I always start math tutoring sessions by determining first how well the student knows basic math operations and how well he/she applies these operations to word problems. In addition, I see to it that he/she is completely familiar with factors/multiples of numbers from 1 to 100. I also require the student to keep in mind the distinction between prime and composite numbers. The reason: these knowledge carry a long way and they make it easier to understand a lot of study areas.
Let us take adding fractions and solve this:
1/11 + 3/5 + 2/7 = .
We know that 5, 7, and 11 are primes; so, the most expedient way to add is by cross-multiplying numerators with denominators. 5 x 1 = 1; 3 x 11 = 33; add 5 + 33 = 38, and we have the numerator for the sum of the !st and 2nd terms. Multiply 5 x 11 = 55 and we have the denominator of the sum of the first two terms which is 38/55.
We cross multiply 2/7 and 38/55 get 376/385. We are done.
Let us take another example: 1/4 + 3/8 + 5/16.
We know that all the denominators will go into 16 without remainder, so 16 is the least common denominator (LCM). What we need to do now is to raise the 1st and 2nd terms into 16th's, add the resulting fractions and we are done- 4/16 + 6/16 + 5/16 = 15/16.
Let's go to algebra and factor the following:
X2 + -15x + 56 (note: x2 is read x squared).
We look at the middle term and see at once ( that is, if we are thoroughly familiar with factors/multiples of numbers) that the only multiples that can give a sum of 15 are 8 and 7. It then boils down to tweaking the signs, and we get the factors (x-8) (x-7). We multiply the x's we get the first term; we multiply -8 and x; -7 and x, add their products, we get the middle term; and we multiply -8 and -7 and we get 56.
If we change the 2nd and 3rd terms so that we have x2 - x - 56, we know that the factors are still 8 and 7, but because the middle term is a negative 1, then we know that 8 must be the negative factor- (x-8)(x+7).
Let's find a missing number in a proportion problem and see how knowledge of factors/multiples can be used almost automatically. In this problem 4/9 = 24/? we see that 4 became 24 by multiplying it by 6; so, if we multiplied 9 by 6, we find the missing number which is 54.
Demonstrably, knowledge of fundamentals carry a long, long way.
As with any math subject I teach, I begin by checking how well grounded the student is in basic operations, how he/she applies this knowledge to word problems, his/her knowledge of factors/multiples, and his/her understanding of prime and composite numbers. If there are weaknesses, I remedy them. For instance, if there is some weakness in the student's understanding of the basic operations, I start with addition by relating addition to counting (most students know how to count before they are in first grade.) I show the student that when he/she is counting by one's, what he/she is doing is adding one to each successive number, so, one becomes two by adding one to one and so on. I then tell the student that he can play around and change his base and his addends. For example, he can start with 2 and add 2 to each successive number that is generated. So he/she quickly realizes that adding is really "putting" things/items together and "counting" them. This is a short step to understanding subtraction which is "taking" away from a pile/collection and "counting" what is left.
But how do we relate counting/addition to multiplication which seems entirely different. Not if we take multiplication as short-cut addition which it really is. Take say, 3 x 4. Reading literally, it means take 4 three times. Then what? Count! And the answer is 12. Giving the student this insight is much better than having him/her memorize the multiplication table by rote. If he/she gets stuck, the way out is addition. Now that the student knows multiplication, it is easier for him/her to understand division which after all is the reverse of multiplication. The problem 30 divided by 5 can be answered by asking the question: how many times do I take 5 to make 30? And the easy answer is 6.
I'll go now to adding fractions to illustrate the advantage of knowing the distinction between primes (factors are themselves and 1) and composites (whose factors are more than 2.)
Take the problem 1/2 + 2/13 + 2/5. Knowing 2, 5, and 13 are primes, we can easily get the answer by cross-multiplication. Multiply 1 x 13= 13; 2 x 2= 4; add the cross products and you get 17, the numerator of the sum of the first and second terms. Multiply 2 and 13 which is 26 and we get the new denominator. Apply cross-multiplication to 17/26 and 2/5, and we get 137/130 or 1 7/130.
