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All the questions about " Middle School Mathematics " that have been answered are listed below. To search for specific questions, enter one or more search terms. There are 87 questions currently posted in the database that match this query. Displaying question 1 through 87 Question: Could you please explain the difference between direct and inverse variation? We were given formulas but I don't know when to use them. Asked by: Expert's keyword(s) : variation, direct, inverse Answer : Variation involves working with two or more variables. When one variable changes, there may be a resulting effect on the other or others. In direct variation, when one variable goes up, the other goes up as well. When one variable goes down, the other also goes down. For example, if you charge $5 per hour to babysit, this amount becomes your constant because it remains the same hourly rate regardless of how many hours you work. However, the more hours you babysit, the more money you earn. Consequently, the variable of hours worked and money earned are related. When the hours worked goes up, the money earned goes up as well. You could use y = kx to show direct variation. Usually, x is the independent variable and y is the dependent variable. In our example, the amount of money earned depends on the hours you work. If you were asked to find how much money is made if you babsat for three hours, you would use: y = ($5) (3) or $15.Inverse variation is just the opposite. When one variable goes up, the other goes down. For example, the faster you travel, the less time it takes to arrive at your destination. The more people there are working at a job, the less time it should take to complete the task. You could use xy = k to show inverse variation. If 4 painters can paint a house in 9 days, how long will it take 12 painters to complete the house? 4*9 = 36, making 36 your "k." Therefore, if 12y= 36, then y = 3. You can see that only 3 days are needed to paint the house if you have 12 workers on the job. Notice, in the example of direct variation that I gave you, the constant "k" was given. It was the rate of $5 per hour. In the inverse variation example, you had to first determine the "k," and then use it to solve the problem. When you come across a variation problem, first determine if the situation calls for direct or inverse variation. Then write down the proper equation. If the constant is given, you can solve the problem immediately. If not, you must first find the constant and then use the constant to complete the solution. Answered by : Question: My daughter's teacher has suggested that we have my daughter tested for dyscalculia. What is this exactly? How can we help her? Asked by: Expert's keyword(s) : dyscalculia, learning disability Answer : Dyscalculia simply means to have difficulty with mathematical calculations. (Many people are familiar with the term dyslexia, whereby someone has language difficulties.) Dyscalculia can manifest itself in several ways. The student may have difficulty working with numbers. He or she may not be able to understand the meaning of signs or symbols that are common in mathematics, such as the "+" or "-" signs. Simple operations like addition and subtraction may not be performed easily. Some students transpose numbers. There is difficulty trying to memorize the times tables. Mental arithmetic could be very difficult to perform. The student could find it difficult to understand and follow the directions in a word problem. Even telling time or counting back money could be problematic.There is no standard level of mathematical ability that a student fails to meet to be diagnosed as having dyscalculia. Neither is there a definitive diagnostic test to confirm dyscalculia. However, students with dyscalculia usually have visual processing and sequencing difficulties. There is difficulty with comprehending abstract concepts. Math phobia and anxiety can develop and add to the difficulty. A learning specialist may help your child deal with the problem of dyscalculia. Concrete manipulatives and the use of graphs or pictures can transform the abstract into something tangible for the student. Have your daughter read word problems aloud so that her auditory skills may be tapped. She should use index cards to set up sample problems and refer to the samples as she does her homework. Let her use real-life situations to help her comprehend difficult theories and concepts. Have her do her assignments on graph paper to keep her work orderly. Ask her teacher if problems on assessments can be spread out (easily done by cutting and pasting from the original assessment) to prevent your daughter from becoming overwhelmed. Help her memorize formulas by setting them to music or rhyme. Inspire a positive attitude and make math seem like fun. Answered by : Question: I recently had a parent conference to discuss a student who was absent on numerous occasions throughout the school year. The student is presently failing the course. The parent has requested that I give the student extra help so she can pass the year and not have to go to summer school. I don't mind helping students, but this particular student has just missed too much work. What should I do? Asked by: Expert's keyword(s) : absences, missed work, remediation, summer school Answer : Students who are absent from school either legitimately or without authorization miss small chunks of the material from a sequential curriculum such as mathematics. When you put these chunks together, they form big holes in a course that eventually serves as the building blocks of future mathematics courses. Trying to patch up these holes with limited remedial assistance may allow the student to pass a final examination, but that student will find it difficult to transition to other concepts in the future.In my opinion, it would benefit the child to attend summer school to assure that she has the proper background to continue in mathematics. Summer courses usually provide extended remediation over several weeks so the student can learn the topics at a comfortable pace. There will be plenty of time to have some fun also. In your question, you seem to recognize that the task this parent has asked is a formidable one. Could you imagine how difficult it would be for the student to catch up on the remedial work, continue with the present work, and prepare for the finals? Answered by : Question: We are moving upstate during the summer and my son will be enrolling in a new school district. Last year, he seemed to have difficulty with mathematics when he first entered his present middle school. I had hoped to help him during the summer so he won't struggle when he enters still another school. Is there a set curriculum or textbook that is used in middle schools across New York State? Asked by: Expert's keyword(s) : standardized curriculum, standard textbooks, changing schools Answer : New York State provides a scope and sequence of topics that should be covered in elementary, middle, and high schools. However, the state leaves the actual curriculum up to the individual school districts. It is the same with textbooks; school districts are free to adopt the books they want to use in their classrooms.In the eighth grade, students take a standardized test in mathematics. The scores from this test serve more as a report card on how well each school is succeeding in teaching mathematics to their students. If a student fails the test, the state does not mandate that the student be retained. However, an individual school district may use the test scores to track the students for the following year. Students failing the test may be placed in a modified curriculum. Likewise, there is no specific curriculum or set of textbooks for the high schools. In high school, the students do need to pass the Math A examination as part of their graduation requirements so it is important that schools teach and students learn the topics proposed by the state for that age level. For your situation, why not contact your son's present teacher and find out the topics with which your son struggled. In addition, try to find out what material is going to be covered in the new school. Is there a placement test that your son needs to take? Do they track students according to their ability? Then you might be able to point him in the right direction this summer. It would be wonderful for your son to walk into the new school in September confident in his ability to succeed. Answered by : Question: My daughter is really struggling in her math class. She is a very talented writer and spends hours of her free time reading books far beyond her grade level. Wouldn't it be better for her to take a basic mathematics course just to learn balancing check books, computing interest, and topics that she could use in her life and leave the topics of algebra and geometry to students who are really interested in mathematics? Asked by: Expert's keyword(s) : Basic mathematics, remedial course Answer :It is wonderful to hear that your daughter writes so well and reads above her grade level, but it also is important for her to continue with a genuine mathematics program. Students have difficulty in mathematics for a number of reasons; however, these would not be grounds to relegate any student to a lower track in mathematics. Every child needs to take the gateway courses of algebra and geometry. By understanding the material in these courses, they can proceed to more advanced courses. Algebra and geometry also have fundamental concepts that can be used in every day life. For example, in algebra there is the concept that if you input something, you will get an output. You put money into a vending machine; you will get candy as an output. In geometry you would need to know how to measure your room to purchase the proper size rug. If children run into difficulty, parents should meet with the teacher to find out what kind of support there is available. Sometimes, it is just the matter of time that helps children improve their understanding of mathematics. As your child matures, the process of abstract reasoning begins, enabling the child to understand the meaning of a variable. To put any child in a remedial class in a middle school is just too premature. Your daughter's biological clock may just be running a little slower than some of the others in her class. The middle schools years are a bridging time in a child's education. It would be much too early for a child to decide what career path he or she would like to follow. Consequently, ruling out mathematics, or any subject, because it seems too difficult at this level can certainly not be done at this time. The best thing is to speak to your daughter's teacher to get her some help for the remainder of the year. Then you can either enroll her in some fun summer activity that would include mathematics or work with her yourself at home. Answered by : Question: I am going to introduce the unit on probability to my math classes but I, myself, am not clear on some of the terminology. Could you explain the difference between the terms independent, dependent, and mutually exclusive events? Asked by: Expert's keyword(s) : probability, dependent, independent, mutually exclusive Answer : Let's take a look at independent events first. Two events A and B are said to be independent if the outcome of the first event does not affect the outcome of the second. If you roll a die and then toss a coin, you would have independent events. Suppose you wanted to find the probability of getting a 4 by rolling the die and a head by flipping the coin. Your probability would be 1/6 * 1/2 = 1/12. Remember, with more than one outcome, the word "and" generally tells you to multiply in probability. Therefore, for independent events P (A intersection B) = P(A) * P(B).Moving on to dependent events, let's do some probability with M&Ms. Suppose you have 5 red and 4 green M&M's. What would be the probability of picking a red and a green if you ate the first red M&M before choosing the second? Here the probability is 5/9 * 4/8. The denominator (or sample space) for the second fraction is reduced by one because you ate the first selected M&M. Therefore, the second fraction is dependent on the first outcome. In a dependent event, P(A intersection B) does not equal P(A) * P(B). For mutually exclusive events, you are finding the union where there is no overlap. N(A union B) = N(A) + N(B); whereas, events that are not mutually exclusive are N (A union B) = N(A) + N( B) - N(A intersection B). Mutually exclusive events are also called disjoint because they have no events in common. If you want to find the probability of picking a jack or queen from a deck of cards, you would have 4/52 + 4/52 for a total of 8/52 because no queens are jacks, since they are disjoint sets. However, picking a jack or a heart would not be disjoint since there is a jack of hearts. To find this probability, you would add 4/52 to 13/52 and subtract 1/52. Answered by : Question: I heard from some parents that their middle school students had to take the SAT test for a Johns Hopkins summer program. What is this all about? Should I have my son review over the