，包括教師的備課(教科書內容的理解與評價)和課堂教學(教科書內容的選取、調適和運用)兩部分，並據此分析了三地數學教師與教科書之間的關係。研究發現，從教師對教科書的滿意度和忠誠度的角度來看，三地數學教師明顯呈現不同的教科書使用風格。研究結論對於課程實施和教師專業發展實踐都具有重要的參考意義。
[Copyright © 2017 湖南師範大學.]

Use of Textbooks: Textbooks, which reflects the curriculum, are an important informant to portray the intended curriculum (Johansson, 2003). They are absolutely a valuable source for teachers (Stodolsky, 1999; Remillard, 2005; Tarr et al., 2006). The research on textbooks benefits substantively to the understanding of a curriculum in a particular region (Thaqi & Gimenez, 2016). Outstanding Performance in Some Regions: Mathematics learning in some Asian regions interests many educators, as the students from these regions have outperformed others in several worldwide mathematical assessments. The PISA report highlights that more than a quarter of students from each of the top-performing regions: Beijing-Shanghai-Jiangsu-Guangdong (China), Hong Kong (China) and Chinese Taipei could mathematically verbalize complicated problems (Organization of Economic Co-operation and Development, 2016). Another renowned international assessment TIMSS also shows that almost half (i.e. 47-49%) of students at Secondary 2 in South Korea, Taiwan and Hong Kong reach the advanced international benchmark (Mullis et al., 2012). The textbooks adopted by these Asian countries or regions are definitely worth referencing. Textbook Content Analysis: This study selected one textbook from each of these Asian regions and their content was decoded according to a proposed framework called Assessment-Background-Content textbook analysis, which was developed from the textbook context analysis of Incikabi & Tjoe (2013), Huntley (2008), Alajmi (2012) and National Research Council (2004). Textbooks were coded according to their lesson components. Pickle (2012) adapted from Jones’s (2004) coding suggested ten lesson components in identifying units in the textbooks. The ten components include description, definition, question, activity, worked example, oral exercise, follow-up practice, highlight, project, and reflection (Pickle, 2012, p.51). Textbooks in Asian Regions: The proposed presentation will compare the

在2012及2015年學生能力國際評估計劃（PISA）考試中，亞洲地區包括上海、香港、新加坡及台灣的學生（平均15歲）的數學能力名列首五名，美國、英格蘭等歐美國家則處於中游位置。從這項考試數據顯示，美、英兩地學生的數學能力不及亞洲地方。硏究顯示，不同地方學生的數學運用差異與身處的學習環境有一定關係；此外，他們如何學習數學、應用數學邏輯亦會影響評估表現（OECD, 2012）。課堂上，教科書是輔助老師教學的主要工具。Nciol & Crepo（2006）指出，教科書對學生數學學習表現有重要影響，亦在教師教授數學時扮演關鍵角色。Stein、Remillard與Smith（2007）研究進一步發現，教科書内容中的排序、組織、焦點會影響數學老師的教學及學生如何學習數學。 因此，本文對照香港、新加坡、美國及英格蘭的初中數學教科書，探討當中異同。透過20多篇探討四地初中數學教科書的文章，檢視學生數學成績差異與教科書編撰的關係。全文將從教科書的課程目的、課題內容、練習題目及使用等四方面分析問題。