- Role and purpose of mathematics education: NCF-2005 has envisioned that the main goal of mathematics education in schools is the mathematisation of the child’s thinking. Clarity of thought and pursuing assumptions to logical conclusions is central to the mathematical enterprise. There are many ways of thinking, and the kind of thinking one learns in mathematics is an ability to handle abstractions, and an approach to problem solving. Please comment on the level of importance you would attach to different stages of mathematics education and your views regarding its role and purpose(s).
Mathematical learning/teaching must deal with three components:
Development of the ability to carry out computations (often without worrying about the context). This is sometimes called Algebra.
Pinpointing one's doubts and analysing them through logical thinking. This is sometimes called Analysis.
Constructing new mathematical objects out of old ones and justifying these constructions. In school this is primarily in the context of Geometry but later on it occurs elsewhere as well.
All three aspects --- Algebra, Analysis and Geometry --- are essential to Mathematics. One neglects one or the other at one's own peril!
- Concerns regarding mathematics: NCF-2005 has identified the following core areas of concern: (a) A sense of fear and failure regarding mathematics among a majority of children, (b) A curriculum that disappoints both a talented minority as well as the non-participating majority at the same time, (c) Crude methods of assessment that encourage perception of mathematics as mechanical computation, and (d) Lack of teacher preparation and support in the teaching of mathematics. (e) The emphasis on procedural skills rather than on the understanding of mathematics (f) There are concerns about the type and quality of the mathematics education that students experience in schools. Systemic problems further aggravate the situation, in the sense that structures of social discrimination get reflected in mathematics education as well. Especially worth mentioning in this regard is the gender dimension, leading to a stereotype that boys are better at mathematics than girls. Keeping these concerns NCF-2005 is in action since last 6-7 years. We would welcome your views on these issues or other concerns that you may wish to raise. Also comments on the practical experiences/ concerns on achievement via NCF-2005 recommendations are invited.
The point (d) about lack of trained teachers is the most important concern. The primary difficulty is not the lack of Mathematical training on the part of the teacher, though that is certainly lacking in many cases. Rather, the problem is that the teacher fails to see the classroom as one where she/he too can learn. Thus, the teacher (and hence the student) fail to learn to learn.
The second most important point is the focus on assessment (c) which is seen as a punishment and reward system rather than as a feedback mechanism by which the teacher and student figure out what needs to be studied next.
The remaining problems stem from these two sources.
- Recent developments in mathematics education: In the past years, a revised mathematics curriculum has been implemented at different stages of schooling. NCF-2005 has recommended: (a) Shifting the focus of mathematics education from achieving ‘narrow’ goals to ‘higher’ goals, (b) Engaging every student with a sense of success, while at the same time offering conceptual challenges to the emerging mathematician, (c) Changing modes of assessment to examine students’ mathematization abilities rather than procedural knowledge, and (d) Enriching teachers with a variety of mathematical resources. The shift in focus NCF-2005 proposes is from mathematical content to mathematical learning environments, where a whole range of processes take precedence: formal problem solving, use of heuristics, estimation and approximation, optimisation, use of patterns, visualisation, representation, reasoning and proof, making connections, mathematical communication. Please comment on the impact of these changes and whether they go far enough to address the problems in mathematics that have been identified. Also comments on the practical experiences/ concerns on achievement via NCF-2005 recommendations are invited.
As mentioned above, no amount of twisting and turning with the curriculum and contents will have a significant impact unless such changes can help motivate the teachers better.
Also mentioned above is the need to retain the computational skill component of mathematics. One learns to ride a cycle before one learns how it is made and one learns to nurture and to grow plants in a garden well before one learns the chemical and biological processes behind agriculture. Of course, mathematics is distinct in that the process of computation can be analysed and re-built from the ground up. This does not mean that one should start with such a (de-)(re-)construction.
A bad teacher can teach computational skills (through hateful drill) which still will be useful, just as a bad programmer can still write ugly computer programs which work. However, one needs a good teacher if one is going to teach conceptualisation. While the suggested aspects like visualisation are important, one should ensure that one does not throw out the baby with the bath water.
