Pencils Down

Undeserving

2011-04-10T16:25:00.002-04:00

Tomorrow I begin the fourth academic quarter of my first full year teaching. Hopefully, by the summer I will have been offered a continuing contract with my district, acquiring at least a small measure of security. In a profession that has been much denigrated of late, it will be nice to have some verification that I am doing the job to someone's satisfaction. Over the last year, I have received a lot of praise. My graduate cohort leader holds me up as an example to her current students, inviting me back to give presentations from my pedagogy. New teachers come to me for advice, believing I must have answers to some of their questions. Administrators write glowing reviews. Students praise me in the halls.

And I don't deserve any of it.

I have been doing this for a grand total of 14 months, counting the past summer where I had no students, but carefully modified hastily constructed lessons from the year before. In June, when I celebrate the end of my first full year, one of my closest colleagues will mark the end of a career spanning decades. She is the one who deserves praise, not me.

Don't get me wrong. Nobody who knows me personally would accuse me of being modest. I believe very much in my own potential for greatness. I just know I haven't made it there yet. Outside of a few memorable lessons and a willingness to experiment outside of accepted norms, I am a pretty typical teacher. There's really no danger of my story being adapted into a screenplay.

But I have promised myself that my best is yet to come. I vow to constantly grow, to adapt, and to evolve into an educator that students never forget. I doubt Disney will ever come knocking at my door. I doubt I'll ever even get much in the way of a thank you. But as long as I can look back someday and know that I fought the fight even when others said it couldn't be won, well then I'll be willing to accept any praise.

Because then I will know I deserve it.

Counterintuitive Discovery

2009-09-21T19:34:00.002-04:00

Today I taught a lesson that went horribly wrong. I was being observed by my cohort leader, so I was already nervous, but some additional time constraints made me feel very rushed. As the lesson progressed and time slipped away, I started talking faster, pausing less between questions, and taking the first raised hand that presented itself. Needless to say, the kids were completely baffled.

After class, I had a couple of postmortem conversations with my mentor teacher and my cohort leader. I came to the surprising discovery that you can actually go faster by slowing down. Had I spoken slower and paused more, I would never have gotten so far ahead of my students. It's as though I was racing them to the end of the lesson. What purpose does that serve to get to where you're headed before the students do?

In the future, I am going to make a concerted effort to linger. I believe that by doing that, I will actually cover more information in less time.

Everything's in Motion

2009-09-14T12:50:00.002-04:00

Today we had an assembly sposored by Honneywell and NASA about Newton's laws of motion. The presenters used hip hop, humor, and interactive displays to keep the kids involved. I'll give a more detailed critique later, but for now check out the handsome sumo wrestler in the blue trunks.

My Education, My Future

2009-09-09T05:07:00.002-04:00

Just watch it.

Excuse Me..But Your Question has an Open End

2009-09-08T20:17:00.002-04:00

I inadvertently sat in on what turned out to be a rather clever little lesson on open-ended questions. While I still think that name could be retooled a bit, the demonstration itself was quite useful, and surprisingly, it was aimed at students in a 7th grade health class.

As teachers we need to get into the habit of asking questions that probe just a little deeper. If your question can be answered with a single word, you probably aren't delving all that deeply into the subject. In the lesson I witnessed, the teacher challenged to students to ask her open ended questions. She made a game of answering with as few words as possible. At first, she was able to deliver yes and no type answers, but gradually, some of the students started to get it. By the end, she was getting a lot more "why do you like that?" and "what do you think about that?"

The teacher may not have known I was eavesdropping, but I am thankful that I did.

Let the Testing Begin

2009-09-02T13:01:00.002-04:00

Three days into the first week of school and already we're talking about state testing. Within the first two months of school, our students will spend 5 days, a full school week, taking norm referenced standardized tests. That's five days right at the beginning of the year, before they've even brushed of the summer malaise, where they won't even have an opportunity to learn new thins.

I haven't had the opportunity to ask kids what they think about all the testing. They are probably used to it by now, since evidently it has started in kindergarten. Still, it seems excessive to me.

Teacher Colleges Beneficial in Only in Theory?

2009-08-30T07:47:00.002-04:00

I just read and interesting article in the New York Times debating the relevance of teacher colleges and education degrees. Sure, getting that Masters will raise your salary, but how much does that degree really help your kids? I am going to take the easy way out and let the article speak for itself, especially since my own school cohort leader peruses this blog from time to time.

My Teaching Stance

2009-08-29T14:54:00.001-04:00

Welcome to Westbrook Middle

2009-08-28T05:54:00.002-04:00

My first day as an intern began with a district wide breakfast at the high school. I say "the" high school because there is only one of them. For some of you, this may be par for the course, but coming from where I grew up, it will take some getting used to. I lived in a suburban sprawl, where entire neighborhoods appeared overnight and new schools were always being unveiled. There were at least ten high schools in my district that I can recall offhand, and my graduating high school class was nearly 600 students. Here in New England, that numbers dwarfs the populations of many schools, including faculty and staff.

Even though it is sometimes surprising to me, and the administrative costs must be astronomical, I find that I really enjoy the tight-knit community structure. Already, I am on a first name basis with umpteen faculty and staff members, which certainly helps assuage any nervousness I may feel as the new guy on the block.

Also helping me gain confidence was the crisp new $20 bill I received courtesy of the state teachers' union. As part of her presentation, the union rep distributed raffle tickets to the audience. When she emphasized a benefit to union membership, she called out a number and rewarded the ticket holder with $20. Mine was the second number called. Frickin' sweet!

Next, there was a series of introductions and gifts for faculty members. The longer you had worked for the district, the better your gift. Five year tenure earned you a corsage; forty year tenure earned you a windbreaker and a totebag. So on your first day as an intern, you get twenty dollars to do with as you please. After 40 years, a totebag. Does anyone else see the irony?

By the end of the day, I was beginning to wonder why my mentor teacher signed on for this. The year is already packed with changes and transitions for her. She is implementing a new math program designed by the NSF. I believe it's called Connected Math. In December, the entire school moves across town to a new facility. With that move comes a new name. Wescott Junior High will become Westbrook Middle. And on top of all that, my mentor teacher announced to our team that she is pregnant and due in March.

I can't wait to find out what revelations today brings.

Gender Bender

2009-08-25T09:30:00.003-04:00

There's a story emanating from the track and field World Championship in Berlin last week that I think has a lot of bearing on our classrooms. You may have heard about Usain Bolt's record-breaking sprints and his virtually uncontested gold medals. While Bolt held center stage on the field, the subject of this post is another athlete who remains in the news even after closing ceremonies.

South African runner Caster Semenya took the gold medal in the woman's 800 meter last week. As challenging as that competition must have been to her physically and emotionally, this week she must face a far greater challenge than she could possibly have imagined. As a reward for representing her country, Semenya must undergo gender testing. You see, she has sort of a deep voice and a lot of muscle mass. So instead of celebrating a hard-earned victory, she must submit herself to a barrage of demeaning tests designed push her firmly into a well-labeled little box.

I almost don't even know where to begin.

Semenya is not accused of trying to cheat, but of perhaps unknowingly having a
medical condition that blurs her gender and gives her an unfair advantage over
other female runners.

Maybe I don't understand what a "fair" advantage is. No one claims we ought to revoke Bolt's medals due to his unusual height. Don't his long legs give him an unfair advantage over shorter runners? Don't all great athletes have something that sets them apart from the rest? How do we decide when things are unfair? I recognize that I have an extremely unfair advantage in life as a white male. Should I give back all the awards or recognitions I have received?

For female athletes, the answer is always the same. Women from Babe Zaharias to the present have had to defend their abilities by defending their gender identities. Not so for men. They may have to submit to drug testing on occasion, but no one asks them to prove their manhood. Why do we continually ask this of women? What would we do to them if they started to win against men, instead of merely proving themselves against fellow women?

I think maybe some of the debate stems from our mistaken notion that gender and sex are synonyms. Gender simply means kind or type. So although our species has but two sexes, it has many different genders. By confusing the two terms, we invite these kind of arguments. I wonder what this might lead to in the classroom. As girls make steady gains in stereotypically male subjects like math and science, should we be expected to call their gender identity into question?

"Great job on the math test, Sarah. Now please pee in this cup. What's that? No, we don't think you cheated, per se. We just think that your brain might be wired differently than the other girls. You may have an unfair advantage."

Maybe some of my readers can point to the flaws in my reasoning, but it seems to me that a persons identity is their own business. We ought to be celebrating the triumph of the human spirit, whether on the track or in the classroom. No one ought to be punished for excellence, and certainly not by a committee with DNA tests.

Ropes Courses and Rain Checks

2009-08-21T19:49:00.002-04:00

The funny thing about experiential learning is that you never know exactly what kind of experience you're going to get. I am currently sitting on my couch when I had fully intended to be enjoying s'mores around the campfire with my ETEP cohort. Unfortunately, impending thunderstorms and threats of tornadoes sent us scurrying for the cars.

Before the sudden exodus, we had been enjoying a fun filled day of low ropes courses and team building type activities. Some I enjoyed more than others and the high ropes course whispered seductively to me throughout the day, but despite that disappointment, I really do feel as though I got a lot out of the exercises. The lessons we learned could probably have been presented as easily through an in class video or a Powerpoint slide show, but I suspect that what we learned today will be more fluidly transferred to other contexts. Often times, epiphanies made in the classroom never filter out to other environments. I really think the ripples of today's lesson will continue for some time.

Still, I really would have enjoyed a good zip line.

Praxis Makes Perfect Part II

2009-08-20T16:41:00.001-04:00

Well, according to the education professionals over at ETS, I am officially qualified to teach. I passed both my math and science content tests with flying colors.

In Defense of Blogging

2009-08-12T05:26:00.002-04:00

I go to bed early. I do my homework. I read ahead in the assigned text. I sit at the front of the class.

And I blog.

These are all things that a younger version of me would have found terribly amusing. I used to stay up late for no particular reason, aimlessly flipping channels from one TV movie to another. I used to coast by on my innate intelligence, ignoring those homework assignments that weren't graded and procrastinating on the ones that were, knowing that my exam scores would save me.

And I used to mock and berate anyone self-involved enough to think the Internet community wanted or cared to read their quaint little journal entries. Even using the word blog was enough to illicit a sneer.

After two and a half years and roughly 250 blog posts, I will now explain with perfect clarity why I am glad I changed my tune.

As a teacher, I can not afford to stagnate. Just as I chastise my mother for not being able to program the VCR (or for still owning a VCR,) my students will mock anyone who isn't on Facebook or Twitter. One of our first assignments in grad school has been to build a wikispace. While many of my peers have been stumped by issues of formatting and functionality, I have breezed through, having experienced this process already. I am not bragging, but I am thankful for having anticipated the role technology would play in my classroom ahead of time.

In another week, I will be assigned to my mentor teacher. I can only hope that he or she is as influential in my life as the edubloggers that have guided my development for these past two years. I hope he or she is as innovative as Dan or as supportive as Jackie. Thanks to blogging, I have mentor teachers all over the world.

Admittedly, i have a long way to go. I still don't tweet and I text in complete sentences with full punctuation. I will never be able to completely keep technologic pace with my students. But as long as I can stay just a few steps behind, I ought to be able to communicate with them at the times when I really need to.

Hula Hoops and Bovine Poops: Day One of Grad School

2009-08-10T17:01:00.002-04:00

Today I began the nine month boot camp that will shed the intellectual flab and give me the superior training that will transform me into a secondary educator. As first days go, it was pretty much what you would expect from a classroom experience, regardless of whether your role as teacher or student. You plan and plan and brainstorm every possible question or expectation, and just when you think you have all your bases covered, reality knocks to firmly on your ass.

I arrived on time and in style, sporting the first of some fancy new digs that my lovely wife Sarah helped me to pick out. She has been lecturing me about my total lack of style for years now, and with a shiny new ring on my finger and grad school teachers to impress, she figured this was the perfect time for a much needed make-over. As I donned my black sweater vest and stylish Chuck Taylors, I thought to myself, I'm glad I will be in an air-conditioned classroom or this would be a really bad idea. Note the not-to-subtle foreshadowing.

For the first hour, our cohort leader laid out the plans for the coming weeks. After some standard Q and A, she turned the reigns over to our exceptionality instructor. Our first week was going to be devoted to special needs learners, a topic which has always intrigued me. I benefited greatly from a TAG pull-out program myself, but it always struck me as odd to remove students from a mainstream class instead of allowing the varied skill levels and learning styles to augment one another.

Within minutes, and much to my chagrin, we were outside. Several games ensued, including one in which we linked hands and passed a hula hoop from person to person without unlinking. It was an excerise in learning, as the latter half of the circle watched the techniques of the first. Before I was completely drenched in sweat, we escaped back to the AC.

Our next and most substantial portion of our day consisted of reading about the former state-run facility at Pineland, ME. Before a person can truly understand where they are or where they might be going, they must appreciate where they have been. When the topic is exceptionality and special needs education in the state of Maine, looking backward means learning a little something about Pineland. The bulk of today’s activity was devoted to reading about the Maine School for the Feeble-Minded, a politically incorrect reminder of eugenics past.

As a relatively new arrival to Maine, I had never heard of the facility or what went on there. Sadly, I suspect the story is far from unique. While I believe it is important to judge men’s actions by the standard of the times in which they lived, it is hard to believe how quickly the road to hell is paved with good and superficially scientific intentions. It is both surprising and unsettling that this institution lasted as long as it did. In the future, should I ever feel hamstrung by exceptionality regulations or frustrated by special needs students, I will remember the shameful mission statement of Pineland and with that will come the empathy and understanding required.

As a demonstration of how things can change, we ventured the 45 minutes northward to visit the Pineland campus. While dining and socializing in the shiny new building atop the hill known as the Commons, it was hard to reconcile the history of atrocities of which we had read with the elegantly manicured landscaping and attractive architecture surrounding us.

After lunch, we meandered down to the Pineland Farm, a working dairy farm adjacent to the main campus. In addition to caring for a herd of show quality cows and numerous other livestock, the staff at the farm maintains a thriving education program. It is precisely the type of program that allows exceptional students to engage in learning activities alongside of their mainstream peers. The variety of sensory experiences and depths of understanding possible make lesson plan differentiation much easier than traditional classroom settings.

And there I was...ninety degree heat, a brand new sweater vest and Chuck Taylors...standing scant inches from streaming cow excrement. A hell of a first day.

Married....with Children?

2009-08-07T08:14:00.001-04:00

Now that Sarah and I have tied the knot, our discussions about the future seem to carry more weight. The topics are the same, but somehow the air around us seems thicker. Still enjoying our honeymoon, we have already begun discussing the pitter patter of little feet. My position on that issue has become cloudier than it used to be.

I have always wanted to become a father. Maybe it’s because divorce robbed me of a traditional relationship with my own dad, or maybe it’s just the strong pull of genetics. Regardless, I have long thought of hiking trips and bedtime stories with sons and daughters. But now that I am about to become a teacher, I wonder how I could possibly balance my passion for education with the family I intended to have.

One of my favorite things about attending school at USM is that class demographics are so non-traditional. Students range in age for 18 to 80, which makes for interesting group dynamics. As much as I love collaborating with the older students, I have noticed that they seem to be much less competitive than their younger classmates. It isn’t that they are less intelligent or less capable; it’s a matter of time. Most of them are parents who simply don’t have any time. Between force feeding recommended daily allowances of green vegetables, parent-teacher conferences, and shuttles to hockey games and dance recitals, there isn’t much time left for writing papers and studying for exams.

I know the kind of teacher I want to be. I know that I will spend as much time outside of class preparing and perfecting as I do in class with students. How can I possibly juggle fatherhood with that? How can I be the kind of teacher my students deserve while being the world’s greatest dad?

This Just In

2009-08-04T17:40:00.003-04:00

My Morning with ETS

2009-07-25T21:13:00.002-04:00

So I got to spend my morning with Henry Goddard's present day disciples (i.e. the good people at the Educational Testing Service.) But that's starting in the middle of the story.

By the time I registered for my tests, all of the local testing dates had passed. The only date available prior to the start of grad school was at a center in Salem, MA. Unfortunately, it was a week before my wedding and on the day of another wedding I was to attend. Nonetheless, I felt it was the best choice in the long run. Plus it was months away, so I had plenty of time to study.

Without too much more adieu, this morning arrived. My alarm went off at 4:30am and I suddenly realized that I hadn't really studied as much as I probably should have. Still, I sharpened my number 2 pencils and hit the road. On the way, I dropped off my lovely fiance at the marine animal rescue center where she volunteers. Google Map print-out and toll money in the passenger seat, I tooled on down the highway.

I had planned to get there with time to spare, but after several wrong turns, I arrived with seconds to spare. If you have ever tried to navigate through New England as a visitor, you know my pain. These states seem to make a game of obscuring road signs. As you are flowing through traffic, you must somehow glimpse the tiny non-reflective sign hidden behind two hedgerows, one giant oak tree, and any one of a dozen quaint New England landscaping features that that seem to exist solely to frustrate visiting drivers. Still, in the end I prevailed and arrived at Salem State College on time.

Four hours and 120 lead-filled bubbles later, I headed back home. I really hope I passed, because I don't want to go through this again.

Praxis Makes Perfect

2009-07-24T07:51:00.002-04:00

I have to drive down to Salem, MA tomorrow to take back-to-back Praxis II tests for math and physical science content knowledge. I've been brushing up a bit on all my trig identities and special factoring formulas and such. I feel pretty well prepared, especially considering that I only have to get 60% of the questions right to pass. Isn't it great that we have such high standards for our children's teachers? I'll let you know how it goes.

P.S. I'm getting married in a week. Woo-hoo!

The Dreaded Word Problem

2009-07-10T14:39:00.005-04:00

All across this great country of ours, math students grapple to the death with their arch nemeses- the dreaded word problems.

Regardless of how they are dressed up or repackaged, most math textbooks are still pretty much the same. A particular lesson or skill set is explained, several examples are given, and then two to three pages of exercises follow. At the very end of these practice problems, buried in the back so they are easily ignored are the much maligned word problems. While the future mathematicians relish with excitement the chance to challenge themselves with these rhetorical abstractions, the average students find ways to conveniently skip over them, like peas being pushed around an otherwise empty plate.

Despite our efforts to reassure the students, to give them problem solving techniques and confidence boosters, we must admit that we are sending mixed signals. Anyone who truly understands mathematics realizes that word problems are not only a key part of math, they are the only part of math. Mathematics is a way of thinking about our world. Seldom does one find themselves confronted by a floating quadratic function demanding to be solved at gunpoint. Instead, we encounter normal, everyday questions or problems that can be illuminated using the tools of mathematics. This means translating the idea into language and translating that language into a mathematical construct. Thus, by hiding these problems at the end, we allow our students to skip the only problems that they really ought to attempt at all.

