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Michal Yerushalmy University of Haifa Israel

 Designed to challenge traditional notions of what school mathematics is and how it can be taught and learned  For over two decades, this curriculum has been implemented in a variety of settings in Israel  Aims at addressing future innovations and analyzing the potential of new technologies  Developed with and for schools while coping with societal and curricular constrains


Lakatos Proofs and Refutations (1976)

A “simple pattern of mathematical discovery” (p. 127)  [Having] a primitive conjecture  An initial proof (a rough thought, experiment, or argument)  Global counterexamples to the primitive conjecture  Proof re-examined  Primitive conjecture improved  Theorems that “might be lying at the cross-roads of different proofs” examined  Turning consequences and counterexamples into examples of a new inquiry 3

     


An exhibition Guidance A few central objects Many galleries (Too) Many exhibits in each Do-It-Yourself stations


TechnologyBased Inquiry Curriculum •Geometry •Function's based Algebra •Modelling •Calculus 6

The design of digital textbooks VisualMath: •Function’s based Algebra Interactive eBook

Mobiles Ubiquitous Learning •Math 4 Mobile Learning •Click 2 go •Augmented Textbook

• • • • •


Technology-Based but Not Technology Driven Driven by the need to support authentic Mathematical Inquiry Evolution rather than Revolution Designed to support long term educational change Done for schools with schools


Foundational software • Software designed to afford substantial inquiry learning throughout secondary school (7-12 grade)


Geometric Supposer

Schwartz, J.L. ,Yerushalmy, M., Shternberg, B. (19902005) The Geometric Supposer II (in Hebrew and English). Developed at the Center for Educational Technology, CET, Ramat-Aviv, Israel.

Algebraic Patterns

Yerushalmy, M & Shterenberg, B. (1992). Algebraic Patterns. (In Hebrew, English and Arabic) for Center of Educational Technology, Ramat-Aviv.

Function Sketcher

Yerushalmy, M & Shterenberg, B. (1993) The Function Sketcher (in Hebrew, English and Arabic) for Center of Educational Technology, Ramat-Aviv.

Move on!

Yerushalmy, M & Shterenberg, B. & Schwartz, J.L. (2001) Move On! Motion Laboratory (software in Hebrew and English) for the Center of Educational Technology, Ramat-Aviv.

Calculus UnLimited

Schwartz, J.L., Yerushalmy, M. (1996) Calculus Unlimited (software in Hebrew, Japanese and English) for the Center of Educational Technology, Ramat-Aviv.

The Geometric Supposer: New reasoning tools The Geometric Supposer is continuously going under cycles of R&D since 1985. Different from most DGEs the Supposer designers chose to limit the content to Euclidean Geometry while broadening and deepening the representations’ structures, the understanding of dragging and the support tools and materials for conjecturing being a core of elementary and secondary curricula.

Recent research: •Talmon, V, Yerushalmy, M.(2004) Understanding Dynamic Behavior: Parent- Child Relations in Dynamic Geometry Environments. Educational Studies in Mathematics. 57, 91-119. •Talmon, V., Yerushalmy, M. (2006) Computer "Knowledge" and Student's Images of Figures: The Case of Dragging. In the Proceedings of the 30th Annual meeting of the International Group of Psychology of Mathematics Education, vol. 5, pp. 241-248 Prague. •Maymon-Erez M., Yerushalmy, M. (2007) "If you can turn a rectangle to a square then you can turn a square to a rectangle…": On the complexity and importance of psychologizing the dragging tool by young students. 10 International Journal of Computers for Mathematical Learning. 11 (3), 271-299.

Algebraic Patterns: Bridging language to conjectures in algebra

Yerushalmy, M., & Nassar, S. (2002) From Algebra to Arithmetic: An analysis of the transition. Multimedia package in Hebrew and Arabic (Video, Learning Materials and Documents). University of Haifa. Yerushalmy, M & Shternberg, B. (1994) Symbolic awareness of Algebra Beginners. In the Proceedings of the 18th annual meeting of the International Group for the Psychology of Mathematics Education (PME), vol. 4, pp. 393-400. Portugal.


The Function Sketcher: Bridging Language to Modeling Intervals

Analysis of Change Verbal Constrains

Iconic Constrains

The Sketcher provides bridging language that includes 7 icons, verbs and visual tools to measure the rate of change. It offers creation of models from stories (to the younger age) and foundations for calculus by offering the Rate-ofchange” to be the leading concept.

