# Math Examples that Keep Students Challenge

By Vic Odarve

Solving math examples in different methods but yield the same answer is itself a challenge.  And it works; keeping students at bay!

This approach allows the students to learn the principles of each method of solving same examples and develop mastery. As they master the different methods, students can readily compare the results with ease and build confidence. As confidence builds up, they challenge math problems by doing themselves; hence math becomes easier and simpler.

Solving math problems by different methods can surprise many      students, even the stupid ones. Math teachers, an expert in this field and speak with a power of persuasion, keep students’ eyes, glue to the board; ears listen to every discussion, and silence throughout the classroom. For example, in Algebra, find the value of x in a quadratic equation by factoring, by completing the square, and by the use quadratic formula. These three methods of finding the solution work differently but yields the same answer. This way the students learn three math principles in just one example. Furthermore, they can spot the differences, and mimic the teacher’s steps in arriving the final answer. Students feel great, satisfied, and enjoyed by learning three different approaches. This develops mastery of the course. Of course, a surprise and challenge!

Working Examples

Solving single math problem and obtain the same result using different methods also allows students to check and compare the final answer; hence the process builds up their confidence. Students turn this great opportunity to learn more methods; thus, increasing their math knowledge. These environments keep them changing, growing, and learning more math; thus becoming them to be an expert. Since all the methods yield the same results, they are motivated and making great grades… one of the top class math performers. The approach does build their confidence. Henceforth, as confidence develops, students challenge themselves in solving math exercises and problems.

What were the results? Some students came to the office smiling and showed their solutions confidently; others wanted to discuss how they got their final answer. Sounds great and fulfilling! It all begins with single math example that keeps the student’s challenge.

# Math Becomes Easy

By Vic Odarve

“Really math becomes easy when the teacher explains well and I follow it up at home”, exclaimed an engineering student.

Competent teacher, conducting a review of the notes every after the class, and practice by reworking the steps, are all needed to understand math. Math professor, like King Solomon, must discuss math in an eloquent and conversationalist way, expert and master on the course topics, and has an ability to challenge the students. Students, on their part, must review the notes after the class to enhance memory retention. Then, actual rework or practice on what was discussed and solved in the classroom. And math becomes an easy prey!

Math is the most difficult course in College

The professor must know very well the intricacies and beauty of the numbers hidden in math; he can explain lucidly the subject matter in a simple way and invite the students who are afraid of numbers to ask and interact during the classroom discussion. The classroom becomes a learning place where students often refer to as their second home. This is the place where they can voice out and share their own opinion with others; and

Math Professor details the steps

learn from their classmates by asking. A competent teacher is like an environmental designer; keeps classroom atmosphere conducive to learning. He establishes redlines; build up student’s confidence, and keep the math course easy and simple instead of intimidating and frightening.

Engineering students at work at Madonna University

Conducting a review on the topics discuss in the classroom proves to retain further one’s memory. The study showed that the first in a two- hour lecture is the most effective period for memory retention. Faced with this dilemma, a review after the class is needed to recover part of the time where memory retention was least. Reading the notes for a few minutes after classroom strengthen further your memory. Any lapses and misunderstood notes can be easily corrected by recalling your memory.

Finally, students must rework the problems and its solution steps. Although simple as it is, this is the most important part of learning math. The practice enables students to understand the whys, how, and when, these steps are done. It keeps their mind working; challenging ideas at will, and keeps them on the state of analyzing and recognizing patterns. Math is like a game of chess; sometimes pondering the board to find the best moves over your opponents; the clearer the picture over the situation, the better your moves and win the games. Reworking the classroom’s lectures makes understanding perfect!

It is not that math is difficult because it is math and students could not get along with numbers; with competent teachers, coupled with regular patience to open your notes after the class, and constant practice, learning math becomes easy and enjoyable. Students and math, then become the very best of friends.

# Practice Makes Perfect in Math

By Vic Odarve

To practice is to learn. That is learning by doing! And so is mathematics! The same principle can be applied to pass the course in mathematics! An old adage “Practice makes perfect” holds true. After the classroom illustrative examples, solved step by step by the teacher, students must strive hard at home to practice solving by themselves with the aid of the notes. Notes will be put aside only if their confidence is building up and that they can do it by themselves. Teachers often told the students “Don’t stop practicing until you can perfect it!” That’s it!

Classroom lecture

Practice working out the solution after classroom can be difficult for some students. But if they continue working on the same examples, chances are they can do it perfectly. It can be developed into a habit later. It is often suggested to read and rework the exercises before going to bed and refresh at early morning before going to school.