Rather than work umpteen practice problems, already laid out in clearly defined mathematical language exactly mirroring the guided examples, the students should skip directly to the world problems. Too often I hear my pupils tell me that they understand everything but the word problems. I politely respond that if they don't understand the word problems, they don't understand anything. Instead, they are confusing familiarity with understanding. They think because they can generate the expected answers with a series of repetitious algorithms, that they are preparing themselves for the exam and beyond. There curriculum is a mile wide and an inch deep.

What I would prefer is this: fewer practice problems investigated with greater depth. I would like the answers to come in narrative form, where the student explains to me and to themselves exactly why they made the decisions they made and what axioms of mathematics allow them to employ the techniques they chose. Take the following example:

An Internet service provider charges $9.95/month for the first 20 hours and
$0.50 for each additional hour. Write an expression representing the charges for
h hours of use in one month when h is more than 20 hours. What is the charge for
35 hours?

I don't want to see 9.95 + 0.5(h-20) = Cost (h) ; Cost (35) = 17.45. I want to see the following:

The Internet service provider is offering 20 hours a month of internet usage for
a flat rate of $9.95. This means that regardless of how many hours we use, our
bill will be at least $9.95. In other words, this value is a constant. If we go
over our allotted 20 hours, we will have to pay an overage fee of $0.50 for each
additional hour. Since the number of hours we use will change each month, we can
represent that value by the variable h. (We label it h out of convenience, since
hours starts with the letter h.) It is important to note that we only pay
overage fees on the hours we use beyond 20. An expression for those extra hours
would be h – 20. Therefore, our total bill will be $9.95 plus $0.50 for every
extra hour, or in algebraic terms,

Total Cost = 9.95 + 0.50(h – 20)

To evaluate the expression where h = 35 hours, we simply plug 35 in for h and solve.

Total Cost = 9.95 + 0.50(35 – 20)

9.95 + 0.50(15) = 9.95 + 7.50 = $17.45

That, ladies and gentlemen is how you solve a word problem, and until our students are able to clearly explain every step, they aren't mathematicians. They are walking, talking abacuses.

Intro to Chem Lab

2009-07-02T07:31:00.002-04:00

Last semester I was forced to endure the most painful chemistry lab. As is typical for labs, the amount of work required is disproportional to the one hour credit earned, but that was not the source of my frustration. A scientific laboratory is by its nature about discovery, but this lab was really just about pedagogy. We were really only there to learn how to run columns or pipet solutions, yet we were to write our lab reports as though we were conducting real science.

Evidentally, some of the faculty members shared my concern. This summer, the entire general chemistry curriculum is being retooled. A new textbook has been selected, one written by the American Chemistry Society and centered around the most abundant and familiar molecule on the planet- water. Subsequently, the labs must be rewritten as well.

I have been working for the chemistry department this summer, researching procedures regarding using titanium dioxide to remediate environmental contaminants. The work was done toward designing some new modules for the analytical chemistry lab. The idea was to give the students all the requisite data to design their own lab procedures instead of following a cookbook recipe. That way, they get to appreciate what science really is. Yesterday, I finished that project. The professor I was working with is also involved in retooling the aforementioned gen chem labs. When I was given the option to turn my attention toward those, I jumped at it.

Next semester, when students download their chemistry labs, they will be reading my words. Pre-lab questions, procedures, post lab review- all written by me. Hopefully, I can spare them from the boredom I experienced.

Writing to Learn Math

2009-06-28T13:21:00.002-04:00

I have a new tutoring student starting tomorrow. She has always gotten A's in math, but she works very hard to do so. Next fall, she begins Algebra I and her parents have opted to put her into the accelerated program, fearing she would be bored in the standard pace. To give her a bit of a head start, they asked me to work with her once a week for the rest of the summer. We got a copy of the actual textbook she will use from the school and I have started preparing some lessons.

Since I have curriculum design on the brain, I am making a concerted effort to design an actual curriculum, rather than just work through the text. Her favorite subject is language arts and she loves to write. As a pre-assessment, in addition to having her complete a skills review, I have asked to read some of her writing. I don't have a lot of time to get to know her and I thought that might help. Additionally, I have purchased a bound sketchbook which I am going to make her math journal. I want her to keep all of her notes and assignments in it, but I also want her to have a place to jot down her thoughts and questions about math. Ideally, it will be equal parts mathematical notation and writing.

I'll let you know how she responds to it.

What's in a Name?

2009-06-26T23:12:00.004-04:00

Je m'appelle Tony. Mi chiamo Tony. Ich heisse Tony. My name is Tony.

And that's my problem. My name isn't Mr. Lucchese. I understand that formal address is meant to be a sign of respect, but it makes me uncomfortable. I prefer to be addressed as Tony, regardless of the age or station of the speaker. I have worked around people of varying ages in a variety of capacities my entire life, and I have never had trouble commanding respect as Tony.

So how do I reconcile my own personal preference with the fact that the administration of most schools is going to frown upon such informality? I've been trying to come up with compromises. On one hand, I could call my students Mr. or Ms. followed by their surname. At least then we'd all be equally uncomfortable. Or maybe the students and I can come up with an appropriate nickname, like teach or chief or oh captain, my captain.

Did anybody else have this problem?

What's in a Name?

2009-06-26T23:12:00.001-04:00

And That's My Final Answer

2009-06-24T15:39:00.005-04:00

Recently, American audiences got an Academy Award winning look at how easily traditional assessment overlooks cognitive gains made outside of formal classroom instruction. Slumdog Millionaire, a film about an improbable winner of the Indian version of Who Wants to Be a Millionaire, provides a kind of case study in multiple intelligences and learning styles, particularly those with cultural components.

The protagonist, Jamal Malik, along with his brother Salim and friend Latika, demonstrates a tremendous intelligence born from the streets of Mumbai. When circumstances call for him to test his abilities in a formal, albeit fantastic environment, Jamal’s surprising success is met with suspicion and accusations. By assigning him the label slumdog, the world has placed restrictions on the paths to which he may aspire. As educators, we often perceive limits to our students’ understanding based on our own cultural paradigms. The damage brought about by these unfortunate assumptions is made transparent when children like Jamal rise above them.

When their parents are killed by an anti-Muslim mob, Jamal and his companions leave their already impoverished life behind. Orphaned and alone, these three musketeers must learn to navigate the physical and political labyrinth of the Juhu slums. The tests they face are far removed from the ruthlessly efficient norm-referenced exams synonymous with Western culture. Their tests are ones of sheer survival, and these amazing youths score in the top percentiles. In a country where potable water is a luxury, accepted codes of ethics offer no particular advantage. Jamal and Samil learn that it is better to steal than starve, just as it is better to kill than to be killed. Though both boys exhibit clear intelligence, their ways of knowing are as individual as students of any classroom.

Samil, only marginally older and physically developed than Jamal, keenly reads the threads of power that hold their violent world together. He knows who is weak and who is strong. He knows when to strike out and when to run. These skills are recognized almost immediately by Maman, who begins to groom Samil for a role in his organization. Later, when Samil is forced to kill Maman, he does so with calculation, not passion. At a young age, he has learned to leave behind corpses, not enemies. In a situation that would break most adults, Samil follows one cold decision with another when he uses Maman’s death to curry favor with Javed, a rival crime lord. This way of knowing might translate to the playground, or even the boardroom, but it would not gain high marks in a classroom setting. In fact, students in Western schools showing similar aptitudes are typically branded as bullies and troublemakers, and despite its obvious merits, interpersonal intelligence is ignored at best and punished at worst.

Despite widely published and often cited research on multiple intelligences, most Western public schools still judge pupils by traditional rubrics. Jamal’s success on the quiz show emphasizes the flaws in this style of assessment. The questions asked by the host represent a statistical sampling of fields ranging from history to popular culture. As with standardized selected response tests, the implication is that performance on sample questions is predictive of overall knowledge. Thus, Jamal is labeled either a prodigy or a cheat. The possibility that he may simply have been extremely lucky is overlooked.

By his own admission, happenstance was a key factor. While he knows whose face appears on the American hundred dollar bill, he would be hard pressed to name those appearing on most rupee denominations. This naturally begs the question, do we judge intelligence by what a person knows or their capacity for knowing. By the first standard, Jamal would likely fall short in most classrooms. By the second, he would be judged a genius.

His ability to mine sense memories for trivial facts gives the impression that he can recall relevant pieces of information at will. Whether triggered by the smell of a dollar bill, the fall of his mother, or a chance encounter with a pop icon, Jamal’s mind makes connections that allow for swift recall of seemingly useless data. It is important to note that his intelligence pushes far beyond rote memorization. He is able to reassemble his knowledge in new and creative ways. When he deceives gullible tourists into believing that he is a guide at the Taj Mahal, he deftly parries questions and clarifications with fabrications of the truth constructed on the spot. This requires gifts for both language and reasoning ability, areas in which he clearly excels, despite having no formal education. While Samil would probably have difficulties in most formal school systems, all evidence points to the possibility that Jamal might have been successful, if only he had been given the chance to do so.

A teacher’s job description is often convoluted. We are essentially charged with the task of making students smarter. To do so, we must understand that intelligence is not a rigid characteristic, but an ever changing capacity to learn. As more knowledge is acquired, more nodes exist for future connections to be made. Nor is any one way of knowing manifestly superior to another. It is possible to take advantage of student’s contextual experience and cultural foundation by building new ideas on top of previous models. The characters of Slumdog exemplify the need to recognize these alternative learning styles and to design classroom assessment with them in mind.

The Odd Answers are in the Back

2009-06-21T12:55:00.002-04:00

I have recently been contracted by a local publisher to proofread some of their math textbooks before they go to print. It is actually a lot of fun. Basically, I am getting paid to do middle school level algebra problems. Occasionally, I recommend sweeping changes to the text, and I wanted to run some of them by my readers to see what you think.

First, on the subject of quadratic functions, if the guided example demonstrates how to find the vertex of a parabola written in standard form, do feel as I do that it is exceedingly cruel to give the students an entire page of functions written in vertex form.

Second, on the same subject, when the guided example shows how to find the vertex from the vertex form, do you think it is unreasonable to give an example like f(x) = 2(3x-2)^2 + 4? Nowhere in the example does it discuss how to handle this case or describe a horizontal compression. I recommended either removing these exercises or added an additional example to show the students what to do.

What would you have done?

Dividing Fractions

2009-06-18T07:08:00.003-04:00

Does anyone know of an applet or animation online that clearly demonstrates the process of dividing fractions? None of the textbooks I've seen make any effort to depict it graphically. They do it for multiplication, but then bail out on division. I can draw it for individual cases, but I would really like something more dynamic, so that students can investigate multiple examples for themselves.

Out with the Old, In with the New

2009-06-17T07:18:00.002-04:00

Yesterday I attended an informal luncheon for recipients of the NSF grant that is funding my graduate degree. It was a chance for the incoming class to pick the brains of those who had just graduated. The details of the program are not terribly complex, so we mostly asked them about how their student teaching placements had gone and what success they were having with the job hunt. Their answers were surprisingly varied. Some had been placed in classes with excited and engaged students who fought to answer as many questions as possible. Others were in classes where students rarely picked there heads up off their desks. Some had been accepted at the first school where they had interviewed. Others had sent out resume after resume with no success.

I suppose I had better cross my fingers.

Cultural Confusion

2009-06-16T19:37:00.002-04:00

Consider the following phrases: "4 by 6" and "3 into 12." To which of the four arithmetic operations are these referring? If you said, multiplication and division, respectively, I can conclude one thing. You are not from India.

I recently began tutoring a young woman who immigrated here from India last year. Through a series of unfortunate events, she has experienced a 6 year gap in her education. At 18 years old, she is only eligible to attend public school for one more year. After that, she must pursue a GED. She is actually a very capable young mathematician, although she needs to build confidence.

During our first few lessons, she appeared to be confusing multiplication with division. After speaking with her father, I discovered that I was the one who was confused. Or rather, we were suffering from a miscommunication. Evidently, in India, the word "by" denotes division and "into" refers to multiplication.

Isn't that delightfully fascinating? I think so.

Certification or Bust

2009-06-15T22:05:00.002-04:00

This past Wednesday I met with my graduate student cohort for the first time. There are roughly 20 of us and it seems we come from all walks of life. There are older professionals changing careers and eager young graduates alike. All in all, I think it's going to be good mix of personalities and perspectives.

Our first meeting was predominantly informational. We had a chance to introduce ourselves, or rather to be introduced by one of our peers. Beyond that, it was mostly paperwork and scheduling for the upcoming semester. We were given a few assignments, which I have begun exploring, and I am hoping to begin posting again at Pencils Down, as a secondary sounding board. I know I've promised that before, but this time I really mean it.

Pythagorean Lesson Update

2009-01-06T10:39:00.002-05:00

I know the last few loyal followers have been holding their breath in anticipation of the results of my first efforts as a teacher. Sorry for the wait.

I was fortunate enough to teach this lesson three different times. As would be expected, each attempt resulted in a unique outcome. For my first effort, I had Sequence 2. This was the group of kids with whom I had spent the least amount of time. Although the three pre-algebra sections are not deliberately grouped by ability, a definite caste system seems to have materialized with Sequence 2 performing near the bottom. Still, I decided to teach my lesson as-is, despite advice to the contrary from my placement teacher. It turned out including the proof was a bad idea, as she had warned. It took way more time than I had estimated and consequently, I was not able to stress key aspects of the lesson. In looking at student work, I found that I had not clearly communicated the fact that c² refers specifically to the hypotenuse and that it always goes in the same place in the theorem. For example, a² does not equal b² plus c². I also discovered that my instructions on the handout could have been clearer, and that the students would have benefited greatly from working more examples together in class.

Next up was Sequence 5. I had worked with a few of these students before, so I was definitely more comfortable and my sentences flowed with greater clarity. I decided to cut the proof from the lesson and spend more time working through examples. Student work greatly improved as a result, but there was still some confusion over the order of the variables in the theorem, with several students jumbling the equation. Exit Slip feedback suggested that I had not stressed that the relationship only works for right triangles.

For the final group of the day, I had Sequence 4, which was the group that I had observed for EDU320. The lesson went much more smoothly as a result of the relationship I had established with those students. The simple fact that I was able to call them by name made things much easier. Despite my disappointment over skipping the proof earlier, I left it out again so that I could work even more sample problems and stress the importance of keeping terms in the proper order. One interesting thing did happen that I had not expected, though. I had made many efforts to make this a culturally sensitive lesson, and I made mention of great geometers from China, Egypt, Greece, and the Mayan culture. When I was mentioning the Egyptians method of using ropes to measure distance, I inadvertently suggested that this was an antiquated technology. One student in the class who is of South American heritage raised his hand to inform me that it many parts of the world, this is still the preferred method. He was pleased at my mention of the Mayas, and even took an opportunity to teach me some Mayan words, but it was clear that I had erred and should be careful of that in the future. He is typically a problem student, prone to gang related discipline problems, but his interest in his Mayan heritage helped him to focus on this activity. He did not finish the entire worksheet, but what he did do was correct and flawlessly organized.

In general, I thought it was a success. The students said that they enjoyed the hands-on activity with the ropes, though in the future, I would eliminate the 5-12-13 rope, as it was clearly too complicated for them. If I had another day, I would definitely revisit the proof, but only after the students had the main idea down cold. Access to a Powerpoint projector would have been helpful, but I know that I can not depend on technology, so it is probably better that I learn to work without it.

Change of Plans

2008-12-07T07:11:00.003-05:00

So there's been a change of plans since the last post. My mentor teacher decided that my lesson on proofs was going to be way over the students heads. I am inclined to disagree, but it's her class, so I'm happy to comply with her wishes. I'm going to teach a lesson on the Pythagorean Theorem instead. As an activity, I made some of those knotted ropes the ancient Egytians used to survey land. We're going to scout out the foundation of a pyramid and hopefully find some Pythagorean triples in the process. After that, I'm going to introduce the actual equation and do the following informal proof.

Each of the four right triangles are of equal size with area equal to

½AB

The A-side angle and B-side angle of each of these triangles are complementary angles, so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C². Thus the area of everything together is given by:

However, as the large square has sides of length A + B, we can calculate its area as (A + B)². We can expand this to A² + 2AB + B². So

4(½AB) + C² = A² + 2AB + B²

2AB + C² = A² + 2AB + B²

C² = A² + B²

Q.E.D.

So that's the plan for now. I teach the lesson this coming Friday. I'll let you know how it goes. Right now, I'm off to take the Praxis I. Wish me luck.

My First Lesson Plan

2008-11-24T22:17:00.007-05:00

So I am about to teach my very first lesson ever. My mentor teacher is giving me the reigns of her 8th grade math class for one day and she's letting me talk about anything I want. Since I am only teaching one lesson which in no way required to link to their current unit, I have made a crazy decision. When I was back in school, the biggest road block for me in math was the sudden and unforeseen appearance of proofs in geometry. It has always bothered me that we wait so long to introduce this subject. I understand that students may not be cognitively ready to handle this concept until then, but I still feel that if we wait until they are 100% ready, then we've probably waited too long. I think it makes more sense to start broaching the subject as early as possible. To that end, I have decided to do just that. The following is a handout that I have prepared for my lesson. I do not intend to read this word for word, but these are the concepts I want to cover and in this order. If anyone is still reading this, let me know what you think regarding scaffolding, differentiation, and all those other buzz words. (Note: The true/false "quiz" at the end will be handed out immediately after the introductory remarks and before the discussion of statements.)

Mathematical Proof

The word proof means something a little bit different to mathematicians than it does to normal people. This is not surprising, since we know that anyone who willingly chooses to study math is odd, to say the least. For the average person walking down the street, if a statement seems reasonable and comes from a reliable source, then the statement is assumed to be true. Let’s say you turn on the evening news and hear that President-Elect Obama has officially selected his Secretary of State. There’s no obvious reason why you shouldn’t believe the story. We know that the newsroom carefully checks all the facts, and we were expecting Mr. Obama to begin choosing his cabinet soon anyway. So we are willing to accept the statement as true.

Other people would be a little more suspicious. Scientists, for example, test a hypothesis by performing the same experiment many times. If the result is always the same, no matter how many times they repeat the experiment, then they say that the hypothesis must be true

Mathematicians are the most stubborn of all. For them, it doesn’t matter how many times the experiment is run or how many supporting examples there are. There is always the possibility that if you run one more test or check one more example, that the hypothesis will prove false. So instead, what they do is carefully build an argument that proves the statement has to be true in all possible cases. This is similar to the way a lawyer might build a case or a film critic might write a review.

