My Math Visitor

By Vic Odarve

Except for private institutions, the country’s federal state universities and colleges were plagued with teachers’ union strike. Students stayed at their homes waiting for trouble to settle and resume their classes afterward. But one student refused to stop her schooling!

So I am lucky to teach in one of the private universities. No teachers’ union strike!

Engineering Mathematics 1

Engineering Mathematics 1

In one of my visit to the university library to read some local newspaper, one of the students approached and requested me to have some explanation on the math solutions previously discussed in the classroom. “Okay, what part of the topic?” I told her. She opened her notes and showed the steps where she got the problems and I did the explanation and discussion. Satisfied, she said, “Thank you, Sir, I understand already”, she said. Then I said,” How was the result of your last quiz?” “I got 90 % score, Sir”, she smiled. “Good, and on next quiz you must get a perfect score”, I replied. Then she slowly walked quietly and slip away. I paid little attention. She got a good score. So nothing to worry!  She may just serious enough and want to do more on her study every day.

Family tour-Chocolate hills,Philippines

Family tour-Chocolate hills, Philippines

Then, the next quiz came. Only 5 out of 234 students in a class got a perfect score. And she was one of them!

It was then that I gave her an attention. I noticed that during the class session, she sat at the end of the last row of the seats in a packed and overcrowded room. She listened intently and focused to my discussion. No friend, no talking even on her seatmates as if she knew none of her classmates! “She must be too shy”, I guessed.

The semester math course was about to finish. I gave the final quiz before the final exam. “This would be the last quiz”, I told the students. Then what was the result? She got perfect score again! She had done it! “She is a good performing student”, I told myself.

The next meeting I returned the quiz scripts. She was absent. For almost two weeks, I could no longer see her. Then I inquired the course representative and her classmates what happened to this girl. To my surprise, none of her classmates knew her. She vanished and I received no word from her and her whereabouts! I am completely blank!

After two weeks my phone rang. The girl informed me that she returned to her school at the federal university since the teachers’ strike was over. She thanked and told me that she learned and enjoyed math during her stint in my class. It was then I knew that she was my math- visitor. She was not my student! She refused to stop schooling! Then she returned to where she belonged!


Nature Teaches Mathematics

By Vic Odarve

Not everybody loves mathematics! “I don’t like numbers. That’s why I did not take up engineering courses”, said one student. This summed up everything on his schooling! Since kinder up to high school, he has performed miserably in mathematics. But one thing is certain; whether he likes it or not, mathematics is a part of his everyday life. Unknown to many, math already existed in nature the moment we took our first breath of life. What we do is simply marvel mathematics!


The “petals” flower form spirals curve describing Fibonacci’s numbers.

Also nature teaches mathematics! Nature and math works in never-ending harmoniously. From the particle world of billions of stars spreading in our cosmic ocean, mathematical laws seem to work keeping the universe in order. From Newton’s law to Kepler’s Laws of planetary motion and Einstein’s law of relativity, math offers our deep understanding on things around us.

Nature has its own way of expressing mathematics! Patterns such as hexagonal beehives, spiraling Fibonacci flowering trees, fractals flowers and plants, fruits and leaves, and many more are actually mathematics!


Fractals are repeating geometric pattern. They are rough fragmented geometric shapes that can be repeatedly subdivided into parts and that each small portion can be viewed as a replica of the whole. These are geometric designs we teach in classrooms. Since repeated, in mathematics is called iterations or algorithms. Newton’s method, simplex programming, Gauss Seidel, Jacobi, and many more algorithms are among few examples. Some plants exhibit simple recursive operations that after a few number repetitions or iterations their shape can be recognized as a repeated geometric pattern. It is mathematics! Fractals in nature are always pleasing to our eyes – clouds, plants leaves and fruits, galaxies, shells and many more. Plants such as sunflowers, ferns, cabbages, and papaya leaves are among a few of them.

Leaves arrangement suggests repetition

Leaf arrangement suggests repetition

Honeycomb structure is also a nature mathematical exhibition. Look at its hexagonal shape. Its cross-section is made of the comb, a latticework of hexagonal cells made of beeswax. It is an expression of geometry in action. The design might be the most economical in terms of labor and wax; hence expressing least cost in economics and linear programming in mathematics!
Fibonacci number is a simple example illustrating the existence of sequence in mathematics. Just like arithmetic and geometric sequences and progressions. But in the Fibonacci sequence of numbers, each number is the sum of the previous two: 0,1,1,2,3,5,8,13,21, etc. And these numbers are exhibited in various “family trees” and patterns of spirals of leaves and seeds. The image of a nautilus shell approximates the sequence of the Fibonacci’s number in nature. Once again, nature teaches math!

