Technology-Enhanced Mathematics Education
Objectives
The primary objective of this project is to increase the number of students successfully completing mathematics courses. In particular, developmental mathematics students need additional support in mastering elementary mathematics concepts and skills.
By including computer technology in the classroom, we anticipate greater participation by students in the learning process. Since the only way to learn mathematics is to do mathematics, students are likely to be more successful if they are actively involved in the classroom than if they are simply scribing notes. (“I understood it when you explained it in class, but when I got home, I couldn’t do the homework”)
In addition to the developmental courses, higher level math courses will benefit from the use of technology in the classroom. For instance, the use of Mathcad or Maple in Calculus will aid students in their understanding of concepts from limits and derivatives to curves and surfaces in three dimensionalspace.
Pedagogy
By definition, pedagogy is the art and science of teaching. The primary objective of effective teaching is student learning. But what is the student expected to learn? And what does it mean to teach a student?
The current emphasis on student outcomes makes it clear that the general public is interested in the specific concepts and skills that need to be mastered. The outcome encompasses both the learning objectives and the expected assessment results. Therefore, any educational pedagogy must include both aninstruction/learning component and an assessment component.
The following paragraphs provide an overview of the pedagogical requirements which technology-enhanced mathematics education must address.
Instruction/Learning
As stated above, students learn mathematics by doing mathematics. By doing
mathematics, I mean
1. Reading the textbook, previewing material before class, and reviewing it after class.
2. Taking notes and listing questions about textbook examples as they read the book.
3. Asking questions during class or during office hours or via email
4. Doing assigned exercises. Students must develop the discipline to solve problems without referring to textbook examples.
5. Reflecting on solutions to problems and summarizingthe solution method
6. Taking a “practice test” (e.g. in textbook), gradingthe test and reviewingthe problems they solved incorrectly.
These requirements are consistent with the recent PK-16 report available at . In this respect, the mathematics community has not strayed from its conviction that students must do mathematics to learn mathematics.[1]
Assessment
A single test is not sufficient to determine a student’s mastery level in any subject, but least of all in mathematics. Several assessments are needed to determine the student’s “working knowledge” of mathematics. By “working knowledge” I mean real time problem solving skills that can be applied at the next academic level or on the job with minimal use of reference material.
Based on the need for working knowledge, cumulative assessments are necessary. The only effective way to measure working knowledge is to test students on a regular basis. Since a key element in all assessments is the time required to accurately solve problems, timed assessments are essential. (For higher level mathematics courses, developing mathematical models for applications is another primary area for assessment)
The benefits of continuous assessment pedagogy are well-documented. The primary benefits to the student are mastery of material and confidence in using mathematics. Benefits to the teacher and student include knowledge of the student’s strengths and weaknesses and development of individualized study plans for achieving mastery.
Technology: A Key Element of this Pedagogy
The use of software like CourseCompass/MyMathLab or MathZone/ALEKS would be useful for implementing an educational pedagogy with the above-mentioned requirements.
1. Students would receive immediate feedback on each problem they attempt during the learning process. In addition, a host of on-line supplements are available for each problem (i.e. guided solutions, links to the textbook, videos, etc.)
2. On-line testing allows greater frequency of feedback than is possible with limited numbers of instructors and tutors.
3. Instructors have summaries of student activity that allow them to guide students to their areas of greatest need. The software also allows for the creation of study plans tailored to each student. This individual study plan is not possible when many students are assigned to one instructor.
4. If students are unable to attend, they can still work on the scheduled topics at home (most students do have computers). Instructors can still monitor the student’s progress and provide assistance on-line. (This a “distance-learning” component which is not currently available)
5. As noted by many, the videos provide mini-lectures for students who may not understand the explanation in the textbook (but these should not replace the required reading)
6. Students can practice on a larger number of exercises since the software regenerates the numerical values for a problem. As a result, particularly challenging exercises can be done repeatedly. In this way, students review the solution process (i.e. reflect on the solution process) until they understand its application to that type of problem.