In this lesson, we will take a closer look at how mathematicians build proofs and why it is important for us to learn them.

How do we decide what to prove?

Before we can set about building our argument, we have to decide what it is we are trying to prove. That usually begins with a statement. A mathematician defines a statement pretty much the same way as everyone else. It has to be a declarative sentence. For example, which of the following are declarative sentences?

What are we having for dinner?
Birds have feathers.
I love math.
Bao, please close the door.
The square root of 9 is 3.

The first sentence is a question. Questions sometimes lead to interesting discoveries, but there is no way for us to prove a question by itself. The fourth sentence is a command, also known as an imperative. It tells someone what to do. Again, there is nothing for us to prove. The rest of the sentences are all declarative.

We also require that the sentence have an objective truth value, meaning it is either true or false. If the sentence states an opinion, there is no way for us to prove it true. Which of the following declarative sentences are statements?

I love math.
The square root of 9 is 3.
September 12, 2027 will be a Friday.
Mr. L’s lesson is boring.

The first and last sentences are matters of opinion. There is no way to prove them true or false, no matter how strongly you may agree or disagree. Both of the middle two sentences are statements. The second sentence is obviously a statement, and we know immediately that it is true. What about the third sentence? This is also a statement, because we know that it must be either true or false, even if we don’t immediately know which one.

Now that we know how to recognize a provable statement, let’s look at how to begin building our argument.

How do we build an argument?

There are a lot of different kinds of proof strategies, but the one you will probably use the most is the direct proof. In a direct proof, you start with something you know to be true and build the argument one fact at a time until to reach the desired conclusion. How many facts you use depends on how complicated the statement is and who your audience is. It is a good idea to design your proof for someone who knows less about math than you do.

The facts you will use to construct your proof will basically come in three forms: definitions, axioms, and theorems. A definition is just what it sounds like. Let’s say you want to prove a statement related to a triangle, you must first know what a triangle is. An axiom, sometimes called a postulate, is a statement so basic that it doesn’t need to be proven. For example, the statement “the product of any number multiplied by one is equal to the original number (x · 1 = x) is an axiom. That’s just the way multiplication works. There’s no way for us to prove this; we just have to assume that it’s true. A theorem is a statement that has already been proven, and math textbooks are full of them. Once a statement has been proven true, we are then free to use that statement to prove other things.

The tricky part is deciding what facts to use and in what order to put them. Let’s look at the example of the next page.

What if you were asked to prove that dogs are mammals? Which of the following sentences might you use to build your case and why?

Dogs have a backbone.
Dogs have four legs.
Dogs have either fur or hair.
A mammal is a warm-blooded animal that has a backbone.
Mammals are the only animals that have hair or fur.
All dogs go to heaven.
Dogs can learn over 200 human words.

Choosing the right facts to use in your argument is trickier than it might seem. Sometimes you will come across a true statement that is related to the topic, which doesn’t help you prove your point. For example, the last sentence is true. Scientists have demonstrated that some dogs can respond to more than 200 different commands, but that doesn’t help us decide if they are classified as mammals. The same is true of the fact that dogs have four legs, since we know that lizards have four legs but are reptiles. Each line of your proof should move you closer to your goal, but don’t be afraid if you hit a dead end. When that happens, just go back a step or two and take a different path.

Once you have some idea of the kinds of facts you will need, you have to figure out where your proof should begin. Sometimes that will be obvious by looking at the statement you are trying to prove. For example, look at the statement “If the student gets an A on the final exam, then the student will get an A in the class.” In order to prove this is true, we can begin by assuming that the student gets an A on the final exam. But sometimes, the starting place is not immediately clear. In that case, it becomes a judgment call. Where you choose to start may depend on your audience. If you are proving something to your teacher, you might choose to start differently than if you were proving something to a classmate. Generally, it is a good idea to assume that your audience knows almost nothing about math.

Let’s look at an example. What if you were asked to prove that ice cubes float in water? Where might you begin? It would probably be a good idea to make sure that you explain what we mean by the word float. So a good starting point would be the following statement: “When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.” Once we have that, we can start adding more details. The following sentences, when placed in the correct order, will complete the proof. In what order do you think they should go?

1. When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.
2. Since 917 is less than 1000, we know that ice cubes float.
3. The density of liquid water is 1000 kg per cubic meter.
4. For objects to float, the object must be less dense than the fluid.
5. The density of water in solid form (ice) is about 917 kg per cubic meter.

We have already said that the first sentence will be our starting point, so we can set that one aside. The second sentence restates what we are trying to prove, so it makes a good conclusion. Now we just have to worry about the other three. According to the fourth statement, in order to prove that something floats, we must first prove something about its density. We don’t even need to know what density is, just that the object’s density is less than water. Statements three and five give us key information about the density of both water and ice. So we may conclude that the statements belong in the following order: 1,4,5,3,2, You could make the argument that statements 5 and 3 could be flipped, which is certainly true. However, it is a good idea to match the natural order given in the previous statements. Since both 2 and 5 mention the density of ice first, we should put statement 5 before statement 3.

What does any of this have to do with math?

So far we have been avoiding dealing with actual math. Now that you know a little more about proofs, we can look at an actual mathematical example. The following proof deals with a topic that is not normally presented until Algebra II. DO NOT BE ALARMED!!!
None of what we have been talking about is usually covered until Geometry, so you’re already way ahead anyway.

Before we can deal with the proof, we must first look at the following theorem.

Theorem A: For any number x , x^a · x^b = x^a+b.

Remember, a theorem is a statement that we already know to be true. Somebody has proven it for us and put it here for us to use. But we still need to make sure that we understand what it is saying. Let’s look at an example.

Example: x³ · x⁴ = x³⁺⁴ or x⁷

We know that x³ = x · x · x and x⁴ = x · x · x · x
So we are saying that x³ · x⁴ = (x · x · x) · (x · x · x · x) = x · x · x · x · x · x · x = x⁷

Once you understand what the theorem is telling you, it isn’t that scary. You have already been working a lot with x², and even though these powers are larger, the idea is exactly the same. Now that we understand how to apply the theorem, we can use it to prove other statements. For example, let’s look at how to prove the following:

For any number x, x⁰ = 1.

You are probably starting to get a headache. You know what x³ means. That’s just x multiplied by itself three times. We can do the same with x⁴ and x⁵, but what in the world do we mean by x⁰? DO NOT BE ALARMED!!! This is exactly the same question that mathematicians once asked themselves. Then they answered it with the following proof:

To Prove: For any number x, x⁰ = 1.

We are told that x is a number. This tells us that we can apply Theorem A to the problem since that theorem is true for all numbers. Begin by raising x to some arbitrary power, say x⁷. Then using the theorem, we can rewrite x⁷ as x⁷ · x⁰ since

x⁷ · x⁰ = x⁷⁺⁰ = x⁷.

We know by equality that x⁷ · x⁰ = x⁷. In other words, if we multiply x⁷ by x⁰, we get back x⁷. There is only one number that when multiplied by another number returns that second number. That number is the number 1, therefore, we have proven that x⁰ = 1. ▪

The little ▪ symbol at the end of the proof shows that you are finished. You do not have to use that symbol. You could simply write “The End” if you like, just as long as your reader knows that you are finished proving whatever you are trying to prove.

So far, all of our proofs have been written in paragraph form. Although there is nothing wrong with this, in school we commonly use an organizational device known as the two-column proof. As you’ve probably guessed, they have two columns. On one side you have a list of statements and on the other you have the reasoning behind them. The following is an example of what the previous proof would look like as a two-column proof:

Argument Reason Why

1. The unknown x is a number. 1. We are given this in the problem.
2. x⁷ · x⁰ = x⁷⁺⁰ 2. By Theorem A.
3. x⁷ · x⁰ = x⁷ 3. By addition.
4. x⁰ = 1 4. By the multiplicative identity. ▪

Are we done yet?

The short answer is “yes.” You now know some of the basic concepts of mathematical proofs, and even though you won’t see any of this again until high school, you can start using some of the ideas now. Whenever you are asked to solve a word problem, you can practice using some what you have just learned to defend your answer. You can also use the idea of proof outside of math class. When you are writing essays in language arts or social studies, you can use the same steps to build your argument.

The remaining pages of this handout are full of examples for practice. You will be asked to pick out provable statements, put statements in proper order, and even to build a few simple proofs. The more you practice now, the easier this will be in the future.

Name__________________ Sequence___

1) Give an example of a statement. (Remember that for it to be a statement, it must be provably true or false.)

2) Put the following statements in order to prove that water is a liquid at room temperature.

Room temperature is defined as 70° F.
The temperature at which a liquid becomes a solid is called the freezing point.
There are only three states of matter: solid, liquid, or gas.
Since 70 is greater than 32 and less than 212, water must be a liquid at room temperature.
The temperature at which a liquid becomes a gas is called the boiling point.
The freezing point of water is 32° F.
The boiling point of water is 212° F.

3) Fill in the blanks with the appropriate letters.

To Prove: If 4x + 10 = 38, then x = 7.

We begin by assuming that 4x + 10 = 38. By the subtraction property of equality, we can subtract 10 from both sides of the equation. (_____) Then using the (______), we can divide both sides of the equation by 4. (______) Therefore, we have proven that if 4x + 10 = 38, then x = 7. ▪

a. multiplicative identity
b. This gives us 4x = 28.
c. This leaves us with x = 7.
d. division property of equality
e. This gives us 4x = 38.

4) Convert the finished proof from Exercise 3 into a two-column proof.

Argument Reason Why

1. 1.
2. 2.
3. 3.
4. 4. ▪

5) Prove that if x + 3 = 15, then x = 12. (Hint: Look at Exercises 3 and 4.)

6) Prove that the area of the inner square is exactly half the area of the outer square.
(Hint: Don’t be afraid to draw more lines if it will help.)

7) Congruent angles are angles that have the same measure. Prove that if the two horizontal lines are parallel, then angle 1 is congruent to angle 8. ( Hint: Use alternate interior angles.

8) Prove the three angles of a triangle sum to 180°. (Hint: Draw a line through C that is parallel to line AB, then think about supplementary angles.)

Name_______________________ Sequence_____

For each question, circle the correct answer.

1) Give an example of an integer. True or False.

2) How do we simplify 5x + 3y – (-2x) ? True or False

3) Portland, Maine is the best city in the US. True or False.

4) The first three questions are impossible to answer. True or False.

The Other Maine: Faces of Homelessness

2008-11-18T16:01:00.002-05:00

The following is a reflection on a series of articles recounting tales of homelessness in the state of Maine.

This is the sixth time I’ve started this reflection. I’m too angry to know exactly how to begin, but the due date is fast approaching, so I’d better get something on paper. The articles on Maine’s homeless children have induced a state of frustration in me so powerful as to dampen my regular flow of wit. The situation seems hopeless to me. One of the best and worst things about being human is that our ability to act deliberately in total disregard for natural instinct makes us think that we can “fix” laws of nature. Try as we might, we can not legislate away the fact that all systems have selection pressures and that not every member of that system is going to survive.

When I moved here to Maine, I had to get a new driver’s license. I also had to get a dog license, which I had never had before, despite have pets my whole life. In the same office, I saw applications for hunting licenses, fishing licenses, business licenses, and marriage licenses. You need a license to start a fire in city limits, to broadcast on a radio station, or to practice law. But anyone with a working set of genitalia can have children. We don’t get to license that. It’s a natural right endowed upon us by the universe. But it’s a right that carries with it a tremendous amount of responsibility and therein lays the problem.

The children are blameless in this struggle. As a social species, we feel compassion for them, knowing full well that they are the victims of their parents’ bad choices. We want to help, so we pass laws or enact assistance programs. But nothing works. The system is simply too big with too many cracks. The price of progress is that our family group is just too extended for us to help one another anymore. We are largely on our own and some of us are bound to fail.

It is a question of inheritance and education, really. For example, by monetary standards, I am neither rich nor poor. I live from paycheck to paycheck, and though I am comfortable now, I am one disaster away from having to renegotiate. But I have a huge safety net underneath, because no matter what happens, I always have my inheritance. Don’t misunderstand; there is no money to be had. I am not to be the beneficiary of some familial fortune. My inheritance is the power than comes from a superior education. It is my firm belief that if I were to be stripped naked, blindfolded, and dumped anywhere on the globe, that I would have the requisite skills to quickly rebuild a life. Nothing short of massive head trauma can steal that from me.

It occurs to me that my exception has proved the rule, that even an eternal “have” like me could be turned into a “have-not” with a quick crushing blow to the occipital lobe. So how do I respond to that realization? Can we chalk these scenarios up to bad luck and count our own blessings or are we obligated to help in any way we can? I suspect our humanity obligates us to the latter course. As admirable as that instinct may be, it dooms us to a certain amount of frustration and failure. I have chosen to combat this social ill, along with all other systemic malfunctions, in the only way I know how- as an educator. A good education can provide a measure of relief that no government assistance program can. It is the only solution I see as being effective, so that is how I choose to do my part.

Whose History?

2008-10-20T07:37:00.001-04:00

Those who can not learn from history are doomed to repeat it.
-George
Santayana

Same shit, different day.
-Steven King, Dreamcatcher

Why do we study history? Scholars assure us that history informs both our present and our future, that while the universe is vast with possibility, mankind tends to tread familiar and well-worn paths. Great generals painstakingly recreate battles waged beyond living memory to better understand the nature of warfare and to prepare for future engagements. Scientists use data gleaned from the past to predict tomorrow’s reality. To be sure, the past constantly nips at the heels of the present. But is that really why we so dutifully record our stories for the historians of tomorrow? Perhaps, it is. Maybe humanity, the only species on the planet known to understand its own mortality, compiles these complex annals for practical reasons. But I doubt it. It strikes me that a far more primal imperative is at work. In short, we love a good story.

Since before history was history, men and women have been telling stories. We tell stories about the gods and about the heavens and about the creatures of the earth. But mostly, we tell stories about other people. In evolutionary terms, we have been singing our own praises since we strayed from the safety of the trees and started roaming the African savannahs. History has mostly been an oral tradition, passed down from generation to generation as campfire tales and bedtime stories. Apprentice bards learned their craft from tribal elders as tales were honed and polished to suit the tongues of the tellers. It was very much more an art than a science, and a certain creative license was expected and encouraged. Then along came the written word, and suddenly that which was ethereal and fleeting achieved a degree of permanence that changed the discipline forever.

Once a story gets written down, it is much harder to edit. Books can be burned and edicts decreed, but some vestige will always remain. With this realization, the study of history took an egocentric turn. To the victor go the spoils, and no spoil is of greater importance than the ability to calcify one’s own version of the tale. This incontrovertible truth lies at the heart of every multicultural historical debate. Truth, like beauty, is in the eye of the beholder. The history textbooks in our classrooms typically have a lot of ground to cover. The authors are limited to one version of each story, and too often, that version is the only one most students will ever hear.

To truly understand history in a way that will move mankind toward enlightenment, we must be willing to listen to all sides, to construct our truth from all the facts at hand. As educators, it is our duty to teach our students to think critically and take nothing for granted. But that does not mean throwing out the historical baby with the bath water. Historians like Howard Zinn would have us simply substitute one half-truth for another. It is the responsibility of a scientific observer to remain as impartial as possible, to acquire objective evidence before drawing any conclusion or backing any agenda. I have read Zinn’s A Peoples History, and objective it is not. There is a clear and unapologetic agenda of tearing down heroes and championing the downtrodden. If history is a popularity contest, Zinn is backing the kid with coke-bottle lenses and acne.

According to Alejandro Segura-Mora, teachers are “cultural workers” who “can, and should, challenge white supremacist values.” I agree with this in as much as I believe we should teach our students to question everything, including ourselves. It is unfortunate that students expend so much effort parsing out the “right” answers. I am lucky enough to be training in fields like mathematics and physics, where subjectivity rarely comes into play, and “right” answers are even possible. For the rest of the world, such a concept is meaningless. For historians and poets, if you think you know the answer, you probably don’t. Here in America, where we argue black and white, it is especially easy to overlook the myriad shades of gray. As the current bearers of the mantle of imperialism the stretches back for millennia, we must force ourselves to take a hard look in the mirror. To those who mock opponents of the PATRIOT Act as overly dramatic, I offer the Alien and Sedition Laws and the Japanese Internment. To those who celebrate the low prices of Wal-Mart, I reply with sweat shops and abusive child labor. These comparative histories are no less powerful for my having heard them before. They are essential to our understanding of civilization and the stories must be told.

What we should not do, what I refuse to do, is to promote any one set of values of another. My job is to teach mathematics, and it is not an easy one. I have no problem with demonstrating how mathematical problem solving can be used to inform cultural debates, but it is not my place to inject my own politics, even if they be the politics of multi-cultural awareness and equality. Our goal should be to strip away bias, not to replace it with our own. As much as I despise the dogma of “white supremacy,” I would prefer that they at least be white supremacists that can model linear equations, follow statistical arguments, and demonstrate abstract reasoning ability. To that end, I will resort to whatever strategies I deem effective, including allegedly “sexist, racist, culturally insensitive, and contemptuous” games like Oregon Trail.

There is little doubt in my mind that this popular simulation exhibits each of these qualities, but then so does history. I freely admit to spending many hours attempting to cross the Great Plains in my digital wagon train, although in the interest of full disclosure, I never made it to the Oregon frontier. On the occasional attempts where I avoided falling victim to dysentery, my adolescent male fascination with violence sidetracked me toward extended squirrel hunting expeditions. Even so, I enjoyed learning through the game for no other reason than that it was fun. That’s why we play games after all-for fun. Part of that fun comes in the challenge of winning. If the game can not be won, the wind slips out of our sails. No one would want to play a game called Trail of Tears. Slave Trade 2 won’t be flying off the shelves. Imagine spending countless hours watching your computer avatar lying side by side with countless others in the underbelly of a slave ship with the ultimate goal of arriving triumphantly in the cotton fields of the agrarian American south. How is that supposed to facilitate the love of learning? It can’t. So I guess we’ll just have to settle for the “contemptuous” Oregon Trail and remember that it is only a supplement to a balanced curriculum, not a primary source.

The issue of cultural bias is unavoidable in history class, but it exists in one form or another in all subjects. Though we can never eradicate those prejudices completely, by discussing them openly and honestly, we can significantly curtail their influence. As teachers, we should content ourselves with sparking that debate, knowing that it is often more important to ask questions than to find answers.

