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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). ...
... Að þessu leyti er staerðfraeði ólík flestum öðrum fraeðigreinum og námsgreinum í skólum, þar sem í til daemis náttúruvísindum leika tilraunir og vísun í efnislegan veruleika og reynslu mun staerra hlutverk. Eins og staerðfraeðingar lýsa sinni grein er staerðfraeðiiðkun lifandi og skapandi ferli þar sem sannanir eru í fyrirrúmi (sjá til daemis Lockhart, 2009;Pólya, 1990;Su, 2020) og Movshovits-Hadar (1988) útskýrir hvernig sérhver setning (það er staerðfraeðiregla, e. theorem) í námsefni skólanna (ef til vill með örfáum undantekningum) eigi rót sína að rekja til undrunar: ef ekkert hefði einhvern tíma verið óvaent við einhverja setningu hefði staerðfraeðingurinn sem uppgötvaði hana varla haft fyrir því að setja hana fram, enginn hefði haft áhuga á henni og setningin hefði ekki ratað inn í námsefni skóla um allan heim. Movshovits-Hadar (1988) telur, eins og Lockhart (2009), að í kennslubókum sé iðulega búið að loka á möguleika nemenda til þess að upplifa nokkra undrun eða tilfinningar gagnvart efninu -hið lifandi ferli hafi verið drepið og bútað niður í mola af dauðum staðreyndum. ...
... Eins og staerðfraeðingar lýsa sinni grein er staerðfraeðiiðkun lifandi og skapandi ferli þar sem sannanir eru í fyrirrúmi (sjá til daemis Lockhart, 2009;Pólya, 1990;Su, 2020) og Movshovits-Hadar (1988) útskýrir hvernig sérhver setning (það er staerðfraeðiregla, e. theorem) í námsefni skólanna (ef til vill með örfáum undantekningum) eigi rót sína að rekja til undrunar: ef ekkert hefði einhvern tíma verið óvaent við einhverja setningu hefði staerðfraeðingurinn sem uppgötvaði hana varla haft fyrir því að setja hana fram, enginn hefði haft áhuga á henni og setningin hefði ekki ratað inn í námsefni skóla um allan heim. Movshovits-Hadar (1988) telur, eins og Lockhart (2009), að í kennslubókum sé iðulega búið að loka á möguleika nemenda til þess að upplifa nokkra undrun eða tilfinningar gagnvart efninu -hið lifandi ferli hafi verið drepið og bútað niður í mola af dauðum staðreyndum. Ef fólk umgengst staerðfraeði eins og safn af tilbúnum þekkingaratriðum og reikniaðferðum til þess að leggja á minnið án þess að reyna að skilja rökin að baki þeim, eða þá upplifa þá undrun sem felst í þeim, vaeri það í andstöðu við gildi fraeðigreinarinnar og baeri ekki vott um umhyggju fyrir staerðfraeði. ...
Í íslenskum framhaldsskólum taka nemendur iðulega mörg próf í sínum stærðfræðiáföngum, bæði hlutapróf og lokapróf, auk þess sem þeir skila öðrum verkefnum sem gilda til lokaeinkunnar. Nemendur og kennarar eru oft mjög uppteknir af þessu prófahaldi. Nemendur hafa áhyggjur af árangri sínum á prófum og kennarar verja miklum tíma í að undirbúa og fara yfir próf. Ég ræði og greini slíka prófadrifna stærðfræðikennslu frá heimspekilegu sjónarhorni sem nefnist umhyggja fyrir stærðfræðinámi. Það sjónarhorn byggir á að leiða saman hugmyndir um stærðfræðikennslu sem umhyggju fyrir nemendum og umhyggju fyrir stærðfræði, auk umhyggju fyrir samfélaginu í heild. Markmið rannsóknarinnar er að setja fram og nota slíkt siðferðislegt sjónarhorn til að gagnrýna námsmatvenjur í stærðfræðikennslu. Ég set fram og túlka átta stuttar atvikasögur úr eigin stærðfræðikennslu í íslenskum framhaldsskólum sem tengjast prófum og einkunnagjöf. Sögurnar varpa ljósi á hvað umhyggja fyrir stærðfræðinámi felur í sér og hvaða möguleikar eru til að iðka slíka umhyggju í prófadrifinni kennslu. Sögurnar draga einnig fram „þrefalda togstreitu“ sem stærðfræðikennarar þurfa að takast á við í starfi sínu: Í fyrsta lagi að reyna að mæta þeim þörfum sem nemendur tjá og rækta tengsl við þá, í öðru lagi að hafa í heiðri gildi stærðfræðinnar um sannleiksleit og röksemdafærslur og í þriðja lagi að uppfylla kröfur skólakerfisins og samfélagsins um mælanlegan árangur af kennslunni. Niðurstöður mínar eru að prófadrifin kennsla hafa grafið undan umhyggju minni og nemenda minna fyrir stærðfræðinámi.
... Teachers are expected to be flexible in responding to students' sudden questions (Leikin & Dinur, 2007). Movshovitz -Hadar (1988) states that teachers' being authentic in their lessons contributes to students' motivation. However, Polya (1963) mentions that it is difficult for teachers who do not have a doctorate or master's degree to direct students' creative activities since they do not have any individual research experience. ...
In this study, it was aimed to examine the views and practices of middle school mathematics teachers regarding instructional creativity and to reveal to what extent the views and classroom practices overlap. This study is qualitative in nature and was conducted using a case study design. The participants of the study consisted of four middle school mathematics teachers. In the process, face-to-face interviews were conducted with the teachers and their views on instructional creativity were obtained, three selected lessons of the teachers were followed and video and audio recorded, and after the teachers were informed about instructional creativity, they were asked to create a lesson plan based on creative teaching and these one lesson were also recorded. Semi-structured interviews were conducted with the teachers at all stages and the interviews were supported by lesson observations. The data were analysed by content analysis method. The findings revealed that in their views and practices, teachers mentioned and applied the flexibility component of instructional creativity in terms of connecting mathematical concepts and interdisciplines, and the originality component in terms of student discoveries in their lessons. On the other hand, it was determined that teachers gave very little attention to the elaboration component both in the interviews and in the practices. The findings indicate that teachers' views and practices regarding instructional creativity do not overlap sufficiently. Especially in the last lessons observed after the information about instructional creativity, it is noteworthy that very few creative practices were observed as in other lesson observations. The findings indicate that teachers need professional development opportunities in terms of creative lesson teaching. The findings are discussed in the light of the related literature.
... Como expone Movshovitz-Hadar [10]: ...
Con la firme convicción de que se pueden hacer matemáticas sofisticadas con contenidos elementales (Dreyfus, 1991) y de que los profesores de matemáticas deben tener un conocimiento profundo tanto de las matemáticas como de su didáctica (Hill et al., 2008), presentamos en este tres teoremas de distintas ramas de las matemáticas que hemos llevado al aula, junto con una descripción detallada de cómo los hemos presentado, qué objetivos perseguíamos y qué han descubierto los estudiantes al trabajar con ellos.
... 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.
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