Selected Publications: •Schwartz, J.L., Yerushalmy, M. (1995) On the need for a Bridging Language for Mathematical Modeling. For the Learning of Mathematics, 15 (2), 29-35. •Yerushalmy, M. (1997) Mathematizing Qualitative Verbal Descriptions of Situations: A Language to Support Modeling. Cognition and Instruction. 15(2), 207-264. •Shternberg, B., Yerushalmy, M. (2004) Didactic Model – Bridging a Concept with Phenomena. In the Proceedings of the 28th Annual meeting of the International Group of Psychology of Mathematics Education, vol. 4, pp. 185–192. Norway. •Shternberg, B., Yerushalmy, M. (2003) Models of functions and models of situations: On design of a modeling based learning environment In H. M. Doerr & R. Lesh (Eds.) Beyond constructivism: A model and modeling perspective on teaching, learning, and problem solving in mathematics education. pp. 479-500. Mahwah, NJ: Lawrence Erlbaum.


MoveOn! Language to Bridge Embodied knowledge & Mathematical models Analysis of kinematic experiment

Synthesis of an Experiment from mathematical objects Selected Publications: •Yerushalmy, M., Shternberg, B. (2006) Epistemological and cognitive aspects of time: A tool perspective. Journal for Research in Mathematics Education Monograph 13. •Botzer, G. & Yerushalmy, M. (2006) Interpreting motion graphs through metaphorical projection of embodied experience. International Journal for Technology in Mathematics Education. 13(3). •Botzer, G. & Yerushalmy, M. (2008) Turning artifacts into meaningful signs through bodily interaction: the case of motion graphs. The International Journal for Computers in Mathematical learning. 13. 13

Function’s Based Algebra: Algebraic skills based on Conceptual operations with Functions Selected Research: •Yerushalmy, M. (1999) Making Exploration Visible: On Software Design and School Algebra Curriculum. International Journal for Computers in Mathematical Learning, 4 (2-3), 169-189. •Kieran, C, Yerushalmy, M. (2004) Research on the role of technological environments in algebra learning and teaching. In K. Stacey, H. Shick & M. Kendal (Eds.) The Future of the Teaching and Learning of Algebra. The 12th ICMI Study. New ICMI (International Commission on Mathematical Instruction) Study Series, Vol. 8. pp.99-152. Kluwer Academic Publishers. •Chazan, D. & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: Research on algebra learning and directions of curricular change. In J. Kilpatrick, D. Schifter, & G. Martin. A Research Companion to the Principles and Standards for School Mathematics. pp. 123135 NCTM, Reston, Virginia.


Calculus UnLimited: Visual tools supporting Differential & Integral procedural thinking

Recent Publications: • Yerushalmy, M., Schwartz, J.L. (1999) A procedural approach to exploration in calculus. International Journal of Mathematical Education in Science and Technology, 30 (6), 903-914. • Yerushalmy, M., Swidan, O. (2012) Signifying the accumulation graph in a dynamic and multirepresentation environment Educational Studies in Mathematics, 80(3), 287-306


VisualMath Textbooks Development:

• Textbooks in Hebrew and Arabic for grades 7,8,9. • First books appear at 1993. • During 1993-1996 a longitudinal study was designed to follow 5 schools • 4 classrooms in two schools, each taught by a different teacher were weekly observed for 3 years. The same teacher taught the class for the 3 years. • 6 task-based interviews, twice a year, were carried with 12 pairs of students: 3 pairs from each observed classroom. They were chosen as the Low25%, High25% and average class level. • Teachers'’ materials were based on this study • Activity books for the high-school appeared throughout 1993-1996.


Assessment According to the goals of the test the same task can be: • considered as exploration or as an exercise • done with or without technological tools • supported by 3 generic types of tools • Emphasize on explanation and argumentation

time Up to 40 minutes


kind of technology  numeric  graphic  symbolic  graphic  symbolic manipulator.