summer so he can take the SAT next year? Will his scores go on his transcript when he goes to high school? Asked by: Expert's keyword(s) : SAT, Johns Hopkins, talent search Answer : Johns Hopkins conducts a talent search to recognize those seventh and eighth grade students who score at or above the 97th percentile on a nationally normed standardized test. The test the students take is the SAT examination, which is designed for older students. This allows those middle school students who typically are at the high end of testing scales to have a higher ceiling with which to compare their scores. The middle school schools, who want to participate in the program, must take the examination only once that year and must take the test by the January test date.Those students who do achieve the 97th percentile score on the SAT are invited to take summer courses either in person or online. Those who do not meet the specified cut-off have had the experience of sitting for the long SAT examination in a low stakes testing atmosphere. These scores generally do not appear on their high school transcripts and do not affect their college acceptance status. Middle school students should not prepare for this examination because the program was designed to find those students who have the natural ability to take on the challenging summer courses. Let your son enjoy his summer. There will be plenty of time for him to prepare for the SATs when he is in high school. Answered by : Question: Each year I teach my students the rules of divisibility. However, when my professor asked us to give a proof for the divisibility by 9, I did not know how to start. Do you know the proof? Asked by: Expert's keyword(s) : divisibility proof Answer : The divisibility of 9 rules states that if the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9. For example, if you have the number 4,185 and want to know if it divisible by 9, simply add 4 + 1 + 8 + 5. Your result is 18, which is divisible by 9. Therefore, the original number 4,185 is divisible by 9.The proof, a rather difficult one at first glance, comes from modulo arithmetic. Every integer n can be expressed as n = a1(10k) + a2(10k-1) + +ak+1 10 + ak where a1, a2, ak+1, ak are integers between 0 and 9. 10 is congruent to 1 (mod 9). Therefore, 10k is congruent to 1 (mod 9) for ever k > or = 0. This makes n congruent to (a1 + a2+ + ak+1 + ak)(mod 9). Consequently, (a1 + a2 + + ak+1 + ak ) is divisible by 9 if and only if n is divisible by 9. To put that in simpler terms, we count from 0 through 9 and then repeat the units digits, for every tens place value. We keep repeating until we change the place values by a power of 10 (tens, hundreds, thousands...) Since the number 1 less than a power of 10 is a long string of 9's, the number is divisible by 9 (and also by 3!). It all falls into place because of the counting system we use. This also fits into the proof of checking addition by crossing out 9's. If you add 467 + 322 + 589, the result is 1,477. If you add each number across and eliminate the 9's in each you get: 4 + 6 + 7 = 17 =8; 3 + 2 + 2 = 7; 5 + 8 + 9 = 22 = 4. Repeat with 8 + 7 + 4 = 19 = 1. Then add the numbers in your answer and cross out the 9's (1 + 4 + 7 + 7 = 19 = 1). Since 1 = 1, the answer checks! Answered by : Question: How do you add and divide with Roman numerals without first changing them into our numbers, finding the answer, and then changing them back to Roman numerals? Asked by: Expert's keyword(s) : Roman numerals Answer : Actually, the addition of Roman numerals is fairly simple. You merely string the Roman numerals along, group them, and convert to a higher Roman numeral symbol wherever possible. For example, if you add XXXV + LXXII (35 + 72), string them along without the addition operation sign and regroup in descending value order: LXXXXXVII. Change the five X's to an L, which becomes LLVII. Again, change the two L's to C, resulting in CVII (107). However, working with a IV (4) or IX (9) can get a little tricky since the I is subtracted from the subsequent numeral. If you add XXIV + XVI (24 + 16), change the IV to IIII before stringing the symbols in descending value order: XXXVIIIII. The five I's become V, and VV transforms to X. You are left with XXXX, which because of the rules of Roman numerals whereby you may not have more than three numerals repeated, the final answer is XL (40).Subtraction is a bit more complicated because of the borrowing that may be involved. A simpler problem would be LXXIV - XX (74 - 20). You can just cross off the matching X's from both terms and be left with LIV (54). When you have LXXXI - XXXIX (81 - 39), you can cross off the matching X's also, leaving you with LI - IX. However, you may not cross out the I's because the I in the first term is added to the L, but in the second term, the I is subtracted from the X. In this case, you need to convert the higher value numeral to lower value numerals. LI - IX would become XXXXXI - VIIII. Cross out one I from both terms and repeat the process of converting a higher numeral into lower numerals to equal its value. You would now have XXXXVV - VIII. Cross out the matching V's, leaving you with XXXXV - III. Perform the conversion once more and get XXXXIIIII - III. Cross out the matching III's to get XXXXII. Now you have to reverse the process and change to a higher value numeral, with is XLII (42). Multiplication and division is also possible, but the process becomes even more involved. I imagine your teacher wants you to work only in the Roman numeral system so you can logically arrive at the solutions. Remember, when in Rome, do as the Romans do! Answered by : Question: Could you explain the difference between a reflection over the origin and a rotation of 180 degrees? Asked by: Expert's keyword(s) : reflection, rotation, transformation Answer : Reflections and rotations belong to the family of transformations. Reflections are flips, while rotations are turns. The general rule for reflecting a point over the origin is (x,y) goes to (-x, -y) . For example, if you reflect (3,2) over the origin, the image would be(-3, -2). The general rule for rotating a point 180 degrees is (x, y) goes to (-x, -y). If you rotate point (3,2) 180 degrees, the image would be (-3, -2). The preimages are the same, and the images are the same, while the transformations are different. Let us look at another case. Suppose we begin with the preimages of capital "I" and capital "L." Reflect each of these letters over a given line of symmetry. The image of the letter "L" can be obtained by reflection. The image of the letter "I" can be obtained by reflection as well as rotation about a point on the line. This works because any rotation is equal to two reflections. A single reflection gives the same result as that reflection followed by a reflection in its line of symmetry when the object has reflection symmetry. This makes it equal to a rotation about the point of intersection of the line in which it was reflected and the line of symmetry. Using the letter "I" example, the reflection and rotation would give the same transformation result. On the other hand, if the vertices were marked, the flipping of the reflection would yield a different result. Answered by : Question: When I use my calculator to find the upper and lower quartiles, I get a different answer from the textbook. How should I teach my students to find quartiles? Asked by: Expert's keyword(s) : quartiles, statistics, data points Answer :Quartiles are found by ordering data and then by dividing the data into four parts. Each quartile contains 25% of the data. The separators are called the first or lower quartile, the second or middle quartile, and the third or upper quartile. There is discussion among statisticians as to whether the quartile values should be data points or whether they should fall between data point. Different textbooks, calculators, and software packages calculate the quartiles differently. John Tukey, who invented the box-and-whiskers display, finds the median of the data first. Then he finds the median of the lower and upper halves of the data points. If there are an odd number of values, he includes the median value in both the lower and upper halves to calculate the quartile values. The Moore and McCable method (M & M) of finding quartile values does not include the median in either half of the data values. To add to the confusion, there are other methods of computing quartiles that have their own unique formulas (the Mendenhall & Sincich method and the Freund and Perles method). Minitab and Excel use different approaches to find the quartiles. Middle school students will most likely use either the Tukey or M&M method. You need to choose one method to teach your class in order to avoid confusing the students. If they will be faced with calculating quartiles on a standardized test, be sure to use the method that the test manual prefers. In deciding on a quartile method, also consider whether you will follow the textbook for your problems or whether the students will be collecting their own data and using calculators to find their quartiles. Answered by : Question: There are several learning disabled students in my class and according to their Individual Education Plans (IEPs), they can test with extended time in the resource room with other teachers. One student in particular cannot answer any questions correctly in class, makes major errors when sent to the blackboard, and does not know his multiplication tables. However, when he returns with his test from the resource room, he always scores in the nineties. I firmly believe that the teachers are actually doing the tests for him so he can pass. As a new teacher, I am afraid to question them. What should I do? Asked by: Expert's keyword(s) : resource room, IEP, learning disabled, testing accommodations Answer :Students with learning disabilities are supposed to be tested by a learning specialist. After a diagnosis is made, an IEP is drawn up to help the student achieve success in accordance with his or her abilities. Often students in need are sent to take their tests in a different room to help them lessen the stress that takes hold in a classroom-testing situation. These students are often given extended time to complete the test. If these students were kept in their regular classroom, their peers would be proceeding with other work as they continued to work on their assessment. Some IEPs allow the students to have a reader and possibly a scribe when they take tests. The reader is supposed to read the question so that the student's reading level would not jeopardize his or her mathematics test scores. Scribes are supposed to write the information exactly as the student relates it to them. These teachers should have a background in special education so they understand the child's needs and then assist the child accordingly. There are some educators who go beyond the limit in helping their learning disabled students. These teachers may feel that the material is above the level of these students and to prevent the students from earning a failing grade, they actually complete the assessments for these students with minimal input from the students. This situation only sets up the student to fail in the future. It sounds as though you are truly concerned but as a teacher without tenure, you may be in an awkward situation. Perhaps you can discuss this situation with the learning specialist in your school and have them do his or her own investigation into this matter. Answered by : Question: We have several parents who are insisting that their children be placed in an honors class next year. Their children are fairly good mathematics students but may not be able to maintain their grades if they are moved to a higher level. Asked by: Expert's keyword(s) : honors classes, tracking, moving classes Answer :Many people attempt to live vicariously through their children. They begin their dreams of having their children attend an Ivy League college when they are registering their children for kindergarten. There are several variables that these parents do not take into consideration and which could possibly affect these aspirations. Students are going through a myriad of changes physically, mentally, emotionally, and socially during the middles school years. Students mature at different rates. Since the study of mathematics involves the idea of abstraction and the concept of abstraction can only be comprehended at a certain developmental level, children may not fully understand the use of variables at the time they are taking a prealgebra course. If students take a course beyond their comprehension, they could possibly pass the course through memorization of algorithms and formulas, but they will find it difficult to bridge their learning with a more advanced mathematics course. Students who are placed in a course beyond their scope of ability may initially do well in the class. However, as the level of the work increases, they are no longer able to scaffold their learning process, and they begin to fall behind. Feelings of frustration can set in, and these students, who once were potentially good students, may decide to give up and shy