Or the problem 1/3 + 2/9 + 2/15. If we are at home with factors and multiples, we can see at once that all the denominators can go to 45, the lowest common denominator (LCM), and all that needs to be done is raise all the fractions to 45th's. 1/3 is 15/45; 2/9 is 10/45; and 2/15 is 6/45 giving the sum of 31/45.
Multiplication of fractions also becomes easier. Take the problem 15/25 x 6/16. We don't want to handle big numbers if we can help it, so we reduce to lowest terms before anything else. This requires knowledge of factors/multiples. Using 5 to divide the numerator and denominator of the first term we get 3/5; using 2 to the second term, we obtain 3/8. Now it is a lot easier to get the product which is 9/40.
Problems in other areas are easier to solve too. Take this proportion problem: ?/9 = 30/54. Since we know that 9 became 54 by multiplying it with 6, then all we need to find is the number which when multiplied by 6 will produce 30.
A good grasp of some basics in math will carry a long, long way.
Over the years, I have taught a very diverse group of ESL students from the basic level to the more advance levels. Some are high school students, most are refugees or immigrants from Cuba, Colombia, Mexico, Iraq, Palestine, Bhuttan, Nepal, Mozambique, Sudan, and the Congo. Because of this diversity, I have tried different methods and techniques and retained and refined those that engage the students' attention and hold their interest.
I use visuals like flash cards, pictures, objects, and places to enable the students to build their vocabulary and understand idioms, phrases, and sentences. For instance, instead of simply defining a kitchen as a place in the house/apartment where food is prepared and cooked, I point to a picture of a kitchen, or if inside a house/apartment, go to the kitchen and ask "What part of the house/apartment is this?" to start a conversation. I use demonstration and pantomime to illustrate things and actions. For example, I can illustrate opening, reading, and closing a book by going through the motions and saying "I open the book," "I read the book," and "I close the book." Since not every word is easily demonstrated or pantomimed, I teach students how to use a dictionary. Most helpful are dictionaries that translate English to the students' native languages.
I read passages along with students as many times as necessary for the students to catch the rhythm, pace, and intonation of the passage. Then I let each of the students read alone until he/she is fluent. I rely on repetition and review to achieve reading fluency.
I use role playing to simulate conversations in the most common situations the students may find themselves in. For example, students who are parents may need to see their children's teacher or they may go see a doctor. After playing the roles of patients or parents, the students can switch roles and play receptionists or teachers. These simulated conversations prepare students and make them more confident in real-life situations.
Whenever feasible, I incorporate grammar lessons in other activities. For instance, when introducing parts of a house/apartment in vocabulary enrichment, I can insert the use of demonstrative pronouns (this, that, these, and those)and personal pronouns (I, you, we, etc.) without making them distractions. While in the kitchen, I say "This is a kitchen," and away from the kitchen I say "That is a kitchen" to demonstrate the use of the demonstrative pronouns. Likewise, I can say "I cook in the kitchen" and "You cook in the kitchen" to emphasize the use of the personal pronouns. The incorporated grammar lesson can be reinforced with a short homework which can be read and corrected in the next oral activities. This way the students do not spend too much class or tutoring time going over and answering grammar worksheets. The more time the students spend using, speaking, and reading aloud the English language, the better for them.
For the more advance students, I require them to write short paragraphs on topics they are most familiar with or interested in, like their favorite food, store, or player, or how to cook a certain food or grow a vegetable garden. Because they are familiar with or interested in the topic, they are more likely to succeed.
As a material source and as a guide to structure lessons I use textbooks. For beginners, I use Taking Off, Beginning English and for the more advance, I use Excellent English. For grammar and writing, I use Writer's Craft and other grammar books for reference.
I am patient, considerate. and caring. I use praise when appropriate to encourage my student. Being patient and appreciative of the students' achievements are a sound foundation for success.
