Accountability

How often have you heard this? The schools have to be accountable for our students' graduating. What does this mean? Most of the things I hear them say make me understand that this statement means the schools, i.e., the school boards, the school administrations, and, ultimately the teachers, must ensure that all students are graduated.

This is absurd if meant one way, but makes sense, if meant another way.

If it means that teachers make goals for their students in their classes, then it makes sense. If it means to ensure that the students understand and know these goals, then it makes sense. If it means the students understand that the nearer to the goal they get the better they do, then it makes sense. If it means that it is necessary the student achieves a certain proximity to this goal for him to be graduated, then it makes sense. If it means, with all of these things communicated and understood, that the student is accountable for his being graduated, it makes sense.

If it means that the teachers must do anything to graduate their students, then it is absurd. If there is no accountability by the student, and I will say that if the ultimate accountability is not the student's, then this notion being promulgated by our leaders, our parents, et al., is absurd.

Which one do we mean? I want to know.

I think, everyone would like the former to be what is meant. Nonetheless, I think the latter is what is in practice. I think there are many reasons for this, the most important of which is fear.

An extra thought: there is a reason that to be graduated in this context is a passive verb. The student's activity is to learn, to learn to a degree to be graduated from that degree to another. the student undergoes graduation, the teacher graduates.

Wonder

The most important thing in education is missing from education. The truth of this can be seen in fact that it is missing from almost all of us. This thing is, wonder.

Someone might think I am speaking dreams. For there is a similarity, in as much as dreams can be lost at some point between childhood and manhood. Some young boy or young girl might have a dream, but somewhere along the way to becoming a man or woman, their respective dream is lost by the person. There are plenty of books, plays, music, and movies about the loss of a dream, the holding on to a dream, the regaining of dream, the fulilling of a dream, etc. But this is not of which I speak. Dreams are about what someone wants to do. Wonder is about someone wanting to know.

Every child, it must be admitted, is born with it. Somewhere along way it is squelched. Wonder is the beginning, for wonder is in its nature a cause of a desire to enquire, and ultimatley to know.

Why is it, how is it, and when is it taken away from us?

Does it exist?

What is it?

How can we stop taking it away, having it taken away, and losing it? Also, for those of us that have lost it, how can we get it back?

I want to know.

Ninety Degrees?

Earlier I said I wanted to speak of mathematics and never got to it. I wish to speak about the whole art and science of mathematics. Mathematics is, in fine, a science, but it is also one of the arts of the quadrivium; this is why I say I wish to speak of the whole art and science.

I do not wish to speak of the whole in one sitting, so I will speak of it in parts by way of examples. I have questions, and I want to ask these questions of all, not only of mathematicians and of math teachers (for these are not necessarily the same; another example of this is that not all priests are theologians though all priest are teachers of theology). I hope many, lovers of math and its haters, come with other questions and answers. I hope that if my questions are not clear or not well asked, that someone will help me better ask them. I think these are questions that should be asked, for we are trying to educate our children. I only ask these to ask, are we doing it well.

I have taught mathematics. I truly enjoy the study of mathematics, the learning of it, and the teaching of it. Through my experience quite a few questions of those writers of textbooks and makers of curricula mathematica have come to mind.

Why is it, for example, in geometry, qua geometry, in the postulate for right-angles there is this number ninety, as in ninety degrees? Is there need for it?

Why ninety? Why not one? Why not one to four? Why not three-hundred sixty? Is it because Ptolemy (or someone else) divided the circle into three-hundred sixty parts, and the quarters of these are ninety? What does a circle have to do with a right-angle or any angle? Can I not know what an angle is without thinking about a circle? Etc.?

I ask these questions for a few reasons: my first is it is an example of other problems that come to mind when I have read and taught from these textbooks that our students use.

