Abstract:
The Problems in Mathematics Education Addressed by College Prep Math
(CPMath)
Today, largely because of technology, math permeates more of university education and life outside the university than ever before. In almost every field of study, math is strikingly more important than it was even twenty years ago. Many fields of study place demands on math that were not even thought about twenty years ago. Today's culture is both math needy and math hungry. But most students arrive at the university poorly prepared for the math they will use in their university education and their work. They may be prepared for the math their parents and grandparents had to learn, but this is not adequate preparation for the math they will actually use in today's world of technology. In fact, most of the students who score at the top on standardized tests are poorly prepared. Some can even manipulate x's and y's with reckless abandon, but have little understanding of how those skills relate to their university courses and their work beyond the university. They do not realize that because of technology such skills are not as valuable today as they were even twenty years ago. With wise use of technology, the CPMath students move from symbol pushing to understanding and see mathematics as worthy of serious attention. On the other side of this problem is that many of the most creative students have been turned off with math courses in algebra and trigonometry that emphasize drill on rote manipulations of seemingly meaningless symbols. These are often the students who continually ask,"What's this stuff good for?" only to be told (often incorrectly) that they will need it later on. This group needs a new way of learning with content appropriate for answering the question "What's this stuff good for in today's world?"
College Prep Math (CPMath) is designed to address both issues. For students who have been enjoying their school math, the course is an opportunity to learn what is important at the university level before they get to the university. For students who have been turned off by school math, this course is an opportunity for a new start and a fast track to the math (in context) that actually arises in science, engineering, technology and in the workplace.
Why CPMath is a recommended alternative to conventional courses called "Pre-Calculus?"
Conventional precalculus courses spend the most of the students' time rehashing the techniques that the students have taken previously. What's worse is that conventional precalculus courses give almost no hint of what calculus is. These courses look backward and not forward. CPMath looks both ways. CPMath students deal with almost all underlying calculus and engineeering concepts through approximate numerical calculations instead of the exact formula manipulation found in calculus. These include the studies of growth and change as well as approximate area measurements and approximate tangent lines. CPMath students experience all the underlying calculus ideas (but not all calculus techniques) and are not blind-sided as they step into university calculus and mathematics.
How learning in CPMath differs from learning in conventional mathematics courses
When you want a youngster to learn about cats, do you give a lecture telling them that a cat is a small carnivorous mammal with rectile claws and distinctive sonic output? Instead, you put a kitty in front of the youngster and let the youngster see, feel and play with the kitty. CPMath students use the CPMath computer courseware to play with the math kitty through dazzling interactive computer graphics (of a quality and a quantity impossible in a conventional textbook), instant calculation and experimentation. (One student called the CPMath format "a mathematical chemistry set.") CPMath students have the freedom to experiment and inspect the results. The interactive visualizations and experiments in CPMath set up the ideas giving students instant access to the ideas. Students in conventional courses attempt to learn the material through a maze of unfamiliar words. They are forced to memorize because they don't always have an idea about what the words mean. In CPMath, the understanding students get through visualization and experimentation minimizes the need for memorization. In CPMath, only after an issue has been set up visually do the words go on. And when the words go on, they are in standard American and not in the stilted language usually associated with conventional mathematics text books. Professional studies show that random learners and sequential learners do equally well in the CPMath format; they use the computer-based courseware in different ways as they see fit. Anecdotal information from University of Illinois at Urbana-Champaign and The Ohio State University indicates that dyslexic, ADD, autistic and brain-damaged students have done well in the CPMath format. Perhaps Lynn Arthur Steen, past president of the Mathematical Association of America, summed it up best when he wrote: "Most students take only one or two terms of college mathematics,and quickly forget what little they learned of memorized methods for calculation. Innovative instruction using a new symbiosis of machine calculation and human thinking can shift the balance of mathematical learning to understanding, insight, and mathematical intuition." CPMath is the first course at the precollege level that delivers on Steen's vision.
Abstract: Dynamic Geometry Systems (DGS) were developed with the intention of allowing the user to "explore" (plane) Euclidean Geometry the same way he would proceed with (graduated) rule and compass. The adjective "dynamic" comes from the fact that, once the sketch is finished, the first objects drawn (points) can be dragged and dropped with the mouse, consequently changing the whole sketch.
Unfortunately, Computer Algebra Systems (CAS) and DGS have evolved independently. Some CAS, like Maple, include powerful packages devoted to Euclidean Geometry, but no one includes dynamic capabilities.
On the other hand, DGS cannot deal with non-assigned variables. This fact prevents the use of DGS in any process involving non-assigned variables, like mechanical theorem proving in Geometry (where it is standard and necessary to consider parameters as certain coordinates of the initial points).
Other authors have solved these lacks in different ways, but all the approaches we know are based in implementing new software (whole DGS and/or CAS): The Algebraic Geometer, GDI (unifying the former Lugares, Discovery and Rex), Geometry Expert and MathXP.