Reasoning & technology  You may give an explanation based just on technology.  You must give an explanation based not just on technology.  You may give an explanation that is not based just on technology.

assignment 3503eng

Teachers Professional Development Materials Major attention to support teachers’ challenges of establishing classroom inquiry: Workshops: “I didn’t see that they were receiving any answers to their questions. ... the teacher went on and on “.. I was losing control in the sense that I was listening to the problem raised …” Multimedia packages: Yerushalmy, M. & Elikan, S. (2000) Discussions in the Mathematics Classroom. Multimedia package in Hebrew and English (Video and Documents). University of Haifa and Center for Educational Technology. Yerushalmy, M. & Shternberg, B. (2000) Mathematizing Reality. Multimedia package in English and Hebrew (Video and Documents). University of Haifa. Research Sample Understanding Teachers’ Understanding of Algebra Taught with the Support of Graphing Technology. Final report submitted to the Spencer Foundation, Small Grants Program. (25 pages). October 2000. final_report.pdf


Teachers’ Guides: Guides are product of users’ groups work. It emphasize classrooms’ observations, authentic students’ work and teachers’ input regarding the nature of the mathematics.

4 editions of “VisualMath: Expressions” Left & clockwise: 1992 – VisualMath first edition 2002- Revised paper unit. Side comments designed to support elaborations and explorations 2003-5 – Interactive unit in the Function’s eBook 2008 – KOTAR - Paper & digital edition, supplemented by the VisualMath applets when read online.


VisualMath Curricular Reform Major Findings Summative analysis in Israel and intensive formative research of student achievements and teacher professional development, revealed successes and weaknesses of the programs and served as a springboard for the next innovative development.

8th graders scored above the national average 9th graders tend to attend the higher level math programs at their high-school studies Successfully performing paper&pencil algebraic manipulations Over-perform comparative students in solving traditional algebra wordproblems Over-perform in solving complex new problems Exhibit rare level of mathematical inquiry Debugged mistakes using multi-representational strategies Lower achievers turn to be successful inquirers Challenging curriculum for Higher achievers

• • • • • • • • •


VisualMath 8th graders scored above the national average

The graph is part of a report of the Israeli National Assessment department 2003. It compares scores of VisualMath students to the average score (yellow ‫ץ‬marks). Red bar provides the overall score. Measured skills (Left to Right blue bars) are: Expressions and variables, single variable word problems, retrieving information from chart, basic geometry, line, exploration task.


Better at Solving complex new problems Could cooking time be less than 2 hours using these two ovens? A cook has a large meat portion at a room temperature which should be cooked as quickly as possible. He has at his disposal a conventional oven, and a microwave. In a cooking trial the cook found out that although the meat temp in the conventional oven is always higher than in the microwave cooking time is 2 hours in both ovens. Shternberg, B. (2002) Understanding the mathematical meaning of phenomena and of the concept of function. Ph.D dissertation (Cum Laude). Faculty of Education, University of Haifa. Under the supervision of M. Yerushalmy


Exhibit rare level of mathematical inquiry along various topics of the school curriculum Maymon-Erez M., Yerushalmy, M. (2007) "If you can turn a rectangle to a square then you can turn a square to a rectangle‌": On the complexity and importance of psychologizing the dragging tool by young students. International Journal of Computers for Mathematical Learning. 11 (3), 271-299. Yerushalmy, M. (1997) Reaching the Unreachable: Technology and the Semantics of Asymptotes. International Journal of Computers for Mathematical Learning. 2(1), 1-25.


Would a parabola have a vertical asymptote?

Better problem Solvers of “school tasks” • They were significantly better than comparable groups. 90.8% vs. 57.2% •

87.2% included a graph describing the situation

• Usually the graph was a sketch 73% • Algebraic equations were the most popular model for formulating the solutions (90.8%) • 70.9% students formulated a correct equation in one variable (F(x)= G(x)) • 96.5% of the solutions that included the above components resulted correct algebraic equation.


Gilead, S. (2003). Solving Algebra Word Problems: Effects of situational and quantitative models. Ph.D dissertation (Cum Laude). Faculty of Education, University of Haifa. Under the supervision of M. Yerushalmy Yerushalmy, M. (2001) Problem Solving Strategies and Mathematical Resources: A longitudinal view on problem solving in a function based approach to algebra. Educational Studies in Mathematics. 43, 125-147. Yerushalmy, M. Gilead, S. (1999) Structures of constant rate word problems: A functional approach analysis. Educational Studies in Mathematics. 39, 185-203.

