From some comments and suggestions on math and education I gather that some people think I have a somewhat negative attitude towards mathematics, so I’ll try to explain my opinion.

I really like math (and physics, my other subject) a lot. I liked it in school, and I like studying it at the uni.

Kris regrettet that I have not met a charismatic person who thought math to be the queen of sciences. This is not quite true. I didn’t encounter a professor at the uni who had this feeling (or who could show this feeling to us students), but in school I had a very good math teacher from eleventh through 13th grade. (Yes, in Germany we have 13 years of school – but only for students who want to continue studying at a university.)

He had studied math as a sciences, not in order to become a teacher, but he did a very good job! He was very enthusiastic about math and managed to pass this on to us students. I think he was able to get some things across better than other teachers because his own attitude towards math was different from other teachers. Since he had studied math for itself, not as one subject among others (as teachers-to-be do in Germany), he probably had a deeper insight than other teachers. At least, that was my impression.

So I really love math and think it is a very important (if not the most important ) subject at school. Otherwise I would not have chosen to become a math teacher.

In fact, I started studying only physics in order to be a physicist, but after two and a half years I found I was not so enthusiastic about that any more. So I thought the whole thing over and decided to become a teacher instead. And there was no doubt about the second subject I had to choose for that: mathematics!

This semester, I took a course about the relationship between religion an education in school. Most of the time, we studied texts of philosophers I couldn’t agree with. I was disappointed in the course because I had expected something more practical.

Anyway, I have to write an essay on a related subject in order to get my “Schein” (the paper which says I have successfully participated in the course). I went to the professor to talk about the subject of my essay, and when he heard I studied math, he suggested that I think about the reason why math is part of our education. That’s fine with me, and it seems to be a lot more useful for my future job than any other subject from the course could have been.

I know a lot of good reasons: Math is useful in everyday life. It is used in physics, computer science etc.. Math helps you to learn logic, reasoning, rational thinking, and last but not least it’s very interesting!

(I have difficulties in expressing myself in English (lack of practice and vocabulary), but I strongly agree with what Dave and others said.)

Unfortunately, what my professor wants is some ultimate reason beyond all that, something like “Since God created the universe to be infinite, you have to learn math in order to get an impression of what infinity really means.” (I admit, this is not a good example.) Otherwise, there would be no justification for math being taught at school.

The big question is: Will I find a reason like that, or will the “worldly reasons” have to suffice? For **me**, they do…

## Mathematik, das Universum und der ganze Rest

Aus einigen Kommentaren bezüglich der Vorbereitungen zu meiner Hausarbeit wird ersichtlich, daß der Eindruck entstanden ist, ich hätte keine hohe Meinung von der Mathematik. Ich werde versuchen, meine Einstellung hier deutlich zu machen.

Ich mag die Mathematik (und die Physik, mein anderes Fach) sehr. Beides machte mir schon in der Schule viel Spaß – es waren auch meine Leistungskurse -, und auch in der Uni studiere ich beides sehr gern.

Kris bedauerte, daß ich offenbar keinem charismatischen Professor begegnet bin, der Mathematik für die Königin der Wissenschaft hält. Das stimmt aber so nicht ganz. Ich habe zwar an der Uni keinen Prof kennengelernt, der diese Einstellung vermittelt hat, aber in der Schule hatte ich einen sehr guten Mathelehrer in der Oberstufe.

Er war Diplom-Mathematiker, kein “gelernter” Lehrer, aber seinen Job hat er sehr gut gemacht! Er war sehr enthusiastisch und konnte auch seine Begeisterung für die Mathematik an uns Schüler – zumindest an mich – ‘rüberbringen. Ich denke, er konnte einiges sogar noch besser als andere Mathelehrer klarmachen, da er die Mathematik um ihrer selbst willen studiert hat, nicht als eins von mehreren Fächern wie andere Lehrer. Vielleicht hatte er daher auch ein tieferes Verständnis von einigen Dingen; zumindest war das mein Eindruck. (Es läßt sich nicht abstreiten, daß man als Lehrer ein Fach nicht so intensiv studieren kann wie mit dem Ziel, in einem Fach Diplom zu machen.)

