1. The Role of REU in Mathematics Education
  2. The Changing Nature of Academia in U.S.
  3. What is Applied Mathematics?
  4. What is a Research University?
  5. Now that is a Really Good Teaching!

I wrote this document for myself in order to preserve from the lost some of my own thoughts about mathematics education that came to my mind during many years of teaching mathematics and through reflection on that experience.

1. The Role of REU in Mathematics Education

Research experience for undergraduates (REU for short) became a buzz word on many U.S. campuses in the past 10-15 years. So what really REU stands for? Can undergraduate students really conduct a genuine research in mathematics for example? Is there a real place for RUE in mathematics education?

Term REU is coined relatively recently but the concept is actually not that new. Most European universities include in their graduation requirements mandatory final diploma work which bares all characteristics of REU. The most noticeable characteristics of REU in mathematics are:

So is REU a real research? Can undergraduate students obtain a genuine research results in mathematics? For most part people (university administrators in particular) who think that undergraduate students can conduct a genuine research in mathematics are delusional. Only in a very rare circumstances exceptionally talented undergraduate students guided by some of the world's best research mathematicians can produce a genuine results. Most universities neither have such students nor employ such faculty. Why is that so? It is due to the nature of mathematics and the very sophisticated language which uses. Unlike other sciences (astronomy comes to mind) it is virtually impossible for any non-expert in the mathematics field even to ask a serious let alone an open question. Only after many years of training one can learn enough language to start asking serious questions. This is completely different situation than in astronomy for example where anybody can ask the question which can not be answered even by the greatest experts.

So, is there a role for the RUE in mathematics education? Yes! REU properly understood can greatly enhance the quality of education and improve students' training. However, it can also easily increase the cost of education.

2. The Changing Nature of Academia in U.S.

This essay is written in desire to document my own observation about rapidly changing nature of academia in U.S.

2.1. First Universities in U.S.

Harvard university founded in 1636 claims to be the oldest institution of higher education in U.S. This claim is logically incorrect since U.S. didn't exist as an independent country until 1776. At the Declaration of Independence Harvard university was definitely not the only institution of higher education in U.S. so logically it can be just one of the oldest institutions. Even more importantly U.S. significantly expended its territories in 19th century through various acquisitions and annexations like Louisiana Purchase, Texas Annexation, and Mexican Cession that U.S. of 1776 and 1905 have practically only name in common. Until mid 19 century American Universities were not the universities in the real sense of that world. They were essentially institutions for religious indoctrination of ruling elites. The first real university in U.S. the Johns Hopkins University became fully functional in the second half of 19th century.

The U.S. universities are true and the greatest benefactors of two great wars that were fought in 20 centry in Europe and they rose to its current prominence thanks to many favorable geopolitical and economic factors which shield U.S. from any destruction in two World Wars.

2.2. A Traditional Role of a University

A traditionally universities are divided in two groups according to their core mission. On one hand we have research university which have primary mission of discovering new knowledge through the research. An extreme example would be College de France. On another hand we have so called teaching universities which are also in business of knowledge transfer like a Sorbonne. Most of 2500 or so institutions of "higher education" in U.S. are misclassified and incorrectly refereed as universities while in reality they are confused about their identity. While most students see them as vocational schools a place where they receive some job related training faculty tend to see them as real universities due to their internal organization and psychological reasons. Namely most faculty have attended real universities.

Historically universities are the places where typically gentlemens (in historical terms women are allowed to study relatively recently) of means were educated and sometime did some research. Their education had nothing or little to do with a job training and future economic activity except in few specific fields like in the field of medicine or engineering where universities provides some initial job training. Studying at a traditional university is a full time occupation even though you will be ill advised to wait your retirement as a student. Attending the university also traditionally had nothing to do with upward social mobility. A phenomenon of using universities as a tool to move from one social class to another is very recent (past 100-150 years) and is related to the greater role of knowledge in the economy post industrial revolution and inability of higher class to provide sufficient number of scientists, engineers, and even doctors.

