Esla/Research Proposal

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!Rough Draft!

Contents 1 curr 2 Background 3 Goal 4 Research methods 5 Existing research in the subject 5.1 Linear Algebra Teaching 5.2 Visualization in Eeducation 5.3 Design experiment/ research 5.4 Misc 6 What I have done so far 6.1 Some background and motivation 6.2 Software developed 7 Future time table 7.1 Till the middle of 2010 7.2 Till the end of 2010 7.3 Till middle of 2011 7.4 Till the end of 2011 8 Computer / Human education 9 Misc 10 Evaluation measures 11 References 11.1 Linear Algbera 11.2 Visualization 11.3 Design Experiment 12 External links

curr

Interactive_geometry_software [1]

• nitsa suggests this:

http://www.effectiveeducation.org/wiki/index.php?title=Experiment:Minesweeper_and_Hypothetical_Thinking_Action_Research_%26_Pilot_Study/Plan

• see educational software in wikipedia (both english spanish and hebrew)
• read about brainpop and see if you can draw any conclusions

• animation in education references articles [2]

תקופה מיוחדת שבה הויזואליזצה האינטראקטיבית נגישה יותר...

• מעבר לטכנולוגיות רשת..
  • HTML5
  • SVG
  • canvas module
  • geogebra

Assesment.. evaluation... מחקרי התערבות.. interversion..

פיתוח מבוקר של תוכנית לימודים ....

• See the latest google keynotes

• google's directory for math educational software [3]

המליצו לי להביא מחקרים על שלושה נושאים עיקריים: ויזואליזציה, הוראת אלגברה לינארית, ומחק עיצוב...

ויזואליזציה זה לא התחום בו אני צריך להתעניין... ויזואליזציה זה יותר לקחת נתונים ולהראות אותם במבטים שונים.

אני יותר מדבר על לומדה אינטראקטיבית

באמת יש חשיבות למבטים השונים על העולם אבל תחום המחקר הוא אחר.. אני צריך למצוא מחקרים על לומדות שמדגישות את ההיבט הגאומטרי של דברים ותוכנות שמאפשרות התנסות ומשחק...

מה בדיוק אני רוצה למצוא? "מחקרים הוכיחו ש.. ככל שמושיבים יותר ילדים מפוחלצים סביב תוכנה אינטראקטיבית אז...

הוראת אלגברה לינארית... בעיות בתחום.. נקודות הכשל המרכזיות .. מדוע הקורס בעיתי.. אולי לראיין תלמידים, אולי לראיין מורים... לקרוא מאמרים

תחום מתאים במחקר מדעי.. איך קוראים למחקר שבו אדם מנסה לפתח שיטת הוראה... בוחן אותה כל פעם ומשנה אותה בהתאם לאפקטיביות שלה...

חוברות לפיתוח תוכנית לימודים מין קוק בוק איך לעשות זאת בצורה מבוקרת...

formative evaluation

Instructional design

נצה אמורה לתת לי לקרוא פרק מהספר הצהוב שמדבר על פיתוח תוכנית לימודים כמו על פיתוח מערכת הנדסית?

• http://en.wikipedia.org/wiki/Educational_technology
• http://en.wikipedia.org/wiki/Instructional_theory
• http://en.wikipedia.org/wiki/Educational_technology
• http://en.wikipedia.org/wiki/Instructional_design
• http://en.wikipedia.org/wiki/Instructional_technology
• http://en.wikipedia.org/wiki/Learning_theory_(education)
• http://en.wikipedia.org/wiki/Teaching_method
• http://en.wikipedia.org/wiki/Constructivism_(learning_theory)
• http://en.wikipedia.org/wiki/Interactive_geometry_software

Background

Basic linear algebra knowledge, is considered essential in almost any scientific and engineering education. It is been taught in elementary courses to millions of students throughout the world.

Although the subject has firm theoretical foundations and is relatively well understood, many students find it intimidating and hard to understand ^[1]. Even students who succeed in basic linear algebra courses, often complain about lack of understanding. In spite of the fact that they manage to master the required techniques, they still feel they are just performing formal manipulations without a deep understanding of their meaning.

Basic concepts like vector-space, subspace, linear combination, span, linear dependency, dimension, linear transformation and coefficient vectors are given a formal - exact definition but frequently doesn't get a clear meaning in the mind of the student.

Goal

The purpose of this research is to seek for educational methods to improve understanding of basic concepts in linear algebra. The focus will be on creating interactive tools for visualization of the fundamental ideas of the field.

As part of the research, I will develop software that will enable the student to play with vectors, move them around, see their span, sense their dependency or independence, apply a linear transformation on them, change the base and watch the coordination's change accordingly, find the intersection of two subspaces and more.

After letting the student using the tools, I intend to interview them and ask them about their experience. I want to understand what was beneficial, what was not and why. According to the input I get from the students I intend to change and tune the software to be worthy for the students to use.

I hope that by working closely with the students and adapting the software to their needs, I will gain a better understanding of the learning process of linear algebra. A good outcome of this research will be a new technique to aid linear algebra student and its reasoning.