away from the more challenging mathematics courses. Parents need to look at the total picture. Some students participate in after-school activities, some have long distances to travel, and others have social problems that hinder them from doing their best in school. These students cannot take on honors courses in every subject without adversely affecting themselves academically as well as physically. Doing what is right for the child has to be the foremost consideration. Parents should work in conjunction with the educators of their children. Teachers have undergone years of training and have experienced many situations as professionals. They may be able to offer solid recommendations as to what is best for the student. Answered by : Question: My teacher said there is a perfect number and asked us to find out which one it is. I could not find perfect numbers in our textbook. Could you explain what a perfect number is? Asked by: Expert's keyword(s) : Perfect numbers Answer : The idea of perfect numbers dates back to Pythagoras' time (500 B.C.) and possibly even before. A perfect number is defined as a positive integer that is equal to the sum of all of its positive factors not including itself. For example, the number 6 is said to be perfect. The factors of 6 are 1, 2, 3, and 6. If you add all of the factors excluding 6, you will get 6. Again, the number 28 is perfect. If you add its factors together (1 + 2 + 4 + 7 + 14), you will get 28. The next perfect number after 28 is 496.About 300 B.C., a mathematician named Euclid came up with a method to determine if a number was perfect. Start with the number 1 and keep adding the powers of 2 until you get a sum that is a prime number. (The sum of 1 + 2 is prime.) You find the perfect number by multiplying the sum (3) with the last power of 2 you used (2). The perfect number is 6. To find the perfect number 28, use 1 + 2 + 4, which sums to the prime number 7. Multiply 7 by 4 (the last power of 2), and the result is 28. With the help of computers, 39 perfect numbers have been found so far. The largest contains more than four million digits! None of the perfect numbers that have been found are odd. In fact, all of the perfect numbers found to date end in a 6 or an 8. Answered by : Question: What I'm looking for is why Intro to Algebra isn't worth teaching at the 9th grade and why Algebra is more important? High poverty high schools that are eliminating lower levels of math - anything below Algebra - and the success they are having - this is probably where I need the most help - eliminating the low levels yields success Asked by: Expert's keyword(s) : Pre-algebra, algebra, advanced mathematics Answer :Algebra I is considered the gateway to higher mathematics. A good foundation in Algebra I is needed to build on concepts and theories that the students learn as they progress through the mathematics curriculum. You cannot have a good Algebra I course without a good introductory course preceding Algebra I. It is in the introductory course that students learn what a variable represents, how simple exponents work, and how to solve simple equations by inverse operation manipulations. Students do need to feel successful in these basic courses so they are confident enough to take on more challenging courses. Too often students just feel they are simply not good mathematics students and give up before they even try. Years ago, many students did not take algebra, which caused a problem when they tried to enter college or technical schools. To fix this problem, schools seem to have gone to the other extreme and offer courses to students beyond their reach without the proper prerequisite courses. High school administrators are being pushed to bring down the higher-level courses to keep up with other schools and the standardized test schedules. If a scope and sequence is compiled, it becomes another meaningless pack of paperwork or transforms into a set of lesson plans. Learning gaps are created because educators are not able to look at the whole picture of each student?s learning track. If students fail Algebra I, educators tend to put the blame on the student. Someone needs to be responsible at looking at the mathematics program from first through twelfth grades. If a prealgebra course can be brought down to the middle school level, then the students could be ready for an Algebra I course in grade nine. However, if the basics cannot be given to the middle school student, then we cannot expect the students to be able to handle the leap into such an important mathematics course. Answered by : Question: I volunteered to work with a science teacher on integrating our mathematics and science curricula for next year. Do you have any recommendations? Asked by: Expert's keyword(s) : integration, team teaching, science and mathematics Answer :Mathematics and science seem to be good disciplines to integrate. However, the attempts at integration often fail, leaving the faculty with a sense of frustration. Some of the problems that affect the outcome include the teachers being unfamiliar with each other?s curriculum. In addition, teachers may not realize that integration needs careful planning. There are several different integration models that can be followed. Team teaching and integration can also be thwarted because of the need to prepare students for standardized testing. Other problems that teachers may encounter with integration include class scheduling, class size, and stability of teaching faculty. One integration model could be based on discipline. You might integrate algebra, geometry, and statistics with physical science if students were given a project that had them drop different items and then measure the length of time or distance from the mark. If you use a content model, you could integrate measurement with the amount of chemicals needed to make a certain solution. You can use a thematic model, whereby the science and mathematics teachers can contribute lessons plans related to a certain theme, such as power. Methodological integration uses similar approaches such as constructivism, to present lessons. Another integration model is based on process, where the students can investigate, discover, or use hands-on learning during their lesson. Do not expect integration to be seamless between your two disciplines. Sometimes topics in the curriculum need to be taught but do not lend themselves to integration at that particular age level. To force integration at this point creates tension for both teachers and students. On the other hand, some topics have a natural setting for integration of science concepts with mathematics skills. Plan to integrate topics on a limited scale the first year. Be sure that students do not feel frustrated because of the different teaching styles and expectations. After each integrated topic, discuss the positives and negatives of the experience. Could the lesson be improved or enhanced the following year, or should it be eliminated? Each year as you become more comfortable, you will be building up your repertoire of integrated lesson plans. Answered by : Question: I always seem to lose points because answers are supposed to be given in a certain format: rational or integers or whole numbers. Could you explain the difference between these types of numbers? Asked by: Expert's keyword(s) : real, imaginary, rational, irrational, whole, counting, natural, integers Answer :All numbers are divided into the real number or imaginary number system. As a middle school student, you will only be involved in the real number system. In this system, numbers can be classified as either rational or irrational numbers. All rational numbers can be put into the form a/b as long as b does not equal zero. On the other hand, irrational numbers cannot be put into the form a/b. Some examples of rational numbers are 4 (4/1), 1/2, .3333333 (1/3), and .4 (4/10). Irrational numbers include the square root of 2 and pi. There are other important number classifications. The counting numbers, also called the natural numbers, start with 1, 2, 3, and go to infinity. To remember these, think of your learning numbers as a child. It was natural to start counting with one. The whole numbers start with the number 0, 1, 2, 3 and go to infinity. To remember this, think of the word hole within whole, and a hole resembles a zero. The integers start with 0 and go in either direction on the number line to infinity: +1 and ?1, +2 and ?2, +3, and ?3, and so on. To remember this, the initial letter of the word integer is an I. If you slide the top bar of the letter I down toward the middle, the letter resembles the plus-minus symbol. It is very important in mathematics to reread the problem and ascertain exactly what the problem is seeking for an answer. If the problem states the set of all positive integers, you would not include the zero or any of the negative numbers. Answered by : Question: Some of our middle school teachers have strong backgrounds in fields other than mathematics. When they are scheduled to teach mathematics classes, they often run into difficulty teaching some of the mathematics topics. Should we schedule the weaker mathematics teachers with the weaker students or schedule these teachers with the stronger students? Asked by: Expert's keyword(s) : Master teachers, teachers for homogenous classes Answer :Educators who teach subjects in which they lack a strong background is a growing problem, with the middle schools being the hardest hit. The Education Trust, a nonprofit education watchdog claims that about 44% of middle school teachers lack either a major or a minor in the subject they teach. In urban areas, where staffing is difficult, the percentage might be higher. This creates a serious problem: teachers who are unable to see the reasons for mathematical theories and concepts are unable to provide thorough explanations to their students. Consequently, you are faced with a dilemma that is a by-product of this problem. Do you assign the most experience teacher to the accelerated class so that the students can sharpen their edge, or do you place the experienced teacher with those students who need the most help to bring them up to standard? The best solution would be to have team teachers so the various levels of students are supported equally, and the stronger teacher can mentor the teachers who lack mastery of the material. However, scheduling parameters do not always allow for this type of arrangement. If you have to make a choice, the weaker students should be assigned the stronger teacher. That educator would be familiar with different ways to explain the material and different teaching styles that can accommodate the variety of students in the class. The mastery teacher could help the weaker students from falling further behind and may even bring them on par with the students' peers. The strong students may be able to use their text to supplement the teaching lesson from the weaker teacher. At the middle school level, they may even be able to understand the reasoning behind the concept without any enhanced explanation. Answered by : Question: We are having department discussions on how to raise our standardized test scores in mathematics. A suggestion was to have single-gender mathematics classes that seem to be a growing trend across the country. Could this change make a difference? Asked by: Expert's keyword(s) : single-gender classes, all-girls' classes, standardized test scores Answer :There have been numerous studies on single-gender mathematics classes over the last few decades. Recently, this idea is getting even more press as educators are trying to come up with different ways to insure that all children succeed. Most of the research seems to be in favor of the single-gender classroom for mathematics. Students at the middle school level are very conscious of the social interactions with each other. Boys tend to show off, and girls tend to sit back, pretending that they do not know anything for fear of being teased. Certain teaching styles appeal more to one gender as opposed to the other. Boys tend to like hands-on projects where they are engaged in learning while doing. Girls want more explanations and practice problems. Cultural expectations of the family may also come into play. Discipline issues within a classroom are also different between genders. By separating classes by gender, teachers are more apt to find what works for a more homogenous grouping. On the other hand, a small number of studies show that students do better in coeducational settings. The mixed gender classroom usually works because the teachers are sensitive to the differences in learning styles between the genders. They encourage all of their students to participate in discussions and work on hand-on projects cooperatively. Limiting class size is also another enabling factor for success as the teacher can give personalized attention to a student in need. If you have the opportunity to separate your students by gender next year in the mathematics classes without disrupting your general school schedule, you should try it and have the teachers keep note of the advantages and disadvantages throughout the year. The change itself