Enrichment Teaching Archives

Helping Students Learn Mathematics

October 23, 2014 Brad Hoffman Leave a Comment

I nostalgically remember how exciting becoming an eighth grader was for me as a student, mainly because that would bring the challenges of Algebra to my academic life. I also proudly remember inheriting that humongous Algebra book that used to belong to my sister and that basically represented the transition to a more mature and relevant academic journey. Sadly, I also remember the frustration that followed after being exposed to some of the more basic algebraic concepts that would basically represent the foundations of a more advanced body of knowledge, fundamental in my future as an engineer. Unfortunately, I learned mathematics in a very robotic and repetitive manner that did not require much the use of problem solving or critical thinking skills, vital in the process of truly understanding mathematics since the early years; memorization, repetition of steps to solve problems, and almost no connection with the real world was my day by day contact with math. Although the process of connecting mathematics with the real world in simple and complex scenarios was time consuming, it was absolutely necessary for me to understand the applications of mathematics in more advanced fields.

But times have changed! Considering all the technology and research in education, we have innumerable tools and resources that could assist us in the process of teaching meaningful content to help students establish that link between mathematics and a better understanding of the world around us. Manipulatives, interactive software, educational games, videos, well design textbooks, free online classes, and other resources have become so important in the teaching of mathematics with purpose not only for children during their first school years but also for college students. As the National Council of Teachers of Mathematics states “Young learners’ future understanding of mathematics requires an early foundation based on a high-quality, challenging, and accessible mathematics education” (NCTM, 2014).

It is rare to listen to someone say “I am so bad at reading” or “I do not know how to read”; however “I am so bad at math” has become extremely common and apparently accepted. As parents and educators we have the responsibility of changing that distorted perspective of mathematics. The likelihood of students disliking math should decrease when they can understand how it is useful and applicable to their lives, as well as their frustration should decrease with the success they can achieve when studying the subject with purpose since their early years.

By Jose Parra, Math Specialist

Physics: What’s the Answer?

October 17, 2014 Brad Hoffman Leave a Comment

When I am discussing a physics problem and a student interrupts me to ask for the answer, I sometimes wait a moment before replying. Not because I don’t want the student to know the answer. On the contrary, I am pleased if he or she can get it right without my help. But if I sense that the answer is the only thing the student wants, then I wait.

Teaching physics is not just about getting the right answer to the problem at hand. It’s also about teaching how to view the natural world, how to think critically, how to approach the unknown so that the student may eventually become a master and solve a physics problem no one has ever solved before.

This sounds idealistic. Of course, the reality of homework assignments, lab exercises, quizzes, tests and exams should not be ignored—like all instructors, I want my students to do well. Furthermore, not all those who study physics in high school or college will go on to study the subject at higher levels. Nevertheless, I think it’s important when teaching physics—and science in general—to convey some sense of the whole enterprise, which goes beyond memorizing formulas or facts, but is aimed at understanding the interrelationships of nature.

And this leads to a paradox. Because the only way I am able to convey any general idea is through specific examples. That is, I work with my students, guiding them through problems and showing them how to analyze situations that are not completely general. These problems—such as throwing a baseball—usually contain some simplification (the ball doesn’t rotate) or approximation (air drag is neglected) that limits how well any result obtained (say, for the speed of the ball just before it reaches the bat) corresponds to what goes on in the real world. And yet, I still hope that the students will catch on and realize that there is a general pattern, so that they will be able to apply what they learned with the baseball to a jet of water coming out of a fountain and maybe one day to designing a missile defense system.

Therefore, after the student has asked for the answer and the wait is over, I say, “I’ll tell you the answer. But first tell me how you solved the problem.” Once the student has told me what he or she did, I’ll probe a little more and ask about the assumptions the student made or any limitations of the calculation, trying to encourage a general view of the physical principles involved. And then I’ll give the answer. Because we all like to get it right.