Developed strategies to locate and overcome mistaken manipulations


Gafni, R. (1996) Using multiple representations of functions to improve semantics understanding of algebraic expressions and equations. Ph.D dissertation. Faculty of Education, University of Haifa. Under the supervision of M. Yerushalmy Yerushalmy, M & Gafni, R. (1992) Syntactic Manipulations and Semantic Interpretations in Algebra: The effect of graphic representation. Learning and Instruction. Pergamon Press Ltd. 2, 303-319. Yerushalmy, M. (1991) Effects of Computerized Feedbacks on Performing and Debugging Algebraic Transformations. Journal of Educational Computing Research. 7, 309-330

Manipulating Creatively How would you think about the equation x+y=2x- y?

y=x/2 x+1=2x-1, x+2=2x-2... f(x,y)=g(x,y)

Yerushalmy, M., Gilead, S. (1997) Solving Equation in a Technological Environment: Seeing and Manipulating. Mathematics Teacher, National Council for Teachers of Mathematics. 90 (2), 156-163. 26

Generalizing key features of Functions 7th graders design a poster to describe prices for rental cars

payment is by day and kms

Yerushalmy, M. (1997) Designing Representations: Reasoning about Functions of Two Variables. Journal for Research in Mathematics Education. 28 (4), 431-466. Yerushalmy, M., Bohr, M. (1995) Between Equations and Solutions: An Odyssey in 3D. In the Proceedings of the 19th annual meeting of the International Group for the Psychology of Mathematics Education (PME), vol. 2, pp. 218-225. Brazil. 27

Mathematical guided inquiry is the core curriculum for all students: The lower 25% • Yerushalmy's long-term study of the lower 25% achievers provided evidence that deep thinking and relational understanding are appropriate challenges for all algebra students.

Yerushalmy, M. (2006) Slower algebra students meet faster tools: Solving algebra word problems with graphing software. Journal for Research in Mathematics Education. 37 (5), 356 – 387. Gorev, D. (2003) Assessment in a computer-based environment: Semantic Calculus knowledge of low achievers. Ph.D dissertation. Faculty of Education, University of Haifa. Under the supervision of M. Yerushalmy Abas, S. (2000). Characteristics of solution processes of low achievers who study algebra in the Function with Technology approach. M.A. thesis (with distinction). Faculty of Education, University of Haifa. Under the supervision of M. Yerushalmy


Mathematical guided inquiry is the core curriculum for all students: Talented Programs

Yerushalmy's long-term study of software and activities designed by her were chosen for the core curriculum of a program for exceptionally talented students (TELEM.)

Yerushalmy, M. (2009). Educational technology and curricular design: Promoting mathematical creativity for all students. In R. Leikin, A. Berman & B. Koichu (Eds.), Mathematical creativity and the education of gifted students. Sense Publishers. pp. 101-113.



Design of the VisualMath Function Web Book (Hebrew & English: Yerushalmy, Katriel, & Shternberg, 2002/4)

• • • • • •


Problem Solving based on visual analysis “Big Ideas” approached in various ways and order Interactive mirror feedback Consistent principles of design Engaging students at different levels Challenging traditional views on mathematics concepts

Linear Text “an exhibition in which the paintings are hung in long corridors through which the visitors must move, following signs, to eventually end up at the exit,” Kress and van Leeuwen


Non Linear Text

“exhibition in a large room which visitors can traverse any way they like… It will not be random that a particular major sculpture is placed in the center of the room, or that a particular major painting has been hung on the wall opposite the entrance, to be noticed first by all visitors entering the room”

Each activity in the eBook designed to include: • Short exposition in which one is invited to try out examples while reading • The randomly generated diagrams (upon defined constrains) provide each student different examples • Pivot problem “writing an essay ..” acting as an guidance for the activity • Problem solving exploration tasks • Exercises’ libraries with mirror feedback

• activity tools designing to support a systematic investigation • generic tools’ box for the whole • Any diagram is an interactive diagram designed to function as: illustrating’ narrating or elaborating diagram 33

1.The Rate of Change tour 2.The Equations tour 3.The Algebraic Structure tour


An illustrating diagram, usually offers a single graphic representation and relatively simple actions Elaborating diagrams designed to provide a variety of occurrences for explorations. Narrating diagrams are designed to set boundaries for the available occurrences .


