Executive Summary of Results

This purpose of this project was to create a mathematics classroom laboratory and to develop a curriculum for the laboratory course where students apply their mathematical knowledge to a variety of disciplines using data collection devices and computer software to collect, analyze, model, synthesize, and report their findings. The curriculum was based on the theory of problem-based learning. The course was developed to respond to the need for new learning environments for students who will not pursue the traditional algebra and calculus route through college. These students will use the problem solving process and computer-generated models to develop mathematical competence and confidence in solving and modeling real world problems. The course has been developed and is currently being offered at IUS for the second time. Student success rates in the one year pilot study and the first course were very high (around 90%), but have been limited to a small number of students so far.

Need for the Project

The goal of the project is to promote quantitative literacy in all college graduates at Indiana University and not just those few who complete college algebra and calculus, which is now less than 10 percent of our student population. Basic mathematics has been identified as the field most in need of assistance in the area of retention. Most remedial and introductory math courses have very high rates of withdrawal and failure. Technology-mediated instruction and project based learning have both been successful in improving student success and retention rates, but have not been implemented in lower level mathematics courses to any large degree. The Carnegie Learning Company has reported research at other colleges using the Cognitive Tutor software where success rates are as high as 85% ¹  With this innovative curriculum, we were able to replicate the high success rate for students in our new introductory math class, M112.

Use of Technology

The Cognitive Tutor software is an intelligent tutoring system based on cognitive theory. Having the tutor run in the background allows the computer to act as a tutor while tracking students’ performance. Students use the computer to solve problems, graph their results, and visualize how these problems apply to the world. These students gain confidence in modeling the world around them with the computer tools they learn to use in our course. The computer is also an effective motivator. Students gain confidence when using the computer. They are able to build models, verify their results, ask questions they would normally not ask, and assess themselves using technology and peer review. They are able to work in mathematics. They learn to solve problems not equations.

We used the Cognitive tutor software and a modeling curriculum to offer students an active learning experience. The classroom was a computer-equipped laboratory where students were actively engaged in a problem solving collaboration with their peers and instructors. The students were able to work at their own pace and the computer performed a “knowledge trace” of the actual objectives mastered by each student. Additional assignments or explanations were easy to implement using Oncourse and the Internet to post notes, examples, or explanations.

Instructional Design Plan

Describe how the use of technology used supported your teaching approach:

• What student learning outcomes were expected?
• How did your teaching plan encourage active learning, provide feedback, address respective learning styles, and build collaboration?
• What resources were used? (campus technology support services, etc.)
• Would your project be easily understood and executed by other faculty?

Students who complete the course should be able to meet the five capabilities outlined in the MAA document on Quantitative Literacy and continue to use and apply their knowledge in upper level courses. The five capabilities are quoted from the MAA website².

1. Interpret mathematical models such as formulas, graphs, tables, and schematics and draw inferences from them.
2. Represent mathematical information symbolically, visually, numerically, and verbally.
3. Use arithmetical, algebraic, geometric, and statistical methods to solve problems.
4. Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.
5. Recognize that mathematical and statistical methods have limits.

The software CD contains content, mathematical tools (similar to Excel’s) and the Cognitive Tutor (an intelligent tutor built on cognitive models that simulate human thinking). With the new technology, a variety of models were introduced. We used Oncourse and the Web to post projects for students to complete.  In completing projects, the focus is on using mathematical models to quantify and understand a situation, not solve equations. Students were required to collaborate on some projects and were also to critique other students’ models. Student work was posted in Oncourse and other students were expected to test previously created models to see if they held up over time. (Example: Summer Olympic data was integrated with previously developed models to see if the predictions from last year were correct.)

The Oncourse management system was used to manage the course. The obstacles to using a different ePortfolio instrument to collect student work precluded us from using other software. The students were required to use too many tools in this course. Oncourse in itself is not hard to master but many students were not able to work in Oncourse, the CD Rom (which had course materials), TI Interactive math software (which was used for projects), and a networked drive where their documents were held. These software programs do not work together so the students do complain of too many different environments.  This is a valid concern.  I have no solution for students or faculty to date.

Potential to Impact Student Learning

Clearly define how your project improved student learning - include specific examples of how your project:

• Fostered depth of learning
• Supported efforts to improve retention
• Fostered the learning of large numbers of students
• Has broader applications that will affect learning K-12 or community environments?

Motivating students to become independent, successful learners is an explicit expectation of the course as well as applying knowledge to solving problems. Student success with the computer is the most powerful motivator I have seen in 15 years of teaching. Students who were initially afraid of math or the computer stay into their lunch hour to finish projects. They write that they never believed they could do any math and now they want more. Math phobia , as self-reported, decreased by one hundred percent. Once students start working problems on the computer it appears that they forget their phobia and they give it to their new ally, the computer. Student confidence is always built up, never knocked down with the Cognitive Tutor. The ability to ask for help without being detected by the teacher or neighbor in the lab is the best feature of the system. Student success rates with this course are very high. We are able to retain and pass about 90% of the students with this curriculum. Large numbers have not yet been reached because of the long process of getting a new course approved and accepted as a requirement.