- Current trends in mathematics education: A crucial implication of recommended shift lies in offering a multiplicity of approaches, procedures, solutions. NCF-2005 see this as crucial for liberating school mathematics from the tyranny of the one right answer, found by applying the one algorithm taught. Such learning environments invite participation, engage children, and offer a sense of success. Please comment on the relative merits / concerns of applicability of such approaches in Mathematics classes at different stages.
The algorithms that one learns have been developed through the ages. This does not make them sacrosanct, but it does make it important to learn to carry them out. For learning is, in good part, the gathering of one's ancestral wisdom. At a certain age, (usually around the time one enters high school) one learns to question this wisdom --- and that should be encouraged, but one can only do that if one has it on hand.
In mathematics, there is almost always only one "right" answer. However, the emphasis should be on learning from the process by which one obtained the answer, rather than reward for the right answer and punishment for the wrong one.
It would be wrong to give students a sense of ambiguity about the correctness of mathematics. An important aspect of Mathematics (to all users of Mathematics) is its sense of universal correctness. This becomes more nuanced as one studies it deeper, but there is no doubt (amongst its practitioners) that this is its primary purpose.
- Teaching of Mathematics:- A VISION STATEMENT: NCF-2005 envisioned that, school mathematics takes place in a situation where: • Children learn to enjoy mathematics: • Children learn important mathematics: • Children see mathematics as something to talk about, to communicate, to discuss among themselves, to work together on. • Children pose and solve meaningful problems: In school, mathematics is the domain, which formally addresses problem solving as a skill. • Children use abstractions to perceive relationships, to see structure, to reason about things, to argue the truth or falsity of statements. • Children understand the basic structure of mathematics: Arithmetic, algebra, geometry and trigonometry etc. • Teachers expect to engage every child in class. Please comment on these visionary statements and on the relationship between learning mathematics and these processes. Please comment on the relative merits / concerns of applicability of such approaches in Mathematics classes at different stages. We would also welcome your views on the students' learning intake and any suggestions you might have for improvement/ updating.
The kind of child-centred learning is difficult if there are a large number of students in each class. However, if the teacher involves students in teaching each other, a lot is possible. At the same time, the teacher needs to move from the role of giver (or imparter) and be more involved in being trained herself/himself. Adopting this role will help students see a live example of continuous learning and thus pick up the habit themselves Such a role will be difficult, if not impossible, in a traditional-minded society such as ours.
- Influence of the assessment: In terms of assessment, NCF-2005 recommends that Board examinations be restructured, so that the minimum eligibility for a State certificate be numeracy, reducing the instance of failure in mathematics. On the other hand, at the higher end, it recommends that examinations be more challenging, evaluating conceptual understanding and competence. Please give us your views on the assessment of mathematics. Also, please comment on the relative merits / concerns of applicability of such approaches in Mathematics classes at different stages.
One of the greatest ills that plagues our education system is that it is largely geared towards certification and eligibility. Given societal needs this aspect of education may be unavoidable and perhaps even necessary evil. We need to think about mechanisms to give the students something more.
The primary purpose of in-classroom evaluation is as a feedback mechanism that helps the student and the teacher improve themselves and move forward. Focussing on grades at this stage is definitely counter-productive. I believe that using this assessment as part of the final grade has reduced its utility for this reason. Thus internal evaluation should remain internal!
Coaching/drilling for certification examinations may need to be separated from this process of classroom learning. This is already happening through coaching classes. While it may be true that coaching only helps students pass the "test of fire" (and then feel "burnt out"), the drill and stamina development is not unimportant if carried out in moderation. In today's society, one does need to learn how to give examinations, appear for interviews etcetera.
- Syllabus levels and Curricular Choices: When it comes to curricular choices, NCF-2005 recommends moving away from the structure of tall and spindly education (where one concept builds on another, culminating in university mathematics), to a broader and well-rounded structure, with many topics “closer to the ground”. If accommodating processes like geometric visualisation can only be done by reducing content, NCF-2005 suggests that content be reduced rather than compromise on the former. Moreover, it suggests a principle of postponement: in general, if a theme can be offered with better motivation and applications at a later stage, wait for introducing it at that stage, rather than go for technical preparation without due motivation. As a practitioner, how you feel about these recommendations in current syllabus and its practices. Also, please comment on the issues, if any and on how they might be addressed within the current review.
