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«5th International Realistic Mathematics Education Conference Using Realistic Contexts and Emergent Models to Develop Mathematical Reasoning ...»

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Friday, September 18th

Onsite Registration available in the Kittredge Central Lobby from 8 – 10 am

Full breakfast

08:00 – 08:45

Kittredge Central Lobby

Welcome and opening Henk van der Kooij 08:45 – 09:00 Kittredge Central N114BC David Webb Plenary 1 An Orientation to Realistic Mathematics 09:00 – 10:30 Kittredge Central Johnson, Peck, et al Education N114 BC 10:30 – 11:00 Break Irma Vazquez & Mieke Abels & Gloriana 11:00 – 12:30 Interactive 1 Sonia Palha Jacqueline Sack Martin Kindt Gonzalez Kitt Cen N114D Parallel Sessions Kitt Cen N114A Kitt Cen N114B Kitt Cen N114C Lunch 12:30 – 01:30 Kittredge Multipurpose Room Kara Imm Jacqueline Sack 01:30 – 03:00 Interactive 2 Frans van Galen Heather Johnson (modeling) & Judith Quander Kitt Cen N114B Kitt Cen N114C Parallel Sessions Kitt Cen N114A Kitt Cen N114D 03:00 – 03:30 Break Plenary 2 Henk van der Kooij A personal journey in the (re)design of 03:30 – 05:00 Kittr

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Sessions are organized below according to the surname of the lead speaker.

Abstracts for Plenary Sessions Beyond flatland in primary school mathematics education in the Netherlands Marja van den Heuvel-Panhuizen, Freudenthal Institute, University of Utrecht The Netherlands has a strong tradition in teaching mathematics following the principles of Realistic Mathematics Education (RME). Characteristic for this approach to mathematics education is that it starts with offering students problems in meaningful situations, of which the contexts can gradually evolve into didactical models that can be used to solve a broader scope of problems, and, through progressive schematization, eventually ends up in understanding mathematics at a more formal level. This process of mathematization can be distinguished in horizontal (from the world to the mathematics) and vertical mathematization (within the mathematics). However, in RME as it is conceptualized in primary school mathematics textbooks (and consequently in classroom practice), there is not much attention for vertical mathematization, which actually is restricted to carrying out plain operations with numbers. This results in a rather ‘flat’ primary school curriculum which does not give students a good basis for developing higher-order thinking in mathematics. Therefore, new research-based design is necessary to make the primary school mathematics curriculum more mathematical.

3 An Orientation to Realistic Mathematics Education Raymond Johnson, FIUS, University of Colorado Boulder Fred Peck, FIUS, University of Montana William Campbell, Ryan Grover, Susan Miller, Ashley Scroggins & David C. Webb, FIUS, University of Colorado Boulder Realistic Mathematics Education is distinguished by a rich history and a number of core principles and design heuristics that distinguish it from other approaches to mathematics education that use context, manipulatives, and real-world phenomena. This opening plenary will orient participants to the conference by surveying the primary theories and concepts that guide RME and detailing some of the exemplary work representing RME research and design.

Fi-gured Algebra Martin Kindt, Freudenthal Institute, University of Utrecht The main image of ‘algebra’ at school is manipulating with symbols and formal expressions. But algebra is much older than ‘algebra’. More than 3500 years ago the Babylonian mathematicians could solve every second-degree equation, without using symbols. They used geometric terms;

width and length for unknowns, rectangle for product. The Pythagoreans used patterns of dots to describe special sequences of numbers like squares and triangular numbers. This presentation will propagate a visual approach of algebra and also discuss some challenging problems. These type of problems provoke a productive use of algebraic rules and promote an understanding for algebra.

A personal journey in the (re)design of teaching and learning math Henk van der Kooij, Freudenthal Institute, University of Utrecht Working as a high school teacher since 1975, I was asked in 1987 to join a small team to redesign a Dutch senior high school program (grades 10 and 11). That was my start of working for the Freudenthal Institute. Several design projects followed until 2010, among them the NSFfunded high school reform project AR!SE (Solomon Garfunkel, COMAP), TechMap (COMAP), TWIN (a Dutch reform project for vocational education in engineering) and other more smallscale projects in the Netherlands and the USA. I want to look back with you at the most important aspects of all that work over the past 30 years. How my ideas about “mathematics for all” changed over time. How “theoretical learning trajectories” (the basis for the design of teaching and learning materials) sometimes turned out in practice as eye openers for alternative students’ thinking. Learning mathematics for students is not learning to imitate your teacher's professional math knowledge. In my opinion, every student deserves enough space to develop his/her own construct of math and opportunities to discuss personal ideas with fellow students and the teacher to improve his/her mathematical conceptual knowledge and (more algorithmic) skills. Be prepared to be challenged to solve intriguing problems from several curriculum projects, because at least 40 minutes of my 90 minutes presentation will be used to make you think about and work on problems I will present!

Post-secondary mathematics: Keeping it real Eric Stade, University of Colorado Boulder When real-world scenarios are introduced at the very beginning of an undergraduate mathematics course, instead of somewhere in the Exercises for the Chapter 5 (for example), great things happen. These scenarios become not (just) applications of the mathematics, but contexts 4 for the development of the mathematics. That is, the scenarios serve the mathematics, instead of (or in addition to) the other way around. Such a contextual approach allows for development of material in a way that's completely faithful to all of the beauty and elegance of mathematics as a "pure" discipline, but is just as faithful to the utility and power of mathematics as an "applied" discipline. We explore all of this primarily through the lens of a contextual first semester calculus course, but touch, along the way, on the relevance of these ideas to other undergraduate courses, ranging from Math for Elementary Teachers to Fourier Analysis.

