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.



 “It was so simple but challenging”, quipped one student after the quiz. Definitely students love this type of quiz. They find the quiz on finding the determinants, inverse matrices, and eigenvalues/eigenvectors give them the opportunity to test how far their learning progresses on that particular course. That is, taking a simple but challenging quiz!

Engineering Mathematics 2 Batch 2012-2013 Second Semester

Engineering Mathematics 2 Batch 2012-2013 Second Semester

What makes it simple? It is simple because the given matrices during the quiz were almost similar to the examples discussed in details during classroom session except a change on one of the entries in every row. This type of quiz is challenging because it is open notes, books, and other related materials. Laptops are allowed where they can use the program MS Excel. But everybody has to observe the” strictly no borrowing materials from the classmates!” rule.

Another word of caution! Solutions must be legibly written and presented in detail or step by step. Laptops are used only for counterchecking or guide as their works in progress. That’s it and nothing more!


Open books, notes, and related references during quizzes are welcomed by engineering students. Mathematics especially dealing on matrices is difficult to work with. As matrix size increases, the difficulty in computation also increases. Rules are sometime complicated and requires complete understanding before our confidence of getting correct answer develops. Memorization of procedures does not work well if without understanding as you will soon forget the procedures as the solution progresses. Most often in matrices a change of a single element or entry will change the whole environment of computation. The solution is very sensitive to the slight changes in the initial condition of the matrices. Just like chaotic theory, any error in a single step can produce large error in the final answer. That’s why open notes, books, etc during quizzes works well with matrices. The main objective is to make students familiar the algorithm on how to get to the final answer.

The reason is clear. Let the students learn the basics of matrices operation. With almost all computation nowadays done by a computer, we are preparing our students to this world ahead. It is then the classroom learning works in the open fields after completing their course. In a real world, computers and engineers work hand in hand in the industrial fields.


As expected silence engulfed the whole classroom as quiz started. Everybody was busy and concentrated on their works. No ogling for solutions on another paper. All eyes were on their own paper. Time limitation for the quiz was enforced. A real world in a classroom!

What was the result on that quiz? Only sixty percent (60%) passed. Failed students came to me with various reasons. Some confided that they had not prepared well and not listening during classroom lectures. In summary, failed students had reasons with the same denominator. They same old stories! Still clinging to their old ways of bad habits! They vowed to change the old ways. They blamed themselves for failing the quiz! One thing is certain…not blaming to the lecturer!

So, what is next? Some wept and admittedly vowed to change for a better! Then I told these poor youngsters, “Go and change your ways for it is not too late, more quizzes are coming. Unless you change your ways, you will not pass this course”

Sounds biblical?  Yes. Students must change their ways in order to pass the course. Jesus at the early part of his preaching often told the crowd to change the ways and believe the good news so that they will not be perished. Everybody must be prepared.

“He said, “The time has come; the kingdom of God is at hand. Change your ways and believe the Good News.””

– Mark 1,15

“I tell you: no. But unless you change your ways, you will all perish as they did.”

– Luke 13,3

That’s it. Failed students must change for a better for they will they will not only pass this course but one day they will also meet the One sitting on the throne for final judgment to all mankind.