To teach is to learn twice
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THE FORTNIGHT PRECEDING the first year university exams was a grueling time for me. It was the sole occasion in my life that I lived for and by mathematics and only mathematics. From early morning to near midnight, I was holed up in an isolated corner of the university library. My task: To decipher the arcane concepts of pure mathematics and decode the mind boggling formulas of applied mathematics. This single minded pursuit was interrupted by three short jaunts to the cafeteria, and a few hours of snooze. Even to close compatriots, I was totally incommunicado.
Not that I was miserable. Quite the opposite. It was an experience to relish, a time of enchanting discovery. Ideas I had but faintly grasped finally made sense. Concepts presumed disparate turned out to have subtle linkages. The outlines of the sublime thread linking the branches of the science of numbers began to emerge. This intense plunge into the abstract arena fostered a feeling of inner joy. To put it in spiritual terms, I attained a numerical nirvana of sorts.
The effort paid off handsomely. I swiftly breezed through the exams, as indicated in the last chapter. My frail physical self, however, did not emerge unscathed from the extended sleep depriving ordeal. In the final days, a sense of unease gnawed me, but I ignored it. Once the last paper was done, it could not be wished away. A sizable boil in a sensitive part of my anatomy confined me to bed with pain and fever. When a course of antibiotics failed to provide relief, a surgical procedure at the University Health Center was in order. Done expeditiously under local anesthesia, the outcome was swift and dramatic. Within a day, the fever had abated and the intense pain was history. The wound being dressed for three days, I was a free bird. It was my luck that the exams were over when the bio-devil reared its ugly head.
The last exam heralded a three month period of bliss for the students: no lecture, seminar or essay. From April to June, they engaged in pursuits of their own choice. One group, however, did not have it as good. For the BSc or BA with Education students, a divisive equation applied. Our vacation was a disjoint interval with the initial and final segments intersected by five weeks of teaching practice at assigned secondary schools.
No sooner was I up and about from my surgery, I found myself on a day long bus trip. Hurtling at a perilous speed on a slippery, bumpy, rain soaked road, it felt like gliding on a sine curve. Was I uneasy? Hardly. Nazir Nensi, Shiraz Ramji and Zubeda Vellani, close buddies and fellow first year UDSM students, sat close by. Their chattiness made time flow fast. We were headed to Tanga for our teaching practice. Rain or shine, we were in ebullient spirits.
We put up at a homely boarding house, some half a mile from the central business area of Tanga. Our landlady doted on us like a mother, serving ample, hot, tasty breakfast, lunch and dinner each day. Sunday lunch was special: meat pilau with marinated salad and madafu (coconut water). We found four other boarders in residence. All primary school children, their friendliness made it feel like a home away from home. An expansive garden enlivened by hibiscus, rose, papaya, lemon plants and sugar cane clusters encircled the house. A thorny sisal plant in the middle lorded over the flora. On weekends, we played catch ball with or read to the kids in its grassy area.
Popatlal Secondary School, our allotted school, was a twenty minute brisk walk. Traversing a lush, semi-forested area over run by coconut palm, we arrived in time. On the way back, it was a leisurely stroll, the expanse of greenery mercifully shielding us from the blazing rays of the tropical sun.
The semi-private school, opened three years back, was run by the Hindu community. In the colonial era, the students would have been exclusively Asian. In 1969, about half were Asian, the rest African. At the entrance, we found Mr Kulwant Chaudry, the headmaster. It was the norm for him to eye his wards trickle in. Our first impression: A man with a genial demeanor. In practice, though, he was a strict, no-nonsense administrator.
He welcomed us warmly. Both sides would benefit, he frankly told us. We would acquire essential skills and his overworked staff would get some relief. While not spending as many hours in the classroom as a regular teacher, we were to abide by the same rules applying to them. Timely attendance was essential. Our main responsibility was to ensure that our students did not lag behind in the subjects we would teach.
After the pep talk, he took us around, introduced us to the teachers, and showed us the staff room and other facilities. An overview of the extra-curricular activities at the school preceded an invitation to take part in any of interest to us. Should we have any concerns, we were free to approach him directly. It was an edifying start for the edgy first-time teachers.
I was to teach mathematics to Form IA and Form IIB. There was but a day to prepare. We had to make a list of topics with corresponding lesson plans. Beside guiding our classwork, the plans would be inspected by our UDSM supervisor.
On the fateful day, I was escorted into the classroom by the Form I teacher I was temporarily replacing. He had briefed me on the topics he had covered, and those I had to cover in the next five weeks. A brief introduction, and he left me to my own devices. Though not jittery, I was not too assured either. It being my first time to stand in such a role in front of a skeptical bunch of thirteen year old faces, I was not sure how it would go. Tongue tied momentarily, I hid it by rearranging my papers. How to begin? After a deep breath, my first word popped out:
After the briefest of a pause, they responded:
And then in English:
How are you?
