ORGANIZE YOUR RESOURCES (CALCULUS)

Since this is my first year focusing on the Calculus curriculum, final exam time is the perfect opportunity for me to refresh my own study skills. For those who struggle with preparation for the cumulative exam each semester, perhaps seeing how someone else studies could give you some ideas of your own.

My situation is slightly different than the students’. I have 4 Calculus students in 3 different schools, 4 different teachers, and 3 different textbooks (one of which is a newer version of another with many of the same problems, some new ones, and all on different pages!). The organization needed to remember where each student was in the various schedules and what homework I had completed in preparation for tutorials for whom got the better of me and was in need of serious revamping even before I began over Winter Break to study for finals. At that time, I collected all homework and piled up assignments by student, then within each student’s pile, I put things in chronological order.

Some teachers had flitted around the text, while others had followed the order given in the book, so when one student was working on Inverse Trig differentiation, another was working on Related Rates. My solution for organizing was to make 4 lines of work (one for each student), assembled by TOPIC. And that gave me the idea to make my own workbook with notes relating to topic and separated with tabs so I could access my backup materials as needed.

MY TIP TO YOU
You already know I expect you to keep all class notes, homework assignments, tests and quizzes (from the first day of the semester until the day you retire to a nursing home). If you’ve been following my suggestions, this material is already organized by chapter or unit according to the chronological order established by your teacher.

That’s a lot of paper and probably difficult to use as reference. I suggest you acquire either the “Stickies index tabs” (mine are the Staples brand but I think 3M has a version also) or actual binder tab sheets. Mark the tabs according to the topics you’ve studied and separate your work by topic. When you come across a final review question that relates to the Mean Value Theorem, for example, you can easily find your past work on that subject.

Here’s a list of the topics in my newly organized, personal reference binder:

FUNCTIONS
LIMITS

RULES OF DIFFERENTIATION (this is my universal sheet with all of the examples including derivatives of Trig functions, e, and all that sum, difference, quotient, chain rule stuff in one location. Not every text provides the handy list on the book cover, so I have my own to use as reference if I forget a rule or just want to verify my work.)

VELOCITY/ACCELERATION
EXTREMA
MEAN VALUE THEOREM
F’’
OPTIMIZATION
NEWTON’S METHOD
IMPLICIT DIFFERENTIATION
RELATED RATES (by far the most voluminous section complete with examples from the internet. I defy any teacher to find a problem I have not researched and copied the algorithm for solution.)

Finals are fast approaching. You may already have your Final Review Packet. I’m thankful to have the organizing completed before trying to work on 4 entirely different sets of problems from 4 uniquely disparate teachers!! Your study plan will surely be less complicated than mine, so take heart in knowing that STUDYING SMARTER, NOT JUST HARDER has many positive rewards.

THE AMBIGUOUS CASE (Geometry and Trig)

Remember triangle congruence in Geometry? In what ways can you prove two triangles congruent? SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), HL (hypotenuse-leg).

But why can’t you use SSA (side-side-angle)?

BECAUSE IT’S THE AMBIGUOUS CASE! (What a great SAT word - ambiguous - meaning “unclear”)

Let’s take 2 sides of a triangle and the non-included angle (SSA).
We know the lengths of the two green legs and angle A, so the dotted line is the path that line AC will take even though we don’t know how long it will be.

But we don’t know angle B, so line BC can swing either left
or right

Notice that angle B could be either acute or obtuse. When you use the Law of Sines to find angle B, the calculator only gives you an acute angle. This will certainly be one value of angle B, OR THE ANGLE YOU CALCULATE MIGHT BE THE REFERENCE ANGLE OF OBTUSE ANGLE B!

How do you know if the obtuse angle is really possible ? This is the AMBIGUOUS part. You have to “think” about it, or rather, you need to "calculate" about it.

Take the angle you know (angle A) and add the angle B that was the result of the Law of Sines. By subtracting the sum from 180°, you will have one value of angle C.

Now use the angle calculated through the Law of Sines as the reference angle to find the other possible value of angle B. Add known angle A to it. Are there any degrees left for angle C? (Is the sum still less than 180°?) If angle A plus obtuse angle B is less than 180 degrees, you have a second possible triangle from the given information.