may make teachers more aware of their teaching styles and the students more comfortable in their learning space. Answered by : Question: Can you recommend anything that can be done to help a severely learning disabled student in my pre-algebra class? Asked by: Expert's keyword(s) : learning disabilities, mainstreaming Answer :With the No Child Left Behind Act, students with physical, mental, or emotional disabilities are being mainstreamed into the academic classes and are expected to take the same academic courses as their peers. Since these students will eventually take their place in society, they should be offered the same educational opportunities as every other student. It would be beneficial to the classroom teacher to have a paraprofessional help these learning disabled students during the regular class sessions. However, school budgets are not always accommodating to this need. Other accommodations that are very supportive include the availability of extra tutoring and having readers and scribes for these students on test days. Allowing the studen who is eligible for extended or untimed testing to finish an assessment in a resource room or return to his class after school is very helpful. However, if the classroom teacher is not available to clarify a question, the resource room teacher may not be able to help the student. Note-takers also make it easier on the learning disabled student. Ask a student who takes thorough and legible notes to copy his or her notes each day and share them with his student partner. If you have access to an electronic white board or blackboard, you can print out copies of the lesson after each class for that particular student. Colored high lighters can help the student distinguish between concepts, algorithms, and sample problems. You can use peer tutoring, but it would be better if you set up the arranged meeting time and place to have the student helped by a peer. Focusing for extended time periods may also be a problem for the learning disabled student. It might be helpful to break your class into cooperative learning groups whenever possible and have all the students work on hands-on projects. More specifically, be sure you keep repeating the proper mathematics terminology during your classes. This is actually beneficial to all your students. Be sure to differentiate between terms and factors. Keep stressing the proper use of parentheses. Most important is to provide support and encouragement so that the learning disabled student feels that he or she can succeed and be motivated to take on my challenging classes in high school. Answered by : Question: I have been considering having some of my stronger mathematics students tutor their peers who seem to be in academic trouble. Do you think that this would be helpful? Asked by: Expert's keyword(s) : peer tutoring Answer :Peer tutoring is the practice of having students help their fellow students to bring up their skills or to update their fellow classmates on work missed due to absence. There have been many research studies on this practice with most reporting favorable outcomes (Van Zant and Bailey (2002), Bergen and Mi (2002), Viadero (2003). Some schools set up peer tutoring as a part of a service program. Other schools have provisions that require the prospective tutors to formally complete an application for the tutoring position and then pay their young tutors for hours they put in. While still other schools may just ask a strong student to help out a weaker student occasionally. Quite often, peer tutoring helps both the tutee as well as the tutor. The tutee receives individualized attention in a safe relationship with a peer. Generally, the examples and explanations that are given are in terms that are understood by the students' age group. The tutee is free to ask questions without fear of feeling stupid. In most cases, mathematical skills improve, as well as the student's self-esteem and attitude toward learning. The tutor, although strong in mathematics at the onset, usually becomes even stronger. Additionally, the tutor is intrinsically rewarded for having helped a fellow student. The most successful peer tutoring models have two important components. The first is structure. As the adult, it would be best for you to set up a convenient time, meeting place, and the topics that need to be reviewed. Then step aside and let the young tutor take charge. Make it clear that if he or she encounters any problems, you will be available. The second component is to set up a monitoring system. This could be as simple as checking to see if the tutee's grades have improved or more complicated as having the tutor routinely complete a follow-up questionnaire. Answered by : Question: Students are always asking for extra credit to raise their mathematics grade before report cards are sent out. It seems that they are only interested in raising their grade and not learning the mathematics they did not understand. How can I keep a balance? Asked by: Expert's keyword(s) : extra credit, retests Answer :It is unfortunate that most students are only interested in the bottom-line grade. If they do not understand a certain concept and fail a test as a result, their reaction is to do some extra credit on an unrelated concept to pull up their grade. Many students who are clamoring for extra credit have not even completed all of their homework assignments. This policy of giving extra credit to increase the end-of-term grade is especially detrimental to a discipline such as mathematics, where most concepts serve as building blocks for future learning. Extra credit should be reserved for those students who wish to go beyond their course of study. They should be encouraged to delve deeper and find practical applications for the concepts and theories they are taught in class. On the other hand, you do want to encourage students who have done poorly on an assessment to revisit the material so they can fill the gap they are missing. Students many not realize the value of correcting their mistakes unless they are rewarded with some kind of credit enticement. You need to devise an extra credit policy to cover both cases and put the policy in writing at the beginning of the school year. This way, students cannot connive to gain extra points that are not directly beneficial to what they are learning. One way to help your students learn the material that is giving them the most difficulty is to establish a retest policy. This encourages students to try again if they did not understand the material and also helps those students who just seemed to have had a bad testing day. The scoring on the retest needs to be scaled back so as not to hurt those students who may have scored a B or a C the first time around and, consequently, are not eligible to retest. The extra credit scoring policy needs to be fair and open to all the students in your class. Your students should also be aware that the policy is primarily to improve their mathematics and secondarily to raise their end-of-term grade. Answered by : Question: My students are learning factorials. They know that they must find the product of all the positive whole numbers less than or equal to the given number. I taught them that 0! is a special case because it equates to 1, but I am not sure of the reason behind this value. Asked by: Expert's keyword(s) : factorials, permutations, arrangements, ordering Answer :The factorial of a number (n!) is the product of all the positive integers from one up to n. 3! = 3 x 2 x 1 = 6; 2! = 2 x 1 = 2; 1! = 1. Factorials are used to show how many different ways n objects can be ordered or arranged. This arrangement is also called a permutation. For example, if you have five students to be seated in five chairs and want to know how many different ways they can be seated, you compute 5! and find there are 120 different seating arrangements. Most scientific and graphing calculators have the factorial button so the computation is made quickly and easily. Another way of computing n! is by using n(n-1)! Using this, we find that 4! equates to 4 x 3 x 2 x 1 or 24, and 5! is equal to 5 (4!) or 120. If n =1, then n(n-1)! = 1! = 1 (0)! And 0! =1. Does that sound confusing? Then let's do the reverse. If 4! = 24, then 3! = 4!/4 or 6. 2! = 3!/3 or 2. 1! = 2!/2 or 1. 0! = 1!/1 or 1. For your students, it might be easier to explain the value of 0! to them by referring to the usage of factorials; that is, finding the different ways a certain number of objects can be arranged. 0! =1 because you can arrange a set of nothing (the empty set) in only one way: as the empty set itself! Answered by : Question: Our principal has asked us to consider the option of having a very bright student skip a grade level next year. He has straight A's in my class, but I am concerned that he will miss certain topics that might be needed in his mathematics foundation. What is your opinion on skipping grades? Asked by: Expert's keyword(s) : skipping grades, advancing in mathematics Answer : There are stacks of research on the question of advancing gifted and talented students. (You may want to read some of the many available articles on the ERIC database, including Tolan (1990), Southern, Jones and Fiscus (1989) and Lynch (1990).) Most of the research has shown that acceleration tends to be a positive experience. It has been found that the important issue is to keep that special student motivated and challenged. Sometimes this can be accomplished through resource work or special honors sections. Sometimes the student can take higher-level courses in certain subjects, while remaining with his or her peers in other disciplines.However, there is no clear-cut answer to your question. Each child is in a unique situation, and there is a multitude of variables that need to be considered. For your discipline, what topics will the student miss by skipping the grade? Obviously, a student cannot miss a year of algebra, the gateway to advanced mathematics courses. However, if there are a limited number of new topics that will be introduced in a year full of repetitious practice work, summer worksheets or extra support during the new school year could be made available so the student will not feel overwhelmed. The child should be made part of the discussion and be made aware of the implications of skipping the grade. By advancing, the student may no longer be the best in the class. Parents also need to realize that they must refrain from pressuring their child, especially while he or she is adjusting to the new class setting. If the acceleration takes place, is there an assessment in place to measure the effect on the student? Is there a way to reverse this decision if an academic, social, or emotional problem arises? Continue discussing the advantages and disadvantages with all of the parties involved until you feel comfortable with the decision. Answered by : Question: As a new teacher, I want to make sure that my students are well prepared for the citywide tests at the end of the year. Besides, following the mathematics topics in the curriculum guide, what else should I do? Asked by: Expert's keyword(s) : standardized test preparation Answer : Standardized assessments come in all shapes and sizes. You need to familiarize yourself with the particular test that will be administered to your students. Sometimes, there are old tests that can be used to review. If this is the case, review several examinations, noting the topics that are repeated and the types of questions asked. Familiarize yourself with the number of parts on the test, and note how many questions are included in each section. Is there a mixture of short-answer questions and questions requiring students to show their work? Check to see if calculators are permitted and if they are, have your students practice with the type of calculator they will be using on the test. Some assessments have punch-out manipulatives like rulers and protractors. Have your students practice with similar instruments. Some tests include tables that the students may pull out and use. Students should know how to read these tables and be able to extract the information they need easily. If the assessment has multiple-choice questions, will students be penalized for guessing? Are the questions worth a certain amount of points, or is there a rubric for scoring the test? Practice scoring some tests beforehand so you can help your students in answering the questions completely. Although it is always essential that students do their best on assessments, how will the results be used?If your students have to take one of those secure assessments that are carefully counted and sent away to be scored leaving no tests to review, you should ask your colleagues, school administrators, or possibly contact the test company to explain the format, types of questions posed, scoring procedure, and samples of good, fair, and poor responses. Are there special strategies or problem-solving methods that are either approved or disapproved? You also should prepare your students mentally to take the test. You do not want them to develop test anxiety and have their state of mind interfere with their performance. On the other hand, you do not