Thoughts on teaching mathtematics, the role of mathematics and other ideas.

About Professor Gross

For those of you who have met Herb Gross, he needs no further introduction. For those of you who have not met him, presented below are a few of the highlights in his professional life. They do not even begin to describe the person but do give some sense of his accomplishments.

Herb Gross has over 50 years of teaching experience in diverse settings ranging from Central Prison's Death Row in Raleigh, North Carolina to MIT's Center for Advanced Engineering Study where he produced the critically acclaimed video course “Calculus Revisited". He has a long list of achievements, especially in the area of community college mathematics. Among these achievements are:

• From 1958 to 1968 he served as the founding Head of the Mathematics Department at Corning (NY) Community College. He became a pioneer in distance learning in 1959 when he taught a calculus course for high school students via Corning's Educational Television Network.

• In 1967 he became the founding president of the New York State Mathematics Association of Two Year Colleges; and in 1974 he became the founding president of the American Mathematics Association of Two Year Colleges (AMATYC).

• From 1968 to 1973 he served as the Senior Lecturer at the M. I. T. Center for Advanced Engineering Study, during which time he developed the “Calculus Revisited” program, consisting of a series of 83 video taped lectures and 17 volumes of study guides. To date, the program has been used by over 50,000 scientists, engineers and technicians on six continents to learn or to review the essentials of calculus.

• In 1973 he became the Founding Chairperson of the Mathematics Department at Bunker Hill Community College in Boston, Massachusetts, where he served until his retirement in May of 2003.

• In 1986 he was named by the American Association of Community College Trustees as the nation's Outstanding Community College Instructor.

• In 1988 he served as a special consultant to the Chancellor of the Minnesota Community College System, playing a role in helping the state establish a Center for Excellence in Teaching.

• Since his retirement from Bunker Hill Community College in 2003, he has been working diligently on developing his many works into a PowerPoint slide show format; while at the same time continuing to give workshops and other presentations.

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.

One 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:

(1) The Videotaped Lectures:
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:
While 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:
If 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 supplementary
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 you
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 extra help.