In Quotations

2008-09-28T21:11:00.002-04:00

The following is part of a class assignment. Quotes are pulled from Ordinary Ressurections by Jonathon Kozol.

“Liberals may think that they can contradict a stereotype,” he notes,” by
walking off into a neighborhood where they do not belong” and may believe
they’ll “be protected” by their ideologies or sentimental loyalties. “It
doesn’t work like that,” he says, “and you’d be unwise to believe it.” (p.167)

There is a famous story in mathematical circles concerning the death of Pythagoras. It is said there was an uprising against his mysterious cult, but that the venerated leader escaped the original attack only to be cut down later, due to his refusal to cross a bean field. It seems amidst their reverence for number and theorems, the Pythagoreans harbored beliefs in the most extraordinary of things. Because of its passing resemblance to human sex organs, the legume was deemed indecent. Pythagoras, according to this legend, was killed because of his steadfast refusal to be touched by such “vile” plants.

Nowadays, our intellectual elites, our modern-day Pythagoreans, are sometimes slain because of where they are afraid not to go. For fear of appearing racist or bigoted, they disregard stereotypes with celebrated flair. With the kind of passionate ignorance of reality that stems from early sequestering within the walls of the Ivory Tower, they literally and metaphorically walk where they do not belong.

Yet they should take note of one of mathematics less superstitious methodologies- the field of logic. It is one thing to say that not all group members fit a stereotype, and another entirely to say that none do. Behind every cultural stereotype lies a probabilistic heuristic that has proven true enough to survive. Our protection comes not from naively ignoring those stereotypes, but from being able to see past them.

He doesn’t just “sustain” the difficulties of existence. He steers around
them in inventive ways that give him the defense he needs…He never seems like
someone who’s agreed to be defeated. (p. 236)

There is much talk of the “resilience” of children among those schooled in the social and psychological sciences. The term conjures images of the downtrodden, of those who are forever being peeled from the tread of society’s boot. We marvel at how these young people are able to persevere in spite of such a relentless attack. In reality, we give them too much credit, and at the same time not enough. We are right to be surprised by their ability to rise up after being hit by such continuous volleys. No human could be expected to “sustain” those difficulties. Yet while we overestimate their recovery strategies, we underestimate their ability to avoid the attacks altogether. Their “resilience” comes not from standing strong before life’s frontal assault, but from their keen awareness of the likely places of ambush and the ingenuity to escape around the danger. In so much as they manage to avoid the obstacles, they demonstrate their ability to play outside the rules. They employ strategies that boggle our minds, because for all our well-meaning ideologies and good intentions, these children are the ones actually playing the game. We are merely spectators.

I know nothing of theology; but it occurs to me that modest hesitations-normal
ones, like those in ordinary conversations- may allow a bit more space than a
relentless speaking style does for people in a congregation who may feel the
world has tried to clip their wings and that the powers and the principalities
of their society might actually prefer it if they didn’t fly too high. (p.246)

Regardless of the chosen medium, a true artist knows that there is as much power placed in what is not said as in what is. The blank patch of canvas, the silent rest signs of a great composer, the soft pause of a seasoned orator- these are the nothings and nowheres that make the lasting impressions. In that little white spot, the artist invites the audience to fill in the blank- to insert a piece of themselves into the work. This is a lesson of which we as educators would do well to take heed. People are willing to be lead, but will steadfastly refuse to be pushed. We must remember that each lesson belongs not just to us as teachers, but the students as well. In order for our artwork to be remembered through the passing of time, we must not be afraid to put brushes in the hands of our audience.

Profiling 101

2008-09-15T07:01:00.002-04:00

Over the weekend, I had to create an academic profile for myself using data gleaned from a series of questionnaires. One was called the GREGORC inventory, which I had never heard of before. Another was regarding Gardiner's multiple intelligences, which I had heard of before. The rest were short answer questions about learning styles and preferences.

I scored comfortable to very comfortable on 6 out of the 8 intelligences. I just missed Bodily-Kinesthetic, but I am very deficient in IntraPersonal. Evidently, I don't care very much about people's feelings. On the GREGORC, I came up Abstract Sequential, which means I value thinking and analysis above all else.

Surprisingly, I found the learning styles sheet the most telling. It consisted of a flowchart of sorts, which laid out all the different learning environments. (noisy, quiet, visual, aural, etc.) I feel quite comfortable with any and all environments save for one- Kinesthetic Mobility Tactile. That category includes such horrors as role playing, mime, and immersion. You can put me in a dark, loud, hot room and I can still concentrate, but five minutes of mime and I'm contemplating murder-suicide.

Well, that's me in a nutshell. (How did I get in this nutshell?)

Where I'm From

2008-09-05T18:52:00.002-04:00

So my first assignment for Culture and Community is to right a poem. It is supposed to be modeled on Where I'm From by George Ella Lyon. Here is a rough draft. Feel free to critique.

WHERE I'M FROM

I am from the streets
(of suburbia.)
From circles to courts and cul-de-sacs.
I am from Big Wheels and lemonade stands,
from broken bones to broken homes,
and always at Pizza Hut.
I am from where the street only goes in one direction.

I am from t-shirts and Bible Belts,
from funnel cakes and Rocky Top.
(Man, I hate that song.)
I am from Frisbee and tenor clefs,
from matinees and tender feet.
I am from sadness and euphoria.

I am from behind the scenes to on-stage,
where everything is backwards.
I am from perennials and postholes,
from what do you want to be when you grow up
to you can't get there from here.
I am from the library.

I am from GA to ME
on foot over mountains.
I am from spotty dogs and smelly socks,
from sunsets to morning dew,
from heavy hearts to ultra-light.
I am from the do-overs to the birds ablaze.
Where the street now goes in both directions.

Culture and Community

2008-09-03T19:35:00.002-04:00

I had my first education class yesterday, CPI211I: Culture and Community. I thought you'd like to hear about it.

Let's begin the story by explaining that my university has two campuses, 8 miles apart. I think they began as separate colleges that merged at some point. I typically attend classes here in Portland, which is within walking distance from my apartment. But the EDU department is on the other campus in Gorham, ME. Being the eco-friendly fellow that I am, I decided to take the bus that the university offers between campuses, especially since I help fund that bus as part of my student fees.

I hopped on the bus along with 40-50 other people. Every seat was filled and the aisle jammed full of late arrivals. The subsequent 8 mile ride set the tone for the rest of the day. It took 40 minutes to make the trip, which was compounded by the miserably sweltering heat. As I felt my internal temp rising far above comfort level, I read with eyes squinting away the sting of sweat pouring down my forehead, a sign that proclaimed the bus a WiFi Zone. You see, at USM, we have the technology to allow students to receive wireless internet while rolling down the highway at 60mph, but lack that required to build a bus with functioning windows.

I arrived with minutes to spare at took one of the remaining seats in the classroom, which regrettably, was just barely cooler than the bus. It is not hyperbole to say that knowing how that class went, I would have preferred to remain on the bus for that 3 hours.

Almost immediately, we were asked to leave our seats and form a mosh pit in the center of the room. We would spend the next 2 hours performing various getting-to-know-you tasks, beginning with the "human atom." We were asked to move about the room as though orbiting a fictional atomic nucleus. At random intervals, a call to "freeze" was given and further instruction given. At each stopping point we were to grab a new partner and perform a task. The first time, we were to join elbows with someone and introduce ourselves. Next time, it was the knees. The next time, we were to invent a three-part secret handshake. Next, find a common letter between your names and think of three ways to form that letter with your body.

It went on, and on, and on.

At the end, we formed a "truth circle," where we commented on how we felt. People stepped forward to proclaim how relaxed they were and how great it was not to be confined to desks. I stepped up and announced that I was far more tense than I had been before we started.

First impressions are important, and my impression of this class is that it is a complete waste of my time and money. It isn't that I don't get the point of these exercises. In fact, I may have been the only one in the classroom that did. The point was to make connections. Before we can teach someone or learn from someone, we must first establish a connection. That connection will be based on mutual experience or memory or goals or history or whatever. That kernel will provide the foundation for the relationship that will allow for learning to occur. Many of those relationships form a community. This kernel need not be some huge thing; it only has to be real.

Nothing about these artificial activities serve that purpose.

So Far, So Good

2008-09-02T08:13:00.003-04:00

Just got back from a 2 mile run. I know that isn't much, but I hate running and I haven't moved that fast in a long time. Anyway, I thought I'd blog a bit to celebrate.

As I mentioned, I have my first two education classes this semester. I've read both of the textbooks already (insert nerd joke here) and I think I'm really going to like one, and really hate the second. The first is all about how to better integrate the knowledge coming out of laboratories with the day to day functions of the classroom. It discusses recent breakthroughs in neural networks, epigenetic theory, and a host of other interesting fields that may one day be useful to teachers. Hopefully, that day will come sooner than we think.

That book is entitled How People Learn.

The second class is more touchy-feely. It seems to be about how cultural differences embedded in society affect the achievement of students in the classroom. I certainly understand how this knowledge will prove useful to me as an educator, but based on the textbook, the class seems to go far beyond that. There is definitely a subtext of massaging and molding the culture via education to curb some larger social ills, and I don't think that is the primary job of schools.

But we shall see. Perhaps my opinions will change throughout the semester. For now, I have to pack my lunch in my A-Team lunchbox and head off to campus.

Promises, Promises

2008-09-01T19:03:00.002-04:00

Look, I know you have no reason to trust me. I've made promises like this before. I'll do better, I say. I'll write more often, I swear. And for a while, I do. It's just like old times. We talk, we laugh. It's like when we first met. But eventually, things change and I'm back to my unreliable ways.

No more, I say. It's a new semester and the buck stops here. I have two education classes and I expect much debate, discussion, and reason for intellectual reflection. I vow to share every thought, every argument, every tiny detail with you.

And I'm going to start jogging as well.

Unequal Pay

2008-07-25T08:41:00.002-04:00

It seems that many school districts are having a hard time attracting qualified math teachers. Gee, I wonder why that is? Could it be that those potential educators are having a hard time turning down the six figure salaries that Google or AT&T are offering them? I seriously doubt there's nearly that kind of disparity between professional historians and social studies teachers.

I can't even count the number of times I have advocated raising teachers salaries. All teachers need to be paid more for what they do, but we ought to bear in mind the realities of the situation. Marquee players bring in a larger chunk of revenue and so they get paid a larger salary. Well in the world of education, STEM grads are the marquee players. I'm not saying this is fair; I'm saying "fair" is irrelevant. If the goal is to better educate young people, then those kids deserve to get the best teacher money can buy.

In the case of math teachers, that price may just have to be a bit higher.

Word Problems

2008-06-05T07:21:00.001-04:00

This is what word problems sound like to mathphobes.

See more funny videos at CollegeHumor

Exceptions and Higher Standards

2008-03-25T09:40:00.002-04:00

I'm not quite sure how I feel about this. An art teacher in Florida is in danger of losing his certification because he cannot pass the minimum general skills requirements due to diagnosed discalcula. His argument is that he doesn't see what math skills have to do with art.

I guess it depends on your perspective. I certainly feel bad for him, but truly, all things being equal, wouldn't you rather have an art teacher who is good at math? I know I would.

Classroom Etiquette

2008-03-19T11:57:00.003-04:00

I have heard it said that there are no stupid questions, and while this may be true, I think it deserves some clarification. A well crafted question designed to invoke a particular response can never be stupid, and nor can the person asking it. The very fact that the question was raised implies that the student has identified a specific gap in knowledge and wishes that hole to be filled. A stupid person doesn't even know what they don't know. Even broad questions like "Can you explain that again?" or implied questions like "I don't understand" can be helpful in determining where the explanation ought to begin. As a teacher, I will strive to maintain patience, addressing all questions no matter how outlandish or repetitive.

However...

There are certainly inappropriate questions. I believe that students ought to keep the following things in mind when asking a question in class.

1. Determine how specific your question needs to be. If you know the specific point that you need explained, feel free to cut to the chase. If you are completely baffled, just say so.

2. Remember that in most cases, you are not the only person in the class. If you feel that you are monopolizing the teachers class time, perhaps you should seek help outside of the regularly scheduled class.

3. Attempt to gauge your needs against those of your peers. If you feel that others are likely to have similar queries, then the question will benefit the whole class. If your are the only one who is confused, save the question for another time.

4. If you know for a fact that the question has been asked before, it should probably be held until after class.

5. Remember that good teachers have a prepared lesson plan that includes not only topics to be presented but the order in which they are to be presented. Do not ask questions about a topic that has not yet been covered. This disrupts the flow of the lesson and can negatively affect the ability of your peers to learn.

Most importantly.

6. Remember when I said there were no stupid questions? Well I lied. There are. There are really, really stupid questions. You should desperately fear being the person who mistakenly asks one of these questions. Before you even think of raising your hand or uttering a single syllable, ask yourself what your classmates will think of you. Will they be grateful that you share their confusion and that you have bravely stuck your neck out to obtain clarification? Or will they scoff, roll their eyes, or berate you.

These simple guidelines ought to ensure that your questions improve the overall quality of the class, rather than detracting from it. Teachers desperately want participation, but there comes a point when enough is enough. And as a student, you ought to know where that point is.

AYTMTB

2008-02-19T07:24:00.003-05:00

Recently, I was forced to add a text messaging package to my cell phone plan. My incoming texts have skyrocketed in the last few months, as more and more of my loved ones rediscover the joys of the telegraph. I must admit, there are situations where I find the technology useful. Perhaps I'm sitting in class and I can't talk or I'm at work and I shouldn't talk. Yet even as I send my conveniently packaged alphanumeric messages sailing through the stratosphere, I am not a real texter.

I write my messages in plain English. Occasionally, I will exchange 2 for to or too, or 4 for for. (That was fun to write.) But that's pretty much it. I never LOL or talk to my BFF. It isn't that I look down on the abbreviation process. Far from it, actually. I admire its speed and efficiency. Fluent texters can condense essays into a few acronyms. Despite what many language mavens or old fogies might think, there is nothing inherently wrong with streamlining communication. Oh, and if you currently find yourself disagreeing with me, I'm going to go ahead and point out your hypocrisy. Laser, robot, sonar, scuba, TGIF, snafu, RSVP. If you have ever used any of these words, heck, if you've ever used a contraction, you too are butchering the Queen's English.

If you are a mathematician, you certainly have no room to talk. Imagine what math would be like without all the symbols and notation. (This isn't rhetorical, I actually want you to imagine it.) There was a time when symbolic algebra didn't exist. There was an era when quadratic equations were expressed plainly in words. For example, I could ask you to draw a square such that the magnitude of its area added to the magnitude of a single side is equal to six. That is equivalent to saying x squared plus x minus 6 equals zero or x^2 +x-6=0. (The square would be 2x2.) In recent years, we've taken to calling these word problems. There are no symbols to manipulate or mathematical abbreviations to remember. It's simply a question written out in our language of choice. Ironically, these simple problems cause math students the greatest trouble.

The difficulty stems from the down side to abbreviation. Sure it's fast and efficient, but you do lose a certain something in the process. Imagine reading a Shakespearean sonnet in text messaging. It probably wouldn't carry the same meaning. Actually, it would probably never happen. The commonly used texting lexicon, while great for casual conversation, is not designed for expressing new or interesting ideas. Symbolic math suffers from the same problem.

Last semester, my most challenging class turned out to be Intro to Statistics. My professor was Egyptian, and there were hidden language difficulties. You wouldn't notice it at first. His accent was thick, but not impenetrable. He was perfectly fluent, at least when it came to his discipline. After awhile, I noticed something. He was essentially reading the math straight from the board, simply pronouncing the rhetorical equivalent to the symbols he was writing. There were no metaphors or personal anecdotes. No stories or comparisons. Because of this, I struggled when it came time to apply what I was allegedly learning. I grasped the equations, but not the math behind them. In effect, I was struggling with the word problems.

And you're telling me this because? (Refer to title of post for irony.) I'm simply saying that the same breakthroughs that made long division and calculus possible, also make math too easy to compartmentalize. It becomes an oversimplified model, so efficiently streamlined that it no longer represents the real world, which is exactly what students complain about. Of course, they complain about the word problems, too, but then those little buggers are never going to be completely happy.

So what's the point of all this? Simply that we have been the architects of our own demise. And the only solution is this. More word problems. Less symbols. And for god's sake, if you have something to tell me, quit doing calisthenics with your thumbs and just talk to me on the phone.

Practice Makes Perfect

2008-01-30T07:42:00.000-05:00

When I volunteered to lead a study group for my physics class, I wasn't reallly expecting anyone to show up. Not a single person asked me for help last semester, even though it was obvious by my test scores that I was significantly ahead of the curve. So I was planning on essentially having a study hall after class each day where I could get other work done. Surprisingly, I have had several customers and at least one new person each day.

I am becoming more and more certain that I love teaching. It gives me such a high when a student leaves the room feeling more confident about the material than when they arrived. In many ways, they are helping me more than I'm helping them.

The Joy of Cooking

2008-01-23T10:26:00.000-05:00

While on your bi-weekly excursion to your local grocer, you notice that there is a new species of mushroom in stock. You've never seen it before, and you're not sure how to prepare it or what exactly it will taste like, but something about it's aroma appeals to you. You place it in your cart an continue on your way.

You run to Target for some personal items, and as you scurry around a child screaming in the center of an aisle, your eyes fall on a birthday card with a curious drawing. You open it, read, and chuckle. Though no one close to you has an upcoming birthday, you buy it and save for an appropriate occasion.

Realizing that your couch desperately needs to be either reupholstered or thrown away, you embark on a trip to the fabric discount store. You find the shade of red that you are looking for, but you also find a bolt of green that you can't resist. It's a real bargain, and it finds it's way home with you.

What do all of these stories have in common? Well, for starters, I know people who regularly do these things, and I'll bet you do, too. People who buy a dress without having anywhere to wear it or squirrel away the styrofoam packing from a computer purchase. These aren't packrats that I'm portraying. They don't save everything indiscriminately. Rather, these are people whose life experience allows them to judge the potential usefulness of a brand new item with reasonable accuracy. They are experienced cooks with a new ingredient, accomplished seamsters with a new cloth, and stylish socialites with closets full of perfect ensembles.

These characters are to their area of expertise what mathematicians are to mathematics. Why do these spmetimes bespectacled bookworms play around with formulas and numbers and patterns, seemingly without purpose? Why do they care about things that don't relate to the "real world?" They play for the same reason we all play. Because it's fun. They care because the shelf life of a piece of mathematics is a whole lot longer than a mushroom. It is not uncommon for a discovery to collect dust for centuries before someone makes a connection or draws a comparison, and suddenly that dusty function springs to life, providing just the thing needed to complete the recipe.