Now the response was faster and louder:
We are fine, sir.
And then, with a typical Asian accent, I asked in Gujarati:
The class responded with chuckles and the ice was broken. After writing my name on the black board and saying a few words about myself, I stated my goals for the class. A stony silence prevailed.
My first question, whose answer I knew, was: what was topic of the last class? A chorus of voices said:
Linear equation in one unknown.
After briefly going over this topic, in an authoritative tone, I declared,
Now we advance to linear equations with two unknowns.
I started by writing on the blackboard:
2x + 5y = 19
3x + 2y = 12
I repeated the equations in words and declared x and y to be the unknown quantities whose values we needed to find. This was done by equalizing the coefficients of one variable in both equations. To make the coefficients of x equal, we multiply the first equation by 3 and the second, by 2 to get
6x + 15y = 57
6x + 4y = 24
Subtracting the second equation from the first gives
11y = 33 or y = 3
Finally, we substitute this value of y in any of the original equations and find that x = 2.
I demonstrated this procedure in a step by step manner for two other sets of equations. Setting three sets of equations to solve on their own, I went around checking their work. At the end, I assigned the homework. The next class covered a variation of this method called the method of eliminating one variable. The graphical method, which better illustrated with the idea of a variable and pictured the solution, followed.
I began in similar way in Form IIB where the topic was from geometry. In a few days, I had settled down into a decent routine with both the classes. I strove to balance chalk with talk, classwork with homework, and queries with instruction. Preparing for class was not an issue; I knew the subject like the back of my hand. What took up a lot of my free time was grading homework. At the end of our stay, it was gratifying to see most students scoring well. Only a handful remained stuck in dire numerical straits.
That is the low key manner in which what was to turn out to be a momentous and exhilarating saga of numerical pedagogy began. I was more than satisfied with what I had accomplished at Popatlal. But after several years of full time teaching, it dawned upon me that my approach had been tainted with a fundamental flaw. I deal with it at the end of this chapter.
Waiting for our supervisor was like waiting for Godot. She was expected any day, but it never seemed to come. Just as we were to enter the final week, Ms Elizabeth Connelly, an instructor in education at UDSM, landed in our midst to assess our work.
Though quite personable, she was all business from the first instance. First there was a group meeting where we outlined what we had done. We were told to hand in our lesson plans and informed that she would observe us in class over the next four days. Each person should expect at least two visits but who would be visited when was not spelled out.
There she was, in my classroom, the very next day, sitting in subdued silence in a corner as I entered. After a curt hello, I was told to proceed as I normally would. I was shaken but not sufficiently to lose my balance. When the hour was over, I felt it had gone rather well. She, however, did not think so. According to her, there were several problems. My pace was uneven; at times too slow and at times too fast. I needed to interact with the class as a whole, not just a few students. My grade for this session was a majestic C. I was not pleased.
The day after, she was back. Her report, reproduced below in full, says it all.
24-4-69, Mr. Hirji, Popatlal S. S., Maths – 1A
A much better lesson. The preliminary work — individuals doing problems on the board, got almost the whole class involved — hence the noise. They were really paying attention. Can you think of a way of retaining their interest without the noise? I still think you helped the pupils too much “That is wrong, correct it” instead of “Is that right?” But you definitely had much more class participation in working out the problems than before.
There was a nice balance between class work and individual work. The pupils were helping each other in the latter part of the lesson, which is very valuable as you can’t see everyone at once & I think they understand each others’ mistakes.
Some finished earlier than others. What can you do about this?
E M Connelly
Despite the laudatory remarks, my new grade was just a tad better: C+/B-. Some instructors are impossible to please, groaned I.
Truth be told, in the past four weeks, not only had I become eminently predisposed towards teaching but had formed an elevated opinion of my teaching ability as well. I felt that the performance of my students in the national exams would be as good as if they had been taught by an experienced teacher. I enjoyed their company. The queries they posed made me view mathematical ideas at a fresh angle. The staff were pleasant and helpful. The atmosphere was conservative, but it reflected the society at large. I covered the set topics on time, and conducted a review session. On the last test, most students met my expectations. I patted myself on the back, supervisor or no supervisor.
In the year 1967, Tanzania adopted socialism and self-reliance as the guiding policy for national social and economic development (TANU 1967). The component of SSR dealing with educational matters was spelled out in the document Education for Self-Reliance (ESR) released around that time (Nyerere 1967).
