FREE ELECTRONIC LIBRARY - Online materials, documents

Pages:     | 1 |   ...   | 118 | 119 || 121 |


-- [ Page 120 ] --

Keywords: intellectual need, Realistic Mathematics Education, mathematizing, bridge course Introduction Harel’s (1998) Necessity Principle states, “Students are most likely to learn when they see a need for what we intend to teach them, where by ‘need’, is meant intellectual need, as opposed to social or economic need” (p. 501). The aforementioned Necessity Principle puts forth a conjecture about how students learn (Speer, Smith & Horvath, 2010) and has been used extensively by Harel as a component of a larger conceptual framework called Duality, Necessity, and RepeatedReasoning (DNR) (Harel, 2001). More recently, Harel, (2011) has refined and expanded these intellectual needs into five inextricably-linked categories: the need for certainty (to establish that a statement is true), the need for causality (to determine why a statement is true), the need for computation (to quantify and calculate), the need for communication (to persuade others of truth and to agree on conventions), and the need for structure (to re-organize knowledge into a logical system). Harel has illustrated each of these categories of intellectual need using both examples from his own research, as well as documented accounts from the history of mathematics, which suggests that intellectual need permeates throughout the discipline of mathematics. However, at the same time, it leaves one to question whether all of these categories of intellectual need can be illustrated in a single context and more importantly, how students exhibit and seek to address these intellectual needs in an actual classroom setting. This research study investigates these questions by examining the role that students can play in articulating their intellectual needs within the context of axiomatizing.

Theoretical perspective Several theoretical perspectives—both in curriculum design and in the preliminary data analysis, influence this research study. First, the group theory curriculum (Larsen, 2004) that was used in this teaching experiment was inspired by the design theory of Realistic Mathematics Education (RME) (Freudenthal, 1991). One of the tenets of RME is the importance of guided reinvention, in which a student is encouraged to “invent something that is new to him, but wellknown to the guide” (Freudenthal, 1991, p. 48). In this particular setting, guided reinvention is the process by which students’ model of an abstract group first emerges as a model of the symmetries of 15TH Annual Conference on Research in Undergraduate Mathematics Education 603 an equilateral triangle and evolves into a model for the abstract dihedral group of order six (Larsen, 2004). A core component of Larsen’s curriculum is the mathematical activity of axiomatizing, which requires students not only to construct a system of rules for operating on their symmetries, but also to refine these rules to a minimal list of axioms that could be used independently of the objects from which they were abstracted (Larsen, 2009). Therefore, the researcher in this study considered students’ axiomatizing as a mathematical activity analogous to that of symbolizing (Gravemeijer, Cobb, Bowers, & Whitenack, 2000), and sought to examine some of the intellectual needs that the students articulated when developing their axiomatic system. Consequently, the DNR conceptual framework was used as an indispensable tool for analyzing the data. Specifically, the categories of intellectual need were used to code instances in which students referred to knowledge that they would need to construct to resolve a problematic situation (Harel, 2011).

Background and research methodology Over the course of nine weeks, the teacher-researcher and his students progressed through a subset of an RME-inspired curriculum (Larsen, 2004) for re-inventing the concept of group.

Extensive written and video data were collected from this teaching experiment, which occurred in an elective mathematics bridge course at a medium-sized, suburban community college. The teacher-researcher was a full-time community college instructor with more than ten years of experience teaching courses ranging from arithmetic through integral calculus. The participants were nine community college students (five female and four male) whose ages ranged from 17 to 35 years. Four of the students were math majors, two were engineering majors, one was a music major, and two students had not yet declared a major. The students’ mathematical experience varied greatly: four had taken courses through differential equations, one had completed calculus III, three had completed calculus I, and one student had only completed college algebra. None of the students had taken a junior-level collegiate math course, but two students were familiar with some group theory concepts from taking bridge course the previous year. A retrospective analysis (Cobb & Whitenack, 1996; Stylianides, 2005) was conducted on the classroom video data collected from this term-long teaching experiment. In the initial pass of the data, the researcher identified instances in which students may have had opportunities to address intellectual needs—specifically, where they were confronted with a problematic situation that was unsolvable by their current knowledge (Harel, 2011). The majority of these problems came directly from the instructional prompts that were part of Larsen’s curriculum, but other problematic situations originated either from the teacher or from students in the class. In a second pass of selected classroom episodes, students’ acts of axiomatizing were analyzed and correlated (when possible) with Harel’s existing categories of intellectual need.

