WME: a Web-based Mathematics Education System for Teaching and Learning

Research-based paper) WME is being developed as a modern Web-based system that fosters a new paradigm for creating, customizing, and sharing, interactive mathematics education materials online. By an innovative combination of standard Web technologies and by creating powerful Web-based tools, WME can deliver classroom-ready lessons that are interesting, inquiry-based, as well as interoperable. The WME system allows easy implementation and modification of lessons or manipulatives so that very little technical ability is required. To-date we are developing the following tools to create manipulatives within WME pages: GeoSVG is a Web-based tool to author and run SVG-based geometry, DMAS an assessment system that supports all types of assessments, and MathEdit to enable mathematics to be displayed and computed on the web. Modifications of manipulatives or new lessons can be published for other teachers using the WME system to view or incorporate into their own lessons and modules. To date we have piloted several WME lessons in area middle schools. We have gathered both technical and educational data that we have used to guide the development and future directions of our work, but more trials and data are needed to measure the total effects of our system on the teaching and learning of mathematics. 1. Theoretical framework guiding the WME system development and use. The WME work has the benefit of an interdisciplinary team at the Institute for Computational Mathematics (ICM/Kent State University). Our group consists of faculty and graduate students from Computer Science, Mathematics, Mathematics Education, Graphic Design, and Middle School Mathematics. The technical part of the system is being guided by the use of a combination of innovative open Internet technologies. Our diverse group of students and faculty have allowed us to work on complex problems and issues involved in teaching and learning mathematics, but it has also created the need to develop a common philosophy of learning and teaching mathematics among this group of individuals. Our philosophy has been evolving over the past 4 years and we have (over many iterations of our work) developed a consistent set of beliefs about what is important mathematics and how middle school students learn specific topics. Our work is guided by current research in mathematics education and we have adopted the view that students need to make sense of mathematical ideas by building rich connections to their existing knowledge and exploring the limits of their mathematical thinking. Another belief of the group is that the context in which tasks are developed needs to be interesting to the students and contain significant mathematical ideas for students to explore. Our activities are constructed in many cases to confront the limitations of students’ thinking. With this in mind, we have adopted the research framework outlined by Silver that theory, practice, and problems should mutually support one another (Silver, E. & A. Herbst, P.G., 2007). That is the manipulatives and lessons are designed consistent with recommendations from current research. Many topics have been identified initially by collaborating middle school teachers to help their students learn challenging mathematics concepts identified in their practice. This design provided not only a consistent theory for building the WME system, but also a specific process to guide the day-to-day work. The mathematics education faculty met regularly with the middle school teachers to discuss where their students were having difficulty learning specific topics. The teachers would communicate both how the students encountered the topic (types of lessons and problems) and specific difficulties (sometimes referred to as errors) they observed in their students’ thinking. This began a discussion about specific research studies (and their implications) for how teachers might provide different tasks for their students to facilitate them making sense of these ideas. Together the middle school teachers and mathematics educator sketched ideas for tasks and tools, in web-based active lessons that would be helpful in creating an environment for the middle school students to overcome their difficulties and make sense of the ideas. The mathematics educator then met with the larger group to discuss the specific needs for the lesson and tools. These interactions and discussions helped to guide the computer science part of the team to determine the most appropriate way to develop what the educators needed. A computer science student developing a specific tool or active lesson presented his/her work by first presenting the goals and programming techniques. These presentations by all members of the group served as a way to critique our work before, during, and after being developed. As tools and lessons were developed they would be made available to teachers for use with their students. When the teachers would implement these WME lessons with their classes at least one member of the WME team were present in the room with teachers and students. We assisted in technical, mathematical, and educational matters during these in-class trials. As these lessons were implemented we gathered observational data about both teacher and student use of the WME lessons, and informal survey data from students regarding how they liked using the system. This data then was used to update the technical aspects of the system and adjust specific educational aspects of the lessons. The next part of the paper will illustrate this process with an individual topic that we built early on in our WME project. Example of initial topic module and lessons The initial teachers for this project both taught 7 grade mathematics to 4 different classes of students each day. After initial discussions of our work the mathematics educator met with these teachers to discuss what topic needed additional support. Both teachers agreed initially that their students had very little understanding (procedural or conceptual) of percents. The teachers indicated that their students were initially taught percentages in 6 grade in a standard manner (introduced the definition, some examples of the 3 basic types of percent problems and then these same types embedded in contextual problems). In 7 grade they were given additional problems where students had to apply their knowledge of percents to a wider variety of problems (involved multi-step problems for example). The resource materials used (mostly the text) presented the data to students by illustrating common types of problems and standard techniques for solving the problems. Teachers reported that their students (then in the 7 grade) scored average or better than average on the 6 grade tests regarding this topic. However, they noticed that a significant number of their students were struggling with this same topic in their 7 grade classes. What knowledge these students had acquired in the 6 grade either wasn’t accessible or wasn’t useful in solving the problems they were facing in the 7 grade. The mathematics educator introduced some research that was conducted regarding the importance of understanding the concept of percents and how even students without formal instruction in percentages had an informal understanding of standard benchmarks (50% for example means 1⁄2 of something)(Lembke & Reys, 1994). We discussed how we might take advantage of the informal notion of percents and try to build some proficiency in solving the 3 types of standard percentage problems mentioned above. We sketched out a set of tools that we thought would help to develop the concept of percent and a few activities with spreadsheet like tools for students to explore and build on their informal notion of percentages. Some of the initial tools developed a stronger notion of percent by presenting pictures of pizza arranged with 10 rows of 10 pepperoni when students clicked a pepperoni it would disappear and keep track of the percent “eaten” and the percent left. This was followed by other tools like the one below where the total number pieces were different factors of 100. These tools were available for students to use while answering questions like, how many different ways can you select 50%,? What about for 25%? How are these representations of 25% the same? How are they different? Figure 1: Example of tools that were developed to help students broaden their notion of percents. Both teachers used WME lessons with groups of their students while part of the WME team observed. These trials lasted for three to five days. We helped with the technical issues that came up during the class as well as observed students using the web-based lessons and tools. We were encouraged by the student interest and their candid comments about the system and their using it to learn mathematics. Because the use of the WME system was intermittent and limited to remediation of topics where the teachers identified as needing supplement, we were unable to assess mathematics learning in any significant way. We did, however, continue this process to develop our lessons, tools, and technical parts of our WME system. 2 The technical aspects of the WME system Figure 2: The WME Concept WME can deliver, via the Internet or a LAN (wired or wireless), classroom-ready lessons that are well-prepared, mathematically and developmentally appropriate, interactive, and effective. In addition to multimedia content and hyperlinks, lesson pages feature interoperable and customizable manipulatives to help students understand and explore mathematical concepts through hands-on activities and a teacher guide that assists in the teaching of mathematics more effectively. We believe a system like WME can help teachers provide a significantly better mathematical experience for their students. The WME system supports, among other features, mathematical formulas through MathML, interactive geometry objects through SVG (Scala