Sato

Hi Hakon,

I do not think your solicitation will be taken too keenly here; it is often seen as a bit "bad mannered" to enter into a community (forum) and immediately request something from the members. The moderators will likely remove your link for this reason, but please do stick around for a while and you may receive other support. There are several professional physicists on here, so if in your ventures you have some problems, you can post them on SFN and expect responses from those resident experts.

Also, to your fundraising effort itself, it would be helpful if you provided more information on the Indiegogo campaign, to make potential funders more confident in the project. Can you somehow verify your knowledge and experience? Maybe list previous patents, publications, or even typed up notes? Or if you've worked in private labs, relay your position? There's been a good bunch of previous work on cold fusion; how do your ideas relate to those? Cold fusion is largely unfeasible not because the experiments fail, but because the physics of producing such a system don't work out practically. If you had something, you could model it mathematically, which doesn't require much capital. Could you outline what you need the $15,000 for? Access to a supercomputer for simulation? This information would greatly increase the chances of a successful campaign.

Several years ago I spent the summer working on something at Princeton. One day, after taking lunch in the eatery, a friend from the maths department pointed out that John Nash was in view. I took interest, having learned about him a few months before, and approached an old man wearing an Aloha shirt. I began, "What's your name?". In a timid voice he returned, "Nash." I went on to note how I'd seen the film—which, I imagine, was undoubtedly something he'd gotten tired of hearing—and he muffled something about it in response. I proceeded to introduce myself and he inquired about the work I was doing, the professor I was working with, etc, and went on to discuss his then present research interests (Cosmology). Finally as I thanked him and took a photo for record, he asked something along the lines of "how many inches do you have left?". I stood confused, and then he gestured some clarification, that he was inquiring about my height. I said "maybe an inch", to which he countered, "I think two or three". I was 5'8" or 5'9" then, and today I'm 5'11"; it seems an apt testament to his mathematical prowess.

From another view, the professor I was working with was an undergraduate at Princeton while Nash was in his affected phase. He noted how he'd walk around the physics department building at night and scribble equations on the walls. He also expressed a more uncanny disposition that Nash no longer did much work and just wandered the campus, retaining his Senior Research Mathematician title. But I think it was well earned, and it seems that he would only go on for some years longer, grimly. I don't think I'll ever forget that small interaction, and in fact thought to myself just a few weeks ago, in a bout of introspection, that I better produce something interesting before Nash dies. Still grim, but without further ado, some light:

The term "hacker" is returning to its original meaning as of recent. Hackathons are becoming more popular in the public eye, especially with millennials like me and others who hadn't been earlier interested in the culture, and so the connotation is leaning towards "cyber-circumventer of problems, practitioner of programming". You see more people describing themselves as hackers with reference to that than you do otherwise; now it's "software engineer, hacker", "information security expert, hacker".

In any case, the game may not be intended to give a rigorous education on computer security, but rather to give an idea of the technical goings-ons behind "hacking" in a gammified context. Previous attempts are either entirely too stoic and technical for the general gamer population, or otherwise excessively inaccurate (a trade-off for ease of experience).

The game might be fun, but if it were at all realistic, it would probably have limited appeal. However, if you're learning game design or software engineering generally, it might be a fun side project regardless of its ultimate success. I would have said something similar about Minecraft before its release, though, so eh. You never know what will catch on.

 

As for the term "hacker," pavel was referring to the original meaning detailed here. Of course, particularly clever methods found for breaking security might justifiably be called hacks, but nowadays any 15-year-old who can work out how to install LOIC might be called a "hacker" by the media (including tech-oriented media) as well as by computing enthusiasts in general. I think that particular battle has been lost, anyway, so it's a minor point.

In infosec hacker lingo that's called a "skiddie" or script kiddie. Don't forget Havij and Cain And Able!

For anyone not keen to read the article, and unaware that this is April fools, it is a joke. Damn.

"Kenobi's seminal paper 'May the Force be with EU'"

Hey there,

I've only just seen this thread, and I imagine a good couple of others haven't yet, due to time-zone conflicts and other such things. This is an interesting topic, and it'd be likely to garner some discussion, but it's best to wait a day or two before bumping. I see you're new to SFN so I'd like to extend a warm welcome.

