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School Mathematics Theorems - and Endless Source of Surprise
Abstract
In this paper I argue that all school theorems, except possibly a very small number of them, possess a built-in surprise, and that by exploiting this surprise-potential their learning can become an exciting experience of intellectual endevour to the students.
... Surprise, additionally, is claimed to be a useful and powerful tool for the improvement of students' mathematics learning in a classroom. According to Movshovits-Hadar (1988), teaching mathematics 'the surprise-way' (p. 35) can serve as a useful route for gaining students' curiosity and interest, which in turn serves as motivation for learning and increases its successfulness. ...
... For the sake of argument, I don't remember any of the other problems you did in that lesson, but this problem I remember till now. Surprise has been recognized in the literature as a factor which can support student learning in class by raising student curiosity and interest (e.g., Movshovits-Hadar, 1988;Nunokawa, 2001). From a complementary angle, surprise has been acknowledged as a contributing element to the creation of a mathematical aesthetic experience (e.g., de Freitas & Sinclair, 2014;Koichu, Katz, & Berman, 2017;Marmur & Koichu, 2016). ...
... The case of Lesson-S was based on a type of surprise Movshovits-Hadar (1988) refers to as an "unexpected existence" (p. 35). ...
... The choice of potentially surprising problems for the study is informed by the work of Movshovitz-Hadar (1988). She conceptualizes surprise as the feeling that emerges in an individual when something occurs in contradiction to expectations, and concentrates on an intellectual surprise associated with "the discovery of some unforeseen truth" (p. ...
... The hypothetical mechanism of this association has been presented above in terms of the gaps between the initial and informed solutions to the problems. This is in line with Nunokawa's (2001) and Marmur and Koichu's (2016) examples of surprising situations as well as Movshovitz-Hadar's (1988) suggestions about the types of surprise in mathematics. In addition, given the nature of the problems used in our study, we can relate our findings to the apparent gap between prototypical and non-prototypical drawings needed in order to solve the problems (Clements, 2003). ...
We investigated aesthetic responses of 60 middle school students as they engaged in a pair of similar looking geometry problems in one-on-one semi-structured interviews. The investigation was driven by three predictions. The first two predictions were about the association between the evaluative aesthetic response and surprise stemming from the solution to each problem. The third, main, prediction was that the problem with more surprising solution would be evaluated as more beautiful. The extent of surprise was manipulated by the order in which two problems were given. The third prediction came to be true in 90% of the cases, in which the first two predictions were fulfilled. The findings suggest that school students’ evaluative aesthetic response to mathematical problems can be stimulated in instructional settings. Implications for research and practice are drawn.
... Such "play" with mathematics relationships offers students opportunities to experience the pleasure of mathematical surprise, such as "Odd numbers hide in squares!" (Figures 1 and 3). Surprise is an important part of mathematics learning (Movshovitz-Hadar, 1994;Watson & Mason, 2007) and the related uncertainty and excitement is part of mathematicians' "world of knowing" (Burton, 1999, p. 138). These are important mathematics teaching and learning experiences that we seek to offer to our teacher candidates. ...
While change has been a constant in the faculty, grappling with a transition from a one-year
consecutive professional Bachelor of Education (B.Ed.) program to a two-year B.Ed. program is a
particular challenge. In addition to the challenges inherent in any process of curriculum reform,
the faculty also strives to we differentiate itself. While dealing with the teething pains that are
inevitable in such a fundamentally seismic and sudden change, University of Windsor Education
looks to seize opportunities presented to provide exceptional learning experiences for teacher
candidates manifested in transferrable and transformative competencies required for teaching and
living in diverse contexts.
This chapter discusses a diverse suite of courses designed to enhance experiential learning,
internationalization and global education, and community service-learning. This reinforces our
commitment to preparing holistic teachers who understand the multiple roles of teachers and the
social, political and moral imperatives of teaching.
... In particular, resist the temptation to break harder inverse problems into manageable (direct) steps.' As concepts become clarified through examining alternative processes, we see many opportunities for productive use of surprise (Movshovits-Hadar, 1988; 'surprise and delight' is one of the criteria for the International Society for Design and Development in Education prize 19 ), and supporting students' satisfaction in learning mathematics (Lockhart, 2009). Within the mathematics curriculum as story framework (Dietiker, 2015), engineering moments of surprise within the narrative arc has the potential to generate interest and insight, since 'enabling expectation allows both surprise if an expectation is violated and relief and satisfaction when an anticipated result . . . is met' (p. ...
