It's great to be doing maths! : Engaging primary students : in a lunchtime club:
做数学真好！：让小学生参与午餐俱乐部：

Kimberley Pressick-Kilborn
金伯利-普雷斯克-基尔伯恩
University of Technology, Sydney Kimberley.Pressick-Kilborn@uts.edu.au
悉尼科技大学 Kimberley.Pressick-Kilborn@uts.edu.au

Maths clubs have been successfully implemented in upper primary and early secondary school contexts with the purpose of engaging students in mathematics learning outside of regular school lessons (Karp & Niemi, 2000; Papanastasiou & Bottiger, 2004). The activities in extracurricular maths clubs can include individual or collaborative problem-solving (Papanastasiou & Bottiger, 2004), or authentic activities that situate mathematics in real-life contexts (Karp & Niemi, 2000). Maths clubs may sometimes target high performing mathematics students and focus on providing extension and enrichment tasks. In such clubs, the focus is on creating an opportunity to engage in learning mathematics content beyond the regular curriculum with other mathematically capable students (Papanastasiou & Bottiger, 2004). Other maths clubs have included students who range in ability, but share an interest in participating in mathematics-related activities outside of class time (Karp & Niemi, 2000).
数学俱乐部已在小学高年级和中学低年级成功实施，其目的是让学生在常规学校课程之外参与数学学习（Karp & Niemi, 2000; Papanastasiou & Bottiger, 2004）。课外数学俱乐部的活动可包括个人或合作解决问题（Papanastasiou & Bottiger, 2004），或将数学置于现实生活情境中的真实活动（Karp & Niemi, 2000）。数学俱乐部有时会以数学成绩优秀的学生为目标，重点提供拓展和丰富任务。在这类俱乐部中，重点是创造机会，让学生与其他数学能力强的学生一起学习常规课程以外的数学内容（Papanastasiou & Bottiger, 2004）。其他数学俱乐部的成员包括能力参差不齐的学生，但他们都有兴趣参与课外的数学相关活动（Karp & Niemi, 2000）。

All pre-service primary teachers engage in professional experience placements in schools in the course of their undergraduate degrees. However it is problematic that the opportunity to plan and implement engaging and hands-on mathematics lessons does not always arise. In response to anecdotal feedback from pre-service teachers that suggested a need for more targeted learning about how to teach mathematics in school-based contexts, we initiated a project in partnership with a Sydney primary school. In consultation with teachers at the school, we decided that the project could best take the form of a lunchtime maths club for interested students in grades 4,5 and 6 , which would be led and facilitated by pre-service teachers.
所有职前小学教师在攻读本科学位期间都会到学校进行专业实习。然而，并不是每次都有机会规划和实施引人入胜的实践数学课程，这是一个问题。针对职前教师的反馈意见，即需要更有针对性地学习如何在校本环境中教授数学，我们与悉尼一所小学合作启动了一个项目。在与该校教师协商后，我们决定该项目最好采取午餐数学俱乐部的形式，由职前教师领导和协助，对象是四、五、六年级感兴趣的学生。