Research methods

As I mentioned in the above section, the research will be conducted as an alternating sequence of developing software/questioning users. My hope is, of course, that this sequence will converge to something of value.

In terms of educational research, my research falls into the category of qualitative action research. I do not intend to use intensive statistical analysis to...

Existing research in the subject

One of Linear Algebra Curriculum Study Group (LACSG) recommendation is: Faculty should be encouraged to utilize technology in the first linear algebra course

Linear Algebra Teaching

Gilbert Strang

... very much inspired by his approach and attitude .. In his book "Introduction to linear Algebra"^[2].. says: " If we teach pure abstraction, or settle for cookbook formulas, we miss the best part" .. in his preface he emphasizes the importance of visualizing vectors ans spaces

Guershon Harel In modern courses of linear algebra, it is often the case that students are presented with very abstract concepts in early stages. Harel emphasizes the need for establishing a firm visual understanding using concrete examples before the introduction of the concepts in their most abstract formulation ^{[1] [3] [4]}.

Harel's studies show that students indeed have a great difficulty in understanding the formal approach of linear algebra modern teaching. He describes students who are able to perform correct manipulations without any understanding of their meanings. Those students learn only to copy techniques shown to them and often apply them in the wrong manner when presented with a new version of the problem or when they are been asked to explain relationship to something else that they have learned.

Instead of following the formal approach, Harel built a course in with several phases. In the first phase, the students see geometrical examples of vectors in R^3. The teachers asks the students various questions aiming to gain intuitive understanding of the basic concepts. For example, one question was: "What are the conditions for a group of vectors to be close under linear combination". Playing geometrically with this question gain intuitive understanding of the concept of subspace. Many concepts gets a clear geometrical meaning in this context. For example, three independent vectors are three non-coplanar vectors.

After "playing around" with the geometrical examples, the course generalize the concepts to N dimensions and then to general vector space.

The main assertion is that this graduate building greatly improved the understanding and the ability to apply the learned material comparing to students that learned in the conventional approach.

My research will hopefully be useful for the first phase of Hael's course. The ability to to draw good visualization on the board is very limited. Drawing good images of three dimensional scenarios is hard. Sometimes even a good drawing is hard to understand without the ability to look at it from several different view points. Even in two dimensional scenarios, it is hard to draw the animation that helps to understand concepts like linear transformation (see the video clip of linear transformation visualization).

David Carlson linear algebra students, which he described "as if a heavy fog has rolled in over my students. and they cannot see where they are? or where they are going." ^[5] ... take articles from first proposal ..

Jane M. Day and Dan Kalman ^[6] [4] Describe the difficulties in teaching linear algebra, some which are common to all abstract theoretical teaching and some unique to linear algebra. One of the main claims is that the subject of teaching linear algebra is not well understood. They assert that we have the means to check whether the teaching process was successful but we do not understand why.

Additional relevant publications:

• Jean-Luc Dorier and Anna Sierpinska, Research into the Teaching and Learning of Linear Algebra ^[7]

• Steven J. Leon, Linear Algebra with Applications^[8]

Visualization in Eeducation

The ATLAST project (Augment the Teaching of Linear Algebra through the use of Software Tools) [5] ^[9] motivation is very simillar to the one of this research, both are trying to aid the learning process through the use of software visualization and interactive tools.

... different approaches .. the student where supposed to use a MATLAB and some additional m files... I try to provide applications and video clips.. less technical knowledge required from the user ..

• EJ348095 - Visualisation in High School Mathematics / For the Learning of Mathematics--An International Journal of Mathematics Education, v6 n3 p42-46 Nov 1986

Design experiment/ research

Misc

• Olofsson, Peter. Probability, Statistics ^[10]

What I have done so far

Some background and motivation

As an undergraduate student I used love linear algebra. I have been learning in the axiomatic-formal approach. First I learned about basic algebraic structures such as group, ring, field and vector sapce over a field. While dealing with vector spaces, I have noticed that although I was impressed by the beauty and purity of the presented theory, many fellow students complained that it all seems just formal definitions symbols manipulations. Some of those fellow student were very bright skillful and talented. Looking backwards, asking myself why I felt so differently about the material, I came to the conclusion that it has to do with the way I was taught algebra at high school. I learned according to an experimental program by Michael Maschler which focused on R^n and emphasized the geometrical aspect of the material. When I started learning linear algebra at the university, I already had a firm intuition about vector in dimension less or equal to 3. Later on, I became a lecturer of software programming in the academic. I used to give my undergraduate students, exercises that relates to the theoretical material that they were learning in the mathematical courses. One of those exercises was to draw a tetrahedron, rotate it and pull it in a certain way. These manipulations require understanding of the concepts of linear transformation and change of base representation. I was surprised to see that many students, in spite of the fact that they have just learned the subject in the linear algebra course, had know idea how to approach the challenge of the exercise. In order to the student cope with the exercise, I found myself giving a lecture about the geometrical aspects of certain basic linear algebra concepts.