During the colonial era, education for the broad masses was severely restricted, both in quality and quantity. The entire system was structured by race. A Form IV certificate was a passport to better living, but only a few had the chance to acquire it. Villagers and town people were hungry for education for their children. Soon after Uhuru, our nation thereby witnessed a vast expansion of primary and secondary schooling. Yet, all was not well. A decade into nationhood, the educational pyramid remained an unfair one. A small portion of primary school leavers entered secondary school and a smaller portion of secondary school graduates entered a college or university. Those not able to climb the educational ladder were in a quandary: the skills they had acquired qualified them for clerical jobs in the public and private sectors. But such jobs were too few and the number of applicants was rising fast. An education system of that mode could hardly facilitate genuine socio-economic development for the nation.
ESR aimed to rectify this state of affairs through introduction of activities that would impart practical skills at all levels of the system. Each school, primary or higher, had to have a farm, dairy or poultry project, wood workshop, sewing project or a similar project. In addition to taking part in these activities, the students would be involved in keeping the school compound and classrooms clean and organized.
It was hypothesized that the students would benefit on four fronts: (i) They would learn the elements of skills that could enable them to engage in income generating activities; (ii) Their attitude towards manual work would become more favorable; (iii) The advantages of working in a cooperative setting would become apparent to them; and (iv) They would come to value socialistic cooperation as a way of life. A youth ingrained with such an orientation would enable the nation to confidently embark on the path of genuine socio-economic development based on mutual cooperation. That, at least, was the grand theory.
Like other schools, Popatlal had received a directive from the Ministry of Education on setting up ESR activities. The policy document laid down the basic philosophy, not practical steps. The ministerial directive lacked detailed instructions. What projects to undertake and how to implement them was left to the school administration.
It so happened that two members from our group had a keen interest in ESR: Shiraz, a physics student, and I. Being members of a radical student group, the University Students African Revolutionary Front (USARF), we were committed to the promotion, through practical and educational activities, of the ideas and ideals of African liberation, Pan-Africanism and Socialism. USARF worked hand in hand with the UDSM branch of the youth wing, TANU Youth League (TYL), of the ruling party, TANU.
The headmaster told us that the school ran a two acre shamba (farm). On the second day, we first had a talk with the teacher in charge of the school TYL branch. Upon getting a lukewarm reception from him, we took matters in our own hands. We visited the area to find a cleared location, but with minimal and not too systematic planting. It was a random distribution of cassava, pineapple and papaya plants. A few sugar cane clusters stood in two corners, but not much else.
Even to our agriculturally stunted brains, it was not a pretty sight. What could be done? The school maintenance worker came to our rescue. Hamisi said that the soil was fertile and the rains were sufficient. If student labor was utilized as required, a productive farm with a variety of crops could be set up. But his views had fallen on deaf ears. Finding us receptive, he showered onto us a ton of suggestions on setting up a viable shamba.
On the first teaching day, as school was to end, Shiraz and I went to the headmaster’s office. Though surprised, he had a sympathetic demeanor:
Ramji, Hirji, already having problems?
No sir, our classes have started well. We want to ask you about the school shamba.
Our unexpected reply puzzled him:
Oh, that. What about it?
We outlined our proposal on developing the school shamba. Would he permit us to take our student to work on it two days of the week? Not all of them, only the willing ones. While others played soccer, basketball, etc., our group would dig, clear, plant and water, two hours at a stretch. We outlined our cultivation plan but did not disclose then that, word by word, it reflected the advice we had got from Hamisi. To this day, I recall the expression on his face: delight mixed with astonishment. Without hesitation, he blurted out:
Sure, sure, you have my permission.
The school was just three years old. The staff had many things to take care of, he said. Hence, the farm had not received the attention it needed. In a hushed tone, he went on:
I will be frank with you. Our teachers do not like ESR. The students and parents hate it. They prefer volleyball or soccer to shamba work. But politics is politics; it has to be done even if it will lower educational standards. The district officer is not pleased at the state of our shamba. I feel ill at ease when I have to show it to visitors. So, if you can improve it, you will do Popatlal a major favor. We have the farming equipment in the store, and I will give you all the help you need
He conversed with us in the spirit of a conservative educator. A nice farm, though, conferred political advantage. His words had a racial tinge too; he talked to us as an Asian would to other Asians. It was implied that when Africans took over any project, they mixed it with politics and ruined it. We disagreed with that view but kept quiet. Our priority was to secure his cooperation in order to enable us to do what we wanted to do. I doubt he would have been as direct if an African was present.
