Mathematics and Population

Beniel Seka

One day, when I was in Form One at Old Moshi Secondary School our teacher of English language took us to the school library. Her name was Mrs Laing from Canada. She was also a teacher in charge of the library.

After a brief introduction about the library and its services, she allowed us to pick a book of interest and read. I picked a book called Mathematics for the Million. When she passed around, she found me holding the book. “Wow, mathematics for the million!” she remarked. She took the book and opened a few pages. She handed it back to me and said,”Congratulations.”

I did not know why she congratulated me but, I learned later that she was also a mathematics teacher. She knew that I had chosen an interesting book because she liked mathematics. One of my school mates told me that she taught them the compound interest formula which is applied in calculating population estimates

On World Population Day in March this year (2012), my son John, an advocate who works with the Tanzania Women’s Bank wrote to me the following message: ‘Today, the world population has reached seven billion. You can write a features article to show how mathematics is involved in population’.

 He had seen how I related mathematics with love and thought I could do the same with population. I had said that love is like mathematics: It adds your pleasure; it subtracts your frustrations; it multiplies your wealth; it squares your peace and it divides your emotions.

“Yes,” I said to myself. “One day I will do it. I thought of how statistics is involved in population. We can calculate means and standard deviations. After population census a lot of statistics will be involved in processing the census data. Statistics is a branch of mathematics. Percentages are also used to a great extent in making population predictions.

The population census in Tanzania has been scheduled to take place in Tanzania on 26^th August 2012. Data collectors have been trained to collect accurate data that will be used to make plans for Tanzania’s future as well as providing information to the world statistics. For example, projections based on prediction made after the last census may be checked. To what extent the predictions were accurate will be revealed by the real data.

In making the predictions, life expectancy, mortality rates and demographics are used. For example, demographic data available in 2012 shows that the global sex rate is approximately 1.01 male to 1 female.  All these use to a great extent, mathematical formulae and techniques. The compound interest formula is a typical example.

The world population has experienced continuous growth since the end of the ‘great famine’ and the ‘black death’ in 1350 when it stood at around 370 million. It is estimated that the world population reached one billion for the first time in 1804.  In1960 it reached three billion. It reached four billion in 1974 and five billion in 1987. However, in 1999 it reached six million.

It is worth noting that in the modern era, during the industrial and agricultural revolutions, the life expectancy increased drastically. For example it is said that the percentage of children born in London who died before the age of 5 years decreased from 74.5% in the period of 1739-1749 to 31.8% in 1810-1829.

Many countries in the developing world have experienced rapid population growth over the past century. For example, China’s population rose from approximately 430 million in 1950 to 590 million in 1953 and now stands at 1.3 billion.

According to the 2008 world population figures, the country with the highest population is China which recorded 1.33 billion living humans. The top ten populous countries in that year, with figures in brackets are as follows:

1.    China (1.33 billion)

2.    India (1.14 billion)

3.    USA (304 million)

4.    Indonesia (228 million)

5.    Brazil (102 million)

6.    Pakistan (165 million)

7.    Bangladesh (160 million)

8.    Nigeria (151 million)

9.    Russia (142 million)

10.                Japan (128 million)

The day of seven billion was declared by the population Division of the United Nations Organisation to be October 2011. How does Tanzania compare its population statistics with other countries? This is the question that will be answered after calculations have been made. Mathematics will be used, will it not?

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Mathematics Teaching is a Career

Beniel Seka

One of the greatest problems facing our schools is lack of teachers’ especially science and mathematics teachers. The problem is rooted in other problems such as low salaries, unsuitable teaching environment and inability to admit enough suitable students in the teachers’ colleges. Mathematics is feared by many pupils especially in secondary schools and records the highest failure rate for the Secondary O-level National examinations. This makes it very difficult to select students to train as mathematics teachers.

A teacher of mathematics is a member of a distinguished profession, and as such he/she has a responsibility to uphold and up-grade the standards of his/her profession and to support and draw help from the professional association called Mathematical Association of Tanzania (MAT/CHAHITA). Every mathematics teacher should join and participate in the activities of this association both at national and zonal level. For example, MAT/CHAHITA plans to hold its annual seminar at Makumira Theological College, Arusha from 17^th to 22nd September, 2012 and all mathematics teachers are invited to participate.

If you have an aptitude for mathematics, and you enjoy it, you can have even more fun teaching it. A good teacher will be learning new mathematics all his/her life. She/he will find that the very best way really to understand mathematics is to teach it to another person. The more you teach the more you know. If you know mathematics and you do not let others know it you are becoming selfish. Share your knowledge with others.

You as a teacher will find it especially challenging to keep abreast of new developments in mathematics. Many people have the opinion that mathematics is a static subject. On the contrary, no field of knowledge is as dynamic as mathematics. New mathematics is being discovered all the time. New teaching methods and techniques and teaching aids are being tried out, Many teachers have gone back to school to up-grade their teaching. Many teachers conduct interesting experiments in their own classrooms. MAT/CHAHITA has developed a scheme to award them through the Mathematics Teachers' Award.

Studies in mathematics education are full of interesting topics at all levels. Whether you are a beginner or a veteran, you can always find something new, interesting and almost unbelievable, in mathematics. As you study new topics in mathematics, you will have the thrill of occasionally discovering a theorem or idea which is new to you. If your idea is new to the literature in mathematics, you have made a significant contribution. If your contribution does not attract attention, it is still significant to you.

Although a number of teachers have been frustrated by the system and situations they find themselves in, a few have enjoyed the profession. These teachers willingly devote extra time to their and avoid being tempted to look for more highly paid jobs. When interviewed by the author of this article, one of the teachers said, “To me, teaching is a challenging and exciting carer. The beginning of each school year is an adventure. I have new ideas to try out.”

Another teacher had the following to say: “One of my students returned to visit me after he had made his mark in life. He thanked me sincerely for my contribution to his success.” This claim shows that good teaching is rewarding. It opens up friendship between you and your learners.

“One of the reasons why I like to teach mathematics is that I love the subject,” said an experienced mathematics teacher. “I am always intrigued by the many ways in which one can have fun with mathematics. I try to pass my enthusiasm for it on to my pupils.”

Our world has become more technical and scientific than ever before. The need for trained minds has rapidly increased. A few generations ago the need was for the academically strong individual to do the job. We can no longer wait for them do the job. After all we are not motivating them. Now that we have discovered almost unlimited sources of power, our priority should be to elevate the brains to direct this power.

The need for trained minds is especially acute in mathematics. You should realise that mathematics is the very likelihood of our modern world. Thus, when you teach mathematics, you have the satisfaction that you are making a significant contribution. So join the profession.

Mathematics was used to capture the illegal fishermen

BENIEL SEKA

“I used mathematics to capture the illegal fishermen in the Indian Ocean,” said the former Minister for Livestock Development and Fisheries who is now the Minister for Works, Dr John Pombe Maghufuli (MP) when delivering a speech during the 6^th Pi-Day celebration in 2009.  Surveys show that East African countries including Tanzania, suffer an annual loss of several million USD through illegal fishing. After his delighting speech, as he was leaving the venue, a number of journalists surrounded him asking him how he did it. He did not disclose the secret because the case was still in court and might interfere with the investigations being carried out.

The Pi Day event in 2009 took place at the Tanzania Institute of Education (TIE) on an open ground near the parking area. Pi-Day is celebrated on 14^th March in many countries of the world. This year’s (2013) celebration has been scheduled to take place at Mnazi Mmoja in the city of Dar es Salaam. The guest of honour is expected to be the Coordinator of Education in the Prime Minister’s Office for Regional Administration and Local Government (TAMISEMI), Mr Bernard Bakari. The celebration will start with a charity walk from Azania Secondary School and will pass through Morogoro Road and enter Lumumba Street Road to the chosen site at Mnazi Mmoja.

PI-Day is a day used to celebrate the mathematical constant pi pronounced as “Pai” to bring mathematicians and their supporters to deliberate on mathematics. Pi, represented by the Greek letter (symbol) π, is approximately equal to 3.14 when taken to two decimal places. The 3 at the left hand side is used to represent the 3^rd calendar month (March) and the number 14 represents the 14^th day of the month. Thus Pi-Day is on the 14^th of March. Thanks to the pioneer of the day, a mathematics teacher from San Fransisco, USA who suggested it in 1988. In Tanzania, the first celebration was held in Tanzania in 2004 when the curriculum Developer for Mathematics introduced at TIE.

A World Federation of Pi has been formed and mathematicians all over the world have acquired a platform for expressing their ideas directly through internet and what have you. Among the activities performed on PI-Day are: singing pi songs, discussing the application of mathematics in daily life and observing mathematics exhibitions which portray mathematics uses. At the climax of the celebration, all participants in the event sing the birthday song: HAPPY PIDAY TO YOU X 2. HAPPY PIDAY EVERYBODY. HAPPY PIDAY TO YOU.

Since 2004, the celebration has been gaining ground in terms of participation and awareness. The inclusion of pre-Pi Day activities, charity walk and exhibitions has made it even more popular. The Mathematical Association of Tanzania MAT/CHAHITA) which took over the organization of the event in 2007 has been appealing to supporters of mathematics education to help in funding the great event.

Why do we celebrate Pi? This is the question many people ask when they learn that there is Pi Day.  We do it because pi has a lot of applications, have a simple definition and can create fun. For any circle, the principle is without exception. It is universal, that is, it I the same whether it be applied to a small ring, to circular tins or bicycle wheels.  Application of pi has spread all over the world and even the universe where circles are involved.

Word wide, we hear of celebrations conducted at the San Francisco Explotorium where Pi Day was created by the phycist Larry Shaw in 1988. Like in Tanzania, many groups have announced their plans to mark the day. For example, the Ford Fussion, a car company, intends to celebrate the 2013 Pi Day. The firm will celebrate by letting its marketing team ‘write’ their hearts out about how much mathematics has influenced the all-new Fusion, making the car more fuel efficient and more technically filled.  The Ford Motor Company product development engineer, Gil Portalatin has this to say: “By celebrating Pi Day, we hope to get people to appreciate the impact that mathematics has in helping them to get from home to the grocery store.”

