Mathematics and Critical Thinking, Part II

In a previous article, I began to tell the story of an unusual high school geometry course that took place at Ohio State University in the 1930s. The course was designed and taught by Professor Harold F. Fawcett, who later published an account in The nature of the evidence (NCTM, 13th Yearbook, Reprint 1995). To quote from a book,

Probably there never was a time in the history of American education when the development of critical and reflexive thought was not considered desirable from high school.

According to Fawcett, geometry was the most appropriate course in high school for teaching critical and reflexive thinking. He provides a decent selection of quotes to support his opinion and explain the source of his dissatisfaction with traditional courses. Traditionally

... the main focus is on the set of theorems that need to be studied, and not on method by which these theorems are established.

As a result

... there is no evidence that students who have studied demonstrative geometry are less gullible, more logical and more critical in their thinking than those who have not followed such a course.

A worthy result for students pursuing a geometric course is not only providing and learning a set of theorems, but also acquiring mental habits that save them from the junk in the behavior of life , Not only students should learn to prove a number of theorems, but also nature of evidence so their analytical ability could be transferred to to non-geometric situations. And how is this achieved? Fawcett quotes a common view.

The transfer will not occur if the material is not studied in connection with the field in which the transfer is required. ... The transfer is not performed automatically. We reap more than sow ...

Fawcett concludes that transmission is provided only by training for transmission , which explains the unconventional opening of its course (see Part I). He then reviews the methods and procedures that are appropriate for such a study. His treatment is so appropriate in modern discussions (taking into account his own logic and individual ways, group discussions, open approach, discovery and research) that Foucetta’s experiment and book deserve better knowledge among mathematics teachers. The point of the initial discussion was to establish the need for agreed definitions that seemed alien to the thoughts of the students For example, from the outside, all students agreed that "Abraham Lincoln spent very little time at school" and no one said that the truth of this statement depends on how the “school” is defined So, starting from the first meeting, students were forced to recognize the importance of definitions, and then the need for vague terms. They were taught to recognize the presence of implicit assumptions. even in the most elementary kinds of life.

Flelener was interviewed by Warren Matthews, a graduate of the course. Mathews & # 39; comments were,

I remember all our work with definitions. When I was the vice president at Hughes, and now in my work with my church, I understand how important the definitions are. It's amazing that when we can agree with our definitions, most conflicts end.

What Flener noticed

In the field of education, we probably argue ourselves more complicated, because we do not have the same definitions.

How correct! And how sad! An exception to the course for teachers of mathematics has no particular reason to feel in this regard. "In the field of education" should be considered as a common design.

But let's try to apply the basics of the course to the course itself. Was it right to design Fawcett? geometry ? What does studying during the school year by a group of students with one or more teachers have a geometric course? Can you think of a suitable definition?

How it shifts with Fawcett's remark [p. 102 of his book]?

While controlling a geometric object was not one of the main goals that students of class A should perform, it nevertheless seemed desirable to compare their achievements in this regard with those students who followed the usual course in geometry.

I think that the description of the “Critical Thinking Course with Applications to Geometry” well serves the purpose, materials and results of the Fosetta course. The transfer of skills occurred in the opposite direction to the declared goal! The Fawcett experiment very convincingly shows that the development of critical thinking skills helps students master mathematics, even if they do not have much sympathy for the subject.

At the end of the course, Fawcett interviewed students, parents. With their parents looking, the course helped 16 students improve their ability to think critically, but only 3 of a little more than 20 students learned to like mathematics.

So what?

In a 1997 article Is math needed? , Underwood Dudley argues that the answer to the question in the title of his report sounds No. He finishes the article with a pun,

Do you need math? No. But this is an enough

This may or may not be the case. But in any case, some things seem more satisfactory than others. (A discussion about what kind of mathematics might be sufficient could fit the requirements of the Fawcett geometry course.) We have just seen how critical thinking skills helped to learn mathematics. Flener too connects the success of the course with the fact that the students who took the course were veterans of a university school for three years and were used for open research.

The following is a more complete quote from the Dudley article:

Can you remember why you fell in love with math? In my opinion, this was not due to its utility in inventory management. Isn’t it because of the delight, the feeling of power and the satisfaction that it excites; theorems that aroused fear, or jubilation, or amazement; the wonder and glory of what I consider to be the highest intellectual achievement of the human race? Mathematics is more important than work. He surpasses them, they do not need them.

Do you need math? No. But this is an enough

Without a doubt, the mathematician Fawcett knew and could appreciate the glory and beauty of mathematics. He was an outstanding teacher and could, if he wanted, do the best work, passing on to his students the feeling of beauty and amazement that all mathematicians share. Apparently, he decided not to do that. His goal was to teach students, interacting with mathematics, critical and reflective thinking But the goals of education are many: the acquisition of useful skills, the development of local and global cultures, the development of innate potential. Course offers can and should be consistent with a variety of goals. Of course, the manner in which the planned and conducted mathematical course should be aimed at achieving a specific goal. There is no one correct way to teach and study mathematics.

Definitions are important. To allow cross-discussion, it is equally important to accept the possibility that the approach may be both right and good, like another, for the other end.