This school year I tutored Algebra I for 98 hours. Several students' grades, I was told, shot up remarkably. I think that it is due to the one-to-one interactions that go on. The student can ask for help with any step he or she is puzzled by, for one thing, and by watching the student as he or she works out a problem, I can catch errors and interject as complete an explanation as necessary, ensuring that the student gains a thorough understanding of the principles and develops good habits. My aim is to assist the student in the immediate coursework and to lay a solid foundation for future work in mathematics.
I tutored Algebra II for 34 hours this past school year. This is the course where algebra comes into its own, with topics from probability to analytic geometry on the menu. I keep George Simmons' "Precalculus Mathematics in a Nutshell" handy for its concise and lucid presentations of most of the topics which puzzle students, and I usually teach a little differentiation to my students, too, because it is just so very useful.
Naturalist for over 50 years. B.S. in Molecular Biology. Ph.D. in Microbiology and Immunology. Biological oceanography and estuarine biology. Co-principal Investigator on NIH grant investigating pulmonary architecture and mechanics. Microscopy: light, electron, and confocal laser scanning. Physiological measurement and instrumentation. Data collection and analysis. Molecular structure, too. Deep and broad experience across much of biology.
Calculus is for me two things: first, an essential tool and, second, an
armature -- an important, supporting armature -- within the physical world,
the bones within the body. I consider it a disservice to withhold calculus
from students until late in high school or even until college: finding the
slope of a graphed function at any point and finding the area under the same
curve between two bounds could be, literally "elementary". (Why not
demonstrate the mechanics at least by ninth grade at the latest?!)
Calculus, though, is more than a convenient graphical technique. It is no
coincidence that Newton's F=ma is a differential equation. Newton saw not
just that the apple fell, but that the farther it fell, the faster it went.
In the lab, we needed integral calculus to find the likelihood of observing
tissue structures in the microscope. Calculus underpins not only motion but
much geometry as well; calculus, or "analysis", models much of the world.
I like to pick apart a working system to see what makes it tick, while
mathematicians tend towards a pure, from-the-bottom-up approach. Students who
find it easier to learn from-the-outside-in may enjoy exploring calculus with
To me it seems that the most important goal is to bring the student to see elementary math as "common sense", plus some rather mechanical procedures. I have caught errors in knowing how to multiply and divide which, left uncorrected, could have made all further math quite difficult. If the student
seems curious, I will show some of the inner workings of arithmetic, such as how, for example, to count with five symbols or just two. Sometimes as the student sees the possible variety, everyday arithmetic becomes less like a mystery and more like an intelligible, conscious choice.
I am a scientist: I chose biology, but the courses that I took, my personal interests, and the paths down which my research led me kept my outlook from becoming narrow. One of my most enjoyable activities was showing "science" to fifth-graders in public and private schools. What questions: insightful and perfectly stated! Some children's questions could bring the class right up to the frontier of knowledge, and I always thought -- and pointed out! -- that that was what we sometimes forget about "science", that the unknown can be just a thought away.
When the first student came to me asking me for help with composition this past year, I recalled a form that one course in college obliged us to observe: the "brief". It was both constrictive and liberating. The idea was to state in three paragraphs the three most important points of the novel or play or short story. The three paragraphs had to fill just one page; that page had to be single-spaced, have pleasing margins, and feature a centered title.
My English students and I both write briefs, and then we compare them. We rewrite them until they are clear and cogent. Afterwards, we find, longer compositions seem almost to write themselves.
To date, I have tutored Geometry for 97 hours. Students came twice a week throughout the school year to work on geometry, and what advances they made in that time! Geometry is useful for its subject matter, certainly, but it can also stimulate imagination and foster an ability to organize and present thoughts logically. It struck me often during the year that a new problem was a problem that the student had solved before but now encountered from a different angle, and it was this realization that I tried to pass on: this is a wheel you already have invented; don't reinvent it, but reuse it. This is no new concept to the people who make up tests, by the way, and if a student improves his or her ability to spot a familiar configuration in a new guise, then test results are bound to improve.
I lived and worked three years in Germany, and I came to understand, speak, write, think in, and dream in German, but I only once or twice tried to teach German to someone else. When a student asked me, through WyzAnt, to teach him more German, I recreated some of the methods I had used to become competent in the language.