Ich bin also ein großer Fan von Mathematik und halte es auch für ein sehr wichtiges Unterrichtsfach (wenn nicht sogar das wichtigste ). Sonst würde ich wohl kein Mathelehrer werden.

Tatsächlich habe ich in den ersten fünf Semestern nur Physik studiert, mit dem Ziel, in dem Fach ein Diplom zu machen. Danach war ich aber davon überzeugt, daß das nicht das Richtige für mich ist… Ich habe also nochmal darüber nachgedacht und mich entschlossen, Lehrer zu werden. Keine Frage, was mein zweites Fach sein würde: Mathematik!

Dieses Semester habe ich im erziehungswissenschaftlichen Begleitstudium ein Seminar namens *Religion und Schulbildung* belegt. (Es gab nicht viel Auswahl, leider.) Die meiste Zeit wurde über Texte von verschiedenen Philosophen und Pädagogen diskutiert, mit denen ich nicht sehr viel anfangen konnte. (Einigen konnte ich auch überhaupt nicht zustimmen.) Im Großen und Ganzen war ich eher enttäuscht von dem Seminar, daß ich mir etwas weniger abgehoben, näher an der Praxis vorgestellt hatte. Dies sagte immerhin auch der Prof selbst, der wohl nicht damit gerechnet hat, in seinem Seminar mehr Philosophie- als Lehramtsstudenten sitzen zu haben!

Wie auch immer, für meinen Schein muß ich natürlich jetzt eine Hausarbeit schreiben. Ich ging also in die Sprechstunde, um das Thema mit dem Prof zu besprechen. Als er hörte, daß ich Mathe studiere, schlug er vor, daß ich mir Gedanken über die Gründe machen könnte, warum Mathematik zur Bildung gehört. Das Thema sagte mir natürlich viel mehr zu als alles, was direkt im Seminar besprochen worden war, und es erscheint mir auch viel nützlicher für meinen späteren Beruf. (Schließlich fragt sich wohl jeder Schüler mal, wozu er den ganzen Mist lernen muß… )

Mir fallen da viele gute Gründe ein: Mathe ist im täglichen Leben nützlich. Man braucht Mathe für Physik, Informatik und andere Fächer. Mathe schult das logische Denken, und zu guter Letzt ist es natürlich auch ein sehr interessantes Fach!

Dummerweise hätte der Prof jetzt gerne eine darüber noch hinausgehende, weltliche oder sogar außerweltliche (sein O-Ton) Begründung, etwas in der Richtung “Weil Gott das Universum unendlich groß gemacht hat, sollte man Mathe in der Schule haben, um eine Idee davon zu bekommen, was unendlich überhaupt heißt.” (Sehr flapsiges Beispiel.) Sollte es eine derartige Begründung nicht geben, hat nach Meinung des Profs das Unterrichtsfach Mathematik keine Daseinsberechtigung.

Die große Frage lautet jetzt: Werde ich einen solchen Grund finden, oder genügen die “weltlichen” Gründe?

Jeremy BowersOne of the things I’ve seen from the suggestions is that they still don’t explain why, say, an art major should still be forced to take math in order to have a complete education.

(This is really late for me and I really should be in bed, but I didn’t want to forget this. A pity that my best new-idea-generation time and my best idea-expression time don’t overlap at

all.)As background, I’m not a math major, I’m a computer science major (senior year), and a music minor. I’d love to take math classes until I had a double major, it’s just that if I did that for all the things I’d love to do that for, I’d never leave school. I really, really, enjoy math; I’m the only wacko in the comp. sci class I know who *thoroughly enjoyed* the mathematical underpinnings of computer science. (The final proof that there are problems that are incapable of being solved nearly took my breath away… you have no idea how hard it can be to suppress that reaction in public sometimes!)