Difficult exercises in mathematics can be visualized and understood easier by

Engineering students-Practice makes math perfect
Madonna University, Nigeria, Africa

breaking down to its simplest forms. But the visualization and better understanding can be achieved only by constant practice. This is the way of life… being an engineering student. Actual practice works best in mathematics because the solution involves several processes before getting into its final form. Its algorithm. The more you practice the more you understand about why the steps are being done. If done habitually you can explore more techniques and better approaches on

Workbooks are designed for practice solving Math problems

what to do with the mathematical exercises or problems. The more you discover its secrets, the more you love mathematics. One will find this learning method quite fascinating just like wormholes… believed to be existed in our planet and were used by alien to travel to the distant universe!

STUDENTS UNCOMFORTABLE AND TEACHERS’ HAPPY

Teachers often told engineering students to do constant practice in solving mathematical exercises or problems whatever written on the classroom. These ways teachers are molding students to create a strong study habits. That is actual practice or hands on! But sometimes students feel uncomfortable when teachers press them to do more practice at home. There were times when teachers reminded students to do constant practice, they could see their eyebrows rose up and their hairs also rose at their backs. It’s something eerie! Teachers know best that by building this habit they are building the students character and lifestyle. Teachers also want to create the student’s landscape and its environment that they are supposed to live in! These teachers are really working hard to help weak learners and by doing so they feel happy to give the students the best education possible! It is happiness in giving your best to the profession you are in. The great teacher Saint Luke, who traveled far and wide with Saint Paul, worked hard in the early expansion of the Christian church. Thus, he once said,

“In every way I have shown you that by so working hard one must help the weak, remembering the words that the Lord Jesus himself said, “Happiness lies more in giving than in receiving.”- Acts 20, 35

STRATEGY THAT WORKS

Constant practice in solving exercises and problems develops the skills on mastery and understanding mathematics. These, too, in turn, are the basic ingredients in developing students to analytical and critical thinking skills. That is, breaking any difficult problems down to its simplest forms. This is the strategy that works. In all human endeavors, practice makes perfect!

# Why Workbook in Math is Important?

By Vic Odarve

Although usually smaller and lighter than textbooks, workbooks serve as an extension in classroom lecture on large class size, determine the quality of teaching and learning topics, and facilitate homework. Math courses with a class size of hundreds of students, which teachers have difficulty to facilitate a good interaction with the students, workbooks effectively filled the inadequacy. Workbooks, being a condensed course outlines from a standard textbook, serve as a teachers’ roadmap in classroom discussion and guide for the students on what topics to follow next. Further, workbook facilitates homework for the students to study at home and allow the teachers to monitor their progress.

Math Workbook

Unlike in most colleges and universities in western and eastern part of the globe, some African schools allow having larger class sizes. Here, schools allowed 200 to 1500 student class size for the general math courses in 100 levels. For engineering, mathematics 200 and up levels, schools usually have 200-500 students per class. Even with the aid of modern available equipments such as power points or slides and sound devices, a number of students which must listen and do some interaction during discussion already pose a problem. Some failed to catch up the lesson discussed on the presented slides or materials written on the board and others may not hear properly the discussion or explanation. With all these problems, workbook seems to fill up by reviewing and reworking exercises at home. Students who were less fortunate to grasp the discussion must look on the workbook and copy notes from their classmates. A scenario common in big African universities!

Engineering Math Workbooks

Most workbook topics are presented chronologically since math is cumulative. It is written in a clear, precise, and easy to understand as compared to textbooks. Each lesson carefully builds on the concepts students learned previously, with the problems and illustrations discussed steps by steps. Thus, this book guides the students on how the course discussion is pacing. The workbook serves as a road map for the teacher and a helpful aid for substitute teachers. Further, it provides information on the quality of teaching materials that lecturers or teachers must inculcate in the minds of the students, Students, too, would notice what is expected to learn for this semester in Math course.

As this book is also specifically designed for student’s activity at home, workbooks are usually filled with illustrative examples from simple to complex problems presented in clear, step by step manner. Practice problems, which usually followed after the completion of each topic, can be worked out and the answers can be written directly in the book. Often times these practice problems are taken as assignments or homeworks where teachers can follow up at classrooms. Workbooks allow students to compare their solution steps from the classroom lectures; use as reference when they got stuck on the topic discussed; and offer an opportunity for easy learning outside the classroom at the own pace. The homework activities motivate and challenge students; thus math becomes useful and enjoyable…never a dull and boring! The workbook is a haven for gifted students where they can enjoy the challenge of solving the difficult math exercises and problems on their own.

What, then if the teacher is the one who wrote the workbook? The results are simply amazing as workbook and teacher are helping the students in unison! The workbook they are using is the language of his mind…a best picture and the most effective means of transfer of knowledge. Everybody can expect the best results; maximizing the best potential for the good of the students. Workbook and teacher are one! Thus, workbook is necessary in math!

# Practice is Needed in Math

BY VIC ODARVE

To practice is to learn. That is learning by doing! And so is mathematics! The same principle can be applied to pass the course in mathematics! An old adage “Practice makes perfect” holds true. After the classroom illustrative examples solved step by step by the teacher, students must strive hard at home to practice solving by themselves with the aid of the notes. Notes will be put aside only if their confidence is building up and that they can do it by themselves. Teachers often told the students “Don’t stop practice until you can perfect it!” That’s it!