Pronunciation is a major point: 'z' is not 'z', 'ch' is not k, and so on. I like to learn a language through its fairy tales; German has "Grimms Kinder-und Hausmärchen", 800-plus pages of fairly straightforward yet academically-reviewed prose, a goldmine for a student. As the hours pass, the student reads and speaks German with increasing comprehension and fluency, English-to-German follows.
It is so satisfying to build sentences. When you know the rules of English grammar, you find that there is probably not another language on earth with such range. When you can produce a grammatical dependent clause, it is as though you have discovered a chain saw in your kitchen drawer. And the
tenses! In English, you have near-complete control of time! You _can_ say what you mean. And it isn't hard at all.
The GRE math portion does not seem to ask for knowledge much beyond Algebra II and Trigonometry; what it does do differently from most other exams is ask for an ability to evaluate graphs and data. Since I am a retired researcher, I find that I am able to point students in the right direction in these "data analysis" questions.
The verbal parts of the GRE are much like those on other standardized tests, in my opinion, and thus I can counsel a student on how to answer these questions efficiently and correctly.
The interesting part of the current GRE is the dual-essay: two 30-minutes essays, one on an "issue" and the other on an "argument".
Writing the "issue" essay is much like writing a newspaper column presenting a reasoned viewpoint on a particular subject, which is something that I have done; this kind of writing can be quite colorful and impassioned, so here is a chance to express yourself!
The "argument" essay is prompted by some statement of evidence. Evaluating this evidence results in a piece of writing similar to what one might find in the "Discussion" section of a scientific article. (The "Discussion" is the final piece of text in an article, and it is where the authors have a chance to point out how the material presented in the article confirms, extends, or contradicts previous work by themselves or others on the subject at hand. It also is where the authors evaluate their own work, possibly setting forward arguments in support of adopting their interpretation of their work. Consequently, it is almost always done in a measured, careful, thoughtful manner and occupies the other end of the style spectrum from the "issue" essay. As the first author on a number of scientific publications, I have considerable experience in this kind of writing.
My typical method of teaching essay writing is to write an essay on the same topic as the student; when we are done, we compare the results and talk about our different approaches and choices
The MCAT tests knowledge and ability in Physical Sciences, Verbal Reasoning, and Biological Sciences. Although I am a Ph.D (took the GRE) and was an Assoc. Prof. (Research) in Brown University's School of Medicine, I also applied successfully to medical school (took the MCAT). I have decades of experience in physical and biological sciences, and I am also an author of scientific articles (verbal reasoning!). Let's go through some sample tests; nothing to fear!
My doctorate is in microbiology. Now, microbiology is not a narrow field, because there is medical microbiology, ecological microbiology, and the subcellular and molecular aspects of the subject, too, but I am able to adjust to the range of subject matter presented in the courses at different
institutions, I have found. What I do not happen to know I can find out, understand, and then explain -- that, after all, is what a researcher does.
In college, a physics major in twenty minutes taught me all the calculus I would need for the next forty years. I like to repay the favor by laying out the "ladder" (or ladders) of functions -- derivatives are rungs below and integrals (antiderivatives) are rungs above. With that concept in hand, classical Newtonian mechanics makes sense at once. (After all, Newton invented calculus precisely in order to express his understanding of mechanics and then push mechanics farther than it had ever been pushed before.)
Another part of my training reappears in high school physics. I took physical oceanography in grad school, and wrangling units was a pervasive theme: salinity, pressure, motion, and more, with the added twist of converting back and forth between measurement systems. I watch my students carefully to be certain that they do not fall off the track when keeping track of units in their solutions to problems.
However, I am neither a physicist nor a mathematician nor an engineer, but I find all three fields irresistible, and I usually, in my free time, am stretching my understanding of these subjects and searching -- if this does not sound too corny -- for the deeper truths. It all is part of a desire to find out the proper way to see the world. Multiple integrals and vector differential equations, I think, lie in the right direction.
I am first author on a series of peer-reviewed papers in The Journal of Applied Physiology; these articles document the results of investigations into the microstructure and micromechanics of the mammalian lung conducted with my colleagues at Brown University School of Medicine and the Harvard School of Public Health. Thus, I am a physiologist by profession, I guess you could say, although my formal training was in microbiology and oceanography. (You never know what will come in handy in research!)