That was the genesis of what soon came to be known as the Mapinduzi Shamba (Revolutionary Farm). Twelve students from Shiraz’s classes and twelve from my classes had volunteered for shamba work. For two days a week, and two hours at a time, we and the students became wakulima (farmers). When our students saw that we did not just stand and issue orders, but jumped into the trenches with them, their spirits rose remarkably. At the end, our brows were as sweaty, our clothes as soiled, and our hearts as content as theirs.
Indeed, shamba work became a fun activity. Loud chatter, jokes and laughter abounded. Those who handled the hoe, rake or spade awkwardly were teased but gently. And work got done; lots of it. Boys worked hard to impress the girls. Shiraz and I recalled our adventures in the National Service a year ago (see GUWTZ).
With a corner reserved for us, a fair amount of the area was cleared. Cassava, sweet potato, tomato, beans and pumpkin were planted and watered. Two students brought kidney bean, small red bean and pumpkin seeds from home. They were planted but to what effect, we do not know. To set up a viable farm with a suitable crop mix needs at least two years of attention and effort. But our stay here was for five weeks only.
Our last afternoon at the school was spent at the shamba. The students were sad to see us leave. Some had brought in their autograph books. For one, I wrote:
Live the life of a candle that consumes itself while radiating light and love to all around it.
And for another:
This world produces 4 times the food it needs yet one half of its population goes hungry. Should we not try our best to change this awful situation?
As we bade farewell to each other, I wondered which teacher would continue what we had started? Would the farm bloom into an oasis of nutritious vegetables? Not being an era of easy communication, we unfortunately do not know if our efforts bore decent fruits or not.
We told the headmaster about the fine assistance given to us by Mr Hamisi and requested that he get a zawadi (present). Mr Chaudry told us not to worry. He would get the sum of twenty shillings for his contribution.
Besides the shamba work, the other progressive activity we undertook at Popatlal was to hold a school-wide written quiz competition. The quiz had forty questions on science, world affairs, African liberation and national socio-political issues. There was a book prize for the three highest scorers. We had got them in advance from the UDSM bookstore. They were: Frantz Fanon’s The Wretched of the Earth, Sembene Ousmane’s God’s Bits of Wood, and Robert Tressel’s The Ragged Trousered Philanthropists. These works promoted liberation of Africa, socio-political interests of workers and peasants and socialism. We had estimated that a secondary school student of above average caliber would understand them. Looking back, I see we had miscalculated with respect to Fanon. He is a demanding read. Possibly, they appreciated the gist of his militant stance.
To our surprise, the headmaster was enthusiastic about the quiz. He may have felt that it concerned siasa (politics). Since political education was accorded a high priority in those days, it would enhance the school’s standing with the district authorities. We did not tell him that some of the questions would challenge the ruling party’s dogmatic approach to political education.
Our request for stencils and reams of paper met with his swift approval. Shiraz and I set the quiz and the secretary typed it onto the stencils. Five hours of work on the cyclostyling press on a hot afternoon enabled us to produce and bind about 100 copies. They were locked away in a cabinet in her office until the day of the quiz.
I recall about eighty students, mostly from Form III and Form IV, turning up. The questions were mostly multiple choice, pairing answers and filling in the blank type of questions. Five questions required a one sentence answer. One entire weekend was spent marking the papers and selecting the winners. We also wrote an answer scheme and placed it on two noticeboards. A typical question was:
Connect the freedom fighters with their nations: (A) Nelson Mandela (B) Amilcar Cabral (C) Agostinho Neto (D) Eduardo Mondlane (E) Malcolm X and (a) USA (b) Mozambique (c) Guinea Bissau (d) Azania (e) Angola.
The top scorer had 50% of the answers correct telling us that socialistic education for the youth had a long road ahead. The prizes were given out in a school assembly during our final week. The head took the opportunity to thank Shiraz and myself for our efforts on the shamba and the quiz, and praised the students who had volunteered for farm work. But when he asked other teachers to follow our footsteps, I do not think it went down well. Some of them had said that we were ruling party stooges spreading political propaganda. Had they told us that directly, we would have countered that propaganda comes in two forms, the capitalist form and the socialist form. In modern society, the former is drummed into people’s heads from childhood; we were balancing the picture by showing that on the other side lay a humanistic vision of society whose aim was to remedy the harsh reality faced by the common folk in Africa through cooperative effort and firm resistance to external economic domination.
Did the students gain from our projects? The time was too short for gaining a lasting impression. Yet, I feel they were impressed with our dedication, our respect for their views, and the fact that we struggled with the soil as hard as they did. Did they acquire a degree of respect for manual labor, and begin to appreciate the value of joint effort? We hoped so. The shamba was run in a democratic way. Open talk about what to do and how to do it was the norm. As we labored, we discussed global and local issues. They were keen to learn about life at the university. I wonder if we managed to convince them that one could be, in Chairman Mao’s words, both red and expert? That a doctor or a mathematician can at the same time be an activist dedicated to the struggle for the rights and dignity of the downtrodden? That politics did not always signify opportunism? Perhaps we did or perhaps we did not.