My second is that I can know what a right-angle is without knowing anything about number. Ninety, or any number for that matter, is not essential to right-angle. Number is not in the nature of right-angle. Angle is not number in its nature. It is true that every angle is a quantity, but not all quantity is number, for example, a line is a quantity, but a line can be two; three, one-hundred three, or some other number. No one number belongs to it. I do not have to cut up a line or apply some other cut-up-line (a ruler) to it to know that it is a quantity and to learn about its other properties as quantity.

My third is that the division of an angle (whether it be divided into two, three, ninety, etc.) is not a postulate but a proof*. There are postulates, and they do have a quality of a proof, but they are not proofs, and proofs are not postulates.

This brings up another two questions: why is it that modern textbooks of geometry call these things postulates that can and have been proved? And the last is, why do these books need so many? I know works of geometry that have fewer postulates but have done more in teaching this art and science than these new textbooks ever have. In fact these new textbooks would not exist were it not for these other works, and all these new textbooks do is make more difficult the learning of these works what was better said by these works.

I will leave this here. I think I have asked enough questions.

This question about the different kinds of quantity (number and magnitude) brings about another distinction that is lost in our speech today. I do not know how many times I hear people of all ilk, from the most uneducated to the most educated, say something such as, "there are less troops...", "there are less customers buying..." etc. "Less" is an adjective that strictly speaking belongs to magnitude, not number. What should be said is, "there are few troops...", "there are few customers buying..." etc. "Few" is the adjective that strictly speaking belongs to number. There are exceptions to this, e.g., the number ten is less than the number thirty-three. This exception is based on the fact that both of these numbers are being considered as one (like a whole, in the manner of a magnitude) and not as the multitude of units.

The point is that this loss of distinction in speech is a sign of the loss in mathematics of the distinction between number and magnitude. Dedekind aside, they are still different. There are still no odd and even lines. There are no prime lines and ones that are not prime. Every number is either even or odd. Every number is either prime or it is not. They are still not magnitudes and magnitudes are not numbers. Etc.

So I give another reason for my above question.

*Proof here is being used generally. There are two basic (some might say three with Q.E.I.'s): ones that show a things property and ones that show that a thing exists. The first usually ends in Q.E.D., the other with Q.E.F. Q.E.D. stands for, "quod erat demonstrandum", which was to be shown, and Q.E.F. stands for, "quod erat faciendum", which was to be done. Q.E.I., by the way, signifies a proof that is to find something; this to me is the same as showing something exists, so I do not make the distinction, though Sir Isaac Newton does. I am still thinking about it. Oh, Q.E.I., as one might have guessed, stands for, "quod erat inveniendum" which was to be found.

A Gift from a Friend

There are two things about which I wish to speak: one, a blessing and gift that a friend gave to me and; two, mathematics.

I recently have been catching up with a friend, to whom I have not spoken in some time. My friends are a blessing to me for many reasons, not the least of which is they do not always let me say something and pretend it was perfectly said.

Recently I received a critique, and a valid one, of what I have written in my first entry which was:

"...It can be interesting to find out who wrote Shakespeare's works. It can be
interesting to speculate about such things, but that it is all it is,
interesting. The works themselves stand alone..."

This could be read to imply: the authors are not important; their different points of view are not important; no author but the original can give a relevant and substantive point of view; and the author is not the source of the idea. These are a reading my friend gave to my words, and I suspect many others can and will do the the same.

Firstly, I do not give my friend's words here, for I have not permission to do so. I hope it will be fair if the paraphrase of them above suffices. If my friend wishes to write in this blog in open forum, my friend is welcome. Nonetheless, I will not introduce anyone to this or anything without their permission, out of respect and manners. There is one phrase in my friend's written thoughts that I will quote for it is a general phrase and nothing that could expose contempt. This phrase is: "the author of a work".