From the University's perspective, the key purpose of the project was to promote a discipline-specific form of professional experience for primary pre-service teachers, embedded in a campus-based final year mathematics education subject. This was a new professional experience context for these pre-service teachers enrolled in a Bachelor of Education degree, and one which other Universities also have embedded in their programs (for example, the Learning for Life Mathematics project, Australian Catholic University Ballarat). While they had at least some experience teaching mathematics to students in primary schools, the support that they received in preparing and evaluating their lessons was mixed and sometimes limited to textbook-based lessons (Reys, Reys, & Chaves-Lopez, 2004). The pre-service teachers who volunteered to be a part of the maths club project were not necessarily the most confident or capable students in their cohort, but were those who saw an opportunity to practise mathematics teaching in a different context. Running the maths club became an alternative assessment task and involvement therefore was optional for the pre-service teachers. In total, eight pre-service teachers volunteered in 2013 and a further eight in 2014. They formed
从大学的角度来看，该项目的主要目的是促进小学职前教师专业经验的学科特定形式，并将其嵌入以校园为基础的最后一年数学教育学科中。对于这些攻读教育学学士学位的职前教师来说，这是一种全新的专业经验背景，其他大学也在其课程中嵌入了这种背景（例如，澳大利亚天主教大学巴拉瑞特分校的 "为生命学习数学 "项目）。虽然他们至少有一些在小学向学生教授数学的经验，但在备课和评课方面得到的支持却参差不齐，有时还仅限于以教科书为基础的课程（Reys, Reys, & Chaves-Lopez, 2004）。自愿参加数学俱乐部项目的职前教师并不一定是他们这批学生中最自信或最有能力的学生，但他们看到了在不同环境中实践数学教学的机会。管理数学俱乐部成为一项替代性评估任务，因此，职前教师可以选择是否参与。2013 年共有八名职前教师自愿参加，2014 年又有八名。他们组成了
collaborative teaching groups of four, with each person taking responsibility for leading one club meeting with the assistance of the other three group members in planning, implementing and evaluating the focal activity. This meant that in total, each teaching group took responsibility for four maths club sessions. Each session ran for 40 minutes allowing for students to arrive and for packing up. The pre-service teachers were given some support from campus-based mathematics education lecturers, through suggested activities, which included explanations of the mathematics concepts. Team teaching while facilitating the maths club provided another layer of support for the pre-service teachers. None of them felt they had to run the whole activity on their own. Their related assessment task also required them to collaboratively engage in and document professional reflection, which is drawn on in this article along with written reflections from teachers and student surveys at the partner school.
四人合作教学小组，每人负责领导一次俱乐部会议，其他三名小组成员协助规划、实施和评估重点活动。这意味着每个教学小组总共负责四次数学俱乐部会议。每节课的时间为 40 分钟，以便学生到校和收拾东西。职前教师从校内数学教育讲师那里获得了一些支持，包括活动建议，其中包括数学概念的解释。在促进数学俱乐部活动的同时，团队教学为职前教师提供了另一层支持。他们都不认为自己必须独自完成整个活动。他们的相关评估任务还要求他们共同参与并记录专业反思，本文将参考合作学校教师的书面反思和学生的调查。

Upper primary students at our Sydney partner school were invited to participate in the maths club. At first the students who were a part of the Australasian Problem Solving Mathematical Olympiad team joined. However, once the maths club was under way, other students expressed a desire to join in the activities and this resulted in a wider ability range of students participating each week. Fifteen to twenty students regularly attended the club, with girls and boys equally represented.
我们悉尼伙伴学校的高年级小学生应邀参加了数学俱乐部。起初，参加澳大拉西亚问题解决数学奥林匹克竞赛团队的学生参加了活动。然而，数学俱乐部活动开始后，其他学生也表达了参加活动的愿望，因此每周都有更多不同能力的学生参加。经常参加俱乐部活动的学生有 15 到 20 人，男女生人数相当。

The pre-service teachers decided on their own activities for the maths club (Table 1) and only once did they double up (egg experiments). While not all of the activities were linked specifically to mathematics, emphasis was placed on giving pre-service teachers choice so that they themselves were comfortable and confident in implementing the tasks selected for students. All of the activities appeared to be engaging for both the students (Figure 1) and the pre-service teachers because each was characterised by opportunities for handson, active collaboration to develop solutions. While it is difficult to link the activities to specific content descriptors, each task linked to one or more of the Australian Curriculum Proficiency
职前教师自行决定数学俱乐部的活动（表 1），只有一次是双人活动（鸡蛋实验）。虽然并不是所有的活动都与数学有关，但重点是让职前教师有选择的余地，以便他们能够自如和自信地为学生完成所选的任务。所有的活动似乎都能吸引学生（图 1）和职前教师，因为每项活动都有机会让他们亲 身参与，积极合作制定解决方案。虽然很难将这些活动与具体的内容描述相联系，但每项任务都与一个或多个澳大利亚课程 能力描述相联系。
Strands — Understanding, Fluency, Problem Solving and Reasoning (ACARA, 2014). The emphasis on proficiency was particularly important for the students' ability to participate; if the activities had relied on content, many of the activities would have been too difficult for them. The focus was thus on the thinking that the activity promoted rather than the content of the activity itself.
这几个方面是：理解、流畅、解决问题和推理（ACARA，2014 年）。强调能力对学生的参与能力尤为重要；如果活动依赖于内容，许多活动对他们来说就太难了。因此，重点在于活动所促进的思维，而不是活动内容本身。