Software Reviewed
Educational software for mathematics should enhance the learning process. Students should be engaged in doing mathematics when they are using the software. The above-mentioned pedagogy demands a dynamic, learning and assessment environment. The two software packages reviewed by the department are MathZone/ALEKS (McGraw-Hill) and CourseCompass/MyMathLab (Addison-Wesley)
Features
A. For Instruction/Learning
When students are doing practice exercises, software features that are useful include:
Feature / MathZone/ALEKS / CourseCompass/MyMathLabSolve similar problem / Hint / View an Example
Solve the problem / Show Me / Help Me Solve This
Step through solution / Guided Solution / Help Me Solve This
Print displayed page / Print / Print
Send email to instructor / Ask My Instructor / Ask My Instructor
Opens textbook in pdf-format / Link to Text / Textbook Pages
Video presentation by instructor / Video Lecture / Video
Audio presentation / e-Professor / Animation
Individualized study plan / ALEKS assessment / Study Plan
Interactive exercises in on-line version of textbook / You Try It
B. For Assessment
When instructors develop assessments to determine a student’s learning progress, the ideal is to test core knowledge and skills as well as the student’s ability to apply that knowledge to new problems. Software packages excel at the skill level testing, but not necessarily at the new problem assessments. To provide for this testing requirement, both software packages allow instructors to design and upload tests written by the instructor.
It should be noted that each software package partitions exercises by the learning objectives for the sections of the chapter. This is an important feature for students and instructors. Students can focus attention to specific problem types they have not mastered and instructors can focus presentations on difficulties the class may be experiencing.
C. For Administration
Both software packages provide grade book capabilities. Instructors can examine the records of individual students or the entire class. Performance on specific exercises enables the instructor to prepare focused presentations on topics the class may be struggling to master.
Graded assignments can be exported to Excel and combined with the results of written tests. Students who stop attending class can be removed from the active roster.
Assignments can be timed.
Hardware and Software Requirements for MathZone
Windows 98 R2, ME, 2000 and XP
Internet Explorer 5.5 and 6.0
Netscape 6.2 and 7.1
Mozilla Firefox
Mac OS 9.2, OS X 10.2.8, OS X 10.3.2
Internet Explorer 5.2
Netscape 6.2 and 7.1
Safari 1.3
Mozilla Firefox
Some MathZone sites require use of the following plugins. Please install these plugins before continuing...
Macromedia Flash Player
Macromedia Shockwave Player
QuickTime Player
Your browser must be Java enabled
Click the following link to check your browser for the required plugins
Hardware and Software Requirements CourseCompass
To work with CourseCompass 4.5, your computer must meet the following system requirements for operating systems, connection speed, and browser versions:
Operating system
CourseCompass is supported for the following operating systems:
Windows® 2000 and XP
Macintosh® 9.2 and OS X Version 10.1 and 10.2
Connections and browsers
CourseCompass requires an Internet connection with a minimum connection speed of 28.8 kbps (kilobits per second) and either of the following Internet browsers.
Microsoft® Internet Explorer, Version 5.x up to Version 6.0
Netscape® Navigator, Version 7.0
If you have an earlier version of one of these browsers, you can download a newer version from the appropriate manufacturer's website:
For Internet Explorer, go to
For Communicator or Navigator, go to
AOL users: You cannot view CourseCompass using the AOL browser. You can, however, use AOL as your Internet Service Provider to access the Internet, then open either the Internet Explorer or Netscape Communicator/Navigator browser within AOL to access CourseCompass.
Browser settings: cookies and Javascript options
CourseCompass uses both cookies and JavaScript technology. Both of these features must be turned on in your browser, and are usually turned on by default. See your browser Help for instructions on how to view or change these browser options.
Additional software
To view the online (PDF) version of the CourseCompass Quick Start Guide, you need to download and install Adobe® Reader®:
Adobe Reader - Needed to view online CourseCompass guides
To use multimedia material provided with some courses, you may also need to download and install additional software. If you're uncertain whether you'll need these resources, you can open your course and see what it requires. Note that some plug-ins are not supported by Internet Explorer V5.5 SP2 or higher.
