International Panel on Policy and Practice
in Mathematics Education: 2001 Report
Issue 5: Depth and Breadth in the Mathematics Curriculum
How does your educational system decide the balance between depth and breadth, that is between insistence on in-depth knowledge of relatively fewer core topics vs. a broad inclusion of topics, with less emphasis on each? How is this decision effected in practice?
India: Sudhakar Agarkar and Shailesh Shirali
India has a long tradition of mathematics education. Vedic mathematics was an essential part of the education system in the ancient Indian Gurukul system. Mathematics education was, however, limited to a certain class of the society. In colonial times, class education continued and emphasized certain arithmetic topics. The majority of the people had no opportunity to study formal mathematics. Folk mathematics, however, enabled them to handle day to day transactions. After independence in 1947, the focus shifted from class education to mass education, and the school system expanded rapidly. This resulted in a shortage of math teachers with the level of content knowledge needed to do justice to depth in mathematics education.
Education is the responsibility of the states. Different states have different ways of organizing the system. The Education Commission Report of 1966 recommended a uniform 10+2+3 arrangement across states. It also recommended teaching mathematics on a compulsory basis up to grade ten. The expectation of math education was that the courses would be taught at a level high enough to provide the base necessary for advanced study in later years. In 1968 the government of India adopted the recommendations of the Education Commission. The NCERT prepared the curriculum and textbooks. They were criticized as being "heavy" and a two-level curriculum was suggested but not implemented. There was a high failure rate on the school-leaving exam. Dedication to the concept of mass education forced the preparation of a "lighter" syllabus. In 1986, the National Education Policy (NEP) was adopted. This policy suggested a shift in emphasis, away from manipulation and towards visualizing math as a vehicle to train pupils to think, reason, analyze, and articulate logically.
As the system of mass education was expanding rapidly, two attempts were made to achieve depth. These included: 1) the inclusion of challenging exercises, and 2) the establishment of minimum levels of learning (MLLs). Many teachers and policy makers believed that focusing on depth would eventually pay off as students would be able, as a result, to handle superficial topics on their own when the need arose. Despite this, the prevailing approach was to cut the depth and add breadth to the curriculum.
Another issue that arose out of mass education was the need to provide the necessary knowledge and skills for students with different plans for the future. The National Curriculum Framework in 2000 states:
While determining the curriculum in mathematics, it must be kept in mind that the majority of pupils would leave education at the end of secondary stage (grade ten). They would need to apply math skills and competencies in their work situation. Only a small number would go on to higher education. The curriculum needs to strike a balance between the requirements of both groups.
The framework also suggests that the history of mathematics, with special reference to India and the nature of mathematical thinking should find a place in the curriculum, and students should be encouraged to enhance their computational skills by the use of Vedic Mathematics.
I teach in Rishi Valley School, a co-educational residential school belonging to the Krishnamurti Foundation of India (KFI). It is located in a hilly and drought-prone region of southern India, and it caters to students ranging in age from 8-17 years (standards 4-12). It is comparatively small in size, with 350 students in all. I have been at this school for nearly two decades, and chiefly teach mathematics at the 11th/12th standard level, and (occasionally) physics and computer science; on occasion, even geography! In addition to my regular teaching, I am closely involved with the Mathematics Olympiad movement in India, and I also do a lot of expository writing (articles as well as books). In this note I shall make a few observations on the teaching of mathematics in India, based on my personal experience-that of teaching at the 11th/12th standard levels.
Brief note on the Indian education system
At the 10th standard level, students in India take their first major public examination, in which they are tested on a wide variety of subjects (English, a second language, mathematics, physics, chemistry, biology, geography, history/civics, plus an elective12). After clearing this examination they do the "Plus-2" course (11th and 12th standard), and it is here that they start to specialize. They are required to take four elective courses, plus Compulsory English. Here are some typical combinations for which students opt:
There are, unfortunately, several different examination boards in the country: two all-India boards - Council for the Indian School Certificate Examination (CISCE)13 and Central Board of Secondary Education (CBSE) - plus separate boards in every state. Each board has its own sets of syllabi, and the different syllabi do not mesh particularly well. An unfortunate fall-out, which sometimes takes place because of this multiplicity of boards, is grade inflation.