Our strategy is completely different: software reuse. Another difference is that we do not try to produce a system that exceeds in a certain task (e.g. mechanical theorem proving in Geometry) but a kind of dynamic Graphic User Interface (GUI) with diverse applications.
We have developed a bridge between existing DGS and CAS: GSP v.3 & v.4 and Maple 8 and Derive 5 that is published elsewhere. Now a new adaptation to the projective case is presented.
KEY WORDS
Projective Geometry, Symbolic Computation, Dynamic Geometry, Mechanical Theorem Proving.
Abstract: Maple's new student package provides tools for supporting the teaching and learning of concepts in first year calculus, linear algebra as well as precalculus. The packages allow computations (like integration, differentiation, limits, Gaussian elimination and others) to be performed step by step such that the solution process is exposed. Visualization routines illustrate key concepts in the material being taught. Interactive tutors (using the Maplets technology) provide an environment for exploring the mathematical concepts in an interactive way. Educators can use all of this material to enhance the delivery of a course. Students will be able to use the package to reinforce what they have learned in class.
Abstract: Why is technology (CAS) dividing mathematics teachers into (at least) two groups? Those (this is our case) who love to use it and think that software like Derive and the TI-92 Plus/Voyage 200 are very good tools for teaching and learning mathematics and those who think that students should learn to do everything by hand, first, before using Computer Algebra Systems. We don't know the answer but we are going to give examples of how both (use of a CAS and paper/pencil techniques) can be happily joined. There is no magic way to teaching but trying to convince our (engineering) students that mathematics are useful comes easier with a good use of technology. Live examples will be performed, originating from students' questions.
Abstract: Authorities in the German state of Thuringia are considering allowing the use of CAS technology in math education in schools. Therefore, a project in 8 grammar schools is being carried out to investigate which effects the use of the Texas Instruments TI-89 in math classes has on math skills. As from grade 10, these calculators are provided free of charge to all students in the project schools for use in math courses (and examinations).
In November 2002 a survey was carried out to find out which views all (about 1000) grade 11 and 12 students in the project schools have on using the TI-89 in math and science. Students of three different levels are in the project: each student has at least to attend a math "Grundkurs", a course that is offered in every grade and communicates the basic math competence. Students with more interest in math opt to attend the math "Leistungskurs" which imparts more advanced material (in more hours per week). Two of the project schools have special classes open only to the most talented math students in Thuringia.
The main part of the one-page questionnaire comprised 8 statements - 7 related to the effects of using the TI-89 in math and science lessons and one general statement about math lessons. Students were asked to mark on a scale how strong they agree or disagree with each statement. In the paper it will also be analyzed how certain characteristics of the students (e.g., which type of course they are in; which grade they got in math in their last report) influenced their answers.
Abstract: The use of computer algebra in the mathematics classroom will be discussed. As an example, a specific computer classroom lecture will be demonstrated, that of the behavior of the Riemann Product.
Last year in Volos, Greece, the tangent exponential was discussed. (A tangent exponential is an exponential function that is tangent to a curve.) This year the same technique used to develop the tangent exponential is turned on integration. The result is the Riemann Product.
Computer algebra is in a unique position to aid students in learning the Riemann Product. A graphing calculator is not enough, since symbol manipulation is required to do the necessary limits. Simultaneously, the presentation will demonstrate by example, the modern methods of presenting a mathematical concept from the numerical, graphical, and symbolic points of view.
Abstract: It is well known that differential equations represent a fundamental tool in the mathematical modelling in many applied fields such as physics, astronomy, economic sciences and others.
During our courses of mathematical analyses, we often use differential equations linked to the daily life problem, discovering in such a way, the real adhesion between experience and mathematical environment and giving motivation to the students for the study of the subject.
In such cases, we have noted that students were used to learning just some techniques, which make them able to solve some equations and when they get the results, they do not relate them to the initial problem or they elaborate some restrictions in order to avoid nonsense solutions. They were used to seeing mathematics as a tool, so they were used to learning just some techniques without "thinking".
We believe that teaching should not be a pure transfer of notions and techniques, but it has to stimulate learning that is reasoning, induction and so on. Thus it is important to change methods, and techniques of learning and teaching. This is more and more true, because otherwise we will have just people able to do standard computations or solve standard problems, but not people able to face new problems.
We believe that the computer support allows to interpret more deeply the mathematical results that in the case of classical approaches we didn't do. We want to stimulate the student to have a more critical attitude towards the solution of the problem described by a differential equation. We are not concentrating on "how to solve a differential equation or a Cauchy problem", but indeed on what is the meaning of some results, stimulating students to be active asking themselves questions, not to accept passively the results they have from their own calculations.
In this work, we propose to research how the CAS can foster a more critical, significant and effective learning.