In Search of a Good Teacher

If we, teachers, only listen to our students, they could teach us a thing or two.

Karidme Rocha and Shawn Klesel are both wonderful students of mine in Geometry. Like a host of my former and current students, they're both on my Facebook friends list.

The following is part of a thread that stemmed from Kari's status that she posted on December 30th:

Kari:      In search of a prince charming:)
Shawn:      im a prince and im charming but i aint no prince charmingXD
Kari:      o wow Shawn lol you need to be both at the same time lolXD
Shawn:      hahaha only to certain ppl at certain timeXD

Upon reading this thread, I reflected. Face to face, I looked intently at this teacher that they call Mr. Jope.

You are a teacher, and you are good. But are you a good teacher?
Sometimes? All the time?

Good teacher TO WHOM?

Classroom Economics: Saving for Rainy Days

Holt Geometry, the textbook my students use, defines postulate as "a statement that is accepted as true without proof." According to Reader's Digest Oxford Complete Wordfinder, postulates, or axioms, are used as "basis for mathematical reasoning." Geometry, in fact, is built on a strong foundation of postulates that include the following: There is exactly one line that passes through any two points. I am not quite knowledgeable in the science of finance, but I would assume that one of its postulates must be the following: There'll be nothing for one to withdraw if there was nothing deposited in the first place. Postulate or not, I use it to strengthen the foundation of my own teaching.

At the beginning of each school year, making deposits is on top of my priority list. Like a hungry eagle looking for food, I scour for and swoop on every opportunity - big and small- to make a deposit.

Giving generous compliments is making deposits. While I strive to get to know my students, I make sure that I give each one of them appropriate and hopefully nurturing compliments. Lots of them. I compliment anything about the individual that I can safely and appropriately compliment on. I celebrate every little positive thing I see.

Greeting and wishing them well on their birthdays is making deposits. So are grieving with them when someone in their lives perished, listening to them when their bffs break their hearts, and celebrating the genius behind each academic mistake.

Shaking it in the middle of the dance floor during school dances and being a kewl teacher the right way are making deposits.

Calling parents to tell them positive things is an example of making multiple deposits. With one single deposit, I get to  increase my deposits in two accounts instead of just one.

Sharing my own life stories, whenever appropriate, allows me to deposit to a host of accounts, not just two, with one transaction.

Although I am a sucker for making deposits, I am well-aware that my deposits, like bank deposits, are limited to certain currencies, and because I'm just a teacher, I cannot just deposit large amounts whose sources I cannot justify. Certain amounts of deposits are certainly going to raise alarms. I certainly do not want my students to feel uncomfortable with me.

We do not like to withdraw monies we have saved, but rainy days are bound to come. It is for this reason that we save in the first place.

Making withdrawals is getting after my students for a host of reasons, such as misbehaving, failing to turn in homework or project, violating school or classroom rules, and failing to perform in class satisfactorily or according to certain mutually accepted higher expectations. Sometimes, I withdraw before it becomes necessary to withdraw.

Just like in real banking and finance, my withdrawals have limits. But unlike real banks, my banks, which in this case are my students, are not "financially" stable and well-founded. They are kids, and they are volatile. Sometimes, I withdraw as much as I needed. But most of the time, I withdraw according to the conditions that my banks are in.

No matter what the state of economy is, it is deemed wise to always save. In fact, it is a value we are encouraged to teach our youth.

My classroom economics is always a winner. I may fail to help all students of mine achieve academic mastery. But with my classroom economics, I fervently hope to touch their lives with mine.

When the Devil Talked Back

It's two days yet before Christmas, but I already received a very nice gift!

Around four this morning, I received an email from an editor of PhilStar, the internet site of the Philippine Star, one of the leading national newspapers in the Philippines. Here's Mr. Dino Maragay's message:

Congratulations! Your story has been chosen for publication in our website. Here is the link to your story:

http://www.philstar.com/community/Article201005.aspx?articleId=642058&publicationSubCategoryId=503

Meantime, please wait for our staff's email on how to claim your contributor's fee. Thanks a lot for joining and please continue visiting philstar.com.

Cheers,
Dino

Dino was referring to PhilStar's writing contest that I didn't know about till just the other night. Then, the contest, which opened on December 1, 2009, was going to end in nine days. I thought I would still give it a try.

I wrote When the Devil Talked Back that same night. The following day, I submitted it online, along with the required signed entry form and photocopies of my passport and Texas I.D. Just about twelve hours later, I received the message above.