I can't cook. I wouldn't know a porcini from a shitake. But I love to eat, and I appreciate the experienced chefs who are willing to try new ingredients and new combinations. Nor will I ever gain fame as a mathematician, but I still love what it is they do. And you should, too.

Cutting the Cord

2008-01-16T15:11:00.000-05:00

Finally, I'm wireless. Let the games begin.

Here We Go Again

2008-01-16T13:15:00.001-05:00

Second verse, same as the first. Almost. This semester, I only have one new professor. The other classes are continuations of last Fall. In a way, that's good, since I know exactly what to expect. I would like a bit more variety, though. I think USM might be too small for that kind of diversity. The Physics Dept only has three professors, if that gives you any idea.

Speaking of physics, my professor walked into class yesterday with a cast on his right arm. Evidently, he had attempted to close a window in his office on Monday and fell, breaking the bone fairly dramatically. He is being forced to revamp his teaching style, since he can't really write on the board very well. I am attempting to turn his misfortune into my advantage by offering to be a pseudo-TA. I would help prepare class materials (handouts,slides, etc.) and gain valuable experience in exchange, and maybe even some work study money. It's still up in the air right now, but I'm hopeful.

In other news, as of about 4pm, I should be the proud owner of a new Dell notebook. AmongHopefully, this should make it easier for me to keep up with my blogging. Perhaps I'll even toss in a vlog or two.

A Steel-Driving Man

2008-01-10T10:48:00.000-05:00

I've been using my holiday break to catch up on my pleasure reading, which to the lay person, would be largely indistinguishable from school assigned reading. I just polished off one about econometrics called Super Crunchers. Econometrics, as near as I can tell, is what actuaries do, only hepped on on some powerful digital 'roids. Evidently, as Moore's Law continues to hold and computing power explodes, decision making that was once left to the "experts" is now being given over to fairly rudimentary mathematical formulas backed up by a whole lot of terabytes.

Chapter after chapter chronicled the successful usage of econometrics to predict things as diverse as good baseball players to Hollywood blockbusters to medical diagnoses. See as it turns out, those experts aren't really all that expert. Time after time, they fail to beat the predicting ability of simple equations, equations which boil down all of life's subtleties to bare bones. As I read, I found myself nodding in agreement. It is well documented how poorly doctors perform on tests of statistical reasoning. Why should patients trust in their abilities to prescribe treatment if they don't really understand the odds. If the use of a simple algorithm can help save lives, then why shouldn't doctors swallow their pride and admit defeat, as it were?

Of course, it wasn't my ox in the grinder. Until the chapter on education. Then suddenly, I was appalled. How can the process of teaching be boiled down to scripted lesson plans? What kind of robotic rote learning could possibly come out of that?

I've since come to my senses. These computer programs are not a threat, but a blessing. They allow us to do the things that we, as fellow humans, are uniquely suited to do. If a doctor is free from having to analyze symptom after symptom, if a diagnosis is less than a Google search away, then that medical professional can focus more on the healing process. They can spend a few extra minutes holding the hand of a scared little girl, or explaining the treatment details to her mother. Besides, the programs are still guided by the garbage in, garbage out principle. They are only as accurate as the information passed into them, and that will still require skilled human doctors. The same is true for teaching.

I have no desire to sit and read from a script all day, and frankly, it's silly for me to do so, even if it proves to be the most effective teaching method. The same computers that analyze the data behind these lesson plans can easily conduct the lessons as well. Sit the kids down in front of some powerful learning software. That's fine by me, because I know it doesn't render me obsolete. It empowers me to do all the "extra" things that I wouldn't ordinarily get to do. I would have the time to really get to know each of my students. I could allow them each to progress at their own pace, and have the ability to work one on one with each of them.

No one alive today would take offense at being beaten by a machine in a test of speed or strength. What John Henry learned the hard way is second nature to us. Yet we cannot seem to accept that certain intellectual feats are now better performed by cousins of those athletic machines. We feel as though our very humanity being stolen, when in fact, the machines are helping to teach us what being human really means.

Don't Cry for Me

2008-01-05T08:55:00.000-05:00

My love of math and science is no secret to those who know me. Friends call me up at all hours, asking me to answer questions or settle bets regarding all manner of things. I am the Phone a Friend. At least, that's how it works with those who have come to love me. The reaction from strangers is quite different.

I've noticed lately, and especially at work, that the discovery of my mathematical predilections is normally accompanied by a wince and/or head tilt. As soon as I mention that I am majoring in math and physics, I am treated as though I've announced a death in the family. "Oh, I'm sorry," they say. Or my personal favorite, "So you're one of those." Usually, I laugh it off, and use it as an opportunity for research. I ask why they feel that way. Why is they're fear and loathing of math so complete that someone else's involvement causes them pain? I find the conversations fruitful, if not more than a bit repetitious.

The respondent almost always remembers loving math as a small child. They can usually pinpoint an exact year or teacher which soured them on their studies. Often times, they remember being told by a teacher that math was simply not for them. It is at that point that I am able to commiserate. In my junior year of high school, after having taken all honors math classes, my teacher told me one day that I simply lacked the "flare for math." It galls me that people who would say such things are allowed to teach any subject at all, let alone such a notoriously tricky one.

I know now, and I continuously attempt to impart to my friends, that math class is not terribly different from shop class. Both are all about tools and toolboxes. Math class is no more about mathematics than wood shop is about craftsmanship and design. Just because you can hammer a nail does not make you an architect, nor does hating long division mean you are cosmically predestined to avoid math. This is a fact that is lost on most students, and too many teachers, and it is one that bears constant reminder. Students must be given a glimpse of the horizon so that they have something to journey toward. Otherwise, we are asking them to practice for a championship game that will never come.

Let the Chips Fall Where They May

2007-12-22T12:54:00.000-05:00

Well I've finished up all final exams and turned in all projects. Now I just have to sit back and wait for the grades to start rolling in. I'm not really worried about any of them. I worked really hard throughout the semester and I had enough of a cushion going into finals, that I would REALLY have had to bomb them to lose my A average. At any rate, I feel like I've learned a lot and that it has been a good start to my second collegiate endeavor.

My biggest regret is that I haven't been very good about keeping up this blog. I had made a lot of lofty goals and promises that seem to have fallen through. I think my schedule for next semester will allow me more time for personal reflection here. If I use the holiday break to get back into the habit of daily posting, maybe that will help.

Definition of Spiraling

2007-12-06T07:16:00.000-05:00

Multiplication, geometry, clock arithmetic, and art.

Numbering from Zero

2007-11-27T06:41:00.000-05:00

I've been noticing a trend lately, and I hope it's a fad. Several popular math books that I have read have begun with a Chapter Zero. I can rationalize several excuses for this, but I still think it boils down to nerds being nerdy.

The first excuse is that in modular arithmetic, a set of N numbers begins 0,1,2,3...N-1 so that it can include the identity. But a book isn't a clock. The chapters aren't going to start over at any point.

The second excuse is that the authors are also modern computer scientists. (Both of my computer science classes began with an Assignment Zero.) After the popular language C, all languages have begun their indexing sequences from zero rather than one. This is a consequence of the decision to have the memory address of an array element be the same as its index. But there's no reason why they couldn't have started indexing at 1 and defined the memory address using an imaginary number equal to one less than the index. So this isn't really any more rational.

The last excuse is that the numbers are meant to serve as markers instead of names, counting forward in a sort of greatest integer function scenario. Since the markers come at the beginning of the chapter, and the first page of the first chapter can be viewed as the origin of the book, it might make sense to label it Zero. However, if books were designed to be read from right to left (I know there are many languages which read right to left across the page, but I don't know if the books themselves read from right to left as well,) we wouldn't expect to start with Chapter Zero followed by negative chapters.

As far as I can tell, there is no good reason other than fashion why a math book should break the convention of using counting numbers to label chapters. I find it irritating and i wish it would stop.

A Letter to a Young Mathematician

2007-11-26T07:16:00.000-05:00

Dear Gina,

I've been thinking a lot about our conversation the other day, particularly your question about proofs. Even though your tone suggested that you had already made up your mind that the entire process was useless to you, I thought I would take a minute to defend the mathematical community.

First, I understand how you could feel blindsided by proofs. After all, you've been getting along quite well in math for years without them, so why start now, right? You might even feel a bit betrayed. Up until recently, school was about getting the right answers, and no subject exemplified that more than math class. Even as English diverges from grammar, and into the realm of essays and theme papers, mathematics remains firmly shrouded in its safe cocoon of black and white, right and wrong. Unfortunately, that security blanket is in large part a lie, and you might as well learn that now. It isn't even your fault that you've gotten the wrong idea. Many of the teachers you've had so far, especially those in elementary school have the wrong idea as well. There are many definitions for the science of mathematics, but however you look at it, it is about a specific way of thinking. It is more about asking interesting questions than it is about finding the correct answers. It is as much a journey as it is a destination, and the concept of proof lies at the heart of it.

Proofs are curious things. They are perfectly ordered step by step accounts, a yet they often hinge on fairly large assumptions. There are proofs which show an answer exists, but give no clues how to find it. There are even proofs which show that there are some mathematical facts, which while true, can never be proven. And to top it all off, there's a proof to show that we have no way of knowing what those proof-less facts are. So it's entirely understandable why they may confuse you.

There are several reasons why your teachers feel it is necessary to torture you with proofs. The most obvious is that they are trying to prepare you and your peers for every possible future. Should you decide to go into on of the STEM fields (science, technology, engineering, or mathematics) you will be required to take upper division math classes and the ability to do proofs with be a prerequisite. By your comments, I think it safe to assume that your path lies along a different fork of the road, so I won't belabor this point.

My second and third reasons fall into what I'm going to call the Karate Kid category. Now because I acknowledge that our difference in age means that this brilliant film reference may be lost on you, I will now provide the key plot elements needed for understanding. In the movie, the new kid in town, Daniel, is being picked on by a band of bullies, who in addition to their snobby upbringing have been trained in karate at a local dojo. The scrawny hero befriends the lovably wise Okinawan janitor, and convinces him to pass on his family karate secrets. The following scenes show Daniel performing a series of menial tasks for Mr. Miyagi, including painting fences, sanding floors, and waxing cars. Daniel grows increasingly angry over his friend's abuse of their agreement, and finally confronts him. At that point, we learn that the repetitive motions of those chores mimic exactly the movements required to defend oneself against an attacker.

In retrospect, this beloved film from by youth is cheesy and not a little bit far-fetched. But it has always seemed like the perfect way to make the following point. In school, as in life, what you are really learning isn't always what you think you are learning. If you can think of your generation's equivalent of the Karate Kid, please let me know, and save my future students the agony of this comparison. Still, the fact of the matter remains that the skills you are learning through doing proofs are useful in more circumstances than you can possibly imagine. When a doctor makes a diagnosis, or a lawyer builds a case, or a football coach draws a play, they are using the kind of analytical thinking that you are practicing through proofs. You first begin with a small pool of facts or postulates, and then you use carefully constructed reasoning to arrive at a sound conclusion. As you continue through school, your skills in other subjects will be improved by your ability to do proofs. Your English papers will be clearer and better supported. Your debating skills will improve. Really, there's no telling how far proofs will reach into your life.

All of this talk of arguments and debating brings me to my last point about proofs. I've already touched on how proofs are built up from first principles with each layer relying on the strength of the one below. Because of this, it is important to be aware of keystone elements of each proof. For example, you have probably proved that every triangle has angles which sum to 180 degrees. This proof follows directly from what Euclid called the Parallel Postulate. Roughly, it states that given a line and a point not on that line, there is only one way to draw a second line through that point so that it is parallel to the first line. Without going into too many details, I want to make an example of this postulate. Like any postulate, it can not be proven. It is an assumption considered so obvious that it can stand alone without proof. The problem with these kind of assumptions, whether they are in math or English or History, is that if they turn out to be wrong, then any argument or proof based on them crumbles as well. In this particular case, there are several systems of geometry that have been shown to both exist and to be invaluable, in which the Parallel Postulate does not hold true. There are spaces and surfaces where it is impossible to draw lines which do not intersect. In these spaces, a triangle may have more than 180 degrees. There are other spaces where they can have less.

After all this, I doubt I have changed your mind much. You probably still hate doing proofs and even after my best effort, you still don't see the point of it all. One of the things you mentioned troubled me more than any other. You told me that your teacher required you to do proofs from memory, naming each Theorem and Corollary as you go. In this one regard, we are on the same page. To many teachers confuse memorization with learning. They think that as long as you have a head full of facts you are better off for it. I disagree. Wrote knowledge without the ability to synthesize and improvise does not in any real way demonstrate learning. To some extent, your teachers can be forgiven their slowness to realize this. You do not remember a time before internet search engines, but I do. Not so long ago, information was hard to find. It could take hours to find the specific piece of data you were looking for, so it was often easier to memorize it once and carry it around with you forever. It was a kind of "be prepared" attitude toward education. Those days are over. There is no longer a need to fill your head with facts on the off chance that they may one day be useful. You can sift out the necessary info in a Google search that takes a blink of the eye. You shouldn't have to remember the names of each Theorem. There are after all quite a few. As long as you can understand them and put them in the right order to build your argument, that is what is important.

I hope some of this has gotten through to you. I know you aren't going to share my love for math, but rest assured, you will never be beyond its sway, so gaining at least a passing familiarity with its methods will prove useful to you.

Love,
Your big brother

Je ne parle pas francais.

2007-11-24T20:28:00.000-05:00

I just got home from a Thanksgiving trip to Quebec City. It was my first time visiting our neighbors to the north, and their hospitality was warm and welcoming. Neither Sarah nor I speak a word of French, at least not beyond the few phrases I printed out before we left. I had made up my mind to make an effort, but I rarely got out more than two words before they broke seamlessly into fluent and flawless English. It's funny that even in the province of Quebec, where French language and heritage is celebrated to the point of revolution, every school child is taught to speak English. Yet I live in a country which is nearly fifty percent Spanish speaking and xenophobic loonies treat Dora the Explorer like she's going to break up the Union.

After overhearing one restaurant owner speaking five different languages. I felt amazed and ashamed all at once. Something has to be done about this.

Lead, Follow,or Get Out of the Classroom

2007-11-16T11:33:00.000-05:00

I don't have much to say lately. Even when I have the time to blog, I don't have anything interesting to report. Today happens to be an exception. First, let me quickly get the auto-biographical particulars out of the way. I am nearing the end of my first semester, still maintaining a 4.0 GPA, and registering for next Spring. There, now we can get on with the show.

Last Saturday I was invited to attend a conference on Teacher-Leadership held at Colby College in Waterville, ME. When I say that I was invited, what I mean is that the opportunity presented itself and I volunteered. (As with the last conference I attended, I was disappointed, but more on that later.) In addition to my desire to master the content I intend to teach, I am very much curious about systemic design of the current education system. From my lay perspective, it looks to me like many schools are still stuck in what is essentially a one-room schoolhouse paradigm. Although, there are more of those rooms under a single roof, the individual classrooms are run autonomously from one another, with little to no coordination between teachers. This self-isolation makes it very difficult for teachers to learn from one another.

Some schools are seeking to change that. They are transitioning into a new kind of structure, where teachers are encouraged to cooperate and "compete" in a less-superficial way. By "compete," I don't mean literally fighting for their very job, as the high-stakes testing/accountability advocates intend. I mean the kind of friendly challenge that helps both parties reach new and unexpected heights together.

That is what the conference was supposed to be about, and it started out well. The opening address was passionate and to the point. The speaker outlined what constitutes teacher-leadership. He pointed out the self-similarity between layers of a school. Just as students learn from one another and drive each other to greater standards of success, so should teachers. Currently, unlike students who are keenly aware of each others methods and their relative successes, teachers are in the dark when it comes to what is happening next door. Instead of the once a year visit from the administration spies, we ought to be volunteering to sit in on each other's classes and welcome peers into our own. Only in this way can we strengthen our own abilities and the cohesion of our departments and schools.

After that, it all went downhill fast. Immediately following the opening address was a panel discussion, only the panel was made up mostly of students with no real understanding of teacher leadership, so the audience questions went largely unanswered. Then we divided up into several break-out sessions which had been decided upon before hand. I had selected one entitled "What a New Teacher Needs," reasoning that I would soon be a new teacher and would like to know what I will need. Unfortunately, that question was never answered. The presenter based the session on her ongoing doctoral thesis, in which she interviewed first year teachers at several points during the year, and made note of their personal success and intent to continue. Now i know that sounds like a great opportunity to learn something, and I would agree, if the sample size had been greater than four.

That's right, four. Out of all the first-year teachers in the state of Maine, she interviewed exactly four. Two of them had never really intended to go into education in the first place and one of those experienced an unfortunate lawsuit during the first month of school. All told, only one of them continued into a second year. Now what can we possibly glean from all this? I was hoping to start with a large sample of eager, well-educated young teachers, find out what obstacles they might face, and some possible ways in which teacher-mentors can recognize and solve those issues. Instead, I got a lot of poetry and rhetorical questions. It was an utter waste of time, and I seriously doubt that her thesis will pass review.

After the session, and on the way to lunch, I fell in with a group of students and their teacher who were likewise disgruntled. It turned out that they were from the ETEP graduate program at my very own USM. I introduced myself and complained along with them. It was the highlight of my day.

Lunch was adequate and featured round table discussions of teacher-leadership from those in the field. It turns out that many teacher-leaders are unsure as to exactly what their job entails. They are also reluctant to take the responsibility, fearing that the nail sticking out is the one to get hammered down.

To summarize, it was a well-intended conference which from my perspective fell short. I am still very much in support of the concept of teacher-leadership. At some point, an organization grows too large and bulky for top-down leadership to work. There must be a combination of bottom-up and top-down for the system to reach optimization.

Gravity Suspended

2007-10-31T15:47:00.000-04:00

Physicists the world over have been scratching their heads this Fall, as our current understanding of gravity has been shattered. The most famous of the fundamental forces, gravity has long been a source of curiosity for the eager young scientist. Newton took the first successful crack at it with his infamous apple and Einstein came along to sure up the theory for later generations. All was going well until recently.

A month or so ago, an extremely focused disruption in the gravity field allowed blogger and student, Tony Lucchese, to literally fall off of the face of the Earth. It is unclear as to what caused the disturbance, but what is clear is that Lucchese has been MIA for several weeks. Scientists from CalTech to CERN are working around the clock to rectify the problem. Until then, we can only wait.