In addition, I’ve been through the first two years of music theory, covering into about halfway to 3/4s of the way into the 19th century. This is the era of Mozart, of Beethoven, of those great musicians who many have heard of, but too few have heard.

The music of that era has

extremelyrigorous rules. Learning those rules gives you even more respect for people like Mozart, who dashed off (legend has it without any eraser in some cases, and my prof said that was fairly well backed up) entire symphonies that not only touch the heart and soul of the listener, but also meet theallthe criteria of this rigorous set of rules.The reason that I can place a time on the end of my education is that as time progressed, musicians began to feel constrained by the rules and found that there were more refinements and elaborations they could develop, using more dissonance and creating more exceptions to the old rules. My education, after the basics were laid, progressed basically chronologically, there being no better way to organize the material.

But my education

onlygoes up to 1850-75, and while I’m only a minor, there’s not very many other theory courses left. Music has dramatically changed since then… listen to some computer music pioneer’s piece from last year and you’ll probably wonder why they’re calling it “music”.Many of the students in the courses wonder why they are being forced to take this, especially the performance majors who never intend to write a song in their life. They are told, correctly, that it will improve their musicianship to understand the theory. They are also told, correctly, that they may someday have to arrange something, and it will be better to know how.

But the real reason, underlying both of those, is that for as far as music has traveled in the past 400 years, the roots are still the same. All but the most far-out of computer music is explained in terms of

how it relates to the old theory, and how it amplifies and corrects it, not how it breaks the old theory to pieces. (Much as Einstien amplified and modified Newton’s Laws of Motion, not destroyed them.) You cannot understand the new without understanding the old. You cannot understandmusicwithout understanding these underlying principles.Math isn’t about ‘numbers’. Math is about the world; that we can manipulate numbers to describe it is a happy coincidence (well, you know what I mean). Education, too, is about the world; it is to teach you easily what millions upon millions of people have learned and distilled down for you.

We teach mathematics to our children because mathematics is the only “division” in the standard hierarchy of “knowlege” (‘math’, ‘science’, ‘art’, ‘experience’, etc.) whose ‘intent’ is exactly parrelled in the intent of education: To teach about the world.

To someone approaching a 1940 symphony with an 1800 music knowlege, it does not always seem the basic rules of consonance and dissonance apply. In the end, further understanding and learning will show they are still there… and you shall be the better off for having had at least that 1800 level knowledge.

And with even the simplest understanding of mathematics, it will prepare you, not just for the ‘logic’ of creating programs to perform tasks, not just the ‘math’ of learning physics, but for the structure of human-created laws, the logical exercises of the philosophers, and the motivation of humans in psychology. How can it do all this? Because even when the underlying order is obscured, and even when we don’t understand the particular way in which the order is obscured, mathematics always at the very least gives us a handle on approaching the problem, a handle that is at the same time infinitely adaptable, yet not so infinitely weak that it is incapable of handling anything. Just as classical music theory gives a handle on approaching newer music, because the newer music always somehow reflects the fundamental principles articulated in the classical theory.

(There are far too many ‘philosophers’ running around who are unaware of the concept of “contradiction”… not just understanding it and rejecting, fundamentally unaware that such a thing exists. One presumes with a better mathematical education, this sort of thing might be avoided, and perhaps better philosophy would result. [If they’re going to reject something, they should at least understand what they are rejecting!])

We may not always understand the exact permutations involved in a particular problem, like what makes one painting beautiful and the next ugly, and we may never understand them, but it is our lack, not mathematics. Math still provides us a handle, even in this case that seems there is no math involved at all; we have the math of perspective, the math of light rays, mixing paints, describing shapes. It is a pity that so many math teachers truly believe that math is about numbers. Math

iseducationismath. Numbers are just a convenient and powerful special case. Other special cases include law, engineering, architecure, classical music theory… and, on one level or another, everything.(I knew that would get rambling…. now maybe if you ask nice, I’ll condense that down to the actually importent stuff after I get some sleep and some time to ponder it…)

