The purpose, the aim and drive of these schools is to equip the child with the most excellent technological proficiency so that the student may function with clarity and efficiency in the modern world. A far more important purpose than this is to create the right climate and environment so that the children may develop fully as complete human beings. This means giving the child the opportunity to flower in goodness so that he or she is rightly related to people, things and ideas, to the whole of life...

J. Krishnamurti

The challenge of creating well-integrated human beings spurs educators in the Krishnamurti schools to constantly churn over questions such as what to teach and how to teach. In particular the dual aim of creating a well thought-out, well-designed curriculum that stretches the cognitive capacities to their potential, while simultaneously creating a space for the mind to remain reflective and be rightly related in the sense indicated above, poses a conundrum.

Over the last few years we have attempted to explore these issues in a more focused manner in our junior and middle school. We started with questions like: ‘How does one nurture a genuine atmosphere of learning?’ ‘How does one bring about a quality of attention in the classroom?’ ‘How can we make students take ownership of their learning?’ ‘How do children actually learn?’ ‘What is the nature and quality of student-teacher interaction in the classroom?’ ‘What prevents someone from learning?’

In this article I will outline an attempt to respond to some of the above questions in a Class 5 Mathematics classroom.

There was a certain amount of discontent with the traditional mode of delivering Mathematics instruction where the teacher introduces the topic, works out some examples, and then gets the students to do some similar problems. It didn’t seem to leave the teacher with much space to observe students closely, and created an artificial pressure on students to keep up with the class pace (especially for the slower ones) and at times encouraged a competitive spirit among the quicker ones. The sequential nature of explanations also resulted in some students losing the thread (either because they had been day dreaming or due to other distractions), and then finding it difficult to pick up the explanation at a different point. In fact in some ways this mode of teaching did not take sufficient account of the children’s natural disposition towards constructing their own understanding and making their own discoveries. So the real question for us was how to tap into the children’s own drive and curiosity, and let that be the motor for their learning.

The Class 5 Mathematics programme

The Class 5 Mathematics programme starts with a story of a brother and sister who get lost in a forest and encounter various characters that help them find their way back, but only after solving dilemmas and puzzles posed by these characters. This is used as a diagnostic tool in a non-threatening context; and at the same time it builds up a picture of what Mathematics is about in its widest sense. Some short-cuts in Mathematics, for example the ‘999 trick’, come across as pure magic. The beautiful patterns in the Chinese triangle (also known as Pascal’s triangle) naturally demonstrate the addition patterns of odd and even numbers. Students encounter the recursive procedure in the ‘missionaries and cannibal’ river-crossing problem. The need for the commutative law comes across in the magic squares and triangles. The understanding of place value is reinforced through a problem involving census of the king’s soldiers as well as ‘octopus’ addition (using base 8). The intention is to loosen up the minds of the children and challenge some of their notions about what Mathematics entails. Apart from enthusing the children and getting them inspired, it is important to give them a wide perspective as well as some experience of how real mathematicians work—where they pose their own questions and pursue different lines of enquiry, each line of enquiry raising further questions. They can then feel the joy experienced in discovering unexpected patterns and the frustrations encountered when coming across a dead end.

The rest of the curriculum is divided into four ‘self-learning modules’—there being two weeks of teacher-led classes in between each module. The modules are largely based on a set of SMP (School Mathematics Project) booklets, along with supplementary material drawn from other sources. Also during the self-learning modules, there is one class a week set aside for whole class teaching—this could be used for clarifying frequently asked doubts, or for attempting something entirely different like an investigation or a thinking problem.

A template for one such module is given in the box in the next section. Its components are explained, while the curriculum and pedagogic approach, as well as shifts in the student’s and teacher’s roles are discussed later.

Design of the curriculum

The curriculum is broken up into smaller components: conceptual play, skill building, problem solving (application of concepts and real-life problem solving where assumptions have to be stated and information required to be identified), investigations and mental maths. Mental maths, the ability to calculate quickly with small numbers, together with an ability to make good estimates when dealing with larger numbers, has been identified as an important pre-requisite for improving computational accuracy.