After the Math practice, go outside and relax. Join the Christmas merrymaking with your old folks at home.

Practice working out the solution after classroom can be difficult to some students. But if they continue working on the same examples, chances are they can do it perfectly. It can be developed to a habit later. It is often suggested to read and rework the exercises before going to bed and will be refreshed at early morning before going to school.

Difficult exercises in mathematics can be visualized and understood easier by breaking down to its simplest forms. The picture looks the same thing for the babies. Solid foods are broken down to its small pieces before taking it whole. But the visualization and better understanding can be achieved only by constant practice. This is the way of life …being an engineering student. An actual practice works best in mathematics because solution involves several processes before getting into its final form. It’s algorithm. The more you practice the more you

Engineering students use MS Excel as practice in solving Engineering mathematics at Madonna University, Elele campus, Nigeria

understand why the steps are being done. If done habitually you can explore more techniques and better approaches on what to do with the mathematical exercises or problems. The more you discover its secrets, the more you love mathematics. One will find this learning method quite fascinating just like wormholes…believed to exist in our planet and were used by alien to travel to distant universes!

Teachers often told engineering students to do constant practice in solving mathematical exercises or problems whatever written on the classroom. These way teachers are molding students to create strong study habits. That is actual practice or hands on! But sometimes students feel uncomfortable when teachers press them to do more practice at home. There were times when teachers reminded students to do constant practice, they could see their eyebrows rose up and their hairs also rose at their backs. It’s something eerie!  Teachers know best that by building this habit they are building the students character and lifestyle. Teachers also want to create the student’s landscape that they are supposed to live in! These teachers are really working hard to help weak learners and by doing so they feel happy to give the students the best education possible! It is happiness in giving your best to the profession you are in. The great teacher Saint Luke who traveled far and wide with Saint Paul worked hard in an early expansion of the Christian church. Thus, he once said,

“In every way I have shown you that by so working hard one must help the weak, remembering the words that the Lord Jesus himself said, ‘Happiness lies more in giving than in receiving.’””- Acts 20, 35

This is the strategy that works. In all human endeavor, practice makes perfect!

# Professor Challenges Math Students

By Vicodar

Students need to be challenged. That is why in almost all my quizzes in mathematics are open notes, books, etc. just to keep them a challenge. But of course only a few got perfect scores. As for most of the professors/lecturers, we know how far are our students could get the correct answers even though we give them every opportunity to do so. Mathematics is difficult and most students hate this course. But if we let them open notes and books during quizzes, they are smiling ear to ear!

Open books and notes quiz keep students smiling from ear to ear.

In addition to open notes and books, I also post the quiz/exam results in the website. It’s another challenge! Posting makes students feel like climbing a ladder. Noting their names on top 3 performers would raise their adrenaline of studying more. Much more, as good performers, students boost their pride such that they invite their parents to visit the website and see their names posted. And seizing this opportunity, students feel they have more bargaining power to secure for more allowance for the next semester. “This is how you would like to be on top 3”, exclaimed one of the students.

How about the professors/lecturers! They will be receiving e-mails that excite their hearts and elevate themselves to the highest level of their profession. Among e-mails they received are citing them as best lecturers, world-class teachers, mathematics genius, the best teacher I ever meet, and many more. Parents sometimes send e-mails for their support of the professors.

Prestigious colleges and universities usually share common denominators; their lecturers/professors are authors of textbooks and journals, etc. This generation of lecturers/professors gives the best education of our youths. It is world-class! A professor who could write books on his field of expertise shows ‘having something’ to impart. You could not give if you don’t have! Thus with their expertise they could lecture well and play the students to the next level of education. They can bring students to dance on their own rhythm or pacing of teaching. It is on this quality of mentors that keep the student’s interested in math. These are the teachers that keep student’s ability to challenge and learn and actively participate in classroom activities. A missing link in most African universities and colleges!

As long as this scenario is present in any higher institutions of learning, students flock to this school and join thousands of others benefiting the best education of the land. One of the lecturers had noticed and told me “Sir, did you know that students are increasing every year and the enrollment is flowing like a river?” Then I replied, ”As long as good/competent lecturers are here; so long live these lecturers, and this gives life to thee”.

This is the picture today, and should remain as is. Father founder philosophy of the institution is simple; this is for the youth the hope of the fatherland, and for those who love education.The day will soon come when these students will be catapulted to be one of the world’s best. Expat –White men lecturers who mould some of these students always say that races don’t exist: there is only the human race, to which we all belong.

# 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, BEEHIVES, AND FIBONACCI’S NUMBERS

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.

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 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

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!

# LEARNING MATH AT CLASSROOM

LEARNING MATH AT CLASSROOM

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!

Students 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

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 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.