(a final thought on the application of this idea: A good math teacher should teach some principle, I don’t know what, then hand out a problem to the class that they are

incapable of solving, and simply ask, “With what you already know about the principles of the universe, as expressed in mathematics, what can you tell me? How far can you get?” The exercise will probably start with some [apparently] inane answers like “The problem exists”… but it can proceed from there, and perhaps surprise both studentandteacher with how much information can be extracted from an unknown object using just the simplest of the ideals of mathematics.)Andrea FrickI knew that would get rambling…. now maybe if you ask nice, I’ll condense that down to the actually importent stuff after I get some sleep and some time to ponder it…Only if you think it’s necessary or useful. It sounds like you actually

wantto do that, though, right?!Thanks!

Jeremy BowersThat message was quite the train wreck! Again, I apologize in some sense for posting it, but I know from experience that if I were to go to bed at that point, the thought would simply vaporize. You’d never know, but I’d be disappointed :-)

The message had two main points, I think. One I can cut down quite a bit, and I think the other one is pretty much as condensed as it can reasonably be.

First, math isn’t about numbers; math is about the study of structure, relationships, and order. Numbers are a particularly potent and useful special case, where we can teach most of the principles, but that’s not the full extent of math. For example, consider a child’s geometry proof that two angles in a diagram are equal. No numbers need show up in the problem… and for that very reason, students everywhere hate those proofs :-)

Education is about the strucutre, relationships, and order inherent in the world, connecting various facts. Without math, you can study the facts, but you’ll get only a dim, fuzzy conceptualization of the relationships. Everybody agrees that an education that only encompasses facts is lacking.

We teach math because it is the only “branch” of knowledge that is completely congruent in goals to the goals of education. In some sense, the question is not “Why should we teach math?”, but “How on Earth can we possibly educate while somehow

avoidingmath?” and the answer is, “You can’t.” It’s a stronger statement then “Math is the foundation of several other discplines like logic and engineering.” I’m claiming “Mathiseducationismath.”The other bit about music I think stands. You can sum it up in one sentance, “Math helps you understand things even in domains where you wouldn’t expect it, such as visual arts, because even when it isn’t apparent, there is always some sort of mathematical handle you can get on the subject to enhance your understanding.” but to expect someone who doesn’t really understand math to buy that without a better explanation is probably expecting too much. Music theory is a great example anyhow; many people don’t even know such a thing exists (such as, say, nearly ever popular musician in my country), but it has exactly the same effects in music that math can have on your whole world. And it’s one of those fuzzy domains that the man-on-the-street probably thinks has no math involved whatsoever. Goodness knows the freshman music students thought the same thing… the musicians consider Theory their hardest class, because it’s as close as a musicians education gets to mathematics. Such a good example should not be stripped, methinks :-)

There… hope that’s a little cleaner.

Andrea FrickYou can sum it up in one sentance, “Math helps you understand things even in domains where you wouldn’t expect it, such as visual arts […] Music theory is a great example anyhow; many people don’t even know such a thing exists (such as, say, nearly ever popular musician in my country), but it has exactly the same effects in music that math can have on your whole world. […] the musicians consider Theory their hardest class, because it’s as close as a musicians education gets to mathematics.I totally agree with you. (I am not able to put it in words as well as you did, though, at least not in English.)

Music is a very good example. I also found out that music has to with math when I learned about music theory (harmony, for example) in school although the teacher didn’t mention the mathematic structures in music. I guess most students didn’t see these structures and thought the whole thing to be boring and useless, anyway.

A friend of me, who is a math and history teacher, also has to teach music at his school because they don’t have enough music teachers. You see: If you are a math teacher, you are fit to teach music too!

OliverA friend of me, who is a math and history teacher, also has to teach music at his school because they don’t have enoughmusic teachers. You see: If you are a math teacher, you are fit to teach music too!

Teaching is 80% psychology and 20% topical knowledge. I guess you could teach most subjects at least up to “abitur” level after a couple of years of teaching. Of course most teacher do not want to because of the extra work but as far as I know it is possible.

