Мирка (inkogniton) wrote,


Как-то совершенно нет времени вздохнуть, то одно, то другое и всё одновременно и всё сразу. Я вздохну (обещаю в который раз сама себе) и обязательно запишу всё, что накопилось. А пока -- вот: эссе, посвящённое преподаванию. Это моё домашнее задание на тему инновационных техник преподавания. В названии эссе присутствует Сократ, потому назвать это инновационным несколько сложно, но тем не менее. Очень захотелось поделиться. Несмотря на то, что это практически из-под палки и скорее в тягость, чем в радость, в общем и целом, мне кажется получилось хорошо.

*эссе на английском*

Socratic dialogue in a large classroom


One of the challenges of a teacher is to engage the students in the learning process, and the challenge is particularly acute for university teachers which often face a classroom of many tens or even hundreds of students. The importance of engaging students is conceptualised by the notion of active learning, which, according to Wikipedia, is “a form of learning in which teaching strives to involve students in the learning process more directly than in other methods.” (Wikipedia, 2017).

The importance of active learning has been the subject of several monographs, such as the works of Meyers and Jones (1993) and Johnson, Johnson and Smith (1998). There is even a journal, published by SAGE Publications, devoted to “Active Learning in Higher Education”.

Here we shall not dwell on the subject of active learning in its generality, and content ourselves, as a working postulate, with a quote from the paper of the famous Hungarian mathematician George Pólya (1887–1985): “For efficient learning, the learner should discover by himself as large a fraction of the material to be learnt as feasible under the given circumstances” (Pólya, 1963).

Instead, we shall focus on a particular method to promote active learning: the Socratic dialogue. The term “Socratic dialogue”, coined 2500 years ago by the contemporaries of the Greek philosopher and teacher Socrates (470–399 BC), refers to the variety of teaching techniques in which the instructor forces the student to contemplate on a problem by asking questions, particularly of a kind that forces the student to reconsider the meaning of notions that he has previously taken for granted.

Its efficiency in one-to-one lessons has been demonstrated in myriads of experiments, the first ones having been conducted by Socrates himself.

It is much less clear whether the method retains its efficiency in large classrooms, such as the university classrooms that we usually face. I will argue in the positive, first recounting an experiment described in the aforecited work of Pólya, and then passing to my own experience.

To conclude this introduction, I would like to mention that the use of Socratic dialogue is not confined to the mathematical sciences: for example, the work of Rhee (2007) emphasises the usefulness of the Socratic dialogue and particular of Pólya's approach in law studies; the use of Socratic dialogue in medical and other studies is discussed, for example, in the article of Brooke (2006). I have also discussed variations of the method with colleagues in humanities (particularly, philosophy), where similar methods turn out to be useful.

In the current essay, I will restrict myself to the teaching of mathematics, which is the area of my competence.

What is the time at noon?

Pólya described a hypothetical dialogue which could be conducted with seventh-grade pupils. He tried versions of it in two classes of high school teachers.

The initial question was “What is the time at noon in San Francisco?” Not surprisingly, the students answered that noon is at twelve o'clock. The conversation (which I reproduce from (Pólya, 1963)) continued as follows:

— And what is the time at noon in Sacramento?

—Twelve o'clock — of course, not twelve o'clock midnight.

— And what is the time at noon in New York?

— Twelve o'clock.

— But I thought that San Francisco and New York do not have noon at the same time, and you say that both have noon at twelve o'clock!

After a sequence of follow-up questions, the students are led to formulate the notion of “astronomical” and “conventional” noon, and to formulate a well-posed problem: what is the West Standard Time when the sun reaches its highest point above San Francisco.

Pólya emphasises the importance of having the students formulate the problem themselves, rather than have it formulated by the teacher, and argues that this is one of the basic principles of active learning in mathematics.

Colouring an interval

I have tried several variations of the following Socratic dialogue in freshmen classrooms. I learned the basic idea many years ago from a colleague in Jerusalem; then I elaborated it and brought to the form described below.

The starting point is a mental experiment: I tell the students that I have drawn a unit interval [0,1], and I have coloured all the rational numbers inside it in blue, and all the irrational numbers — in yellow. Then I ask them, what is the colour of the drawing? Some of the students believed that they will see both colours, while others — that they will see a green mixture.

A first sequence of follow up questions leads the students to understand that if blue and yellow dots are mixed in even proportion and placed very close to one another (for example, the even pixels of a screen are coloured in yellow, and the odd ones — in blue), the eye sees the picture as green.

The next sequence of questions emphasises the importance of proportion: if blue and yellow dots are mixed in different proportion, the eye sees different shades of green. At this stage one can pass to infinite sets: instead of a finite array of dots, the screen is covered with small yellow and green spots (of the same intensity). The students realise that the crucial quantity is the proportion between the total area of the blue spots and the total area of the yellow spots. For one-dimensional picture with intervals instead of spots, the corresponding quantity is the proportion of the total length.

After these preparations, the students are able to pose the problem: what is the proportion of the length covered by the rational numbers in [0,1] and the length covered by the irrational numbers?

The continuation depends on the background of the students. Students that are familiar with the notion of cardinality of an infinite set can be led to realise that any countable set, and particularly the rationals, has zero length, and therefore the colour of the drawing will be yellow. Students that are not familiar with this notions can be led to a discussion of cardinality.

From my experience, this experimental lesson lasts about an hour, and is usually a great success.


The example I chose illustrates the use of Socratic dialogue in a university classroom. I have used versions of it several times, and found it a rather efficient way to lead students to a new notion (in this case, the notion of Lebesgue measure). An obvious drawback is that it takes much more time (with respect to the traditional methodic) to introduce the notion, and the required time increases with the size of the classroom. Still, I confidently conclude that even in the large modules of 100—150 students it is feasible to use the Socratic methodic for the introduction of one or two key notions in a term (with approximately an hour allocated for each).

The method is significantly flexible to accommodate students of various backgrounds and levels. In fact, the flexibility is an inherent feature of the method: the next question is constructed on the basis of the answers to the previous one. Accordingly, a significant amount of improvisation is required, and this is the main challenge faced in the application of the method. An additional challenge, specific to the larger classrooms, is to keep the discussion focused when several students present different answers. Both my experience and the discussion presented in the work of Pólya show that this is not impossible.

Among the numerous benefits of Socratic dialogue at large, as seen both in the example of Pólya and in my own, I will emphasise two. First, the students get an opportunity to formulate the problem themselves, rather than having it impressed on them by the instructor. In this way, the nuances in the setting of the problem and all the notions involved in it become clear to the students. This feature is essential in mathematics and in any other discipline involving a significant element of abstraction (such as law studies, as discussed in the aforecited work of Rhee (2007)). Second, the students get an opportunity to exercise critical reasoning, which is crucial in mathematics but also very important outside it.

These considerations lead us to the conclusion: Socratic dialogue, properly used, can be a very efficient teaching devise even in large classrooms.
Tags: математика
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