Random Samples: Making an Ass of You and Me

2007-10-03T12:56:00.000-04:00

I first heard the colloquial definition of assume from my middle school band teacher. He relayed it during one of his notoriously crimson-faced tirades as a part of his tireless crusade to get us to practice our instruments. The mere mention of the word ass brought a cacophony of pubescent snickers, as you might well imagine.

Twenty years later, I find myself enrolled in school once again. This time around, I'm not taking any fine arts, and my musical stylings are reserved for my shower head and the lonely walls of the Pottery Barn stockroom. Still, the trite but true words of Mr. Danner stick with me. Daily I am confronted with a seemingly innocuous request to assume. Assume a friction-less surface. Assume no air resistance. Assume a random sample.

Of course, for the purposes of class, I play along. I understand that in order to proceed, a beginner like myself, must temporarily set aside the more confounding components of mathematical problems. Yet I do not take these assumptions lightly, and they hover somewhere in my psyche underlined, in both italics and bold-face. I fear that many people become far too accustomed to these kinds of assumptions, never truly revisiting their ramifications at that later date. Even seasoned professionals fall victim to this permanent credulousness.

My physics teacher relayed the results of a now famous statistical experiment whose findings declared that cats stand a greater chance of survival falling from higher windows than from lower ones. My hand shot up with such velocity that it nearly dislocated my shoulder. You see, that study has a huge hole in it. The sample space isn't really random at all. It originally appeared in a 1987 issue of the Journal of the American Veterinary Medical Association and was authored by two doctors at the Animal Medical Center of Manhattan. They published based on a small sample of cases they had seen at their practice, all feline victims of high-rise free-fall. On the surface, it all seems on the up and up, until you consider the one gaping hole in their methodology. In order to be a random sample, the study must include all instances of the event in question, which is a cat falling out of a window and hitting the ground. Ask yourself this, if your precious Fluffy were to fall victim to accidental defenestration, and upon reaching the street level, you found myriad biological parts where your furry friend was hoped to be, would you scoop up the pieces and take them to the vet? No, and neither do most of the pet-owners in Manhattan. The 1987 study is based on the assumption of random sampling that did not actually occur.

As I have said, these school room assumptions are necessary steps to greater understanding in any and all STEM related fields. But we must never forget that they exist. Our future failures hinge upon them.

Choking Down Technology

2007-09-29T19:20:00.000-04:00

I'm going to skip past the part where I apologize for my long absence. I've been busy, let's not over-analyze it.

The most blog-worthy event in recent happenings was my attendance at a T3 conference for pre-service math teachers. I'm not sure "conference" is the right word; in truth, it was a two day sales pitch. I was invited to participate by the faculty adviser for the math ed program. Mostly I was doing it for the free TI-84+ graphing calculator I would allegedly receive at the end of it. ( I still haven't received it, by the way.)

There was only one other pre-service teacher there. Everyone else came from local middle and high schools. The presenter/salesperson is a full-time professor at Drake University in Ohio, but I'd wager that she gets a healthy stipend from Texas Instruments, as well. She spent two days demonstrating all of the fancy things you can do with their various products. The central focus was on the Navigator hardware, which allow all calculators to be networked together. The teacher can view all activity on each student's machine at the press of a button. That image can be projected onto a screen or better yet, onto a Smartboard. It was all glitz and glamor, and the
audience ate it up. I was less than impressed.

The stuff she was showing us was neat, I'll admit it. But it wasn't as revolutionary as everyone seemed to think it was. There are any number of softwares available that could do the same thing with desktop computers. And those machines would still be able to perform a multitude of other tasks, rather than merely being graphing calculators. I asked if TI produced similar software, and was told that they did. Guess what. It costs exactly the same as a calculator.

The slogan of the event was "TI Cares." That's even their phone number: 800-TI-CARES. According to the presenter, Texas Instruments makes all of this technology available to schools because it wants to improve the world, and help train future employees. The fact that they are making money hand over fist doesn't factor into it at all.

If you know me, you know I love technology. I will use it in the classroom, and use it abundantly. I was just hoping to learn something new about how best to use it. Instead, what I got was a tutorial on several of the latest models. There was no discussion on where low-tech might still be superior. No statistics on the efficacy of technology heavy programs. The equipment she brought didn't even work consistently enough for us to do all of the activities that she had planned. All in all, I think I got out of it exactly what I had intended to- a new calculator. Except, I don't even have that yet, so I guess the joke's on me.

Politics of the Professoriate

2007-09-20T07:23:00.000-04:00

I have read a few articles from the right wing recently complaining about the politicking of liberal professors. Until yesterday, I didn't really understand what the were getting at.

It makes sense to me to ground your teaching in real-world examples. If you're teaching a statistics class, then analyzing welfare statistics seems reasonable to me. If you're teaching biology, then debating the ins and outs of stem cell research seems entirely appropriate. Though they are common battlegrounds of today's political realm, as long as they tie in with the lesson, I think any controversy can be excused.

In my physics class, we are covering motion in two dimensions. Basically, ballistics. In a simple example involving range-finding, my professor segued into a review of the documentary Why We Fight and the Iraq War. He repeated a quote from the film regarding the success rate of so-called Smart Bombs and without much transition at all, stated that "we shouldn't be killing people."

It was an awkward moment, I thought, even though I happen to completely agree with him. I didn't see how the side note appreciably increased our understanding in any way, especially since Smart Bombs have on-board guidance systems and are not simply launched projectiles.

I have emailed him about the incident, and have yet to receive a reply.

Choosing "Choice"

2007-09-18T07:19:00.000-04:00

I know I've been promising to write some posts explaining some of the concepts I am covering in class. Unfortunately, all of the things I have thought about discussing involve me being much more familiar with the Latex formatting syntax than I currently am. For now, you'll just have to read another one of my extemporaneous ramblings.

My schedule has kept me off of the blogosphere lately. The only discussion I have really been able to participate in lately is one about school choice over at Mindless Math Mutterings. I thought I would take a few minutes to elaborate on that issue here.

There are a few misconceptions about school choice and what it can and can't do. The idea is that simply creating a competitive environment will force schools to fix all their problems and become successful. They will either sink or swim. Here is the problem with that reasoning. The choice model is patterned after good old-fashioned natural selection. Whether you prefer the metaphor of a biological system or the business world, the simple fact is this. The vast majority of species and businesses that have ever existed have failed. They are extinct. They caved under the competitive pressures.

Selection pressures do force the cream to rise, but it happens through a process that can best be titled "creative destruction." If you implement that kind of system, you have to accept the fact that most of the schools will fail and be forced to close. Most of them will not be able to magically adapt overnight. That will leave legions of children looking for a new educational institution.

That population overflow will cause the second major problem. Will the few remaining schools be able to scale up their success? Most of the schools that are doing really well are small schools. Whether they are private, public, or charter, they have small teacher to student ratios and they are able to cater to specific demographics. They do creative things like using students to clean up the physical plant instead of hiring full-time janitorial staff. But will they be able to do the same things with a student body that's been increased tenfold?

If you have any experience in the kitchen, you know that some recipes can be doubled or tripled, while others can't. What was exquisitely delicious when prepared for two is revolting when prepared for two hundred. Or a political example, if you prefer. Communism works extremely well in small groups. Contained familial styled villages can live quite happily using that political system. But I think we all remember what happened when that compound was scaled up to the size of a country.

Choosing "choice" would provide our nation with better schools over the long run, of that I have little doubt. But in the meantime, we will be closing a lot of schools and failing a lot of kids. I just don't think it's the answer.

I Pity the Fool

2007-09-14T22:07:00.001-04:00

If you have a problem,

if no one else can help,

and if you can find them,

maybe you can hire....

A few weeks ago, my co-workers asked me if I had organized my Trapper Keeper for school, yet. I told them that I was all set except for my A-Team lunch box.

Today they gave me this belated birthday present.

Please Stand By

2007-09-14T16:35:00.000-04:00

Sorry posts have been light this week, folks. I'm still trying to get used to the new schedule. I plan to post the first "lesson" tomorrow. I think it will be a short one on probability. It should be simple enough that a total beginner could understand it, so even if you don't "get" math, try to follow along. That means you, Mother.

Happy Birthday to Me

2007-09-12T07:21:00.001-04:00

I received this lovely image from a friend today to commemorate the start of my fourth decade upon this Earth. Many people have asked me how I am handling this milestone birthday, and my typically nerdly response has been that it only appears significant due to our use of the base-10 counting system. Seriously, though, I see it as just another day in a life that is being lived well. I have no real regrets and I have been steadily crossing things off of life's "to do" list. My plan is to devote the next thirty years to helping kids understand how math can help them cross things of of theirs.

So here's to the big 3-0.

Politics on the Brain

2007-09-10T07:15:00.000-04:00

Occasionally, Pencils Down strays away from its math education mission statement and ventures into the world of politics. I don't see any way to avoid this, since part of being sentient is having one's own opinions and part of being a blogger is voicing them. My own position on the political continuum is roughly at the midpoint. I tend to be fiscally conservative and socially liberal. I am a registered Independent and have happily voted for members of both major parties and several third-party candidates. I have friends who fall to the far right and far left of the field, and I argue with all of them.

Curiously, one of the most common arguments is of what exactly defines a Liberal and a Conservative. Many people choose to define it by what positions they take on certain issues, but that doesn't speak to controversies yet to be unveiled. Others use those political labels and the names of their party interchangeably, despite the fact that the party platforms have slid up and down the spectrum, even flip-flopping over the years. I read definitions in a political science textbook once that have stuck with me for both simplicity and accuracy. In short, a to be conservative means to look for the solution to a problem in the past. When faced with a new dilemma, they will attempt to apply the solution that has always worked with similar issues before. A liberal, when confronted with a new problem, will tend to dream up an entirely new solution. They prefer the untested to the status quo.

There is nothing to stop people for employing both methods in differing areas of life. I have already admitted to doing so. But what exactly causes a person to be one way or the other. It's the age-old nature vs. nurture argument, and nature has recently released some new results. Some psychologists at New York University have employed a simple test to investigate political persuasion. Test subjects were first asked to rate their political persuasion, 1 being conservative and 5 being liberal. Then the were each shown a different series of two letters, M and W. Regardless of the pattern, one letter was always more prevalent, showing up 80% of the time. The researchers found that when asked to match letters with the computer, subjects identifying with conservativism were more likely to "incorrectly" choose the dominant letter even when shown the other one. Liberals had a slightly "better" results, showing a greater ability to choose the "correct" letter even though it appeared far less frequently.

It is important to note that even if this experiment can be repeated, it doesn't imply that one position is evolutionarily more adaptive. There have been plenty of times in history when the combined selection pressures have favored either a liberal or a conservative approach, so one can not be said to be more adaptive than the other. I suspect that applying a healthy dose of both is the most pragmatic way to go about it.

Great Expectations

2007-09-09T19:42:00.000-04:00

If you recently felt an inexplicable tremor in the tectonic plates of the blogosphere, and are curious as to exactly what might have occurred, wonder no longer. The shifting was caused by the much anticipated arrival of newcomer and reformed lurker, Jackie. Her comments have brought insight to numerous blogs over the last year, and she's finally ready to unleash the full measure of her intellectual fury. Continuities is sure to make its mark and I recommend that edubloggers go ahead and get in on the ground floor. I'm putting her in my blogroll and RSS feed immediately.

Back to School Week: Day 4

2007-09-07T11:57:00.000-04:00

This will be the last post in this series. I'm getting sick of writing a diary, and frankly, I don't know how people do it. I've grown bored with myself after only four posts.

After I posted yesterday, I downloaded the first two Calculus lectures off of iTunes. It was basic algebra and I fast-forwarded through most of it. This class will be a review the biggest challenge for me is going to be scheduling times to take the tests.

This morning, I experienced my first extended use of the clicker. After we answered the questions, our results appeared on the screen. Then we argued with each other for a few minutes, and voted again. We continued until one answer received 100% of the vote. It was really quite fun, and definitely a useful teaching tool.

Now, I am off to work. For the first time since school began, I will be working a full shift.

Back to School Week: Day 3

2007-09-06T19:11:00.001-04:00

In the traditional Tuesday/Thursday repetition, this morning found me once again in general Physics. Today was our first foray into the use of the infamous "clicker." I was still flying high from my fortuitous clicker purchase yesterday. I had planned on making a quick stop at the bookstore before my Java class, expecting it to be a fairly simple transaction. The powers that be had decided to trump my hand by packaging the clicker with the textbook rather than stand-alone. Just as I was cursing the gods, a classmate who already had the clicker from last semester arrived hoping to buy just the book. It was one of those happy little accidents that rarely occur outside of Bob Ross paintings.

Anyway, as I said, we used the clickers to answer some simple questions. I kept expected Regis Philbin to ask me if it was my final answer. We quickly moved on to some basic definitions and set the stage for kinematics in one dimension.

Statistics is moving along at a slow and steady pace. I expect we'll be into simple probability by by next week. Today we were defining events and sample spaces and variance, etc. Still no end in site to the stupid questions from the geriatric brigade. I suppose I'm just going to have to learn to deal with it.

On an unrelated note, I found a new apartment today. It is everything I was looking for-3rd floor, utilities included, off street parking, pet friendly, and less than I'm paying now. There's even free wireless and the landlady will let me paint. I'll have a two week overlap, so I can decorate and move at my leisure.

The future's so bright, I gotta wear shades.

Back to School Week: Day 2

2007-09-05T18:43:00.001-04:00

Only one class today. Java programming. Seems like it's going to be pretty easy. The teacher is aiming the class at people with zero programming experience. Personally, the last time I wrote a program it was in BASIC on my Commodore VIC 20, so I don't mind taking it slow. Things got off to a rocky start when our classroom door was locked. We decided to commandeer the adjacent room, and began the lecture. It soon became apparent that the lesson plan was dependent on the LCD projector locked next door, so we waited on security to come with a set of keys.

In addition to the aforementioned snafus, there were many stupid questions asked by my fellow students. I need someone to tell me how I can get over my immediate and transparent physical response to these inane queries? As a teacher, I can't afford to react this way.

Back to School Week: Day One

2007-09-04T13:25:00.000-04:00

For the next week, my posts will be purely autobiographical. Feel free to tune back in next week if you begin to grow bored with me.

Today began with Physics (Calculus based.) There are about 80 people in the class and there's a good chance that I'm the oldest. Most of them are either Physics majors or engineers of some kind. The teacher seems pleasant enough, insisting that we call him by his first name, Paul. His teaching style is somewhat discombobulated. He pulls examples out of the air, rather than preparing them ahead of time. Because of this, he often confuses himself. Much of the class consisted of us struggling to follow his running monologue. I did a lot of erasing, and in the future, I probably will let him get a bit ahead of my note-taking in order to preserve rubber.

I was not surprised that technology is going to factor heavily in the course. For starters, people don't answer questions with raised hands anymore. I have to purchase a radio-frequency clicker that will allow me to answer multiple choice questions. It feels a bit like Who Wants to Be a Millionaire. Also, all homework is done online. Since I purchased my textbook via Amazon, I did not receive the passcode for the homework database. I will have to buy it stand alone from the website.

After class, I watched as he answered a handful of questions. He was patient and willing to restate himself. This is something that I'm going to have to work on as a teacher.

Between classes, I got my student loan refund and deposited it in the bank. I also timed out the trip from school to work. It took me about 20 minutes, and even with a new apartment, I expect to be able to do it in 30.

I had learned in Physics that lab classes will not meet until next week, so Statistics would be my last class of the day. The demographics are very different from the first class. Out of roughly 20 people, 6 of them are obviously much older than me. I can tell those older people are going to annoy me. Several of them talked nonstop, in that lonely, awkward sort of way that people advancing in years tend to do. I'm sure one day I will fall victim to this verbal diarrhea, but for now it is irritating. There are also many more women in this class, close to 50%. This tells me that the class is probably required of many majors, including biology and history. I know that sounds chauvinistic, and I hope my regular readers know that I wish that were not the reality. Like it or not, women are not currently flocking to the STEM fields. I hope to change that, but for now, I think it's a fair assessment that the math in this class will be geared to a wider audience.

So that was Day 1. I'm still on cloud nine, and part of me really can't believe that I'm finally back in school. Tomorrow, I have Java programming and that's it. Hopefully, my deposit will have cleared so I can buy my clicker and do my physics homework. If not, I could be off to a bad start.

Twas the Night Before College

2007-09-03T22:03:00.000-04:00

Visions of derivatives dance through my head.

Math for the Cosmo Girl

2007-09-01T20:38:00.000-04:00

I wrote so many posts (here, here, and here) in anticipation of Danika McKellar's Math Doesn't Suck that many of you may be wondering why I haven't reviewed it yet. I have no excuse other than that I've been busy. But I finally got around to reading it tonight and it's about what I expected.

It looks and reads exactly like an issue of Cosmo. The front cover sports a sexy photo of McKellar with superimposed titles in a variety of fonts and sizes, exactly like the glossy covers of fashion mags. Inside, there are explanations and examples of math concepts, broken up by testimonials and horoscopes. I read it pretty much the way I read Cosmo*. I scanned through the testimonials, skipped over the quizzes, scoffed at the horoscope, and put it back on the shelf.

It is definitely geared toward the girly-girl and I think that's just fine. Boys have plenty of other books directed at them. Although, the quotes from school aged girls got a bit redundant, and Danika is not nearly as funny as she seems to think she is, the explanations were presented in a clear, concise manner with lots of great tips and short-cuts. The book will definitely be a powerful weapon against math anxiety for young girls.

*My girlfriend has a subscription and I like to know what manner of rubbish is going to somehow get me into trouble each month. And as an aside, she always manages to stack the issues out of order, which bugs me because it jumbles up the Guy Without His Shirt on the spines.

Metamorphosis

2007-09-01T06:25:00.000-04:00

After several years of plotting and planning, only one holiday weekend stands between me and my second collegiate experience. That being the case, I thought now was as good a time as any to discuss some changes to Pencils Down.
Up until this point, I have been sharing my own assorted layman's ramblings concerning math education. All that I know has come from either first hand experience as a student or from various books I have found at the local library. Many of my posts have seemed somewhat outlandish, even to me, but they have all stemmed from my sincere belief that math education in this country can be improved. There are still way too many people who cower in fear when presented with even the most vaguely math related conundrum. We can do better.

