By Paul Chika Emekwulu
When I was younger I visited a classmate who I thought then was just a classmate and a friend for a study session. When I came back I reported to my mother by saying that I went to my friend’s house.
“That’s your brother, not your friend,” she corrected me. Without knowing why, I didn’t ask for any explanation. Maybe because I was too young! Unfortunately, she didn’t take it up from there and treat it as a teachable moment. She probably thought I wouldn’t understand, for as I said, I was too little and tender. Anyway, that would have been an opportunity for a genealogy class. Because of catching up with the world and perhaps for other reasons, the school paid more attention to teaching us about Mungo Park and Christopher Columbus who as they taught us discovered America in 1492 (always a Friday General Knowledge test item) in addition to the Lander Brothers (John and Richard), the European explorers who, in 1830 went on an expedition to find out the source of the River Niger, than teaching us about our roots as well. This was when the white missionaries were in charge of the mission schools. The school probably left that to individual homes to handle. It would have been better if the home had been involved by making family genealogy class a take home class assignment.
Now having said this, there is something I want to point out here. In our society today, compared to previous decades. The literacy rate is increasing astronomically. Almost every community in Igbo land has a boys’ and or a girls’ secondary school or a co-educational community secondary school. This has helped the literacy rate to be on the rise. One would have expected the reading culture to be on the rise as well but unfortunately, this is not the case, for the reading culture is below expectation.
People don’t read novels anymore. The cellular phone has taken over completely. Because of this, Nigerian publishers, quite unlike their American counterparts have abandoned publishing trade books (books sold in bookshops or bookstores) in favour of school books. This was even long before the coming of cellular phones. With the coming of cellular phones, it seems as if reading generally has been put to death. Because of this rise experienced in literacy rate, most offices in various mens’ and womens’ groups like CMO, “ụmụada”, “inyemọna”, CWO etc. are sought for, won and handled by men and women with at least First School Leaving Certificate which is the basic tool for measuring literacy in Nigeria. Any parent or guardian with First School Leaving Certificate is therefore a literate parent. Literate parents or guardians should be able to help their children with homework class assignments quite unlike their uncles, cousins, parents, grandparents or great grandparents etc. for lucky ones among us. Such parents or guardians should be able to recognize teachable moments and use such to reinforce the concepts the child was taught in school or use such to provide the child an entirely new learning experience. To do this, literate parents or guardians can start with subjects of their interest or as situations arise.
Doing so is not equivalent to homeschooling. It doesn’t necessarily has to be in mathematics. It could be English Language, like when a child says, “We are going back to the school for an after school evening program by 3 o’clock”, when a child means exactly 3 o’clock.
This should be a teachable moment where a literate parent or guardian has to jump in and let the child know that such an English expression is wrong. “By 3 o’clock” means “before 3 o’clock” and not 3 o’clock prompt. This is not being done enough in our time because the parent or guardian himself or herself is not free. He or she is not free because he or she is caught up in the same wrong use of word and doesn’t even care about it. Literate parents or guardians should do better and do their best with these teachable moments which are everywhere and here is one from personal experience.
Just a day or two after December 25, 2022, three boys of primary school age visited me at my house. They knocked at the gate and when I got closer I recognized two familiar faces. They are friends and I’ve had one or more interactive sessions with one of them in the past. An English proverb says, “A good turn deserves another.”
An Igbo proverb literally puts it this way, “A journey that turns out good is usually repeated.” in Igbo this becomes: “Ụzọ dị mma, a gaa ya ugboro abụọ.”
He liked his experience and now came back with two of his friends. I didn’t want to open the gate, so I decided to talk to them from inside. I’m I did for the interaction that would have taken place might not have taken us to the level we got to. The boys came for a specific purpose and from the looks on their faces they were determined not to have no for an answer. It was Christmas time and we all know how it goes at this time. I therefore knew why they came.
“Gbaara anyị Christmas,” one of them said in Igbo, an expression I don’t remember using when I was their age.
Feigning ignorance I said, “How is it done?”