At the shallow end of a pool one can learn not to fear water, and one can learn how to push it around to get an idea of the "theory" of swimming. However, one cannot learn swimming by wallowing in shallow waters!
Numerous examples of the above kind have already been provided before to say that there does not seem to be a reason to delay the development of skills until one has learned the concepts behind their operation. This does not mean that one should neglect analysis and construction. However, it is acceptable for the latter two to lag behind the pace at which one develops skills. Just like singing, dancing and playing football, it is easier to pick up a facility with numbers and symbols at an early age. It is only a few highly motivated individuals who show the courage and dedication required to learn these skills when they are beyond their adolescent years.
- Student achievement in mathematics: In previous questions considers the evidences of learning Mathematics. How effective, in your view, would each of the following measures be in improving the performance of students in mathematics assessment?
Remark: In question (i) below, this additional class time should not be used to introduce more material!
(i) allocation of more class time to mathematics
effective
(ii) better pre-service and inservice education for teachers of mathematics
very effective
(iii) improved mathematics textbooks and other learning resources
effective
(iv) provision of learning support for students who are experiencing difficulties with the subject
very effective
(v) provision of ‘general’ as well as ‘specialist’ mathematics courses
effective
(vi) increased emphasis in examination questions on the application of mathematics to real-world problems
not effective
(vii) the introduction of additional forms of assessment, such as coursework
not effective
(viii) improving the perception of mathematics among parents and the general public
very effective
- Teaching and learning in mathematics: What do you observe that currently Mathematics classrooms indicates that mathematics is taught and learned in a ‘traditional’ manner, mainly involving teacher exposition or demonstration of procedural skills and techniques for answering examination-type questions, followed by student practice of these techniques (in class or as homework) using similar questions. There appears to be little or no emphasis on students understanding the mathematics involved, or on its application in different or unfamiliar contexts. Please comment on the strengths and weaknesses of this approach. We would also welcome your views on change in teaching and learning approach in Mathematics, the degree to which syllabus change, assessment change, teacher professional development and support would contribute to bringing about changes in teaching and learning.
The current mathematics classrooms are limited, but what they are teaching is not unrelated to mathematical education. Thus the current approach needs to be supplemented with additional work on discussing the concepts, analysing doubts and attempting to construct new concepts. For this to work, the teacher must be also be encouraged to learn in the classroom along with her/his students. The teacher can learn to teach better and also learn to analyse and evaluate any mathematical ideas that are discussed.
A certain amount of change in the syllabus (say about 10%) is required to "keep up with the times". Even though Mathematics is eternal, tastes and utility of mathematical ideas changes with time.
In terms of assessment, it seems to me that involving internal assessment in the certification process (of big competitive examinations) is a failed experiment. It would be far better if the former is kept separate from the latter so that classroom assessment can serve its true goal of self-improvement.
- Attitudes to and beliefs about mathematics: NCF-2005 – and research papers on international trends in mathematics education – raises, on a number of occasions, issues surrounding the perceptions, attitudes and beliefs that exist in relation to mathematics, such as • the view that mathematics is a difficult subject • negative attitudes towards mathematics including, for some, a ‘fear’ of the subject • the perception and advocacy of mathematics, particularly Higher level mathematics, as an elite subject for only the ‘best’ students • research findings that suggest a connection between teachers’ views of mathematics and their approach to teaching it. We would welcome your views on these or other issues associated with mathematics.
Since I have never feared mathematics myself, it is difficult for me to pronounce judgement on why someone may fear or hate it. However, I did fear and hate Biology and History in school since these subjects seemed to rely excessively on memorisation and elaborate descriptions, in place of analysis and accurate summaries. I have since come to realise that my vision of these subjects was myopic. I cannot with certainty say that it was due to bad teaching either!