For us, it was a class in agriculture. Students from farming families educated us about plants that should be placed in proximity and those that should be apart; watering schedules for different crops; and improvisation when what you need is hard to find. We came to know them close up. We learned what was overlooked in the educational psychology lectures at UDSM. Above all, we worked like a family; they respected us, but as elder brothers, and not as aloof authority figures.
RACE AND RACISM
Like other nations in East Africa, our recent colonial past was structured on racial separation, inequality and prejudice. Some strides had been made in confronting racial barriers in education but many areas of society were still plagued by them. The Asian community, comprising 1% of the population, still occupied a privileged position in the social and economic hierarchy. From the colonial days, and with justification, popular opinion in Tanzania held that it was an inward looking, money minded community.
To our students, we formed a counter example. They saw two Asians who, in flesh and blood, punctured that typical image. I hope it made our students learn to judge people not from the color of their skin but from their behavior and actions. One thing I was pleased to note was that by the end of our stay, our students interacted with us as fellow Tanzanians rather than in terms of race.
Yet, we remained mired within racial boundaries. There were six UDSM students in teaching practice at Popatlal, three Asian and three African. Though we interacted well in school, the Asians traveled and lived separately from the Africans. Sadly, those divisions persist to this day (see GUWTZ).
A ROMANTIC TRIANGLE
We were young, energetic. Our hopes and dreams stretched beyond infinity. But it was not just socialism and teaching that fired our spirits. A fragrance of romance floated in the air as well.
From our days in the national service, Shiraz and I had been friendly with Zubeda. She was in the Biology and Chemistry with Education stream at UDSM, and good happenstance for us, her school for teaching practice was the same as ours.
We were ecstatic. Her petite, well-formed figure, sparkling eyes and irresistible smile ruled our inner thoughts. For the two grandly smitten fellows, to talk to her daily was a dream come true. While on one front, Shiraz and I were comrades in arms, on this front, we were irreconcilable rivals. It was an unspoken rivalry. The European hippie radicals of the day promoted free love. We did not share that outlook. For now she was amiable to us both. But ultimately, it had to be either him or me, and nothing in between.
Yet, what a majestic illusion it was. After we returned to DSM, we saw her only sporadically. Two months on, as the new academic year began, a stupefying sight awaited us. Zubeda walked hand in hand with another guy. She still favored us with a smile but the two of them were as inseparable as we were inconsolable. In no time, it became a solid bond. They got married upon graduation from UDSM and migrated to Canada to live happily ever after.
The budding proponents of scientific socialism had their heads in the clouds. We did not appreciate that there was a key sticking point between her and us. She was devoutly religious, going daily to the prayer house. Our secular, humanistic stand had placed a different set of priorities on our plates. She asked us now and then: Why do you just talk about politics?
Instead of taking her to the Amboni caves or the ocean front or the cinema during the weekends, we marked the quiz or did other school work. Perhaps our efforts reflected a desire to impress her. If they did, they served a good purpose. Should we have toned down our ideals to comply with her wishes? If one comes short in the days of youthful idealism, what can the future augur?
We just had to nurse our loss. Humming sentimental songs from Hindi movies helped. We did not talk about it, but I guess Shiraz was mired in a similar mood. Gladly, our comradeship and activism did not abate one iota. Had one of our students sent us this saucy vision of algebra, it surely would have consoled us.
Stop asking us to find your X
She is not coming back
Despite our versatility in the discipline of quantification, the margin of error in our romantic calculations was spectacular. Our emotional equations did not fathom that matters of the heart do not operate on a planar world where simple rules of Euclidean geometry prevail. They unfold on a wobbly spherical surface, where what is just ahead of you disappears on the horizon without warning. On a plane, the angles of a triangle sum to 180 degrees. On a sphere, they can add up to 270 degrees. The additional degrees of freedom converted our simple binary projections into a maze of implausible outcomes. We had mistaken wild dreams as representations of objective reality; a hazy mirage, an oasis of sweet date palm; an enchanting smile, a sign of true love. It was too late when we realized that in the emotional arena, the Darwinian law turned from the survival of the fittest to that of the flirty-est.
My second teaching practice was a low key affair. Lasting two weeks in September 1970, it was at the Azania Secondary School in DSM. I taught mathematics to Form 3A and Form 3B. This time my supervisor was Mr SR Nkonoki, a lecturer from the Department of Education of UDSM. He observed my teaching on two occasions.