Areas of difficulty: personal observations
There are very strong socio-economic pressures on students to register for mathematics at the Plus-2 level. As a result, enrollment is high. Often one sees students in the class who do not possess the requisite aptitude for the subject. This results in wide differences in ability in the class, and classroom teaching begins to lose its effectiveness. Closing the door on such students is not a satisfactory answer, as they have their future career at stake. For instance, those who wish to study economics are required to do mathematics at the school level. The net effect is that teaching becomes much harder, as individualized instruction is difficult to accomplish, given the pressures of completing the syllabus.
Also, the Indian education system seems to be more examination-driven than any other country. The examinations include not just the school-leaving public examinations but also admission entrance examinations conducted routinely by institutes offering courses in engineering, medicine, law, architecture, management, computer applications, fine art, etc. These entrance examinations are taken by a very large number of students, and pressures for admission are very intense. Inevitably, private institutions ("coaching centres," "tutorial colleges") have sprung up over the country, offering professional help (at a stiff price, of course) for examination preparation. This in turn has led to a general rise in the level of sophistication of the examinations, leading to a still greater demand for tutorial colleges, and so on. The cycle is a vicious one, and becoming steadily more vicious with time. The obvious fallout for students is that they are divided between competing demands. The syllabi for entrance exams do not completely mesh with their regular school syllabi, and the styles of examination also differ in many ways. The effect of such stress on motivation may easily be imagined. Another inevitable fallout is that teaching tends to focus on training rather than on education. Without question, the examination-driven nature of our educational system has had a serious negative effect on the country.
For further consideration of the question of depth versus breadth through four case studies, see Appendix E.
As mentioned earlier, I have drawn from my own experience in writing the above comments. However, I have also received some comments from other teachers of mathematics. For instance, a colleague who teaches at the 10th standard level reported to me that "the syllabus seems to be very wide but not too deep." The comments that I made, independently, are about the syllabus for 11th/12th. The example he quotes is that of the unit on commercial arithmetic, wherein students are expected to acquire an understanding of insurance schemes, taxation (personal income tax as well as sales tax), shares and dividends, etc. Much of this material lies totally outside student experience and, thus, is thrust on them, and of necessity the coverage is very superficial. Other examples may be given. Pythagoras's theorem is studied but only its statement and usage in straightforward application; the proof is not learned. After a brief glimpse of transformation geometry (the use of reflection to solve a certain problem), one moves to another topic.
It may seem that I am highlighting only the portions of the syllabus with which I have some quarrel. This is certainly so! However, it is not as though I have such feelings for every segment of the syllabus. Fortunately there are many portions that receive fairly good coverage: coordinate geometry in two and three dimensions, differentiation, integration, differential equations, vector algebra, complex numbers, determinants, etc. (Interestingly, these are all traditional "old-fashioned" topics; their coverage in high school/college must have been much deeper during the second half of the 19th century and the first half of the 20th century.) Here the depth of coverage is adequate, and students tackle a variety of problems.
I should add here that I have often attempted to go "beyond the syllabus." For instance, in the group theory segment I have brought in the mathematics of the Rubik cube, and in the segment on the mean value theorem, I have shown how these results help in arriving at some nice inequalities [e.g. for the sine function, or for the function: square root of (1 + x)]. These initiatives have, however, received mixed responses. This is discouraging at first encounter, but one must recall to oneself the tremendous pressures which students face because of the public examination.
The more traditional areas receive good coverage (see sample exam questions in Appendix D). However, much more could be attempted, and that is what I have tried to focus upon in this essay.
Participants identified the similarities and differences represented in Table 6.
Table 6 India: Similarities and Differences Among Participating Countries with Respect to Depth and Breadth in the Mathematics Curriculum
Depth is appreciated if it helps to enhance students' mathematical thinking (creativity and heuristic insights included.) Gifted students are exceptions; let them go higher. A small number (about .0005%) will make good mathematicians. Their mathematical thinking power could be acquired by learning theoretically advanced topics. The role of the teacher is difficult; it is the teachers' role to locate these gifted students, work with them, and get mathematicians to guide them.
Depth in math education is different. For general students, the top 20 percent and majority, mathematical thinking power should be enhanced through challenge against properly hard problems, through problem solving, and open-ended problems.
