designing and developing a digital mathematics curriculum
TRANSCRIPT
Designing and Developing a Digital Mathematics Curriculum
Dr. Judith OlsonCurriculum Research & Development Group
University of Hawai‘i at MānoaUnited States
Dr. Melfried OlsonCurriculum Research & Development Group
University of Hawai‘i at MānoaUnited States
ABSTRACT: This paper is a report on the work to create a digital curriculum for strugglingstudents. The Curriculum Research & Development Group developed A Modeling Approach toAlgebra, a curriculum designed and developed to support ninth-grade students taking Algebra I.Technology that provides the opportunity to interact with dynamic representations of concepts forclassroom instruction is integrated throughout the curriculum both for teacher and student use. Inaddition to the inclusion of technology within the curriculum, the curriculum package is availabledigitally. Specifically, PublishView™ created by Texas Instruments™, was used as the vehicle toprovide access to student pages, annotated student notes, teacher notes, and links to technology forexplorations outside of the classroom.
INTRODUCTION
To successfully complete the mathematics requirements created by adopting the Common Core StateStandards in Mathematics (CCSSM) (National Governors Association Center for Best Practices, Council of ChiefState School Officers, 2010) raises the bar for students as well as for their teachers. This is of particular concern forAlgebra I, the traditional entry point in the high school mathematics curriculum. Such is the case in Hawai’i wherepre-algebra is no longer a high school course even though there are high school students who most likely willencounter difficulties when enrolled in Algebra I. Partially as a result of these policies, approximately one in threestudents do not succeed in high school Algebra I (Gottlieb, personal communication, Spring 2011). To address theAlgebra I failure rate, a course Modeling our World I (MOW I) focusing on modeling and providing opportunitiesfor students to learn mathematics in a more investigative manner was established. Although designed for strugglinglearners, it is intended that MOW I be taken concurrently with Algebra I. As such, MOW I is not viewed as aremedial course but as one in which students learn content that complements that of Algebra I. The curriculum anddigital materials are briefly described below followed by a short summary.
CURRICULUM MATERIALS
The Curriculum Research & Development Group (CRDG) at the University of Hawai’i was contracted bythe Hawai‘i Department of Education to design and develop the curriculum materials, A Modeling Approach toAlgebra (AMAA) (Olson, J., Olson, M., Slovin, Venenciano, & Zenigami, 2013), for the MOW I course. TheCCSSM together with materials from CRDG’s previous curriculum research and development projects, e.g. AlgebraI: A Process Approach (Rachlin, Matsumoto, Wada, & Dougherty, 2001), Reshaping Mathematics forUnderstanding (Slovin, Venenciano, Ishihara, & Beppu, 2003), provided a research base from which to begin thedevelopment for AMAA. The modeling standards embedded in the CCSSM were established as the frameworkaround which to build the curriculum materials.
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The AMAA curriculum materials are designed around the premise that there is more to learning algebrathan memorizing formulas and following rules to determine answers to questions. The lessons, built aroundproblems as compared to exercises, emphasize the use of models, promote the investigation of open-ended problemsolving tasks, and provide multiple opportunities for students to develop concepts, generalizations, and skills. Inaddition, there is heavy emphasis on the CCSSM eight Standards for Mathematical Practice throughout theinvestigations to develop positive student dispositions for learning mathematics, encourage student perseverance,promote classroom discourse, and emphasize communication of mathematical ideas. Students are asked to model,represent, graph, write about, and discuss their strategies for investigating and solving problems as they begin tointernalize algebraic ideas and develop an understanding of algebraic techniques. The Teacher Notes that accompanyeach lesson give guidance for instructional strategies for enhancing the efforts of the students, helping students learnthe content and communicate their thinking. The development of the teacher materials is reflective of extensive priorwork in professional development projects conducted by CRDG (Olson, M., Zenigami, Slovin, & Olson, J., 2011)and feedback from teachers who piloted the AMAA materials during the 2012 – 2013 school year.
The AMAA content is organized according to the five critical areas identified in the CCSSM for theTraditional Pathway: High School Algebra I. These areas are organized into five units, Unit 1 Relationships BetweenQuantities and Reasoning with Equations, Unit 2 Linear and Exponential Relationships, Unit 3 DescriptiveStatistics, Unit 4 Expressions and Equations, and Unit 5 Quadratic Functions and Modeling. Because it was thoughtthat the modeling approach together with the emphasis on problem solving would be new for students, a preliminaryunit was also created. Unit 0 Getting Started introduces students to the problem solving investigations and processesused throughout the course. Because students for whom this course was intended often do not have experienceconducting mathematical investigations, Unit 0 starts with problems to highlight features of modeling and introducesstudents to the specific modeling cycle suggested by the CCSSM (2010, p. 72–73). This cycle is shown in Figure 1.Teachers are encouraged to have students use the modeling cycle to write solutions to open-ended problems.