Perhaps it is pessimistic to say so, but it seems there will always be some students who will fear and hate mathematics (or any subject) just as there will be some who love it. This will be independent of the quality of their teachers, the curriculum, the contents or the books.
Just as it would be wrong to focus on the best students, it is wrong to focus on the weak ("no child left behind").
Our primary job is to ensure that the large (97%) that sits around the middle of the class does get a good mathematical education---algebra, analysis and geometry/construction in roughly equal parts.
- Other influences: The discussion paper draws attention to a range of other cross-cutting themes or issues that affect mathematics education in schools: • cultural issues related to the value of education in general and mathematics education in particular • equality issues (gender, uptake and achievement; socio-economic factors; educational disadvantage; students with disabilities or special educational needs) • recent developments in, and availability of, information and communications technology (ICT) in schools. Please comment on any of these issues, or on other factors that impact on mathematics education in schools.
To the extent that computers provide us with (yet another) example of the effectiveness of a mathematical approach, and to the extent that they can help us handle computations that cannot be done by hand, it would be good to involve them in mathematical learning.
At the same time, it must be said that excessive exposure to excellence and power can discourage. Just as it may be bad for a youngster to watch too much IPL during the hours when she/he should be playing cricket, it will hurt a young student of mathematics to use a computer to carry out calculations which are instructive to carry out "by hand".
Conclusion The purpose of this review is to map out the direction that must be taken in planning curriculum and assessment provision for mathematics education at different school stages in the years ahead while reviewing the NCF-2005. Please use the space below to make any additional comments on current issues in mathematics education or to give us your views regarding its future.
There is certainly scope for continuous improvement in the teaching of Mathematics. Some mathematical skills which were thought "advanced" 200 years ago could even be considered "essential" for the generations to come.
Elsewhere, I have pronounced that mathematics is "conscious" abstraction in the following sense. We make abstractions unconsciously all the time. However, these are often very individualised. In order to teach these abstractions to others we must understand our own thought process and go through the steps consciously. Only then can we turn these processes into theorems and algorithms ... and thus into mathematics.
]]>Academics and academia are themselves responsible for a lot of issues.
General whining: Academics tend to whine a lot. This does not (necessarily) mean that they are unhappy. A good part of academic training is learning how to spot mistakes. Thus an academic tends to find flaws all around her/him.
Teaching/learning: Research is not something you can teach in a classroom. It has to be done. As a result, research students are increasingly annoyed at their "teacher"'s inability to teach. Most students never make the transition to "learning" rather than "being taught".
Excluded Middle: While it is still true that the "genius!"-types continue to be found in academia, the middle tier of bright people are increasingly going elsewhere. This is not sustainable in the long run. There must be a "We are the 99%" movement for people to take control of their science/knowledge --- but who will lead it?!
Exponential growth: Research requires sustained mental growth. This is a steep curve and many feel like stepping off. There are careers where one can solve problems from day to day or week to week without worrying about becoming obsolete. Academia is not one of them.
Specialisation: Getting a PhD can be described as "becoming the worlds foremost expert on almost nothing". This can also be described as digging a very deep well which is only wide enough for one person (read under-nourished graduate student).
As a result of one or more of these, someone who completes a PhD thesis often feels disheartened. However, here are some things to look forward to:
General whining: People in academia remember (and embellish) their stories as a way of substantiating their whining. Many of these stories are entertaining and almost all are educational. This form of anecdotal learning about one's workplace has no parallels in the startup culture of today. Perhaps working for some of the dinosaurs like IBM, AT&T will be similar --- even those can't compare with 400 year-old oral histories.
Teaching/learning: There is no better place to learn a subject than in a classroom --- as a teacher. More seriously, preparing to teach a class or preparing exercises is one of the ways to learn something really well. As someone once said, you have not learned something properly until you have taught it.
Excluded Middle: "A cat may look at a king." Being in academics allows one to challenge and bring the lofty to earth. A "genius" may (and often does) ignore those who are not academics when they pose uncomfortable questions. A middle-level academic is not so easy to dismiss.