(There was silence, and exchange of glances among them.)
“Give us money so that we can buy something,” another finally said.
As if what one of them said was not enough or was falling on deaf ears, they all uttered in Igbo saying, “Anyị bịara ka ị gbaara anyị Christmas,” thereby reinforcing what their friend had already said. One outstanding quality I noticed among these three friends is that they speak Igbo Language fluently despite the fact that they are private schooled.
If you are an Igbo man or woman you don’t only have to understand the Igbo Language when spoken, you have to also speak it well and write it well as well. The Igbo Language is in distress and being defaced by both the older and newer generations. The Igbo Language is more in distress because of a subtle movement that I call the Anglicizing movement. The Anglicizing movement can be defined as any formal or informal deliberate act, explicit or implicit aimed at belittling the Igbọ Language in favour of the English Language or any other foreign language.
Among others, examples if such acts include:
Spelling and writing of Igbo names using English alphabets and pronouncing such names in Igbo.
Preference for English names or any other name to Igbo names at baptism.
Speaking English or another language in places where Igbo Language should have been more appropriate.
When in public, you as an individual prefer being identified with your baptismal name over your Igbo name and so on.
These are all forms of the Anglicizing movement. For those who spell and write their names in English, the goal is to structure their names first, to look English and second, to sound English. One interesting thing about the movement is that there was no conference or meeting, formal or informal calling for the launching of this insult on the Igbo Language which can in another way be described as a preference for English Language alphabets over Igbo Language alphabets. It was and still an individual decision that has affected many lives and still affecting many lives. It is not about the vowels only but the entire array of Igbọ Language alphabets. The Anglicizing movement is just among other problems facing the Igbo Language. There are others like the Mixed Spelling Syndrome and what I call vowel interchange. I’ve written at least two published books about these problems and these are titled,
Somebody Should Help Me: Asụsụ Igbo Na-Ada Ụmị
When Spell Your Name is Equivalent to Spell Your Igbo Name in Igbo – A Mathematical Slant
The problems being addressed in these books may not be syllabus items in the Igbo Language curriculum in our schools but it doesn’t diminish their relevance. Moreover, it has been my strong opinion that any syllabus item or related item in any school subject that presents enough information should be given a chance to be a book and exist as a supplemental material, any argument against such is faulty.
Now, whether the subject is Igbo Language, mathematics, or just any school subject, when students are restricted to the syllabus certain things are evident.
The syllabus becomes their world.
Their freedom to think is jeopardized.
They loose the eagerness and curiosity to look beyond the syllabus
Mathematics for example, shouldn’t be about what was taught, how and where it was taught to us in the past. We’ve to go beyond what was taught, how and where it was taught. At times we think that learning has to take place only in the four walls of a classroom. Learning can take place anytime, anywhere and everywhere.
What presented itself was a teachable moment and wouldn’t have waited for Christmas to be over. A teachable moment at times is like water under the bridge, once it is gone it is gone. Teachers parents and guardians have to recognize teachable moments. Also, Christmas or no Christmas, school children on their part have to avail themselves of the golden opportunity to learn, the learning environment notwithstanding.
Back to my friends. Next I asked them what class each of them was in. Two of them are classmates and in primary 4 while the third is in primary 5. Now continuing I asked, “How many of you are really good in mathematics?”
(There was an awkward silence and no hands went up)
“So, what’s the problem?” I asked (to break the silence).
With a frowned face each of them seems to have an individual story to tell and I wanted to hear them all. They said that mathematics is not taught in such a way that it can be understood. Intuitively, I knew it could be a gender issue. Our primary schools (it doesn’t matter where), are fastly experiencing absence of male teachers who are exploring opportunities elsewhere and this is affecting certain school subjects including mathematics.
“Are you taught by a female or a male teacher?”
“Female,” frustratingly they all responded. This shouldn’t be the concern of these students alone. It should be the concern of parents and guardians, it should be the concern of the Ministry of Education, it should be the concern of the PTA, it should be the concern of the teachers themselves and that of the society, for something has to be done.