A wide variety of topics are covered, some of which tend to get neglected in a traditional Mathematics curriculum. This includes, for instance, visual/spatial development through model making, visualizing perspectives and orientations. The first section of each module is based on playing around with concepts, with plenty of hands-on material to facilitate this, as well as games to reinforce the concept. The concepts are coded into red for number concepts, blue for shape and space concepts and green for mensuration and pre-algebra concepts. This is followed by a mental maths component, where children practise memorising basic facts for the four operations and other important number facts. The skill building part develops computational skills and simple applications of concepts learned. Then there are teacher cards, which—depending on the need of the student—could contain enrichment work covering some new areas of study (e.g. networks), puzzles, problem solving and investigations or reinforcing basic skills through patterns and investigations. A student moves between these activities in a certain sequence, for example he chooses and works on a ‘red concept’, then does a review, followed by mental maths, skill cards and teacher cards, which completes one cycle. In the next cycle a different coloured concept would be chosen. Within the same module, concepts are independent of each other and therefore can be tackled in any order.

Class 5 Maths Self-Learning: Module 2

Self-directed learning

The whole programme is structured so as to give students more responsibility and ownership of their learning. They are given a template (such as the one in the box) at the beginning of each module, and have some choice in the order in which they work on the various topics. This gives them a sense of being in charge of their learning. Materials required for the module are displayed around the classroom and students are free to pick up whatever materials they need. At the end of the class everything is put back in its right place. Students also have to take initiative to clarify their doubts when they are unable to understand any aspect. Unlike in the teacher-led classes, where the confident and vociferous dominate the proceedings, using this structure, those who require more attention from the teacher are provided with more teacher support (not necessarily just those who come for clarification quickly—since some of them may not have read the instructions properly). A teacher is not required to start the class and it is often the case that the students have already begun working before the teacher has entered the room. If classes are missed, it is easy to pick up from where the student left off. The teacher no longer provides the yardstick for their progress, and students soon start looking at their own progress more keenly. All this, of course, doesn’t happen at one shot and initially many discussions on the intentions of the programme are conducted with the students as this is the first time students may have been asked to think in such a way. However, because they begin to enjoy this particular way of working they are quite receptive and willing to reflect on their actions. The time period for the module is stated at the beginning and is kept fairly generous even for the slower ones, while the pace of the quicker ones is modulated with the more challenging teacher cards. There is also an option to take some of the work home to complete.

The teacher-led classes are used to expand on some of the finer points, clarify frequent doubts (though some children may not yet have finished with that particular topic, this exercise may still be carried out in a way where everyone understands the problem), tie up seemingly different ideas and sometimes introduce investigations not connected with any particular topic.

The spiraling curriculum

The curriculum is designed in a spiraling manner—one part of the concept is done at a time and then revisited, and given a slight twist and complexity at each succeeding level—as opposed to completing one topic fully and then moving to the next. Understanding is something that grows the more one revisits the concept, and becomes firmer when approached through different angles. Initially it is good to start with an intuitive understanding, deepen it with playing around and discovering connections and only then formalize it through rules and algorithms. For example, the topic of decimals is first intuitively approached through breaking a whole into ten parts through measurement in centimetres and millimetres, and through a fraction of ten parts. Formal notations are then introduced and consolidated through a game, where the idea of one and one point zero being equal is reinforced. In the second module, an intuitive idea of hundredths is introduced through money, and ideas on decimals through measurement are further explored by comparing heights. The teacher-led classes are used to tie up the ideas of tenths and hundredths in the decimal system and the magnitude of these numbers is brought alive through some visual displays. In this manner the whole concept of decimals is developed and the final module ends by exploring multiplication and division by powers of ten, introducing an intuitive sense of decimal multiplication and division which the students will actually encounter in Class 6. Similarly the topic of solid shapes is explored through faces using 2 2 2 D shapes, edges represented by straws with connectors in 3 3-D models, as well as perspective views.