His longer reports first noted my strong points: (i) I knew the subject well; (ii) Each time I began with a review of the last session; and (iii) My blackboard writing was clear and to the point. On the negative side, he noted that: (i) I did not beforehand ensure that chalk was at hand: (ii) I did not give enough time for the class to copy the material on the board; (iii) I talked too much: (iv) I did not go over issues that had not been understood well; (v) I rushed the class; (vi) I did not have a detailed lesson plan; (vii) More pupil activity was needed; and so on and so forth.
His grades are not stated on his reports, but I gathered that they were on the low side. A `good potential mathematics teacher,’ that was his verdict. It was clearly implied that reaching that potential would require time and effort, and even then, the desired outcome was not guaranteed
But that is not how I saw it. By the end of the two weeks, I felt I had done well with my class. As they say, when the man has spoken, he has spoken.
Azania School was a short walk from my house. My good high school friend Elias Kisamo was posted there at the same time. He taught physics. We spent breaks together over a cup of tea or soda and roasted ground nuts. I did not take up any extracurricular activity of the sort I had done at Popatlal. Not that my socialistic activism was on the wane. At this time, my priority was to work on an upcoming issue of Cheche, the USARF magazine. As its principal editor, I spent many hours after school and during weekends at the UDSM campus on this unpaid job.
Cheche gave space to prominent activists to air their views on burning social, economic and political issues of the day. It took the ruling party to task for its half-hearted implementation of socialism. It did not come as a complete surprise that two months later it was banned by the government of Tanzania (see Cheche for details).
REFLECTIONS ON TEACHER TRAINING
My undergraduate lectures on education at UDSM covered varied courses like child development, educational psychology, sociology of education, philosophy of education, curriculum development and teaching methods. Yet apart from Pavlov’s experiments with dogs and some of John Dewey’s ideas on pedagogy, there was hardly anything I retained from these courses. The presentations were too general and the material bore little relevance to the concrete conditions of Africa. Much of it looked as if it was lifted straight out of American textbooks and sources.
I found the education courses unattractive in the extreme. With their shallow ideas, the lecturers did not inspire. Their basic concepts projected and rationalized the tenets and values of capitalism. Usually, a couple of lectures into a course, I would develop an adversarial relationship with the main instructor. My class essays often contradicted his or her views. Even though they were better researched and longer than the essays of most students, all except one instructor marked them down. In the mathematics courses, I was at the top. In the education courses, I was lucky to get a passing grade.
My teaching practice reinforced my stand: these courses had not prepared us to teach, let alone to do it well. What I found most lacking was a subject oriented approach to teacher training. There are admittedly common issues and skills each teacher, whatever his focal area, must imbibe. But that is not the whole story. Teaching chemistry has challenges distinct from those encountered in teaching literature; teaching history is a far cry from teaching mathematics. Apart from being deficient in the basics of teaching, my training implied that in the deep, hazardous waters of mathematics instruction, I would have to swim on my own and without any life support.
Students typically regard a teacher as a monster out to make them sweat, whether they like it or not. The first barrier a teacher has to overcome is to impress upon them that she is not of that mold, that she is a sympathetic, friendly companion on a common journey. If that is accomplished without compromising her position of authority, she has gone a long way towards enhancing the receptivity of her words.
But that is half the story. A typical student also regards a subject like mathematics a monstrous entity. Hence, it is as critical to convey to the students that the subject she is teaching is as worthy of attention as say the music of the latest pop star or antics of their favorite soccer hero. I do not exaggerate. Her second primary task is to make them love the subject she teaches, be it history, English or mathematics. If at least a few of them continue to take a keen interest in it long after she has left the scene, she could not have asked for a bigger prize. She has ignited mental flames that will blaze on for decades.
The mind is not a vessel to be filled
But a fire to be ignited
This is especially apropos for mathematics. In Tanzania, as in nations across the world, it is the subject with the worst scores in the national exams. Moreover, it is a subject openly detested by a majority of the students. When of adult age, most of them are not in the least embarrassed to declare: I flunked mathematics, or I despise mathematics.
This is less related to what it is taught but more to the pernicious manner in which it is taught, the topics taught, and crucially, topics left out. As I have discussed these issues at some length in GUWTZ and Cheche, I refer you to these and other texts for elaboration. Suffice to say that my training to be a teacher was more in line with suppressing youthful fires, and producing robots who manipulated mathematics symbols effectively so to get good scores in the exams. And even in that regard, it did a poor job.
Whatever I learned about how to teach well, I did it on the job, through trial and error, and by learning from at times quite serious mistakes.
My teaching practice supervisors were smart, friendly, devoted, knowledgeable individuals. Sympathetic to the socialistic policies of the nation, they sought to raise the standards of instruction in schools. But both were locked into a traditionalist, standardized approach to teaching, and especially, teaching mathematics.