As the most faculty who were educated at traditional university one of the most frustrating things as a new faculty were students' expectations of me to provide some job related training and no education. My current employer Augusta State University is neither equipped nor properly stuffed for any meaningful job training. On the another hand most students come with expectations to leave the place with the same level education that they came with. On the top of it all they expect to "accomplish" that while working one sometime two minimal wage full time jobs to support themselves and often their families.

2.3. New Roles for the "Universities" in U.S.

If three traditional roles:

  1. education
  2. research
  3. field specific job related training

were not enough for U.S. "Universities" which were already dealing with confused identity the society have more recently pushed on us "new roles".

2.3.1. K-12 education quality control

The first and the most important new role for U.S. universities in the 21st century is the ultimate quality control for the failing K-12 education. The most often in recent times the university degree is not an indication of one's competence in the real subject but rather indication that a person is not functionally illiteral as many high school graduates.

2.3.2. Job training

To add an insult to injury over the past 20 years U.S. corporations have gradually phased out all job related training genuinely expecting from U.S. institution of "higher education" to pick up the tab and provide job training without additional cost to their future employees. It is needless to say that without additional funding and re-tooling U.S. institutions of "higher education" have been unable to meet this new expectations further alienating their core "customers" also known to some faculty as students.

2.3.3. Miscellaneous social services

Even more disturbingly past 2008 U.S. institutions have been heavily used to mask sky high unemployment rate among U.S. population. As a native of former Yugoslavia where this method was perfected by the late dictator and his hairs in the last century I was taken aback how easy was to sell this old trick supposedly to more sophisticated U.S. populations. At present time it is fairly common to encounter recently laid off middle aged women and men at U.S. college campuses who are supposedly getting job retraining even though we have never provided any job training to begin with. Once they wake up from that daydream they will find out themselves just in deeper debts than they had when there were laid off.

2.3.4. Mental health services

Finally as all of the above was not enough I as a college instructor deal with an increasing number of people who need counseling and special education. As someone who has been hardly trained just like most college professors to do anything beyond research and some transfer of knowledge in my core subject I am just not trained to council people who are attending college because "they recently got divorced" or students whose social and intellectual IQ requires medical help.

3. What is Applied Mathematics?

There is no such thing as applied mathematics. There are just applications of mathematics in other sciences. Actaully exceptionally broad range of its applications is one of the fundamental characteristic features of mathematics as observed great Russian mathematician A.D. Aleksandrov in his article A General View of Mathematics published in the first volume of "Mathematics Its Contents, Methods and Meaning".

To be continued!

4. What is a Research University?

This small essay was motivated by the recent "merger" of my employer Augusta State University with the Georgia Health Science University and lots of fluff talk about new R1 university. Since most regular faculty in both of those institutions hold terminal degrees from real R1 universities we were stunned by the amount of fluff talk that came out of mouths of university officials who were trying to sell Board of Regents mandated consolidation to students, stuff, and faculty.

Let me immediately state for the record that I am not against consolidation in-spite of the fact that my own job might be on the line in a very near future. The consolidation is creating lots of new opportunities for students, consolidate resources, and unties hands to university management to prune deadwood among faculty. However merger of a second rate (no offense to some first rate faculty) research medical school and a for all practical purpose junior college is no way to create a research university even if the state had financial recourse to sustain new institution which it doesn't have.

A pure R1 research university (no more than 40-45 Universities in U.S. could be classified in this category) is a university which is in business of creating new knowledge (doing research). Transfer of knowledge in particular to undergraduate students at a research university is considered a secondary activity if not just a distraction from the primary mission.

The faculty are recruited and promoted purely based on their research accomplishments. Since there has been some heavy misuse of the term research by new university officials as well I would like to clarify that research in academia mean creating new knowledge. It has nothing to do with grants, awards, even employment. One of the greatest recent research accomplishments in mathematics a proof of Thurston Geometric conjecture which in turn implies Topological Poincare Hypothesis in dimension 3 has been accomplished by Grisha Perelman who was officially unemployed for over ten years prior to posting his three ground braking papers on the Archive of Mathematics web site. There is a very wide margin of tolerance for poor not necessary bad teaching at a real research university. On another hand no amount of great teaching, service, even grants can save a job of a faculty whose research is not up to the standards. A hiring, promotions, and even department evaluations at R1 universities are done by external committees consisting of the greatest world experts in the field which are hired as consultants.