Looking back on my experiences with students in prealgebra this year, I see two things. The first is that I kept waiting for the other shoe to drop: they've brought up this topic, so why not finish the thought and show the student where it leads? The second is that students who do have trouble often are confused by the "soft" explanations provided. Their difficulties can usually be dissolved in one of three ways: 1. see whether there is some procedure which simply was learned incorrectly and fix it, 2. present a technique as something useful and effective that can be learned mechanically (Have we forgotten how very good children are at learning?), or 3. strip away any "cute" half-explanations and proceed to a clear and fairly complete presentation. Or, the fourth. A bored student may underestimate the "ferocity" of homework problems or test questions; this often leads to skipped steps or faulty reading, and the problem "wins" and grades go down. The good
news is that all four blocks can be cleared away.
I keep returning to George Simmons' "Precalculus Mathematics in a Nutshell", which covers geometry, algebra, and trigonometry in about a hundred pages of concise and lucid text and figures. A student who knows the material in that small book ought to step easily into the realm of calculus. (There I
recommend Thompson's "Calculus Made Easy", the Gardner revision; it presents differential and integral calculus in the older yet valid infinitesimal style without recourse to limits. Those can be found elsewhere when needed.)
What I see in some precalculus syllabi are topics more at home in college algebra, rather in precalculus, and those have to be handled as they arise: some are quite unexpected and arcane, but not impenetrable.
In grad school, my department chairman tapped me to proofread outgoing manuscripts with his secretary, and it has not let up since then. I have edited and proofread manuscripts for grants, articles, and books, and I will be able to help you with logic, style, construction, spelling, grammar, and punctuation.
I have accumulated two binders full of archived Regents examinations, one binder for mathematics and the other for the sciences and history. They deserve at least a booklet of commentary, but here are a few thoughts.
Integrated Algebra: a pleasant romp through beginning high school algebra.
Geometry: previously also a pleasant romp, but now exhibiting a tendency towards obfuscation, that is, stuffing twice as much information into a problem as will be needed to solve it and so imposing another sort of burden on the test-taker. Nevertheless, quite consistent over the past few years in
the topics covered and so in that sense predictable.
Algebra II/Trigonometry: challenging in parts, but the archived exams are a good study aid, as the content seems consistent from one version to the next.
Living Environment: biology is a big subject, but the Regents exam gamely tries to test a student's knowledge from ribosomes to ecosystems to experimental design. The extraordinarily wide-ranging subject matter makes it difficult to devise a "crash course" for the test. I have had good results,
though, using the questions as stepping-off points for short, impromptu lectures: I'll ask a student what they know about the topic behind a question, and if they are uncertain of their command of it, I begin to talk and sketch, laying out its context and sometimes the history -- there's an
advantage to having been in biology for more than fifty years!
The SAT was once known as the Scholastic Aptitude Test ("Aptitude", not "Achievement"), and the mathematics portion of the test still asks questions which more prod the test-taker to be clever than to summon up obscure or advanced facts and techniques. The Math Regents tests will try to find out what you know; the SAT will try to see what you can figure out.
The word "insight" is probably much overused, but SAT Math problems do ask for "insight" into the situation described; can the test-taker add or remove information so as to see the question that is really being asked? Example: a bill with a twelve-digit account number has been torn so that the last two digits of that number have been lost; how many possible values are there for the account number? The "real" question is: how many numbers are formed when you count from 00 to 99? The answer to that "real" question is simply: count from 1 to 99, add 1 for zero, so, 100 numbers. (Equivalently, 2 digits in base-10, so, 10^2 or 100.) The "trick" was not in answering the question; the "trick" is to see the question!
Sometimes the test-taker has to "decorate" the information given in a question. For example, a question was accompanied by a table with two columns and terse headings; the last entry in the first column was unknown. The descriptive text said that the number given in the left-hand column was distinctive, a minimum, say, but the column heading did not mention this fact. Jot this special condition into the heading and sketch an empty third column on the right edge of the table, and the solution presents itself: write the product of the first two columns into the third, where possible, notice that these entries form an arithmetic series, write in the next member of the series, and quickly solve for the answer. Here, the test showed a framework for the data that was too skimpy to show the test-taker the way to the answer. (In effect, the data were not presented helpfully.)
