Appendix E: The Question of Depth Versus Breadth - Four Case Studies from India

Shailesh Shirali
Rishi Valley School

As a teacher, I am not involved in any way in curriculum design, so I cannot authentically comment on the reasons why syllabi have taken particular forms. However, my own leanings are clear: I value depth over breadth. I feel that an understanding of the usefulness of and inherent elegance of a concept, which leads to empowerment in problem solving, has far greater value than exposure to a large number of concepts, the intent in this case being appreciation and enrichment. (It goes without saying that these considerations have to be weighed against pedagogical factors such as mental growth patterns, what concepts are suitable at what age, and so on.)

Four case studies related to depth and breadth follow. They are all from the 11^th/12^th standard level, but similar case studies could be made concerning the 10^th standard portions.

Group Theory

The group theory component of the syllabus is very small indeed; it barely goes beyond the definition. Examples are given of infinite and finite groups, classifying them only as abelian or non-abelian. The notion of cyclic group is not mentioned, nor that of subgroups, or isomorphism of groups. Crucially, there is little scope for showing the relevance or usefulness of the group concept or for showing its real elegance.

Until recently the topic of geometrical transformations was part of the syllabus (studies as an application of matrix algebra rather than "pure geometry"). This allowed the possibility of showing how groups occur naturally, in actual application; but the option is not available to us any longer. It is interesting to speculate on why this portion got axed. I feel that it is not a coincidence that geometrical transformations are not part of the standard syllabus for engineering entrance examination.

Here are some typical questions from the 12th standard public examination:

1. Show that the set {1,2,3,4} forms a group under x (mod 5).
2. Define an operation, , by the rule a b = a + b - ¹/₂ab. Show that the set of real numbers excluding two forms a group under .
3. Show that the inverse of an element in a group is unique.

Typically, the group theory segment takes a week to cover. Which is better a one-week exposure to groups, going only as far as definitions and examples, or problem solving in areas already being covered (for example, optimization; challenging problems in trigonometry and coordinate geometry)?

Boolean algebra

Much the same comments may be made for the segment on Boolean Algebra. The syllabus merely specifies that Boolean Algebra is to be presented as an algebraic structure, with a listing of axioms and proofs for some of the basic theorems. Students are expected to know how to simplify Boolean expressions using axioms of Boolean algebra, and application to switching circuits is part of the syllabus. The portion takes roughly one week to cover.

As in the case of the group theory segment, not much depth of coverage is possible; but the situation here is certainly more encouraging-simply because many students work with computers. Many of them take Computer Science as an elective and, therefore, see Boolean algebra at play in a natural setting.

A typical examination question would be to find the Boolean functions for a given circuit, to simplify the function using the axioms, and then to construct a simplified but equivalent circuit. Students are also asked for proofs of identities; two examples are given below.

1. Prove the identity 1 + a = 1 in Boolean algebra.
2. Prove that a * a = a in Boolean algebra.

Data analysis

I consider the data analysis segment of the Indian curriculum to be highly unsatisfactory: much more so than in the two case studies just described. A heavy emphasis is placed on computation-of mean-mode-median, standard deviation, quartiles and percentiles, index numbers, different coefficients of correlation (Pearson's product moment coefficient r; Spearman's coefficient of rank correlations p, Kendall's coefficient of rank correlation ), moving averages, equations of lines of regression. The emphasis on computation is easily seen in this listing. This may be contrasted with the attention given to data interpretation.

Now there is nothing stated in the syllabus that excludes data interpretation. However, it must be emphasized that the Indian education scene at the high school level is very highly examination driven, and it is the style and tone of the examinations that largely determines what students actually learn. In the data analysis segment, the questions met in the examination are exclusively in the area of computation, the weight given to interpretation being nil for all practical purposes. The following questions are typical. (In the first one, the command "interpret the result" is, I'm afraid, given only lip-service.)

1. The following table gives two kinds of assessment of the work of ten students. Find the Spearman coefficient of rank correlation and interpret the result. [The table follows.]
2. The table given below shows the daily attendance in thousands at a certain exhibition over a period of weeks. Calculate seven-day moving averages and illustrate these and the original information on the same graph using the same scales. [The table follows.]

It should come as no surprise that in surveys amongst students soliciting feedback on teaching styles and how much they enjoyed the various topics, the statistics segment invariably ranks at the bottom. Most students perceive it as a unit where "one simply applies a formula;" as a topic having nothing to do with anything interesting or relevant and requiring no mathematical expertise.

Is an alternative possible? Yes, the data analysis segment could be made more open-ended, dealing with live topical data; say data relating to environmental problems, or to economic disparities, the emphasis being not on computation of test statistics but on interpretation.

Until about ten years ago, sampling and hypothesis testing were on the syllabus (at a fairly rudimentary level), but these components have since been deleted.

Mean value theorem

Included in the "higher analysis" segment are Rolle's theorem and Lagrange's mean value theorem. The statements are presented as facts, without proof, and the examination tests only whether one can verify the theorem in a particular setting, e.g. (these are from the examination):

1. Verify Lagrange's mean value theorem for the function f(x)=ln x defined over the interval [1,2].
2. Verify Rolle's theorem for the function f(x)=sin x defined over the interval [0, ].

It is hard to ascertain what students actually make of such theorems. They seem rather obvious, geometrically, and a typical reaction is, "Why bother to enunciate them? What could be more obvious than that a curve which goes up and comes down must at some instant have a slope of zero?" As the statements are not shown to have significant consequences, they are merely islands in a vast sea of facts.

Sample examination questions

1. Find the angle between the straight lines represented by the equation x² +2 (sq.rt 2)xy + y² + 4x + 2(sq.rt 2)y + 2=0
   
   
2. Prove by the method of mathematical induction that 3^{2n + 2} - 8n + 9 is divisible by 64 for all positive integers n.
   
   
3. Using the properties of determinants, prove that [matrix]
   
   
   
   
4. A plane meets the coordinate axes in the points P, Q, R. Given that the centroid of PQR is the point (a,b,c), show that the equation of the plane is bcx + cay + abz = 3abc.
   
   
5. A variable plane passes through a fixed point P = (a,b,c) and meets the axes at the variable points A,B,C. Show that the centre of the sphere ABC traces out the surface whose equation is a/x + b/y + c/z = 1.
   
   
6. Find the largest and least possible values of [the absolute value of] z given that z satisfies the condition [absolute value of] z + 1/z = 4.
   
   
7. Triangle ABC is right-angled and has a given area. Find the sides of the triangle for which the area of the circumscribing circle is the least.
   
   
8. Show that if x^y = e^x-y then dy/dx = ln x/ (1 + ln x)².
   
   
9. Evaluate the integral (from 0 to /₂) sin 2x * ln tan x dx.

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