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Paul's Dilemma: Is This a Polyhedron?
Journal of Humanistic Mathematics Pub Date : 2017-07-01 , DOI: 10.5642/jhummath.201702.05
Bethany Noblitt , Shelly Harkness

Teachers play the believing game when they honor students’ mathematical thinking, even when it means they must suspend their own mathematical thinking momentarily [2, 3]. The study reported here tells the story of what happened in a university mathematics classroom when one student did not think that a particular figure satisfied the definition of a polyhedron and the instructor chose to play the believing game. The result was a very rich discussion, where both students and the authors grappled with their own mathematical understanding. One author served as the instructor of the course and the other author was an observer, taking field notes and video recordings that provided evidence. Definitions of geometry terms are important in order for students to classify figures. When considering a set of figures, teachers want their students to be able to distinguish between triangles and pyramids and to identify which figures are rectangles and which figures are squares. But definitions are important for another reason. They can prompt rich mathematical discussions in the classroom. The narrative we portray in this paper occurred when a student described a particular three-dimensional figure and asked if it was a polyhedron. Simply asking if a figure satisfies the definition of a polyhedron is not in itself a high-level task sufficient to facilitate meaningful discussion. What can an instructor do to encourage meaningful discussion of a mathematical definition? The mathematical discussion described here was the result of taking advantage of a teachable moment created by a perfect storm Journal of Humanistic Mathematics Vol 7, No 2, July 2017 Bethany Noblitt and Shelly Sheats Harkness 73 consisting of a difficult definition, a nonstandard example, and an instructor willing to participate in both the believing and the doubting games. Bethany was teaching the course, Mathematics for Elementary and Middle School Teachers, and Shelly was taking field notes as she videotaped the classroom discourse. We were collecting data to answer the following research question: How does a teacher play the believing game [2, 3] in a mathematics classroom? Elbow [2, 3] contended that we can improve our attempt to understand by using two opposing processes: methodological belief and methodological doubt. In terms of the art of teaching, believing is an endeavor to find virtues and strengths, no matter how unlikely students’ ideas, solutions or answers might seem and doubting is an attempt to find flaws or contradictions. Unfortunately, methodological doubt is essentially the automatic mode of logic that some mathematics teachers use. As teachers, we may doubt too much. However, we should make conclusions only after considering the results of both believing and doubting when students’ ideas, answers, or solutions are deemed incorrect or wrong. When teachers play the believing game they honor and respect students’ mathematical thinking [4, 5, 6, 7]. Honoring students’ mathematical thinking while at the same time keeping the mathematical goals of the lesson in mind and prominent can be challenging [12]. Thus, the believing game is a tough game to play, but one that has great payoffs. At the time of this study, Shelly had participated in research on the believing game in the mathematics classroom, but Bethany had not. Shelly’s previous research on the believing game in the mathematics classroom focused, in part, on identifying when a professor played the believing game, see [4, 5, 6]. In [7], Bethany and Shelly described the different teacher and student actions in a classroom that prompted the professor to play the believing and doubting games. This article elaborates on one of those specific instances of when a professor played the believing game and describes in great detail the rich whole-class discussion that took place as a result. One aspect of mathematical discourse that a balanced use of methodological doubt and belief can facilitate is that of building. In building, students respond to each other’s mathematical ideas and use each other’s ideas as a basis for their own mathematical thinking [11]. A student providing additional insight into another’s idea or creating his or her own mathematical conjectures based on another’s ideas are both evidence that building has occurred in the classroom ([11]. 