Selected publications from the Interactive Diagrams project Yerushlamy, M. (2005). Functions of Interactive Visual Representations in Interactive Mathematical Textbooks. International Journal of Computers for Mathematical learning, 10. Daher, W. (2005). Semiotic tools for evaluating electronic mathematical texts. The ICTMT7窶的nternational Conference for Technology in Mathematics. Bristol, UK. Naftaliev, E. & Yerushalmy, M. (2009). Interactive Diagrams: Alternative Practices for the Design of Algebra Inquiry. In the Proceedings of the 33rd Annual Conference of the International Group for the Psychology of Mathematics Education. Thessaloniki, Greece. Yerushalmy, M. & Shubash, N. (2009). Experiencing examples with interactive diagrams: two types of generalizations. In the Proceedings of the 33rd Annual Conference of the International Group for the Psychology of Mathematics Education. Thessaloniki, Greece. Yerushalmy, M. & Naftaliev, E. (2007) Learning Mathematics with Interactive Diagrams. In the Proceedings for "Cognition & Exploratory Learning in Digital Age" (CELDA), December 2007, Algrave, Portugal. 36


R&D mission: Understanding what would make the mobile phone a deserving pedagogical tool • Study the designs of inexpensive technology to increase students' commitment and creativity within the framework of the curriculum • Design the innovations that engage all students with mathematical and scientific ideas • Define new learning opportunities • Answer the demands of the real world of schools • Study the associated socio-cultural aspects


Design Principles for Mobile Environment Yerushalmy July2010 Presentation at IIT Mumbai

• Variety  relatively well-known applications  have been successfully tried in various learning settings  useful for school and as personal tools • Added value  extended communication options  personal tools included inside the phone  mobility • Comply with the hardware limitations  offline use  simple graphics  small screen  phone keyboard • Augment with school traditional technology  Classroom Interaction System  Augmented Textbooks 39

Design of Content Applications


Useful to a wide range of users – Graph2Go

Studied and found to be contributing to learning – Quad2Go, Solve2Go

Useful and motivating learning out of the classroom Fit2Go, Sketch2Go

Take advantage of communication features SMS center

Fit to limitations and constraints less text typing more control of visuals

Users of Math4Mobile • Figures of software downloads • Users demonstrating their work with Math4Mobile in uTube: • Schools of the Arab sector experimenting the innovation for the 3rd year.



SMS link

Barcode download PC (zip) download

other direct Direct download mobile download

80,713 74,983 28,609 40,800 34,633

282 104 47 92 57

127 84 34 28 26

53410 57735 18097 29470 19794

21383 13748 9373 9908 13367

1932 1047 431 493 501







3579 2265 627 809 888


Graph2Go Solve2Go Quad2Go Sketch2Go Fit2Go

8,168 Total

Recent research related to the mobile VisualMath of Math4Mobile • Botzer, G. & Yerushalmy, M. (2007) Mobile Applications for Mobile learning. In the proceedings to the Cognition and Exploratory Learning in the Digital Age (CELDA). December. Algarve, Portugal. • Yerushalmy, M. & Botzer, G. (In Press) Teaching secondary mathematics in the mobile age. In Zaslavsky, O. and Sullivan, P. (Eds.) Tasks For Secondary Mathematics Teacher Education.


Relating the mathematics to their personal interest

Dana used the Amman Videotaped two kids walkingsketch2go to construct graphs showing the boy and sent the video and the girl walking and to her mate via sent to Amman via MMS SMS

Supporting Whole Class Inquiry Learning Design of Learner Response System Demonstration

Lessons learned from long study of the challenge of whole class inquiry had led to the special design of Click2Go

Click2Go facilitates:

•Teaching large groups •Online interaction and feedback on discussed problems in large groups •Close and open questions

Student Mobile phone

•Simultaneous presentation on a board and mobile phones •Using WiFi communication

Teacher Web application 43


Augmented Textbook

The above two book pages are almost identical, however, on the right page a 2D barcode that can be read by any mobile phone camera. An identical diagram will appear on the mobile screen but it will be an Interactive Diagram that Has all the capabilities of the Graph2Go application. 44

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Visual Math - 시각적 기술을 활용한 수학교육

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