Mathematics was used to capture the illegal fishermen

BENIEL SEKA

“I used mathematics to capture the illegal fishermen in the Indian Ocean,” said the former Minister for Irrigation and Fishing who is now the Minister for Construction, Dr John Pombe Maghufuli (MP) when officiating the 6^th Pi-Day celebration in 2009. The event took place at the Tanzania Institute of Education on an open ground near the parking area. Pi-Day is celebrated on 14^th March in many countries of the world. This year’s celebration has been scheduled to take place at Mnazi Mmoja in the city of Dar es Salaam.

PI-Day is a day used to celebrate the mathematical constant pi pronounced as “Pai” to bring mathematicians and their supporters to deliberate on mathematics. Pi, represented by the Greek letter (symbol) π, is approximately equal to 3.14 when taken to two decimal places. The 3 at the left hand side is used to represent the 3^rd calendar month (March) and the number 14 represents the 14^th day of the month. Thus Pi-Day is on the 14^th of March. Thanks to the pioneer of the day, a mathematics teacher from San Fransisco, USA who suggested it in 1988. In Tanzania, the first celebration was held in Tanzania in 2004 when the curriculum Developer for Mathematics introduced at TIE.

A World Federation of Pi has been formed and mathematicians all over the world have acquired a platform for expressing their ideas directly through internet (piday@pidayinternational.org) and what have you. It was established for all who enjoy mathematics, to foster communication about mathematics and unique worldwide activities. Among the activities performed on PI-Day are: singing pi songs, discussing the application of mathematics in daily life and observing mathematics exhibitions which portray mathematics uses. At the climax of the celebration, all participants in the event sing the birthday song: HAPPY PIDAY TO YOU X 2. HAPPY PIDAY EVERYBODY. HAPPY PIDAY TO YOU.

Pi, represented by the Greek letter π, is defined as circumference of a circle divided by its diameter. It approximately equal to3.14 or 22 over seven as many people know it. The value of pi has been worked out to millions of decimal places but no pattern about its digits has been established up to now. Some people have been able to memorise pi to a certain number of decimal places.  Or instance pi is 3.141592 to six decimal places. Can you remember it? You probably need assistance, don’t you?

A mnemonic will do. Consider the statement ‘HOW I WISH I COULD CALCULATE PI’.  Count the number of digits in each word. The number obtained will be used to help you write pi to the specified number of decimal places. In this case the word ‘how’ has 3 letters; ‘I’ has 1 letter; ‘wish’ has 4 letters and so on in that order. Then you write the number of digits obtained in the same order. No doubt you see that the number formed is 3141592. You finally place the decimal point after 3 and obtain the value of pi to six decimal places as 3.141592. Create your own mnemonics to assist you remember more.

Pi has been given special priority among other mathematical constants because it has lot of applications and fun. Many objects around us have a circular shape. Many decorations and designs use circles to make them more attractive. Circular shapes involve areas, arcs (circular distances) volumes and surface areas. All these use pi in their calculation. These also have direct implication on cost. One needs to do some planning in order to minimize cost or maximize profit. In view of these things, pi has been offered a day for celebration.

On Pi-day participants take part in a charity walk chanting mathematical messages and a number of them carry placards expressing the same. Examples are: ‘mathematics is a queen science’; ‘Mathematics is a service subject in other disciplines’; and ‘Mathematics is fun’. Eventually the participants settle for a programme arranged in a particular venue. As mentioned earlier, this year’s Pi-day celebration will take place at Mnazi Mmoja. This is the first time that the event is being held outside an institution. The intention is to spread the activity so that many more people may access it. Schools and colleges outside Dar es Salaam may follow suit.

A prominent person is usually invited to address the participants. For example the 2009 celebration invited Dr John Maghufuli as guest of honour. During his address, Dr Maghufuli reiterated the importance of mathematics in industry and life in general. He cited his own example where he used mathematics to arrest the fishermen who were conducting the illegal fishing in Tanzania coastal water. This created some fun but his message was understood. Join them in this year’s encounter and subsequently promote mathematics in your country.

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Bado Watanzania wengi wanatumia futi

Beniel Seka

Futi ni moja ya vipomo ambayo vimeendelea kuwepo katika mfumo wa upimaji tangu tupate uhuru hadi sasa. Kipimo hiki kilichokuwa katika mfumo wa Kiingereza uliojulikana kama “imperial units” bado kinatumiwa kwa kiasi kikubwa na mafundi uashi, useremala na mafundi bomba.

Baada ya uhuru Tanzania iliamua kutumia mfumo wa metriki ambao kigezo kikuucha urefu  ni meta(m). Vigezo vingine vinavyoambatana na meta ni milimeta (mm), sentimeta(cm) na desimeta(dm) ambavyo ni vidogo kuliko meta. Katika mpangilio huo milimeta ndio kiwango kidogo kuliko vyote. Vingine  ni dekameta(dm), hektometa(hm) na kilometa(km). Katika mpangilio wa ukubwa kilometa ndio kiwango  kikubwa kuliko vyote.

Mfumo huu wa metriki hutumiwa na nchi nyingi duniani. Kinachoufanya ukubalike na nchi hizo ni urahisi ulioko katika kuhama kigezo kimoja kwenda kingine. Kutoka kigezo kimoja kikubwa kwenda kingine cha chini kinachofuata, unazidisha kwa kumi. Kutoka kigezo kimoja kidogo kwenda kingine cha juu kinachofuata, unagawanya kwa kumi. Kwa mfano, kubadili kilometa kuwa hektometa,  unazidisha kwa kumi, yaani kilometa moja ni hektometa 10. Kubadili milimeta kuwa sentimeta unagawanya kwa kumi kwa kuwa  milimeta 10 ni sentimeta 1.

Shule za Tanzania hufundisha mfumo huo wa metriki. Mifumo mingine hutajwa tu kwa ajili ya kulinganisha, lakini si kwa nia ya matumizi. Utafiti umeonesha kuwa wanafunzi hawana tatizo na mfumo wa metriki. Walimu wengine wamekwishawapatia hata njia za kukumbuka mpangilio ili kurahisisha kubadili kigezo kimoja hadi kingine. Kwa mfano, sentensi: ‘Mimi sili dagaa maana daima hupata kichefuchefu (mm,sm,dm,m,dam,hm,km)’. Pamoja na sentensi hiyo kuwa kejeli, lakini husaidia kukumbuka uhusiano wa vigezo hivyo.

Tatizo ni nini hasa? Tatizo lipo kwa jamii. Bado watu wengi katika jamii ya Tanzania hawatumii mfumo huu. Kama ulivyokwishaona hapo mwanzo mwa makala hii, watu wengi wanatambua na kutumia futi badala ya meta. Utasikia fundi seremala akizungumzia kitanda cha futi sita kwa sita au bati la futi 10. Vigezo vingine katika mfumo huu ni yadi, cheini, maili na inchi. Utasikia kwa mfano, fundi akisema “nipe misumari ya inchi nne” au “bomba la nusu inchi”.

Matumizi hayo humchanganya  mwanafunzi ambaye hakufundishwa vipimo hivyo. Mwanafunzi anaona kuwa mambo ya shuleni hayana matumizi ndani ya jamii. Anafikiri anayojifunza shuleni ni kwa ajili ya taaluma tu. Mazingira ya namna hiyo humfanya mwanafunzi asifurahie masomo kwa kuwa yanaishia darasani tu.Baadhiya mafundi waliwahi kuomba mtindo wa vipimo vya Waingereza vifundishwe sanjari na vile vya metriki.

Tatizo jingine analokumbana nalo mwanafunzi ni kutokijua kipimo. Hajui yadi ina urefu wa kufananishwa na kitu gani. Anajua meta inakaribia urefu toka sakafuni  hadi kiunoni anapokuwa amesimama. Hajui inchi ina urefu unaofananishwa na nini. Anajua sentimeta inalingana kwa urefu na ukucha wa kidole gumba. Hajui futi, lakini walau hii anaiona kwenye rula zilizotengenezwa kwa urefu huo na zinatumika darasani. Hajui maili lakini anajua kilometa moja ni umbali kutoka wapi hadi wapi. Anajua kutoka nyumbani kwenda hadi shuleni ni umbali wa kilometa ngapi. Yote hayo amejifunza shuleni.

Waingereza walipotengeneza mfumo wao wa upimaji walihusisha mambo yao. Kwa mfano, wana kizio kinachoitwa dazeni chenye dhana ya 12. Ndio sababu wakaweka kigezo cha inchi na kusema inchi 12 ni sawa na futi moja. Neno ‘futi’ ni neno lililotoholewa kutoka neno la kiingereza ‘foot’ ambalo lina maana ya unyayo, lakini tusiingie huko. Ishirini ni kizio kingine walichokitumia Walikuwa na kigezo cha uzani walichokiita ‘hundredweght(cwt)’. Tani moja ilikuwa na cwt 20.

Waliteua yadi kuwa futi tatu. Kwa nini tatu ikatumika wanajua wao. Pengine inatokana na misahafu.  Katika mantiki hiyo ukikokotoa unapata yadi moja ni sawa sawa na inchi 36, yaani  12 x 3. Halafu wakaja na kigezo cha cheini; pengine kwa sababu maalum. Cheini ni yadi 22. Kisha waliteua maili kuwa na cheini 80 ambazo ni sawasawa na yadi 22 x 80 = yadi 1760. Pia kulikuwa na kigezo kilichoitwa ‘furlong’ ambapo furlon moja ilikamilisha urefu wa futi 660 au cheini 10.

Wakati wanatutawala walitufanya tuone kwamba vipimo vya mfumo wao vilikuwa bora zaidi. Walitukaririsha vipimo hivyo vikaendelea kudumu ndani ya fikra za wengi waliosoma enzi hizo. Matumizi yalifanikiwa kwa kiasi kikubwa kwa kutumia mfumo huo kwa sababu mfumo wa metriki haukupewa kipaumbele.