Modal logic, to my understanding, studies truth with respect to a statement's world(s) of interpretations; most commonly by "necessity", that is, it is true in every world, and "possibility", that is, it is true in some world. In non-paraconsistent (consistent?) logics, a contradiction in a theory results in every theorem being true, and this is so by the standard rules of inference of classical logic. Paraconsistent logics aim to serve as logical systems which do not "explode" upon a contradiction and so can maintain some sort of usefulness.

Here is a very relevant paper that may come in handy to you: http://sqig.math.ist.utl.pt/pub/MarcosJ/04-M-ModPar.pdf

Cheers,

Sato

I think it might be useful to refer to the answers in this, by the way, identical topic. The phrase "businessmen going on safari" evokes a memorably bizarre image.

From the definitions within Wikipedia, it is close but not really what I am talking about. What I am talking about is not allowing the connected devices to have direct access, but the requests sent to the device that is acting as the hotspot and the hotspot device will essentially download the webpage to itself and then send the information requested for to the user asking for it. It doesn't give the user access to the Internet access, simply access to the information. It would be limited, but possible.

I think you are not understanding. In any case of downloading (that means accessing a web page here), information is passed between devices and at some point stored on each one in the process, the particularities of which are those of the protocol and software implementing it. A system which does all this, but restricts the communication to HTTP and downloads the web page and then sends it to another device, is not different. It might help to give those articles a more thorough read; they apply here directly, esp with the implication that someone is not completely aware or condoning of the connection, but still so otherwise.

I'm tried and about to pass out, but hope this helped.

Satout.

It looks like you're thinking of piggybacking or tethering, depending on the relationship between the routing and dependant devices / users. What you described is essentially how most implementations work, and in fact, how "downloading" works (**note, your router is a computer**).

You can access your blog administration page by clicking on the "Dashboard" link situated on the grey navigation bar at the top left of most SFN blogs you visit (while logged in). You can create and manage your blog from there. There's no link to this on the main SFN site.

I wouldn't recommend thinking about it purely in terms of hours spent if you're not studying under the constraints of an academic degree; what matters is really how you spend your time. That is, you may spend a good chunk of that series of one-hour study intervals working on topics or problems which are 1) of no actual interest to you, 2) not actually necessary to achieve your goals, and/or 3) requisite of a continuous stream of several more hours of work. It would be important for you to find where your interests really lay first, and given by how you described your goals ("Quantum physics (Theoretical physics?)", "I want to learn 'pure mathematics' (if that's even a thing)"), I imagine you still have some work to do on that end.

Familiarize yourself with the broad strokes of the specific facets of those fields, and then choose what (even if vaguely) you want to pursue, and make sure you're not simply making decisions on what to spend your time on because of what appears most often in television, magazines, or otherwise culture as of high intellectual status. That is not to say that all of the fields you've expressed interest in studying aren't of great interest and profundity, which they are, but chances are that putting efforts towards clearer and possibly more particular goals will be more fruitful for you.

