龔鵬程對話海外學(xué)者第一百零六期：在后現(xiàn)代情境中，被技術(shù)統(tǒng)治的人類社會，只有強(qiáng)化交談、重建溝通倫理，才能獲得文化新生的力量。這不是誰的理論，而是每個人都應(yīng)實踐的活動。龔鵬程先生遊走世界，并曾主持過“世界漢學(xué)研究中心”。我們會陸續(xù)推出“龔鵬程對話海外學(xué)者”系列文章，請他對話一些學(xué)界有意義的靈魂。范圍不局限于漢學(xué)，會涉及多種學(xué)科。以期深山長谷之水，四面而出。

加布里埃爾·斯蒂利亞尼德教授（Professor Gabriel Stylianides）

牛津大學(xué)數(shù)學(xué)教育教授，教育系博士研究項目主任。

龔鵬程教授：您好。您的研究真是學(xué)生們的福音。您知道的，學(xué)數(shù)學(xué)，曾是很多地區(qū)學(xué)生的共同噩夢。特別是推理證明，很多小孩會感到困難。您認(rèn)為這該怎么處理？

加布里埃爾·斯蒂利亞尼德教授：龔教授，您好。您講的沒錯，數(shù)學(xué)學(xué)習(xí)對許多兒童來說是困難的。這主要與學(xué)校的數(shù)學(xué)教學(xué)方式有關(guān)，傳統(tǒng)上強(qiáng)調(diào)記憶事實和應(yīng)用程序，而這對學(xué)生來說沒有什么意義。這種傳統(tǒng)的數(shù)學(xué)教學(xué)方式，使學(xué)生在學(xué)習(xí)數(shù)學(xué)時感到沮喪：數(shù)學(xué)的不同領(lǐng)域，在學(xué)生的頭腦中仍然是脫節(jié)的，學(xué)生缺乏重要的工具來理解他們所學(xué)的數(shù)學(xué)。

這就是推理與證明的概念的意義所在。讓學(xué)生參與推理和證明，可以把學(xué)習(xí)數(shù)學(xué)從一個令人沮喪的活動變成一個有意義的活動。但讓我解釋一下我所說的推理與證明是什么意思，為什么它很重要。

推理與證明，指的是一系列的活動，這些活動經(jīng)常是數(shù)學(xué)中新知識的產(chǎn)生和驗證的一部分：識別模式，提出猜想，并提供論據(jù)和證明來決定是否接受或拒絕猜想。

我所說的"新知識 "是指對某一特定社區(qū)的成員來說是新的知識，例如小學(xué)課堂。因此，推理與證明的概念，對所有學(xué)生的數(shù)學(xué)學(xué)習(xí)都是相關(guān)的和重要的，因為發(fā)展這方面的能力，可以幫助學(xué)生理解他們正在學(xué)習(xí)的數(shù)學(xué)，在不同的數(shù)學(xué)領(lǐng)域之間建立聯(lián)系，并發(fā)展一些技能，使他們不僅能夠發(fā)現(xiàn)新的（對他們來說）數(shù)學(xué)知識，而且能夠不依賴教師或教科書的權(quán)威，來決定其有效性。

但遺憾的是，推理與證明在日常數(shù)學(xué)實踐中很少受到關(guān)注，這就使學(xué)生不僅失去了將數(shù)學(xué)作為一種感性活動來參與的寶貴機(jī)會，也失去了發(fā)展推理與證明能力的機(jī)會，因此會覺得的推理與證明有難度。

我們有大量的證據(jù)表明，在支持性的課堂環(huán)境中，即使是幼兒也能成功地進(jìn)行推理和證明。我的研究的很大一部分集中在如何幫助支持大量的教師成功地讓他們的學(xué)生了解推理和證明的關(guān)鍵要素。

You are right in saying that children’s learning of mathematics has been difficult for many students. This has mostly to do with the way in which mathematics has been taught in schools and the traditional emphasis on memorization of facts and application of procedures that make little sense to students. This traditional way of teaching mathematics has made learning mathematics frustrating for students: the different areas of mathematics remain disconnected in students’ minds and students are lacking important tools to make sense of the mathematics they are learning.

This is where the notion of reasoning-and-proving becomes relevant. Engaging students in reasoning-and-proving can help turn learning mathematics from a frustrating to asense-making activity. But let me explain what I mean by reasoning-and-proving and why it is important.

Reasoning-and-proving refers to a family of activities that are frequently part of the generation and validation of new knowledge in mathematics: identifying patterns, making conjectures, and providing arguments and proofs to decide whether to accept or reject the conjectures. By ‘new knowledge’ I refer to knowledge that is new to the members of a particular community, such as an elementary school classroom. So the notion of reasoning-and-proving is relevant and important for all students’ learning of mathematics as developing competency in this area can help students make sense of the mathematics they are learning, make connections between different areas of mathematics, and develop skills that will allow them to not only discover new (for them) mathematical knowledge but also decide on its validity without relying on the authority of the teacher or the textbook.