On Tuesday, I will take one step closer to my goal, and I fully expect that phase transition to manifest itself here. As you know, I will be working full-time while carrying 16 hours my first year. I will be busy to say the least, and many of you have expressed understanding, should the frequency of my posts diminish. I thank you, but I doubt that is going to happen. The primary reason that I began blogging was to get a jump on my own education. I have learned so much already from teachers like Dan, Dave, and IB, that despite my real life instructors, I can't imagine cutting myself off from the free communal knowledge I can get online.

I am, however, going to structure my posts a bit differently. In addition to my random musings, I am going to add two regular weekly elements. Once a week, I am going to give a lesson based on some concept I am covering in school. I want to practice explaining math skills to others and now is as good a time as any to start. Please be critical of them, so that I can learn from your experience. Another weekly theme will be a kind of meta-analysis. While I am in class, I will not only be listening to the teachers, but also studying the other students. I will be in there with a bunch of 18-20 year old kids. This group isn't too much older than the students I plan on teaching, and I feel that a little sociological journaling might help me prepare for my own classroom. I will change names when necessary, but I plan to write once a week about how my younger classmates are responding to the teaching methods of the instructors.

You'll still get a healthy dose of the random; I can hardly help that. But since you've been at my side for the journey so far, I thought I ought to bring you with me on the next step.

Tit for Tat

2007-08-30T14:58:00.000-04:00

An old high school pal of mine has recently returned to the blogosphere. Reading Allen's blog was part of what got me interested in posting my own ramblings in the first place. He has been on a hiatus for a while, having become a first-time dad this past Fall, but he has triumphantly returned. Pencils Down was the subject of one of his first articles.

In a follow-up post, Allen referenced one of my posts regarding the famous Monty Hall Problem. He goes into great detail about his personal peccadilloes in the set-up of the problem, discussing the different strategies the game show host might take, and how they would affect the statistical outcome. As he says, the host's prior knowledge of the prize locations is a key element. This reminded me of some recent argument I've seen online as to whether Deal or No Deal comes down to a Monty Hall Problem. Any contestant lucky enough (or ballsy enough) to make it to the final two boxes is always offered the chance to switch by Howie Mandel. The general consensus seems to be that Howie has no prior knowledge of the game and so his offer can not affect the contestant's odds.

I'm not sure why Howie is believed to be any less duplicitous than Hall. Perhaps it is because he is Canadian.

Birds, Bees, and STEM

2007-08-27T18:29:00.000-04:00

The National Science Foundation's (NSF) just released the results of their Research on Gender in Science and Engineering (GSE) program. It found that while things have markedly improved for women over the years, some myths still endure. But there is still hope for improvement, they say, and better education for women will translate into gains for both genders and all levels.

Thou Shalt Not Covet Thy Neighbor's Job

2007-08-27T06:50:00.000-04:00

It's back to school time, and everywhere I turn, I find an edublogger lamenting some problem or concern they will have to face this year. While I certainly empathize with their worries/fears, I also would like to grab them by the shoulders and shake the hell out of them.

Good teaching comes with a sense of responsibility that eclipses many other professions. To stand before a classroom, means to tilt against an impossibly powerful opponent. It is an endless battle, and one which is predominantly beyond your control. You will be blamed for every failure, by critics at large and the one within. You will ask yourself, "Did I do enough? Did I ask the right questions? Could I have pushed harder? Did I push too hard?" You will beat yourself up over everything, agonizing over each lesson plan, focus in on excruciating details, in the hopes that the self-flagellation will make you a better educator. And when it's all said and done, it really isn't, because you get to do it all again in a few months.

What kind of self-loathing lunatic would sign on for this? Well, me for one. I know it's easy for me to be critical, safely on the outside looking in. Maybe I'll feel differently in a few years, but right now I am desperate to charge full speed into the fight.

I am a sucker for cheesy sports movies, especially underdog stories. I would say to my edublogger friends what those coaches say to their teams at half-time, when the deck is stacked against them, and winning seems impossible. The other team will always be bigger and stronger, more talented, better equipped, and have many more reserves. They will inevitably win 99 times out of a hundred. But that still leaves the one time. That one student on the verge of dropping out, the kid who doesn't think college is for kids like her, the child with the undiagnosed learning disability. A good teacher gets to win big every once in a while. They get to point to a child and say," There, that one right there. I helped that one." They may not earn a decent wage or get the thanks they deserve, but they know in their hearts that the world is a little better because they were willing to fight a battle when others said it couldn't be won.

That sounds like the job for me. Put me in Coach. I'm ready to play.

Uhhh....What?

2007-08-26T08:50:00.000-04:00

Good thing she's pretty.

Nod to IB a Math Teacher.

I'm Talkin' Bout the Funk

2007-08-24T08:07:00.000-04:00

Off pitch with some horrible fake guitar playing. But funny nonetheless.

The Best Laid Plans

2007-08-22T20:39:00.000-04:00

School starts in just over a week. At that point, my plan was to cut back to a maximum of 30 hours a week at Pottery Barn. I figured that would allow me to keep my health insurance as well as help pay back the student loans I will be receiving. Most of my first year will be review, so I should be able to handle the load.

That was my plan, and like many plans it has gone awry.

Two weeks ago, my immediate supervisor put in his notice. If you've ever worked retail before, you know that the holiday season begins soon, and that put us without a stockroom manager for the busiest time of year. As it turns out there is already an intelligent, hard working fellow familiar with the stockroom. I think you see where I'm going with this. The promotion means that I will have to work a full 40 hour week, but I will be making half again as much as I am right now. The offer is just too good for me to pass up. My boss has known about my school schedule for some time, and she is perfectly willing to work around it. She has even given me the first week of classes off, so that I can acclimate myself to college without worrying about work. So come the second week of September, I will be working full-time and carrying a 16 hour course load.

Don't tell my adviser.

On the Same Page

2007-08-22T07:26:00.000-04:00

Recently, I had the opportunity to work with a bright, creative student on improving his writing skills. We spent a great deal of time discussing how to assess the intended audience, and how diction, tone, and detail must be adjusted accordingly. In everyday conversation, a staggering amount of background information is assumed to be shared. When you write, you typically reach a larger audience, and you can assume far less.

Just for fun, I have selected a random reading comprehension sample from the net to see just how much core knowledge is required for real understanding. The test is aimed at students on the fourth grade level.

How many things can you see in the night sky? A lot! On a clear night you might see the Moon, some planets, and thousands of sparkling stars.

You can see even more with a telescope. You might see stars where before you only saw dark space. You might see that many stars look larger than others. You might see that some stars that look white are really red or blue. With bigger and bigger telescopes you can see more and more objects in the sky. And you can see those objects in more and more detail.

But scientists believe there are some things in the sky that we will never see. We won't see them with the biggest telescope in the world, on the clearest night of the year.

You might find it hard to imagine that stars die. After all, our Sun is a star. Year after year we see it up in the sky, burning brightly, giving us heat and light. The Sun certainly doesn't seem to be getting old or weak. But stars do burn out and die after billions of years.

As a star's gases burn, they give off light and heat. But when the gas runs out, the star stops burning and begins to die.

As the star cools, the outer layers of the star pull in toward the center. The star squashes into a smaller and smaller ball. If the star was very small, the star ends up as a cold, dark ball called a black dwarf. If the star was very big, it keeps squashing inward until it's packed together tighter than anything in the universe.

Imagine if the Earth were crushed until it was the size of a tiny marble. That's how tightly this dead star, a black hole, is packed. What pulls the star in toward its center with such power? It's the same force that pulls you down when you jump — the force called gravity. A black hole is so tightly packed that its gravity sucks in everything — even light. The light from a black hole can never come back to your eyes. That's why you see nothing but blackness.

So the next time you stare up at the night sky, remember: there's more in the sky than meets the eye! Scattered in the silent darkness are black holes — the great mystery of space that's because they're invisible. They're the mysterious dead stars called black holes.

I can certainly see how this would be an intriguing passage for a fourth grader, but there is a lot that the author has assumed. For example, the reader must know what the difference is between the Moon, planets, and stars. Although context clues such as "the" instead of "a" and the capitalization of "Moon" might be enough to suggest that Earth has only one natural satellite, I suspect that this fact needs to be understood ahead of time.

The fact that these things are visible each night requires an understanding of periodicity, if not necessarily rotation. The student must know that a telescope somehow magnifies images, and that the larger the telescope is, the more powerful its magnification.

They must have an understanding of scale as it pertains to the decimal system of measurement. "Billions" is a lot of years, more than most adults can conceive of, let alone a child.

There needs to be knowledge of the three common phases of matter, as well as chemical combustion. (Although it should be noted here that the flammability of certain gases in our atmosphere has nothing to do with the nuclear reactor of the sun. I don't know if the author is ignorant or finds it easier to massage certain key facts.)
Students also need to understand gravity. They need to know that it is often related to the size of objects, which for the purposes of this paragraph, serves as an indirect measure of mass.

All this needs to be firmly embedded in the child's brain before any of this passage will really make sense. We can teach all the tricks and parsing techniques we want, but unless students have a lot of background knowledge, they are still going to have trouble comprehending what they read. I think this is why we are having so much trouble raising our reading scores on standardized tests. There has been too much focus on "context clues" and not enough on the shear quantities of information that must be shared to even get young readers on the same page.

Fossilization and Memory

2007-08-20T21:29:00.000-04:00

It occurred to me today that human memory is a lot like the fossil record. Those few vivid recollections we each possess are few and far between when compared to our lifespan, and there is almost no consistent recipe for making a memory. The right combination of sensory input, context, and repetition cause some snapshots of time to stick better than others. Even when we do remember an event, we are more likely remembering isolated fragments instead of a coherent whole. Then, like puzzle-loving scientists, we attempt to piece the tiny bones back together into a probable design. Often times we get it wrong, and our finished skeleton doesn't match those of other similar finds.

But like the fossil record, our memories, though incomplete, are all we have to go on. Even with the advent of audiovisual recording technology, the odds of having a truly important event caught on tape are slim. So we are forced to use recollections we know to be flawed to guide us in predicting future outcomes.

Oly Oly Oxen Free

2007-08-20T06:54:00.000-04:00

It seems as though there is an endless game of tag running rampant through the blogosphere. At least once a month, one of my blog friends punches me in the arm as they zip past, shouting "tag, you're it." These memes still remind me of chain letters, which I have always hated. Therefore, I will respond to Lost Clown as I have with past rounds of the game, by posting my random list without tagging anyone else. The buck shall stop with me.

Four Jobs I Have Had
US Census Taker
Set Designer/Technical Director for Community Theatre
Tenor in a church choir (funny since I'm an atheist)
Salesman at baby/pregnancy supply store

Four Places I Have Lived
Fallston, MD
Knoxville, TN
the Appalachian Trail
Portland, ME

Four of My Favorite Foods
cheese ravioli
beef with broccoli stir fry
chicken Parmesan
Hot and Spicy Chex Mix (I make it a meal, believe me.)

Four Places I'd Rather Be Right Now
Grand Teton National Park
Times Square
the Outback
Anywhere in New Zealand

Four Movies I Can Watch Over and Over Again
Dead Poets' Society
Braveheart
Die Hard
Napoleon Dynamite

Four TV Shows I Like to Watch
Smallville
Prison Break
So You Think You Can Dance
Mythbusters

Four Websites I Visit Daily
Netflix
Dy/Dan
Angry for a Reason
Technorati

Four Early Musical Influences
the Beatles
Billy Joel
Sting
Aerosmith

Four Computers I have Had
Commodore Vic 20
Tandy PC
Dell PC
iMac (soon)

And there you have it, folks. Yet another glimpse into the intricate and disturbing psyche of me.

I Love Cats

2007-08-18T07:03:00.001-04:00

This post in an experiment, designed to assess how friendly felines affect the popularity of blog posts. You see, I have grown somewhat obsessed with my Google Analytics profile of late. I visit it religiously each morning to determine how successful my blog is becoming. As the page loads each day, I am like an investor checking the financial ticker, praying for my stock to go up, preparing for it to go down. I celebrate every spike in ratings as though it were the ultimate tipping point, the threshold that stands between me and total internet domination.

The largest spike I have received to date was my comical take on Oscar, the death-sniffing cat. Though I was elated to have the readership, I was a bit saddened that it was thanks to a pet. Even my deliberately controversial post submitted to the Carnival of Education did not warrant as much attention. My girlfriend overheard me grumbling about what I felt was a peculiar disparity, and she matter-of-factly provided me with the hypothesis of this little experiment. "People love cats," she said.

So in an effort to test just how much the web-surfing community prefers felines to mathematics, I am writing this warm, fuzzy post about my own pets.

Zoe is an exceptionally small gray female with little white socks. She wandered into Sarah's house and despite the best efforts, could not be persuaded to leave. Sarah had recently lost two cats that had been with her for nearly 15 years, and little Zoe helped to fill the void in her heart. She and I became fast friends, but there has been a bump in our relationship which she has not as of yet gotten past. I left her to hike the Appalachian Trail for five months and she has never forgiven me for it. She will tolerate my attention now, but it isn't like it once was.

Oz is our gentle giant. Sarah added him to the mix while I was hiking so that Zoe might have a friend. It was a risky venture, since we had already attempted to add a second cat the year before with disastrous results. Zoe had attempted to kill that cat. I'm not talking about your standard hissing, swatting, cat-fight. Usually they pin back their ears and box faces for a few seconds until the loser runs away. This was something else entirely, something I had not seen before nor hope to see again. This was a no holds barred cage match with blood and fur flying. Naturally, we expected a similar ordeal with Oz. Instead, she spit out only the slightest little hiss, and they quickly became pals.

Oz is an exercise in counter-intuitive psychology. When Sarah rescued him from the adoption agency, she told me he was the ugliest cat she had ever seen. He had been abused, and I mean seriously abused. He had been set on fire and still has a BB embedded in his left side. To add insult to injury, his size made him an ideal candidate for blood donor, so he was shaved in patches all over. Despite all this, he is the absolute sweetest cat I have ever known. I regularly wake in the night to find him bathing my head and he meets you at the door like a dog.

Now that I have completed the experiment, I thought I would toss in one more bio. This is Freckles. Freckles was meant to be a gift for Sarah, to keep her company while I was hiking. She is a Disney fanatic and had wanted a dalmatian since she was a girl. I had been apprehensive about owning a dog while living in an apartment, but one day while perusing the online edition of my local paper, a pop-up appeared for the county shelter. It was his face. I placed a call to the shelter and was told he was still available. I wanted to meet him first, in case it wasn't going to work out. He peed no fewer than twenty times between his run and the visiting area. I walked him through the cat room to test his demeanor and he seemed unusually calm for a dal. Finally, I gave Sarah the call. She hurried down to meet him, but after less than a minute with him, she declared, "I don't like him." I assured her it was just her nerves, that the idea of her not liking any dalmatian was ludicrous. We filled out the paperwork and took him home, and you know what, Sarah was right. Freckles immediately bonded to me and has not left my side since. He hiked the entire Appalachian Trail with me and is unquestionably the best friend I have ever had. Sarah has grown to love him, but he is definitely my dog.

So there you have it, the menagerie de Tony. I don't know what I would do without them.

The Jungles of the Amazon.com

2007-08-17T16:10:00.000-04:00

The first of my textbooks arrived today: Physics for Engineers and Scientists and Building Java Programs. I've already started reading the first one, having decided that I would like to be one of those nerds that stays a chapter or two ahead of the syllabus. It is ridiculous how giddy I feel about all this. I know I ought to be embarrassed, but I am just too excited about school to care.

Nerds of the world unite!

They Don't Know Numbers

2007-08-16T07:25:00.000-04:00

"He doesn't know numbers; he only knows beads." -Matthew Broderick as Richard Feynman, Infinity

It seems the Arab world is attempting to take a step backward in order to go forward. They are adopting an adding technology that goes back millenia.

Mental arithmetic is a form of calculation that does not involve the use of any physical or external gadgets, such as calculators or computers. The skill is developed early through the use of the abacus. Eventually children are trained to calculate large numbers in seconds with accuracy and speed.

Though the popular image of an abacus is correctly attributed to the Chinese, the actual pebble replacement system was used first in Babylon. It still is the fastest pre-electronic way to do simple arithmetic, beating even today's standard algorithm. Though, the last time I checked, it is both a physical and external gadget. The above program, known as Universal Concept of Mental Arithmetic System (UCMAS,) allegedly uses the abacus to improve mental arithmetic. For the life of me, I don't see how the abacus could possible be useful for this. It only requires students to be able to count as high as the base of their numbering system (i.e. ten) and then to understand the concept of place value. Sure it's really fast by analog standards, but if it's speed you're looking for, go digital and electronic. If you want to teach mental math, teach some form of pencil/paper algorithm and provide memory drills.

Strong Men and Bearded Ladies

2007-08-15T15:08:00.000-04:00

The 132nd Carnival of Education is up over at Education Matters, and it includes my recent post "M" for Mature. I am excited to be included, even though I know no submissions were rejected this month.

With regards to that post, I have this to say. There is absolutely no evidence for or against the efficacy of my theory. I'm certainly not married to it, and no full well that it will probably never happen. It stemmed from my willingness to do whatever is necessary to eradicate math phobia, no matter how outlandish the idea. Take it with a grain of salt, as they say.

Monkey Math

2007-08-15T13:08:00.000-04:00

Elsa Addessi, a researcher at the Institute of Cognitive Sciences and Technologies in Rome, Italy, has demonstrated a surprisingly advanced concept of numerosity in capuchin monkeys. The test subjects were given coins of differing "value," which could be traded for quantities of peanuts. While some of the animals showed preference for color or quantity, several were able to consistently maximize their payoff. While the monkeys are not technically adding, they are showing both the ability to understand the use of symbols and concepts like "more than" and "less than." This is still pretty impressive considering they diverged from us 65 million years ago.

Redundancy?

2007-08-15T07:34:00.000-04:00

This was staring me in the face at the grocery store check-out yesterday.

The jokes just write themselves.

Sleeping Around

2007-08-14T07:12:00.000-04:00

There was an article in the Chicago Tribune the other day that called into question the results of nearly every sex survey from Kinsey to the present. It is popularly believed, and supported by surveys, that men have more sexual partners on average than women. Mathematicians know this is a logical fallacy.

He provided a proof, called the High School Prom Theorem:

"Each girl is asked to give the number of boys she danced with. These numbers are then added up, giving a number G. The same information is then obtained from the boys, giving a number B. Theorem: G=B. Proof: Both G and B are equal to C, the number of couples who danced together at the prom."

Researchers speculate that one of two things is happening, either the men are going outside of the interviewed population for their sexual escapades or both genders are simply lying. I'm sure it's a bit of both.