want to have your students shrug the assessment off as just another annoying standardized test. Explain to them the importance of the test and have them practice enough questions so they will feel that the actual test is just one more practice session. However, do not become so intent on your students' outcomes that you begin teaching to the test. You will be missing the beauty of mathematics, and your lessons will become empty training sessions. Answered by : Question: According to the state standards for middle school mathematics, students are supposed to be able to work with spreadsheets. I am not proficient with technology myself. How can I help my students? Asked by: Expert's keyword(s) : spreadsheets, statistic, software Answer :Even though you are not confident teaching with technology, you can introduce your students to the basics of spreadsheets. Students should know the difference between rows and columns and how to enter data. They can learn how the use of spreadsheets facilitates data analysis. They should understand that by using a simple formula (for example, adding 5 to each entry in a column), a new column could be automatically generated by the software. At the start, your students can use graph paper for simple spreadsheet work. Have your students divide the paper into columns, labeling each with a letter. They should use numbers to label the rows. For a simple assignment, have your students enter a set of numbers representing weights of a group of people. Have them calculate the mean, median, and mode. Explain that this group of people is on a special diet, and each plans to lose ten pound in the first month. In the next column, have your students enter the new weights by subtracting ten pounds from each of the original entries. Let your students investigate the effect of the decrease on the mean, median, and mode. If you have computers available, you can introduce your students to Excel or the spreadsheet in Claris Works. There is also special software for beginning students such as The Cruncher. It might be fun to work along with your students and improve your own skills at the same time. Once your students realize how much fun spreadsheet projects can be, you can find many others spreadsheet projects on the Internet for them to build their expertise. Answered by : Question: My students have learned to do the typical conversions using the metric system. They are able to convert millimeters to centimeters, milliliters to liters, and milligrams to grams. However, when they solve word problems involving metric measurements and are asked to estimate an appropriate answer, they have no idea what a good metric answer would be. Can you offer any suggestions to help them make good conversion estimates? Asked by: Expert's keyword(s) : Metric estimation, Metric conversion, estimating solutions Answer : Students recognize units of measure that are found in their daily life experiences and then are able to offer a sound estimate when a problem calls for units of measures with which they are familiar. However, it is important for them to be able to make approximate conversions to the Metric system so they are confident that their solutions to word problems make sense. There are many sites on the Internet that offer instantaneous conversions. Allow your students some class time to play with these conversion tables. You can possibly make a game of it and have students on different teams vie for the fastest (and most accurate) conversions.Your middle school students should also be comfortable with the formula to convert between Fahrenheit and Centigrade by using C = 5/9 (F-32) or F = (9/5*C) +32. Most students know that 0 degrees Centigrade = 32 degrees Fahrenheit and 100 degrees Celsius = 212 degrees Fahrenheit. In addition, they should also know that 20 degrees Centigrade is approximately 70 degrees Fahrenheit or room temperature. For linear measurement, your students should become familiar with one kilometer being approximately .6 of a mile. An inch is about 2.5 centimeters, and a meter is a little longer than a yard. They should know that one liter is a little bigger than a quart, and 1,000 mL of water normally weigh about one kilogram. One kilogram is also approximately equal to 2.2 pounds. These few conversion approximations should be enough for them to make their estimates. Have your students solve word problems with the different conversion measures often throughout the year, and the conversion will surely become a part of their thought processing. Students recognize units of measure that are found in their daily life experiences and then are able to offer a sound estimate when a problem calls for units of measures with which they are familiar. However, it is important for them to be able to make approximate conversions to the Metric system so they are confident that their solutions to word problems make sense. There are many sites on the Internet that offer instantaneous conversions. Allow you students some class time to play with these conversion tables. You can possibly make a game of it and have students on different teams vie for the fastest (and most accurate) conversions. Your middle school students should also be comfortable with the formula to convert between Fahrenheit and Centigrade by using C = 5/9 (F-32) or F = (9/5*C) +32. Most students know that 0 degrees Centigrade = 32 degrees Fahrenheit and 100 degrees Celsius = 212 degrees Fahrenheit. In addition, they should also know that 20 degrees Centigrade is approximately 70 degrees Fahrenheit or room temperature. For linear measurement, your students should become familiar with one kilometer being approximately .6 of a mile. An inch is about 2.5 centimeters, and a meter is a little longer than a yard. They should know that one liter is a little bigger than a quart, and 1,000 mL of water normally weighs about one kilogram. One kilogram is also approximately equal to 2.2 pounds. These few conversion approximations should be enough for them to make sound estimates. Have your students solve word problems with the different conversion measures often throughout the year, and the conversions will surely become a part of their thought processing. Answered by : Question: What is a concave polygon? Is there a formula to find the area of such a polygon? Asked by: Expert's keyword(s) : Concave polygons, convex polygons, area of polygons Answer : The polygons you generally work with in school are convex polygons that include the triangle, rectangle, trapezoid, and parallelogram. You learn special formulas to calculate the areas of each of these. If you draw a line to connect any of the vertices of these convex polygons, the line you draw is either inside or on the polygon.On the other hand, a concave polygon has a side that caves in. If you draw a line to connect the vertices, one or more of these lines run outside of the polygon. We don't study standard formulas for the areas of these polygons. Instead, we can divide these polygons up my adding lines to make common convex polygons within the figure and then to calculate the sum of these areas. Another way to calculate the area of concave polygons would be to circumscribe a rectangle about the concave figure. Shade in the areas that are outside of the concave polygon but inside the rectangle. These figures are usually right triangles, for which their areas are easily calculated with the formula ½ base times altitude. Subtract the sum of all the shaded areas from the area of the circumscribed polygon. The difference is the area of the concave polygon. Answered by : Question: We are going to begin a unit on triangles. How can I help my students remember that there are 180 degrees in a triangle? Asked by: Expert's keyword(s) : Triangles, angle measure, sides of triangles, exterior angle Answer :Triangles can become an exciting discovery lesson for your middle school students. You can introduce the triangle as a polygon having three sides and three angles. The sum of the three angles always equates to 180 degrees. Have each student draw a triangle and then have the students cut off the angles. Tell the students to tape their three angles side by side. Have them place a straight edge along the outer sides, and they will discover they have made a line or a straight angle. They should already know that there are 180 degrees in a straight angle. Next have the students look at the relationship that involves the length of the sides of a triangle. Have the students attempt to form triangles by putting together three sticks of different lengths. Can they always make triangles? They will find that sometimes their figures do not close, and at other times, one of the sides has an extra tail. Put measures of the sides that do not form triangles on one side of the blackboard, and the measures of the lengths that form triangles on the other side. The students should discover that triangles could only be formed if the sum of the two smaller sides is greater than the third side. While the students are working on triangle measurements, have them extend one of the sides and measure the exterior angle that was formed. Have the students cut off the two angles that are not touching the exterior angle and tape these two angles into the interior of their external angle. They will discover that the sum of the two nonadjacent angles is equal to the exterior angle. Answered by : Question: Yesterday in my middle school mathematics class, we were discussing the different types of polygons and how prefixes help determine the number of sides. The students learned that a quadrilateral has four sides. Then in the class I take in the evening, we were discussing quadratic equations that have only two answers. Why is the prefix quad used instead of bi in that case? Asked by: Expert's keyword(s) : quadratic, quadrilateral, polygons, polynomials Answer : Yesterday in my middle school mathematics class, we were discussing the different types of polygons and how prefixes help determine the number of sides. The students learned that a quadrilateral has four sides. Then in the class I take in the evening, we were discussing quadratic equations that have only two answers. Why is the prefix quad used instead of bi in that case?The prefixes of the various types of polygons do describe the number of sides. A triangle has three sides; a quadrilateral has four sides; and a pentagon has five sides. In middle school, students generally learn the names of the polygons through the ten-sided figure, which is called a decagon. It is also good to give your students other words with the same prefix to help them remember the mathematical terms. For example, a decade has ten years. With polynomial equations, a quadratic has two roots, a cubic has three roots, while a quartic has four roots and a quintic has five roots. The quadratic equation involves the second power or the first variable in standard form being squared. In Latin quadratum means square, and quadratus means squared. Using the quadri prefix that means squared, we have the term quadratic equation, which is an equation of the form ax^2 + bx + c = 0. Answered by : Question: I was thinking about using the stock market for a mathematics project. Can you recommend any ready-made projects that would hold the attention of middle school students? Asked by: Expert's keyword(s) : stock market, activities, math projects, numeration, statistics, probability Answer : Stock market projects can be so much fun while so much learning is taking place. Students can sharpen their skills working with fractions, decimals, and percents. You can use the stock market with graphing, simple statistics, and probability. The project can extend for any duration.There are many stock market games on the Internet that you can join at any time. Some can be worked individually, while others offer class competitions. Some good web sites include www.smg2000.org, www.game.marketwatch.com, and www.archieve.ncsa.uiuc.edu/edu/rse/RSEyellow/gnb.html. However, since this is your first attempt with this type of project, it might be easier to set up your own program with your own parameters. You can choose two dozen companies, whose names your students would recognize. Give each student a set amount of play money to invest. You might want to have them track their stock on a daily basis, but give them the opportunity to buy or sell just once a week. Focus on different concepts each week. For example, the first week the students could compute their gains and losses. The second week could involve graphing. Another week could be used to examine trends and make predictions. You might want to offer prizes for a variety of categories that include the largest investment growth overall or the biggest increase in a week. Good luck! Answered by : Question: All of my students are scheduled to take mathematics seven times in a six-day cycle in order to improve their basic skills. I have noticed that several students are often taken on extended vacations during the school year. Although a full day was scheduled for the Wednesday before Thanksgiving, there were several students absent on Tuesday and Wednesday. How do you prevent these students from falling even farther behind? Asked by: Expert's keyword(s) : extended vacation, missing classes Answer : You do, indeed, have a tough job. You are responsible for planning your work to bolster your students' skills while continuing on with the topics in your curriculum. The school has given you an extra period in your schedule to help you toward your goal; however, parents pull their children out for whatever reason. Of course, it would be understandable if there were an emergency that included illness or death. It might be reasonable for students to begin their vacation early if this meant a substantial savings in travel fare. However, if the extension is merely to extend the vacation by getting an early start, the parents have decided that education can take a back seat in their child's learning process. They often expect the teacher to give their child extra help with work missed when their child returns.If this situation occurs infrequently, it might be wise to make up spare packets for students to take away with them. Parents should be asked to supervise the completion of the work. This is an added burden for you, as you need to prepare the packets and then correct the completed packets upon the students' return. Would there be a consequence for unfinished packets or packets that were done haphazardly? If this situation occurs often, especially with the same students, it might be wise to bring the problem to the attention of your supervisor or administration. There might be a school policy in place that deals with this problem. If not, one should be designed. It is important that parents are made aware that in order for their child to be successful in the learning process, he or she must attend school on a regular basis, stay on task while in the classroom, and review or practice at home. Parents and teachers need to work as a team to make the plan work. Answered by : Question: In a recent question, you mentioned that either the traditional algebra course or an integrated course is good for an eighth grade mathematics course. In our school, we still teach a pre-algebra course with some geometry and probability included. Are we holding our students back? Asked by: Expert's keyword(s) : eighth grade mathematics, algebra, pre-algebra Answer : Students are generally able to understand the concepts of an algebra course by the time they are in eighth grade. If algebra is one of the curriculum strands in the elementary and middle school scope and sequence, teachers can help form the building blocks that are needed to take an algebra course before entering high school. Pre-algebra can be taught in sixth and seventh grades.The National Council of Teachers of Mathematics (NCTM) recommends that students have a solid algebra background by the end of eighth grade. In their Principles and Standards for School Mathematics, NCTM provides guidelines for introducing algebraic concepts in elementary school and then strengthening the concepts as the students move on. In fact, the Grades 6-9 Standards focus on algebra. NCTM believes that students should take solid mathematics courses during their four years of high school. If you are considering changing your middle school curriculum, include all the grade levels in the transition. Students need to move smoothly in the direction of the advanced mathematics that they will be taking. Putting in a full algebra course in your eighth grade without the proper support of pre-algebra in the grades below will only result in a frustrating experience for both the students and teachers involved. Answered by : Question: My students often seem confused with what is asked of them on tests. They especially mix up the following directives: simplify, evaluate, factor, and solve. How can I make these terms clear to the students? Asked by: Expert's keyword(s) : Mathematical terminology, simplify, evaluate, factor, solve Answer : It is not uncommon for students to become confused with what is asked of them on the different questions on an assessment. To make certain that they give a complete answer, they often go further than they are asked or create an incorrect method to arrive at the solution. When instructed to simplify an algebraic expression, they often condense it and then set the expression equal to zero to arrive at a solution. They believe that only a single number could actually be the simplest form of an expression. When asked to factor, many times students are afraid to walk away from the problem with algebraic factors; they prefer to turn the original expression into an equation and find the zeroes. This way, they have the factors and the zeroes in hopes that the teacher will give credit for the portion of their answer that is correct.Explain to your students what the terms mean and give them leaning helps. For example, to evaluate means to come down to a single value. To simplify, you are making the expression simpler; however, since you do not know the value of the variable, you cannot go any farther. When factoring an expression, answers should have multiplicative factors. Use these terms as often as possible during class. The repetition will help. Students need to use accurate mathematical terms when asking questions or giving answers. Don't be afraid to correct your students if their answers are correct, but they used incorrect terms in their explanations. Also, have your students write out their own questions and answers. Students who can communicate mathematically will truly understand mathematics. Answered by : Question: Our honors eighth grade class is taking an integrated mathematics course that is primarily algebra with a touch of geometry, probability, and statistics topics. A colleague of mine from another school is teaching her honors class a traditional algebra course. Our students will eventually have to take the Regents A examination. Which is the better route? Asked by: Expert's keyword(s) : Integrated mathematics, algebra, A examination, curriculum Answer : I have taught both courses for many years. The integrated mathematics course, as you stated, combines topics from algebra, geometry, statistics, and probability. The following year, the focus is mainly on geometry, but other topics are included as well. On the other hand, the traditional algebra course presents all of the Algebra I topics during the year and is generally followed by a traditional geometry course. If taught properly, both courses can be very good courses. In the integrated course, students become aware of the way different types of mathematics topics are weaved together to provide a better understanding of real-world situations. In the algebra course you may go into greater depth of the topics to satisfy the curious minds of those honors students.The decision on which course to teach actually depends on the students' subsequent mathematics course. If all of the students taking the integrated course continue their secondary school mathematics doing integrated mathematics, they will have a strong background to enter higher mathematics. If the algebra students continue in the traditional geometry sequence, they too will have a solid foundation for their advanced mathematics studies. Both curricula will prepare the students for the Regents Mathematics A Examination. However, since you teach middle school, your students may go onto different high schools that may offer a variety of mathematics programs. Students moving onto a school that offers a different sequence from the one they had started may have some topics repeated but miss other important topics causing gaps in their understanding. When these students go onto more advanced mathematics courses, they might not understand why they are having difficulty and become frustrated. Answered by : Question: I recently returned from my second field trip taking students outside of the classroom to see how a lesson in mathematics has real-life usage. The students were using simple trigonometric functions to find the heights of trees in the park. (Last time we had a geometry trip.) The following day, the students were asked some questions on a test pertaining to the trip, and most answered them incorrectly. Was the field trip just a waste of time? Asked by: Expert's keyword(s) : field trip, discovery learning, real-life experiences, applications in trigonometry Answer : You should be applauded for connecting classroom lessons with real-world activities. It is a great way to answer the common question of how will we ever use this lesson in life.In my experience, discovery learning is not always the best approach. Sometimes students discover the incorrect answer, and this learning has to be undone. Other students may take a circuitous route to their discovery and not be able to think through a more efficient procedure. Still other students have a difficult time transitioning their learning experience back to the classroom, and they find themselves unable to make the connection on assessments. The next time you plan an outing, spend a few days introducing the material and have your students work through theoretical problems. You might even give them a short quiz to measure their comprehension. Then take the students out and let them recognize that the same lesson you taught in the classroom can be applied to their outdoor activity. I am sure that you will have much better results when you test them on the material. Answered by : Question: As a student in education, I recently spent a day observing different mathematics classes. I noticed that all the teachers started with checking homework and then proceeded with their new lesson. One teacher even gave a short quiz in addition. The teachers seemed to have accounted for every minute of their class periods. Are lessons always so structured? Asked by: Expert's keyword(s) : class period, school day Answer : Teachers have the dilemma to provide balanced mathematics classes that involve their students in a variety of problem-solving situations in addition to their lessons presenting mathematical theory. Teachers must cover a set curriculum during the school year if their students are to be successful on the end-of-year standardized city and state examinations. Since secondary school mathematics requires a strong middle school background, teachers must also cover all of the topics that will serve as these building blocks. Teachers ordinarily assign homework to reinforce the lesson that was taught in class, and this homework needs to be checked the following day to insure that the students have an understanding of the old material before presenting the new material. Of course, teachers also must give quizzes and tests to assess their students' progress.All of these needs must be met within 45-minute class periods that may be interrupted by fire drills and class assemblies or canceled totally due to inclement weather. The successful teacher becomes a juggler of both tasks and time and accomplishes all of the goals for the year. Careful scheduling and good lesson plans play integral roles in this achievement. Experience hones this process by giving teachers baselines on how long it generally takes to complete a particular topic. Changing the routine with projects or group work keeps the students interested. As you witnessed, a lot goes into a teacher's school day. Answered by : Question: My daughter, who is in the eighth grade, earned a low score on her standardized test. However, she seems to be doing fine in class, and her teacher says there is no problem. Should we be concerned? Asked by: Expert's keyword(s) : Standardized test scores, classwork, assessments Answer : There could be several reasons for your daughter scoring low on a standardized test but having no problem with her present schoolwork. Some children are just not good test-takers. A standardized test is probably the toughest kind of assessment for these children. They are not able to study for the test, so they come to the test fearful that they are unprepared. Then they might be confused on the testing strategy. Sometimes it is better to guess at answers on standardized tests, and at other times, the student is penalized for an incorrect answer. Often students are confronted with a concept or notation that is totally unfamiliar to them, and they become flustered. Sometimes, a student just has an off day, possibly due to illness, and the standardized test provides a false snapshot of the student's ability.Middle school students mature at different rates. As a result, some students do not do as well as expected on standardized tests because they are not able to fully comprehend some of the mathematical concepts that are being tested. To be sure that there is no problem, check your daughter's homework to see if she is having difficulty with practice work. Review your daughter's school quizzes and tests to see if she is consistently making the same type of errors. However, since your daughter's teacher has told you that your daughter was doing fine in the classroom, I would not be too concerned with the recent standardized test score. Answered by : Question: I am trying to teach my students to convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit. However, the students are never sure when to use the 5/9 rather than the 