It is interesting to know that your maths assignment actually originates in the psychology department. Would you care to explain how that came about?

Cheers,

Oliver.

Andrea FrickI guess you could teach most subjects at least up to “abitur” level after a couple of years of teaching. Of course most teacher do not want to because of the extra work but as far as I know it is possible.By “possible”, do you mean “legally allowed”?

I would not like to teach geography or history, for example, because my knowledge of these subjects is restricted. It may exceed the knowledge of the students, but I would fear that I could not explain things or show connections well enough because I don’t know enough about them myself. Students loose respect for teachers easily if they get the impression that the teacher hardly knows anything himself. I’m not saying teachers ought to know

everything, but they should know enough to be able to answer even the more challenging questions of the students.And for obvious reasons I would not like to teach Chinese or Japanese… or sports.

It is interesting to know that your maths assignment actually originates in the psychology department. Would you care to explain how that came about?I never said it was a

mathassignment, did I?It was like this: I took this course about religion and education (I had no choice since it was the only one that was available). I have to write an essay on any subject that was related to the topics of the course, so I decided to write something about LER (

Lebensgestaltung, Ethik, Religionskunde) since it was the least horrible subject.I went to see the professor in order to ask for suggestions on literature. He happened to ask me what subjects I study, and I said, “physics and mathematics”.

Then

hesuggested I should write about why math is part of education. Criteria for deciding which subjects belong to education and are to be taught in school was one of the course’s topics, hence the question whymathis part of education.garret p vreelandi’ve been thinking about your challenge for a couple of days now. i keep visualizing the opening sequence of stanley kubrick’s ‘2001’, trying to figure out which developed first, language or numerical skills, and why.

this linkmay throw some interesting twists in your paper.a quote:

‘Ms. Brannon and Prof. Terrace believe that arithmetic and language evolved separately, and that number skills preceded human speech. “Language is a complex social skill, whereas counting can be learned by the individual,” Prof. Terrace said. “Counting is useful in foraging for food, assessing a group of predators or ordering the number of dominant males in one’s group.”‘

interesting! thanks for sharing your challenge with us. i’ve had a kick thinking about this.

btw, your professor’s comment about chess and ‘logic, reason and rational thinking’ got me a little crazy. if you ever get the chance, check out john von neumann’s manuscript, ‘games and economic theory’ and learn why

pokeris the game to be playing.Andrea Frickthanks for sharing your challenge with us. i’ve had a kick thinking about this.You’re welcome! When I wrote about my assignment here, I didn’t think that anyone would be so enthusiastic about this. I mean, if you listen to

mostpeople, the web is lots of fun, pictures, games and the like, and nobody thinks of pondering philosophical questions like mine, but it seems we’re notmostpeople here!Thanks for the link; I’m still looking around there…

garret p vreelandi love the occasional philosophical brain twister. i could easily justify language … but i feel i

should knowhow to justify math!i can’t remember the book, but the quote runs something like this:

“in the book of knowledge it is no less important to count the sands than it is to name the stars.”

i like that concept. good luck!

OliverI would not like to teach geography or history, for example, because my knowledge of these subjects is restricted. It may exceed the knowledge of the students, but I would fear that I could not explain things or show connections well enough because I don’t know enough about them myself.I think it is definitely a problem of todayÂ´s teachers education that a lot of confidence in being able to teach is based on “knowledge of these subjects”.

The future of learning will be quite different: A teacher learns together with his pupils. You see this happening at schools today: The kids know MUCH MORE about computers and the internet than their teachers. So the kids become teachers themselves. I think that is the way to go. Most teachers of course do have a problem admitting that they still can learn from their pupils.

OliverAnd for obvious reasons I would not like to teach Chinese or Japanese…Imagine a world where learning were no shame.

Obviously, you could not “teach” students how to speak Chinese, but you could teach them how to learn chinese and in the process learn it yourself.

Todays schools are “debuggers”, devices for finding and eliminating mistakes. That is why they fail at many levels. Schools should be about discovery and learning. A place to connect to the world. Not about being told your shortfalls.

