Haroutune Dadourian - Make Mathematics Attractive

In August 1959 Haroutune Dadourian delivered the lecture "How to make mathematics more attractive. Some suggestions for the teacher who is attempting to help his students find mathematics as interesting as he does himself" to the Institute for Teachers and Professors of Mathematics, held at the University of Rhode Island. The lecture was published in The Mathematics Teacher 53 (7) (1960), 548-551. We give a version of the first part of the lecture.

How to make mathematics more attractive.

The reaction of the student to the study of mathematics depends on the student himself and the totality of his environment, past and present. However, in this discussion we are interested only in that part of the environment created by the presentation of the subject in textbooks and in the classroom.

After many years of classroom experience and observation, I ascribe the troubles some students have with mathematics to four sources:
  1. The average student's notion that the study of mathematics requires a special kind of talent, and that mathematics, beyond the elements of arithmetic, is of little or no value to him unless he is planning to become a scientist or an engineer. These two notions produce fear of the subject and apathy toward it.
  2. Failure to associate mathematics with experience and common sense, which leads to mystification and befuddlement.
  3. Lack of knowledge of how to study effectively, which results in waste of time and effort, and consequent discouragement.
  4. Disorderly ways of attempting to solve problems, which expose the student to blunders and errors.
The first of these sources of trouble is created mainly by the general public, including teachers in fields other than mathematics, engineering and the natural sciences. For the other sources, teachers 1 From a lecture delivered at the Institute for Teachers and Professors of Mathematics, held in August, 1959, at the University of Rhode Island. and textbooks of mathematics are largely responsible.

I want to make a few suggestions of general character and follow them up with specific suggestions which relate to different subjects.
  1. Approach the abstract through the concrete, with an appeal to the student's experience and common sense.
  2. Instead of introducing a new concept by a formal definition, use the definition to crystallize and epitomize the explanation of the concept.
  3. Avoid topics which do not fall in the main line of progress of the subject. Also, avoid topics which can be treated more effectively in a higher subject or which are untimely. Introducing the symbolism of symbolic logic into a first course in algebra constitutes a good example of bad timing.
  4. Do not underestimate or over estimate the student's capacity.
  5. Since problem solving forms the major part of the study of school and college mathematics, the following suggestions should be helpful:

    a) Do not require the student to solve a great many problems of the same type. While studying a mathematical topic or subject, the student has to make use of most of what he has learned before. This fact makes understanding of principles and methods of greater importance than the development of skills.

    b) Solve illustrative examples in such a way that the analysis forms a straight chain of reasoning, not a tangled mass of links.

    c) Solve numerical problems as literal problems first, and introduce numerical values after the required magnitudes have been expressed in terms of the symbols of the given.

    d) Discuss the expressions of the required magnitudes with the object of checking them, and of obtaining special or otherwise interesting cases.

    e) Impress upon the student that the object of solving a hypothetical problem is to carry out a chain of reasoning which leads to the answer in a logical and orderly way, and brings out the underlying principles - that the purpose is not merely to find the answer.

    f) Train the student to rephrase a verbal problem in its bare outlines and to form a strategy before starting the operation of solving it.

    g) Whenever possible, denote geometrical and physical magnitudes by single letters, preferably by the first letter of the name of the magnitude.

    h) In applied problems see to it that the terms of an equation have the same dimensions.
Arithmetic. Any program designed to make mathematics attractive should begin with arithmetic, for the student's reaction against mathematics starts in elementary school.

Three years ago the principal of an elementary school came to see me. He asked me if I could do something about a group of bright fifth and sixth graders who were two years behind in arithmetic compared with their work in English. I met with fourteen of these children, six times. During the first meeting it became perfectly evident that they were not doing well in arithmetic because they were bored. I was appalled to learn that arithmetic is taught for eight years, using eight different books, and after examining the texts used in the fifth and sixth grades, I was not at all surprised that they were bored.

I should like to make three suggestions which should help to make arithmetic more attractive:
  1. Do not require the solution of a great many single operation problems. The solving of such problems is comparable to learning how to button one's coat. Constant repetition is not necessary in either case.
  2. Introduce problems which involve observation and measurement, as well as counting.
  3. Introduce the rudiments of algebra into the lower grades, say the fifth grade.
The solution of an equation such as x+2=5x + 2 = 5 is no more difficult that that of the problem, "I hold five pennies in my hands; two of the pennies are in my left hand, how many are in my right hand?" In fact, this problem could be presented by actual pennies in the hands of the teacher as a preliminary to the solution of the corresponding equation.

During my sessions with the fifth and sixth graders, I gave a number of examples of the uses to which they could put their knowledge of mathematics. One of these was the determination of the height of a tall object, such as a flagpole, a tree, or a building, by measuring its shadow and that of a yardstick held upright. I introduced the concept of similar triangles and the proportionality of their sides. Those fifth and sixth graders understood what was being done in spite of the fact that they had been doing very poorly in arithmetic. It is important to note that I used H,h,LH, h, L and ll to denote the magnitudes involved instead of the mysterious x,x,y,yx, x', y, y'. Such a problem as an actual project for pairs of children would be not only interesting to the children, but would give an opportunity to the teacher to explain some of the precautions which have to be taken in order to make the determination of the required height fairly accurate.

Algebra. Most of my suggestions of a general character are applicable to the presentation of algebra. I want to stress here the importance of solving verbal numerical problems as literal problems first. When a verbal numerical problem is solved literally, all problems of the same type are solved, irrespective of differences in the nature of the magnitudes and their numerical values. This is the main characteristic of algebra which distinguishes it from arithmetic.

I have found that college students act as if the main object of solving a numerical problem is to obtain a numerical answer, whether the problem is in physics, mechanics, or any other subject. They plunge into arithmetical operations, and when an answer is obtained, they follow it with the designation "Ans." and stop.

With the introduction of the elements of algebra in lower grades, it becomes practicable to include some of the topics of college algebra in a tenth grade course.

Geometry. I should like to see plane geometry given in the first semester of the eleventh grade, with its scope reduced to explanation of the mathematical concept of truth and illustrations of inductive reasoning by means of the proofs of a relatively few selected theorems. I would drop solid geometry from the curriculum. I am taking for granted that mensuration and presentation of solids on a plane would be taken care of in connection with courses in arithmetic and algebra.

Trigonometry. I should like to see trigonometry taught in the second semester of the eleventh grade. I would include in it a chapter on the addition and subtraction of vectors and their application to physics. In a text of only 98 pages, I have devoted a chapter to these without sacrificing any of the essentials of plane trigonometry. I was able to do this by such devices as re placing multifarious reduction formulas with a very simple rule of action.

Last Updated July 2020