A couple of things bothered me about the article. First, the article interchanged the words median, mean, and average continuously, which I find quite irritating, since they are not necessarily the same. Second, although I can clearly see why the arithmetic mean number of copulations must be the same for both genders of a heterosexual population, I'm not sure that really tells us anything about relative promiscuity.

Let's say we define promiscuity as having more than one sexual partners in relative simultaneity. Then we imagine a graph of heterosexual pairings that reflects the common alpha-male situation. There are 10 men and 10 women. One studly/slutty dude hooks up with 4 of the women. Two other men and women engage in monogamous coitus. The rest go home alone. If we use a simple arithmetic mean, then both genders engage in 0.6 sexual encounters. However, that one guy accounts for 2/3 of the men's numbers. If you randomly selected a guy from the room, there is a 1/10 chance of him being promiscuous, by my definition. The same cannot be said for the girls, as none of them had more than one partner. So in this sense, men can be said to sleep around more.

You could translate this into a weighted average as well. If you assign weights of 0 to not promiscuous and 1 to promiscuous, then the men achieve a weighted average of 0.4 and the women of 0.0. This ranking puts the men ahead in the slut department.

It's important when applying mathematical computations to real life situations that the math is not only correct, but non-trivial. And the method you use can significantly affect the outcome, as is evidenced here.

Look Before You Leak

2007-08-12T14:51:00.000-04:00

Proving once again that mathematical thinking may be employed in the most surprising of places, a statistically minded blogger at Guns Guns Guns Bikes Bikes (seriously, that's the name,) has worked out exactly where each gender stands (or sits) in the age-old toilet seat debate. I wouldn't recommend citing this source during an argument with your significant other, but it sure is good for a laugh.

Mathematics is Rated M for Mature

2007-08-11T07:02:00.000-04:00

Pay close attention. I am about to suggest something that will make many of my math contemporaries put down their protractors and take up pitchforks against me. Before I get to the point, I'm going to attempt to outline the path that has led me to this heresy.

The following is a collection of ideas that have been espoused at one time or another on this blog.

The math education requirements set for aspiring and current elementary school teachers are far too relaxed.
Calculators are the latest in a series of tools, each adopted in turn for there superiority.
"It takes a certain maturity level to comprehend certain types of math." (Comment from Andy)
When engaged in the design process, sometimes weak links can simply be removed.

Now for some elaboration.

1. The Chinese say that in order to give each of your students a cupful of knowledge, you must have a pitcherful. Clearly, the people who determine the educational standards for elementary school teachers disagree. I recently had an opportunity to peruse a Praxis II practice test for elementary ed, and I was astounded at the difficulty level. The hardest question on the test involved little more than correct application, not derivation, but application of the Pythagorean Theorem. I think it is important to point out that the ancient Babylonians had already mastered this much. I appreciate the fact that there is much more that goes into teaching this just content knowledge. There is all the pedagogy and psychology, especially with the little ones. But the knowledge of how to teach becomes useless without mastery of what you are teaching, and in many cases, what they are teaching is how to hate math. And their students are learning it well.

2. Every time a new technology edges out an old one, traditionalists cry foul. What of the information that will be lost? What if this new technology is suddenly unavailable? This is the argument that naysayers employ against the use of digital calculators today. It is a valid argument, which is merely to say that it is not an outright lie. If a student is taught to perform arithmetic primarily by calculator, than that student forfeits the ability to use the "standard" pencil and paper algorithm, should the need arise.

As I type this, I am within sight of three calculators. The first is the built in application on the computer itself, the second is on my cellular phone, and the third is an actual hand-held with a total of 24 buttons. This machine, which probably retails for two dollars, has the ability to perform 5 arithmetic operations, can store values between steps, and can perform any calculation that would be required of the average person. Calculators are so ubiquitous that to suddenly be without them would mean one of two things, either society has collapsed or you are stranded on a desert island. In the first situation, I suspect there would be more to worry about than the ability to do long division, and in the second, simple finger calculation should suffice for survival.

Progress requires that we give up knowledge that our parents and grandparents depended upon. For example, can you start a fire without a match, can you even start a fire with a match, can you identify edible or poisonous plants, can you drive a stick shift, and the list goes on. When we trade that knowledge, it is with the understanding that we get something more from the deal. Maybe that is a dangerous assumption, but it has brought us safely down from the trees and into the modern era.

3. Before Andy made this comment, it had never occurred to me that the ability to understand math might depend on the maturity of the student. I have read so many stories on prodigies like Gauss, that I had assumed even the most advanced math could be grasped by a child, would that they had the right teacher. Now I am starting to see this may not be true. I have said many times that mathematics is the science of patterns. In order to see pattern, you have to be able to make connections between often disparate things, and that requires a healthy base of facts and experience from which to draw. Maybe children struggle with math simply because they do not have the mental and emotional background necessary to bring meaning to the algorithms.

4. The design process is just that, a process. Ideas seldom spring fully formed from the minds of their creators. Instead, there is a tedious and painstaking struggle to turn the initial concept into the finished product, and there are often heart-wrenching decisions to make along the way. As you watch the deleted scenes on any DVD, imagine how the director felt as the cut was made. You will notice that sometimes a different variation of the scene appears in the final cut, but often times it has simply been deemed unworthy and removed in it's entirety. It just wasn't working, and the faulty part had to be removed for the good of the whole.

Now for the synthesis.

Brace yourself. I propose that math education be delayed until the secondary level. I know that sounds crazy, but the more I think about it the more I love the idea. The two reasons we teach arithmetic are practical application for its own sake and as a precursor to later concepts. As I mentioned earlier, the practicality issue can be solved with a rudimentary explanation of the various operations followed by a brief tutorial on the use of a calculator. The issue of laying a foundation is much trickier. I can't even begin to argue that concepts touched on in arithmetic will not carry over to algebra and beyond. The latter is just a generalized version of the former. What I am suggesting is that school children lack the emotional maturity that makes that transition work. They have no concept of delayed gratification. They do not see that they are working toward something which may not become clear for several years. All they understand is that they are being forced to agonize over multiplication tables and long division and fractions, when they could just punch in the numbers on a calculator and be done with it. To them, it must seem like torture, and who's to say it isn't.

The other factor that conspires to defeat students from enjoying math is the poorly prepared elementary teachers. They often times don't understand themselves exactly how what they are teaching is laying the framework for what is to come, so all they can do is drill the lesson as it appears in their workbook. Reform math programs, which are well intentioned, often make the problem worse, because they require a greater mastery of subject matter from the instructor, not less.

Rather than spending those elementary years teaching students to despise math, we could devote that time to other ventures. Whether the extra space is filled up by music or reading or recess is a question for another day and another blogger. When the students reach the secondary level, then we can begin teaching real mathematics. It's true that they will lack the aforementioned foundation, but they will also lack the ingrained aversion to math. It should be a simple matter to teach long division algorithms along side polynomials or multiplying fractions with rational functions. Students will then be in a position to appreciate what the are learning and why they are learning it.

The math education system is broken. Certain links in the chain have rusted with time. Opinionated cognoscenti from all sides are locked in heated debate over how to repair it, but I think perhaps the solution may instead require total removal of faulty parts.

Or have I gone crazy?

A Brief History of Luddism

2007-08-09T21:50:00.000-04:00

Pre-History: "Boy lazy. He no wait sky fire. He rub sticks. Make flame."

Stone Age: "Man-child is lazy. He uses bow-drill to start fire. He is losing the old ways."

Medieval Era: "Methinks yonder child dost laze about. He summons flame with flint and steel."

Age of Enlightenment: "My son has grown complacent from the new technology of sulfur matches."

Victorian Era: "The kid can't even be bothered to strike a match. He just flicks his lighter."

Modern Era: "Kids today are dependent on calculators. The can't do math the old-fashioned way."

Simple Arithmetic

2007-08-08T18:29:00.000-04:00

Most of you probably think arithmetic is easy. All you computer programmers and tenured professors out there can add,subtract, multiply, and divide in your sleep, and you scoff at people who can't make change quickly in their head. Certainly, our nation's public schools think arithmetic is easy. They require elementary school teachers to have hardly any math background at all. And yet our elementary school students are using math technology 50,000 years in the making.

How Mathematics Happened by Peter Rudman explores in depth why and how civilized man came to depend on math. He focuses on the early years, beginning well before writing was invented, and giving an intriguing account of the birth of the science. I am in the process of reading it and it is truly fascinating. This isn't the first early math history I have read, but it is by far the most involved. Check it out. You'll never underestimate the power of the third r again.

A Blessing and a Curse

2007-08-07T07:08:00.000-04:00

Have you ever worked with someone that had been promoted to his/her level of incompetency? This idiot that you're thinking of was once a model employee. It was proven excellence at a lower level that got this person promoted in the first place. Unfortunately, for both of you, this person just tried to meet one too many challenges. When I took Calculus III in college, that incompetent was me.

I have always liked math, and I fancied myself pretty good at it once. I sailed through Algebra, Geometry, and Pre-Calc without having to do much homework. As I moved into Calculus, things started to break down. My intuitive understanding was slowly eroding and I found myself struggling to grasp what looked like simple concepts. I kept getting A's, but I was working harder for them than ever before. I remember the first week of calculus in infinite dimensions. It was the hardest I have ever slammed into an intellectual wall.

Since I am going back to school in less than a month, I have started thinking about what may have caused this roadblock for me, and I have come to two conclusions. First, students determine their feelings toward and perceptions of a subject very early on. I felt math was easy and should require next to no effort on my part, because that's the way it had always been. Once it started to get tough, I began to have low-grade self-esteem issues that affected my work. Perhaps I should have been challenged more in earlier grades, just enough so I knew that some work was necessary.

The second revelation is more intriguing to me, and it has come through several years of reading popular math books, the ones with no formulas or equations, just a lot of metaphors and lay-person explanations. They have helped me learn some things that I didn't know in school. Now I understand the difference between applied and pure mathematics, terms I didn't hear of in school. Pure mathematics doesn't have to have any practical applications, as Hardy was fond of pointing out. It is very common for tools of pure math to sit on a shelf for decades or centuries before someone finds a good use for them, and sometimes one is never found. In many ways, the correspondence between chalkboard and reality is accidental. For example, it is a fortunate coincidence that Euclid's geometry so strongly correlates to life in flat space. At the scale the Greeks were used to working, it was flawlessly accurate. For space-traveling moderns like us, Euclid will not suffice. We live in a world where space and time curve, and we have had to use other non-Euclidean geometries, geometries fortunately constructed long before Einstein took his mind-trip on a beam of light.

If you've ever tried to explain non-Euclidean geometry to someone, you may have experienced how much a metaphor can help and hinder understanding. That's exactly what mathematical constructs are; they are metaphors or models for reality. If you are too wrapped up in the similarities between metaphor and reality, you may be blind to differences. That's what happened to me back in college. I had grown accustomed to thinking of all math spatially. Whenever I heard the word dimension, I was thinking of height, length, width, etc. Most of the examples in my textbooks applied the lesson to measurement of space, so when I got to Calc III, and the dimensions grew beyond the familiar three, I was lost.

I am not suggesting that we abandon spatial examples and metaphors. Student's inherent understanding of space is strong and math education is wise to piggy-back of of it. But maybe there ought to be more examples in the texts that have nothing to do with space. Comparisons of color to light/heat absorption, or age to bone density, or whatever. This way students will begin to understand that dimension can refer to any variable characteristic, not just space. This realization has certainly helped me, and I can't wait to get back into class and prove my competence.

Scene of the Climb

2007-08-03T12:31:00.000-04:00

Tomorrow will mark the one year anniversary of the completion of my AT thru-hike. I've decided to celebrate by summitting Katahdin again. I'm going to head up tonight and do some trail magic for any of this year's hikers, although it's still pretty early in the season, so there might not be any. This time, I will have my girlfriend, Sarah, at the top with me. She had to sit at home for five months while I pursued my dream and she isn't going to sit this one out.

Tilting at Windmills

2007-08-01T06:34:00.000-04:00

I wanted to respond to the following comment that appeared on a recent post.

The following authors have been banned in some public school classrooms:

1. Alvin Schwartz
2. Judy Blume
3. Robert Cormier
4. J.K. Rowling
5. Michael Willhoite
6. Katherine Paterson
7. Stephen King8. Maya Angelou
9. R.L. Stine
10. John Steinbeck

If you stick to private schools you will not only be able to teach the classics but be called by your first name!

The list of books is the only part of this comment that I do not take issue with. It comes from the American Library Association, and it is actually the list of challenged authors. According to the ALA, a challenge occurs when a book is recommended for removal, whereas a ban is a successful censoring. I searched their website for a breakdown of public vs. private schools, but they don't separate the data that way.

The distinction seems trivial to me, especially considering the fact that it in no way records books that were never made available in the first place. I suspect that if fewer books are banned in private schools, it is only because private schools give parents much more control in curriculum selection. I found many stories of parents switching to private schools because they had lost a challenge to a book in a public school, and especially considering that most private schools are still religiously affiliated, I am willing to bet that as a group, they offer far less intellectual freedom.

I will assume that the schools referenced in this comment are purely secular. Even so, I have no desire to teach at those institutions. Since I decided to devote myself to education, many people have recommended that I go the private school route. I don't see the point. The types of students that attend private schools are as varied as their reasons why, but they all have one thing in common- parents that care about their education. This is a better predictor of success than any IQ test. Private school students are probably going to succeed in life regardless of how competent the particular school is. To me, it would feel like a fireman rushing to save a baby from a building that isn't burning.

As far as the allegedly superior success rates of private schools, I think the evidence is dubious at best. These schools have the luxury of hand-picking students. Whether specializing in troubled students, gifted students, or religiously conservative ones, the schools get to choose the niche in which they will be most effective. Public schools teach everyone. They turn no one away, and their finished product suffers accordingly.

I am getting into this to make a difference, and the place where I can affect the most change is in the trenches of public education. Sure I could find a school that perfectly compliments my convictions, where my career would be trouble-free, but again, I really don't see the point.

Conservatives Sleep with the Lights On

2007-07-31T21:46:00.000-04:00

This one goes out to all my right-wing pals out there, and you know who you are. If you're tired of the inflammatory Liberal rhetoric embedded in your child's favorite picture book, then I have the thing for you.

Help! Mom! There Are Liberals Under My Bed is the book conservative parents have been seeking. This illustrated book the first in the "Help! Mom!" series from Kids Ahead is perfect for parents who seek to share their traditional values with their children, as well as adults who wish to give a humorous gift to a friend.

Hailed as "the answer to a baseball mom's prayers" by talk radio host Melanie Morgan, Liberals Under My Bed has already been the subject of coverage in The Wall Street Journal and Harper's magazine. Written by a self-proclaimed "Security Mom for Bush" and featuring hilarious full-color illustrations by a Reuben Award winning artist, it is certain to be one of the most talked about children's books of the year.

Regardless of where you fall on the political continuum, the idea of Ted Kennedy under your bed should scare the bejesus out of you.

A Rose By Any Other Name

2007-07-30T22:50:00.001-04:00

According to my birth certificate, I am Anthony Mario Lucchese. My father is Catholic and when my parents had trouble conceiving, they prayed to Saint Anthony for a miracle. I can only assume that he is the patron saint of awesome, because here I am. Mario is my father's name and his father's name. My mom flat out refused to have a III, and for that I thank her. I get enough Nintendo induced flack as it is. Lucchese, of course, is my Old World Italian familial. It is also one of the big names in organized crime and boot manufacture, though I'm sure not by the same folks. My friends call me Tony.

What's my point, you might inquire? Simply that I am going to be standing in front of a classroom in a few years, and I'm wondering what name to write on the board.

When my parents divorced (evidently not all their prayers were answered,) I was but six years old. Within a few years, I had two new step-parents, whom I love dearly. From the beginning, I called them by their first names. When I was twelve, I got my first summer job, and I worked predominantly with adults. I called all of them by their first names. For as long as I can remember, it just seemed natural to call all people, regardless of age, by their first name, assuming at least a passing familiarity. I know many people find this disrespectful. My father, for one, still has my friends call him Mr. Lucchese, even though they are all adults. But I am not a Mister; I'm just me.

As I grow older, and find myself reaching the other side of the generation gap, it still feels unnatural to be anything but Tony. I have worked with high-school age kids for several years now, and I allow them to call me by my first name. I have never had any trouble commanding respect. Do I really need to add what feels like an artificial title now? Do I have to be Mr. Lucchese, or can I leave that to my dad?

I understand that some students might feel uncomfortable calling me Tony, and for those I'll accept a more formal address. But for the rest, I'm inclined to allow anything that isn't vulgar or disrespectful. Coach, teach, chief, oh captain my captain, or just plain Tony. Will my school allow me to do this? Are there rules that I must follow? What does everyone else do?

Just something on my mind.

When Good Science and Bad Reporting Collide

2007-07-30T07:23:00.001-04:00

The University of Virginia and Harvard recently released the results of a study comparing student science achievement in high school and college. My news feed-reader has been full of articles, in which reporters skilled in selling newspapers take a crack at scientific reasoning. This recent one from the Washington Post is particularly irritating to me, as it suggests that debates over secondary science curricula are now superfluous.

I love math. I believe mathematical education is absolutely imperative for all students at all levels. But I've got to give some props to science, too. The study simply found that students' experience in a particular science (biology, chemistry, physics, etc.) was not a strong predictor of success in another of those sciences, whereas math excellence was a predictor for success in all science fields. Excluding the fact that science is little more than applied math, let's look at the logic of this finding. The sciences have always been taught separately from one another. The distinction is somewhat arbitrarily related to scale. These smaller chunks are easier to teach and to learn, but all of them are necessary for a complete understanding of nature. Mathematical thinking is necessary to all three, but then so is reading comprehension, so these findings seem somewhat immature to me. While the results may prove useful to college admittions departments, I don't think they ought to be used to determine high school science curricula.

As of 2005, the percentage of high school graduates who enrolled in post-secondary the Fall after graduation had risen to 70%. That's high compared to our parents' generation, but it still leaves 30% who don't attend. Of course, there is some fuzziness in the numbers. Some students will take time off in between and some of those who attend will drop out after only a semester. But the point stands. Not all kids go to college. I would really like those people to have enough science reasoning to make their way in the world. They need to have enough bio to make sound health decisions, or to pass judgment on stem cell research and the like. They need the chemistry to understand the harmful effects of pollution. They need the physics to understand why to keep a safe following distance and not to drive too fast.

Both science and math are all around us. Our public education system is supposed to give kids all that they will need to be good citizens, and that has to include a healthy dose of math and all sciences. Excluding the latter in favor of the former may help out college professors, but it does our students and our nation a great disservice.