Waingereza hao na washirika wao kama Marekani, bado wanatumia mfumo huo. Walijua fika kwamba Watanganyika hatimaye wangekutana na huo mfumo wa metriki. Waliandaa katika majalada ya madaftari, orodha za kubadili vipimo vyao kwenda vipimo vya metriki. Kwa mfano:

Inchi 1 = sentimeta 2.54 Meta 1 = inchi 39.37 Maili 1 = kilometa 1.6 Yadi 1 = sentimeta 91.44

 

Ukokotoaji huo kwa ajili ya kubadili ulikuwa mgumu. Hii iliwafanya baadhi ya watu kuridhika na mfumo huo wa kiingereza na kutotilia maanani mfumo rahisi wa metriki. Hili ndilo tatizo kubwa tulilorithi kwa hao wakoloni. Kuna wakati baadhi ya mafundi waliomba mfumo huo urejeshwe maana wanafunzi waliomaliza darasa la saba walikuwa wanashindwa kumudu mafunzo ya fani kadhaa za ufundi yaliyokuwa yanatumia mfumo wa vipimo vya Waingereza. Ombi lao lilikataliwa na sasa wanalazimisha tu. Tuendelee namna hii mpaka lini?

Kwa wale waliozoea vipimo hivyo vya Waingereza wanaweza kuanza kulinganisha ifuatavyo: Futi 10= meta 3; futi 8=meta 2 na nusu; futi 5= meta 1 na nusu; inchi 4 = sm 10 na maili 5 ni sawa sawa na kilometa 8. Badala ya kutengeneza kitanda cha futi 6  ambavyo ni karibu meta 2, mafundi wangetengeneza vya meta 2, na kadhalika.

Viwanda vingi hapa nchini sasa vinatengeneza bidhaa kwa vipimo vya metriki. Hata Waingereza na washirika wao wameanza kutengeneza bidhaa zao kwa vigezo vya metriki kuwahi soko la kimataifa. Kwa nini wewe uendelee kung’ang’nia vipimo vyao? Nashauri uachane nao. Nenda na wakati.

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UhuruMiaka 51: NiniKimebadilikakwenyeHisabati?

BenielSeka

Hesabunisomotuliloendeleanalobaadayakupatauhuru.Katikashulezamsingidarasa la kwanza hadi la nne, somolilijikitakatikakuhesabu, kujumlisha, kutoa, kuzidishanakugawanyanamba. Piahesabuzafedhazikiwapoankra, vipimonawakativilikuwavinafundishwa. Mkazouliwekwakwenyekukaririorodhayakuzidisha (Tebo) pamojanahesabuzakichwa.

VipimovilihusumfumowaKiingerza (British Emperial System of measurements). Katikamfumohuo, kulikuwanavipimovyaurefuambavyonipamojanainchi, futi, yadi, ‘cheini’ (chain)namaili. Vipimovyauzanivilikuwawakia, ratili, ‘hundredweight (CWT)’ natani. Vipimovyaujazovilikuwanipamojanapanti (pint), ‘kwati’ (quart), galoninadebe.

Uhusianokatiyakipimokimojanakingineulikuwamgumu. Kwamfano, futimojailikuwanainchi 12. Yadimojailikuwanafuti 3 au inchi 36. ‘Cheini’mojailikuwanayadi 22 namailiilikuwanayadi 1760 au chain 80. Wakia 16 vilifanyaratilimojawakatiratili 112 zilikuwasawana hundredweight moja. ‘Handrediweiti’ 20 zilifanyatanimojaambayoilikuwasawanaratili 2240.Nalodebelilkuwalinajazwakwagaloni 4.

Vitabuvilivyokuwavinatumikawakatihuovilikuwavimeandikwanawazungu.Vitabuhivyovilitumikahatabaadayauhuru.Hiiilitokananakutokuwepojitihadamaalumuzakuhamasishawazalendokuandikavitabuvyakitaaluma.Mfanommojanikitabukiitwacho ‘Misingiya Geometry’ (Foundations of Geometry) kilichoandikwana Kenneth Anderson mwaka1963. Kitabuhichokilitafsiriwakwa Kiswahili mwaka 1964.

Baadhiyenumtakumbuka vile vitabuvilivyoitwa ‘HesabuzaKikwetu’ villivyoandikwanaE.Carey Francis nawenzake. Hivivilikuwanapichayatwigakwenyejalada lake kuashiriakuwavilikusudiwakwa Tanganyika. Vingineni vile vilivykusudiwakwashulezilizokuwazinafundishwakwalughayaKiingereza. Kwamfano, ‘Highway Arithmetics’ vilivyokuwanapichazatwiga, simba, tausinamashuaviliandikwakwaajiliyashulezaAfrikayamashariki.

Baadaya Tanzania kuzaliwa, mabadilikoyalianzakujitokeza.Shuleza Kati (Middle schools)zilifanywasehemuyaelimuyamsinginakuitwa ‘Upper Primary Schools’. Shulenyinginezamsingizilianzakuingizadarasa la tanonakuendelea. Shulehiziziliitwa ‘Extended Primary Schools’.Kuanzia 1966 darasa la nanelilianzakuondolewahatuakwahatua. Kwahiyo, elimuyamsingiikawainaishiadarasa la saba. Mabadilikohayoyalisababishamabadilikokadhaayamitaala.

MojayamabadilikokwenyeHisabatini ‘hesabumpya’ zilizofanywakwamajaribiokatikabaadhiyashule .Hesabuhizizilitokananamkutanowawanahisabatiuliofanyka Entebbe, Uganda. MkutanohuouliofadhiliwanampangowaElimuwaAfrika( Education Services, Incorporated) ulisababishavitabuvyaHisabatikuandikwakwaajiliyamajaribio.Waandish 30 kutokaAfrika, MarekaninaUingerezawalihusika. Shule 16 zikihusishamadarasa 17 zilishirkikwenyejaribio.

Kuwepokwajaribiohilokulifunguaukurasampyakwabaadhiyawaandishiwazawa.Waandishihaonipamojana Raphael Mwajombena Michael Kinunda (wotesasamarehemu). Mwajombealiandika ‘Setininini?’ naKinundaaliandika ‘FurahiaHeabu’. Baadayajaribio, kitabu cha kwanza cha ‘Hesabu Tanzania’ kilitayarishwana ‘Mahematics Unit’ ambachonikitengokilichoundajopo la wandishikatikaWizarayaElimuyaTaifamwaka 1967.

Kusudi la mpangohuompyawahesabuulikuwakuwasaidiawanafunzikumuduhesabuza ‘kisasa’ (wakatihuo) kwanamna bora ilivyowezekananakuwasaidiawagunduemawazonadhanazakihesabuwenyewe. “Tunatakawanafunziwajifunzenawafikiriejuuyahesabu bora.” Hayanimanenoyaliyondikwanawaandishikatikadibajiyakitabu cha kwanza cha Hesabu Tanzania, Kiongozi cha Mwalimu.

Darasa la kwanza lilitawaliwanamawazomapyayaseti. Dhanayaungano lasetiilifundishwasanjarinatendo la kujumlisha. Dhanayasetitupuambaposifurisiosetitupuiliwachanganyasanabaadhiyawalimu. Baadhiyawalimualiwapotoshawanafunzikwakuwaambiakuwaunganonakujumlishanikitukilekile.Hiyosikweli.

Miakayasabinimwanzoni, Baraza la Kiswahili la Tanzania (BAKITA) liliundaneno ‘hisabati’ kuwatafsirisanifuyaneno ‘mathematics’ ambalonipanazaidikuliko ‘hesabu’.Hesabuni ‘arithmetics’ wakatihisabatiinajumuishahesabu, jometrinaaljebra. Hatahivyo, kunabaadhiyawatuhadisasawanaoendeleakuliitasomo la hisabatikwaneno‘hesabu’.

TaasisiyaElimuiliyoundwa 1975 kamashirika la ummachiniyaWizarayaElimu, ilipewajukumu la kukuzamitaalakiitaifa. Mwaka1979, TaasisihiyoilifanyatathminiyaufundishajiwaHisabatikatikashulezamsingi Tanzania Bara. Tathminihiyoilioneshakuwawanafunziwalikuwawanamuduvizurimadayaseti.Madazamlandanonayamkinizilikuwangumukwangaziyamsingi. Tathminiilipendekezamadahizoziondolewenaileyasetiibaki.

VitabuvipyaviliandikwanaTaasisiyaElimukwakuzingatiamapendekezohayo. Kablayavitabuhivyokusambazwa, WaziriwaElimualivionanakustukakwambamadayasetiilikuwepo. WabungewaJamhuriyaMuunganowalikuwawamehojiuhalaliwamadahiyokatikangaziyamsingi. Hivyoaliagizamadahiyoiondolewe. Hiiilisababishavitabuhivyokuchelewakutumikanakusababishawanafunzikuendeleakutumiavitabuvyazamani.

Badiliko la wazilinalojitokezakatikavitabuhivyoniutumiajiwavipimovyametriki. Tangumwanzowanafunziwanajifunzavipimovyameta,kilometa, sentimetanamilimetakwakupimaurefu. Kilogramu,gramunamiligramuvimeelezewakwaajiliyakupimauzani. Kwenyeujazo, litaimekuwandiokigezokikuu.Piautumiziwaalamayamkatoumeondolewakwendanaagizo la viwangodunianikuachakufanyahivyokuondoautataunaojitokezakwanchimbalimbali.

Hatimayevitabuhivyovilitokamwaka 1983.Vitabuhivyoviliendeleakutumikashulenihadi sera yavitabuvingiilipoanzakufanyakazi.Kwa sera hiyo,vitabuvilivyoandikwanaTaasisiyaElimu, ambayoilishabadilishwajinanakuitwaTaasisiyaUkuzajiMitaala (TUM), vilichukuliwanakampunizauchapishajikwakuvisasishanakuviboresha.

Vitabuvingi, vikiwapovyaHisabatikwaVitendo,vilipunguzatatizo la upungufuwavitabushuleni. TUMilibadilikajinatenanakuitwaTaasisiyaElimuTanzania (TET). Mwaka 2005 TET ilibadilishamitaalayashulezamsinginakusababishamabadilikokadhaayamhutasariwaHisabati.Muhtasariulitajaujuziunaohitajikakukamilishamalengoyaliyokusudiwa.

Mabadilikohayayalihitajimhamowaruwazaambapomwalimuanatakiwaamfanyemwanafunzikuwakitovu cha ufundishajikwakutambuauzoefuwamwanafunzinaafundishekwakuwajengeawanafunzimaanayadhana. Piawanafunziwanatakiwawajifunzekwavitendoiliwajijengeeujuzi.

