Why Our Grading System is Important
Contrary to the belief of Arthur Lean, author of the article The Farce Called Grading, students are naturally, stupid dolts who must somehow be coerced, cajoled, persuaded, threatened, strong-armed into learning. Those few however, who are not, can go to college where an honor code is in place, such as New College of USF or Harvey Mudd, and the farce called grading will be of no encumbrance to the individual. The rest of the students need a system by which academic achievement can be measured. After observing many a high school student, it is apparent that without a system of scholastic comparison very few would strive to learn. What motivation would there be to read that extra page or two the night before the big examination? In the world of grading, the extra page could mean 20 points on a physics test. In the unrealistic world of Lean, however, that one page which could result in the absolute epiphany of the reader, could mean nothing, save the written reports of descriptive comments dependability, intelligence, and honesty.
Arthur Lean claims that it would be more beneficial to an employer to have written reports regarding certain character traits of job seekers. This, he states, is more helpful to the employer than say, a B-plus in college algebra. This idea is altogether untrue. Working as a computer technician, I was informed by my employer that the most advantageous part of my application was my advanced level of high school classes and standardized math test scores. Letters of recommendation were disregarded in his statement. In short, any job that requires high levels of thought and logic can be matched with individuals who present high test scores. Any Microsoft employer would quickly argue that evidence of strong computer programming (i.e. grades, original written programming code, previous jobs) are better indicators of expected performance than written letters of characteristic traits, or history of parent-teacher conferences. Arthur Lean is wrong in believing that grades are poor indicators of academic achievement. There is no scenario to support his view that grading is unfavorable to students, except for different professors issuing different grades. However, he failed to mention that grade curves in college classes do well to take care of this problem. Also, Lean easily slanders our process of student comparison, but gives no reasonable alternatives to grading.
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Why Discrete Math Is Important
by David Patrick
Most middle and high school math curricula follow a well-defined path:
Pre-algebra → Algebra 1 → Geometry → Algebra 2 → Trig / Precalculus → Calculus
Other middle and high schools prefer an "integrated" curriculum, wherein elements of algebra, geometry, and trigonometry are mixed together over a 3-year or 4-year sequence. However, both of these approaches generally lack a great deal of emphasis on discrete math. topics such as combinatorics, probability, number theory, set theory, logic, algorithms, and graph theory. Because discrete math does not figure prominently in most states' middle or high school "high-stakes" progress exams, and because it also does not figure prominently on college-admissions exams such as the SAT, it is often overlooked.
However, discrete math has become increasingly important in recent years, for a number of reasons:Discrete math is essential to college-level mathematics and beyond.
Discrete math—together with calculus and abstract algebra—is one of the core components of mathematics at the undergraduate level. Students who learn a significant quantity of discrete math before entering college will be at a significant advantage when taking undergraduate-level math courses.Discrete math is the mathematics of computing.
The mathematics of modern computer science is built almost entirely on discrete math, in particular combinatorics and graph theory. This means that in order to learn the fundamental algorithms used by computer programmers, students will need a solid background in these subjects. Indeed, at most universities, a undergraduate-level course in discrete mathematics is a required part of pursuing a computer science degree.Discrete math is very much "real world" mathematics.
Many students' complaints about traditional high school math—algebra, geometry, trigonometry, and the like—is "What is this good for?" The somewhat abstract nature of these subjects often turn off students. By contrast, discrete math, in particular counting and probability, allows students—even at the middle school level—to very quickly explore non-trivial "real world" problems that are challenging and interesting.Discrete math shows up on most middle and high school math contests.
Prominent math competitions such as MATHCOUNTS (at the middle school level) and the American Mathematics Competitions (at the high school level) feature discrete math questions as a significant portion of their contests. On harder high school contests, such as the AIME, the quantity of discrete math is even larger. Students that do not have a discrete math background will be at a significant disadvantage in these contests. In fact, one prominent MATHCOUNTS coach tells us that he spends nearly 50% of his preparation time with his students covering counting and probability topics, because of their importance in MATHCOUNTS contests.Discrete math teaches mathematical reasoning and proof techniques.
Algebra is often taught as a series of formulas and algorithms for students to memorize (for example, the quadratic formula, solving systems of linear equations by substitution, etc.), and geometry is often taught as a series of "definition-theorem-proof" exercises that are often done by rote (for example, the infamous "two-column proof"). While undoubtedly the subject matter being taught is important, the material (as least at the introductory level) does not lend itself to a great deal of creative mathematical thinking. By contrast, with discrete mathematics, students will be thinking flexibly and creatively right out of the box. There are relatively few formulas to memorize; rather, there are a number of fundamental concepts to be mastered and applied in many different ways.Discrete math is fun.
Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring. Rarely is this the case with most discrete math topics. When we ask students what their favorite topic is, most respond either "combinatorics" or "number theory." (When we ask them what their least favorite topic is, the overwhelming response is "geometry.") Simply put, most students find discrete math more fun than algebra or geometry.
We strongly recommend that, before students proceed beyond geometry, they invest some time learning elementary discrete math, in particular counting & probability and number theory. Students can start studying discrete math - for example, our books Introduction to Counting & Probability and Introduction to Number Theory - with very little algebra background.
Also see our article The Calculus Trap which discusses the pitfalls of rushing into calculus too quickly and/or with inadequate preparation.