Figure 1
Unit 0 lessons also provide opportunities for the class to establish norms for an environment critical forproductive classroom discourse. These norms may include how to conduct classroom discourse, working together tosolve problems, listening to the ideas of others, and developing a willingness to share thinking. It also helps studentsdevelop important dispositions necessary for success in a mathematics classroom. Unit 0 further initiates the focuson standards for mathematical practice (CCSSM, 2010) that empathize actions expected of mathematicallyproficient students. It is expected that students will engage in and embrace these practices with greater proficiencyas they progress through the materials.
DIGITAL MATERIALS
In the development of the curriculum materials and associated Teacher Notes, technology was consideredimportant in two ways. First, as much as possible, the curriculum is built around ideas that could be dynamicallyexplored with spreadsheets, graphing, or links to internet explorations. Second, the curriculum materials areavailable digitally for the teachers and students. TI-Nspire™ technology has been used to accomplish this. The TI-Nspire™ Teacher Edition PublishView™ feature has been used to digitally format the interactive curriculummaterials. Teachers can display Student Pages to use with lessons or digitally access Teacher Notes and AnnotatedStudent Pages that can be accessed for planning and instruction. The software has also been used to create TI-Nspire™ .tns documents that consist of interactive spreadsheets and graphs teachers can use in a demonstration withthe whole class or send to TI-Nspire™ handhelds via networked classrooms using TI-Nspire Navigator™. Links to
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these .tns documents are provided within the Teacher Notes and Annotated Student Pages so they can be easilyaccessed at appropriate times during instruction. Using TI-Nspire Navigator™ features, the screens of the students’handhelds can be captured and individual students can be selected as class presenters as they explain their work.The teacher can also use TI-Nspire Navigator™ for real-time formative assessment through quick polls and forcollecting students’ graphs, equations, and spreadsheet lists. The features of TI-Nspire technology are discussed inmore detail in the sections that follow.
TI-NSPIRE PUBLISHVIEW™
TI-Nspire PublishView™ provides layout and editing features for presenting mathematics concepts in a document where TI-Nspire™ applications can be interactively and dynamically linked with supporting media. Documents can be created that include formatted text, TI-Nspire™ applications (Calculator, Graphs, Lists and Spreadsheets, Data and Statistics, Geometry, Notes, Vernier DataQuest™, and a Question App), images, hyperlinks, links to videos, and embedded videos in a format that is suitable for printing on a standard piece of paper, publishingto a website or blog, or for use as an interactive worksheet. Multiple dynamically linked pages can be embedded in the same document. This allows for all Student Pages, Teacher Notes, and Annotated Student Pages in the AMAA curriculum materials to be formatted in TI-Nspire PublishView™ documents. These are intended for teachers’ instructional purposes including both planning and conducting lessons. These documents are linked so teachers can use the TI-Nspire Teacher Edition™ software to present a problem to students, link to Teacher Notes or Annotated Students Pages for assistance during instruction, link to an interactive TI-Nspire™ .tns document for whole class discussion or, if available, sent to students’ TI-Nspire™ handhelds. See Figures 2 – 7 for example of the Table of Contents in Figure 2, list of lessons in Unit 2 in Figure 3, Teacher Notes page showing links to the student page for the Painted Cube lesson on the top left and the student annotated page on the top right in Figure 4, Annotated Student Pages in Figures 5 and 6, and an example of a link to a TI-Nspire™ .tns document in Figure 7. Occasionally,links to websites are provided for introducing a problem or for background information on the topic being explored.
Teacher Notes were designed and developed as educative materials (Davis & Krajcik, 2005). As educativematerials, the Teacher Notes are written to expand upon the mathematical ideas within the lesson so teachers have adeeper knowledge of the mathematical ideas within the lesson. Where appropriate these explanations include a lookat mathematics beyond what the students are expected to pursue. Occasionally, external links for furtherexplanations of mathematical ideas are included. Teacher Notes are intended to help with lesson planning byproviding a summary of the content and objectives for the investigation, highlighting opportunities to model withmathematics, and anticipating student thinking and possible responses, including common misunderstandings. TheTeacher Notes also list materials needed and provide guidance for the use of technology by describing ways the useof technology can enhance student learning or provide an alternative approach to understanding the relationshipswithin the lesson. The Annotated Student Pages expand the material from the Student Pages and provide teacherswith notes for managing the investigation and suggest questions teachers can use to prompt classroom discussion.Questions are intended to indicate topics and ideas important to the investigation. As students become familiar withthe instructional approach, they are expected to raise these issues themselves or pose the questions spontaneously toextend a problem or probe its mathematical content.