If you’re a fan of math, the rigorous explanation might appeal to you:

Given 2 sides and the non-included acute angle and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), this is the ambiguous case and two different triangles can be constructed.

Given: AB
Given: BC
Given: angle A, where angle A < 90°

If BC > AB(sin A)
and BC < AB,
the you have an AMBIGUOUS CASE.


Here are a couple of questions to add depth to the concepts relating to the Ambiguous Case:
1. The triangle congruence rule AAS (angle-angle-side) is really a special application of another triangle congruence rule. Which one is it? Explain why. (Hint: Think of the "no choice" rule.)
2. Why does the calculator only give you one possible value of angle B? (Hint: Think about the quadrants of inverse sine, inverse cosine, and inverse tangent.)
3. What unique characteristic of the unit circle (and inverse sine) applies when you use acute angle B as a reference angle? (Hint: Think about the sum of the degrees in a triangle and the degrees in a straight line or semicircle.)
4. The rigorous explanation emphasizes that BC must be shorter than AB. Why? (Hint: Look at the diagrams of BC swinging left and right. What would happen if BC was longer than AB?)
5. Why does the rigorous explanation stress that the known angle must be less than 90°? Why couldn't it be 90°? (Hint: Think about HL.) or obtuse? (Hint: Think about questions 1, 2, 3, and reference angles.)

When you think about it, there really isn't anything new in your math study......just more ways of using what you already know!!

WHAT MOTIVATES THIS STUDENT?

Sometimes students lack the success to which they are capable, not because they can’t learn, but because they just “don’t wanna.” There are many distractions from learning for teenagers and the trick to getting better performance is to find the right motivators.

Parents often struggle with advise from one reliable source that is diametrically opposed to advise from another, equally trusted colleague. One family uses financial rewards while another tries strict discipline. One parent supports free will and natural consequences while another prefers rigid controls. The problem with selecting motivators comes when they are imposed upon a student rather than springing from the wants and needs of the student him- or herself.

To begin assessing the options available for making the learning process easier and more effective for our students, let’s look at what motivates the teen in the first place. In later blogs, we can examine perception and processing avenues, but for now, let’s work off the premise that our kids will prevail if we can arouse the desire to invest as heavily in academics as in their non-scholastic endeavors.

In terms of motivation, people can be placed in 4 general categories based on the desire to be active or passive (sometimes referred to as the demand for results where active means outcomes are vital and passive means "whatever") and the relative necessity for strong relationships.



Two cautionary comments must precede assessment of the student’s classification:
1. Although it is frequently postulated that teens are social creatures (witness the phone bill, social networks, and the like), that generalization is not appropriate when dealing with an individual student.
2. While one category might be significantly stronger than the other three, no person is monochromatic. Each student will be a conglomeration of all 4 styles, which
3. might be manifest in different situations. My own father, for example, was a highly dominant figure at work but in a family situation tended toward the steady personality. A student will also vary in his or her identity, disposition, and nature based on the situation. In a class where the teacher or subject is a favorite, the student may be extroverted while favoring a compliant approach if the class is high risk. In the first case, the motivator may be approval and recognition from the teacher or other students, but the second instance may require clear-cut rules and time to organize.

THE MOTIVATION PICTURE
Step one is to determine where the student falls within each of the 4 categories. The student should complete the Adjective Checklist:

ADJECTIVE CHECKLIST

Read each adjective listed and check ALL ADJECTIVES you feel describe you.

Score the Adjective Checklist:
Every eighth row marks the delineation between categories. Draw lines all the way across the three columns between the 8th and 9th, 16th and 17th, and 24th and 25th rows. Count the number of checks in each category. The first group is D, the second is I, the third is S, and the fourth is C. These totals indicate the relative strength of each of the categories, at the time the student completed the checklist. (Remember that this could change for any given situation. If the student is having difficulty in one particular class, it would be worthwhile to complete the checklist again with that class in mind, just to see how things change.)