It wouldn’t have taken a rocket scientist to notice that our languages were changing. My language wasn’t the language they wanted to hear. To them it was about Ecclesiastes 3:1-8 and not about mathematics. In Ecclesiastes 3, the word of God says, for everything there is a season, a time for every activity under the sun. And of course a time for a math lesson and a time for Christmas which is December 25 of every year. Yes, there is time for everything and special cases have to be recognized and Christmas is one of them. To these children, the school is now over with its stress and responsibilities, it is Christmas and not time for math lessons. They had probably expected candies, money etc. Food is not on the list. To them, Christmas is a time they’ve been waiting for and now instead of getting gifts for Christmas, they are getting mathematics for Christmas. To these children, a math lesson is the least thing to expect as a Christmas gift.
Within me I said, “You don’t have to tell me for I already know why you’re here.”
I said, “You know I’m a good dancer. What kind of music do you want?” (They had earlier said, Gbaara anyị Christmas.”
“Yes, I know you are a good dancer,” one of them said.
“How do you know?” I asked him.
“I see you dance at the Church.”
I’ve ears for good music and I respond whenever one comes along. The dancing version of Glory be to God in the Highest with the various translations in Latin, and Igbo is my most favourite.
“Now I’ve a question for all of you,” I said
“What’s the last thing you did in mathematics?”
With a smile they regarded each other for a few seconds. At this point the tone of the conversation changed.
Casually, one of them mentioned LCM and HCF while others just confirmed.
“Are you sure you’ve all been taught LCM and HCF?”
“Yes,” they responded.
I wanted to make sure my questions are class appropriate rather than age appropriate since some children are delayed for some reasons. One of them is in primary 5 while the other two are in primary 4. One of them in primary 4 felt like,” Yes, I know your questions will be on mathematics.”
I said, “If you answer my questions correctly, then I will listen to what you are all saying about Christmas.”
Again, they looked at each other. I guess they were confirming my seriousness. At this point I could sense them saying, “This is not why we are here. Is it?” The body language said it all. The excitement on their faces was gone and I was determined to bring it back
I asked, “Again, have you all really studied LCM and HCF in class?”
“Yes,” they all answered in unison.
“What does LCM stand for and what does HCF stand for?”
For LCM they all shouted, “Least Common Multiple.”
Similarly, for HCF they all shouted, “Highest Common Factor.”
Since knowing what the abbreviations stand for is not the same with knowing what LCM or HCF really is, LCM and HCF can be compared to a mathematical formula, like the formula for finding each of the measures of the interior angles of a regular polygon.
There is a popular mindset among students in every generation that every conceivable mathematical problem has a formula for its solution. That is why at times all you hear is: “All I is need is the formula to get the answer.”
That’s not true.
This attitude is not new and is still gaining currency because of the way teachers structure their math evaluation test items. Students and teachers alike have this erroneous belief that every conceivable math problem must elicit a numerical response that has to be worked out or has a formula for its solution. That, again is a wrong mindset.
If a student can state an algebraic or a geometric formula needed in solving a problem, he or she is not necessarily in possession of the appropriate computational and logical thinking skills required in solving the problem.
In mathematics, if teachers start moving things arọund, students will start to drop the general belief that a formula is the ultimate tool needed for the solution of every mathematical problem.
Having the knowledge of a formula doesn’t necessarily equip a student with the necessary thought processes required in applying the formula for desired results.
With this in mind, I asked the following question:
“How will I be convinced that you actually know what LCM and HCF mean?”
“By solving problems based on LCM and HCF,” they responded.
“Exactly,” I said.
“But then just solving problems based on LCM and HCF is not enough.”
“By saying that what exactly do I have in mind?”
“The problems have to be solved correctly,” two of the boys responded.
“Yes, the problems have to be solved correctly,” I confirmed.
That is still not enough but we continued anyway.
“What is the LCM of 2 and 3?”
From the answers it was clear that the three boys didn’t understand what LCM is all about.
“Which is greater: LCM of 2 and 3 or HCF of 2 and 3?”