Unfortunately, though, reasoning-and-proving has received little attention in everyday mathematics practice, and this deprives students of valuable opportunities not only to engage in mathematics as a sense-making activity but also to develop competency in reasoning-and-proving thus finding reasoning-and-proving difficult. We have a wealth of evidence that, in supportive classroom environments, even young children can successfully engage in reasoning-and-proving. A big part of my research focuses on ways in which we can help support large numbers of teachers to successfully engage their students in key aspects of reasoning-and-proving.

龔鵬程教授：大學(xué)領(lǐng)域的數(shù)學(xué)教育研究，當(dāng)與中小學(xué)教師在實際課堂上碰到的問題不同，請問，您們現(xiàn)在研究些什么呢？

加布里埃爾·斯蒂利亞尼德教授：中小學(xué)教師遇到的一些問題與大學(xué)教師遇到的問題不同，有些則類似。當(dāng)我在前面說推理與證明的概念對所有學(xué)生的數(shù)學(xué)學(xué)習(xí)都很重要時，這并不限于中小學(xué)學(xué)生，它也適用于大學(xué)生。

盡管不同的學(xué)生群體需要有針對性的教學(xué)支持，但中小學(xué)生和大學(xué)生在推理與證明方面所面臨的許多困難是相同的。事實上，教師和大學(xué)導(dǎo)師都需要幫助，以成功地讓他們的學(xué)生參與推理和證明。這種幫助可以包括發(fā)展他們自己關(guān)于推理和證明的知識的方法，適合他們與學(xué)生一起開展的實現(xiàn)特定學(xué)習(xí)目標(biāo)的活動，以及開展這些活動的適當(dāng)方法。

在過去近二十年中，我在所有這些領(lǐng)域都進(jìn)行了合作研究。我目前參與的項目有兩個。第一個項目是由約翰·費爾資助的，重點是中學(xué)階段，與劍橋大學(xué)合作進(jìn)行研究。第二個項目由教育捐贈基金會資助，側(cè)重于小學(xué)階段，與我在牛津大學(xué)的同事共同進(jìn)行。

這兩個項目都旨在開發(fā)以理論為基礎(chǔ)、以實踐為導(dǎo)向的模式，以幫助擴(kuò)大與推理和證明有關(guān)的干預(yù)措施，我們發(fā)現(xiàn)這些干預(yù)措施在我們先前研究中的當(dāng)?shù)乇尘跋率怯行У摹?/p>

Some problems encountered by teachers at the school level are different and some are similar to those encountered by instructors at the university level.When I said earlier that the notion of reasoning-and-proving is important for all students’ learning of mathematics, this was not restricted to school students; it applies to university students as well. Although different groups of students need tailored instructional support, many of the difficulties that school and university students face with reasoning-and-proving are the same. Indeed, both teachers and university instructors need help to successfully engage their students in reasoning-and-proving. This help can include ways to develop their own knowledge about reasoning-and-proving, activities that are suitable for them to implement with their students to achieve specific learning goals, and appropriate ways to implement those activities.

I have conducted collaborative research in all of these areas over almost the last two decades. I have two projects I am involved currently. The first project is funded by John Fell, focuses on the secondary school level, and is conducted together with the University of Cambridge. The second project is funded by the Education Endowment Foundation, focuses on the primary school level, and is conducted together with my colleague at Oxford. Both projects aim to develop theory-based and practice-oriented models that can help scale up interventions related to reasoning-and-proving that we found to be effective in the local contexts of our prior research studies.

龔鵬程教授：數(shù)學(xué)的推理證明，主要在幾何部分嗎？它如何延伸到生活場域或哲學(xué)思考？

加布里埃爾·斯蒂利亞尼德教授：推理與證明適用于數(shù)學(xué)的所有領(lǐng)域，而不僅僅是幾何學(xué)。許多人把推理與證明與幾何聯(lián)系起來，因為在傳統(tǒng)上，這一概念幾乎只在中學(xué)幾何的背景下受到關(guān)注，并側(cè)重于證明的發(fā)展，而不是推理與證明的所有方面（識別模式，提出猜想等）。

讓我們考慮一個基于實際課堂事件的課堂情景，以說明幼兒在非幾何背景下參與推理與證明會是什么樣子。

想象一下，二年級的學(xué)生（6-7歲）正在研究兩個奇數(shù)相加會發(fā)生什么。他們檢查了幾個例子，并注意到一個規(guī)律，即在他們檢查的所有情況下，總和是一個偶數(shù)。這讓學(xué)生們猜想，他們注意到的模式是否會擴(kuò)展到所有的奇數(shù)對。不過，奇數(shù)是無限的，所以他們無法檢查每一對奇數(shù)。但學(xué)生們有了推理這種情況所需的工具，老師可以幫助他們。為了證明任何兩個奇數(shù)之和是偶數(shù)，學(xué)生不需要使用代數(shù)。例如，教師可以幫助學(xué)生將奇數(shù)表示為兩組的瓦片，都有一塊瓦片剩余，而將偶數(shù)表示為兩組的瓦片，都沒有剩余。當(dāng)兩個奇數(shù)相加時，剩下的兩塊瓷磚可以組合在一起，組成一個偶數(shù)。