Creating and running a real R1 university is very expensive business. Only mightiest world nations can effort to have such institutions. A typical R1 university in U.S. cost no less than 1.5 billion dollars on annual basis excluding university hospitals, satellites, very specialized laboratories. Currently U.S. has over 50% of all world research universities. This percentage is likely to stay in the near future in-spite the fact that absolute number (40-45) is likely to shrink in the near future due to global economic crisis.

5. Now that is a Really Good Teaching!

I started writing this particular essay out of my frustration with the prevailing criteria used to evaluate instructional quality and ultimately my job performance as an instructor as well as a cumulative reflection on instructions I received as a students throughout my undergraduate and graduate studies in three different countries on two different continents.

5.1. Student's Evaluation: The Holy Grail of Teacher Evaluations

Currently in U.S. student's evaluations are used as the most important and sometimes as the only assessment tool to evaluate instructional quality by majority of institutions of "higher education". I am too young to remember when student's evaluations were introduced in U.S. academia and completely uninterested in motives that lead to the their adoption due to their current misuse as the assessment tool. However, I nor anyone else for that matter who is making leaving in part by teaching can not be indifferent or chose to ignore them. It is all too common for a job add to require a "proof of teaching effectiveness" from potential candidates which is usually a code phrase for asking applicants to share their student's evaluations. The hypocritical nature of student's evaluation has been best exposed in the Onion's article "Professor Deeply Hurt by Student's Evaluation" from April of 1996.

In their current form student's evaluation have no value as a teacher assessment tool. This claim is based on several empirical observations. Many of the questions commonly found on student's evaluation forms can be easily and completely accurately answered by the use of technology. For example teacher tardiness is easily identified by requiring instructional faculty to swipe their university ID cards upon entering and leaving classrooms. I used similar trick to ridicule the value of my students' answers to the question about prompt return of graded assignments to students including exams. Namely on the scale of 1.00-5.00 (5 being the best) my students have awarded me grade 4.2 for the prompt return of graded work. The only problem I had with that grade was that the course was hybrid (part on-line) where all assignments including all exams and the final were computer given. The students were receiving their scores immediately after submitting their answers. Any grade on that particular question except 5 was disingenuous.

5.2. Fundamentals of Teacher Evaluations

As a graduate student and a member of graduate student committee at the University of Arizona I was appalled by the fact that some of the best instructors (not necessary the most active researchers) were penalized in their student's evaluations just for upholding the most rigorous academic standards, having an audience unable to rise to their intellectual level, or merely for having certain types of personalities. Even more fundamentally there is a big disconnect how the faculty are hired, promoted, and instructional quality.

A typical research or wonna be research university is hiring faculty based upon their research promise. During the job interviewing process faculty usually give a colloquium lectures about their current research. It is very seldom that the job candidates are asked to teach a class and even if that was the case interview lecture performance could be hardly any indication of future teacher quality due to the reasons which will be explained in the third paragraph of this short essay. So the first thing is to clearly decouple research component from teaching of the job and weight the two and clearly define margins of tolerance. Even the Fields level mathematicians are not necessary the greatest and most comprehendible teachers in-spite of the fact that most of those whose lectures I had privilege attending came close to that level. On the another hand a second or even a third rate mathematician can be a great instructor and even a valuable asset to a top notch research university probably hired on the level of lecturer.

Any university below top 60-80 research universities in U.S. should carefully assess their needs before trying to hire a research trained mathematician and clearly define the fole for a new faculty.

5.3. A Look at the Foundations of Great Teaching

Good teaching is an expensive business. Let me repeat that! Good teaching is an expensive business. If you are a college administrator trying to hire contingent faculty and genuinely expecting great teaching you are delusional at best or fool at worst. It takes years of repetitive teaching a particular course and occasional teaching of related courses to develop course syllabus, assignments, projects, and various assessment tools. Even under the best of circumstances only few among us will rise to a prominence and reach the stellar level of teaching. By that time usually one will author one of the finest testbook(s) in the field and might not even need regular employment.

Five Short Essays on Mathematics Education - Aug/2012