Several answers were given including 1 and none was correct.
(The HCF of two or more numbers is always less than the LCM of the same numbers.)
This is another way of saying that the LCM of two or more numbers is more than the HCF of the same numbers.
We looked at a few examples of how LCM relates to HCF.
We then spent more time in making sure the boys really understand the full meaning of LCM and HCF.
“How would you explain to your little sister or brother who should know what LCM and HCF mean but doesn’t?”
Several answers were given and none was correct. It was Albert Einstein, the Jewish – American philosopher who said that: “Insanity is doing the same thing over and over and expecting a different result.”
It was time to do something different otherwise, the result wouldn’t be different. I separated one of them from the rest and then said, “What’s the least number of oranges that when divided among two people, there will be a remainder of zero and nobody will feel cheated?”
“Two,” they answered.
“Exactly,” I confirmed.
“What will be the next number of oranges that can be divided equally among 2 people?”
“Four,” they answered.
“Exactly.”
The other boy was called back to join the rest. When he joined the rest, the question was repeated as follows:
“What is the least number of oranges to be divided among a group of two and then another group of three?” In other words, there are two separate groups of boys. One group has 2 boys and the second group has three boys.
“Do we have enough oranges?” I asked.
“We need more oranges,” one of them suggested.
“Do we really need more oranges, and if yes, why?” I asked.
“We need more oranges because we need a number of oranges that will be divisible by a group of 2 and then another group of 3,” one of them volunteered.
“Yes, we nerd more oranges.”
Unfortunately, when I was in primary school my teacher didn’t teach LCM this way by using concrete objects. Rather, he taught us how to find the LCM of whole numbers using prime factorization. Of course I didn’t know that then, but I know now
Concrete objects should have been used first, while abstraction which prime factorization represents comes next. Prime factorization is the breaking of whole numbers into prime factors.
For example, the prime factorization for 6, 12 and 24 are as follow:
6 = 2 x 3
12 = 2 x 2 x 3
24 = 2 x 2 x 2 x 3
(The factors are all prime factors).
We jointly defined a prime number as a number divisible only by itself and 1.
At the level we were our teacher never mentioned prime factorization.
That’s by the way.
Anyway, for 2 people the smallest number of oranges that can be shared equally among them is 2. If we leave the number if oranges open and leaving it open means we need 4 oranges, 6 oranges, 8 oranges, 10 oranges, 12 oranges and so on, it means we can give oranges to more boys. But then we need to stop otherwise, we’ll continue like a car without brakes. We can only stop by using the word “least” and we know what it means. The numbers 2, 4, 6, 8, 10, 12….. in that order are all multiples of 2. Here we have to remember what the word of God says in Matthew 22:14. The word of God in Matthew 22:14 says, “For many are called, but few are chosen.” We need to limit ourselves for we cannot choose all the multiples and that is what the word “least” will do for us. We cannot really find the least common multiple of 2 and 3 until we find the common multiples of 2 and 3. We cannot find the common multiples of 2 and 3 until we also find the multiples of 3 as we did for 2. It is only when we have found the multiples of 2 and 3 that we can now find the common multiples of 2 and 3. When we have found the common multiples of 2 and 3, we can now find the least of these common multiples. Wow! I got the message.
“Next is finding the multiples of 3 as we did for 2.”
“Exactly,”, let’s do that.
Doing so means we need 3 oranges for 3 people to share without being cheated if we leave it open again as we did for 2, we’ll have a similar situation. Then that means we need 3 oranges, 6 oranges, 9 oranges, 12 oranges and so on.
We know how to engage our brakes. Right!
Yes, we do!
Otherwise, again, we’ll look like a car without brakes.
What’s next?
You cannot talk of common multiples without talking about multiples. It is only when you’ve talked about multiples and common multiples that you can now talk about the least of the multiples, and that is the LCM.
When the lesson ended, they all left with smiles on their faces because they got their wish. Next, they left without looking back, running off with a lightning speed. They enriched my life and hopefully, they gained from the experience. Happy Christmas everyone!