Their comments dealt with minutiae, mostly minor matters relating to style. What they stressed teachers generally acquire from experience. They took the contents as given. I was to stick to the syllabus. Details were discussed but big issues like what to teach, relevance and basic approach. were not broached. How can I make my students enjoy and love mathematics? That essential question was not on the horizon.
To gain positively from experience, one has to start with a sturdy base. Lacking that, we pursue the try-angle for too long, and stagger from error to error. A beginner finds herself demoralized in no time as her frustrations pile up. The litany of errors begin to fray the bonds that tie her to the students. Teaching turns into a series of routine motions one needs to perform to get the pay check. When an avenue opens up, she abandons teaching altogether. In the setting of abominable pass rates, the teachers of mathematics who persist are those who have nowhere else to go. They do not strive for academic excellence.
And the numbers say it. Mathematics teaching and performance in Tanzania today are at levels worse than was the case in the 1960s and 1970s. Fewer than 20% of the candidates sitting for the national Form IV examination tend to get a pass level grade in mathematics.
Assessing teaching ability through a couple of short classroom visits is dicey at best. While one can weed out the truly incompetent fellows from those doing a satisfactory job, further gradation is not of much value. Good teaching comes in many forms. Uneven progress, noisy classrooms, students staring into the air are not necessarily signs of ineffective teaching. Gauging teaching ability requires dwelling into the longer term bond the teacher forms with her students and the inner sense of respect they develop towards her.
The guidance provided by my supervisors was of lackluster quality. They were more attuned to nit-picking than to giving advice that would impel you to become a life-long, competent molder of young minds. When I today go over the outline of a model lesson plan given to me by Mr Nkonoki, I am aghast. Even then, I did not follow it. It was too rigid and excessively detailed. A teacher was made into an actor who has to memorize every word she will utter at specific times in the lesson.
I would not recommend such a plan to any would be teacher. I favor planning but abhor rigidity. Improvisation, flexibility, changing contents according to the mood and response of the class, and going backward when called for are essential tools of successful instruction. The path can vary but the key thing is that at the end of the day, you land where you had planned to land, and with most students beside you. These key precepts apply whether you are teaching at the primary level or the university level.
A discussion of the teaching of mathematics in Tanzania needs to include the introduction of Modern Mathematics (or Modern Math in the American parlance) that occurred in the late 1960s.
Modern Math originated from the work of the School Mathematics Study Group led by mathematician Edward Begel in the USA. Under the sponsorship of the US government, from 1958 to 1972 it produced school textbooks and training material relating to the teaching of mathematics in schools. The group was highly influential in its heyday affecting mathematical curricula and retraining thousands of teachers across the nation (Phillips 2015).
One key reason why this group gained such wide traction was that it promoted its work as a central pillar in the urgent task of overtaking the perceived technological and scientific advantage gained by the Soviet Union. This Cold War rationale not only raised its popularity and funding but also shielded it from initial critical scrutiny (Phillips 2015; Sawyer 2001).
Achieving a balance between rote learning (drill) and conceptual understanding (explication of general ideas), between facts (specifics) and ideas (theory), between the concrete (intuitive) and the abstract (logical) is an important issue in the teaching of any subject. While attainment of conceptual understanding and ability to apply it to a broad set of specific conditions is an ultimate goal, the path towards this is not a straightforward one. This issue is especially germane to the teaching of mathematics in elementary and secondary schools.
A principal justification for the development of Modern Math was that traditionally, the teaching of mathematics had too heavily relied on drill, intuition and specifics. Modern Math remedied that deficiency by placing general and abstract concepts right at the beginning. Ideas like sets, relations and functions were placed ahead of exercises in arithmetic, algebra and geometry. It was claimed that the conceptual understanding gained at the outset would make it easier for students to grasp the more concrete topics introduced subsequently.
The tension between the two poles of the discipline was a reflection of the tension between the professionals and the educators:
A good mathematician feels that teachers continually distort the subject,
and a good teacher feels that mathematicians continually obscure the subject.
But the outcome of introducing Modern Math turned out to be the opposite of the hypothesized one. In practice, it could not deliver the goods, as one teacher after next discovered. In stressing the abstract, it had gone to the other unworkable extreme. Despite the initial ballyhoo and wide adoption in the USA, Modern Math had died a natural death in that nation by the mid-1970s. While some of its topics persisted in mathematics curricula, as a basic philosophy of teaching the subject, it became an object of derision and humor.