Seminambalimbalizimefanyikakuwaelimishawalimukuhusuufundishajiunaozingatiaujuzi.AidhaTaasisiyaElimu Tanzania imeshaandaamiongozoyakuwasaidiawalimukufanyahilojukumujipya. Mambo hayoyamefanyikakatikajitihadazakuinuakiwango cha uelewawasomo la Hisabatihapanchini.

END

Mtihani wa Kuchagua Jibu sahihi ni Tatizo Kwa shule za Msingi

Beniel Seka

Mtihani wa kuchagua jibu sahihi una faida na hasara. Baadhi ya wadau, wkiwepo wanachama wa Chama cha Hisabati Tanzania (MAT/CHAHIA) tayari wameshalalamika kwa viongozi wa Elimu kuhusu mtihani huo bada ya Baraza la Mitihani la Taifa kuanza kutahini Hisabati kwa kutumia mtindo huo wa kuchagua jibu sahihi. Aidha baaadhi ya waliokerwa na jambohili wameshatoa maoni yao kuhusu muundo huu wa kutahini.

 Mtihani wa aina hii una madhara mengi ukilinganisha na faida.  Baadhi ya  ya hasara ni kuwepo uwezekano wa kubuni majibu na kuruhusu mazingira ya kuibiana au kupeana majibu. Tayari  matokeo ya mtihani wa aina hii yameshasababisha kizaazaa kwa wanafunzi na walimu wa baadhi ya shule.

Hivi karibuni tumepata habari kuwa wanafunzi 41 wa Wilaya ya Newala ambao hawajui kusoma na kuandika wamechaguliwa kujiunga na shule za sekondari. Katika ziara yake wilayani  humo, Waziri wa Nchi katika Ofisi ya Rais inayoshughulikia maadili ya viongozi, Kepteni Mstaafu, George Mkuchika, aligundua kuwa jambo hilo limefanyika katika wilaya hiyo. Alihoji inawezekanaje mwanafunzi asiyejua kusoma wala kuandika achaguliwe kuingia sekondari. “Inawezekanaje mwanafunzi akae miaka saba  bila kujua kusoma  au kuandika halafu akachaguliwa kwenda kidato cha kwanza! Kwani walimu hawatoi testi wakagundua hilo?”

Akizungumza kwa lugha kali, Waziri huyo aliwashutumu walimu kwa kuhusika na jambo hilo. Alisema kuwa hatua kali zitachukuliwa dhidi ya walimu ambako wanafunzi hao  walitoka. “Inawezekana walimu wa shule hizo walihusika katika tatizo hilo. Lazima wawajibishwe.” Waziri huyo  alitoa shutuma hizo alipokuwa anahutubia wakazi wa wilaya hiyo mwishoni mwa ziara yake.

Akihojiwa na Nipashe, mmoja wa wazazi wa mwanafunzi aliyechauliwa bila kujua kusoma au kuandika alisema, “ Mimi najua kuwa mwanangu hajui kusoma wala kuandika. Kinachonisikitisha ni kwamba nimeshtakiwa kwa kutompeleka mtoto huyo sekondari baada ya kuchaguliwa. Nimeona hakuna sababu ya kumpeleka mtoto huyo sekondari kwa kuwa hatafaidi chochote. Hiyo ni kupoteza ada. Hata hivyo bado nasumbuliwa. “

Katika mazingira tofauti, Naibu Waziri wa Elimu na Mafunzo ya Ufundi, Mheshimiwa Phillip Muluge,  akiwa ziarani mkoa wa Kigoma alifahamishwa kuwepo wanafunzi waliochaguliwa kuingia sekondari wakati hawajui kusoma wala kuandika.  Naibu Waziri huyo naye aliahidi kuchukuwa hatua kali kwa wahusika. Hivi karibuni hali kama hiyo imeripotiwa katika wilaya ya Ludewa. Inawezekana kuwa wako wengi ambao bado hawajatajwa kwa wilaya nyingine na hili kizaazaa na hatari sana kwa Taifa.

Tumekuwa tunashuhudia katika televisheni zetu tangazo la shirika la Hakielimu likieleza kuwa wako wanafunzi wengi waliko shule za sekondari ambao hawajui kusoma wala kuandika. Aidha wanafunzi wengi wameonekana wana mwandikombaya usiowezaa kusomeka kirahisi. Matatizo haya yameendelea kuwepo kwa kuwa hakuna hatua za makusudi zinazoelkezwa dhidi yao. Mtihani a kuchagua jibu sahihi umeongeza ugumu wa tatizo hilo kwa kuwa matokeo yake yanaaminiwa kwa ajili ya uchaguzi kwenda sekondari bilamchakatowa ziada.

Mwanafunzi  aliyemaliza darasa la saba huchaguliwa kuingia kidato cha kwanza baada ya kufaulu Mtihani wa Kumaliza Elimu ya Msingi unaotolewa na Baraza la Mitihani la Taifa. Baraza hilo lina kiwango cha ufaulu ambacho mtahiniwa akishapata alama  hiyo  (jumla ya maksi zake kwa masomo yote) au kukivuka, huhesabiwa amefaulu. Kwa mujibu wa sera ya serikali, mwanafunzi anayefaulu anastahili kwenda shule ya sekondari. Hatujasikia sera tofauti.

Mtihani huo unaofanywa mwishoni mwa darasa la saba ni wa kchagua wanafunzi kungia sekondari na si wa ufanisi. Mtihani huo hauna cheti. Cheti cha kumaliza elimu ya msingi kinatolewa kwa utaratibu mwingine. Kwa hiyo, mtihani unakuwa umemaliza kazi iliyotakiwa baada ya kutoa matkeo. Kinachobakia ni kuchagua wanafunzi kuingia shule mbalimbali zikiwepo za vipaji maalumu.

Mtihani huo wa Baraza hutumia muundo wa maswali ya kuchagua jibu sahihi. Tumekwishaona madhara ya mtihani huo hasa kwa madarasa ya chini. Mtihani wa aina hiyo unafaa zaidi kwa watu waliokomaa ili watumie muda wao kuchambua majibu. Mwanafuzi anaweza kuvuka  kiwango cha ufaulu kwa kubuni au kwa kuibia. Kama vigezo vingine havitatumika , basi mwanafunzi ana kila sababu ya kutumia mwanya huo kwenda sekondari.

 Mawaziri wetu wameahidi kuchukuwa hatua kali kwa wahusika. Hilo ni jambo zuri lakini linapaswa kufanyika kwa uangalifu. Sina uhakika kama kuna sera inayomkataza mwanafunzi kutofanya mtihani huo kama hajui kusoma wala kuandika. Kama ipo, mwalimu mkuu na wahusika wengine wana hatia. Kama haipo, inafaa kuiingiza ili tatizo hili lisitokee tena. Mwanafunzi asiyejua kusoma au kuandikwa hapaswi kufanya huo mtihani.

Katika ngazi ya sekondari, Baraza la Mitihani la Taifa hutoa mtihani wa  “maarifa” (Qualifying Test) kwa watahiniwa binafsi (wale wasiokuwa kwenye shule za sekondari zilizosajiliwa). Lengo ni kuhakikisha kwamba kabla ya kufanya mtihani wa kitaifa wa kumaliza sekondari na kutunukiwa cheti, mtahiniwa awe amefikia kiwango cha kuridhisha cha elimu hiyo. Wale wanaoshindwa kufaulu mtihani huo hawaruhusiwi kufanya mtihani huo wa cheti. Chuo Kikuu mwanafunzi haruhusiwi kufanya mtihani ikiwa itathibitishwa hajahudhuria idadi kadhaa ya mihadhara.

Kuna uwezekano pia walimu wasio na uaminifu au watoto wa sekondari, kumrubuni msimamizi kuwaandikia wanafunzi hao majibu kwa kuwa ni rahisi kuyanakili. Kama mtihani ungekuwa na maswali ya kujieleza ingewawia vigumu kufanya hivyo. Kabla ya mtihani wa Hisabati kutumia mtindo huo tulikuwa tunashuhudia wanfunzi hao wasiojua kusoma wala kuandika wakipata sifuri. Hilo limepungua sana katika mtihani wa mwaka jana (2011).

Jambo jingine la kuzingatia ni mkazo ambao mtihani huo unapewa. Bado kuna watu wengi ambao wanafanya kila wanaloweza ili watoto wao waingie sekondari. Pia viongozi wa wilaya ambazo shule zao hufanya vibaya katika mtihani huo huwajibishwa. Hofu hii huweza kuwaingiza baadhi yao kushiriki katika mbinu chafu za kupata matokeo mazuri. Vilevile shule ambazo hupeleka wanafunzi wengi sekondari huonekana bora zaidi ya zile zinazopeleka wachache.

Athari nyingine ambayo inaweza kulikumba Taifa ni walimu kupunguza ari ya kujifunza na wanafunzi pia kutomakinika katika kujifunza wakielewa kuwa mtihani ni wa kuchagua majibu. Hivi umewahi kufikiria kwamba ukimpa mwanafuzi wa darasa la nne anaweza kufanya vizuri hata kumpita wa darasa la saba, mradi tu aweke alama kwenye visanduku husika? Itakuwa balaa.

Kama nilivyoeleza mwanzoni mwa makala haya, mtihani wa kuchagua jibu unaweza  kusababisha kizaazaa na kufunika zile sifa za muundo huo wa mtihani. Waziri Mkuchika alipouliza kwamba wanafunzi hao walikuwa hawafanyi testi au kuandika matini (notes), alikuwa ameleta hoja nzuri. Sisi tunahoji kwamba  ni lazima mtihani huo uwepo ukizingatia gharama za kuuendesha na kuchakata matokeo? Lazima tukubali kubadilika.