Exponential Growth: There is tremendous opportunity for doing "new stuff" in academia. As compared with any other career, it is easiest to justify spending time on "non-core" material in academics---like Alice, we have "to run very fast to stay in the same place".
Specialisation: If training for a PhD can be seen as training oneself to become a specialist, then there is no reason one cannot iterate this and become a specialist in many things. On the other hand, some others choose to "widen the well and let other people in"!
There is no doubt that academia needs to break out of its slumber, but who better than young, disgruntled PhD students to do so?!
]]>It is easy to dismiss these as cases of "burnout" or "frustration". It is equally easy to dismiss these dismissals(!) as "status-quo-ism" and "defense by the entrenched".
On the one hand, as a mathematician who does not depend on extensive funding like some other sciences, or on an army of graduate students to help me carry out my research, much of what is described has not been personally experienced by me. That said, there are issues raised in these articles --- like the problem of evaluation of "merit" --- that cut across all academia. Moreover, one has heard first-hand or second-hand accounts of research in a number of sciences which seems to mirror what is written --- about the "slavery" of PhD students, for example.
Confession time: I enjoy the entire experience of academic life --- right from the hours spent in the library cracking one's head over a one or two pages, and the hours spent in the laboratory tinkering with stuff to get it to work (of late this was more often about debugging code on a computer), and the hours spent discussing theory with colleagues and students, to being part of campus life in various other ways.
So when I hear people describing their experience with academics as being bitter, it is perplexing to say the least.
Are the problems that face academia in the nature of "bugs" that can be fixed by "re-factoring" or are they serious enough that one needs a "re-write"? Evolution or revolution? Most articles and posts that describe the problems are like honey-pots for those who would like to revolutionise or replace current practices. Phrases like "paradigm shift", "holistic research" and "relevance and utility" are bandied about.
On the other hand, senior academicians try to "explain away" the problems by saying that academia should be thought of as an industry and each university, institute or research centre as an enterprise. Any enterprise has its own social, economic and political strucures. The anarchic pursuit of knowledge, whether for its own sake or for the common good of mankind, may seem to be lost in personal or organisational goals that seem far more crass and mundane. The "defenders of the faith" point out that this no different from any other human activity that grows beyond a certain size. Some may also say that compared with other large human undertakings --- for example, the banking and finance sector --- the academic community offers more room for diversity.
It may very well be the case that the questions asked and solved by "big (money) science" cannot be unraveled by small, informal, essentially anarchic organisations. Can the lack of sufficiently many motivated and competent teachers, that currently plagues education, really be solved by MOOCs and "Khan Academy" clones?
Questions like these are difficult and each one of us will have to make choices about whether to "change the system from within" or start on a radical new approach. The first often looks like a compromise that will only strengthen existing institutions along with their known flaws, while the latter runs the risk of fading into irrelevance. Neither risk must prevent those who belive in these approaches from trying to correct existing wrongs.
]]>As adults, we are expected to take responsibility for our decisions and our actions. At the same time, we are social beings and depend on the work done by others to carry out our own tasks. Thus, we need to trust others if we are not to re-invent the wheel all the time. However, the decision of whom we choose to trust and to what extent is our own. We need to take responsibility for such decisions.
In reverse, when people repose their trust in us, what is our responsibility towards them? Primarily, we should be clear on what is being promised. For example, in the context of providing software, it is our responsibility to put a best effort to:
I don't see anything there about catering to some users' desire to dump their responsibility for their own machines on software developers.
If you want to be treated like a child, there are companies that will ``parent'' you. Be warned that you may have abandonment issues later unless you use the opportunity to grow up!
]]>Many of you seem to be confused about the role of classes/labs and examinations at IISER Mohali.^{1}
The above statement is based on the fact that almost all of you come for examinations and many of you miss a large number of classes/labs.
The purpose of examinations at IISER Mohali is for you to get a measure of how much you have learnt, and for the instructor to get a feedback on how much that has been taught has got across and what percentage of students have "got it". The marks and grades that result from examinations are a side effect!
