Visual Communication for Mathematical Confidence

A.Klinger, UCLA

C. Project Description

Introduction

This program adapts a computer-based system designed to develop high school and undergraduate students’ knowledge of college chemistry, and implements this tool to mathematics content. The system employs student writing. This project will convey topics needed for mathematical courses by that approach. Students will develop writing skill and use it to demonstrate mathematics knowledge. The project adapts a second set of high quality materials to the purpose of engaging all students in mathematics. This involves an extensive set of visual web-based materials about mathematical concepts, supplemented by readings, book chapters, and assignments developed to guide writing about them, to create comprehensive change for non-science majors’ college courses in Pre-Calculus and Finite Math. Our effort emphasizes play, games, history and social aspects of mathematics. The comprehensive nature of the activity is supported by the national base of interested faculty. The reading/writing effort depends on developing lessons that access files posted to the world-wide web. Those files quickly convey key mathematical ideas through animations or static visuals. The project begins with adapting the existing computer materials to the writing program. That will be the first stage of a new approach that uses the world-wide web to empower people in mathematics. Support and training for students to compose and evaluate writing assignments are directed at building a base: leaders at employing computer technology in mathematics instruction. Writing efforts by students using the material created should empower individuals in mathematical areas. Both kinds of student participants, those who create the overall writing goals and lessons, and users following them to acquire knowledge, will accomplish steps toward expanding the work force capable of success in environments dominated by science, engineering, technology and mathematics.

Visual Knowledge

There are many mathematical topics that first arose in connection with something that people see. Hence both static and dynamic (animated) images have a major place in any program to ease anxieties and promote mathematical achievement. Traditional mathematical sources tend toward concise exposition. There are many static and animated visuals created by the investigator and students familiar with computer tools. This material can be helpful as brief presentations that become the basis for writing and interchange among people needing to gain confidence in their ability to learn mathematics. Such visuals and animations are already in the web site of the principal investigator, (w1). The site (w1) itself displays other visual elements, two of mathematical character. It also leads to materials that contrast with definitions (w2) factorial., as in the cartoon (w3) and many items that contrast strongly with dictionary mathematical items (w4).

Written Reactions

The project will adapt and implement a well-established world-wide web, computer-based system to guide knowledge acquisition through writing and review by fellow students, to the visual mathematical materials we describe here. This system, Calibrated Peer Review or CPR was developed in the Chemistry Department at UCLA. It has already been adopted at many educational institutions: two hundred twenty-six are registered as users and are in the list reachable from the CPR web site (w5).

This project has a fundamental task enabling implementation of a new system for mathematics instruction based on visual communication of mathematics-related images, and the guidance of learning through writing and review that is the essence of CPR. The effort includes adapting the writing and guidance aspects of CPR to develop confidence about specific mathematical concepts and facilities. While it begins with the existing web materials, both of visual and management character, there is already an example of how this is done. A paper (Decker 2001) by one of the investigators describes exactly what we will be developing in this project: specific statements indicating how a learner is to evaluate and report on a mathematical topic. Arlene Russell and Orville Chapman, and their staff are available to the project. Their assistance as CPR originators, and their experience in having successfully adapted CPR to diverse educational institutions, is central to our attaining the goal of this project initiating fundamental and comprehensive change regarding mathematics instruction. There are many necessary activities to improve teacher training in mathematics at UCLA. They will be based on items developed in this project. Other schools are involved from the beginning of the effort. They include three two- and one four-year institution. There are new CPR accounts (allowing authoring) for Allen Klinger, Bob Decker, June Decker, Miguel Moreno, Ron Kendis, Larry Gurley, Lewis Felton, Concepci'on Valadez, and Jim MacQueen.

Experiencing Motion

Sports, games, even stock market valuations offer realistic situations where interest is kept because of change and the inherent dynamism in the activity. Using dynamic visual images in the form of several animations in (w1) has the same property: stimulating viewer attention. Games like checkers, chess and go provide a natural mode for Calibrated Peer Review exercises involving more than the game itself. Cartesian coordinates in mathematics and array indexing in computer science become subjects for writing exercises if one from places the origin of coordinates at such game boards’ lower left corners and uses (row, column) notation. An animation process can locate stones placement in the game of go or by other images convey game concepts. They are then starting points for mathematical topics such as area and volume measurements described in Steen 1990, all suitable CPR writing assignment starting points. Other board and card games can be added using the world-wide web source (w6). CPR exercises will require written analysis of game rule and mathematical fundamentals relationships.