Preliminary results of the research At this point in the analysis, a few themes have emerged. First, the data lends credence to one of Harel’s claims about the need for computation—that it is indeed a robust intellectual need.

Eight of the nine students in the teaching experiment seemed to be motivated to invent rules that aided in computation and for one student in particular, the associative axiom seemed completely unnecessary because he saw no computational need for it. Secondly, the data suggests that axiomatizing is a mathematical activity that could provide students with opportunities to address a variety of intellectual needs. For example, globally there existed a constant tension between the need to create rules that made students’ computations more efficient, while at the same time, keeping the list of rules as small as possible to avoid redundancy. This tension provided an opportunity to discuss the differences among mathematical terms such as definitions, axioms, theorems, and lemmas and to point out the advantages and disadvantages of lengthening or 604 15TH Annual Conference on Research in Undergraduate Mathematics Education shortening the list of rules. As the students’ model progressed from a model of toward a model for, decisions about how to state certain axioms appeared to be influenced by their needs to communicate, compute, and structure. In fact, throughout the term, the students formally axiomatized five different versions of their list of rules, which provides strong evidence for the existence of the need for structure. Finally, there is evidence in the data to support re-examining Harel’s initial category of the need for elegance, which he described as “what we associate with mathematical beauty, efficiency, and abstraction” (1998, p. 502). In making decisions about notational conventions and which rules to keep or discard, students’ choices may be motivated not only by the existing categories of intellectual need, but also by an intellectual need that is epistemic to the discipline of mathematics—the need for elegance. One of the students in the teaching experiment seemed to be periodically motivated by this need and used a powerful metaphor to

describe the need for elegance of an axiomatic system, as this excerpt illustrates:

Chris: It’s like you know, you got a hammer sitting at home…you get a blue hammer. You go out and get a blue hammer, so you hammer in nails with a blue hammer instead of a red hammer. Cuz we already got the red hammer and the red hammer works just as well to solve the problems as the blue hammer…and we already have it.

Later, Chris acknowledged that the creation of a new axiom would make certain computations “faster,” but he stated that such an axiom did not make the system “stronger.” Sinclair (2004) adds to the importance of this need by stating, “In terms of the aesthetic dimension of mathematical judgments, the emphasis placed on the aesthetic qualities of a result implies a belief that mathematics is not just about a search for truth, but also a search for beauty and elegance” (p. 269).

Questions to further future research In traditional mathematics curriculum, students are rarely given opportunities to develop their own notations, conventions, or axioms, so examining the role that students’ intellectual needs plays in designing and enacting RME-inspired curriculum may be very useful for the field. In particular, Harel (2011) claims that “DNR’s Necessity Principle is an analogue of the RME dictum that students must engage in mathematical activities that are real to them, for which they see a purpose” (p. 23). If that is the case, then how do other acts of mathematizing correlate with DNR’s categories of intellectual need?

Another area that might be worthy of future investigation concerns the function and role of bridge courses. If one of the primary functions of bridge courses is “to ease the transition from lower division, more computational [emphasis added], mathematics courses to upper division, more abstract, mathematics courses such as modern algebra and advanced calculus” (Selden & Selden, 1995, p. 135), then it seems reasonable that students in bridge courses should engage in mathematical activities that give them opportunities to address intellectual needs other than those necessitated by computation. Arguably, proof and the activities associated with it attend to this larger goal, so it is not surprising that much of the research on bridge courses has centered upon proof. However, in addition to proof, what other elements could or should be included in bridge courses to support student learning of more abstract mathematics?

–  –  –

Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30(3), 213-228.

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Norwell, MA:

Kluwer Academic Publishers.

Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling, and instructional design. In P. Cobb, E. Yackel, & K. McCain (Eds.), Symbolizing and communication in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Lawrence Erlbaum Associates, Inc., 225-273.

Harel, G. (1998). Two dual assertions: The First on learning and the second on teaching (or vice versa). American Mathematical Monthly. 105(6), 497-507.

Harel, G. (2001). The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction. In S. Campbell & R. Zaskis (Eds.). Learning and Teaching Number Theory. New Jersey, Ablex Publishing Corporation, 185Harel, G. (2011). Intellectual need and epistemological justification: Historical and pedagogical considerations. In K. Leatham (Ed.), Vital Directions for Mathematics Education Research. Manuscript in preparation.