End (Mwananchi)

HISABATI DARASA LA SABA: Ufundishaji wa Kipeo na Kipeuo

Kipeo na Kipeuo Drs 7

HISABATI DARASA LA SABA

Ufundishaji wa Kipeo na Kipeuo

             Beniel Seka

             sekabeniel@yahoo.com

                 0784498893

Kipeo na Kipeuo ni mada zinazowasumbua baadhi ya wanafunzi na hata walimu wa shule za msingi. Wengi wenu mtakumbuka lile tangazo la Hakielimu kuhusu yule mwalimu aliyekuwa  anababaika darasani wakati anafundisha namna ya kutafuta kipeuo cha pili cha 144. Alitumia njia ya kutenga ambayo hakuielewa. Vilevile kuna tangazo jingine linalohusu mwalimu mwingine akitumia njia shirikishi kuwafanya wanafunzi waelewe njia hiyo  ya kutenga na hatimaye wakalifurahia somo.

Muhtasari wa Hisabati Shule za Msingi uliorekebishwa mwaka 2005 umepunguza upeo wa mada hiyo na kuwataka wanafunzi watumie njia ya vigawo kutafuta vipeuo vya pili. Vilevile wanatakiwa watafute kipeuo cha pili cha namba mraba tu. Kwa hiyo walimu wanapaswa kujua kwamba mada imeshasahilishwa ili kuleta nafuu ya ufundishaji na ujifunzaji. Sasa wanafunzi na walimu wanatarajiwa waanze kufurahia mada hiyo.

Baadhi ya walimu wamekuwa wakifundisha bila wanafunzi wao kujua kama mada wanayofundishwa ina matumizi katika maisha yao. Ufundishaji unakuwa ‘mkavu’ hivyo wanafunzi hawapati hamu ya kulifuatilia somo. Ufundishaji unaozingatia ‘mhamo wa ruwaza’ unasisitiza kutumia uzoefu wa mwanafunzi katika kujenga mawazo mapya. Lazima mifano ya vitu halisi vilivyopo katika uelewa au mazingira ya mwnafunzi vitumike.

Kabla ya kujifunza vipeo, wanafunzi tayari wamejifunza eneo la mraba. Miraba ipo sehemu nyingi kama vile vyumba, kuta za nyumba, sakafu, bustani na mapambo. Dhana ya eneo la mraba huhitaji namba kuzidishwa kwa yenyewe. Kujua eneo kuna manufaa kwa shughuli za kiufundi, kiuchumi na kielimu. Kwa hiyo ni dhana yenye matumizi katika maisha.

Namba inayopatikana kwa kiuzidisha yenyewe  kwa yenyewe inaitwa namba mraba. Wanafunzi wanaweza kufanya zoezi la kuandika namba mraba zote kati ya 0 na 1000 kwa kwa kufanya: 1x1=1;2x2=4; 3x3=9; 4x4=16; 5x5=25 na kuendelea. Hivyo watambua namba mraba hizo kuwa ni: 1,4,9,16,25,36,49 na kuendelea hadi 961 ambayo ni 31x31.

Kuzidisha namba kwa yenyewe ni kuipa kipeo cha pili. Kwa mfano, mbili kipeo cha pili ni 2x2=4. Inaandikwa  2², mbili ya kipeo ikiwekwa juu kulia ya namba. Wanafunzi sasa wanaweza kufumbua mafumbo ya kutafuta eneo kwa kutumia kipeo.

 Kutafuta kipeuo cha pili ni kufanya kinyume chake, yaani namba gani izidishwe kwa yenyewe izae hiyo namba husika. Kwa mfano, 144=12x12. Namba iliyozidishwa kwa yenyewe ni 12. Kwa hiyo kipeuo cha pili cha 144 ni 12. Kipeuo cha pili kinaandikwa kwa alama √, kwa mfano, √144=12. Kama mwnafunzi anamudu vizuri tebo, hatapata shida kwanammba mraba. Kwa mfano, 36=6x6; 64=8x8; 81=9x9 na kadhalika.

Vigawo vya namba vinaweza pia kutumika kwa namba mraba. Kwa mfano, kwa kutumia vigawo 625=5x5x5x5=25x25. Kwa hiyo kipeuo cha pili cha 625 ni 25. Hali kadhalika 25 kipeo cha pili au 25x25=625.

Dhana ya kipeo inaendelezwa  kwa sehemu nadesimali. Kwa mfano, (⅓)x (⅓) ni ⅓ kipeo cha pili ambbacho ni sawa na 1/9. Vilevile (⅜)x(⅜) ni ⅔ kipeo cha pili sawa na 9/64. Kwa namba zilizoko katika desimali, unazidisha kama kawaida. Kwa mfano, 0.6x0.6 ni 0.6 kipeo cha pili sawa na 0.36.

Kipeo cha tatu kinaweza kufundishwa kwa kutumia dhana ya ukubwa au ujazo. Tayari wanafunzi wanajua mchemraba na kuwa ukubwa wake unapatikana kwa kuzidisha urefu mara upana mara kimo. Kwa kuwa urefu, upana na kimo vimelingana unazidisha urefu mara urefu mara urefu. Namba inazidishwa kwa yenyewe halafu kwa yenyewe tena. Namba hiyo hupewa kipeo cha tatu. Kwa mfano, nne kipeo cha tatu ni 4x4x4=64. Inaandikwa 4³=64.

Kufikia hapo wanafunzi wanaweza kupewa zoezi fupi la kutafuta kipeo cha pili cha namba  ili iwe namba mraba na hapo hapo watafute kipeuo cha pili cha namba hiyo mraba ili wazoee huo mchakato.

_______________________________________________________________________

Baada ya kuwapa taaluma hiyo ambayo ina matumizi katika maisha yao tunaweza sasa kutumia hadithi kukazia maarifa na kufanya somo liwe na mvuto ziadi. Tunaweza kuwasimulia hadithi ya Kipeo na Kipeuo Mahakamani. Kama huna hicho kitabu, ufuatao ni ufupisho wake;

Hapo zamani za kale, namba zilikuwa watu wa nchi iitwayo Hisabati. Kipeo ambaye alikuwa msichana na Kipeuo ambaye ni mvulana waliishi kwenye nchi hiyo. Kazi yao ilikuwa kubadilisha thamani za namba. Sifuri na Moja tu ndio hawakuwezekana kubadilishwa. Namba zilichoshwa na tabia hiyo ya kubadilishwa bila ridhaa yao. Hivyo, waliamua kwenda mahakamani kuwashtaki Kipeo na Kipeuo. Hatimaye Kipeo na Kipeuo waliitwa mahakamani kujibu mashtaka dhidi yao. Wote walikana mashtaka.

Hakimu aliwataka mmojammoja kueleza alichokuwa analalamika. Kwa mfano, Mbili alisema, “Mimi  ni namba tasa. Kipeo cha pili anapokaa  juu kidogo mbele yangu nageuka kuwa nne.”

Upande wa washtakiwa walimhoji, “Huoni una bahati ! Wenzako wote wanaogawanyika kwa mbili bila baki ni shufwa. Wewe peke yako tu ndiye tasa. Si bora ujiunge na wenzako  kina  Nne, Sita, Nane na Kumi. Wewe sio namba witiri bwana.” Mbili hakuwa na maelezo ya ziada.

Mwingine aliyejieleza ni Kumi. Yeye alilalamika kuwa kipeo cho pili anamgeuza mia. “Mimi nimeumbwa kwa sifuri moja mbele ya moja. Nina kipindi kinachojulikana kama ‘muongo’”

Walalamikiwa walimhoji kwa nini ajisikie vibaya. “Kwani sifuri mbili mbele yako zina ubaya gani. Hujasikia kuwa kuna kipindi kiitwacho ‘karne’?” wakili wa Kipeo na Kipeuo aliuliza.

“Nakataa. Siyo swali halali” alikataa wakili wa namba.

“Endelea. Jibu swali,” Hakimu alisema.

“Ndiyo, Mheshimiwa Hakimu. “Sina la nyongeza.”

Kilichowashangaza wengi waliohudhuria pale mahakamani ni jinsi Moja na Sifuri walivyosimama kidete kuwatetea Kipeo na Kipeuo. Wao walisema vijana hao hawawabadilishi thamani. Kipeo  cha pili cha moja ni moja na kipeuo cha pili cha moja ni moja. Hali kadhalika kwa Sifuri. Sifuri kipeo cha pili ni sifuri na kipeuo cha pili cha sifuri ni sifuri. “Hao hawana madhara kwetu. Tunaomba waendelee kudumu,” walisisitiza Sifuri na Moja.

Mashahidi wa washtakiwa, kina Jometri na Aljebra  baada  ya kuapa kusema ukweli na ukweli mtupu, walieleza jinsi Kipeo na Kipeuo walivyo muhimu katika kanuni ya Paithagorasi. Walionyesha vielelezo vyao vya pembetatu-mraba na jinsi Kipeo na Kipeuo wanavyohusika katika kukokotoa pande zake. “Wakifungwa kanuni hiyo itakufa,” walionya.

Baada ya kuchambua hoja za pande zote mbili, hakimu aliwaona hawana hatia na kuwaachia huru.

Kwa hadithi hii, unahusisha Hisabati na shughuli za mahakama ambayo pia inawafumbua macho kuhusu taratibu zake. Wanafunzi pia wanaweza kusoma mfululizo wa vitabu hivi kwa mfano Kipeo na Kipeuo Waoana na kipeo na Kipeuo Wapima Ukimwi kwa ajili ya burudani na kupata maarifa kutoka Nyanja nyingine.

END

Carl Friedrich Gauss: Wagunduzi 2

Beniel Seka

Kuna wakati kunatokea tukio la kustaajabisha. Watu wanabakia wakiulizana kulikoni? Jambo moja lilitokea huko Ujerumani miaka mingi iliyopita. Mtoto mmoja aitwaye Carl Friedrick Gauss akiwa na umri wa miaka 10 alimshangaza mwalimu wake kwa uwezo wake wa hesabu. Alionyesha kipaji chake ambacho kilishughulikiwa mapema.

Hadithi ya mtoto huyu ilisimuliwa hivi: Siku moja mwalimu wa darasa alilokuwa anasoma huyu mtoto, alikuwa na kazi ya kusahihisha madaftari ya wanafunzi. Aliamua kuwapa wanafunzi kazi ya kufanya hadi amalize kusahihisha. Aliandika ubaoni hesabu ifuatayo: 1+2+3+4+5+6+…+98+99+100=. Kwa maneno mengine, tafuta jumla ya namba za kuhesabia hadi 100. Maana yake ni kwamba 1+2=3, na 3+3=6; 6+4=10 na kuendelea hadi ujumlishe 100 ambayo ni namba ya mwisho katika mfululizo huo.