In preparing students for the GED examinations, I usually start out with Language Arts, Reading. I want to make sure that the student has the comprehension skills necessary for success in the subject. I supplement the textbook we are using with a book entitled "Six Way Paragraphs" by William Pauk. The book consists of 100 short passages, usually 3 to 6 paragraph long, on a wide variety of topics from history to science. By concentrating on 3 to 4 passages of interest, the student is afforded an opportunity to develop and sharpen his/her comprehension skills like identifying main ideas, recognizing clue words, making inferences, etc. I believe that sharpening the student's comprehension skills makes him far better equiped for success in other GED subjects like social studies and science. Not only this. The passages the student choose are themselves excellent writing samples that show well-focused main ideas, and their development in a clear and organized manner, using well structured sentences.
Next, I guide the student to math. I start math with a simple exercise: finding the multiples/factors of numbers from 1 to 100. This exercise is a good way of gauging the student's knowledge of multiplication and division and introduces or reintroduces him to primes and composites. This knowledge carries a long way. Take for instance the operation 1/2 + 2/5 + 1/13= ?. Knowing that 2, 5, and 13 are primes, the student can easily get the answer 127/130 by cross-multiplying the numerators and denominators of all the terms. Likewise, factoring a polynomial like (x squared +19x + 42) is a lot easier if the student knows that 2 and 21 are multiples of 42. The solution (x-2) (x+21) is easily obtained by tweaking the signs of the multiples 2 and 21 to get the middle term 19x.
I also discuss test taking tactics/strategies with students, like how to spot obviously wrong answers in a multiple choice format, not leaving any question unanswered, having patience and reading through and understanding a question before answering etc. and most important of all, learning to relax.
I have been teaching ESL and preparing students for the GED examination for the past eleven and a half years, and as part of the curriculum for both, I teach English grammar.
To give students a good understanding of the subject matter, I rely on textual and syntactical analyses, that is , I break down phrases, clauses, and sentences into component parts; identify the parts (nouns, pronouns verbs, etc.); then show their relation to one another ( noun/subject, verb/predicate; preposition, object of preposition, etc.). I also rely extensively on the use of rules as a tool to better understanding and as a guide to constructing grammatical structures like phrases, clauses, and sentences. I give students tips so they can remember how to apply a particular rule. For instance, an indefinite pronoun that ends with one, thing or body like someone, is always singular. So, when it is used as subject, the verb should be singular in form (Someone is knocking.)
I use textbooks like Fundamentals of English Grammar, Grammar Dimensions, How English Works, etc. and augment them with exercises I have created to emphasize or give more coverage to specific topics like subject-verb agreement or use of prepositions and the like.
I'm thorough, patient, and considerate.
Before starting pre-algebra tutoring sessions in earnest, I make sure that the student has mastered basic operations in addition, subtraction, multiplication, and division; that he/she has some understanding of basic vocabulary like difference, sum, product, etc. which enables him/her to solve word problems. I also require that he/she knows the distinction between primes and composites.
If the student possesses these basic skills and knowledge, he/she will be better prepared to perform certain tasks moving forward and the tutoring can proceed smoothly and expeditiously. For instance, reducing fractions or changing percent to a fraction becomes easier if the student knows multiples of numbers. The fraction 27/63, for instance, can be reduced to 3/7 easily if the student knows that 9 is a common factor of both 27 and 63. Likewise, the student can easily change 75% to 3/4 if he/she knows that 25 is a common multiple of 75 and 100.
I use props, cut-outs, student's own experiences and familiar objects to clarify, explain or demonstrate math concepts/formulas. For example, I explain the relationship between pi, diameter, and circumference by using a string, a ruler, and a plate. With a ruler, the student and I can lay a ruler through the center of a plate to measure its diameter and multiply diameter with 2.1416 (pi) to obtain the circumference of the plate. The student and I can verify the circumference by running the string around the edge of the plate and measuring its length.
I use a chess board is to explain the formulas for finding areas of quadrilaterals and triangles. The board shows graphically why the formula L x W produces square units. And by letting the student draw a diagonal line from one corner to the other, he/she clearly sees that a triangle is actually half a quadrilateral. This gives the student a clearer idea of how the formulas work.