Learning Resources

Both UCLA and other curricula in mathematics include general courses on Pre-Calculus and Finite Mathematics, material for non-science majors. These are frequently taken by individuals who did poorly at the secondary school level. Excerpts of items in online versions of catalogs for Los Angeles Trade-Technical College, Los Angeles City College, and the University of Hartford are at (w7). Prerequisites and number of courses offered to remedy such deficiencies are shown by excerpts in (w7). Equivalent UCLA descriptions for two courses offered non-science majors, Pre-Calculus [Math 1], and its successor or alternative, Finite Mathematics [Math 2, (w8)] follow:

The purpose of Math 1 is to give students a strong preparation for taking calculus. … Math 1 covers material that is ordinarily addressed in the high school curriculum. … In order to enroll in Math 1, students should have a good background in intermediate algebra. …. Math 2 satisfies requirements for students majoring in Psychology and in Sociology. …The main topic of Math 2 is the theory of probability. … Many facets of everyday life involve probabilities …

Many people who need to take courses like Math 1 and Math 2 can explore a calibrated peer review exercise and acquire general probability knowledge. The project will bring instructors and responsible faculty together to devise means to credit learning done via CPR. The CPR approach can clearly be adapted to mathematical content - see lesson on “Measurement and Significant Figures” at (w5):

Guiding Questions …

1.What … rules regarding precision … in the lab?

2.How is the last digit of a measurement usually determined? …

8.Why …scientific notation useful for … number of significant figures?

Writing Prompt In a logical, grammatically correct paragraph …describe the way that measurements are determined to the correct precision. …

Mathematical Language

Richard Feynman, the Nobel prize winning physicist, commented that “ … mathematics (conveys the) deepest beauty of nature. … it is necessary to understand the language that she speaks in" (1967). The statement echoes the Dantzig (1930) book title. People believing in their mathematical inability can benefit by listening to the sounds of mathematical terms, so developing auditory files is part of the project, one of particular value to linguistic minorities. Current mathematics sound resources are the following terms: product, ratio, factorial, exponent, algebra, arithmetic, probability, boundary condition, fraction; in each case there are voiced versions available at web locations given by (w1) followed by forward slash, the word and then the extension “.wav”. Existing world-wide web resources, including animations by the principal investigator and his students are also available. Such material can supply the basis for a comprehensive approach to revising non-science majors’ mathematics curricula. Some sites [(w10), (w13)] already use the sound source files to augment such animations.

The project is to: 1) enable expanded use of visual, dynamically evolving graphics, and accompanying auditory files to assist students to acquire a level of confidence in speaking about or working with mathematics; and, 2) extend and augment currently available resources to create a new comprehensive system based on combined activities of faculty participants. Because images are a natural way to communicate with individuals from different cultures this approach can create a vehicle for education across cultural/geographic lines. Some files are now available translated into Spanish; (w21) presents cross-cultural issues. The project will adapt existing utilities and animations - birthday digits in an irrational number (w14); Pythagorean theorem (w15); number bases and octal counting (w16); factorial (w10); and basic probability (w13) - to stimulate explorative activity by students.

Developing added material along the same lines would allow integrating mathematical laboratory experiments for K-12 pupils into the CPR writing process. Animations can rapidly present mathematical concepts. The project includes several, that already exist and can be used. One, the Pythagorean theorem falls within the scope of Pre-Calculus. The UCLA Finite Mathematics course is well-able to use another two animations that respectively demonstrate key ideas in probability and in combinations/permutations. Other animations can be readily constructed under this project. There is a very large audience for animations: individuals who need to cover mathematical material needed to be able to take college courses.

Many students find it difficult to remain in a mathematics course because the material is remote from their personal motivation in attending school. For example, a pre-engineering student could be good in working with physical (mechanical, electrical) things. Animations can be very helpful in reaching out to such students, whether they are in community colleges or in primary schools. The project will employ and financially reward undergraduate students at several college institutions. They will work in places and ways that are in addition to composing guides for calibrated peer reviews, new animations, and other computer resources. A valid effort by students will be the expenditure of time in service to public schools. College students can demonstrate mathematical concepts through resources now available. All they need is to be access the world-wide web. Student demonstrations of web-based materials to K-12 students will be part of this project.

Communication

The following material (excerpted from Decker 2001) contains a writing process though it is not yet in the form of a Calibrated Peer Review module. Since it presents numerical experimental results, specifics about probability, and curve-fitting data analysis, it demonstrates how a comprehensive approach with essays required of students fits into the proposed project.

Casino Night

A charity has decided to have a casino night to raise funds. They have come to

you for advice on how to design a game that they want to use. The room they will be using has 9 inch by 9-inch squares on the floor; the game will be to toss a disc onto the floor, and if the disc does not touch an edge of any square, the player wins. It will cost $1 to play the game: if the player wins, she or he gets back their dollar and another dollar as well (thus on each play the player either loses $1 or wins $1). The charity would like advice on the following: 1) What probability of winning will make for a good game? … 2) What diameter disk (would) attain the probability decided on in part 1)? 3) How much money can the charity expect to make? For the past three years the number of people attending the event has been 1475, 1525, 1580, and each person has played each game 5 times on average.