When the Devil Talked Back is about a traumatic teaching experience of mine in the spring of 1996. I was then an 8th grade mathematics teacher in another school district. Read it here. Don't skip the readers' comments at the end of the article. They're quite interesting.

Words to Live By

I was already late to our faculty Christmas party that was being hosted by our quite endearing school principal at her home for the first time since I joined our school less than two years ago. I had a vague idea where she lives, but I was pretty sure I'd get there.

Got a great excuse for being late. Check. Got my "Grinch Better Have My Presents!" shirt on. Check. The trays of Filipino-style beef lumpia (eggroll) and puto (rice cake) that I ordered from Cabalen restaurant. Check. My videoke microphone. Check. The Christmas carol lyrics sheets? Hhmmm...

As I quickly browsed this pile of papers on the second ledge of my white book case to look for the lyrics sheets, I chanced upon this piece that looked familiar. I had not seen this in so many years. I didn't even know I still have it. I thought my colleagues were probably waiting impatiently for me, but I just had to read it one more time. As soon as I had put on my reading glasses, the words touched my heart anew. This was the short speech delivered by class valedictorian, Alam Quintero, who was born and raised in Mexico, in a banquet meant primarily to honor the top ten students of Hidalgo High School's class of 2003.  

"There is this special individual I would like to recognize tonight. A unique human being whose hard work, intelligence and integrity as a man impacted the life of many students around him as well as my own. His honesty, loyalty, and always-appropriate advice are only a few characteristics I have come to praise here. The respect I owe this teacher, this better called doctor in life education, is something I cannot possibly describe, for in fact it can only be experienced rather than seen. This individual I am speaking of tonight, has passed through several tribulations. Yet no matter what hardships have befallen him, he always surpassed them all with courage and maturity. That is certainly his way of standing out from everyone else."

"Regardless of where life may take him, he will undoubtedly influence those who can be reached by his grace, and those who can be touched by the kindness of his being. Yes, there may be quite a few individuals who can fulfill some of the above-mentioned qualities. Yet there is only one who meets them all and who in fact has been close to me throughout these past high school years. He has been my coach, my teacher, and my friend. This is the individual who has helped me not only the most, but also the best. Mr. Jope."

I carefully placed the sheet back. I grabbed my jacket and was finally on my way sans the lyrics sheets. It was easy to find Mrs. Cavazos' house. When I finally showed up, it didn't matter that I was late. The greetings were warm. The most important teacher in the world had just arrived from a glorious chapter of auld lang syne.

Mitsuo's Magic

Before after-school tutoring commenced yesterday afternoon, I found freshman Mitsuo at my door. I had left it open to let the cool December breeze in. He hesitated for a moment when he realized I was chatting with Ashley, one of my talented sophomores. I quickly sensed that he was anxious to tell me something. Before he had the chance to excuse himself and turn away, I asked him to come in.

Although Mitsuo, who's still taking Algebra 1, is not a student of mine, he has warmed to me. A few months earlier, while I was on after-school duty, the boy, then a stranger to me, approached and impressed me with his magic card tricks. He left a really good impression when he thanked me profusely for indulging in his magic.

When I realized that his concern was about a math idea he had conjured up, I motioned him to approach the white board. He grabbed a marker and wrote the following:

[90^(1-3) + 80] /4

I initially didn't understand the part that appeared to be the exponent of 90, but I assumed that it had something to do with the first, second and third six-week periods. Earlier that afternoon I had discussed semester grades with some of my students. The semester grade is the average of the three six-week period grades and the semester exam.

I also wondered why he didn't multiply 90 by 3.

"It looks like you want to determine your semester grade." He said yes.

As if he read my mind, he said, "I'm sorry, Sir, I know I didn't write it right, but I want to show that this 90 is the average for the three six-week periods."

I decided to not call his attention yet to his failure to multiply 90 by 3.

"You see, Sir, I was told that, to get the average..." He stopped. Suddenly I had a feeling that he had shown his idea to other people and it was dismissed. "But I was explaining that the way it's done is harder for me, because it's harder when you multiply and then divide. Right?"

"What do you mean?"

I thought that he probably just didn't like to work with bigger numbers.

"I have an easier way to get the semester grade." He now looked more determined. "All I do is get the difference between 90 and 80, which is 10. Then I divide 10 by 4, which I then multiply by 3."

I felt relief when he mentioned multiplying by 3!

He wrote 10/4 x 3 = 7.5.

"It's right, right?" I nodded. I now realized that the boy was on to something.