MyMathLab Installation Wizard - Needed to install plug-ins (such as MathXL® Player, InterAct Math Plug-in, the TestGen Plug-in appropriate for your course and more) specific to your MyMathLab course. The MyMathLab Installation Wizard is typically found inside your course via a course announcement or a course menu button, or you can access it directly from:
Apple® QuickTime® - Needed to view full-screen video and streamed media, or hear audio files in any of 30 audio, video and image formats, including Flash
Java™ Plug-in - Needed to view the Virtual Classroom and Lightweight Chat sessions in CourseCompass
Macromedia® Flash™ - Needed to improve viewing of high-fidelity websites
Macromedia Shockwave® Player - Needed to run animations in some courses
RealNetworks® RealOne™ Player - Needed to hear music or watch streamed media animations in some courses
TestGen Plug-in - Needed to view and take online TestGen tests in CourseCompass
Implementation of Educational Program
Implementing a technology based program requires a plan. The details below are minimum requirements for any hope of successful implementation.
Instruction/Learning
1. Partition material into units.
2. Give student a unit pretest[2] to establish an Individualized Study Plan for that unit (A course pretest could be given prior to the start of the semester to determine the student’s cumulative knowledge of the course material.)
3. Help the student develop a plan to work on the material identified in the software’s pretest results. (i.e. guide the student through the software generated study plan)
4. The student will do the practice exercises (as identified above). The student may use any or all of the following:
a) Detailed solution
b) Step-by-step solution (student enters answers for intermediate steps of the solution)
c) Use video or audio presentation for further explanation
5. Do the practice exercises until the software indicates the topic is mastered.
6. Retake the pretest
a) If the grade equals or exceeds the minimum grade for approval to take the unit test, take the unit test.
b) If the grade falls below the approval threshold, return to 3 and follow the steps back
to 6. (Need to establish a policy for taking the unit test if the student is not ready after the second pass through the learning process)
7. Based on the results of the learning process (steps 2 through 6), identify student as candidate for one of the following three options:
(1) 10-12 week plan for early completion (student still could extend it to full semester)
(2) 15-week plan for traditional completion of course
(3) 20-week plan for IC and completion within first 5 weeks of the following semester[3].
i) If the student completes the course under this plan, the student should register for the follow-on course in the same semester
ii) If the student does not complete the course within this time period, the student should register again (i.e. pay again) for that course. This time the student should be put on a maximum 15-week plan to complete the course. If the student cannot successfully complete the course during this second registration, the student should be referred to counseling for alternate educational options.
Assessment
1. The pretest, post-test (i.e. 2nd pretest at completion of unit learning process ), and the unit test[4] are designed to measure the student’s ability to learn and retain/recall mathematics with respect to the current topics.
2. A multi-unit cumulative assessment is designed to measure the student’s long-term retention/recall of mathematics. Two multi-unit assessments should be given:
i) The first assessment should be given when the student has completed the first half of the course curriculum units.
ii) The second should be given at the end of the course. This 2nd cumulative assessment should be broken into two parts:
a) The 1st part should be based on the units completed in the second half of the course.
b) The 2nd part should be based on the units completed in the first half of the course.
3. Mastery should be based on all activity in the course (including practice exercises, pretest scores, unit test scores, and cumulative assessment scores). A heavier weighting can be given for unit tests, but the other information is very important for both the student and the instructor.(?)
1
Joseph N. AllenWednesday, 12/28/06
[1] “Recommendations for the Preparation of High School Students for College Mathematics Courses”, Joint Statement of MAA and NCTM, The American Mathematical Monthly, Vol.85, No. 4, April 1978, pp. 228-231
[2] See Footnote 4 next page.
[3]Rob Melucci suggested this during one of our discussions of current policies.
[4] This concept of pretest/post-test/unit test was recommended by Beverly Pepe and is consistent with the early research on mastery learning