74 Paul’s Dilemma: Is This a Polyhedron? When teachers purposefully balance the use of methodological doubt and methodological belief, building can occur and rich discourse can transpire. In fact, “[m]athematics and science education reforms encourage teachers to base their instruction in part on the lesson as it unfolds in the classroom, paying particular attention to the ideas that students raise ... and adapt instruction in the moment . . . ” [13, page 1]. Teachers must help their students engage in mathematical practices such as providing explanations, building on one another’s contributions, making connections, and using representations; and in the same moments teachers’ facilitating moves should be dependent on what their students say and do [1]. The demands on teachers who engage in these practices in fleeting amounts of time between hearing, listening to, and then orchestrating the discourse are enormous. In fact, we deem hearing and listening as separate practices. Listening implies hearing and then delving deeper to understand. Kastberg, in [8], described a classroom episode: I was working through my plan and choosing to answer for them [the students] something that made sense to me. A little voice challenged me to change my direction because an answer I had not even considered made sense to a student . . . and all I did was listen, stop what I was doing and allow them to explore their thoughts. (page 158) We claim that the voice Kastberg heard challenged her to play the believing game. Most teachers have heard this same voice in the classroom asking, “Will you believe? Or will you doubt?” Answering yes to the question, “Will you believe?” can be very difficult. It is a decision that must be made quickly and one which will have consequences for the rest of the classroom discussion. In this paper Bethany tells the story of what happened in her mathematics classroom when she balanced doubting with believing, when she heard that voice asking, “Will you believe?” and Bethany answered, “Yes.” Shelly and Bethany were studying their research question, “How does a teacher play the believing game in the mathematics classroom?” by collecting classroom observation data, which included field notes and video. We considered the a priori categories of believing and doubting, as described by Elbow [2, 3]. Bethany Noblitt and Shelly Sheats Harkness 75 The class in which this discussion took place was Mathematics for Elementary and Middle Grades Teachers which had 28 students, about one-fourth of whom were planning to become middle grades teachers and most of whom were sophomores and juniors. The content of the course was geometry and included twoand three-dimensional figures, measurement, congruence, and transformations. The overall goal of the course was for students to develop deep understanding of mathematical concepts important to the teaching of elementary and middle grades mathematics. The first several class meetings of the course were devoted to two-dimensional geometry and establishing terminology and relationships. Before the discussion on three-dimensional figures students had defined and analyzed polygons in general and specifically they had looked closely at different types of quadrilaterals and triangles. They also had some experience with geometric proof; for example, they had proven alternate interior angles are congruent and the sum of the measures of the angles of a triangle is 180◦. Our story focuses on what transpired when students in Bethany’s class tried to determine whether or not a certain figure was a polyhedron. In the text for the course, Mathematics for Elementary Teachers: A Contemporary Approach [10], the definition of a polyhedron is “the union of polygonal regions, any two of which have at most a side in common, such that a connected finite region in space is enclosed without holes” (page 655). This is the definition that prompted a rich discussion that led to balanced playing of the believing game and the doubting game. Additionally, there was another unexpected aspect of the believing game present during the class discussion on polyhedra. Not only was Bethany intentionally playing both the believing game