Some practical points:

• If the class is bigger than 20 a support teacher may be required.
• A teacher needs to be familiar with the materials as varied demands are made within a class.
• Organizing materials well, to enable easy access is important.
• The teacher should have a view of all the children and even engage those who do not make much demand on teacher time.
• There is an inbuilt flexibility in the system to vary the approach and sometimes modules can be collapsed, some work can be omitted, and more teacher-led classes can be introduced depending on the needs of a particular batch.

Creating an atmosphere of learning

Breaking up the classroom into sections with different children working on topics of their choice creates a very different ambience from a class where everyone is expected to be doing the same work. There is a release from the implicit expectation of having to keep up with the others, veering them away from comparing and using their peers as yardsticks for their growth/sense of success. The non-linear format of the programme, reviews attempted individually, and judicious use of teacher cards—all of this helps in making it difficult for the students to compare themselves with their peers. The structure is also able to deal effectively with difference in pace and ability, with those requiring further support being given specific teacher cards without the students realizing that this is what is being done.

The teacher-led discussions are also more effective; the starting point for these is drawn from the work the students have done, thus creating a more ‘level playing field’ which encourages even the quieter ones to be more participative. Moving from one type of thinking to another—number work to shape and space to spotting patterns in sequences—keeps the mind more alert and attentive and reduces the possibility for becoming habituated and mechanical.

Most significantly, since it is now the student who is directing her own learning, it is possible to develop a relationship of deep trust with the teacher which is independent of the student’s capacities or inclination in the subject. This non-judgemental way of relating is important in overcoming a child’s fear of the subject, and encourages her to take risks and try out new methods.

Observing the child

The ‘looseness’ in the structure allows the students to be more natural in their behaviour, and enables the teacher to observe them at leisure, yet fully. Their interests, inclinations, fears, need to please the teacher, all come to the fore and provide opportunities to reflect on and observe these tendencies together with the child. Those who have relied on the teacher or their peers to gauge their progress, initially find it difficult to adjust to this new way of working. The acknowledgement and the reward come from the enjoyment one gets from the work. Sometimes the blocks to their learning are removed gradually, sometimes it is quick. The intention is to help the students develop the right relationship with the subject, the teacher and with their peers.

Assessment and feedback

The concepts are mainly developed through the SMP booklets, and answer booklets are available for students to correct their own work. If they have made any errors, they are encouraged to first try and correct them on their own, while if they have difficulties in understanding they seek help from the teacher. Once they are satisfied that they have understood the booklet well, they then attempt the review, which is corrected by the teacher (outside class time). The corrections from this review could reveal different aspects of an individual—a tendency to rush through the work, not knowing when to ask for assistance and in some rare cases dishonesty (having skipped some chunk of work)—this becomes an entry point for exploring the student’s attitude towards learning. This kind of space is available since the students are working independently, and only making a demand on the teacher’s time when they encounter some difficulties. Sometimes it also becomes evident that a student is not ready to grasp a concept; because of the spiraling nature of the curriculum she has the chance to wait till the next module to pick up the concept. The skills cards and the teacher’s cards are also corrected by the teacher. The corrections are not too onerous as each day there would be only five or six pieces to correct as the children are working at different things. The students track their own progress by getting their template signed when they complete an activity. The teacher is able to get a more nuanced understanding of the student, either in terms of the development of a concept across the modules or in terms of identifying areas that require help e.g. mental maths, computational skills, applying concepts, problem solving, investigative work, pattern spotting, spatial awareness, confidence with hands-on-work and such like.

Conclusion

Our experience leads us to believe that learning of concepts and skills, and observing the movement of one’s mind, need not be mutually exclusive, and a space can be created for both. However, the role of the teacher is quite crucial, and needs to be redefined in some significant ways. There is a need for the teacher to be alert to the movement of his/her own mind; a lot of insecurities might come to the fore when yielding control of the learning process to the students. The need to be in charge and in control of the proceedings may come in the way of objective observations. Weaning children away from teacher dependency is not easy, as the teacher may also get some pleasure from it. Then there is the genuine fear of not being able to develop a relationship with the student without the subject as an intermediary. Also, there is now a greater responsibility in terms of being alert and observing keenly. Attention, in whichever way one teaches, is essential; through this method, space is created for attention to take root.