Although I already knew that 7.5 must be added to 80 to get the correct average of 87.5, I had to ask how he's going to use it.

His last step turned out to be wrong, but I did not point it out to him. Not the right time.

I knew in my heart that, despite the careless mistake at the end of his solution, Mitsuo deserved to be congratulated for bravely pursuing his unconventional yet intelligent method of determining the semester grade. He made a careless mistake, but his investigative mathematics is the sort that we hope to see in our students. Clearly, he's been empowered!

To assure him that his mathematics was right, I asked him to demonstrate the conventional solution, i.e. [90x3 + 80]/4, and then pointed out to him where I see his 10, the 10/4,and the (10/4)x3 in said solution.

It was only then that he realized his last step was wrong!

But even to him, the little mistake didn't matter. He was just too happy to know that, after being dismissed for his unconventional method, he got validated.

I promptly attended to my older students when Mitsuo left. There were seven of them already who needed some assistance with their circumcenter constructions.  It didn't matter that I was really tired and hungry. Mitsuo got me under his math spell!

Despised Babies Are Eaten by Illogical Crocodiles

Lewis Carroll, who wrote Alice in Wonderland, was a mathematics professor at Oxford University in England. Besides his popular books, he is also known for his puzzles.

Today, our second and last day of the THSP mathematics conference, Job for the Future's Robert Knittle, our presenter, included sample Lewis Carroll puzzles in a booklet he gave us this afternoon. Puzzle #1, supposedly one of the famed professor's simpler ones, aroused my interest.

The puzzle has three implications:

1. All babies are illogical.
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.

Using the three implications listed above, what is the "inescapable" conclusion?

The same booklet offered a solution, but I had the urge to work on it myself. I teach my geometry students about conditional (if-then) statements. Instinctively I thought that the so-called implications are but conditional statements waiting for a proper makeover. I would love for them to know that everything that I had taught them on conditional statements was enough to solve a Lewis Carroll puzzle. :p

Here's how I played it:

I rewrote all three implications into the following if-then statements:

4. If it is a baby, then it is illogical.
5. If you can manage a crocodile, then you are not despised.
6. If you are illogical, then you are despised.

I applied the Law of Syllogism on statements 4 and 6, and it resulted to:

7. If it is a baby, then it is despised.

I compared statements 5 and 7. Since the contrapositive of a conditional statement is logically equivalent to the statement, I decided to write the contrapositive of the former:

8. If you are despised, then you cannot manage a crocodile.

I applied the Law of Syllogism, once again, on statements 7 and 8 to arrive at my "inescapable" conclusion:

9. If it is a baby, then it cannot manage a crocodile.

It is the same as the one presented in the booklet. Yayyy!

In Lewis Carroll's words, that would probably be rephrased as:

No despised baby is spared by illogical crocodiles.

Not really! :D

Collaborative Group Work

To model collaborative group work, our THSP math conference presenter divided us into groups of three or four and gave us the following math problem:

"Farmer Brown sent her children to the field to count the number of horses and ducks. Her son returned with the count of 30 heads while her daughter came back with the count of 94 legs. How many horses and how many ducks does Farmer Brown have?"

The objective was for each group to, not just solve the problem but, determine multiple solutions and to make sure that each member will be able to explain correctly all solutions that the group finds. Our presenter, Robert Knittle (University Park Campus, Worcester, Massachusetts), had indicated earlier that research supports activities such as this that require collaborative group work, one of six components of THSP's common instructional framework.

We all thought we knew the problem well, so we proceeded with less interest.

Tonya Milburn of Early College High School, Galveston, Texas suggested that we solve the problem using a system of equations and solve it using three different methods: by graphing and by using both processes of elimination and substitution.

I volunteered to do the latter. Angelica Vega, the dean of instruction of East Early College High School in Houston, Texas said she will use guess-and-check in tabular form.

While we were all preoccupied with our individual tasks, our fourth member, Canary Branch Bui of Empowerment Early College, also in Houston, remembered something from her younger days back in Vietnam.

"I have solved a similar problem when I was in the elementary," she announced.

Well, I wasn't the only one who got intrigued. Everyone listened intently as Canary explained her unconventional solution.

"Assume that all thirty heads are horses. Okay?" The three of us nodded.

"Since each horse has four legs, then we have 120 legs altogether. But there are only 94 legs according to the farmer's daughter."

Canary proceeded to subtract 94 from 120 to get 26. Then she announced that these 26 legs mean that there are 13 ducks, and there are 17 horses.