Pure mathematics (my favourite topic) is the study of abstractions, and to avoid any confusion, by abstractions I mean "things that exist purely in your understanding/imagination". For example, a common class of abstractions that we as humans access and experience almost invariably is that of shapes and spaces that have position and length and can be twisted and turned and manipulated, and those are which are studied under the mathematical field of "geometry". Another related abstraction, and similarly common, is that of shapes and spaces, but with a focus on properties that remain the same without regard to bending, moving, and twisting of them, like the idea of closeness, openness, connectedness, dimension, and continuous deformation (eg, imagine a coffee cup turning into a doughnut; they're essentially the same object under this interpretation), and this is the field of mathematics called "topology". Consider the idea of a collection of things; this abstraction is studied and formalized as "set theory" and is often considered in fact to be a foundation of the other topics, as a geometric or topological object can be considered to be a set of points endowed with some extra rules/meaning. Consider the idea of a collection of things, but combined with a something, call it *blop*, that relates all of those things to each other; like the collection of planets with a *blop* that represents a planetary collision, and where the results of that collision (a *blop* between planets) is a bunch of smaller or bigger planets, which are also part of that original collection. That's called an algebraic structure, the abstraction studied in the mathematical discipline of "algebra", which can be considered a set equipped with some operation *blop* that operates on its members; to appreciate its generality, note that whenever you see any sort of equation, it is really a structure of operations relating a collection of objects, for example, as the set of real numbers with addition, subtraction, multiplication, and division is really an algebraic structure called a "field". These are often combined to study the algebraic properties of shapes and spaces in algebraic topology and algebraic geometry, as well as the geometric properties of algebraic structures in geometric algebra. There are many other topics studied in pure mathematics, which generally correlate to different abstractions, but those are some of the most general / developed ones. I'd also like to give a mention to category theory, which essentially zooms out from any specifics and considers all of these different abstractions to be sets of objects related to each other, and characterizes them by finding valid relations between different abstractions / kinds of mathematical objects; by this, some ideas from mathematical logic (the mathematical study of the abstraction of reasoning), geometry, algebra, and topology, have been shown to be either, in a sense, the same, or very related—it's sort of a meta-algebra where the collection is that of all mathematical objects (at the highest level). Applied mathematics is the application of the results of pure mathematics to non-mathematical problems, as well as any results arisen from the applications themselves.

I am not familiar enough with theoretical physics to describe the field in such detail as I did pure maths, but a good and wide-scoped roadmap of studies is given by Gerard 't Hooft as linked to by Strange. Theoretical physics as you've shown interest in is really the mathematical modelling and thinking about physics without direct regard to experimentation; however, elements of experimentation, such as lasers and detectors, are often based upon theoretical developments too, like those original of electromagnetism. Cosmology studies the universe's development on a large scale, in its evolution from the dawn to now to so on; particle physics studies the properties and behaviour of the particles which compose matter (and light) and give substance to the universe, and is, to my knowledge, an essential component of cosmology; astrophysics studies the properties and behaviour of interesting objects or phenomena we observe in space but generally do not have access to, such as black holes, stars and supernovas, unexplained behaviours like those that lead to the idea of dark matter, the movement of galaxies, and so on, which, similarly to particle physics, is of significant importance in cosmology. Quantum mechanics is the name given to a large, well developed, evidently accurate, and apparently paradigm shifting with a spritz of philosophical implication, mathematical model describing much of particle physics; it also has a cool name, which was of course, the primary motivation for many researchers in the field! Mathematical physics, which one might consider a branch of theoretical physics, is the study of physical ideas directly as mathematical structures, where results are made by mathematical proofs and derivations (as mathematicians do); as opposed to just describing the physics by the application of mathematics, it is studied as mathematics itself, and is where string theory and M-theory belong, which are really just mathematical structures closely related and possibly giving rise to the mathematical descriptions that we gave to particles in particle physics. Note that I am far from learned in these topics and this is just a general understanding I've gathered from readings / the internet over a while, which other members like ajb may do better to clarify.

To study pure maths you can go from bottom-up foundations (like set theory) or pedagogically introductory material (like number theory, introducing maths by the study of numbers) or somewhere in the middle, and anyway which you (+ an advisor would be helpful) find best for yourself.

To study physics you can go by the pedagogically (and probably logically) sound method described in most university course pamphlets, though I've spoken to people who spoke of starting at a higher level being already familiar with much of the maths prerequisite. You can also study physics from a mathematical perspective, taking on theoretical physics from a view of pure maths, learning what you need to studying whatever specific facet of physics you're interested in that way.

In either situation, it would probably take at least a few hours a day, and maybe a few days of rest, to really grasp and work through all of the concepts, if you're looking to understand rather than just recite some definitions.

I know this is thick, but I hope it's of some use for your pursuits.

Putin is an idealogical left-over from the era of the KGB in the USSR and is a pernicious threat (but sometimes so blunt/obvious that it's mocking) to stability and freedom in his own country and others, especially in the recent case of Ukraine. It is also relevant to note here that Nemtsov was a physicist and engineer who, apparently, made several contributions to QM research and developments in acoustics.

He was, from what I've read, adamant for democracy and freedom (both of speech and action), and has put serious efforts to promulgating those ideals throughout his life, eventually and unfortunately beat out by the one who epitomized all that he had been fighting.

