In the previous article I went to tell the story of the unusual high school geometry course look at Ohio State University in 1930. The course has been designed and taught Prof. Harold F. Fawcett later published account of The Proof (NCTM 13 Yearbook reprint 1995). To quote from the book,
There has probably never been a time in the history of American education as the development of critical and reflective thinking is not known as a desirable outcome college.
Given Fawcett is, geometry was the most suitable courses in college to teach critical and reflective thinking. He provides a respectable selection of quotes to support his opinion and to explain the origin of their dissatisfaction with traditional courses. Traditionally
… the main emphasis is placed on the shoulders of the sentence to be learning rather than on method these theorems are established.
As a result,
… there is little evidence to show that students who have studied visible geometry are less gullible, more logical and more decisive in their thinking than those who did not follow such a course.
The worthy outcome for students taking geometry courses is not only true and learn to put the sentence, but the acquisition of intellectual habits of save them from floundering in the implementation of . Not only students should learn the true number sentence, but also to realize character evidence , so that analysis technology can them could be transferred to non-geometric conditions. And how is this done? Fawcett cites prevailing View Point
No transfer will occur unless the material is learned in the course of the field to which the transfer is requested. … Transfer is not automatic. We harvest no more than we saw …
Fawcett concludes that transfer is secured only by training for transport , which explains the unconventional opening his course (see Part I). Next he discusses methods and procedures which are suitable for such a study. His treatment is so much relevant to contemporary debates today (minding its own logic children and unique way, group discussions, open-ended approach, detection and investigation) that attempt Fawcett and the book deserves to be better known among math teachers. The point of the opening of the debate was to assess the need for agreed-upon definitions seemed alien thinking of students . For example, at the beginning of all students agreed that “Abraham Lincoln was very little time in school” and no one raised the point that the truth of this statement depends on how “school” is defined . So, from the first meeting, students were led to recognize the importance of definitions and later needed some unspecified way. They were invited to recognize the existence of the implicit assumptions even in the most elementary activities of life.
Flener interview Warren Mathews, graduate courses. Comments Mathews’ was
I remember all of our work with the definitions. When I was the head of Hughes, and now my work with my church, I realize how important definitions. It is amazing when we can agree on our definitions of the most conflict ends.
As Flener word
In the field of education, the argument probably cross purposes more because we do not have the same definitions in mind.
How true! And how sad! Unless of course math teachers have no special reason to feel singled out in this regard. “In the field of education” should be considered as a generic name.
But let’s try to apply course descriptions basics of the course itself. Is it right to appoint an attempt Fawcett is geometry right ? What makes a school year long interaction between a group of students with one or more teachers a geometry course? Can you think of an appropriate definition?
How does it jibe with comment Fawcett’s [p. 102 of his book]?
Although control geometric content was not one of the main purpose to be accomplished by students of Class A, however, it seemed desirable to compare their achievements in this regard with the students who had followed the normal course geometry.
I think that the description “Critical Thinking Course with applications to geometry” serves the purpose, procedures and results of course Fawcett is. The skill transfer took place in the opposite direction of the stated goal! What experiment shows Fawcett is very convincing to the development of critical thinking helps students learn math, even when they feel no particular taste for the subject.
At the end of the course, Fawcett interview the parents. In view of their parents, of course, helped 16 students improve their ability to think critically, but only three of more than 20 students have learned to like math.
And then what?
In a 1997 article Is Mathematics Necessary? , Underwood Dudley argues that the answer to the question in the title of his paper is sound No . He ends the article with a pun,
is mathematics necessary? No. But it is enough.
This may or may not be so. But in any case, some things seem to be more abundant than others. (A discussion of what mathematics can be enough for, could have fit right in with Fawcett’s geometry course.) We saw just how critical thinking helped to learn math. Flener ties of the success of the course to students who took courses were veterans of the University for three years and was used to open ended research
Following is a more complete quote from the paper Dudley :.
Do you remember why you fell in love with math? It was not, I think, because of its usefulness in managing inventory. Was it not because of joy, a sense of power and the pleasure it gave; the theorems that inspired fear or jubilation or amazement; wonder and glory which I think is the supreme spiritual achievement of mankind is? Mathematics is more important than jobs. It started then it is not them.
Is mathematics necessary? No. But it is enough.
No doubt mathematician Fawcett knew about and could appreciate the splendor and beauty of mathematics. He was an excellent teacher and could, if he would do better things in their students this sense of beauty and wonder shared by all mathematicians. He chose clearly not. His goal was to teach students about relations with mathematics, critical and reflective thinking . But the aims of education are many: the acquisition of useful skills, the absorption of local and international culture, the development of innate potential. Programmes could and should be matched with a variety of objectives. It stands to reason that the way in which mathematics courses are organized and managed should aim at a specific target. There is no one right way to teach and learn math.
Definitions are important. To solve cross discussion purposes, it is also important to accept the possibility that approach can be just as, or as good as another -. Maybe the other end