Fractals, Fibonacci‘s shells, and beehives are iconic images that capture the beauty of nature displays in mathematical expressions. These displays are like ornaments to the eyes of the mathematicians. A living math in motion in order for us to love and appreciate nature! Indeed, nature teaches mathematics!

Smartphones Boost Your Math Skills

By Vic Odarve

It is a wonderful world. Just a tip of your finger and the world will go round and round. Almost everybody loves this gadget… a cell phone! The cell phone is one of the most versatile electronic gadgets in the market today.This gadget becomes a necessity for everybody. Latest cell phone features include a camera, internet access, storage for pictures, audio, and video files, and many more. These things can help college students with all aspects of their studies. Just press the call button and you are connected to your loved ones and friends throughout the world. Talking and chatting! Young and old people alike… cell phones are their indispensable partners wherever they go. It seems our lives are difficult without this gadget.

Mathematical Formulas from these textbooks are stored in smartphones.

Mathematical Formulas from these textbooks are stored on smartphones.

But college students use these cell phones in other means… boosting their math power! It is a learning tool to develop math skills. Take for example, a student in Calculus! While studying under the shade of the old oak tree at the university campus, he forgot to recall the formula in the Differential Calculus. The product and quotient rules of differentiation! What else did he do? With a cell phone, he simply scanned the picture files and refreshed the formulas on the screen with ease. That’s how it is. Instead of pictures, mathematical formulas are put there! And instead of audio for love and rock songs, recorded audio files on discussion or explanation of the mathematical formulas are put there. Storages for picture and audio files on his phone are filled with mathematical clips. Cell phones become a storage file for his studies. That keeps boosting his math understanding and skills! The student is the one who does the recordings and arranging the formulas for his easy access and reference.

It is his real world with phones… maximizing his gadget to boost his math power. “In my

Smartphone boosts your Mathpower

Smartphone boosts your Mathpower

spare time, Sir, I spent listening to my recorded audio and helped me a lot to remember the previous mathematical theorems, principles, and formulas which are often used in mathematics”, he added.

This way he familiarizes the mathematics by listening to the audio discussions and scanning thru pictures of formulas instead of memorization. No wonder this student is one of the top achievers in my class in Calculus!

It is now a different world. It is different from our college days. Technology really works wonders! Cell phones are now a partner with students in studying mathematics. Like Higgs particle that gives mass to everything that will interact with it, cell phones give boost your math power to those who make it as a tool in mathematics. It seems this gadget is inseparable to our lives. Perhaps someday society has come to a point where the idea of living without cell phones seems absurd. With this gadget, developing the student’s math skills becomes more exciting. We are living in this digital world. And undoubtedly, smartphones boost students’ math skills and understanding!




Let us learn from plants! It is the nature’s way. Just like learning math at classroom! Plants make use of the quantum physics superposition theory. They harvested packets of light energy and converted to chemical energy with remarkable efficiency. Experts found that photons come in and see many energy paths and the effect called “coherence” helps determine which path is the most efficient to follow. That is it! Plants, the receiving end, maximize the conversion process!

IMG-20131003-00161Students studying mathematics are like plants harvesting the light energy. If students do not find the most efficient way to capture the wisdom of the classroom discussion, they are not converting efficiently the best knowledge that supposes to be imparted by the lecturers. Everything they do from the start to the end of the lectures has something to do with their own learning method.
Classroom lectures work on the same idea as plants do. Students, being the recipient of knowledge, must learn it with remarkable efficiency. Teachers always find the most efficient means to impart the subject to the students the best that they can deliver. Teachers feed them different ways how to solve mathematical problems by breaking these into its components slowly

Engineering Mathematics 1 students batch 2013 First Semester

Engineering Mathematics 1 students batch 2013 First Semester

and slowly. Then finally problems are transformed to its basic operations. This is basically the features of the classroom lectures!
Still classroom lectures are the best means for students to learn fast in mathematics. Teachers are there to instruct, discuss, assist, and show different approaches in mathematics by working out examples and illustrations. Classroom lecture offers opportunities for students to ask questions, offers suggestions and presents ideas, and shares knowledge with others.
In order to help students, the following are tips on how to maximize learning on classroom lectures;

1. Take an advanced reading of the topic for discussion before going to the class. Just read and picture out the new subject matter even you do not have clear understanding. Just a glimpse of the new topic! The idea is to let you feel as if you are in a review mode when you are in the classroom.