The computer allows students to model data and work with models that they normally would not see until their third semester of college math in a traditional sequence of classes. This does not mean that they understand all of the algorithms and definitions and equation solving methods. It does mean that they are thinking at a higher level. They are involved in building models, solving rates of change, asking intelligent questions about the properties of the model, and applying a problem solving process that they can carry forward to other disciplines and into the workforce. In summary, the students are working at a higher level of thinking. They are doing mathematics – not memorizing routines to solve problems. They are becoming confident problem solvers and learning skills they can transfer to other disciplines.

The Cognitive Tutor has been used in other software products and it will be offered to a new market in Fall 2005. The company plans to release a new pre-algebra curriculum for middle school math. It may be possible for teachers in the K-12 math community to visit and learn from our program, but we currently have no plans with any of the local school districts for professional development. We will continue to offer training for any faculty member at IUS who wants to teach with the new curriculum and we will be sharing the data collection devices and software in the new math classroom laboratory with other faculty.

Assessment Plan

Briefly explain the effectiveness of your assessment plan:

• What measure of assessment was used?
• How did your approach obtain measures of performance?
• Did you feel your measurement was accurate and effective?

Student work on modeling projects are the primary assessment instruments for the course now. The first two pilot course offerings used the departmental final exams for our intermediate algebra course as a comparison measure. (Initially we were concerned about content but students in these courses exceeded our expectations in their performance on traditional process skills. After the pilot years we dropped the comparison.) The computer tracks student progress through the 12 units of the course where students experiment with different models and their graphs. Students are given credit for completing each of the units but there is no standard here. The application of the models to laboratory based activities using problem solving and collaboration with other students or instructors is what is measured.

The problem solving process is learned by doing. We collect data, graph, look for patterns, discuss, conjecture, model, apply, test, and repeat all while keeping track of our progress. Students write and post and correct their models and report their findings in different ways (sometimes a technical report, sometimes a report to a company, sometimes just a statement of findings.) These reports are posted and shared. The development of good activities and projects, and fair rubrics for grading are the time-consuming tasks for the teacher now.

Student responses and reactions are gathered for each major performance assessment. At the middle and end of each semester students are asked to discuss each performance task and tell what they learned from that task, how much time was involved, and whether we should include it for future students or replace the task because it was too easy or too hard or whatever they want to share. Students who complete the course have amazing abilities to discuss their work in detail and they don’t mind telling you that something was a waste of their time (after they have the credit for it). Each semester I find myself reorganizing the course and adding and subtracting assessments based on student responses. The students want more challenges. With the computer creating the models, the first assignments I created were way too easy for them. They needed more in depth problems than I was used to creating. The creation of assignments remains my main challenge. Once assignments are posted is it impossible to re-use them again, so the task of generating good problems is a major task.

Plan for Colleague Development

Describe your role and activities as a mentor:

• What mentoring activities were implemented to assist colleagues?
• How did you help to develop colleague's instructional technology skills?
• How do you think your contributions to the Knowledge Base will help colleagues planning similar projects?
• Is your project approach potentially applicable to other contexts or disciplines?
• What tips would you offer?

I offered a two-day workshop in summer 2002 for other instructors at IUS.4 May, 2007tics resource laboratory as tutors for the course. I have had two instructors do peer visitation and come to the laboratory classroom in preparation for their course. We share activities on our website and have borrowed extensively ³ (with permission) from the curriculum of other like-minded institutions. We visited the University of Delaware for a problem based learning institute to study the problem based learning approach.

We still need a problem solving database for mathematics on the Web. Current problems are scattered and not housed in any one place where access is easy or consistent. My goal of creating the database of problems that are classified, categorized and searchable has not happened. The development of that collection is a necessity. Instructors currently rely on textbooks as much as they always have and teach how to solve equations (not problems) because there is no wonderful collection of problems.

Final Comments on Project Results

• Do you think you met the goals of your project?
• What, if anything,would you do differently?
• Did you have any unexpected outcomes? (positive or negative)
• Would you recommend this teaching approach to colleagues?
• What other comments or suggestions would you offer to complete this report?

We believe we met the goals of the project. We have developed the new course and it has been approved and offered for two semesters. Students in both the pilot year and this year have been successful in the course at a very high rate (around 90%). We currently have a laboratory classroom (shared), data collection devices and a set of activities for performance assessments. We have a repository of student work to share with other students. We have had the experience of trying to create a technology beyond our means and the failure of a collaborative project to bring about our national collection of problem based learning activities. We have learned the limits of technology and had many down days in the classroom. But we have a successful project that we can continue to offer to students on our campus.

The project was limited for several reasons. New courses take a long time to develop and become accepted. This course must be offered in a computer classroom where students have access to the Internet. The limits of technology must be accepted and contingency plans always ready.  The instructors of the course must be comfortable with the concept of problem based learning ⁴ (where they are not always in control of the answers or the outcomes.)  The need for a repository of good problems will always keep the instructors in search for more good problems or adapting old ones to fit the current need. These are challenges not all faculty will want to accept. Concerns about faculty workload are valid. Using technology and problem based learning are rewarding but labor intensive activities.

Bibliography

1. Carnegie Learning Company, Quantitative Literacy Through Algebra, http://www.carnegielearning.com/curricula/qlta/
2. Mathematical Association of America (MAA Online) Quantitative Reasoning for College Graduates: A Complement to the Standards http://www.maa.org/past/ql/ql_toc.html
3. Problem Based Learning at Maricopa Community College http://www.mcli.dist.maricopa.edu/pbl/info.html
4. Problem Based Learning at Southern Illinois University http://www.pbli.org/

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