Initially, I didn't understand her explanation why the 26 legs mean that there are 13 ducks, but I thought that out of those 30 four-legged animals, there will be 13 of them that should lose two legs each. These 13 must be the number of ducks.

We were all excited to have found a "new" solution to an extremely familiar problem situation.

Angie, whom I met at the Park City Mathematics Institute in Park City, Utah about five years ago, promptly raised an intriguing question. "Will it work if all 30 heads are assumed to be ducks, instead of horses?"

Here's how I convinced my group mates that said method works:

Thirty ducks give us a minimum number of 60 legs. Since all 30 animals have a combined total of 94 legs, then there are 34 legs that are unaccounted for. This means that there are  17 (34/2) heads that require two more legs each. Note that only horses have four legs, therefore there must be 17 of them and the rest are ducks as our other conventional solutions prove.

Canary was requested to show her crowd-pleasing method to everyone in the room, and I volunteered to show my sweet "corollary" to her "theorem."

A problem-solving exercise meant for our classroom use with our students turned out to be a tremendous learning experience even for us!

Without a doubt, subjecting our students on a regular basis to collaborative group work activities such as the one given above is not such a bad idea after all.

Cursing Students

Today, the first day of the two-day Texas High School Project (THSP) mathematics conference here in Dallas, Texas, our presenter, Robert Knittle (University Park Campus School, Worcester, Massachusetts) read to us the children's illustrated book "Math Curse" (1995, Viking Press)  written by Jon Scieska and illustrated by Lane Smith.

Here's the book's plot according to Wikipedia: "The book, which is told by a nameless female student, begins with a seemingly innocent statement by her math teacher- "you know, almost everything in life can be considered a math problem." The next morning, the heroine finds herself thinking of the time she needs to get up along the lines of algebra. Next comes the mathematical school of probability, followed by charts and statistics. As the narrator slowly turns into a "math zombie", everything in her life is transformed into a problem. A class treat of cupcakes becomes a study in fractions, while a trip to the store turns into a problem of money. Finally, she is left painstakingly calculating how many minutes of "math madness" will be in her life now that she is a "mathematical lunatic." Her sister asks her what her problem is, and she responds, "365 days x 24 hours x 60 minutes." Finally, she collapses on her bed, and dreams that she is trapped in a blackboard-room covered in math problems. Armed with only a piece of chalk, she must escape-and she manages to do just that by breaking the chalk in half, because "two halves make a whole." She escapes through this "whole", and awakens the next morning with the ability to solve any problem. Her curse is broken...until the next day, when her science teacher mentions that in life, everything can be viewed as a science experiment."

After the book reading, we were asked to write how we "curse" our students.

In my table, there were two gentlemen whose responses caught my interest.

Gustavo Alvarado (Cotton Valley Early College High School in El Paso, TX) wrote: "Everything in your daily lives is a function."

Andrew Waxman (Mission Early College High School, El Paso, TX) wrote: "Life is (full of) word problems."

I didn't understand the question then, so my response was totally off. But now that I do, I say that I "curse" my students by telling them, "Finding something difficult to do means that you are about to learn something new."

How do you "curse" your students?

I Beg Your Pardon, Sir?

I had sworn to myself many years ago that I'll never make the same mistake again, yet, with so much regret, I still do. I guess it's a hard habit to break.

Although just occasionally nowadays, I still read numbers in decimal form improperly.

For example, I still read 0.34 as "point thirty-four" instead of "thirty-four hundredths." And I admit it is easier to read 0.2345 as "point two, three, four, five" than to read the number as "two thousand, three hundred forty-five ten-thousandths." The longer the decimal, the easier it is to read it improperly.  

This practice seems harmless at first glance but unfortunately it's not! 

We teachers who have the habit of reading decimals improperly are not helping our students mathematically by not helping them master their vocabulary.

As we all know, students, who know very well that 0.34 is thirty-four hundredths, have better facility at being able to express the number as 34/100, its fractional equivalent which they could easily simplify into 17/50.

This failure in number sense is prevalent even with native high school kids. Add that to the fact that the number of non-native English speakers keeps on increasing only makes it even more urgent to change this practice. These immigrants have lots of potential and they need our unwavering commitment in helping them learn the English language so that they can already promptly engage in learning with academic rigor and achieve mastery.

I now know better, but I wouldn't be able to say I've done the right thing until I've completely stopped doing this unfortunate teaching malpractice myself.

Reading decimals improperly is not a good practice at all. Not to stop doing it is downright irresponsible.