Mwalimu aliwaagiza waanze na kila mmoja afanye peke yake. Sekunde chache baadaye mwalimu aliwatazama wanafunzi. Alimuona Gauss ameketi na haandiki. Alimkaripia na kumwambia: “Nimesema ufanye hiyo hesabu, mbona hufanyi? Umeanza utukutu wako sio?”

“Hapana Mwalimu. Nimekwishapata jibu,” Gauss alijibu huku anatabasamu.

“Nini? “ Mwalimu alimuuliza kwa kumkazia macho.

“Hili hapa mwalimu,” Gauss alimuonesha. Mwalimu alishikwa na butwaa. Hakuyatarajia hayo.

 “Hebu nioneshe ulivyofanya,“ Mwalimu alimwagiza.

“Niliandika namba zote 100 katika mstari,“ Gauss alieleza.

1+2+3+4+5…+98+99+100.

“Kisha niliziandika tena kuanzia 100 hadi 1 kwa mstari chini ya zile za kwaza ili 100 ikae chini ya I; 99 iwe chini ya 2 na kadhalika mpaka 1 iwe chini ya 100. Ziko namba 100.

    1+  2+   3+…+98+99+100

100+99+98+… + 3+   2+    1.

Nilijumlisha kila safu ambayo kila moja ilinipa 101,” Gauss alieleza kwa kujiamini. “Kwa kuwa jumla ya safu ni 100, nilizidisha kwa 100  yaani 101x100 kupata 10100. Nilijua kuwa kwa kupanga mistari miwili nimejumlisha namba zote mara mbili. Ili nipate jibu linalotakiwa, niligawanya 10100 kwa mbili na kupata 5050.”

“Wou!” Mwalimu alimshangaa. “Sikuwezi.”

Inaelezewa kuwa mwalimu aliwaacha watoto wale wengine waendelee kufanya kwa njia ya kawaida, lakini hadi kipindi kinaisha, hakuna aliyemaliza. Mwalimu aliwashauri wazazi wake kumpeleka mtoto waokwenye shule za wenye kipaji.

Kazi ya mtoto huyu ilizaa kanuni moja ya Hisabati itumikayo kutafuta jumla ya namba mfululizo zenye tofauti inayofanana. Yaani, katika mfululizo huo, ukitoa ya nne kutoka ya tano ni sawa na kutoa ya tano kutoka ya sita. Kama namba ya kwanza ni a; tofauti sawa ni d na ikiwa namba mfululizo ziko n, kanunni hiyo inasema jumla ni  nusu ya {2xa+(n-1)xd}. Kanuni hii hufundishwa katika kidato cha tatu.

Carl Friedrich Gauss alizaliwa mwaka1777 Brunswick huko Ujerumani. Alikuwa mtoto wa fundi matofali. Baba yake huyo alipenda mtoto wake amrithi kwenye kazi hiyo. Tofauti na Isaac Newton aliebobea kwenye sayansi na hisabati, Gauss alionesha uwezo wake tangu utotoni kama ulivyoona kwenye simulizi. Vilevile inasimuliwa kuwa akiwa na umri wa miaka minne aliweza kugundua kosa baba yake alipomlipa mfanya kazi fedha zaidi ya alivyostahili.

Akiwa shuleni, umahiri na ujanja wake uliwavutia wengi na hatimaye akajulikana kwa ‘Duke’ wa Brunswick ambaye alivutiwa naye. Duke huyo alimpeleka Collegium Carolinum japo wazazi wake hawakupenda aende huko. Mwaka 1795 alienda chuoni huko  Gottingen. Akiwa chuoni hapo hakujua asome Hisabati au Filosofia. Hatimaye Hisabati ilimkolea zaidi, hivyo alichukuwa muda wake mwingi kukuza Hisabati na kuwa bingwa wa somo hilo.

Mwaka 1807, Gauss aliteuliwa kuwa Mkurugenzi wa jumba jipya la uchunguzi (Observatory) pale Gottingen ambako aliishi maisha rahisi yenye mazingira ya kujifunza. Alijaliwa afya njema hadi siku chache kabla ya kifo chake.

Gauss aliandika  majuzuu mengi ya ugunduzi wake. Aliandika kwa ufupisho na kuwaacha wasomi wajazilie. Aliandika Uhakiki wa Namba Tasa (Prime Number Theorem) ambayo ilithibitishwa miaka ya hivi karibuni. Huu uhakiki ni moja wa sehemu ya somo la siku hizi liitwalo ‘Uchambuzi wa Nadharia ya Namba (Analytic Theory of Numbers).

Gauss alisababisha Hisabati kuwa na matawi zaidi  ya yale ya kawaida yaani Aljebra na Jometri. Matawi hao ni pamoja na ’Elliptic  Geometry’ na ‘Hyperbolic Geometry’. Gunduzi za Gauss zilisaidia sana kubadili mtazamo wa Hisabati hasa kwenye kusaidia Fizikia kurahisisha dhana zake.

Gauss alifariki mwaka 1855 akiwa na miaka sabini na nane kama ilivyokuwa kwa Galileo Galilei mwanasayansi mwingine mkongwe.

END.

Galileo Galilei: Wagunduzi 3

Beniel Seka

Sayansi ya kisasa ilianzishwa na watu waliouliza maswali a yaliyohitaji uchunguzi kuliko ya wale waliotangulia. Kulikuwa na mapinduzi ya fikra za sayansi karne ya kumi na sita na ile ya kumi na saba. Mwanasayansi maarufu alikuwa mmoja wa wale walichangia sana katika mabadiliko hayo.

Galileo Galilei alizaliwa mwaka 1564, mwaka aliofariki mtu marufu aitwaye  Michelangelo. Alikufa mwaka 1642, mwaka aliozaliwa mwanasayansi mkongwe, Isaac Newton. Baba yake alikuwa mtu maarufu wa Florence nchini Italia. Galileo alipata elimu bora,kwanza katika uganga katika Chuo Kikuu cha Pisa na baadaye alisoma Hisabati na Fizikia.

Ugunduzi wake wa kwanza ni ule wa ‘pendulum’. Ukitaka kujua zaidi, funga jiwe dogo wa kamba na unin’ginize kwenye sehemu ambapo linaweza kujongea pande zote. Sukuma jiwe kidogo ili licheze pande zote hadi litulie kutokana na ukinzani wa hewa. Kama kusingekuwa na hewa tukio hilo lingendelea bila kukoma.

Ugunduzi wake wa pili uliomfanya maarufu ni uvumbuzi wa msawazo majituli (hydrostatic balance). Pia aliunda saa ya pendulum na kipimajoto (thermometer) cha kwanza. Alitengeneza mrija wa kioo angavu ambacho upande mmoja ulifunikwa kwa kioo cha mviringo mithili ya balbu. Aliweka maji kama kimiminika cha kuhisi joto. Siku hizi tunatumia zekaki. Vilevile, kwa kutumia elimu ya kupinda kwa mwanga unapopita maada angavu tofauti, alitengeneza moja ya hadubini na darubini za kwanza. Hata hivyo utambuzi zaidi alipewa Mdachi Jan Lippershey kwa ugunduzi wa vifaa hivi. Huenda Galileo alichelewa kuutangaza.

Kuanzia mwaka 1592 hadi 1610, Galileo alikuwa Mwenyekiti wa Hisabati Chuo Kikuu cha Padua. Alikuwa mhadhiri aliyependwa sana. Ilibidi ukumbi wa kuchukua watu takriban 2000 utengwe kwa ajili ya wahudhuriaiji wa mihadhara yake.

Kionambali chake kilimfanya awe na wasiwasi na nadharia ya Ptolomy iliyodumu miaka mingi. Nadharia hiyo ilisema Dunia ni kitovu ambapo sayari, nyota , jua na mwezi huizunguka. Alianza kukubali nadharia iliyopendekezwa na Nicholas Corpenicus iliyodai kuwa jua liko katikati na huzungukwa na dunia na sayari. Mwaka 1604 alitangaza rasmi kukataa nadharia ya Ptolomy.

Aliikarabati darubini yake ya mwanzo na kufanya gunduzi zilizofungua ukurasa mpya katika historia ya astronomia. Aligundua miezi (setilaiti) 4 kati ya 11 ya sayari ya Sumbula (Jupiter); madoa ya jua na mabadiliko ya mwonekano wa sayari ya Venus kama yaonekanavyo yale ya mwezi wetu. Hiyo pia ilisaidia kudhihirisha wazo alivyokuwa ametabiri Corpenicus.

Mwaka 1615 aliondolewa Padua na kurejea Florence. Kuunga mkono mawazo ya Copernicus kulianza kumsababishia mgogoro na kanisa lake. Alipewa onyo lakini aliruhusiwa kuandika kwa uhuru, ila alipewa masharti kuwa asihusishe masuala ya mambo matakatifu katika maandiko hayo. Alifanya safari mbili Rome kueleza msimamo wake kwa Papa Mtakatifu. Safari ya pili alikutana na Papa Mtakatifu Urban  VIII ambaye alimkaribisha vizuri.

Alipochapisha andiko lake liitwalo ‘Dialogue on the Great World Systems’ (Mazungumzo ya Mifumo ya Dunia Kubwa), mwaka 1632, aliwasha moto mkali wa kimahusiano kati yake na kanisa. Aliitwa Rome na kushtakiwa kwa kueneza uvumi. Alilazimishwa kukiri kosa na kupewa kifungo cha nyumbani cha maisha. Alirudi kwenye nyumba yake karibu na Florence kutumikia adhabu yake. Aliishi pale maisha ya upweke sana. Mwaka 1637 alikuwa kipofu na akafariki miaka mitano baadaye akiwa na miaka sabini na nane.

Getting Old is to Grow

Beniel Seka

The above statement was uttered by one retired female officer of the Ministry of Education and Vocational Training (MOEVT) in 2012. Her name is Mrs Urassa aka Mama Meta. She mentioned those words when we met at Professor Isaria Kimambo’s residence in Makongo Juu, Kinondoni District. Both of us had gone to see Mrs Mary Kimambo, the professor’s wife who was sick and bed-ridden. She had been invalidated and could not walk.