Such themes are repeated throughout the test.
Another theme, and a valuable one, is moving from a diagram to an equation --- subdivide a square into a triangle and two trapezoids as shown: what is the area of one trapezoid?
And another theme is to translate a situation not into a formula but into an "imitation spreadsheet", old-fashioned rows and columns, in other words. With the data organized, the answer pops out.
Many day-to-day habits and techniques that I used in a career in research turn out to be relevant to solving the SAT math exam's probems, and I enjoy passing these skills on to students. They are not only helpful when taking the test, but they stand by you into the years beyond
Here again, having a good test preparation book has become a near-necessity, but my experiences this past year make me wonder at what seems to me to be a wide disparity in the quality of the questions in the different brands of books. There should be one and clearly, unarguably only one correct answer to a question.
There is no "magic bullet" for the SAT Reading test. With some students I talk our way through each question; with other students, I work quietly in parallel and then after each question compare notes on how we each came to our answer. Where the student errs, I fix the student's view of the sentence or passage or standard English usage. Usually I end up switching back and forth between these methods, reacting to the student's progress and energy.
As with composition in general, I find that writing a "brief" is the most liberating mode of preparation. Think of the three most important points to be made, and then state and develop them clearly, a paragraph to each one. The trigger for the SAT essay tends to be a short passage or a quote, and thinking about writing a brief in response concentrates one's attention on the
meat of the statement, and once one sees that clearly, the rest usually falls readily into place. The result, too, is a piece of writing that is about the right length for an SAT essay and which conveys an argument that the scorer -- hopefully -- will appreciate and grade favorably. Most importantly for the student, it presents a manageable task that can be practiced until the skill
is second nature. I write a brief in parallel with the student, and I often send a completed version by email as an example.
No one has come to me specifically for help with spelling, but I think I would be able to assist a student should one appear. The patterns of often-misspelled words are second nature, and I would be able to improvise exercises to help a pupil, but realistically, spelling is usually part of a
larger endeavor such as writing.
Help with statistics in the world of tutoring usually means help with a statistics course. This may seem obvious, but in research, help with statistics is more likely to mean advice on how to decide how many observations are sufficient to make a case or how to decide whether a set of data can be matched by a model or another dataset. I have taken courses in statistics, written software (but not invented the techniques!) to answer both of the above questions in my research, and learned to use statistical suites on various calculators and computers. Thus, if I were to see topics from, say, a college course in statistics, I would be able to figure out which way to turn and also -- an important point -- be able to explain my reasoning.
As noted elsewhere, my preferred text is George Simmons" "Precalculus Mathematics in a Nutshell", in which the essentials of trig are presented succinctly and clearly in about twenty pages. (Prof. Simmons remarks that it occupies one fifty-minute lecture!) A number of students have taken the
"Simmons tour" with me quite happily and with good results.
I personally notice that one useful goal to try to attain is to be able to recognize when a trigonometric expression, especially one that is rather complicated, can be restated more simply. There is no shame in looking in a manual for an expression's simpler twin; the trick is recognizing that such a twin exists! Do you recognize "sin(theta)*cos(phi) - cos(theta)*sin(phi)" as
"sin (theta + phi)"?
It is important to know how to write. If you are a student, you know that you are asked, time and again, to write reports, term papers, and presentations, and later, at the higher levels, you will have to produce a thesis. Outside the academic world, there are letters to the editor, minutes of meetings, and perhaps your first -- or second -- novel. All these are easier with some guidance, encouragement, and feedback, which I can provide.
Most writing emerges from a collective effort, actually: colleagues become co-authors, copy editors shuffle paragraphs and excise verbosity, and the author alone in a room with pen or keyboard hears the rhythms and diction of the last hundred authors he or she has read. In my books, articles, and grant
applications, I was fortunate to work for years with skilled mentors. Now let me help you.
I have studied zoology at Haverford College and the University of Washington, and I have worked in fish physiology and marine field studies. My main research dealt with mammalian respiratory physiology and microanatomy in a variety of species. I have for years had a broad range of interests in the natural world -- birds, mammals, entomology, botany, mineralogy -- and in the health of habitats, and I am familiar with ctenophores, annelids, and rotifers, that is, invertebrate zoology. Tell me what you need help with, and I will give you my best advice on how to proceed.