By Neer Asherie, Private Tutor

5 Music Activities to Connect to the Ear (and Brain and Hand)

October 10, 2014 Brad Hoffman Leave a Comment

Impulse and Improvisation

Much of the language with which we discuss music composition has been passed down to us by Beethoven’s discussion of his own music. “Music,” said Beethoven, “is the wine which inspires one to new generative processes, and I am Bacchus who presses out this glorious wine for mankind and makes them spiritually drunken.” A lot of great music has been written by people who think of composition in this way, as a process of filtering the raw material of sound. But more often than not, composition involves just as much of the tools influencing the composer’s choice of sound as it does the sound influencing the composer’s choice of tools. To stick with Beethoven’s metaphor, a winemaker may choose a tank based on the grape, but may also choose a grape based on the type of tanks available to him or her. In music composition, the tank and the grapes affect one other in real time, constantly changing shape as the compositional process unfolds.

Beginning composers often feel most comfortable allowing the sounds in their heads to dictate the tools they choose, rather than the other way around. These 5 exercises instead encourage a composer to derive sound from a tool. The primary purpose of these activities is to get the student feeling more comfortable allowing tools to influence sound, on the way to developing a mental space in which sound and tools may influence one another freely. Because these activities stress system and reaction over impulse and improvisation, they may also be thought of as creative ways to catalyze and develop executive functioning. The functions which these exercises emphasize range from the more basic, such as organization and planning, to the more complex, such as working memory and flexibility. Next to each exercise is listed the executive function or functions it will most help to develop.

Play a Floor Plan (Organization, Planning)

Draw a basic floor plan of your living space. Choose a major key (e.g. G Major), and assign each room of the floor plan one of the six common chords in the key (I; ii; iii; IV; V; vi). If you have more than six rooms in your living space, then several adjacent rooms may share a chord. Track your motion from room to room throughout the day. Translate this motion from room-to-room into motion from chord-to-chord, and use this layout as a harmonic basis for a short piece. Only the harmony is fixed; everything else is a compositional decision. Keep in mind, when initially assigning the chords, which rooms you are in most frequently, and, consequently, which chords you want your piece to center around.

Adapt-a-melody (Flexibity)

Choose a simple melody (for example, a riff or chorus from a pop song). If you play piano, then you may accompany yourself. In your right hand, use only the notes from the melody, in the order in which they appear. Improvise freely with your left hand. By shifting style, accent, rhythm and harmony, improvise a piece that extends this melody as long as it can while remaining convincing. The activity may also be done with two people. In this case, one person should improvise freely while the other adapts the melody to the shifting accompaniment.

Fill In the Blank (Goal-Directed Persistence)

In a single piece of music you enjoy, pick at random two measures. One of these measures will serve as the beginning of your piece, and the other will serve as the end. The goal of this exercise is to find a convincing way (other than the way the original composer did) to link these two snippets of music together. Ideally, the piece should be no more than 10-20 measures in total. Each compositional decision should be made with the conscious goal of leading towards the end-point.

Working with Found Sounds and Objects (Working Memory, Abstract Thinking)

Choose a household object you’d like to try using as an instrument (e.g. pot or pan). Trying not to be influenced by this choice, pay attention throughout the day for a sound that you find striking. The activity will work at least as well, if not better, if the sound you choose does not seem to correspond to the object you chose (e.g. a bird call and a pan). Compose a short piece in which you attempt to capture something about your memory of the sound that struck you using the instrument you chose first. This activity is worth trying several times with different combinations of sound and instrument.

Composing with One Chord (Sustained Attention)

Sit at a piano and rearticulate one chord for an extended period of time. You may change the inversion, volume, range, rhythm and articulation of the chord, but not the chord itself. Attempt to focus on both the subtle shifts in your playing of this chord and the consistency of the chord’s sound. After doing this for some time, attempt to write a short piece with one chord or pitch that will keep a listener’s attention as much as possible. For an example of someone doing this exercise shockingly well, look up Scelsi’s “Four Pieces on One Note”.

By Chris Ruenes, Private Tutor

Learning How to Study

September 17, 2014 Brad Hoffman Leave a Comment

In my experience, the hardest part for most students (including myself) when getting ready for an exam is not actually learning the material- the real problem is having, and spending, enough time on all the material to be tested. I’ve seen friends and students study almost endlessly, but also aimlessly, only to realize when it’s too late that they haven’t prepared themselves for the exam after all.

The long-lived rule of Sun Tzu’s Art of War is as follows:

“If you know your enemies and know yourself, you can win a hundred battles without a single loss. If you only know yourself, but not your opponent, you may win or may lose. If you know neither yourself nor your enemy, you will always endanger yourself.”