9/5 and when to subtract 32 rather than adding 32. Can you offer any help in keeping the formulas straight? Asked by: Expert's keyword(s) : Temperature conversion, Fahrenheit, Celsius Answer : The formulas found in most textbooks may seem confusing to students. When changing a temperature to Celsius the formula is C = 5/9 (F-32). When converting to Fahrenheit, the formula is F = (9/5) C + 32. Using these formulas, students would find that a reading of 32 degrees Fahrenheit is equivalent to 0 degrees Celsius. A temperature of 212 degrees Fahrenheit is equal to 100 degrees Celsius.You can eliminate the confusion between the fractions by using 1.8. A Fahrenheit temperature would equal 1.8C +32. A Celsius temperature would equate to (F-32)/1.8. If you would rather have your students study one formula, you could use the ratio C/(F-32) = 5/9. Then if they are given a Celsius temperature, they can just plug it in for C; if they are given a Fahrenheit temperature, they can plug the number into F. However, this ratio falls apart if given either 0 degrees Celsius or 32 degrees Fahrenheit because of the problem when we try to divide by zero. Another conversion method would be to add 40 to the given temperature. Then if the given temperature is in Fahrenheit, multiply the result by 5/9. (The pneumonic would be the temperature begins with F for Fahrenheit, and the fraction begins with F for five.). If the given temperature is in Celsius, multiply the result by 9/5. Then subtract 40 from the result of your multiplication. A short cut to change Celsius temperature to Fahrenheit is to double the Celsius reading, subtract 1/10 of the number from the reading and add 32. I hope one of these options help your students with their temperature conversions. Answered by : Question: I have a question as a concerned grandmother. While babysitting, I noticed my grandson, a seventh grader, was solving algebraic equations. What happened with the notion of waiting until children are able to understand the idea of abstraction before presenting certain material? Asked by: Expert's keyword(s) : abstraction, algebraic concepts, maturation Answer : You are correct in thinking that since mathematics has different levels of abstraction, it is difficult to teach certain concepts to children who have not reached a certain level of maturation. For example, in algebra, a letter such as x serves as a variable and has the job of representing a number. This concept of representation deals with abstraction that could not be processed by younger children. To teach algebra to a student who has not reached what Piaget calls the formal operations stage, one would have to make the concepts concrete by which the problem would no longer be algebraic.Many educators (Elkind, for one) feel that although we are limited to what a child may learn due to his or her maturation level, we still need to encourage the child?s thinking. Since middle school students are at different levels of the maturation process, middle school teachers especially need to modify their lessons to reach the needs of their entire class. Hands-on work with manipulatives would be valuable in explaining some concepts and connecting concrete with abstract ideas. Because children mature at their own individual rate, it is very important to avoid placing students in exclusive honors classes at an early age that preclude the late-bloomers from joining. Answered by : Question: Is it better to teach students to use 3.14 for Pi rather than 22/7? Asked by: Expert's keyword(s) : Pi, irrational numbers, circumference, diameter Answer : If students are taught to use 22/7 or 3.14 for the approximation of Pi, they will come to think Pi as being a rational number. (A rational number is a number that can be put in the form of A/B, as long as B is not zero.) It might be more helpful to explain that first Pi is a number and not a variable and then that Pi is an irrational number. No matter how many places you carry out Pi, there is no termination nor is there a repeating pattern. If your students use calculators, most calculators have a Pi button for the approximation. If your students do not use calculators, vary the approximation of Pi from problem to problem so they become aware of Pi's irrationality.It would be good if you demonstrate what Pi actually is. Have your students measure the diameters of several different sized circular objects and record their measurements. Then have the students wrap their measuring tapes around the outside of the objects and record their measurements. Next, have the students divide the circumference measurement by the diameter, and the answer will be Pi every time! Another fun activity is to make horizontal stripes on the floor with each stripe being one unit apart. Then take a number of needles each one unit in length and drop the needles on the floor. The ratio of the needles, which cross a stripe on the floor to the total number of needles, is approximately 2/Pi. The more needles you use, the closer the number approaches 2/Pi. Answered by : Question: What is the best way to teach students to find square roots without the use of a calculator, other then by a guess-and-check method? Asked by: Expert's keyword(s) : square roots, guess-and-check, perfect squares Answer : By definition, the square root of a number is a number that when multiplied by itself, gives the original number with which you started. You can start your lesson by having the students find the square roots of numbers that are perfect squares. Have the students find the square root of 25. Be sure they understand that there are actually two answers: 5 and ?5 because 5^2 is 25 and (-5)^2 is also 25.Next have the students find the square root of 36 and 49. After they arrive at 6 and 7 respectively, have the students make an educated guess for the square root of 40. They should realize that their answer must fall somewhere between 6 and 7. Now it is time to show them the three-step method: guess i(n the first trial), divide, and average. Have them go through the three steps and then keep repeating steps two and three using their previous answer as the guess number until they have a good approximation. For example, to find the square root of 40: Steps First Trial Second Trial Guess 5 6.5 Divide 40/5 = 8 40/6.5 = 6.15 Average (5+8)/2 = 6.5 (6.5+6.15)/2= 6.325 The square root of 40 is 6.32455? Have your students practice this procedure several times until they can flow through the process easily. Answered by : Question: Now that calculators are set up for computing following the order of operations, should we begin to de-emphasize this topic in our curriculum? Asked by: Expert's keyword(s) : order of operations, PEMDAS, calculators Answer : You are correct in saying that even many of the most basic calculators are now able to give the correct answer by following the order of operations or PEMDAS. This is the procedure where we take care of parentheses first, followed by exponents. We next do the multiplication and division portions from left to right, and finally the addition and subtraction from left to right. Years ago, if students input an expression without regard to the order of operations, the calculator would display an incorrect answer.Now students simply need to type in the correct expression, and the calculator will do the rest. This, of course, is very helpful for speed and accuracy in performing arithmetic operations. However, the topic of order of operations still merits emphasis in your curriculum. Students need to know the proper set-up for expressions. When they graduate into using a graphing calculator in upper level mathematics courses, the calculator uses only sets of parentheses. Brackets and braces are not used. Students have to be very careful to input the correct number of parentheses in the proper places following the order or operations in order for the calculator to display the correct solution. If the calculator window displays an error message, the student needs to be able to backtrack through the input screen and make the corrections. Answered by : Question: I am teaching a chapter on exponents, and I have told my students that any number raised to the zero power is automatically one. However, I don't know why this rule does not hold when you raise zero to the zero power. Asked by: Expert's keyword(s) : Zero power rule, exponents Answer :It might be helpful if you explain to your students the reason why any number being raised to the power of zero equates to one. This works because when you divide numbers with same bases, you subtract their exponents. For example, (4^4) / (4^2) = (4^2) or 16. (5^5) / (5^3) = (5^2) or 25. The students should also know that when they divide a numerator by a matching denominator the quotient is one. (6^7) / (6^7) = 1 because the numerator and denominator are the same. However, (6^7) / (6^7) is also equal to (6^0) using the exponent rule for division with same bases. Consequently, (6^0) must equate to one.
Answered by : Question: I am looking for way to teach base number systems to low level high school freshmen. Asked by: Expert's keyword(s) : base number systems, clock arithmetic Answer : First, you can have your students explore how our base ten system actually works. You can use the example of a car odometer. Each wheel has ten digits ranging from 0 through 9. After reaching 9, the number returns to 0 and turns the wheel to its immediate left one place. You can explain to your students that at this point, they have counted ten more and will need to keep record of the number of tens as the car continues to travel. After the tens wheel reaches 9, it too will next go to 0. This action tells the wheel to its immediate left to count another set of 100.When you move on to other bases, introduce your students to clock arithmetic. Here it is easiest to start in clock 12 since they are familiar with time. Show them how you can add or subtract simply by moving through the clock's digits. (For example, 10 plus 4 = 2.) You could try a clock 7 next and have your students add or subtract days of weeks on a calendar. Another tip is to explain to your students that numbers are simply symbols that represent quantities. You cannot put four in my hand; however, you can put four pennies or four cookies in my hand. Then try a base four where you have different symbols: a circle to represent zero, a stick to represent one, a triangle to represent two, and a box to represent three. Have them try simple addition and subtraction problems using these symbols to encourage them to think how different bases systems actually work. Answered by : Question: During the year, I would like to include some short-term and long-term projects that are designed to assess my students' skills. I'm sure I can find project ideas on the Internet, but how do I go about making a rubric? Asked by: Expert's keyword(s) : rubric, assessment, authentic assessment, alternate assessment Answer :A rubric is a set of scoring guidelines used to evaluate performance. It is also designed to give feedback to the individual student. Rubrics can be flexible but generally fall under two categories: holistic and analytic. Holistic rubrics assess performance on a task combining all the criteria together. However, it is often difficult to fit student performance neatly into the categories. Analytic rubrics assess levels of performance for each criterion, which may be easier to use the first time around. If you use only three levels (limited, acceptable, proficient), you won't have to make fine distinctions.
Answered by : Question: Some of our weakest students will be having an extra period of mathematics twice each week. Our math team has been asked to come up with a range of topics that will bolster their skills and bring them on board with the rest of the students. Where should we start? Asked by: Expert's keyword(s) : remedial class, resource class, Standards Answer :You might want to look at the National Council of Teachers of Mathematics Standards for middle schools (www.standards.nctm.org) to assist you in developing an outline of topics. (The topics in the Standards include numbers and operations, algebra, geometry, measurement, data analysis and probability, problem solving, reasoning and proof, communications, connections, and representations.) You can select one topic each month and have the students explore the topics through different means. Discovery learning would be a good approach to hook the students.
Answered by : Question: In our school, I find that the heterogeneous placement of students makes it difficult to plan for the start of school. I usually begin with a test to see where the students are with their basic skills. What do you recommend? Asked by: Expert's keyword(s) : start of year, entry test, skills assessment Answer :There are other ways to distinguish the different skill levels of your students at the beginning of a new year. The quiz or test given to determine how much they remembered from the previous year can be a real turn-off to students, and the result may not be all that reliable. Even the best of your students can become rusty over the summer. Tests at the start of the year may also bring on heightened anxiety levels.