To obtain a pictogram of the student’s style, enter the highest and lowest scores on the grid. Calculate the average and graph each score by category. (This mathematical step is optional, but meets my personal commitment to visual input and organization! A layer of mean-median-mode or box-and-whiskers could be added for those even more compulsive than I am. Email me if you want directions.)

The next step is to identify the motivators which will inspire the student to devote sufficient effort in the process of learning. The “wants” are what the student expects in return for the endeavor and the “needs” are external stimuli and personal improvements needed to work more effectively.






PROVIDING APPROPRIATE MOTIVATORS:
Now that we have the edification provided by just one of many assessment devices, how can the information be put to constructive use? There is no avoiding the rigors of trial and error. But here are some practical approaches which have worked with the students at Tutoring Resources.

If your student is highly DOMINANT, he or she wants the freedom to make independent choices and the benefits of immediate feedback. If not kept busy, this student can find a plethora of ways to “push the envelop.” Expected outcomes should be clearly defined and guerdon awarded expeditiously. Checking the answer to a math problem immediately upon completion (look in the back of the book) is an example of timely feedback as well as a productive study strategy. Younger students might respond to checking off duties on a task list or daily “chips” for completed work. Added up at the end up at the end of a week, the "chips" can satisfy a desire for longer term gratification.

The INFLUENCER student could flourish in a group situation where his or her prowess can be recognized. The caution is to provide sufficient direction so group work does not regress into just play. Published Honor Roll lists are a form of reward for this student. “Refrigerator” recognition -- the A+ paper posted on the frig for the whole world to see and admire -- is a classic motivator for the Influencer. This student will appreciate Mom or Dad “checking” the work or proofing the essay, provided it is accompanied by a healthy dose of “good job” in appraisal.

STEADY students could benefit from predictability of scheduling but need to be reminded that there is an expected productivity outcome. An established time and/or place to complete homework is comforting for this student and presents the opportunity to set time limits and express approval when tasks are completed. Remembering that there was a History test today and asking how it went can open the doors of communication.

The high COMPLIANT student needs well-defined rules and predictability in order to self-assess results. Threats are rarely effective since they raise the risk level which the Compliant is trying to reduce. Encouragement is a strong, positive motivator which comes naturally to many parents who have adopted the mantra, “You can do it,” which can be heard frequently at almost every sports event. Goals for this student should be set in small steps in order to provide frequent recognition of success, and failure should be immediately mitigated so as not to inhibit further effort. Getting right back on the bike after a fall is an example of overcoming failure through subsequent success. Correcting errors on a test, especially when accompanied by the possibility of extra points, is a teacher’s paradigm for motivating the Compliant student.

No matter which category is paramount at this moment, on this day, in this situation, a combination of motivators should address the secondary and even tertiary styles that the student may exhibit. If YOU are the student, help you family, friends, advisors, confidants, and various significant others to be effective motivators by sharing with them the “wants” that have been identified here. They can support you in STUDYING SMARTER, NOT JUST HARDER!

THE ACT SUGGESTS DEPRESSINGLY LOW BENCHMARKS FOR COLLEGE READINESS

ACT Inc suggests "benchmark standards" which are intended to indicate a student's readiness of college. Using these criteria, the Chicago Tribune reported recently on how Illinois high schools measure up. In a nutshell, the information was not positive. The standards themselves are unbelievably low, leaving the impression that kids don't have to know much in order to "succeed" in college, and results of the ACT component on the Prairie State Achievement Exam cast a shadow on any school that hopes to prepare more than 75% of their students for at least a 50-50 chance of earning a respectable grade in freshmen level college courses.

Not wishing to be the bearer of only bad news, I’ll start by saying that entry level scores on college entrance exams, like the ACT component of the Prairie State Achievement Exam (PSAE), are simply a starting point. There are many reasons a student may not achieve his or her best score if the test is taken “cold.” Some of these reasons are academically founded, but others hinge on knowledge of the test’s format, structure, and expectations.