The curriculum resources used for teaching secondary mathematics vary considerably from school to school. Some schools base their teaching largely on a single published scheme, while others design their own schemes of learning, curating their resources from a range of (often free) online sources. Both approaches seem problematic from the perspective of experiencing the mathematics curriculum as a coherent story, and neither seems likely to take best advantage of the accumulated body of knowledge in the education research literature about effective didactics for mathematics. In this position paper, as we embark on the collaborative, research-informed design of a complete, fully-resourced, free-to-access mathematics curriculum for students aged 11–14, we use the conceptual framework of mathematics curriculum as a story to draw out five key curriculum design principles. A mathematics curriculum should harness and develop the skills and expertise of teachers; balance the teaching of fluency, reasoning and problem solving; give explicit attention to important errors and misconceptions; compare and contrast alternative methods; and engineer coherence through strategic use of consistent representations and contexts. We use these five principles to set out our vision for the next step in research-informed mathematics curriculum design.
... Analogous reasoning is another promising direction. (For a few examples, see also Movshovitz-Hadar 1988, Hadar and Hadass 1981a, b, 1982a A set of valid mathematical tasks for various levels of school mathematics, calling for intellectually courageous acts, needs to be compiled, tested, analyzed, and disseminated by the research community to those who assume the daily challenge of, and responsibility for, dealing with students identified as mathematically talented and inventive 4 . An alternative approach might be to identify students who possess intellectual courage and teach them the curriculum in a way that is more conducive to their interests. ...
... Such "play" with mathematics relationships offers students opportunities to experience the pleasure of mathematical surprise, such as "Odd numbers hide in squares!" (Figures 1 and 3). Surprise is an important part of mathematics learning (Movshovitz-Hadar, 1994;Watson & Mason, 2007) and the related uncertainty and excitement is part of mathematicians' "world of knowing" (Burton, 1999, p. 138). These are important mathematics teaching and learning experiences that we seek to offer to our teacher candidates. ...
The recent change in teacher education in Ontario, moving from a single year to a two-year program, has offered us an opportunity to rethink and redesign our Kindergarten – Grade 12 (K-12)teacher education programs. A major shift has been happening within and outside of education due to a renewed focus on different mathematical ways of thinking, including computational thinking (CT) (Grover & Pea, 2013; Wing, 2006, 2008, 2011; Yadav, Mayfield, Zhou, Hambrusch & Korb, 2014). In this chapter we discuss how CT has been integrated into teacher education programs at two Ontario universities and its connection to mathematics education.
... The evocation of surprise was identified as a KME in two lessons in relation to three instructor actions: (a) Reaching a dead-end in the solution (not declared in advance); (b) Presenting an unexpected non-routine solution; and (c) Reaching an unexpected result. The student interviews supplied further evidence to previous research claims on PME 42 -2018 surprise serving as a factor that supports student learning by raising curiosity, interest, enjoyment, and mathematical aesthetic experience (Marmur & Koichu, 2016;Movshovits-Hadar, 1988). This is exemplified by the following excerpts: ...
The paper focuses on student learning experiences during large-group undergraduate Calculus tutorials. We identify eight types of Key Memorable Events – emotionally loaded events that are meaningful for the learning process in class from a student perspective. The findings are predominantly based on stimulated-recall interviews with 36 students, corresponding to 7 filmed lessons. Implications are drawn in relation to both the learning and teaching in the undergraduate mathematics classroom.
With reference to some personal discoveries using dynamic geometry, the personal need and role of proof is reflected upon. It is then argued that given the high level conviction obtained with the software motivated the search for proofs which in these cases involving generalizations of Van Aubel's theorem and 'Equilic Quadrilaterals' served more the purpose of explanation, discovery, etc.
In this article, we investigate the artistic puzzle of designing mathematics experiences (MEs) to engage young children with ideas of group theory, using a combination of hands-on and computational thinking (CT) tools. We elaborate on: (1) group theory and why we chose it as a context for young mathematicians’ experiences with symmetry and transformations; (2) our ME design principles of agency, access, surprise and audience; (3) the affordances of CT that complement our design principles; and (4) three ME variations we tested in grades 3–6 classrooms. We then reflect on the ME variations based on our design principles and the affordances of CT, and consider how the MEs may be further adapted and improved.
Abductive inferences, which are the only types of inference that produce new ideas, are important in mathematical problem solving. Such inferences, according to Peirce, arise from surprising or unexpected situations. Therefore, one way to improve student problem solving may be to provide them with environments that are designed to evoke surprise. In this paper, we examine the potential of dynamic geometry environments (DGEs)to foster surprise. We conjecture that the ease with which students can explore configurations, along with the immediate feedback, may lead them to encounter surprising situations. We analyse three different examples of student problem solving featuring surprised-provoked abduction, and identify the specific role that the DGE played.
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