To repeat, the grades that you get from examinations are not the primary purpose of the examinations.^{2}
If you do not attend classes or labs, then there is no feedback to the instructor --- since she/he can think "Of course, the material I taught has not got across. The students are not attending."
Each faculty member at IISER has subject expertise that gives a unique perspective on the subject that is being taught. Missing their lectures is a way to ensure that you do not gain their insight. Reading the same material from a textbook is not the same thing as attending lectures. (Anyone, anywhere can do that --- even sitting at an internet cafe downloading books and courses ... and perhaps even reading them!)
So attending classes (not just being in the audience!) is an integral part of this process. Asking questions of the instructor/tutors is an equally important part.
The way the curriculum and the programme are structured, the emphasis is on learning not on grades. A student who meets the minimum CPI^{3} requirements is given a degree after doing all the required courses; The degree does not reflect the CPI!^{4} This means that if you tried to learn in some courses and failed to learn, but on average you learnt something substantial from many courses, you will still get a degree.
In fact, if you learn something substantial, you get a chance to show that in your final-year project which is weighted at 32 credits -- twice than the weightage of all courses in your least favourite subject in the the first two years!
In summary, take an interest in learning stuff at IISER Mohali ... the rest will follow.
This mail is prompted by the low attendance in classes before and after the mid-semester examinations.↩
This is not to say that grades are irrelevant. However, learning can compensate for lower grades since you can demonstrate your skill, while grades can never compensate for something not learned.↩
Given that there are more grades (A,B,C) above 5 points than below, it is clearly "easier" to get above 5.0 than below 5.0.↩
Of course, the CPI will be on your grade card.↩
The core program at IISER is multi-disciplinary. There will be courses in Physics, Mathematics, Chemistry and Biology as well as some courses in Humanties and Social Sciences and few other inter-disciplinary courses as well during the first two years.
The INSPIRE and KVPY scholarships are provided by the DST as an incentive to students to take up studies in science. This means that students are expected to put in a "best effort" to learn what the program puts before them. "Scientists are those who face problems head-on rather than run away from them". In particular, students who do not put in such an effort ("bunk" labs and classes) should expect to lose their scholarship.
The course structure described in the courses of study booklet, is the norm. Students are permitted to drop courses during a particular semester and take them later; this way they volunatarily create a "backlog course". However, the complete course requirement of the core and each major must be met in order to graduate. They can take up to 3 years to complete the core programme and up to 7 years to complete the whole programme.
The graduation requirement is a CPI of 5.0 after completing all the requisite courses (core, core elective, major mandatory, major elective, open elective, seminar and project courses). Completing a course means obtaining a letter grade A,B,C,D,F in the course. (Note that a CPI of 4.0 must be maintained to stay in the programme.)
In particular, there is no "repeat" requirement attached to an F grade. However, a student who is on probation is often required to repeat courses in order to raise her/his CPI above the threshold of 4.0. Only a student with an F grade can repeat a course. While registering for a repeat course during a regular semester, a student should drop some other course of similar or greater credit weightage so that the total weightage does not exceed the prescribed weightage for the given semester. At most two courses can be registered for during the summer semester.
There is no concept of "promotion" from year to year in the BS-MS programme. Students who repeat courses or drop courses or drop semesters, will find themselves "behind" their batch in terms of the number of requisite credits obtained; such students have a "backlog" of courses to complete. Students are permitted to take up to 7 years to complete the 5-year program in order to complete "backlog" courses.
In order to assist such students in completing their program in time, summer courses are sometimes offered by faculty members during a specified 8 week period in summer. The offering of such courses is voluntary on the part of the faculty and enrollment by students in such courses is also voluntary on the part of the students. Moreover, only students with an F grade in a course may register for a summer course.
The course structure has a number of electives and a final-year project in two parts. Students must register for such courses by choosing an elective and providing their choice of possible project supervisors. However, such registration is subject to approval. The instructor of an elective course may ask a student to provide proof that they are capable of handling the course contents. This may include questions about courses they have already done. A faculty member may similarly ask searching questions of a student wishing to do a final-year project with her/him. Students may want to keep this in mind while acquiring F grades or dropping courses or semesters.