Figure 1. Students at the maths club.
图 1.数学俱乐部的学生。
The following activities are indicative of the types of activities in the maths club in that they cross strands in mathematics as well as linking subjects across the curriculum.
以下活动表明了数学俱乐部的活动类型，这些活动跨越了数学领域，并将各学科与课程联系在一起。

The pre-service teachers chose this topic for the first week because it was content free and they were unsure about the level of mathematics of the students. Both pre-service teachers and students were excited by the fact that fractals are a relatively new area of mathematics. The pre-service teachers also believed that it would stimulate the children's interest in the maths club. The children individually created their own Sierpinski Triangle (Figure 2) then joined them to make a very large Sierpinski triangle.
职前教师之所以在第一周选择这个主题，是因为它没有内容限制，而且他们不确定学生的数学水平。分形是一个相对较新的数学领域，职前教师和学生都对此感到兴奋。职前教师还认为，这将激发孩子们对数学俱乐部的兴趣。孩子们各自制作了自己的西尔平斯基三角形（图 2），然后将它们拼成一个非常大的西尔平斯基三角形。

Figure 2. Sierpinski Triangle (http://en.wikipedia.org/wiki/ Sierpinski_triangle).
图 2.Sierpinski 三角形 ( http://en.wikipedia.org/wiki/ Sierpinski_triangle)。

Table 1. The activities implemented at the maths club.
表 1.数学俱乐部开展的活动

The students started by creating tessellations using single shapes (Figure 3) and then, inspired by Escher drawings and a video explaining how to create the flying horse tessellation, modified a square into their own unique shape that tessellates. Students used their newly created shape to create a tessellation artwork. This activity provided students with the opportunity to explore and discover shapes that tessellate, integrating mathematics and visual arts.
学生们首先用单一形状制作出了魔方（图 3），然后在埃舍尔图画和一段解释如何制作飞马魔方的视频的启发下，将一个正方形修改成了自己独特的魔方形状。学生们用新创建的形状创作了一幅魔方艺术品。这项活动为学生提供了探索和发现魔方形状的机会，将数学和视觉艺术融为一体。

Figure 3. Using a single shape to tessellate.
图 3.使用单个形状进行镶嵌。

The in-service teachers at the partner school who observed the primary students were able to see broad benefits for participants in the maths club. The teachers commented on the disappointment of maths club students when the pre-service teachers were undertaking their own professional experience and could not attend. As a result of the powerful message that maths club was fun, many other students were keen to take part. This meant that although the original group was chosen because of their mathematical ability (to take part in the Australasian Problem Solving Mathematical Olympiads), the ability of the participants changed as many students decided to join in the activities because it looked like an enjoyable way to spend their lunchtime. This change led one in-service teacher to say "even some less capable children can excel with support from adults, from the opportunity to work with a group and when being engaged by the activity". Allied with this was her comment that "I can see how engaged students are with hands-on activities and it reminds me how important they are. This was evidenced by many 'non-mathematicians' volunteering to do maths at lunch time!" For many students, the problem solving element of mathematics can be quite difficult because they want to get to the right answer quickly, especially for the more capable mathematicians. "So I think one thing that maths club did was to support and engage the process of problem solving, to allow students to think and verbally justify their answers" (in-service teacher).
观察小学生的伙伴学校在职教师能够看到数学俱乐部参与者的广泛受益。教师们评论说，当职前教师正在进行自己的职业体验而无法参加时，数学俱乐部的学生会感到失望。由于 "数学俱乐部很有趣 "这一强有力的信息，许多其他学生也热衷于参加。这意味着，尽管最初选择这批学生的原因是他们的数学能力（参加澳大拉西亚解题数学奥林匹克竞赛），但参与者的能力发生了变化，因为许多学生决定参加活动，因为这看起来是度过午餐时间的一种愉快方式。这种变化促使一位在职教师说："即使是一些能力较差的孩子，在成人的支持下，在有机会与小组合作和参与活动的情况下，也能取得优异成绩"。她还说："我可以看到学生们是如何参与实践活动的，这提醒我他们是多么重要。许多'非数学家'在午餐时间自愿做数学题就是证明！"对于许多学生来说，数学中的解题元素可能相当困难，因为他们希望快速得到正确答案，尤其是对于能力较强的数学家来说。"因此，我认为数学俱乐部所做的一件事就是支持和参与解决问题的过程，让学生思考并用语言证明他们的答案"（在职教师）。