That said, let’s think about the recent Chicago Tribune article highlighting Illinois statewide statistics on student readiness for college. (Friday, November 12, 2010, section 1, pp1+) Tutoring Resources’ primary service area includes North Cook, Lake, and McHenry counties. For the most part, our students are among the highest scoring in ACT’s recent survey of “college readiness benchmarks.” The creators of the popular entrance exam estimate that meeting the standard indicates a student has a 50 percent chance of earning a B or higher and a 75 percent chance of earning a C of higher in typical freshman courses. (In the spirit of full disclosure, I am of the opinion that a C in a college course is not acceptable, especially in the core courses required of a freshman. As a parent responsible for tuition and other fees, I would be less than pleased by only a 50% chance that my pecuniary investment would have a reasonable return.)

Benchmark scores established by ACT Inc for purposes of comparison are stunningly low in my opinion: English 18, Math 22, Reading 21, and Science Reasoning 24. I’m bewildered that English and Math are the lowest standards, while Science Reasoning is set at well above the average composite score nationwide. An average of the four benchmarks is lower than a national average composite score and just slightly above the state composite average. Yet, at Tutoring Resources' highest ranking high school, only 51.6% of the students met all four benchmarks.

The ACT researchers do not imply that there is blame to lay on our high schools, our teachers, or even our children. I would agree with this restraint and could, if asked, provide proof that our schools are doing better than these statistics might suggest if used in the wrong context. I DO believe that awareness of the criteria used by test developers, knowledge of the testing format, experience with actual test materials, and other nonacademic issues can have a significant impact on the scores each student can achieve and should strive to attain.

It is my experience that almost every freshman will be required to take a composition course through the college’s English department and those professors have every right, in my opinion, to expect students from our high schools to know when to use a semicolon, how to punctuate between independent sentences, that its’ is not a word, the difference between their/there/they’re, and other common grammar rules tested on the ACT English section. To achieve an 18, the student needs to answer only about 54% of the questions correctly. To propose that a student who knows only one-half of the standard grammar concepts will succeed without tremendous effort is wishful thinking.

Similarly in Math, to earn a score of 22, a student needs to find correct solutions for 32 to 35 of the 60 questions (53% to 58%). Considering as many as 5% of the questions might involve Trigonometry and/or higher level concepts not covered in some high school courses, but understanding the statistical probability that careful guessing can be rewarded with as much as a 20% success rate, setting a “benchmark” so low is anticipating that our college-bound students will be able to sustain an enormously challenging effort level in college math classes.

My tutors and I take every ACT that is made public and are constantly evaluating the concepts tested and looking for patterns of study which will help our students. So I tend to be positive about the test itself and the publishers and creators. I’m sure that researchers at ACT are cognizant of the limited ability of a single test to predict college success. Perhaps these “benchmarks” are intended to acknowledge the other factors which play into a student’s achievements at the college level: motivation, maturity, and personality. But I’m not willing to send my kids off to distant parts with only a 50% chance of succeeding.

I actually like the ACT test, not so much in terms of college admission (which is a fact of life not within my control), but as preparation for college itself. I expect my students to arrive on campus with a significant understanding of English rhetoric and grammar and with a thorough comprehension of high school level Math concepts. I want them to be prepared to focus on style and content in their first English course, not where to put a comma. I want them to obviate the need for remedial math courses, at full tuition for no credit, before qualifying to enroll in 2 semesters of required college level math. While I cringe at the low scores suggested as adequate, the ACT test itself and the English and Math concepts included in the testing are without doubt important elements for reviewing what we learned in high school and should take into the world after graduation.

Every student can use preparation for the ACT test as a foundation for success in college. Review the necessary concepts; the ACT authors test common mistakes which should be avoided in a composition class. Create study guides for use in college classes because elementary arithmetic really is necessary when solving a complex calculus problem. Raise those individual section scores to the highest possible level to demonstrate the student’s true potential and maybe even qualify for a great scholarship, but most importantly, to reinforce the foundation for academic prowess in post-secondary education. Treat the ACT as a handy checklist for packing to go off to college:
√ toothpaste
√ pillow
√ computer
√ Tutoring Resources’ “25 Grammar Rules”
√ a note card of those pesky math equations

By using the ACT test as the "benchmark" for what curricular concepts need to be firmly in place before starting college, a student is using the required entrance exam to its fullest potential and greatly increasing the 50% probability of earning outstanding grades in the ubiquitous Freshman English Composition and College Algebra classes.