There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in Mathematics? I understand that it is a research question, but may be somebody can hint me the direction to look on so that I would see sensibility of semigroups, if you see what I mean (so some replies like look for wikipedia are not working as they are anti-answers).
asked Feb 18 '11 at 18:09
Am slightly surprised no one has mentioned the Galvin-Glazer proof of Hindman's theorem via the existence of semigroup structure on $\beta<\mathbb N>$, the Stone-Cech compactification of the positive integers (see, for instance, part of this note by Hindman .
The relevance to the original question is that knowing that ``compact right topological semigroups have idempotents'' may sound recondite, but it is just what was needed to answer Galvin's original question about translation-invariant ultrafilters, which was itself motivated by a "concrete" question in additive combinatorics.
On a related note, while it is in general not possible to embed a locally compact group as a dense subgroup of something compact (the map from a group to its Bohr compactification need not be injective), you can always embed it densely into various semigroups equipped with topological structure that interacts with the semigroup action: there are various of these, perhaps the most common being the WAP-compactification and the LUC-compactification. Unfortunately this often says more about the complicated behaviour of compact semitopological semigroups (and their one-sided versions) than about anything true for all locally compact groups, but the compactifications are a useful resource in some problems in analysis, and the semigroup structure gives one some extra grip on how points in this compactification behave. (Disclaimer: this is rather off my own fields of core competence.)
answered Feb 25 '11 at 23:42
Victor, I don't understand your claim that $C^0$-semigroups aren't really semigroups. You are not free to decide for all the mathematical community what is a semigroup (I guess that you are interested only on discrete semigroups, aren't you ?).
$C^0$-semigroups are fundamental in PDEs (in probability too as mentioned by Steinhurst). The reason is that a lot of evolution PDEs (basically all parabolic ones, like the heat equation, or Navier-Stokes) can be solved only forward but not backward. In linear PDEs, this is a consequence of the Uniform Boundedness Principle (= Banach-Steinhaus Theorem). There is a nice theory relating operators and semigroups, the former being the generator of the latter. In the linear case, a fundamental result is the Hille-Yosida Theorem. Subsequent tools are Duhamel's principle and Trotter's formula. A part of the theory extends to nonlinear semigroups.
Edit. John B. expresses a doubt on the fundamental aspect of semigroups, compared with the evolution equations from which they arise. Let me say that semi-groups say much more, for the following reason. Evolutionary PDEs have classical solutions only when the initial data $u_0$ is smooth enough, typically when $u_0$ belongs to the so-called domain of the generator. This result can never be used to pass from a linear context to a non-linear one via the Duhamel's principle. In other words, in order to have a well-posed Cauchy-problem in Hadamard's sense, we need to invent a notion of weaker solutions ; this is where the semi-group theory comes into play.
answered Feb 19 '11 at 10:35
While you say that $C_0$ semigroups are fundamental, isn't it true that all can one can do with them can be done by using the evolution families that give rise to them? I would call these evolution families fundamental, but I have difficulties agreeing with the same for the consequent semigroups. – John B Dec 20 '15 at 23:01
An important application of semigroups and monoids is algebraic theory of formal languages, like regular languages of finite and infinite words or trees (one could argue this is more theoretical computer science than mathematics, but essentialy TCS is mathematics).
For example, regular languages can be characterized using finite state automata, but can also be described by homomorphisms into finite monoids. The algebraic approach simplifies many proofs (like determinization of Buchi automata for infinite words or proving that FO = LTL) and gives deeper insight into the structure of languages.
answered Feb 18 '11 at 19:03
Circuit complexity. See
Straubing, Howard, Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Birkhäuser Boston, Inc. Boston, MA, 1994.
If you want a research problem relating circuit complexity with (finite) semigroups, there are many in the book and papers by Straubing and others. See also Eilenberg and Schutzenberger (that is in addition to Pin's book mentioned in another answer) - about connections between finite semigroups and regular languages and automata.
answered Feb 18 '11 at 19:42
Though you say that $C_0$ semigroups are not really semigroups, the structure of compact semitopological semigroups plays an important role in the investigation of their asymptotic behaviour. For example, Glicksberg-DeLeeuw type decompositions or Tauberian theorems are obtained such a way, see Engel-Nagel: One-Parameter semigroups for Linear evolution Equations, Springer, 2000, Chapter V.2.
answered Feb 18 '11 at 20:39
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Why the study of Mathematics is important?
Mathematics is an important part of human culture. You can find it as a tool and language in every field of science like engineering, commerce, computing, physics, etc. The purpose of mathematics is to visualize the world around us. But why the study of mathematics is important? Mathematics has played a vital role in many aspects of modern life. For example, you may find it in every field like credit card details, modelling of instruments in civil and mechanical engineering, prediction of stock market in economics, decision making in capital budgets and space travel, etc.
The current society is becoming more and more technical and the requirement of mathematicians is increasing. Analytical skills plus quantitative skills is the ultimate result of mathematical modelling sought by most of the employers. When you acquire a degree in mathematics, you will actually acquiring a broad range of skills in logical reasoning, algorithms, problem solving and modelling techniques. This will lead to the exciting careers that are not only diverse, but also challenging.
This is because of the generic nature of the subject of mathematics that all industries are hiring mathematicians. They not only work in finance, business and government offices, but also education and management industry is also waiting for them. There are also a large proportion of students who use their bachelor’s degree in mathematics to acquire postgraduate and doctorate studies. Additional courses in the subject increases the chance of employment.
Mathematics is not only challenging but also an exciting subject. The excitement and the need of the subject can be seen in the research and development area. The proof of this development is the rapid growth rate. When you start working on any perspective like any real word problem, you will see new and new areas start to open. So, whatever you field of study
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