I wish well on Dr. Nemtsov's corpse, to, gracefully, Decay in Peace.

A line with length 0 is only a point; a "line" is defined as something with length, and so if it does not have length, it isn't a line. Similarly, a necessary condition for some object to be a circle (by its definition) is that it has a circumference, and if the diameter is 0, then so is the radius, in turn the circumference is nil, and so it is not a circle, just a point. A point is defined as an individual with no magnitude (besides maybe relative position), and so it cannot be stretched to infinity. You can, however, consider the distance between two points, or all the points equidistant from some particular point, and you will have a line or a circle.

I think it is for social / nurturing reasons, at least in my locality; this is the context of an upper middle class community. There is a trend of males going to business/management/finance majors and girls pursuing majors in psychology or speech pathology. There are many, though not as many as described in the previous sentence, students who pursue medicine, usually following their parents' paths; this entails usually majoring in chemistry or biology, and there are about as many girls as boys end up following this. There are pockets of poverty in my community as well, and I've noticed a trend of girls coming from these households commonly pursuing careers in cosmetology, art, or nursing (the boys often do not go to college, at least immediately).

I have spoken to some girls who show interest in STEM, at least in the secondary scholastic environment, and have asked them why they chose to pursue careers as clinicians rather than scientists; usually the rationale is that it is a more difficult and less stable venture to be a researcher, and they would like to have and support a family, so the primary reason of their up-to-then-and-onward pursuits in STEM is to eventually enter into a regarded medical program and acquire a job as a doctor or start a medical practice. Certainly, I have spoken to many guys who express the reason for their pursuing a similar major, or even CS, more explicitly as the fields' growing and their monetary potential, but what I guess could be called the maternal pull in females appears to be much more pervasive than the monetary focus of males (there are a good more few who follow purely their intellectual interests).

Both genders are generally equally capable of working, thinking, and reasoning to the extent required by most STEM disciplines, but it appears that there is a persistent, and possibly innate or very finely socially ingrained, quality among women that conflicts with their view of a STEM career path (that is, not including medicine). In fact, being acquainted with many young students / researchers in such fields, one who I respect among the most for their sheer intellectual ability and work is a girl. Coincidentally, that person, and another, the only girl in my school who's shown interest in purer STEM pursuits, are both androgynous in their demeanors, even in having an oddly deep voice or masculine hair, since I've known them.

To conclude, I think that females know they are able in STEM, and are familiar with the potential pursuits, but choose, based on common priorities, to follow other paths. I especially don't think it's fair to blame male scientists or educators for the trend, as many have done or implied so in this thread; at least, to no more extent the teachers, largely women, who in fact drive whole populations of students away from STEM, because teaching was a stable career path. Both sides drive both sides away and have deep issues in their culture.

In terms of the stature of their CS programs, I'm only familiar with the University of Michigan's. I was just there last weekend for MHacks, their biannual university hackathon, which is the largest hackathon I think there is. Since popularizing such events a few years ago (after which other schools soon followed), they appear to have received lots of funding, namely for their CS and engineering departments, and have so built great new facilities and established their CS/tech programs. One cool aspect of it is their [relatively] new maker center, which gives undergrads funding, work space, and supplies to work on projects, which, if I recall correctly, are later usually entered into competitions and sometimes turned into start up companies. Another thing, which you may not care much for however, is the researchers there, like Dragomir Radev, who is the founder of the NACLO (which you can probably take this coming January 29th), and a giant in the field of computational linguistics and natural language processing.

I did not apply, but several students from my school did, and I believe 6 or 8 are presently deferred, so it may be somewhat difficult to get accepted into.

I do hear it is expensive though, so if you can get a significantly better deal in either scholarships or financial aid at another school I'd recommend taking it.

It appears to be a silk headdress according to http://www.pastellists.com/Articles/Handmann.pdf.

 

Euler,

which has decorated postage stamps, is a good

example of the characteristic attention paid to

the silk headdress and striped robe. This interest

in accessories makes it unsurprising that he

turned to the 1714

Recueil de cent estampes

représentant différentes nations du Levant

(assembled

by M. de Ferriol, French ambassador to the

Porte) for the inspiration for Esther Mutach-

Steiger as la Sultane Reine.

