History of the "Gateways
to Mathematics” Video Courses
Development of the "Gateways
To Mathematics" video course indirectly began in 1967 when Harold Mickley chose Herb Gross to be a Senior Lecturer
at the MIT Center for Advanced Engineering Study (CAES). In that
role, Herb taught "Calculus Revisited" to scientists and engineers who were coming to CAES to being a sabbatical
program to update their professional skills. The course proved so successful that in 1969 Professor Mickley asked Herb
to transform the "Calculus Revisited" program into a video course that could be used even by scientists and engineers
who were not coming to CAES to further their studies. The video course was completed in 1972. MIT has chosen to
put Calculus Revisited on its OpenCourseWare site. Please visit http://www.youtube.com/watch?v=8rAo0cN-b2w&feature=channel.
of the companies that purchased the video course was Control Systems Engineering whose president
was R.A. (Anthony) Moore. Mr Moore was so pleased with the program that he called Herb to see if Herb knew anyone who could
do for basic arithmetic what he had done for calculus. Herb, who had been teaching developmental mathematics for several
years at the community college level, offered to produce such a course himself. So in 1983 Herb began writing a textbook
and study guide for a cours entitled "Gateways to Mathematics". The text and study guide were completed in 1985,
at which time Herb began to produce video-taped lectures to supplement the written materials. The entire program was
finished by the end of 1985.
The course has been used
both as a supplement for other developmental math courses and as a self-contained program in its own right. It
has bound extensive use, especially in prisons in Massachusetts and North Carolina. It is this course that now appears
on our web site under the name of "Classic Arirthmetic Course". It is called "classic" not just because
it is 25 years old but also because the written material was typed on a manual typewriter and the video tapes were produced
in black-and-white using no audio or visual aids other than chalk, a blackboard and Herb talking. Note that the written
material is labled "prepublication edition" and has never been edited!
However in the 25 years since the course was produced Herb has been refining the material and has supplemented the course by producing slide shows that upgrade and expand much of the materials that defined
the original course. To
obtain the full flavor of how the progam was productd, you should begin by viewing the "preface" video tape.
The video tapes and/or the slide shows complement the textbook and the study guide. You are invited to study the course
using whatever components that best suit your learning profile. We look forward to you enjoying the course.
Our "Gateways to Mathematics" (GTM) arithmetic course can be approached from
several different points of view:
For most people viewing a lecture is easier to internalize than reading
a traditional textbook. For this reason our suggestion is to begin the study of each lesson by viewing the videotape that
discusses the lesson's content.
(2) The PowerPoint Presentations:
The videotaped lectures were
produced in 1985. A quarter of a century has transpired since that time. So it is understandable
that Professor Gross found time to refine and fine tune the material that was contained in the
videos. These changes are reflected in the PowerPoint presentations that were produced 25 years
later. For this reason it is advisable to look at the PowerPoint presentation of the lesson as well.1
(3) The GTM Textbook:
Even with new technology, there are people
who still learn best by reading. If this is the case, you can begin by reading the textbook which
is available on our site in pdf format. You should feel free to download it and use it as the
text for the course. The written material is presented in the form of a series of connected Illustrative
Examples. The solution of each of these example, along with enrichment commentary, accompanies each example.
You may read the textbook interactively by doing each example on your own before reading our solution. An additional facet
of the textbook is that it contains a second column on the right hand side of the page so that Herb can present asides and
other interesting comments that might otherwise disrupt the flow of the textbook.
(4) The “Check the Main ideas” Section In the Study Guide
In going through the textbook example by example there is the danger of failing to see the
forest because of the trees. The GTM Study Guide, which is also available in pdf format, gives
you a chance to see if you have internalized what you have seen and/or read. So you might benefit
from seeing a qualitative summary of the main points of the lesson, Herb has written his own summary of the lesson but replaced
certain key words by blanks (with the correct answers written in the margin). The resulting "fill-in-the-blank”
format is referred to as "Check the Main ideas”. In this way you have an interactive way to review the lesson
conceptually. Take the time to read what Herb has written and see of you can fill in the blanks
correctly. If you do not know the correct word to put in the blank it isn't cheating if you look
at the word that appears in the right hand column and write it in the blank space. When you are
finished, simply read the completed essay and you will have reviewed the main qualitative points
of the lesson.
(5) The “Mastery Review” in the GTM Study Guide:
it is nice to have a conceptual understanding of the content in the lesson, it is also important
to be able to perform the computations correctly. With this in mind the Study Guide contains a
section called " Mastery Review” in which
a series of exercises is assigned and the answers are also provided. The “Mastery Review” is a self-prescriptive
way to help you check on where you might need supplementary help. Namely the exercises are numbered
to correspond with the Illustrative Examples that appear in the textbook. So, for example, if
you are unable to get the correct answer to Exercise #6, you need only go to the textbook and
look for Illustrative Example #6. The exercise in the Mastery Review is word for word the same
as the corresponding Illustrative Example in the textbook, except for the fact that the numbers are different.2
(6) The “Exercise Sets” in the GTM Study Guide:
you can answer all the questions in the Mastery Review correctly, it indicates that you have
internalized the material that was presented in the lesson. However, the ultimate test for determining
how well you understand the material is whether you can use what you've learned to solve problems
that you have not seen previously. So at the end of each lesson in the study guide there is an Exercise Set
with ten problems that go “one step beyond” the material that was presented in the textbook.
In fact there are three forms of each Exercise Set. Each of the three forms contain exactly the
same problems with the only change being the actual numbers that are used in the exercises. The
answers for all three forms of the Exercise Set are supplied in the Study Guide. If you correctly
do all of the exercises in Form A, you should move on to the next lesson.
It is possible that your
answers to one or more of the exercises are incorrect. If this happens, all solutions, including
commentary, for Form A of the Exercise Set
are contained in the Study Guide. You can read and study our solutions to the exercises for which
obtained the incorrect answer.
Once you understand how to do an exercise for which you obtained the incorrect answer in Form
A, do the correspondingly numbered exercise in Exercise Set Form B. It is exactly the same problem but with different numbers.
Hopefully you will get the correct answer this time. However if you don't, the solution for every
exercise in Exercise Set Form B is presented in the second videotaped lecture for the lesson.3 If you still obtained an incorrect answer after trying both
forms A and B, do the exercises in Form C. The answers for Form C are in the Study Guide but
not the solutions. If after doing the exercise in Form C go to a “live resource" (such as a teacher) to get