When the students were asked about what they liked about the maths club, many mentioned particular activities but other answers included "It was a lot of fun and challenging", "the variety of working with other students as well as the teachers", "[the pre-service teachers] were more fun than teachers", "the teachers were very enthusiastic and interested in teaching maths", "Doing fun maths stuff!" When asked what they did not like about maths club, most of the answers were "Nothing" but a couple found it difficult to eat lunch while doing the maths club activities.
当问及学生喜欢数学俱乐部的哪些活动时，许多学生都提到了一些特别的活动，但其他答案还包括："很有趣，很有挑战性"、"与其他学生以及老师一起工作的多样性"、"（职前教师）比老师更有趣"、"老师们非常热情，对数学教学很感兴趣"、"做有趣的数学事情！"当被问及对数学俱乐部有什么不喜欢时，大多数人的回答都是 "没什么"，但有几个人认为在开展数学俱乐部活动时很难吃到午餐。

The question about what they would always remember about the maths club elicited responses about particular activities, but also responses about maths club being fun "even though it's maths". Several believed that the experience of the maths club was what they would always remember. The students all recommended maths club to their friends because "you learn, practice and experience maths", "it's good to get your mind moving", "it was mind twisting and entertaining", "it's fun and you learn new things". In summary, the students benefitted from the opportunity to see mathematics in a new light.
在 "他们会永远记住数学俱乐部的什么 "这一问题中，学生们回答了一些特别的活 动，但也有学生回答数学俱乐部很有趣，"虽然是数学"。一些学生认为，数学俱乐部的经历是他们永远难忘的。学生们都向他们的朋友推荐数学俱乐部，因为 "你可以学习、练习和体验数学"、"让你的思维活动起来是件好事"、"它让人思维活跃，寓教于乐"、"它很有趣，你可以学到新东西"。总之，学生们受益于从新的角度看待数学的机会。

As student attendance at maths club was voluntary, there was not necessarily regular attendance from all participants, but the total numbers increased over the life of the club. It appeared as though word-of-mouth amongst the students, who informally shared with their peers their enjoyment of belonging to this group, was important. The pre-service teachers learned to plan to accommodate latecomers and students eating their lunch i.e., joining existing pairs or groups. Importantly, those leading the session also learned more about building discussion time into the activity itself, as well as sharing student outcomes and achievements at the end of the session. One in-service teacher observed the usefulness in keeping a maths journal so that the students could document their maths club activities for future reference and reflection.
由于学生参加数学俱乐部是自愿的，因此不一定所有参与者都定期参加，但总人数在俱乐部活动期间有所增加。学生之间的口碑似乎很重要，他们会非正式地与同伴分享他们对参加这个小组的喜悦。职前教师学会了如何安排迟到者和正在吃午餐的学生，即加入现有的对子或小组。重要的是，带队教师还学会了在活动中安排更多的讨论时间，以及在活动结束时分享 学生的成果和成就。一位在职教师发现，写数学日记非常有用，这样学生就可以记录他们的数学俱乐部活动，供今后参考和反思。

In their discussions about whether they would run a maths club in future in their own schools, the pre-service teachers recognised the benefit of working with other enthusiastic teachers to run the club, which acknowledges the value that they perceived in co-teaching. They also recognised
在讨论今后是否会在自己的学校开办数学俱乐部时，职前教师们认识到与其他热心 教师合作开办俱乐部的益处，这说明他们认识到了共同教学的价值。他们还认识到
that they might be the teacher responsible for running the club. This would mean planning activities carefully so they were manageable because some very engaging, hands-on activities may require more resources, adult support and student management. They suggested that such activities should be dispersed throughout a term, perhaps recruiting another teacher to assist for one or two activities. Other suggestions they made were:
他们可能是负责管理俱乐部的教师。这意味着要仔细规划活动，使其易于管理，因为一些非常吸引人的动手活动可能需要更多的资源、成人支持和学生管理。他们建议，此类活动应分散在整个学期中进行，或许可以聘请另一名教师协助开展一到两项活动。他们提出的其他建议包括