Larsen, S. (2004). Supporting the guided reinvention of the concepts of group and isomorphism: A developmental research project. Unpublished Dissertation, Arizona State University.

Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. The Journal of Mathematical Behavior, 28(2-3), 119-137.

Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.

Sinclair, N. (2004). The role of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6(3), 261-284.

Speer, N., Smith, J., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. Journal of Mathematical Behavior, 29, 99-114.

Stylianides, A. J. (2005). Proof and proving in school mathematics instruction: The elementary grades part of the equation. University of Michigan.

–  –  –


This theoretical report aligns itself with Arcavi’s (1994) work and the tradition of ontosemiotic research in mathematics education (Font, Godino, & D’Amore, 2007) and is situated in

the context of statistics education. This report will:

• articulate a notion of symbol sense in statistics

• explain the importance to student understanding of the development of symbol sense.

The goal of this work is to guide both research and curriculum design efforts for introductory undergraduate statistics courses.  The  paper  begins  by  describing    statistical  analogs  of   Arcavi’s  algebraic  symbol  sense,  then  furthers  this  by  noting  the  importance  of  reading   symbols  generally,  reading  symbols  through  the  context  of  the  question,  and  the  reading  of   symbols  related  to  the  visualization  or  selection  of  the  display.    Finally,  the  paper  briefly   explores  how  the  understanding  of  symbols  becomes  more  difficult  and  important  in  the   use  of  the  Central  Limit  Theorem  and  estimation  of  parameters.

Keywords: Statistics, symbols, symbol sense, semiotics

0. Introduction and Motivation While there have been investigations of students’ understanding of measures of center (Mayen, Diaz, Batanero, 2009; Watier, Lamontagne, & Chartier, 2011), variation (Peters, 2011;

Pages:     | 1 |   ...   | 118 | 119 || 121 |

Similar works:

«KISS OF THE SPIDER WOMAN PRESS KIT FOR MORE INFORMATION, PLEASE VISIT www.kissofthespiderwoman.com OR WRITE TO kissofthespiderwoman@earthlink.net KISS OF THE SPIDER WOMAN BACKGROUND NOTES In a prison cell somewhere in Latin America, two very different men warily confront each other. Molina (William Hurt) is first seen wrapping his head in a towel, in the shape of a turban, while Valentin (Raul Julia), bearded and classically macho in appearance, watches with a mixture of fascination and...»

«By Emily Cameron September 2010 Once upon a time, there lived a young lady named Mrs. Johnson and her beautiful nine-year-old daughter, Mary. They lived in a grand house with stained-glass windows and lovely golden walls which they had moved into just a few weeks ago. The Johnsons' house was a rather fancy place fit for a princess, with its polished furniture and glittering crystal chandeliers. Not only was the house pretty, it was safe too. The firm, strong plaster walls protected Mary and her...»

«Annual Report 2007-08 BOOKS AND BOOK-CHAPTERS PUBLISHED Aerospace 1. Fundamentals of Combustion, Prentice Hall of lndia, 2008, Edited book : A. K. Ghosh and D. P. Mishra Proceedings of 21st National Convention of Aerospace Engineers, 2007, D. P. Mishra. Biological Science and Bio-engineering 2. Affinity Cell Separation: Fundamentals, analytical and preparative methods. Editors: Ashok Kumar, Igor Yu Galaev & Bo Mattiasson; Springer-Verlag (Heidelberg, Germany) 2007. 3. Smart polymers in affinity...»

«The Georgia Social Studies Journal Fall 2012, Volume 2, Number 2, pp. 38-45. Georgia Council for the Social Studies We the future: Teaching students about presidential elections. Jeremiah C. Clabough The University of Alabama at Birmingham Thomas N. Turner The University of Tennessee Gary W. Cole Sequoyah High School (Madisonville, TN) This article explores four approaches and multiple activities for teaching about presidential elections. The activities presented focus on getting to know the...»

«presents 5 BROKEN CAMERAS A FILM BY EMAD BURNAT AND GUY DAVIDI France-Israel-Palestine / 2011 / Color and B&W / 90 mins. / Hebrew and Arabic w/English subtitles A Kino Lorber Release from Kino Lorber, Inc. 333 West 39 St., Suite 503 New York, NY 10018 (212) 629 – 6880 Press Contacts: Rodrigo Brandão VP of Publicity and Promotions rodrigo@kinolorber.com 917.434.6168 Julia Pacetti President, JMP Verdant julia@jmpverdant.com 917.584.7846 Kino Lorber, Inc. | 333 West 39 St., Suite 503 | New...»