To answer these questions, … collect and analyze … data that relates disk diameter to the probability of winning ... Play the game 100 times each for several different disk sizes and record the number of wins and losses … a table of data points that relates disk diameter to probability of winning. … results are … :

diameter 0.875 probability 0.84 … diameter 4.75 probability 0.25

… decide which model makes the best predictions, … use that model to estimate the size disk that you need. …1) Explain how you decided on a probability for winning the game. … 3) Discuss what assumptions you needed to make in estimating the profit for the charity, and how you came up with your estimate. …

Form of your report: … an introduction to the entire experiment … (imagine writing a newspaper article). Follow the intro with your results, and finally address the discussion questions. Your report should be written in complete sentences with correct spelling and grammar. …

Evaluation

The project includes establishing a national panel of student reviewers from multiple departments at UCLA, and others nominated by faculty at academic institutions distributed around the nation. These individuals will create new material for CPR writing exercises. They will be able to enter a competition for new sound, animation, and still-image files to communicate about mathematics. Their materials will be exposed to individuals at K-12 schools, community colleges, and in teacher training programs at four year institutions including UCLA.

Merritt College, University of Hartford, Los Angeles Community College, Los Angeles Trade Technical College and Three Rivers Community College faculty members will review the materials and assessment procedures. The project will develop new materials to govern student writing to ensure that documents guarantee learning some new mathematical notions. The participants will examine the actual outcomes from written communications and develop evidence that this approach builds mathematical confidence.

Innovation

The project will involve students at every level of expansion. It will undertake evaluation of fundamental mathematical skill through the CPR process. Students will be allowed to submit their new items (sound, still visuals, animations) for review and possible inclusion in future materials. [Figure 1-5 all credit the students who created them.] Well-established mathematical curricula studies will be made sources for such new items. The many useful suggestions in National Council of Teachers of Mathematics (1989), Osen (1974), and Steen (1990) will guide faculty participants in making suggestions to students for new items. All faculty, investigators, reviewers, and interested parties, will be able to present their new web materials concerning mathematics, at a workshop conference. Faculty participants will recommend selected items from the materials developed here for relevant contemporary computer and mathematics education.

Work Plan

There are three main functions that this project proposes to undertake. The first is: use existing technological resources. The resources consist of web items: calibrated peer review as a system for stimulating activity and learning; animations, static images, text and sound files about diverse and unconventional topics often covered in Finite Mathematics and Pre-Calculus courses. Several of the items there include recent research results, e.g., an Image-Analysis or Geometric Semi-Regular Solid (w17); A New Large Number (w18); and, Large Number Dialogs (w19).

The second function is to establish working relationships among many kinds of individuals currently part of academic organizations. Many people, all able to contribute to the integration of practical and research issues into a renewal of mathematical and computer education, are kept from interacting by institutional and disciplinary barriers or inertia. There is a need for broadening the effectiveness of mathematical education by reflecting on changes introduced through expanded use of communications, digital and information technology. That can best be done by a multi-disciplinary effort that uses the strengths of the American system, where there are significant resources for many different educational levels and purposes. The third function is to engage students to establish a program and set of activities that will be continuing, and develop new teachers well-able to communicate about mathematical and computing fundamental concepts. The proposal is to address all three functions through a multi-year program of activities. These elements involve review of materials described in this proposal and the web locations cited, construction of calibrated peer review items focused on specific aspects of college mathematics courses for non-science majors, and communication among the participants (electronic mail, telephone conferences, preparation for and conducting a workshop meeting).

Participants and Schedule

The following faculty will be part of the effort: Allen Klinger, Concepcio’n Valadez, James MacQueen, Lewis Felton, Orville Chapman, and Arlene Russell (all UCLA); Robert Decker (Univ. Hartford), June Decker (Three Rivers Community), Jun Li (Univ. Oregon), Miguel Moreno (Los Angeles Trade-Tech), Ronald Kendis (Los Angeles City) and Lawrence Gurley (Merritt). Other individuals will circulate opportunities for student participation. They include Peter Reiher , Songwu Lum Jason Cong, Carey Nachenberg, Glenn Reinman, David Smallberg

The schedule involves six segments. In the first we will acquire at least six specific sets of CPR programs for conveying mathematical ideas through writing and critical evaluation. We will cooperate with numerous individuals in the math departments of schools, particularly the six listed above, to ensure the usefulness of these items to guide student learning. A letter indicating the willingness of the University of Hartford to participate follows.

The second segment will involve regional meetings and conference calls. We will have two meetings bringing individuals together who are normally far apart physically to coordinate our approaches.

The third and fourth segments correspond to evaluation phases. We will conduct meetings among faculty and student participants. These meetings will include a contest for best new materials, with travel support for a paper presentation as the prize.

The fifth and sixth segments are to begin with a workshop meeting at the Asilomar Conference Center bringing all key faculty personnel and an equal number of invited mathematics educators together. These segments will conclude with preparation of a report detailing the recommendations of the people involved.