I use familiar objects or places around the house or apartment to illustrate concepts like the slope of a line. I find that a student can easily understand the "run" or the "rise" of a slope, a negative or a positive slope when he/she can connect it to a concrete object or thing.
I use rules to guide and structure a student's thinking process. For instance, I tell students that every even number is divisible by 2, or that any number that ends with 0 or 5 can be divided by 5 or that any multi-digit number can be divided by 3 if the sum of the digits is a number that can be divided by 3. Some examples of the last rule are 27, 51, and 171.
Reminding a student that multiplying or dividing signed numbers with the same signs always results in a positive number and multiplying and dividing numbers with different signs results in a negative number, likewise has beneficial effects.
These rules allow the student to work through some tasks quickly and accurately, and the ability to do so surely builds his self-confidence, too.
The student reads for various reasons. He/she reads for pleasure, for information, for guidance or direction, or for improvement. Whatever his/her reason is for reading, one thing is sure. He/she needs comprehension skills to enable him/her to understand what he/she is reading. Without understanding, reading is boring and difficult, and the student soon loses interest. My job as a tutor is to see to it that the student develops into a fluent, proficient, and independent reader.
Comprehension begins with decoding and recognizing printed words. The student who can not read fluently and spends too much time figuring out words will most likely lose the meaning of the sentence by the time he decodes the last word of the sentence. Lack of reading fluency hinders understanding. For the student who has a fluency and word recognition problem, I have devised decoding exercises that break down words into syllables and use illustrated keywords to facilitate word recognition and recall. For the fluent student, I bring him/her to the next step: building a working vocabulary based on grade level.
I teach the use of a dictionary, but rely on other methods to get the meaning of words because a dictionary has an obvious limitation. It gives the student other words to define or explain another word. If the student does not know a word in the definition or explanation, he is back to square one. So, I also rely on what I will call the definition in use of a word. For example, I point to a picture or representation of an object or demonstrate or pantomime an action or idea when possible. This gives the student a clear understanding of the meaning of the word.
I also employ other techniques and exercises to build an appropriate vocabulary. For instance, the student can finish a sentence by supplying the missing word (cloze) or chose the right word from a group to finish the sentence. For the more advance, I let him/her use vocabulary words in his/her own sentences. I make the student match words with words of the same or opposite meaning. Synonyms and antonyms very often facilitate recall.
Yet, it is not enough to know word meanings in isolation. Often times, the meaning of a word is in context; it is in the word's relation to other words or sentences that gives it a clear meaning . Take the words "go" and "stop." As a toddler, these are probably among the very first words the student learns. Without context, however, their meanings are unclear. "Go," for instance, may be part of a command, like "Go to school." Or it may be part of an idiom like, "Go fly a kite." So it is with "stop." It is a verb that commands: "Stop!" Or, it can threaten with "or else"; "Stop or else...!"
I alert the student that context may be provided, not just by other words or sentences, but by paragraphs, the passage as a whole, or the title. The setting-the time and place of the story- may provide context clues to meaning words.
Armed with the ability to comprehend, the student can now employ it to identify plot, setting, characters or climax, or make inferences and predictions (literary skills) or use a dictionary, or table of contents or index a guide or an outline, or follow instructions (study skills). In short, the student, having developed strong comprehension skills, is well on his way to becoming a fluent, proficient, and independent reader.
To augment grade level tutoring materials I use Walter Pauk's "Six Way Paragraphs", which is a collection of 100 short passages of 3 to 6 paragraphs on a wide variety of topics. It is an excellent multi-level material for enhancing reading comprehension. For the more advance student, I use Ann Raimes's "How English Works". It is a useful grammar and reading book.
One of my minor studies in college was English. As course requirements, I wrote and submitted many, many essays, speeches, term papers, book reports, etc. At this time, I was also a contributing editor to the school paper and published several articles on religion and ethics.
In 1994, as a culmination of years of research, I developed, wrote and published a reading primer entitled "EZ Reader Learning to Read the Phonics Way." It has helped learners of all ages read or improve their reading skills.
This background prepared me to teach students writing at the AAMA Adelante Adult Education Program for seven years. Many students who studied with me obtained their GED certificates