As an extreme example, suppose a to-be brilliant Mathematical Physicist joins the programme and gets A in all Math and Physics (non-lab) courses and an 'F' in all other courses. Let us calculate her CPI/SPI from semester to semester.
Semester I
BIO101:F, BIO111:F, CHM101:F, CHM111:F, PHY101:A, PHY111:F,
MTH101:A, IDC101:A, HSS101:F
CPI=SPI= 80/19 = 4.2
(IDC101 is programming which is mathematical enough!)
(Loses scholarship as CPI < 6.0.)
Semester II
BIO102:F, BIO112:F, CHM102:F, CHM112:F, PHY102:A, PHY112:F,
MTH102:A, IDC102:F, HSS102:F
SPI= 60/19 = 3.2; CPI = 140/38 = 3.7
(Gets a warning and is put on probation.)
Semester III
BIO201:F, BIO111:F, CHM201:F, CHM211:F, PHY201:A, PHY211:F,
MTH201:A, IDC201:A, IDC211:F
SPI= 80/18 = 4.4; CPI = 220/56 = 3.9
(IDC201 is Astronomy/Astrophysics which is math/physics enough!)
(Gets a termination letter and (say!) is allowed to continue
on probation.)
Semester IV
BIO202:F, BIO212:F, CHM202:F, CHM212:F, PHY202:A, PHY212:F,
MTH202:A, IDC204:A, HSS202:F
SPI= 80/19 = 4.2; CPI = 300/75 = 4.0
(IDC204 is Theory of Computation which is math enough!)
(Survives in the program (only just!) and chooses Math Major)
Semester V
MTH301:A, MTH302:A, MTH303:A, MTH304:A, PHY302: A, IDC351:A
SPI= 210/21 = 10.0; CPI = 510/96 = 5.3
Semester VI
MTH305:A, MTH306:A, MTH307:A, MTH308:A, IDC402: A, IDC352:A
SPI= 210/21 = 10.0; CPI = 720/117 = 6.2
(IDC402 is Non-linear dynamics which is math/phys enough!)
(Scholarship is restored as CPI has crossed 6.0!)
Semester VII
MTH401:A, MTH402:A, MTH416:A, MTH411:A, PHY622:A, PHY301:A
SPI= 250/25 = 10.0; CPI = 970/142 = 6.8
Semester VIII
MTH406:A, MTH407:A, MTH408:A, MTH410:A, PHY635:A,
SPI= 210/21 = 10.0; CPI = 1180/163 = 7.2
Semester IX
PRJ501:A, HSS302:F
SPI= 160/20 = 8.0; CPI = 1340/183 = 7.3
Semester X
PRJ502:A, HSS304:F
SPI= 160/20 = 8.0; CPI = 1500/203 = 7.4
The student graduates with Mathematics Major and enough Physics courses as well! (Note that the student has a total of 27 F grades!)
Can it be done with Mathematics Courses only? In other words, all the PHY/IDC open electives to be replaced by MTH electives and get F in PHY theory core as well. In that case, the CPI = (1500-4310-2*10)/203 = 6.7 (since the IDC201 course is ``too much physics'') which is enough to graduate. However, she won't escape the core years with successive CPI's of 50/19 = 2.6, 80/38 = 2.1 110/56 = 2.0, 160/75 = 2.1!
]]>Numbers play multiple roles:
For natural (finite) numbers, these different senses of the use of numbers coincide. (Though one can have some doubts when the numbers are really large!)
In the same way, mathematical objects can be considered in different ways:
Even for finite sets, these notions do not coincide! For example, there are many different finite groups of the same size.
Coming to the problem of "infinity". The simplest notions of infinity are:
Each of the above have associated arithmetic and algebraic operations. For example, with counting numbers we have addition and as a consequence multiplication. With ordinal numbers we have the notion of a successor which can be used to define a notion of addition. The corresponding structure in sets is that of Boolean or sigma algebra of sets. Category theory also has its own notion of algebra called ``universal algebra'', which is like (but not quite the same as) the sigma algebra of sets (infinite sums and products need to be defined and may not exist!).