2. Listen carefully the discussion done by the teachers. Make notes on important points! This keeps you alert for the whole session. This avoids feeling sleepy or daydreaming. Sometimes teachers could share their own knowledge on things that may not be found in your textbook. Be an active participant.

3. Ask question if you fail to grasp some important points during class lectures. This keeps you from guessing and spending time later on your review. By asking questions, you learn and solidify important points. Understand the why behind every steps of the solution.

High-achievers undoubtedly do more efficient way to maximize their learning in classroom. Though not all students are endowed with wisdom to learn quickly like King Solomon, above suggested tips will help the slow learners in Mathematics. Remember there is no day in our life without mathematics!

Working Math Exercises; Tips and Benefits

           By Vic Odarve

Mathematics texts/workbooks have something in common: exercise sets are presented at the end of the chapters. That’s why engineering students often wonder what are they for? More so, lecturers often told students to solve some of these exercises and submit the solutions on the next meeting. The message is clear: they are there to be worked out! These exercises are like a passport before students can go ahead to the next chapter. That is it!

Engineering Mathematics 1

Engineering Students

Like a wormhole, it screens whether students are ready for the next topics; also meant to wrap up everything what students are supposed to learn the whole chapter under study.  If the teachers do not give an assignment on the exercises, solve them by yourself for your own benefit. Following are tips on how to work most out of the exercises;

1. As starting point, select one exercise that is similar to your notes or workbooks or the one discussed by your teachers.

2. If you find still difficult and you could not solve anyone of the exercises, go back to the examples from your notes and textbook you are using. Review your example notes and solve without looking the solution. Check if you have a similar answer. If along the way you get stuck up, uncover the solution for a while and see where in the way you get lost. Repeat this process until you can master and understand how it is done. If you are convinced that you comprehend the process, go back to the exercise which is similar to your notes and solve it.

3. Make it a habit of reviewing or checking your work until your final answer. Since most exercises have answers, look and compare your answer only after you thoroughly check yours. If you get one exercise with a correct answer, solve the remaining. This way will build your confidence.

Working math exercises proves beneficial to engineering students. It relates to academic success. Following are common benefits derive from working math exercises;

1. Students who work math exercises perform better on tests or examination. By working exercises, students are reviewing what they cover in class, thus increasing their understanding of the subject concept better.
2. Students develop self-discipline and management skills too.
3. Students feel enjoyment and challenge in mathematics.
4. Students put their new acquired skills into practice. Practice makes perfect.


Math textbooks- a legacy for the next generation of students

Later, students will realize that what they learn in that particular chapter will be used in the next chapter. Mathematics or workbooks subject contents are presented in sequence. Master the first one before the next. Do not skip. If they could not solve the exercises at the end of the chapter, most likely they will be in trouble the next chapter. Follow the tips. Do it now, do not delay! Solve the exercises! Geniuses in mathematics are a gift from God, but skills could be developed!

Simply by following the tips, academic success and other benefits are yours.

Ask and Learn from your Classmates

By Vic Odarve

It is the simplest thing to do for the students to learn more and pass the course.

Last semester, I had a quiz in Mathematics consisted of four problems. The last problem was the most difficult. Thus, it required an extra mile solution – deeper analysis- so to speak! As expected, the results after marking: only 2 out of 380 students got perfect scores. No wonder, it was difficult. But one of these students who got perfect scores caught my attention. It was intriguing! This student had never been in my list as top performers of the class! After rechecking, I found her script was perfect from the initial steps of the solutions until the very end. The solutions were well presented. Presentation was so detailed that no appreciable error can be seen. When the class resumed, I returned their individual papers or scripts and let her solved the problems on the board. To my surprise she explained very well; discussed  solutions perfectly that the class applauded her. “Thank you, well done”, I commended her after the discussion. Oh my God, I was awed and shocked in disbelief?

Engineering Mathematics

Engineering Mathematics

After a week, I happened to meet and talk her in the library how she was able to solve the last problem. “Yes, very difficult sir but I was able to ask help from my classmate few days before the quiz and we discussed together”. That’s it. It was her secret! Asking help from her classmates! She was referring to the other one who always got perfect score in all of my quizzes –a gifted student in Math!