That is precisely my take on taking an exam- know yourself and know the test.

So how to study?

Know yourself- it is crucial to get this out of the way first. You need to know if you are a person that needs a lot of time to prepare so you never feel like you’re in trouble, or someone who needs pressure to focus. Look over how you prefer to learn- are you a visual, auditory or tactile learner, or some hybrid of the above?
Know your opponent- you should never just study by reading material and then taking the exam. Always start by knowing what the exam consists of, both in its scope of material you need to have prepared, and also the manner in which it will test you. You need to do practice questions. If you know you can answer the questions when it’s practice, you can remind yourself when it’s crunch time during the exam that you’ve been here. It is a HUGE boost.
Define winning. There’s no point in taking an exam and not knowing why you’re there in the first place. Different exams have different importance- some hold the pressure of admission into a program or school, others are just another grade that will average out for your final. Define your win, then shoot for a score 10% higher than that- this means you’ll have to study 10% more (material learned and time committed) than you would to just get the score you’re shooting for. Figure out how far you are from that goal score. Budget time accordingly.
Win. In the days leading up to the exam, the day of the exam, and while you are taking that exam- know you are there to win. When something goes wrong, when there’s a question you freeze up on, when you are not certain if you can finish- remind yourself that you studied like you know you best could, you’ve done enough practice questions, you’ve won before and you can do it again. It’s not positive thinking; it is thinking rationally in stressful situations and using the experiences you’ve earned and accumulated through appropriate study and practice.

The above is a very concise and pretty abstract approach to preparing for an exam. I’d be happy to field questions or help make it work specifically to your situation.

You don’t need to be forced into situations to cram- choose to by being intentional about studying in the time you want to, and you’ll find you can really learn more information than you usually would have.

It gives you more time to doing the things you would want to, and get to do them without feeling guilty for having not studied yet!

By Denny Simon, Private Tutor

Unlocking Difficult Students

September 4, 2014 Brad Hoffman Leave a Comment

Tutors dream of being tasked with a cheerful, bright, dutiful young student, one who follows every direction and has an answer for each question. But the nature of our position often involves assisting students who aren’t the apple of their teacher’s eye.

And, that’s ok. It’s why we became tutors in the first place. Our job is to connect with the student and understand what makes him/her tick.

I’ve found that being able to shed the standard open-your-book -and-begin-the-assignment attitude is imperative when dealing with unorthodox students. For a variety of reasons, students suffering from depression or other behavioral issues do not take to standard classroom instruction and most likely won’t take to your tutoring prompts either. I’ve therefore found it necessary in these cases to go on the offensive and get creative.

Marc was one of these kids. An enormously intelligent adolescent who suffered from severe depression, he had difficulty with social interaction and frequently stayed home from school. When I arrived for my initial session with him, he was not in the best of moods.

Within five minutes he excused himself and I could hear him complain to his mother that he didn’t want to do this anymore. I heard violent screams from the other room: “Tell him to leave! I hate this! I can’t believe you’re doing this to me!” His mother apologetically came back into the living room and told me this probably wasn’t going to work out. But I asked if I could just talk to him once more and she agreed. “Hey Marc,” I whispered, “Can I come in to ask you a question?” “NO! GET OUT!” I realized I needed to come up with something to defuse the situation. “Why don’t we head to the coffee shop I passed by on the way here? Maybe a walk outside will help clear your mind. We’ll get some hot chocolate–we don’t even need to study.”

He was not expecting this. His mood changed instantly. He came out of his room as if nothing had happened (which I thought was a bit odd, but I went with it) and began putting on his coat. His mother shrugged her shoulders with a look of let’s-not-jinx-it and we were off. Our session that day was essentially Yu-Gi-Oh breaks interspersed with Biology, but I succeeded in forming a rapport with him, one that seemed unthinkable an hour prior.

Coming up with interesting games on the fly, creating scenarios that pertain directly to their lives, and making an occasional joke or two are just as important as the actual study material. I’ve found it produces better academic results and helps them adjust to new environments outside their academic lives as well. We would be well-suited to place a higher value on these tutoring techniques.

By David Hollander, Private Tutor

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