I don't see the reason why there shouldn't exist a math domain where 2+2=5, for instance.

Someone posed a similar question earlier, and I tried to elaborate here; maybe that will be of use. If you redefine the addition operation or the symbols '2' and '5' and so on you might arrive at the result "2+2=5", but in our system of natural numbers base-10 and their arithmetic, applying 2 and 2 to the addition operation, 2+2, or +(2,2), will result in 4. This is because we define 2 as the second successor of 0, s(s(0)), and applied to itself is s(s(s(s(0)))), which is symbolized short-hand as '4'. That's the basic structure of what we call the natural numbers, however you define the symbols and all.

The math as we know now is built on a number of self-evident axioms. The problem with these axioms is that they are seemingly correct with our everyday experience. However, that does not grant these axioms any immunity or validity. If we are as small as an electron, I’d bet we will be accustomed to a completely new type of math, a math where calculus is the heredity.

Our abstractions (of which the mathematical objects that we study are a subset) are extrapolations from our perceptions, so how we present or interpret the different facets of mathematics is deeply influenced by our vantage as humans. And that's alright, because mathematics is the generalization, formalization, and study of our abstractions, and so we don't need to be concerned with what some other sentiences in some other circumstances would conceive of; unless for some reason it would be fruitful for our own use and application. Nonetheless, we do have different schools of thought and proposals for foundations of mathematics; from some who deprecate of the real numbers (Wildberger) to some who think/thought all mathematics could be derived from formal systems (Hilbert) to those who support types as a foundation of mathematics and do away with sets in that respect (HoTT movement). An excerpt from a blog post I read recently may interest you:

 

...Now imagine that this hypothetical industrial society also skipped the hunter-gather phase of development. That’s the period that gave birth to counting and natural numbers. I know it’s a stretch of imagination worthy a nerdy science fiction novel, but think of a society that would evolve from industrial robots if they were abandoned by humanity in a distant star system. Such a society could discover natural numbers by studying the topology of manifolds that are solutions to n-dimensional equations. The number of holes in a manifold is always a natural number. You can’t have half a hole!

 

Instead of counting apples (or metal bolts) they would consider the homotopy of the two-apple space: Not all points in that space can be connected by continuous paths

,,,

P.S. Your idea that mathematics can exist as and be founded on other forms is sound, but sometimes you throw out hokum like "...nothing more than hubris, a distinctive characteristic of mammals.", "It certainly can reduce computation complexity", "In the realm of speed of light world, we do not have addition either"; remarks that are either incoherent, unsubstantiated, or just wrong. Unnecessary stuff like that plagues your posts and dilutes the discussion. I think it would be best for you to look into and develop your ideas further first, but I hope this post assists you in someway anyway.

I don't understand why 2+4=6 is considered false, jibba jabba.

2 is 1 and 1. 4 is 1 and 1 and 1 and 1. 6 is 1 and 1 and 1 and 1 and 1 and 1.

It is just as logical as 2+2=4.

The natural numbers with addition is an algebraic structure with a formal definition, the numbers defined as 0 and successors of 0 (so, 1 is the easy-to-read symbol for s(0), 2 is for s(s(0)), and so on). According to that system, the application of 2 and 2 to the addition function—that is, +(s(s(0)), s(s(0)))—will yield s(s(s(s(0)))) or 4. This is something that is simply understood as counting, but the point is that there is a rigorous, logical system that asserts 2 + 2 = 4, as well as 2 + 4 = 6.

Edit: Apologies, I misunderstood your post, which I skimmed initially, as saying that "2 is 1 and 4 is 6 and 6 is 1 and etc, is just as logical as 2+2=4".

I think you are misunderstanding the mental ability needed to do mathematics and physics, and other such intellectual work; most people have sufficient memory, concentration, and visual-spatial skills to understand and practice even the highest level (more aptly, most obscure) maths and physics.