and the doubting game, the students joined her in these games. They believed and doubted Bethany and each other. Their doubting of her understanding and/or explanations greatly increased the richness of the discussion. Their believing and doubting of each other’s arguments gave them the opportunity to engage each other in mathematical discussions. It was during the reflection of this classroom discussion that Shelly and Bethany invented the new terms, reserved believing and reserved doubting [7]. Reserved believing appears on a continuum of play where the teacher does not believe her own mathematical understanding to be the only understanding, yet, she does not fully doubt her own mathematical understanding, either. 76 Paul’s Dilemma: Is This a Polyhedron? When a teacher is participating in reserved believing, she tries to find merits in students’ mathematical understanding. During the class discussion, Bethany wanted to play the believing game with her students. She wanted to find strength in their arguments, but she needed more from them before she was willing to completely suspend her own beliefs and trust their understanding. Reserved believing was a mechanism to motivate students to convince Bethany of the merit of their thinking. Likewise, reserved doubting occurs when a teacher balances her own mathematical understanding with an attempt to find flaws in students’ understanding. In this class discussion, Be

中文翻译:

保罗的困境:这是多面体吗?

当老师尊重学生的数学思维时,他们就发挥了信念游戏,即使这意味着他们必须暂时暂停自己的数学思维[2,3]。此处报道的这项研究讲述了一个大学数学教室里发生的故事,当时一个学生认为一个特定的人物不满足多面体的定义,而讲师选择玩这种相信游戏。结果进行了非常丰富的讨论,学生和作者都在努力应对自己的数学理解。一位作者担任该课程的讲师,另一位作者是一名观察员,他在现场做笔记和录制视频以提供证据。几何术语的定义对于学生对图形进行分类很重要。在考虑一组数字时,老师希望他们的学生能够区分三角形和金字塔,并确定哪些数字是矩形,哪些数字是正方形。但是由于另一个原因,定义很重要。他们可以在课堂上引发丰富的数学讨论。我们在本文中描述的叙述是在学生描述一个特定的三维图形并询问它是否为多面体时发生的。简单地问一个图形是否满足多面体的定义,这本身并不是一个足以促进有意义的讨论的高级任务。老师可以做些什么来鼓励有意义的数学定义讨论?这里描述的数学讨论是利用完美风暴带来的可教的时刻的结果,《人文数学杂志》第7卷第2期 2017年7月,Bethany Noblitt和Shelly Sheats Harkness 73包括一个困难的定义,一个非标准的示例以及一个愿意参加相信游戏和怀疑游戏的教练。贝瑟尼(Bethany)正在教授《小学和中学教师的数学》课程,而雪莉(Shelly)在给课堂录像时给她做现场笔记。我们正在收集数据来回答以下研究问题:教师如何在数学教室中玩“相信游戏” [2,3]?Elbow [2,3]认为,我们可以通过使用两个相反的过程来改进理解的尝试:方法论信念和方法论怀疑。在教学艺术方面,无论学生的想法多么不可能,相信都是努力寻找美德和长处,解决方案或答案可能似乎出现,并且怀疑是试图发现缺陷或矛盾。不幸的是,方法论上的疑问本质上是某些数学老师使用的自动逻辑模式。作为老师,我们可能会怀疑太多。但是,只有在认为学生的想法,答案或解决方案被认为是错误或错误时,才应考虑相信和怀疑的结果,然后才能得出结论。当老师玩有信仰的游戏时,他们会尊重并尊重学生的数学思维[4,5,6,7]。尊重学生的数学思维能力,同时又要牢记本课程的数学目标并突出其表现可能是具有挑战性的[12]。因此,有信心的游戏是一款很难玩的游戏,但收益却很高。在进行这项研究时,Shelly曾在数学课堂中参与过关于相信游戏的研究,但Bethany却没有。Shelly先前在数学课堂上对信仰游戏的研究部分集中在确定教授何时玩信仰游戏上,参见[4,5,6]。在[7]中,Bethany和Shelly在课堂上描述了老师和学生的不同行为,促使教授玩一些相信和怀疑的游戏。本文详细介绍了一位教授玩相信游戏的具体情况之一,并详细描述了由此引发的丰富的全班讨论。平衡使用方法论的疑问和信念可以促进数学论述的一方面是建立。在建筑物里 学生互相回应对方的数学思想,并使用对方的思想作为自己数学思想的基础[11]。如果学生对他人的想法有更多的了解,或根据他人的想法创建自己的数学猜想,则既可以证明课堂上已经发生过建筑([11]。保罗的困境:这是多面体吗?当老师有目的地平衡使用时)关于方法论的疑问和方法论的信念,建设会发生,丰富的话语可能会流传。事实上,“ [m]数学和科学教育改革鼓励教师在课堂上逐步发展他们的授课内容,并特别注意学生提出的想法……并在当下适应教学……” [第13页,第1页]。老师必须帮助他们的学生从事数学实践,例如提供解释,建立在彼此的贡献上,建立联系并使用表示法;同时,教师的便利行动应取决于学生的言行[1]。对从事这些练习的教师的要求是短暂的,从听,听到编排演讲之间的时间短暂。实际上,我们认为听和听是分开的做法。聆听意味着聆听,然后再深入了解。卡斯特伯格(Kastberg)在[8]中描述了一个课堂情节:我正在制定计划,并选择为他们[学生]回答一些对我来说有意义的东西。有点声音挑战我改变方向,因为我什至没有想到的答案对学生来说是没有道理的。。。我所做的就是倾听,停止我正在做的事情,让他们探索自己的想法。(第158页)我们声称Kastberg听到的声音挑战了她玩这种相信游戏。大多数老师在教室里都听到过同样的声音问:“你会相信吗?还是您会怀疑?” 对问题“是,您会相信吗?”的回答是。可能非常困难。这是一项必须迅速做出的决定,并将对其余的课堂讨论产生影响。在本文中,Bethany讲述了在数学课堂上发生的事情,当她在怀疑与信念之间取得平衡时,当她听到那个声音问:“你会相信吗?” 伯大尼回答说:”雪莉和伯大尼正在研究他们的研究问题,“老师如何在数学课堂上玩相信游戏?” 通过收集课堂观察数据,其中包括实地记录和视频。正如Elbow [2,3]所述,我们考虑了信仰和怀疑的先验类别。Bethany Noblitt和Shelly Sheats的烦恼75进行讨论的班级是中小学数学老师,有28名学生,其中约四分之一计划成为中级老师,其中大多数是大二和大三。该课程的内容是几何,包括二维和三维图形,测量,全等和变换。该课程的总体目标是使学生对对初等和中级数学教学很重要的数学概念产生深刻的理解。该课程的前几次班级会议专门讨论二维几何并建立术语和关系。在讨论三维图形之前,学生已经大致定义和分析了多边形,尤其是他们仔细研究了不同类型的四边形和三角形。他们还具有几何证明方面的经验。例如,他们证明了交替的内角是全等的,并且三角形的角度之和为180°。我们的故事集中在Bethany班上的学生试图确定某个人物是否是多面体时发生了什么。在该课程的课文《面向小学教师的数学:当代方法》 [10]中,多面体的定义是“多边形区域的并集,其中任何两个区域最多具有相同的一面,因此相连的有限区域在空间中被封闭而没有孔”(第655页)。这个定义引发了广泛的讨论,导致了相信游戏和怀疑游戏的平衡。另外,在关于多面体的课堂讨论中,相信游戏还有另一个意想不到的方面。Bethany不仅故意玩了相信游戏和怀疑游戏,而且学生们也加入了她的游戏。他们相信并怀疑伯大尼和彼此。他们对她的理解和/或解释的怀疑大大增加了讨论的内容。他们对彼此论点的相信和怀疑使他们有机会彼此参与数学讨论。正是在这次课堂讨论的反思中,雪莉和伯大尼发明了新术语,保留了相信并保留了怀疑[7]。保留的信念出现在一个连续的游戏中,在该过程中,老师不相信自己的数学理解是唯一的理解,然而,她也不完全怀疑自己的数学理解。76保罗的困境:这是多面体吗?当一位教师参与保守的信念时,她会尝试从学生的数学理解中寻找优点。在课堂讨论中,Bethany希望与她的学生一起玩令人信服的游戏。她想在他们的论点中找到力量,但是在她愿意完全中止自己的信念并相信他们的理解之前,她需要他们提供更多的信息。保留信念是一种激励学生说服Bethany思考的优点的机制。同样,当教师在寻找自己的理解缺陷时平衡自己的数学理解时,也会产生保留的怀疑。在课堂讨论中,
更新日期:2017-07-01
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