I had arrived at the house a few minutes earlier and led to a room where Mrs Kimambo was bed-ridden. When Mrs Urassa joined us, I immediately recognised her. “I presume you are Mrs Urassa,” I said without waiting for introduction. She looked puzzled. She was wondering where she had seen me. I read her thoughts and continued, “I met you at the Ministry of Education in 1977.”

“Wow!” she exclaimed. “So many years ago! You have very good memory.”

“Yes. Thirty six years ago. But I remember you because you signed my clearance to enable me to join a masters degree programme at the University of Dar es Salaam in 1977. Every time I look at the clearance certificate in my file, I remember you. The Ministry was reluctant in giving such a release because it usually caused a shortage of teachers in the school. You helped me to obtain it in good time.”

“Aha” I remember it now. That was when I was in charge of the section dealing with higher education. I am glad you were successful. I retired a few years ago and started an NGO for the senile known as ‘Getting Old is to Grow (GOIG)’.”

I immediately recalled Goig in Mbezi Beach along Bagamoyo road or Mwai Kibaki road, to be more up-to date. Conductors of commuter buses always call out passengers disembarking at that bus stand. ”I know the place,” I assured her.

“You are welcome to join,” she expressed her wish to let learn more about the organization. “And who are you?” I can’t recall a specific thing about you,” she confessed.

“My name is Beniel Seka. After obtaining a Master of Science degree, I was employed by the Institute of Education at the main campus of the University of Dar es Salaam as a Curriculum Developer. Mrs Kimambo had been at the Institute much earlier and served as Curriculum Developer for Home Economics previously called Domestic Science.”

“He is the founder of Pi Day in Tanzania,” said Mrs Kimambo, who had been following our conversation closely. “He has written a book called ‘Siku ya Pai (Pi Day)’ published by Kapsel Educational Publications.”

“What is pi day?” inquired Mrs Urassa.

“Nothing very strange,” I said with confidence. “You will recall when you were in school you learned about pi. Your teachers must have told you that pi, represented by the Greek letter π is 22 over 7 or 3.14. The 3.14 version has been used to create a day to celebrate the mathematics constant and it has been named Pi Day.”

“I did not like my teacher because he wanted us to cram formulae which involved pi. He did not tell us where pi came from. He just forced us to remember the formula for calculating areas of circles and volumes of circular tins or tanks,” she complained.

“Don’t worry, it is quite simple. Pi means circumference of a circle divided by its diameter. You probably know that circumference is the distance around a circle and diameter is the length of the chord through the centre. You can do this practically to verify that pi is approximately equal to 3.14. You can use various circular objects to do this. At the end you can find an average to obtain your pi. I’m sure you will enjoy the activity because the value you will obtain will be yours.”

“I am enjoying it now. I wish I could go back to school. I would surprise my teacher,” she said with a sign of relief.

“You have learnt something new, haven’t you? Remember that you said getting old is to grow and therefore to learn new things as you grow,” I reminded her.

“Yes I proclaimed that. You said that 3.14 were used to set a day for pi. How was it done?” she inquired further.

“The figure 3 is used to represent the third calendar month, that is March and 14 represents the 14^th day. So March 14 is Pi Day and it is celebrated worldwide.”

She could not hold her happiness about what she heard. “What do you do on Pi Day?”

“We sing piday songs, conduct charity walk chanting mathematics slogans, watch mathematics exhibitions and listen to speeches given by mathematics educators. At 1:59 we sing the popular birthday song: ‘Happy Pi Day to you x2; happy Pi Day everybody; happy pi day to you.’ Pi calculated to 5 decimal places is 3.14159. The number 159 is used to mark the climax of the celebration and is called the pi hour,” I explained. “Pi Day celebration in 2013 will be held at Mnazi Mmoja.”

“I will be alert. I will also inform those who are not aware of this.  I have grown haven’t I? Keep it up and don’t hesitate to join us.”

 “I agree with you. Getting old is to grow,” I said in appreciation.

The 2013 Pi Day celebration took part as scheduled. It was preceded by pre-Pi Day activities conducted at Azania Secondary School. The guest of honour at the Pi Day was the Assistant Director of Education in the Prime Minister’s office for Regional Administration and Rural Government (TAMISEMI), Mr Maulid. Pupils and teachers marched from Azania Secondary School to Mnazi Mmoja, singing and chanting mathematical messages to the onlookers and the public in general. It was a successful celebration.

“I hope I will live longer to see more of these activities. I want to learn more,” she concluded as she said prepared to leave.

“By the way, I almost forgot,” I told her. “There is another occasion for pi. On 22^nd July is Pi approximation day. This day has been set for those who think twenty two over seven deserves recognition. Thus 22 is for the day and 7 is the seventh month of the calendar year.”

“I will be ready for it,” she said and left.

Very recently I met Mr Oseno of Tanzania Institute of Education. He always addresses me as a senior citizen to honour my retirement. “How are you senior citizen?” he greeted me.

“I am fine. Actually I am growing. Remember getting old is to grow,” I told him jokingly.

“Goig,” he said indicating that he was aware of the organization. “Join them in Mbezi Beach”, he suggested. “I know you have come to inform me about pi approximation day. I know this year it is on Monday 22^nd July. I am aware of it.”

“Well, I hope you will enjoy the day,” I told him.

End.

.

SOMO LA HISABATI SIYO GUMU

NA BENIEL SEKA

Somo la Hisabati limeendelea kuwa somo linalowatesa wengi. Hali ya wanafunzi kuliogopa somo hilo na kuendelea kufanya vibaya imeendelea. Tufanye nini? Hebu kwanza tuangalie simulizi hii.

Siku moja msichana aitwaye Eva na rafiki yake aitwaye Roda walikuwa wanaongea. Mwalimu mmoja aitwaye Vesika aliwakuta na kuwasalimu. Wote wawili walikuwa wanamjua Mwalimu Vesika kuwa ni mtaalamu wa Hisabati. Walionekana kumfurahia kwa namna walivyomsalimia kwa uchangamfu.

Baada ya salamu, Eva alimuuliza, “Hivi mwalimu naweza nikajua Hisabati?”

Mwalimu Vesika alimtazama kwa shauku ya kujua kulikoni. “Kwa nini usiweze? Kwani Hisabati imekufanyaje?” alimuuliza kwa huruma.

“Nimeliona kuwa ni somo gumu. Nilipokuwa shuleni sikuwahi kulipenda. Hata mwalimu nilikuwa simpendi maana alikuwa ananigombeza nilipokosea. Matokeo yake sikujiamini hata kidogo nilipokuwa nafanya mtihani.”

Roda aliyekuwa anamsikiliza mwenzake kwa makini aliongeza, “Mimi niliachana nalo nilipokuwa kidato cha pili tu. ‘Majometri’ yake na ‘makongruensi’ yake yaliniweka mahali pabaya. Nilifeli Hisabati kidato cha nne na hadi sasa sijalitafutia dawa. Kwa sasa nasikia kuna kazi ya kujiendeleza lakini inahitaji Hisabati. Sijui nitaimudu vipi.”

Mwalimu Vesika aliwatazama kwa huruma huku akikuna kichwa. “Itabidi mbadilishe msimamo wenu kuhusu Hisabati. Lazima mjenge urafiki na Hisabati. Somo la Hisabati si gumu kama baadhi ya watu wanavyodai. Kama ungekuwa umefuatilia mtaalamu mmoja wa Hisabati akielezea katika redio, televisheni na magazeti kuhusu kauli hiyo potofu, ungekuwa umeanza kuchukua hatua.”

Hapo Eva hakujizuia kutamka, “Kwa wale wanaoziweza wanasema si ngumu. Sisi ndio tunaopambana nalo, ndio tunauona ugumu wake. Naomba uniambie siri ya kufanikiwa?

“Hakuna siri. Ni suala la kulipenda, kusikiliza na kutii maelekezo na kufanya mazoezi mengi. Kwa kufanya mazoezi mengi utapenda kusoma vitabu mbalimbali ambavyo vitazidi kupanua uwezo wako wa kufikiri na kubuni njia mbadala za kufumbua matatizo. Wakati mwingine unaweza kumzidi hata mwalimu wako. Kumbuka, ulimwengu wa sasa una mitandao yenye taaluma nyingi. Unaweza  kujikuta unaenda sanjari na utandawazi. Unasemaje, Eva?”

Eva alisema, “Mwalimu hapo unatuvika vilemba vya ukoka. Mtu kama mimi niwe mtaalamu wa Hisabati. Hata intaneti inikome! Si itakuwa moja ya maajabu ya dunia, kitu ambacho hata mlima Kilimanjaro pamoja na umaarufu wake haukuweza kupewa!”

“Hapana,” Mwalimu Vesika alimkatalia, “Haitachukuwa muda mfupi, lakini inawezekana. Katika historia ya binadamu, wako watu walioanza kwa hali duni lakini kwa kujitahidi wakaweza kufanya mambo makubwa na kuishangaza dunia. Hata nyie mnaweza. Unajua mimi nilipokuwa darasa la kwanza niliwahi kurudishwa nyumbani inende na mzazikwa kuwa nilikuwa sina maendeleo mazuri katika somo la Hisabati. Mtaa mzima ulijua jambohilo. Nilifedheheka sana.”

“Ikawaje?”  Eva aliuliza kwa shauku.

“Kutofanya bidii kwangu kulitokana na mwalimu wangu kunipiga sana nilipokosea Hesabu. Hii ilinisababisha niwe mtoro. Hali ikazidikuwa mbaya.” Mwalimu Vesika alifafanua. “Ilibidi nibadilike. Nilianza kujitahidi na nilipofika darasa la nne, tayari nilikuwa nafanya vizuri kuliko baadhi ya wenzangu. Nilipofika darasa la sita nilianza kuwa tishio kwa wenzangu baada ya kuwa wa kwanza darasani kwenye mtihani wa muhula katika somo la Hesabu.”

Mara mvulana mmoja aitwaye Mika akajiunga nao. Aliwasikiliza na kusema “Mnaongelea Hisabati? Mimi kwa sasa hainipi taabu. Mwanzoni ilikuwa kasheshe. Kuna siku mwalimu alisema, “nusu mara nusu” ni “robo”. Nilishanga na kusema tangu lini ukazidisha kitu kikapungua? Ukizidisha unatakiwa kupata namba kubwa zaidi. Nusu ni kubwa kuliko robo, Hapo nilianza kufikiri hisabati inadanganya.”