The Challenge and Promise of RME David C. Webb, FIUS, University of Colorado Boulder The evidence is quite clear. Research points to improved access and equity in the learning of mathematics when students are engaged in activities that are more relevant and authentic, when they have opportunities to propose and discuss strategies and representations with peers, and when the teacher develops a classroom in which sense making and problem solving are valued.

Yet, in spite of the promise of improved engagement and learning teacher implementation of Realistic Mathematics Education is confronted by several challenges. We will explore these challenges and discus recommendations for supporting productive implementation of RME.


for Breakout Sessions What to do with fractions?

Mieke Abels & Martin Kindt, Freudenthal Institute, University of Utrecht The way of teaching fractions in primary schools nowadays differs strongly from the teaching of some decades before. Now there is much more attention for the use of different visual models and fraction operations in meaningful contexts. Rules are not longer drilled endlessly. In secondary schools teachers often expect that their students have enough skills to operate with fractions in a formal way, but it often happens to be a ‘miscalculation’. Particularly when algebraic fractions appear many things are going wrong. How to repair these failures? In secondary education there should be paid more attention to an insightful approach of, and challenging exercises with fractions. This may lead to generalization and formalization and it will be a base to work with algebraic fractions. At the Freudenthal Institute we developed not only some units in this spirit, but also a series of activities in the DME project (Digital Math Environment). In our presentation we will show examples of this work.

Indigenous RME Activities for Mathematical Literacy Grade 10 – 12 Riana Adams, Uplands College, Mpumalanga, South Africa I come from an inclusive, co-ed, multi-national school in South Africa. I will take you into the minds of our learners who come to us from enormously diverse backgrounds and with a large range of challenges. My workshop consists of 10 Space, Shape and Measurement - adapted indigenous activities that will engage you in realistic mathematical scenarios. These IEP compliant activities are designed to promote a love for learning and boost confidence in the real world of numbers.

5 Designing RME activities for tablet devices Peter Boon, Freudenthal Institute, University of Utrecht In this working group we use the DME (Digital Mathematics Environment) of the Freudenthal Institute to design interactive activities for students that can be used on ipads and other tablets.

Participants of this working group will get the opportunity to design their own web-apps that will be available on the Internet immediately. The DME offers a powerful authoring tool to design both explorative activities and exercises that give hints and immediate feedback. Moreover, other educational resources available on the Internet, for example the PhET Sims developed at CUBoulder can be embedded in the DME web-apps.

Students’ Use of a Computer-Programming Environment as a Realistic Context for Learning Algebra Anna F. DeJarnette, University of Cincinnati I will present on an after–school computer–programming club designed to examine how students use concepts from algebra when solving programming tasks. Students used Scratch to complete a sequence of tasks related to designing and beating a car–racing game. I will present examples of the tasks and of students’ discussions and solutions, as a basis for considering how programming tasks can serve as realistic contexts for students to make sense of mathematics.

Introducing Realistic Mathematics Education in the Cayman Islands Frank Eade, Cayman Island Government The presentation will provide an account of the implementation of Mathematics in Context with low and middle attaining students in grade 6. We will examine the changes in behaviors and outcomes for the students and the support provided for the teachers to enable positive change.

Although experienced high school teachers’ practices are not easy to influence we will argue that with sustained and appropriate support and a very well researched scheme it is entirely possible to have strong positive impact on both practices and beliefs Using Pictures to Support Student's Developing Understanding of Multiplication Sherri Farmer, Purdue University Visual media is a common mechanism for engaging and communicating information with students. This study describes students’ mathematizing their world while developing unique mathematical insights using photographs as mathematical referents. Using RME foundations and Presmeg’s (2006) method of semiotic “chaining,” this study discusses connections between media and student’s emergent mathematical ideas. It further demonstrates use of photographs as a referent object representing mathematical ideas, such as multiplication, can support the growth of children's mathematical understanding.

Designing for Realistic Mathematics Education: Learning about graphs as an example Frans van Galen, Freudenthal Institute, Utrecht University, the Netherlands ‘Guided reinvention’ is a core concept in Realistic Mathematics Education. In my presentation I want to address the question what it means if we say that children have to reinvent graphs and graphing. Textbooks for mathematics tend to focus on the interpretation of graphs, not on the construction of graphs, which does not help students to understand the basic principles underlying graphs. My examples of alternative learning trajectories will come from grade 4 to 6.

6 Change of Representations as a Design Principle for Initiating Learning Processes for Quadratic Functions Maximilian Gerick, TU Dortmund University, Institute for Development and Research in Mathematics Education For a suitable development of the concept quadratic functions students must establish viable connections between graphical and symbolic representations. They often use graphical representations geometrically instead of functionally which can lead to unviable connections.

Based on multiple cycles of design experiments an iterative development of a learning and teaching arrangement will be presented that helps students to use graphical representations functionally when linking them with symbolic ones.

Teachers’ Understandings of Realistic Contexts for Capitalizing on Their Students’ Prior Knowledge Gloriana González, University of Illinois at Urbana-Champaign The theory for realistic mathematics education establishes that realistic contexts framing mathematics problems provide opportunities for guided reinvention. I use data from a geometry study group to examine teachers’ understandings of what constitutes a realistic context for using students’ prior knowledge during a Lesson Study cycle. Overall, participants increased their attention to students’ prior knowledge and identified relevant contexts for their students when planning and implementing a lesson about finding locations on a map.

What Might RME Look Like in a First Year Algebra Course?

Pamela Weber Harris, The University of Texas at Austin In our project, Focus on Algebra, we designed 3 sets of sequenced tasks to teach major first year algebra topics based on algebra landscapes of learning and the lesson structures of truly problematic situations and problem strings. This session will briefly describe our project, engage participants in tasks, show video of students and teachers at work, discuss design principles, and offer suggestions for further research and collaboration.

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