No doubt! She learned by asking help from her classmates during study and propelled her way to get perfect scores! A simple act yet effective in Mathematics! Asking help from your classmates or teachers is one of the options to learn and pass this course.

Every student runs into problems in Mathematics courses from time to time. Even the bright ones! This is a part of learning process. But students who ask help from teachers or classmates appear to move and recover the lessons easily. Problems are likely getting past by asking help. Resourceful students are doing this way.

Classroom lecture!

Classroom lecture!

By asking, students are benefited! The students may ask help from their classmates and get perfect scores during exams! But every student must know the art of asking either from her teacher or classmates in order to learn most out of it. The following tips help students maximize the benefits of asking from their classmates;

1. Do the explanations. Show the way how you come up the solutions until the steps you lose. The reason is clear: the one who does the explaining will learn the most.
2. When you are working in group or sharing some ideas, share your opinions with others. Don’t let only one person do all the talking. Participate!

Just follow the above tips and it works!

The Joy of Helping Others

               By Vic Odarve

My office is open to all students except on some special occasions. And one day, a student came to my office asking assistance on how to derive the famous Einstein equation E= mc2. Upon recognizing that he was not my student, I asked him “How do you know that I can derive the equation?”My friend told me, Sir”, he replied warily.

Final Exam

              Fr. Edeh’s Arch Exam room

Then on another occasion, students came to my office bringing laptops and asking more tips on solving problems on matrices, statistics, and linear programming using MS Excel. The same thing: most of them were not my students! Then I asked them, “How do you know that I can help you?”Our friends told us, Sir”, they replied in unison.

“No problem”, I said to myself. I love to help students. A little help would mean a lot to them. These students are resourceful and they are making the best of things while I am still in this university.


                  Seminar Workshop

Then one day somebody knocked my office door. I was surprised. He was not a student! He was a university teacher and an owner of a mall in the community. He requested for a website design offering assistance for campus students! I was overwhelmed! I was wondering how he got the information. Without asking questions and hesitation, I made one for him! Later I found out   that students told him that I am making a website for free.

That was it! I designed a website for free. It was a good website design and it attracted students’ attention. It became a big hit in the campus! The news spread like a wildfire in the entire university. Students were asking who design the website! The word was spreading.

Madonna University, Elele Campus, Rivers State, Nigeria

Madonna University, Elele Campus, Rivers State, Nigeria

After a few months and upon their request, I conducted a seminar on website design inside in the campus. The seminar drew much attention to students and community people alike. There I found myself speaking in a huge crowd that I never expected. My heart filled with joy. I could see on their faces how happy the students and community people were!

That’s how the things started. From a student who came to know the E= MC2 and ended on a seminar engagement on website design!

This is life. A little help makes a big difference to our neighbors. Just by helping others freely without expecting something in return! Helping others by sharing your knowledge gives you a happiness that money can’t buy. The events become the opportunity for growth. It motivates us to do more and make our life fulfilling. You feel that you are adored and loved by the students as well as the community people. Sharing our knowledge to the less fortunate makes you always feel happy. It also brings joy to our neighbors, too! You have done something good to your brothers and pleasing to the eyes of our Maker. The bible says,

“Let each of us bring joy to our neighbors helping them for the good purpose, for building up.”
– Romans 15, 2

I did it for the glory of God the Father! And to the one sitting on His right hand where no angels and prophets were ever admitted!



Below is the performance profile of the students taking Engineering Mathematics 2 (EMT 202) for the Second Semester 2012-2013 March batch at Madonna University, Elele campus, Riverstate, Nigeria, Africa.

 Table 1 shows the students performance that in a class of 399 students, only 310 or 78 % passsd the course.

 Table 2 also shows the grade distribution profile and the grading system corresponding to the marking ranges. Surprisingly, number of students who got A grade is almost the same as the number of students who failed to pass the course. That is, 88 students got A grade and 89 also failed to pass the course. Further, 53 students garnered B grade, is nearly equal to those who came up with D grade which is 50. Only 64 students or 16 % got C grade.

 Table 1 – Students performance Profile

















Table 2- Grades Distribution Profile







70 – Up




















below 40







 As in the previous semesters, passing percentage falls at 70 – 80 % of the class. So, this class performs at the same level.


By Vic Odarve

  Do students want a participative kind of lecture in a classroom? Yes, and it is more entertaining in Mathematics! Lectures in mathematics are usually done by presenting problems or exercises. Just by having an illustrative example that catches the students’ attention, all eyes will be on the board and draw their wandering minds to focus on the new lesson for the day. The mind will be put in focus just like a parabola; the focal point will be on the lesson.