我希望這個例子能讓您了解在非幾何學(xué)背景下，幼兒參與推理和證明的情況是怎樣的。此外，這個例子還說明了這種參與可以幫助學(xué)生深入理解他們所研究的數(shù)學(xué)思想，在本例中就是偶數(shù)和奇數(shù)的屬性。

Reasoning-and-proving applies toall areas of mathematics, not only geometry. Many people associate reasoning-and-proving with geometry because, traditionally, this notion received attention almost exclusively in the context of secondary school geometry and with a focus on the development of proofs, out of all the aspects that comprise reasoning-and-proving (identifying patterns, making conjectures, etc.). Let us consider a classroom scenario, which is based on actual classroom events, to illustrate what young children’s engagement with all aspects of reasoning-and-proving can look like and in a non-geometric context.

Imagine the students in a Year 2 classroom (6–7-year-olds) investigating what happens when they add two odd numbers. They check several examples and notice the pattern that, in all the cases they checked, the sum is an even number. This makes the students conjecture whether the pattern they noticed will extend to all pairs of odd numbers. After all, there are infinite odd numbers, so they cannot check every single pair. But the students have the tools they need to reason about this situation and the teacher can help them with that. In order to prove that the sum of any two odd numbers is an even number, students do not need to use algebra. For example, the teacher can help the students represent odd numbers as tiles of groups of two with one left over and even numbers as tiles of groups of two with none left over. When two odd numbers are added together, the two left over tiles can be grouped together and make an even number.

I hope that this example gives you an image of what young children’s engagement in reasoning-and-proving can look like in a non-geometric context. Also, the example illustrates how this engagement can help students develop a deep understanding of the mathematical ideas they investigate, in this case properties of even and odd numbers.

龔鵬程教授：不同種族，對于數(shù)學(xué)學(xué)習(xí)會有什么差別嗎？歷史上，中國、印度、阿拉伯似乎都較精于算數(shù)，幾何學(xué)的發(fā)展則不如或晚于希臘。

加布里埃爾·斯蒂利亞尼德教授：東亞國家在國際數(shù)學(xué)比較中經(jīng)常表現(xiàn)優(yōu)異，例如在2019年TIMSS（國際數(shù)學(xué)和科學(xué)趨勢研究）中，五個東亞國家在四年級和八年級的數(shù)學(xué)測試中都以很大的優(yōu)勢超過了其他約50個國家。

這種一貫的表現(xiàn)模式并非巧合，但作為一個領(lǐng)域，我們?nèi)杂幸欢温芬�，以充分了解東亞國家的成功可能有哪些文化或其他因素，以及這些因素如何幫助其他國家的數(shù)學(xué)教育。

一般來說，數(shù)學(xué)和文化（廣義上包括社會中任何可識別的文化群體）之間的關(guān)系是復(fù)雜的，在數(shù)學(xué)教育中有一個完整的研究領(lǐng)域，稱為民族數(shù)學(xué)，它探討的是這個問題。民族數(shù)學(xué)概念導(dǎo)致了大量的數(shù)學(xué)教育工作和研究視角的發(fā)展，這些研究對數(shù)學(xué)教學(xué)和學(xué)習(xí)環(huán)境的社會、文化和歷史特征非常敏感。

You are raising an important issue that falls outside of my main line of research or area of expertise in mathematics education. East Asian countries are frequently top performers in international mathematical comparisons, such as in the 2019 TIMSS (Trends in International Mathematics and Science Study) where five East Asian countries outperformed about 50 other countries by substantial margins in both the fourth- and eighth-grade mathematics tests. This consistent pattern in performance is not a coincidence but, as a field, we still have a way to go to fully understand what cultural or other factors might account for East Asian countries’ success and how these factors might help inform mathematics education in other countries. In general, the relationship between mathematics and culture (broadly conceived to include any identifiable cultural group within the society) is complex and there is a whole field of study in mathematics education, calledethnomathematics, which investigates that. Ethnomathematics has led to a large body of work in mathematics education and the development of research perspectives which are sensitive to social, cultural, and historical characteristics of the contexts where the teaching and learning of mathematics takes place.

龔鵬程，1956年生于臺北，臺灣師范大學(xué)博士，當(dāng)代著名學(xué)者和思想家。著作已出版一百五十多本。

辦有大學(xué)、出版社、雜志社、書院等，并規(guī)劃城市建設(shè)、主題園區(qū)等多處。講學(xué)于世界各地。并在北京、上海、杭州、臺北、巴黎、日本、澳門等地舉辦過書法展�，F(xiàn)為中國孔子博物館名譽(yù)館長、臺灣國立東華大學(xué)終身榮譽(yù)教授、美國龔鵬程基金會主席。