This phrase affords a distinction I do want to make. It is a distinction that needs to be made because my friend shows me that I did not make the distinction clearly in my first entry. The distinction is found in a phrase the above forces into being: "the work of an author". When I wrote: "what is not the aim of these thoughts is to speak about personality, mine or any one's", I was implying that I want to look at the work of the author, and thereby look at the different authors. I do not want to look at the personality of each author and thereby look at the work of each author. I want to look at the work of Aristotle, the work of Plato, the work of Shakespeare, etc. What I do not want to look at is Aristotle's personality, Plato's personality, Shakespeare's personality, etc.

If I look at the first, the work of the author, I can see the different points of view that each has about the same idea; I can see where they are the same, where they differ, and where they do not coincide at all; and I, from the ideas that offer comparison, can compare these ideas in the works.

If I look at the personalities of each or of any (granting something very difficult to grant and difficult to do: that I can know their personalities) I cannot make any true comparison of their ideas, for ideas do not have any direct connection or correlation with the respective personalities or characters of the authors.

Where my friend wrote that Shakespeare might look at the same idea that Aristotle and Plato have had, but look at it differently (no matter the cause of the difference) and suggested that it is the author that gives the "unoriginal" idea a new perspective, I agree. I do not argue and was not trying to make the argument that the author is not the source of the new perspective on the "unoriginal" idea. My argument was that it is not the author's personality that is a source of my understanding of the truth or falsity of the idea. I do not come to know the truth of the idea of reason given by Shakespeare, for example, because I found out he was effusive and bubbly, because he was sanguinary, because he was having an extra-marital affair with some woman, etc.

My true intent of this introduction was to try to say that these thoughts that I am wishing to introduce in this forum are not about me. I know that many use these sights on the web to talk about themselves, to show themselves in video (driving, brushing their teeth,or doing some other mundane, truly inane, quotidian activity), etc. Let this be so. But also let this author's blog (if it even deserves the term) not be this.

I wish to thank my friend for this: to make clearer my intent. Shakespeare, Aristotle, Plato, et al, are only names allowing me to speak about the general idea that this blog is a forum for ideas to be discussed and not a forum for ad hominum. It is popular today in the media for media stars and personalities to become the centre of attention. So called "reality TV" is aimed at this love affair with personality. It is this that I do not welcome here. I welcome disagreement among ideas. I welcome agreement among ideas. I welcome constructive arguments. I welcome contest, but, as in war, it should be done with honour. I wish to discuss education, as I have mentioned above, and the parts of education. I went on to give a point of view on grammar. I do not lay this out there as though it is definitive. I welcome disagreement and argument. I want to learn too. I put it out there as a welcome, not a welcome-mat, but a welcome and invitation to any who might be interested in learning with me.

Again, I thank my friend.

Grammar

Leaving this question for now, I wish to speak of two things: one is on grammar (a lost art); and the other is something I heard on the radio (NPR).

When teaching grammar to many students, I found it interesting how few of them knew their parts of speech. I am speaking of high school students. It seems as though the confusion brought about in university and in teaching colleges by these "new ways of thinking" have made it unnecessary and impossible to teach grammar. Realising that linguistics is the new lover (thank you Bloomfield and Chomsky), it must be realised that no one, other than the "experts", uses linguistics to judge what is spoken or written; no one at schools, at work, with whom you are speaking or in correspondence uses linguistics to say that you have an incomplete sentence, that your verb and noun do not agree in number, etc. They do use their knowledge of grammar.

Grammar is the art of making a sentence complete and fitting. It is an art that comes naturally from our natural ability to speak. Every language has a grammar. Not all grammars are the same, but all have one. A few things that each has in common is that all have parts of speech and way of arraying (syntax) these parts of speech into a and in a sentence. Every grammar has these two principles: parts of speech and syntax. It is from these that every grammarian (this is anyone judging the sentence making of someone) judges the sentences. Here is a notion that is lost: grammar firstly belongs to speech and secondly to the written sentence. So, no matter whether a people have a written alphabet or not, they have a grammar. This is also why the art of punctuation belongs to writing and not speech. The art of pausing belongs to speech.