So to re-phrase the question, we are asking if the ordinary notion of arithmetic and algebraic operations extends to infinity.
At first glance it does. We can certainly perform Boolean operations with infinite sets. The problem is that the usual statements about these operations are sometimes no longer true and our intuition about algebraic identities would fail us.
For example, it is usual to say that multiplication is the operation of repeated addition. When the number of additions is infinite, it is not very evident what this means. We define the product of sets AxB which clearly explains what this operation (multiplication) is for sets.
Similarly, it is natural to think of addition as repeated successor operations, but it is not always clear what this means for the infinite successor operation. Again, ordinal succession is defined in a way that such an operation is meaningful through the notion of limit ordinals.
However, in each case some "obvious" results from the finite case are no longer valid.
It is worthwhile to extend notions from the finite to the infinite when this is useful in giving us expectations regarding questions (about finite sets!) that we could not have arrived at otherwise. (For an interesting example, have a look at the Goodstein sequence.)
As an addendum, I would like to add the naive re-statement of Skolem-Lowenheim.
Since language consists of countably many sentences, we can only hope to define countably many things and from a practical point of view we can only define finitely many things.
Thus, infinity is a notion that mathematicians handle with care, limiting the roles that it can take, so that playing around with infinity gives meaningful (and correct!) results about the finitely many things that we will actually encounter! Mastering this way of handling infinity with care is what a lot of mathematical training is about.
An important aspect of science is to look for patterns in the data and use that to grasp some simple underlying principles on the basis of which that data can be organised, studied, etc. Such a search for patterns is what mathematics (in its broadest form) is! Hence, it is no surprise that I would like to underline its importance.
Note that we are looking for simple principles. How does one understand simplicity? Is "simplicity" in the eye of the beholder? Indeed it often is! By the time we reach our teens we are looking at the world through thick layers of glasses of preconceived ideas --- some of them put there by our teachers --- and sometimes simplicity involves removing some of these filters/lenses.
The purpose of teaching is also simplification. Some may say that our purpose is to pass on the accumulated knowledge suitably distilled. However, "distillation" is insufficient to arrive at a learning time exponential lower than the time taken to collect the knowledge (note that what we teach in 12-15 years is based on 3000 years of data collection), unless this distillation involves simplification in a central way.
In other words, one of the functions of a teacher is to simplify what the teacher already knows --- and one measure of simplicity is that it should take the learner less time than it took the teacher to learn the same thing!
The bottom line for teaching-researchers is this: Do not tell your students that they need to spend years to learn something since it took you that long --- instead, do some research and try to simplify the material! [*]
[*] | Students reading this should not automatically assume that this means that they can complain about long hours calculating and/or spent collecting data in a lab. There no short-cuts for acquiring skills! |
What if everyone believed that law-abiding citizens should use postcards for their mail? If a nonconformist tried to assert his privacy by using an envelope for his mail, it would draw suspicion. Perhaps the authorities would open his mail to see what he's hiding. Fortunately, we don't live in that kind of world, because everyone protects most of their mail with envelopes. So no one draws suspicion by asserting their privacy with an envelope. There's safety in numbers. Analogously, it would be nice if everyone routinely used encryption for all their email, innocent or not, so that no one drew suspicion by asserting their email privacy with encryption. Think of it as a form of solidarity.
In other words, encrypting mail is a form of solidarity with the person who needs to say something private and confidential. It is also a form of insurance for the time when you are that person.
Twenty years on, this is not as commonplace as Phil Zimmerman imagined. The primary reason is that the public key infrastructure (the web-of-trust) that the authors of PGP thought would lead to an exponential spread of the use of PGP (or tools like it) has failed to grow in the manner envisaged.
The (to my mind) no-so-important reasons cited for the lack of encrpyted mail are the lack of computational power and the lack of security of mail contents. It is true that encryption creates a small overhead. It is also true that encrypted mail does not ensure security in any absolute sense. (Absolute security is absolutely impossible!)
So I find it amazing that, in institutions where this infrastructure is already in place, there are sensible people who argue that we should not do it.
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