After the students manage to solve the exercise, many of them told me that it gave them a great insight to the meaning of linear algebra. In addition, they said it was the first time they had really see the usefulness of the theory.

The experience I have just described, lead me to think that a good visualization software can be a great companion for a linear algebra course. I have searched the web in order to find such software and came across many small tools (mainly applets and matlab scripts) that shows various examples of geometrical visualizations. The tools I have found were all very basic and the interactivity with the learner was very limited.

talk to friends .. people .. leon strang .. need for...

tried to think of good tools... escort a course .. helped students... students confused concepts like span, base (a small group of vectors.. versus a group of a generated vectors ..) I've been trying to help them understand proofs.. the switching lemma .. had problem with the formulation but also because they didn't have an intuitive grasp of the concepts..

Software developed

• Vector space tool

An educational software that aims to help the student gain a better understanding of the following basic concepts:

• A linear combination of vectors
• The span set of a set of vectors
• Linear dependency/independence
• Base (as minimal spanning set or maximal independent set)
• Dimension

• Linear transformation tool

• Bases for additional tools

• installer and friendly installation ..

• Course escorting
• Student interviews and help

Future time table

The time table describes the planning for the following two years in which I hope to finish the research.

Till the middle of 2010

• Finish escorting 3 courses.. collect many interviews... change the software accordingly

Till the end of 2010

• After being a teaching asitance of a basic LA course and combining the software with the course assignment and clips with the teaching.

Till middle of 2011

• Some written results of the educational method
• Maybe an article ..?

Till the end of 2011

• Submitting the thesis and completing any other requirements associated with it.

• Vector space software:
  • Add a representation of a vector according to space
  • Subspaces intersection
• Linear transformation software:
  • Eigenvalues - as a field of arrows

• Additional tools:
  • Gaussian eliminator
  • Complex number visualization
  • Polynomials manipulator

• Course escorting - summer & first semester 2010
• interviewing

• additional screencasts .. educational clips ..

• think about a better integration of the software with the basic LA course .. how to combine in in their weekly assignments.. ?

Computer / Human education

Computer benefits .. Human benefits..

Misc

There is no mean to replace the frontal education with the software tool, the tools are meant to be an additional help...

talk about geogebra and Pitagoras

Formative_assessment (or may it be Summative assessment) - is that what I'm really doing? is this the category of my research?

very relevant article [6] .. can take many conclusions like about the difficulties with linear algebra.. see the references there.. might be helpful

Evaluation measures

References

1. ↑ ^{1.0 1.1} Learning and Teaching Linear Algebra: Difficulties and an Alternative Approach to Visualizing Concepts and Processes/Harel, Guershon. Focus on Learning Problems in Mathematics, v11 n1-2 p139-48 Win-Spr 1989
2. ↑ Gilber Strang, Introduction to linear Algebra, Third Edition
3. ↑ Harel, G. (1989). Applying the principle of multiple embodiments in teaching linear algebra: Aspects of familiarity and mode of representation, School Science and Mathematics, 89, 49-57
4. ↑ Harel, G. (1997). The linear algebra curriculum study group recommendations: Moving beyond concept definition. In Carlson D., Johnson, C, Lay, D., Porter, D., Watkins, A, \& Watkins, W. (Eds.). Resources for Teaching Linear Algebra,. MAA Notes, Vol. 42, 107-126.
5. ↑ David Carlson, Teaching linear algebra: must the fog always roll in? College Mathematics Journal 24, (1993) 29-40. Also in this volume.
6. ↑ Jane M. Day and Dan Kalman "Teaching Linear Algebra: Issues and Resources" College Mathematics Journal, Volume 32, Number 3, May 2001, pp 162 - 168.
7. ↑ Jean-Luc Dorier and Anna Sierpinska "Research into the Teaching and Learning of Linear Algebra",Springer New ICMI Study Series, Volume 7, Section 3, 255-273
8. ↑ Steven J. Leon "Linear Algebra with Applications" 7th ed.
9. ↑ Steven J. Leon, "ATLAST Computer Exercises for Linear Algebra", Second Edition
10. ↑ Olofsson, Peter. Probability, Statistics, and Stochastic Processes. Wiley-Interscience, 2005. Page 8.

• International Linear Algebra Society

Linear Algbera

• EJ797267 - Motivating the Concept of Eigenvectors via Cryptography

• EJ775674 - Embodied, Symbolic and Formal Thinking in Linear Algebra

• A First Course in Linear Algebra by Robert A. Beezer

• EJ797280 - Using Technology to Facilitate Reasoning: Lifting the Fog from Linear Algebra

• EJ811008 - Constructive Learning in Undergraduate Linear Algebra

• Gilbert Strang online course

Visualization

Design Experiment

External links

• Linear Algebra Education Web Service

• [http://teamat.oxfordjournals.org/cgi/reprint/27/2/53.pdf Motivating the concept of eigenvectors via

cryptography]

• The Effectiveness of the Online Algebra Course

• Learning and Teaching Linear Algebra: Difficulties and an Alternative Approach to Visualizing Concepts and Processes. (EJ389514) / Harel

• geogebra