Those here seem to agree that depth is better than breadth; it is better to probe deeply into a selected set of topics and problems, rather than superficially do several topics. This also seems to be the opinion among accomplished teachers. TIMSS characterized the U.S. curriculum as "a mile wide, inch deep." In Japan, a country that performed well, typically an entire class period would be used to focus on one or two problems. Why do we not practice instruction this way? One guess might be the role of assessment. Higher order skills are difficult to measure with short items on exams. When you want to implement assessment on a national scale, there are questions of cost. The technology of assessment leads to having great confidence in what is being measured; the whole methodology of assessment points toward measuring many things in shallow ways. Assessment pushes curriculum toward shallowness. One solution is to better integrate the communities of classroom practice and assessment experts-then assessment could better support the curriculum.
Themes That Emerged From the Discussion
The notion of depth in the learning of mathematics can be interpreted in many ways. Depth can mean detailed study of many problems at different levels. Depth might be about the ways that students address the problems we give them. Depth could mean knowing more and more about the subject. Or, it could mean being able to use ideas in relation to other ideas and to understand ideas this way. The group agreed that to know deeply means you know more and more about the subject and are able to use it to solve a different problem. Deep thinking is different from deep knowledge. When you have a deep knowledge, you understand it in relation to other mathematical ideas. To know something deeply is to feel empowered by that knowledge
"The issue of 'depth and breadth' have different meanings, which are associated with different thoughts regarding math and math ed. It could mean having details and different levels of problems (the common 'traditional' understanding)." (Mina)
Theme 2: Factors related to achieving deep understanding
Assessment can have an impact on the nature of the curriculum and on the depth of understanding expected of students in their learning of mathematics. Depth can be assessed with different types of questions, but often exams have "busy" questions. It is difficult to have challenging problems on a national exam because of issues such as scoring, finding the problems in the first place, managing the administration of such tests. In addition, different boards in the same country may have different standards.
One way to achieve depth might be by integrating knowledge, but integration of knowledge is not done by students. They have many separate ideas but need help to integrate them. A spiral approach might also help students look more deeply into a concept. Another way is to consider topics that are not necessarily treated deeply but that can be such as the long division algorithm. How do classroom teachers treat an idea? Bass noted that the aim of education should be to have students learn about mathematical practices or to think mathematically or deal with knowledge by themselves. Is the question of depth a consequence of the way the students address certain problems that we give them. Do we give students questions about which they have to make conjectures and look for connections to other knowledge with the depth at the end of the process rather than at the beginning? If we go into depth, we should have a reason to do so, and suggest ways to deepen the ideas.Illustrative quotes
"It is difficult to reach depth in a spiral curriculum, because the tendency is to repeat the same topics at the same level each time around." (Shirali)
"We might think of depth as a 'helix'- a sort of spiral in three dimensions." (Hashimoto)
"The minimum tends to become the maximum." (Bodin)
"A spiral curriculum makes sense only in a setting where you are certain that students will continue their education. If you are not, then decisions about what mathematics must be covered take on a different significance." (Mina)
Theme 3: Balancing breadth and depth
When a country is committed to education for all, the depth-breadth challenges become even more complex. In many of the countries the curriculum contains far more topics than teachers can possibly cover. In the U.S., each of the 50 states writes its own curriculum; national standards such as the NCTM standards are available as a guide but there is no requirement that they be followed. This phenomenon of rewriting and adapting the national curriculum is not widely practiced in the other countries represented; and when it is, there are still constraints (e.g., in India, states can modify the national curriculum but they must use 75 percent of it.)
Teachers find it very difficult to contend with a crowded curriculum and to also aim for depth. The result is that they can only do minimal treatment of many important ideas. Teachers may think students do not have a good background, and so they teach to lower expectations. They decide to have fewer topics but still do not teach with depth. The classroom teachers in the seminar tended to favor depth over breadth.Illustrative quotes
"What is the balance between depth and breadth - which is more important - depends on the objectives." (Hashimoto)
"If the teacher does not know what to do with the student's question, the teacher won't go deeply. A teacher can go deep and have students go deep only if the teacher knows the subject deeply." (Sackur)
Key questions about depth and breadth related to:
12Available choices for electives include Fine Art, Economics, Computer Science, Classical Music, Environmental Science, Home Science, and so on.
13A copy of the CISCE syllabus for Mathematics Classes 11 and 12 is in Appendix. My own school is affiliated to the CISCE.
PCMI@MathForum Home || International Seminar Home || IAS/PCMI Home