Unfortunately that requires a high level of personal confidence on the teachers part. Having been thoroughly debugged themselves, most of them believe that being faultless is their only way to gain respect. My personal experience shows that young teachers need a long time to figure this out.

OliverOne last thing for now:

It would be fun to have a part about why maths SHOULDNÂ´T be part of the education.

Along the lines of: “I refuse to proof that I exist”, says god, “because proof denies faith and without faith, I am nothing.”

(from The HitchhikerÂ´s Guide)

Andrea FrickI think it is definitely a problem of todayÂ´s teachers education that a lot of confidence in being able to teach is based on “knowledge of these subjects”.I’m not saying that a teacher has to know everything. But I think that every subject should be taught by someone who has knowledge of it – if such a person is available. If not, as will surely be the case in the future, of course it is better to learn together with the students instead of just saying, “Sorry, we don’t have an expert for this, we can’t do this in school”.

The future of learning will be quite different: A teacher learns together with his pupils. You see this happening at schools today: Thekids know MUCH MORE about computers and the internet than their teachers. So the kids become teachers themselves. I think that is

the way to go.

Of course, this is true fur computers and the internet, but it’s not for most foreign languages, math, history.

But I agree with you that the most important thing that should be taught at school ist

how to learnbecause it is the skill you will need most throughout your life.Most teachers of course do have a problem admitting that they still can learn from their pupils.Is that your impression? I know a lot of teachers who have no problem admitting that they are not living dictionaries.

Andrea FrickIt would be fun to have a part about why mathsshouldn’tbe part of the education.I guess many students would come up with lots of good reasons!

OliverI know a lot of teachers who have no problem admitting that they are not living dictionaries.Since I live in Bavaria, it is possible that I am used to a different definition of “authority”. ;-)

It is hard to keep a discussion going if you agree on so many things… I must say I am happy to hear that my kids (if I should ever get some) will be tought by people with the right attitude!

Cheers,

Oliver.

OliverI guess many students would come up with lots of good reasons!Well, that is the idea. Then you go and argue against those reasons.

Cheers,

Oliver.

Andrea Frick(also in response to 49)

That reminds me that I wanted to write something else yesterday…

Imagine a world where learning were no shame.I hope this is not as unbelievable as you make it sound! When I got my Abitur (almost six years ago), a teacher as well as a parent pointed out in their speeches that this was not the end of learning but that we probably had to learn for the rest of our lives. The idea of life-long learning wasn’t so well-known back then, so I guess this came as a surprise to most of my classmates!

Unfortunately that requires a high level of personal confidence on the teachers part. Having been thoroughly debugged themselves, most of them believe that being faultless is their only way to gain respect. My personal experience shows that young teachers need a long time to figure this out.Have you ever thought about what the

parentswant the teachers to be? Or, for that matter, most people?I haven’t encountered this yet (since I haven’t even finish studying), but my father is a teacher, too, and I have experienced many times that people think teachers have to know everything!

Some expect teachers to know about just everything, while some others are so “realistic” to expect teachers to know only everything concerning their subjects.

So I guess not only will we have to change the schools, the things that are taught there and the ways they are taught, but we also the attitude towards school, knowledge, learning, and teachers.

Andrea FrickMy response is here.

garret p vreelandmathworldand a real good one …

mathematically correct.just happened to skim by these, and saved ’em.

gary thomsonIt has taken me a little time to find the following quote but I think that it gives one answer to Andrea’s essay question:

The advancement and perfection of mathematics are ultimately connected with the prosperity of the state. Napoleon Bonaparte

In other words the reason Mathematics is given such a central role in the curriculum is that it would be economic suicide to do otherwise.

I do not necessarily concur with this; Mathematics should be taught for it’s own sake, and without compulsion. However I am cynical enough to recognise that this is the most likely reason for it being given the importance it has in education.

Andrea FrickI like that one!

Could you tell me where you found this? I might want to use it in my essay, and I need a source (book or whatever) for it.

Thanks!