PAYING FOR COLLEGE

It's the first of the month and I'm in the midst of bill paying, so my thoughts are ranging toward parents who are looking for new ways to assuage the high cost of college tuition. Here are three resources which may prove stimulating:

www.payforcollegeblog.com
This website has many ideas for saving on expenses, generating student income, and related subjects in an ever-expanding venue. Plan on spending significant time perusing the site.

www.fastweb.com
A very professional site with comments and tips on a variety of subjects including scholarships, grants, and loans.

Debt-Free U: How I Paid for an Outstanding College Education Without Loans, Scholarships, or Mooching off My Parents by Zac Bissonnette is an entertaining self-help book that might appeal to both students and parents. Published just this year, Zac's ideas are timely and innovative. It is available in paperback at many bookstores including www.barnesandnoble.com

USING TESTS AND QUIZZES TO MAXIMIZE YOUR STUDY PLAN

We’re coming through the four-week mark in the semester. This is an important benchmark in terms of studying and grades; almost everyone has had at least one quiz and most likely a test in the past week. These grades are needed because teachers are generally required to identify failing students halfway toward mid-term grading periods and prior to parent night when Mom and Dad will probably ask how their student is doing.

This is also a great time to review the steps needed to take control of your grades in every class. Here are some tips on using quiz and test grades as learning devices, not just third party assessments over which you are powerless.

1. Save all tests and quizzes that are not recollected by the teacher**.

• a. You should already have highlighted class notes and homework prompts in studying for the test. Add any concepts from the test which do not already appear in your personal notes.
• b. Now, with a new color, highlight the information which was tested. I like using yellow and pink as my two highlight colors because the combination is a distinct orange that makes it easy to see the intersection of what I thought was important and what actually was.
• c. Analyze the thinking that went into your study for the test and determine how to prepare more effectively next time. I know my weaknesses, so I’m especially vigilant to watch for names and dates, and I use mnemonic devices for anything that requires memorization.

2. Get into the teacher’s head.

• a. Look at the actual questions on the test. Try to figure out where the questions came from. Some teachers will take test questions directly from homework assignments. (This is a no-brainer study notice -- redo old homework.) Others might use the questions from the textbook, but only those which were NOT assigned as homework. (Again a no-brainer -- answer the NOT ASSIGNED questions.) Still others use the published assessments supplied by the textbook makers. (A little more challenging, but usually the tests will mirror the Chapter Reviews in the text.) Many teachers make up their own questions, but will probably have a unique sentence structure similar to their speech pattern and will design questions that mirror the topics and emphasis expressed in classroom lectures. (The most difficult challenge -- as the framework for study, rely on topics the teacher addresses in the classroom . As an example, if the History teacher is constantly telling little stories about the PEOPLE involved in an era, I would suspect THAT as the major test issue also.)
• b. Look specifically at the instructions on the test. Identify similarities and differences between test directions and homework assignment prompts. Are the homework answers essay format, short answer, fill in the blank, multiple choice? How does this match with the test questions? (Check out other blogs about how to study for the specific types of questions that could appear on a test.)

3. Don’t neglect error corrections. Find your errors and plan to fill in any concept gaps and avoid any silly mistakes. (In Math, for example, rework incorrect problems until you can complete them quickly and without error. Also try identifying the algorithm and you might discover that certain problems are always solved by using identical steps.)

4. What if you can't keep the tests?
This can happen, but don't let it rob you of the opportunity to use every assessment to improve your study plan. Get out pencil and paper. Copy down questions that were wrong. Write down a list of concepts tested. Make notes about what you did RIGHT, especially in questions that you thought were especially difficult. In another week or so, ask to review the test paper again so you can spend more time analyzing the issues suggested above.

Using tests, quizzes, and homework assignments to prepare for future assessments will get easier with practice. By the time you are in college, you should be able to "read the instructor" and go into any test fully prepared and confident. If you can predict what will be on a test and center your study on the truly important issues, your grades will reflect a mature, effective study plan.