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Figure 2 Figure 3 Figure 4
Figure 5 Figure 6 Figure 7
While the materials are considered complete as related to AMAA, the PublishView™ feature allows theteacher to build upon the materials for future use. A valuable PublishView™ editing feature is the “add space tool”that teachers can use to digitally record notes about how they used a lesson. For example, a teacher can make notesabout what was successful by inserting a text box and use that information later. A teacher may also insert anappropriate hyperlink or create a TI-Nspire™ spreadsheet document that has proven successful in teaching. Teacherscan also add pages or problems to create lesson plans from existing lessons or TI-Nspire™ documents, link torelated lesson plans or documents, build or interact with other TI-Nspire™ applications directly from the lesson planor embed explanatory text, images, and video.
If students have access to TI-Nspire™ Student software they can use the PublishView™ features to createreports or projects containing data playback, curve fits, pictures, and videos. They can also print and turn inassignments. Students essentially have access to one tool to create a document that contains problems, text, images,hyperlinks, or videos, interactive TI-Nspire™ applications, screen shots, and layout options needed to print adocument.
TI-NSPIRE™
Technology that enables the opportunity to interact with dynamic representations of concepts for classroominstruction is integrated throughout the curriculum. Prepared documents in TI-Nspire™ Teacher Edition softwareprovides opportunities to enhances and extends algebraic concepts throughout the curriculum. Most lessons includea specific TI-Nspire™ .tns documents that primarily consists of spreadsheets and graphs along with suggestions inthe teacher materials for ways to use these during instruction. See Figures 8 – 10 for samples from the Painted Cubelesson. This use of technology focuses on using graphical representations for data, encourages conjecturing andvalidation, and emphasizes relationships between quantities. The .tns documents can be used as classroomdemonstrations that can be led by the teacher or by students. If students have TI-Nspire™ handheld devices, the .tnsdocument can be sent to the students who can work on the document individually.
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Figure 8 Figure 9 Figure 10
TI-Nspire Navigator™The TI-Nspire Navigator™ system is a classroom network of TI-Nspire™ handhelds for instant assessment
and instruction consisting of features such as Question Capability, Document Transfer, Quick Poll, Screen Capture.Live Presenter, Portfolio and Auto-grading. While the AMAA curriculum does not require the use of TI-NspireNavigator™ the digitally formatted materials in PublishView and the TI-Nspire™ .tns documents are ready to use inthe TI-Nspire Navigator Teacher Software™. With the software all the AMAA content can be accessed, additionallessons created, and formative assessment student data can be collected with teacher generated questions.
When teachers send students the TI-Nspire™ .tns documents included in the AMAA PublishView™ theycan use screen capture to make sharing results in real-time displays possible and students can share their work usingthe class presenter feature. This feature affords opportunities to build discussions, to collaborate, and to examineothers’ work on a public screen while performing one’s own activity. The public screen is a dynamic space ofmutually exchanged information that becomes a source from which teachers can generate questions. Studentsreceive immediate feedback and learn by comparing, sharing, and discussing.
TI-Nspire Navigator™ also allows teachers to obtain immediate formative assessment data from allstudents on their understanding of mathematics concepts through actions such as, collecting lists, placingcoordinates on a graph, specifying graph and equation representations for functions, providing explanations for openresponse questions, as well as responding to more traditional uses such as multiple choice, true/false, and Likert-scale questions and explanations.
SUMMARY
This paper describes how the development of the AMAA curriculum materials and ideas were enhanced by the use of digital technology. As such, in the minds of the curriculum development team members was a guiding principle that these materials and ideas would be made available digitally and the format of the digital materials would be useful for both teachers and students. The curriculum materials including Student Pages, Teacher Notes, and Annotated Student Pages, have been successfully piloted and are distributed statewide. Over 40 teachers and 1800 students throughout the state are using AMAA materials during the 2013 – 2014 school year and it is anticipated this will continue for the near future.
REFERENCES
Davis, E. A. and Krajcik, J. (2005) Designing educative curriculum materials to promote teacher learning, EducationalResearcher 34 (2005) (3), pp. 3–14.
Gottlieb, D. (Spring 2011). Personal Communication to Curriculum Research & Development Group researchers whendescribing the need for the course Modeling Our World.
National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core statestandards for mathematics. Washington, DC: Author.
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Olson, J., Olson, M., Slovin, H., Venenciano, L, & Zenigami, F. (2013). A Modeling Approach to Algebra. Honolulu, HI:Curriculum Research & Development Group.
Olson, M., Zenigami, F., Slovin, H., & Olson, J. (2011). Educative curriculum materials in professional development. In Wiest, L.R., & Lamberg, T. (Eds.). Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group forthe Psychology of Mathematics Education. Reno, NV: University of Nevada, Reno.
Rachlin, S. L., Matsumoto, A. N., Wada, L. T., & Dougherty, B. J. (2001) . Algebra 1, A process approach. Honolulu, HI:University of Hawai‘i, Curriculum Research & Development Group.
Slovin, H., Venenciano, L., Ishihara, M., & Beppu, C. (2003). Reshaping mathematics for understanding: Getting started.Honolulu, HI: University of Hawai‘i, Curriculum Research & Development Group.
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