Roda alidakia na kusema, “Ulibadilika lini?”

“Sikubadilika kwa muda mrefu. Nilianza utundu. Walimu walibidi wanivumilie, la sivyo ningeishia pabaya. Hatimaye waliniita Ziro, maana nilikuwa natoa majibu yasiyokubalika, yenye uchizi. Siku moja mwalimu wetu aliuliza ‘nane gawanya kwa mbili unapata nini?’ Mimi nilijibu, ‘inategemea umegawanyaje? Kwa wima au kwa ulalo?” Wanafunzi wote darasani walicheka. Wao walijua jibu ni rahisi yaani nne. Walimshngaa ‘Ziro’ alivyokuwa anamsumbua mwalimu. “Una maana gani kugawanya kwa wima?” Mwalimu alimtaka atoe maelezo.

“Nilikwenda ubaoni na kukata mstari katikati ya tarakimu hiyo 8. Kwa ulalo unapata sifuri mbili. Kwa wima unapata tatu mbili zilizogeukiana nilieleza. Tangu wakati huo walimu wakaanza kuuliza maswali kwa uangalifu. Hii ilinifanya nitumie mantiki kukabiliana na changamoto za hisabati. Nikawa nalifurahia somo.”

“ Kwa kweli kuna njia nyingi za kulipenda somo. Wengine mpaka muitwe ziro. Hii nitaisimulia popote,” alisema Eva huku akicheka.

“Njia pekee ni kuongeza kujiamini. Hisabati hufuata mantiki. Ukishaielewa dhana ni rahisi kujijengea kanuni mbalimbali zinazohusiana nazo. Vilevile unaweza kugundua njia mbadala za kufumbua matatizo. Pengine njia utakayogundua ikawa rahisi kuliko zile ulizojifunza. Utajisikia mwenye furaha sana.”

“Ni kweli kabisa, mwalimu.” Yule mvulana alikubaliana naye. “Nilipoelewa maana ya pai, kwa mfano, niliweza kuunda mwenyewe kanuni ya kukokotoa mzingo wa duara (mduara) kwa kuzidisha pai mara kipenyo. Baadaye nilikuja kuikuta kwenye kitabu. Hii iliniongezea kujiamini.”

 Kama ulivyoshuhudia mazungumzo hayo, upenzi wa somo la Hisabati ndio nguzo kuu ya kulimudu somo hilo. Ukifanya mazoezi mengi utaweza kugundua njia mbalimbai za kukabiliana na ugumu wa somo hilo na hapo hapo ukawa mgunduzi. Unangoja nini? Chukua hatua sasa.

Kisha aliendelea kusema, “Kitu cha ziada cha kufanya ni kuhudhuria kongamano, midahalo au semina mbalimbali za Hisabati. Ni muhimu kuzitafuta ili ufaidike na mawazo yatakayotolewa na wataalamu. Kwa mfano, chama cha Hisabati cha Tanzania (MAT/CHAHITA) kilifanya semina ya Hisabati huko Chuo Kikuu cha Tumaini tawi la Iringa kuanzia 12 Septemba hadi 17 Septemba 2011. Je, ulibahatika kuhudhuria? Mwaka jana iliafanyka tarehe 17 Septemba hadi 22Septemba katika Chuo cha Makumira, Arusha. Mwaka huu inafanyiha Mbeya mwezi huo huo wa  September.  Ni vyema basi ukafanya jitihada ya kutafuta ufadhili ili uhudhurie. Kazi kwako.

How I Surprised the Malawians

Beniel Seka

In June 1997, the Tanzania Institute of Education (TIE) offered me a study visit to Malawi Institute of Education (MIE). By then I was the Head of Department at TIE. This was a programme of exchanging experience and expertise among members of East and Southern African Curriculum Organisation (ESACO). Such visits were conducted under a memorandum of understanding called ‘ESACO’ spirit. In this spirit, the host institution would incur costs of accommodation and up-keep while the visitor would meet travel expenses.

This offer was also given to the Financial and Administrative Manager of TIE, Mr Patrick Komba. Both Patrick and I were used to very hard conditions of life. We wanted to make an adventure and therefore decided to go by bus because there was a bus that operated between Dar es Salaam and Lilongwe. From Lilongwe we would take a bus to Blantyre that would drop us at Domasi near Zomba where MIE is located. We inquired about this venture and were supported by our administrators.

The MIE officers knew about the trip. However, no communication was made available to them about our means of travel. They knew for sure that we would travel by air. They also had an idea of the date of travel. They kept waiting for confirmation so that we would be collected at Lilongwe International Airport.

We purchased tickets for the bus to Lilongwe,  Royal Twiga International Coach Trans that was scheduled to leave the city of Dar-es Salaam at 5:30 am and reporting time was 11:30 am. The bus terminal was at Kisutu. I lived in Sinza while Patrick lived in Msewe. We agreed that we should be at Kisutu by quarter to five. We knew that a slight delay would cost us. I woke up quite early and arrived at Kisutu station at ten minutes to five. I was disappointed to learn that the bus had left a few minutes earlier.

Sympathisers advised me to board a Mbeya bus that was leaving immediately. “This bus is very fast. It will catch up with the Malawi bus before it reaches Iringa,” they shouted. They helped me get into the bus and we left immediately. It drove at a high speed that scared me. We drove past Msamvu in Morogoro but the Twiga Trans was beyond sight. A stand-by passenger told us the bus had just left. Our bus driver drove even faster and the Twiga bus was seen passing Mazimbu.

Our driver hooted continuously to alert the Twiga bus to stop. When it slowed down, our bus overtook me and covered the road to prevent it from a possible revenge. The Twiga Trans made a complete stop to avoid collision. The conductors of our bus helped me carry my bag as I disembarked and jumped into the other bus. There were cheers from my previous bus indicating that they were happy about the success. At last their driver would drive at a lower speed.

When I entered the bus, I noticed that it was full of cargo. No doubt it left earlier deliberately to allow more space for the cargo at the expense of passengers. I tried to locate Patrick but could not see him. “Patrick Komba,” I called. Nobody replied. I realised soon that Mr Komba had also been left behind.

The bus drove up to the border at Songwe where we stopped for a number of hours before crossing the Songwe Bridge to enter Malawi. My eyes were on all vehicles entering Songwe from Mbeya. I thought Patrick might have boarded any of them to attempt a connection. But Patrick was not in any of them. So I would go to MIE alone. What a menace. I even   decision to travel by bus. What would our hosts think of us?

After entering Malawi we drove several kilometres of rough roads in North Malawi and entered Mzuzu at the middle of the night. We managed to enter Lilongwe early in the morning. Monday had found me in Malawi. The Twiga bus was destined to end in Lilogwe. I made a connection at Lilongwe by boarding a bus to Blantyre the former capital of Malawi. I asked the bus conductor to drop me at Malawi Institute in Domasi. “We don’t stop at MIE but Domasi,” they informed me.

When I highlighted, I found a man at the station. I asked him where MIE was and how to get there. “There,” he pointed. “I can take you there by a short-cut. It is not very far. I agreed involuntarily. We crossed a small tributary and entered MIE campus. He took me through the back side of MIE until we reached one of the administrative buildings. I paid him some token and thanked him for his assistance. At last I had arrived.

I had known a little bit about MIE. The Director for the Research and Evaluation Unit had visited TIE some time ago. We had also met in Nairobi for a conference in qualitative research method. As I stood there waiting for the, an MIE officer greeted me. He noted that I was a visitor but why did I not have a host? He inquired about my visit and I said that Mr Jere should be informed about my arrival. He took me to the Director’s office to wait for hi m. He was informed by the secretary that the Director was attending an urgent meeting.

The officer followed him in the meeting room and delivered the message. “What!” shouted some members at the meeting. “A Tanzanian from TIE has arrived?” Three was a sigh of relief. The meeting had been called to discuss how the visitors would be received at the airport. No flight or arrival time had been received and this was bothering them a lot. MR Jere was relieved to attend his visitor. Meanwhile, the remaining members continued to finalise the visitor's visiting schedule.

When the Director joined me in his office we hugged and chatted for a while. I informed him about Patrick’s failure to catch the bus and that he would join me a day or two later. He introduced me to the head of the Research and Evaluation Department, Mrs Kishindo with whom I would work for the entire visit of two weeks. I was provided full board accommodation at MIE rest house.

I learned from Mrs Kishindo’s briefing that my former teacher, Mr George Michongwe was a member of her department. He was my History teacher when I was in Form Three at Old Moshi Secondary School. I was among his best student and he was disappointed when I dropped History in favour of Additional Mathematics. Unfortunately it was not possible to take both subjects simultaneously. Mr Michongwe had served as Minister in Dr Kamuzu Banda’s regime and had fled to Tanzania after misunderstanding between him and Dr Banda.

We met again at the University of Dar s Salaam in the Education Foundation course. He still blamed me for not studying History that would have made me famous. However, he praised me for perseverance and working hard. I worked briefly with him when I was conducting research at Klerruu Teachers’ College where he was a tutor.

Patrick arrived four days later. He joined me in visiting the Examination and Certification Board and University of Malawi in Blantyre where we visited the Centre for Social Research. We were also given an opportunity to visit Lake Nyasa which they called Lake Malawi. At the end of the tour, Mrs Kishindo did not hesitate to say,” I did not know that you were so cooperative, meek and understanding. Mr Michonwe had threatened us by saying: You’d better prepare yourselves. If he is Beniel Seka I know, then he is going to challenge you. But I have found you are ready to help in looking for solutions to pertinent problems.”

 MIE allowed us to spend a night at the prestigious hotel called Capital Hotel along Chilembwe Road in Lilongwe at its own cost.  We were very grateful for that. When bidding us goodbye at Lilongwe after escorting us, Mr Maliakini, on behalf of MIE administration said, “You really surprised us. But you have also shown that it is possible for us to come to TIE by bus.  In that way, we shall be able to make more frequent visits. We wish you a nice journey back home.”

END