This means that naturally all who speak have a beginning way to judge speech. What one does when learning grammar in school is give names and definitions to these natural principles and to use them knowingly when judging a sentence, rather than just instinctively.

There are three arts of speech (I might even say four, but some might argue the fourth belongs more to music than to speech): grammar, logic, and rhetoric. The first of these is essential to an educated person, for it is through this door that the other two are introduced and the rest of the arts and sciences are introduced. It is the beginning.

I exhort teachers to learn grammar again so they can teach it. You do not have know it as Martin of Denmark or Thomas of Erfurt knew grammar, but learn it. There is a wonderful book on English grammar (I have seen it in print again; this is due to its beautiful order and the knowledge of the author). The author's surname is Nesfield. Find it; bring it back into practice. Bring back the teaching of grammar, and you will bring back into being a population of educated students. Learning and teaching Latin and Greek would not do any harm.

Enough on this for now. The item that I heard on the radio is this. Scene: a street in Paris with Starbucks and other American like businesses, and then onto another street with a Parisian cafe, and a voice over telling us that the French artists and philosophers have been replaced by American and Japanese tourists.

My first thought: nothing has changed by the substitution.

A prooemium

My aim is to speak about something that is dear to me. Dear at one time meant expensive, with out peer, and one of a kind. There is no subject more dear to me than education, the educating of our children. Many years ago, the founder of the laws of Sparta, Lycurgus, said that the most important subject with which the state (and therefore statesman) must contend is the educating of the children of that state. Plutarch, in his Life of Lycurgus, said that he agreed with Lycurgus. I do not agree with how Lycurgus put into practice this principle, but I do agree with the principle. Being a constitutional democracy makes every citizen in some way a statesman; therefore every citizen is in some way responsible for thinking about education and putting into act measures so our children are educated.

What is not the aim of these thoughts is to speak about personality, mine or any one's. It can be interesting to find out who wrote Shakespeare's works. It can be interesting to speculate about such things, but that is all it is, interesting. The works themselves stand alone; this is because they are substantive without the author. Whether it be William Shakespeare, the the Earl of Oxford, Francis Bacon, et al., is not truly important. This is because the author's works, thoughts, and ideas live without him. This is because these thoughts found in the writings of Shakespeare are to be found inchoate, nascent, or mature elsewhere in his predecessors' works, e.g., the definition of reason found in Hamlet is to be found in Aristotle and Plato, and therein found with a reason for it being the definition. Therefore, the ideas did not and do not belong to Shakespeare. They were gifts. They were gifts to him from his predecessors. They were gifts to them from theirs. These ideas belong to all and should be looked into by all to the degree that one's capacity allows. So read Shakespeare, Aristotle, Plato, Confucius, Lao Tzu, et al., but read them for the ideas. Do not fall into the fallacious thought that these ideas are the man. Just ask, are they true?

To speak about the ideas is what the aim is. There are many parts of education about which one could speak: parents, students, teachers, classrooms, books, subjects, pay, buildings, administration, unions, libraries, computers, etc. All of these subjects are relevant to an overall understanding of the current state, good and bad, of our children's education. I wish all to be subject to discussion here. This being true, I wish it to be in due order. There is an order in learning and in discussions. These are not always the same order, but there is an order. I think the first thing to find out is whether there is an "education". This is for two reasons: one, a discussion about an infinite number of educations, as such, is impossible; and two, I know from first hand experience and from having read many things that there are many if not most that believe that there is no one definition of education.

Without now making any statements about these here, I simply wish to ask, is this true. Is education what anyone wishes to think it is? Therefore, is it true that there is no one definition, and therefore, no definition of education? For an example of the thought of there being no definition of education go to www.teachersmind.com/education.htm. There is an opening essay called The Meaning of Education. I only bring this as an example of which I speak. It is a good example of the idea that the definitions of education are personal, environmental, etc., and the definition is not a universal.

I hope anyone who wishes will come to this discussion.