Figure 4: On-task learning in the maths club.
图 4：数学俱乐部中的任务学习。

The maths club project benefited the learning of primary students, pre-service and in-service teachers, and teacher educators by providing an authentic and highly engaging context that supported all participants in developing new insights into the nature of mathematics education. For primary students, the main purpose of the mathematics club was to provide an opportunity to enjoy and have fun with mathematics with peers who shared their interest. In this context, the importance of students 'playing' with maths in their lunch break was paramount. The lunchtime maths club raised the profile and visibility of maths in the school. The whole school in fact experienced the enjoyment and wonder of maths when they observed the maths club participants testing their parachutes in the egg experiment. Although this activity could be considered a science or 'design and produce' task rather than a mathematics one, the pre-service teachers were rewarded by the enthusiasm and excitement that the activity generated. The whole school watched the successes (and a couple of failures) with many cheers when the egg survived the fall and support when the egg did not.
数学俱乐部项目为小学生、职前和在职教师以及教师教育工作者提供了一个真实和极具吸引力的环境，帮助所有参与者对数学教育的本质有了新的认识，从而使他们的学习受益匪浅。对于小学生来说，数学俱乐部的主要目的是提供一个机会，让他们与有共同兴趣的同龄人一起享受数学，并从中获得乐趣。在这种情况下，学生在午休时间 "玩 "数学就显得尤为重要。午间数学俱乐部提高了数学在学校的地位和知名度。事实上，当全校师生观察数学俱乐部的参与者在鸡蛋实验中测试他们的降落伞时，都体验到了数学的乐趣和神奇。尽管这项活动可被视为科学或 "设计与制作 "任务，而不是数学任务，但职前教师却因这项活动所激发的热情和兴奋而收获颇丰。全校师生都目睹了这一成功（也有几次失败）的过程，当鸡蛋在坠落过程中幸存下来时，全校师生欢呼雀跃，当鸡蛋没有坠落时，全校师生表示支持。

Participating in the maths club supported pre-service primary teachers to develop deeper, broader content knowledge in specific topics in mathematics, which sometimes took them outside of the syllabus documents that they regularly used to plan classroom-based maths lessons. In addition, the pre-service teachers' confidence was increased in designing, teaching and evaluating hands-on activities with children in a partner primary school, with the main aim of experimenting with mathematics. As one pre-service teacher reflected, "Gaining real life mathematical teaching to engaged students in a supportive environment, provided us with experience no written assignment could ever do." Through their involvement in the maths club, pre-service teachers were equipped with critical skills and understandings to enable them to implement innovative programs in primary schools. They gained new insights into the possibilities of hands-on activities for mathematics learning. They were provided with an opportunity to experience membership of an authentic professional learning community, as they interacted with the teachers at the school and the mathematics teacher educators.
参加数学俱乐部活动有助于职前小学教师在特定的数学主题方面发展更深、更广的内容知识，这有时会使他们超出教学大纲文件的范围，而教学大纲文件是他们经常用来规划课堂数学课程的。此外，在与伙伴小学的儿童一起设计、教授和评估以数学实验为主要目的的实践活动时，职前教师的信心也得到了增强。正如一位职前教师所反映的那样："在一个支持性的环境中向参与其中的学生传授真实的数学知识，为我们提供了书面作业所无法提供的经验"。通过参与数学俱乐部，职前教师掌握了关键技能和理解能力，使他们能够在小学实施创新计划。他们对数学学习实践活动的可能性有了新的认识。在与学校教师和数学教师教育者互动的过程中，他们有机会体验加入一个真实的专业学习社区。