The distinction, I think, between who appears to be a "genius" and otherwise, is that to excel one needs to spend lots of time studying the necessary material and understanding it. I've also read some studies that purport that people, given a set of tasks requiring intellectual activity, performed noticeably better after being shown a comedy routine and laughing, and so it is reasonable to infer that a strong intrigue or passion with the subject of study can weigh strongly in one's creative ability in it.

That was in regard to your question about the place of cognitive psychology, but I do not believe that genetics plays much of a role, unless in the case of someone with a significant mental disability.

Also, the scientists you mentioned vary strongly in their mathematics and physics prowess and, besides being scientists, only share one common quality: fame. (Albeit maybe with the exception of Dirac). Einstein derived his fame from his contributions to physics in his pioneering relativity and quantum mechanics; Hawking derived his fame from his stature as a figure of perseverance in the way of great physical adversity; Kaku did so by way of his science popularization and books (as did Hawking partially). Although each of them of course studied physics and made some contribution, it appears that this was the primary method of your selection; there are many mathematicians and physicists who have been just as or more productive in obscurity.

I've been using Academia.edu for a while now and know that many of my research-keen friends also use it, and often papers that I look for happen to have been posted there, so I am guessing it's pretty well known. However, I certainly don't find most papers I read via their website; I think that googling the paywalled papers' titles or a set of relevant keywords followed by the search modifier inurl:.pdf yields the most results.

Hello,

I have a somewhat-working familiarity with set theory, and have recently been reading (naively) about category theory and type theory, each proposed as potential autonomous foundations for mathematics.

To someone who has a better-than-vague understanding of the three, what is the discrepancy or motivation for one over the other?

Why isn't first-order or higher order logic considered the "foundation"?

When considering set theory as a foundation, can all discussion generally be deferred to ZFC; for category theory, can it be deferred to the category Set or topoi; for type theory, can it be to homotopy/univalent type theory?

Thank you,

Sato

It's entirely possible, but more than just an interest or even passion for some particular topic in math and physics, you'll need discipline and motivation for whatever you're studying.

Around four years ago, I came on Scienceforums looking to learn theoretical physics. That is, the string theory and black hole physics and all that I'd been interested in and reading about at a layman level for some years. Before that, I'd love to read about it as much as I could, and write, summarizing what I'd read, heaps too. Then I starting speaking to someone here, a physics student, and declared that I wanted to build a "general relativity simulator". It was the start of summer, and he recommended that I learn calculus through multivariate. At this point I hadn't even completed Algebra 1 and had no motivations for the calculus besides that it was prerequisite to physics. The sources I studied from were great, and I was passionate about my goal, but because I at the time did not have any motivations for the topic or personal discipline, I wasn't able to do it, and it took a few years before I even got the proper mathematical motivations to learn calculus, and at that point my interests had moved on.

Again, you certainly can learn everything, but it will be difficult. There are certain topics, and further subtopics, like differentiation techniques and basic mechanics, that will seem mundane unless your learning source is extremely good at illustrating the motivations or you force yourself through them. That's one reason it is good to go to university though, because you will have the gained motivation of your peers and professors, and your intimacy with others studying (or teaching/researching) the topic will be much greater than if you're studying in isolation from your computer, possibly getting less out of the interactions. If you dedicate yourself to this, take everything here into account, make sure you're disciplined to get through much you might not particularly want to, and possibly reach out to your local college to see if any professors would be interested in helping/interacting with you.

I was ~14 and tried to work through a mathematically rigorous text "Introduction to Vectors and Matrices", and it took me a few hours to get through each page. More than just being ignorant to facts, I realized how little grasp I had on formalism and the idea of mathematics itself.

At the time I was primarily interested in theoretical physics and was browsing the physics/astronomy/cosmology section of the library, for the most part filled with pop-science books, and spotted a book a few feet away in the mathematics section. It had a cover that looked like a bunch of green numbers streaming down in vertical lines over a black space as well as the word "Matrix" in the title, so I took a look. I thought that the content, after skimming through the first few pages, looked interesting, at least to whatever extent I could appreciate it at the time (maybe the idea of grids of numbers sounded cool). Unfortunately that seemed to not be an introduction, so I found a book beside it with a similar title, and took a jab at it. And, well, the first paragraph of this post describes how that went.