«This book is made into digital form by AJ Entertainment. All rights reserved. Copyright ©AJ Corp. Modifying, renting, and selling of this book is illegal and may give rise to criminal offence. You can distribute is freely to anyone. CONTENTS Part One Who He Was 1. The Jesus I Thought I Knew 2. Birth: The Visited Planet 3. Background: Jewish Roots and Soil 4. Temptation: Showdown in the Desert 5. Profile: What Would I Have Noticed? Part Two Why He Came 6. Beatitudes: Lucky Are the Unlucky 7....»

«Deutsche Bank Markets Research Asia Date Periodical China 24 March 2016 Made in China Strategy Michael Tong Luka Zhu Econ. Update; Alibaba; Results: China Research Analyst Research Associate (+852) 2203 6167 (+852) 2203 6173 Life/GCL/Guangdong/CTIH/DSBG/ENN (+) michael.tong@db.com luka.zhu@db.com Daily Market Statistics 23/03/2016 THEME OF THE DAY Performance Price 1D% chg 1M% chg Update China: Capital outflows slowed further in March (Zhiwei Zhang) HSI 20,615 -0.2 5.8 The SAFE released...»

«The Minutes of Virginia Presbytery Of The Associate Reformed Presbyterian Church Called Meeting Virginia Cottage – Bonclarken Conference Center June 9, 2015 East Flat Rock, NC Called Meeting Timber Ridge ARP Church July 7, 2015 Fairfield, VA Stated Meeting Ebeneezer Associate Reformed Presbyterian Church October 17, 2015 Lexington, VA Virginia Presbytery Of the Associate Reformed Presbyterian Church Fall Stated Meeting October 17, 2015 Ebeneezer Associate Reformed Presbyterian Church...»

«Ministry of Foreign Affairs of Denmark Danida NEPAL COUNTRY CASE STUDY CITIZENS’ VOICE AND ACCOUNTABILITY EVALUATION July 2008 July 2008 Production: Royal Danish Embassy, Kathmandu, Nepal Cover Design: Designgrafik A/S, Copenhagen, Denmark This report can be ordered from the Royal Danish Embassy: ktmamb@um.dk The report will, together with the four other Country Case Studies from Bangladesh, Democratic Republic of Congo, Indonesia and Mozambique, be placed on the CD-ROM inserted in the main...»

«RIGHT HO, JEEVES  By  P. G. Wodehouse To  RAYMOND NEEDHAM, K.C.  WITH AFFECTION AND ADMIRATION CONTENTS:  1­ 2­ 3­ 4­ 5­ 6­ 7­ 8­ 9­ 10­ 11­ 12­ 13­ 14­ 15­ 16­ 17­ 18­ 19­ 20­ 21­ 22­ 23­ 1­ Jeeves, I said, may I speak frankly?  Certainly, sir.  What I have to say may wound you.  Not at all, sir.  Well, then­­­­  No—wait. Hold the line a minute. I've gone off the rails. ...»

«THE SUTTON TRUST S Primed for Success? The characteristics and practices of state schools with good track records of entry into prestigious UK universities A report on research carried out for the Sutton Trust Andrew Curtis, Institute of Education, University of London Sally Power, School of Social Sciences, Cardiff University Geoff Whitty, Institute of Education, University of London Sonia Exley, Institute of Education, University of London Amanda Sasia, Institute of Education, University of...»

«2009 2014 EUROPEAN PARLIAMENT Committee on Budgetary Control 2010/0807(NLE) 25.2.2010 DRAFT REPORT on the nomination of Szabolcs Fazakas as a Member of the Court of Auditors (C7-0019/2010 – 2010/0807(NLE)) Committee on Budgetary Control Rapporteur: Inés Ayala Sender PR\805785EN.doc PE431.020v02-00 EN EN United in diversity PR_NLE_art108 CONTENTS Page PROPOSAL FOR A EUROPEAN PARLIAMENT DECISION ANNEX 1: CURRICULUM VITAE OF SZABOLCS FAZAKAS ANNEX 2: ANSWERS BY SZABOLCS FAZAKAS TO THE...»

<<  HOME   |    CONTACTS
2017 www.thesis.dislib.info - Online materials, documents

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.