[Introduction to Series: Dr. Anjum Altaf, former Dean of the School of Humanities and Social Sciences at LUMS, is writing a page-by-page review of the model textbooks (Pre-I to Grade 5) accompanying the recently implemented Single National Curriculum. These detailed reviews intended to involve parents in the education of their children will appear as a series in Sindh Courier. Parents would benefit by having a copy of the primer under discussion in front of them while reading the review.]
SNC Model Textbooks: Pre-I Mathematics Primer
By Dr. Anjum Altaf
I am devoting an entire article to the treatment of “nothing” (zero) for a reason. The introduction to maths in Pre-I should lay the foundation for learning the subject and, as mentioned before, one of the explicitly stated objectives of the SNC is “to lay a strong foundational basis for further learning in the time ahead.”
The introduction to zero in the Pre-I primer offers a test case for whether this objective is being met or not. Zero is without any doubt the most important number on the number line. Its centrality is easy to establish: Enter ‘History of Zero’ into Google search and count the hundreds of book titles that come up including over a dozen for children between the ages of 5 to 7. By contrast, it would be hard to find too many books about the numbers 1 to 9. Clearly, there is something special about zero. In fact, its “invention” is celebrated as one of the major achievements of the human mind.
Let’s see how zero is dealt with in the Pre-I primer where it makes its first appearance on page 81 sandwiched between the numbers 9 and 10. The poem for zero is shorter than for all the other numbers: “Zero means nothing at all, / That’s why, zero it’s called.” The take-away balloon asks “What does the digit “0” mean?” The teacher’s instructions are as follows: First, “Discuss the given picture [an empty plate with an half-eaten apple lying beside it] and ask question (sic) about it. Tell them that zero “0” means nothing.” And Second, “Place 4 baskets and 3 pencils on the table. Then put a pencil in each basket [Note: How would that be possible if there are only 3 pencils?]. Ask the students how many baskets have balls [Note: The pencils turn mysteriously into balls here] and how many baskets are empty. Let them know that 3 baskets have pencils and 1 is empty. The empty basket shows that there is no ball in it, it means that there are zero “0” balls.”
This is the only discussion of zero — the next two pages (pp. 82-83) are devoted to tracing and writing the number 37 times. Can this be called a stimulating or intelligent introduction to zero? First, the number is not called zero because it means “nothing at all” and this is an important point although an impatient person might consider it pedantic. “Zero” is just an assigned name which could have been anything else. In fact, there are a whole lot of substitute names — śūnya in Sanskrit, pūjyam in Tamil, sifr in Arabic, null in German, etc., etc. It has peculiar names in slang like zilch and nada; in sports, “nothing” is conveyed by duck (cricket) and knot (tennis). In Pakistani student slang the term knotta is often used to convey getting no marks in a test. When children make secret codes, they can invent entirely new names for numbers to confuse elders. So, clearly it is grossly misleading to befuddle students with the tautology that “Zero means nothing at all, That’s why zero it’s called.” There is a concept of “nothingness” which can be given any name. It is the concept and not the name that is important.
Let me identify a problem that occurs immediately and would perplex an intelligent child while his/her intelligence lasts. The very next number after “0” is boldly written as 10 (p 84). If “0” is nothing at all, how does appending it to a lowly 1, turn it into a mighty ten — exemplified by the ten-Rupee note on the page that every child knows is much more than one Rupee? What this reveals is that the number 10 is being taught mechanically as a unique number to be memorized instead of being interrogated and understood. And this, in turn, makes every number in the primer up to 50 a unique number to be similarly memorized by repeated tracing because the beautiful logic of the number system is completely ignored.
Some might claim that this logic is too complex for 5-year-olds and would be taught later. I have no doubt that it would be taught, but I am not convinced by the argument of complexity. It is better to stretch a child’s imagination by challenging it than to dull it with misleading tautologies. Nor am I convinced it is a good idea to lay a poor foundation and hope to build a sound edifice on it later. It might be better not to broach a topic at all if it is going to be taught wrong.
Let me suggest an alternative. Every 5-year-old is familiar with currency coins and bills and most can count to twenty Rupees before they get to Pre-1 (that’s less than what a packet of chips or a bottle of soda costs these days). What if a teacher placed a 10 Rupee bill and a 1 Rupee bill on the table and asked how many Rupees lay on it. Someone is bound to answer 11. The next question would be how come just 2 bills make 11 Rupees? That would trigger a discussion that moves towards a conceptual understanding of the number system. It would not matter if a child does not get it entirely — a puzzled mind encouraged to think is better than one lulled into a false certainty. The child would be prepped for the next year when the topic comes up again with more rigour.
The poor treatment of zero stems from a pedagogic failure — a failure at the very beginning to explain the association of quantities with symbols and the assigning of arbitrary names to them ((aik, ik, hik, yek, yau, and one are all equivalent). Just as children are given individual names to distinguish one from the other, so are quantities. And this naming is needed for the purpose of communication. When I have to convey to someone in another location how much of something I have and I can’t do so by raising and showing my fingers, I have to devise some other way.
This, in fact, is the genesis of writing and there can be a fascinating discussion around naming numbers which will overlap with the naming of the alphabets in English and Urdu. In the maths class, one can start with how the Romans indicated quantities, how cumbersome the system was (as cumbersome as it is in the SNC Pre-I primer) and how the brilliant discovery of the zero meant that we can write the biggest number that can be conceived by a child with just ten symbols. The children don’t have to write every number from 1 to 50 eighteen times each. All they need to know is how to get to any number with just the symbols from 0 to 9 and they would be off and running.
If done right, the children would be excited by this discovery. They would also be amazed to learn that despite the fact that they are being made to learn maths in English, not all wisdom has come from England. In fact, the brilliant invention of the zero as we know it today occurred right here where we are now — 1,600 long years ago by Aryabhata who represented it not by a circle but by a dot and named it śūnya. And that’s not the end of the fascinating history of the zero. It was taken west from India by the Arabs who called it sifr which is actually where the name zero is derived from and which also gives us the word cypher. It’s equally interesting that the Romans couldn’t grasp the concept of nothingness, couldn’t assimilate the zero, and continued to struggle with their archaic systems for another few centuries. What would it do to a child’s confidence to know that our modern numerals belong to the Indo-Arabic number system?
Nothing provides an opportunity like the zero, so summarily dismissed, to make mathematics alive and interesting and, at the same time, to fire the imaginations of children turning them into eager, curious, and active learners. I would much rather children just explored the numbers 1 to 20 in Pre-I and spent a lot more time engaged in the kinds of exciting discussions I have outlined above. I wouldn’t want to be introduced to maths in the dull manner it has been done in the SNC with its meaningless poems and pointless exercises devoted to eating things and washing hands. I wonder how parents allow their children to be miseducated in this way that would make them lose interest in a subject for life.
Does anyone wonder at the dearth of Pakistani mathematicians of international repute? The Fields Medal is considered the equivalent of a Nobel Prize in Mathematics — it has been awarded to individuals of Iranian, Indian, Chinese, and Vietnamese origins. No Pakistani has ever come close to being considered. Why? On that, I have a story to narrate that I will pick up in the next part of this review.
Click here for Part-I , Part-II