Madonna University, Elele  Campus,Riverstate, Nigeria, Africa

Madonna University, Elele Campus,Riverstate, Nigeria, Africa

It puts the students’ engagement to the lesson at high concentration level. This modified traditional teaching method works well if the teacher knows very well the subject matter. Then how to transfer them to the minds of the students in the easiest way is another thing. There are teachers gifted with wisdom and skills to capture one’s mind and power to facilitate understanding. Geniuses are born but skills can be developed. Thus, this method can be developed and implemented in our classroom!

Following are basically the steps on how this method works;

1.    Make an illustrative example of the topics. Usually in mathematics, an exercise or a problem illustrating the discussed principles or theorem can be an example. Make an example or problem that requires the necessary ingredients of the subject matter and its working solution needs deep analyses.

2.     2. Step by step solution. Slowly explain the details clearly on how to solve the problems in a step by step manner. It must be step by step. Step by step discussions must be accompanied along with the reasons and significance. The step must be the prerequisite to the next process. Just like algorithms. Just like climbing a ladder. No shortcut! Complete your discussion until the final answer is done. Remember that we have to number every step! Remember that students learn more by observing how we do the process and imitate it later.

 3.  Entertain questions to further the knowledge if something along your discussion students does not understand. Just like open forum. All are free to ask questions.

4.     4.  Highlight the critical steps that must be taken with attention. Your experience will tell you which of the processes the students most likely find difficulty. Review them.

 5.      5.Test if they can do it by themselves. Go to your favorite and brightest student. Let him come and discuss the steps written on the board. Add more explanation to emphasize the significance of the step. If you think no more hitches along his discussion go to the next step.

 6.      6. Final step.  Make another example and let all the students involve in doing the step by step solution. It is now their turn. Remind them to follow the pattern by following the steps. Start on the students sitting in the first row. That is, first student sitting in a row will be the one to go to the board and work on step 1 of the solution. Then the next one sitting on the same row will be the next to go to the board and do the next step. The process of going to the board and continue the next step by next student is repeated until the final steps are completed or the final answer is achieved. Each student will do each different step in succession! Teacher is now reduced to a mere facilitator. The ball is now rolling to the students. They must do their share.

A two hour period in a course can complete 2 or more topics with the same procedure. In a class of 30-40 students, all can participate by going to the board and do one step each until a problem is completely solved.

It is worth seeing to see students with all the eyes on the board and ears to the teacher. It is because everybody waits his turn to go to the board and write something for the solution or step. All can participate! No student is left behind. Once the teacher is skilled on this system, he can get the attention of the students and add more tricks to make the activity challenging. The class is under his spell when the time of the day is done. Students as well as the teacher are satisfied.

This way teacher and students are doing like a teamwork. It is entertaining and challenging and a learning engaging activity! Students feel the importance of being there by having accomplishment. Perhaps the introduction of this method may result to their lifetime pleasure. We do not know that some students are touched. That is by participation! It is learning by doing… encountering new approach in an exciting ways! A whole new world!



 There is nothing better than a good preparation for the upcoming final exam on July 1, 2013. Since topics are too wide, you must concentrate on the following areas of concentration as shown in the examples below. Study them thoroughly for similar exercises will come out on the final exam. Workbook/textbook and notes will be your references.

1.      1. Find the inverse matrix of       C= 1        2      4

                                                                             2       1       4

               3        2       2

2.  Solve by Cramer’s rule




 3.    3.  Solve by Gaussian elimination method




 4.      4.Find the eigenvalues and eigenvectors of matrix

                         A=   2      4

1                               1       2

5.     5. Find matrix P and diagonal matrix D with D = P-1 A P of matrix

A                        A =    3        4

                                      2          1

6.      6. Solve the coupled differential equations:

 F1’(x) = 2 f1(x) – 3 f2(x)

F’2(x) = 4 f1(x) + 3 f2(x)

 7.      7.Find the dot or inner product a.b if a= [6, -3, 5] and b= [ 2, 4, 8 ] T, so that b is a column vector.

8.     8.  Find the cross product of vectors U= (3, 2,1) and V= (1, -2, 2).

9.     9. Transform vector V =[  with transformation T(V) =[ . Graph.

10. 10.  Find the image of the line y-x =4 under the translation. Graph.