**A brief word about recollecting tests and quizzes. Did you ever wonder WHY? If you are not allowed to keep scored evaluations, it’s probably because the same assessment devices are given over and over, year after year. In a few cases, it might be the result of several teachers of the same course using the same tests but working on different schedules. If the latter is the case, you should be able to acquire you old papers in a week or two, so remember to ask for them before the next test. If yours is a class that is taking last year’s tests, you might be able to review your work under teacher supervision, on your own time, and in a secure location.

THE DIFFERENCE BETWEEN LEARNING AND STUDYING

At the risk of stimulating the philosopher that lurks in the farthest recesses of your mind, I’d like to first describe briefly what is covered in a philosophy course called “epistemology.” It is the study of the nature of knowledge especially with regard to its limits and validity. In epistemology, we think about questions like, “If you know, does it follow that you know that you know?”

And that’s the difference between learning and studying. In learning, you come to know something; in studying, you ensure that you know that you know it and can use the information when appropriate -- on a test for example.

So you’ve done your duty and competed every homework assignment. You might even have checked your answers, found your own mistakes and corrected them. You’ve LEARNED the material. Now comes the test and it’s time to actually STUDY. Here are a few tricks to help design a smart study plan.

1. REVIEW PAST ERRORS. Look at your Math homework, for example. Rework any problems that you didn’t answer correctly the first time. If you’ve been correcting mistakes on daily work, you should have learned the algorithms and now is the time to check to make sure that the steps have been effectively stored in long term memory. You’re asking yourself, “CAN I SOLVE THAT PROBLEM OR ANSWER THAT QUESTION CORRECTLY NOW?”

This is a cyclical process. Any mistakes put the problem back in the cycle until you can correctly answer without error. It’s like when you were in grammar school and had those weekly spelling tests. To help you study, your mom may have quizzed you on the words, eliminating each as you could spell it correctly but coming back to any “misses” until you got it right.

Homework assignments are just the beginning. Save and review all related quizzes and class notes also.

2. LOOK AT A QUESTION BACKWARDS. Actually, you’re looking at the answers and thinking of questions which would result in those answers. This is an especially useful tool for anyone who is thinking in only a straight line. It happens frequently in Algebra. If you see a quadratic equation, you instinctively know to factor, set the factors equal to zero, and solve for X. But what happens if you’re given the binomial factors? Normally the instruction would be to FOIL, so you do. But if you were supposed to solve the problem, your straight line of thinking would have led you down the wrong path. By looking at problems “through the back door” so to speak, you’re developing a deeper understanding of the role each solution step plays. You’ll be able to jump into a question at any point and be sure you’re working through it in the right direction.

3. PREDICT TEST QUESTIONS. By looking at a Math problem backwards, you are predicting the kinds of questions which could be asked on the test. But this strategy works well in other disciplines also. Take History as the example. If you have learned a series of dates, you can predict that a question might ask “What happened next?” -- or “What caused this event?” -- or “How is this event similar to or different from another?” Thinking about what might be asked puts you in better position during the test because you’ve already considered how to answer.

It is also useful to look at questions that weren’t covered in homework. In History, look at the chapter review questions that weren’t assigned. In fact, in EVERY class, check out all of the textbook resources (the questions, study stimulators, discussion starters, and reviews) that were never assigned. Some teachers create the tests from these assets.

4. LIST TOPICS AND RELATED ISSUES. If you did any “webs” in grammar school, you understand this process. Start with a central idea and branch out to related concepts. Some elementary schools use KWL: what do you Know, what do you Want to know, what did you Learn? Start with a central idea and list every thing you already Know about it. Same strategy, one using a visual diagram with words and the other using a list of words. (Refer to the blog on learning styles to discover which approach fits your style.)

5. BE PHILOSOPHICAL. Ask yourself the question, “Just because I know the material, do I know it in such a way that it will be useful to me on the test? Do I KNOW that I KNOW?” Your grade on the test will provide an answer for you, but too late for you to do much about it. So study what you’ve learned and be truly prepared to achieve the "A" you deserve.