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Sunday, June 28, 2009

Writing to Learn Math

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.

Friday, June 26, 2009

What's in a Name?

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?

Wednesday, June 24, 2009

And That's My Final Answer

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.

Sunday, June 21, 2009

The Odd Answers are in the Back

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?

Thursday, June 18, 2009

Dividing Fractions

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.

Wednesday, June 17, 2009

Out with the Old, In with the New

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.

Tuesday, June 16, 2009

Cultural Confusion

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.

Monday, June 15, 2009

Certification or Bust

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.