Unfortunately, though not surprisingly, as Modern Math was on the wane in its country of origin, it was being introduced in Africa. With funding from the US Agency for International Development, the Department of Mathematics and the Institute of Education of UDSM spearheaded the acceptance and dissemination of the subject in Tanzania. Textbooks which were virtual copies of their American counterparts were printed, curricula hastily written but hardly any teacher retraining was done. My education lectures at UDSM did not dwell into why this transformation was needed here. The older experienced mathematics teachers as well as the new ones were essentially left on their own to deal with the strange expressions on the faces of their students as they expounded the abstract concepts of sets and one-to-one and onto functions.
Even as the US was recovering from this Cold War inspired disaster, mathematics education in Tanzania was plunging into a deep ravine from which it has not as yet escaped.
Today it is fashionable to lay most of the blame for the woes of the education sector in Tanzania onto the socialist policies of the past. This ideologically driven charge misses two key points. In the case of mathematics, a significant part of the blame lies in the uncritical adoption of a deeply flawed approach from the citadel of global capitalism.
Furthermore, it overlooks the paradoxical character of the `socialist’ policy of that era. While the ruling party document championed education for self-reliance, our experts and the Ministry of Education were going in the opposite direction. As long as plenty of funds were at hand, they had no qualms in letting the external funding agencies call the tune.
Such contradictory features so extensively affected all the sectors of the economy and society in those days that calling Mwalimu Nyerere’s policy a policy of socialism and self-reliance is akin to calling a lion a giraffe. In critical areas, it was a muddled policy of state capitalism based on dependency on the entities from the capitalist world to formulate, fund and implement most of the projects undertaken. I further discuss the policy of Education for Self-reliance in Chapter 6 and dwell on the character of these pseudo-socialist policies in several other chapters.
Teaching mathematics in ways that effectively instill its basic concepts into the developing mind on a long term basis requires starting from the concrete plane, ascending gradually to the conceptual plane followed by a return to the concrete, and repeating this cycle in a progressive fashion throughout. As an early critic of Modern Math observed:
In the early teaching of mathematics,
there is no danger of making the subject too concrete.
The danger is rather that the subject gets so far from the concrete
that it comes to mean nothing at all.
Telling tales is an essential pedagogic tool for mathematics and statistics. Students love stories, interesting stories. These include historic and contemporary episodes told in an enticing garb. The rewards of a story-based approach are lasting. It can make students who dislike these subjects change their attitude towards it, creating a new learning atmosphere in the classroom.
Take the case of how I introduced linear equations in two unknowns during my teaching practice at Popatlal Secondary School. I could have done better by starting with talking about, say, fruits; those they like best; those they ate yesterday; the fruits in season; and how much they cost.
After some noisy banter, I tell a story:
Juma and Salima go to the market. Juma buys two bananas and three mangoes.
From the same seller, Salima buys three bananas and two mangoes. Juma has paid
one shilling and ninety cents but Salima pays one shilling and sixty cents.
As they nod their heads, I ask:
If you want to buy four bananas and two mangoes, how much will you have to pay?
A scratching of heads ensues. A smart girl gets the correct answer by guessing. But she cannot explain how she got it. Then I say let us put x cents as the price of a banana and y cents, the price of a mango. And then we formulate the two equations in the two unknowns. Now the equations have a down-to-earth meaning. This is followed by graphing the equations and finding an approximate answer. The next hour is spent on the traditional methods of solving the equations.
I take the exercise a step further by reading aloud a short story about a father who buys a bag of oranges for his children on the way home from work. They love the fruit. His funds are stretched but the image of the young ones running excitedly upon seeing the oranges in his hand drives him on. He is tired, the bag is heavy. Along the way, he has to cross a busy highway. The fast moving traffic is blind to pedestrians. It takes him half an hour to cross the road, only to realize that the bag of oranges remains on the other side. Too exhausted to turn back, he trudges home benumbed and empty handed (Salaam al-Mundhri 2008). Reading takes five minutes, an unusual activity in a mathematics class. Yet, it brings tears to their eyes, and cements the lesson of the day firmly in their minds.
For changing the attitudes towards mathematics and improving the performance of the students, we must mobilize a veritable battery of tools — stories, poems, humor, history, puzzles, sports and games, anecdotes from the lives of famous mathematicians, instances of practical use and misuse of the discipline. Field trips, practical tasks and experiments should also be utilized when opportune (see GUWTZ for details).
Foremost, it is the mathematics teacher who has to be versed in the diversity of the educational methods. Yet, her training is sorely deficient in that respect. A typical teacher in secondary school is not aware of the intrinsic beauty or history of mathematics. Having come to it as a routinized discipline, she teaches it accordingly, through rote learning and drill. Is it a surprise that her students end up hating the subject? I reflect further on these and related issues in the final three chapters.