Both in-service teachers and teacher educators contributed positively to the professional learning of pre-service teachers through involvement in this project, however the professional learning was bi-directional. The creativity and responsiveness of the pre-service teachers in designing and
在职教师和教师教育者通过参与这一项目，都对职前教师的专业学习做出了积极贡 献，但这种专业学习是双向的。职前教师在设计和实施项目中的创造力和反应能力
implementing the maths club activities were a source of inspiration for the teachers at the school and the teacher educators, who were, in turn, encouraged to reflect on their own practice, especially in relation to hands-on tasks. The in-service teachers appreciated the resources shared by the pre-service teachers, some of which they now incorporate into their classroom teaching, and the opportunity to observe their own students, some of whom they would not previously have considered as mathematics-oriented. For us as teacher educators, the choices of activities made by pre-service teachers possibly need greater guidance, so that the focus of the maths club is firmly in mathematics. The pre-service teachers' written evaluations of the sessions that they designed and implemented, and their feedback about the perceived benefits of participating in the maths club for their own learning about primary mathematics teaching, were of particular value. These have provided encouragement for us to continue to develop and refine our partnership with the school, but also to seek further opportunities to embed pre-service primary teachers' learning in mathematics-specific professional experiences in schools.
数学俱乐部活动的开展为学校教师和教师教育者提供了灵感来源，反过来也鼓励他们 反思自己的实践，特别是与实践任务有关的实践。在职教师对职前教师分享的资源表示感谢，其中一些资源现在已被他们纳入课堂教学，他们也有机会观察自己的学生，其中一些学生以前并不被他们认为是以数学为导向的。对我们教师教育工作者来说，职前教师在选择活动时可能需要更多的指导，这样才能使数学俱乐部的重点牢牢地放在数学上。职前教师对他们设计和实施的课程的书面评价，以及他们对参加数学俱乐部对他们自 己学习小学数学教学的益处的反馈意见，都特别有价值。这些都鼓励我们继续发展和完善与学校的合作关系，同时寻求更多的机会，将小学职前教师的学习融入到学校的数学专业经验中。

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澳大利亚课程、评估和报告管理局（ACARA）（2014 年）。澳大利亚课程：http://www.australiancurriculum.edu.au/mathematics/ Curriculum/F-10?layout=1
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Teachers interested in starting a maths club with students at their own school or with other local schools may find it useful to draw on the resources developed by Rhodes University in South Africa, which can be accessed at http://www.ru.ac.za/sanc/mathsclubs/. Another useful website is the University of Cambridge's Nrich site, which can be accessed at http://nrich.maths.org/frontpage. Further information about the Learning for Life Mathematics project can be accessed at http://www.acu.edu. au/about_acu/faculties,_institutes_and_centres/ education_and_arts/about_the_faculty/news/ learning_for_life_mathematics_project
有兴趣与自己学校或当地其他学校的学生一起创办数学俱乐部的教师，可以借鉴南非罗得 大学开发的资源，网址是 http://www.ru.ac.za/sanc/mathsclubs/。另一个有用的网站是剑桥大学的 Nrich 网站，网址是 http://nrich.maths.org/frontpage。有关 "为生命学习数学 "项目的更多信息，请访问 http://www.acu.edu. au/about_acu/faculties,_institutes_and_centres/ education_and_arts/about_the_faculty/news/ learning_for_life_mathematics_project 。

Thank you to the staff and students of the partner school. Thank you to the UTS pre-service teachers who volunteered to be a part of the maths club project. In particular, thank you to pre-service teachers Alexandra Amor-Ryan, Benjamin Barker, Natasha Martin, Eleanor Pundyk, Tracy Reid, Madison Walker, Joanna Wilson, and Grace Wong. This project was supported by a UTS VC Learning and Teaching Grant in 2013 and we are grateful for assistance from Margie Christowski, Terry Fitzgerald, and Rick Stevens.
感谢合作学校的师生。感谢自愿参与数学俱乐部项目的UTS职前教师。特别要感谢职前教师亚历山德拉-阿莫尔-瑞安（Alexandra Amor-Ryan）、本杰明-巴克（Benjamin Barker）、娜塔莎-马丁（Natasha Martin）、埃莉诺-庞迪克（Eleanor Pundyk）、特蕾西-里德（Tracy Reid）、麦迪逊-沃克（Madison Walker）、乔安娜-威尔逊（Joanna Wilson）和格蕾丝-王（Grace Wong）。该项目于2013年获得了UTS VC学习与教学拨款的支持，我们在此感谢玛吉-克里斯托夫斯基（Margie Christowski）、特里-菲茨杰拉德（Terry Fitzgerald）和里克-史蒂文斯（Rick Stevens）的协助。