The few PhD's I know personally who went into teaching went sixth form colleges or private schools. I only know one who decided that was the way forward during his PhD. I myself am associated with a sixth form college, part time. I think that just teaching would kill me, and especially in a typical high school in the UK. I have respect for those that have done it, but for sure they have more b*lls than I do. Now I have no idea about US high schools, what we see on the news is not encouraging, but the news always reports, quite naturally, extreme events and not the norm.

Ah, what do you do there then, if not teach? I thought you were affiliated with a research institution.

Teaching's not so bad, at least, I personally enjoy teaching whenever I find someone interested in topics I'm familiar with. Wouldn't you too, say, with a small group of undergrads who are interested in your area of algebraic geometry?

I agree though, if we're discussing such standardized/regulated teaching.

It's not always rape and school shootings, from what I've seen. Of course, there are some students who excel very well and are in the honors/gifted classes, usually going off to become doctors/lawyers, but for the most part they don't have much interest in the actual content. There are many more who don't care and spend much of their time partying and smoking marijuana. There are also some between the two ends of the spectrum. I imagine it's the same as in the UK.

 

This is a problem not just for mathematics.

 

But with specific regard to mathematics, all the teachers will have a degree in mathematics or some very close field and then went on to teacher training. I have only met one undergraduate who said he wanted to be a teacher in high school, that was in physics, and we both knew he would not make a good PhD student. I wonder how many actually enter undergraduate degrees with teaching at high school in mind?

 

I do not know the numbers, but how many of these teachers have only 3rd class degrees?

 

Those that are really bright, interested and hard working will not typically go into high school teaching. That said, there are always some Drs about in high schools, which I imagine fell into teaching rather than unemployment. Once there you get stuck.

 

I know that the Institute of Physics and the London Mathematical Society are offering money to more talented people to encourage them into teaching. This may help a little, and if there are fewer postdoc and research positions in the future your high school teacher maybe a disgruntled scientist who was let down by the system... a sure fire guarantee of a great teacher...

From my experience, over many years in a typical American high school district, it is a problem mainly in mathematics.

I recall one of them noted that they were a cosmetologist, and then rushed through school to get a degree in accounting and education, and got the job. Another, my algebra teacher, actually had a masters degree, I think. It would not be surprising if the degree wasn't in mathematics, but education.

We don't have any PhD's, except for one who recently completed his part-time in our social studies department.

I don't know if that remark was sarcasm, but I think it's true. If he was really talented and passionate about mathematics, but was simply let down by the system, then he'd be inventive enough to figure out a way to pursue research/high-academia-like-thing, even on school hours. Like, say... start a mathematics research program for the students and pitch it to the board of education as a way to get the district money and recognition in competitions. Like my school's science department did.

I am in the US, currently in secondary school, and many of the teachers seem to be interested in or care about what they teach, and do so to the extent that the curriculum (there is often a final exam that must be prepared for) or their class (eg, most English classes enjoy apathetic students, however interesting the book/topic at hand) will allow.

The one exception is mathematics. I have never come across a maths teacher who is interested in or cares about math. In fact, maths is the only class where at the beginning of the year, instead of presenting a motivation for the topic like is done in each other subject, they present information about the final exam that will be prepared for. Two years ago my geometry teacher introduced the class by angrily lamenting the difficulty of the class and how hard everyone will have to work to earn a good grade. Last year my algebra / trigonometry teacher did so by noting how much more intensive the work would be than for any previous year and how extremely important passing the final state-mandated exam was. When I brought my algebra 1 teacher an algebra question, she noted that this specific question was something that was to be studied two years later in algebra 2, and she could not help, and when I pushed, she deferred to an unhelpful class textbook. When I showed a proof I developed of the area of a circle (pi*r^2) to my geometry teacher, she laughed and said "you did circles in the third grade".

These people are like kids who were trained to do well in high school, focus on mathematics classes because they're the most straight forward/mechanical at that level, memorize a pedagogy, and repeat it to new students, with no regard to the actual content; the cycle goes on and more such teachers/students are produced.

I would not trust these people's teaching. They are presenting real mathematical results, but the rote and plug and chug method of teaching is something that could be harmful if gotten used to, even hindering an interest in maths.