Answered by : Question: EXPERT'S UPDATE: Ways to raise standards while lowering anxiety Asked by: Expert's keyword(s) : raising standards, lowering anxiety, teaching techniques, technology in the classroom Answer :Mathematics teachers on all levels are faced with the same dilemma. They are mandated to raise standards while trying to interest the reluctant learners. Bringing mathematics to higher levels can lead to math anxiety and math avoidance unless the classroom teacher is able to use techniques that improve students' attitude and confidence.
Answered by : Question: EXPERT'S UPDATE: Literature can help clarify math concepts and support mathematical thinking Asked by: Expert's keyword(s) : literature support, language arts support Answer :Books can help students think mathematically. However, math teachers lack the experience of using literature to teach mathematics, and language arts teacher are seldom able to coordinate reading material to support mathematics. In a recent article,* the authors propose a possible solution to this dilemma by using text potential. This allows readers to construct multiple meanings from a single text.
Answered by : Question: EXPERT'S UPDATE: Need for students to learn vocabulary to communicate mathematically Asked by: Expert's keyword(s) : Math communication, verbalizing Answer :For years the National Council of Teachers of Mathematics has expressed the need for students to be able to communicate in mathematical language. This would allow the student to understand the information that is given in a word problem and then discuss the solution both orally and in writing.
Answered by : Question: EXPERT'S UPDATE: The need to develop an algebraic thought process before teaching algebra Asked by: Expert's keyword(s) : Algebraic thought; thinking mathematically, abstract reasoning Answer : The current standards call for all students to learn algebra. At the very least, the teaching of pre-algebra studies is begun in the middle schools. However, many students in secondary school still do not understand algebra thoroughly and avoid taking higher-level mathematics courses later on. It seems that many students cannot make the jump from arithmetic and solving word problems with fundamental algorithms to working with abstraction.In a journal article,* the authors feel that algebraic thinking should be encouraged in the early years. This can be accomplished by providing a pictorial representation of both the known and unknown quantities. Students are taught to look at the sequence of facts, make conjectures, and consider algorithms and processes that they may use for that particular situation. Algebraic reasoning develops when the model approach and verbal problem solving are welded together. This approach allows students to transition from pictorial representations to abstract variables with a better understanding of the algebraic process. *Carter, J, Ferrucci, B, and Yeap, B. (2002). Developing algebraic thinking. Mathematics Teaching, 178, 39-42. Answered by : Question: EXPERT'S UPDATE: New York State Regents Policy on Middle School Education — July, 2003 Asked by: Expert's keyword(s) : NYS Regents Policy, middle school policy Answer :The New York State Board of Regents just issued a policy that concerns middle school faculty. The policy entitled Supporting Young Adolescents Regents Policy Statement on Middle-Level Education July, 2003 can be found at www.emsc.nysed.gov in its entirely. The educational program portion is included below.
Answered by : Question: EXPERT'S UPDATE: Effective Homework Asked by: Expert's keyword(s) : Homework, assignments Answer : Most people agree that mathematics homework is essential to reinforce the lesson that is learned in class. However, when individual students encounter difficulty with their homework assignments, teachers are unable to redo every problem in class the next day because of time constraints. If the majority of students understand the work, the teacher must move forward to cover the curriculum. The troubled student may be using an improper algorithm, and consequently, that method is reinforced at home.Stephen Alexander* feels that homework assignments should complete and support the work that was done in class and should not be an expectation of more of the same. The author believes that students should not be given extremely difficult problems because they can be the root of frustration and failure. Homework assignments that go beyond routine may allow teachers to have different objectives in mind rather than simply performing similar problems. Features of useful homework are the same features that would make it a good classroom activity. The author stresses that teachers should realize that the environment outside the classroom is quite different to the one inside. Help and the required resources may not be readily available. On the other hand, exercises with real-world applications can be given. Assignments should vary according to the purpose they are designed to fulfill. Alexander, Stephen. 2002. Making the most effective use of homework. Mathematics Teaching, 178, 36-39. Answered by : Question: EXPERT'S UPDATE: Helping Students Overcome Math Misconceptions Asked by: Expert's keyword(s) : math misconceptions, teaching strategies Answer : Students across the globe develop similar mathematics misconceptions during their middle school years. In a journal article* a study was undertaken in England to analyze these misconceptions in order to give teachers insight into the way students think. The authors have come up with several teaching strategies, some of which are listed below.1. Draw attention to the mathematical voice by looking for conflicting or ambiguous meanings. Have the students rephrase the question to make the mathematical focus more apparent. 2. Ask students to paraphrase the problem and formulate a plan of action. Often the misconceptions become apparent. 3. Creating cognitive conflict can draw students' attention to inconsistency. 4. Vary a condition of the concept. 5. Use the number line with different forms of a number (2.5 and 2 ½) and extend the number line beyond zero. 6. Ask students to read number sentences to uncover the process conception. *Williams, J. and Ryan, J. (2002). Making connections at key stages 2 and 3. Mathematics Teaching, 178, 8-11. Answered by : Question: EXPERT'S UPDATE: Standards-Based Mathematics in Middle Schools Asked by: Expert's keyword(s) : Standards-based curricula Answer : Standards-based mathematics curricula are intended to motivate, interest, and challenge students to deepen their appreciation and understanding of mathematics. There is an enhanced focus on certain topics promulgated by the National Council of Teachers of Mathematics.In a recent journal article*, the authors outlined the effective path to implementation of such curricula. They pointed out that changing a curriculum is not the same as replacing a textbook. A curriculum change needs administrative support, parental support, and teacher buy-in. There needs to be on-going teaching training and built in time for colleague collaboration. Assessment tools need to be tested and developed. Standards-based curricula can be a wonderful opportunity to enrich the learning experience of our students. Implementation of new curricula is often begun with enthusiasm and promise, but the support and collaboration is not sustained largely due to time and budget constraints. The students suffer as a result. *Bay-Williams, J.M., Reys, B.J., Reys, R. E. ( 2003). Effectively implementing standards-based mathematics curricula in middle schools. Middle School Journal, 34, 36-40. Answered by : Question: Several of my friends have enrolled their children in summer math classes. They feel that this will help their children move into an honors class in September. I wanted my daughter to take the summer off. Did I make the wrong decision? Asked by: Expert's keyword(s) : summer class, honors, advanced class Answer : If schools use a homogeneous grouping, students are generally separated into advanced, moderate, and remedial levels. The idea behind grouping students in this matter is to give the more advanced students a higher level of mathematics or to provide a faster-paced instruction. The remedial level gives the students more support so they will be better prepared for more advanced mathematics as they proceed through the grades.Some parents feel that their child will get more out of the course if he or she is in the advanced or honors class. Others feel that the advanced or honors class looks better on their child's transcript. To insure placement in these special classes, they send their child to special summer classes. Sometimes the extra classes coincide with the child's maturation and the class does actually help the child place in a class that is most appropriate to his or her achievement level. However, more often the child begins to struggle during the school year and either needs a tutor during the year or begins to falter in the course. It is a good idea to let your child have a chance to relax and unwind during the summer. Sometimes we forget that they are still children and need their time to play in order to develop as a whole person. If anything, a few math games or software might prevent your child from becoming too bored during the long summer months. Answered by : Question: My daughter is in an accelerated mathematics class in her middle school. She took the first two semesters of course A this year as an eighth grader. She will do her third semester next year when she enters high school and take the exam in January. It appears that this June's New York Regents examination was a real fiasco. My daughter is very concerned that she will have to take the test after completing the course in a new school and with a new teacher. Can you offer her any comforting words? Asked by: Expert's keyword(s) : Math A, Regents Exam, Standardized test, Test validity Answer : In New York State, passing the Regents Math A examination is one of the requirements for graduation. However, Math A is not a course; it is the name of the examination, which includes selected topics from algebra, geometry, and trigonometry. After the students took the Math A examination this June, a hue and cry was raised after a large percentage of the students failed the test. The commissioner of education sent the test for independent analysis, and upon receiving the results, he pulled both this examination and the test scheduled for August.With the problems that have surfaced, a special commission will be assembled to review the test and make recommendations. Your daughter should feel relieved that she would be taking an examination that will most likely be carefully reviewed for its validity and reliability before it is administered. This should make the examination fair for all students who are well prepared. Your daughter has nothing to fear if she did her work during the past year. Actually going to another school and having another teacher might be an advantage by getting a broader perspective on the material that could be included on the January assessment. During the summer, it would be great if she would practice some problems from the Math A review books that are readily available. Answered by : Question: I have identical twin boys who had done very well in all of their academic subjects in elementary school. However, now that they have completed their first year of middle school, one of my sons seems considerably behind in mathematics. Should I be worried? Asked by: Expert's keyword(s) : twins, self-esteem, remedial summer help Answer : Identical twins make look the same physically, but they develop their own unique talents and interests. In elementary school, where the rudiments of mathematics are taught, your sons could have kept pace with each other. In middle school, mathematics includes much more reasoning and less mechanical work. Consequently, one of your boys may enjoy the challenge of sorting out the pertinent information in word problems, while your other son might find it overwhelming.Another important point to consider is that the learning of mathematics has a developmental component. Again, just because your boys have a similar appearance, they may not be developing at the same rate. Some students cannot understand the concept of abstraction until much later on in their teens. You should explain to your boys, that one may be better in a subject than the other, but somewhere else, the other son will be more successful. It is important that their self-esteem is not harmed at this early age, which could become another barrier as they move on in their studies of mathematics. Another option is to have both boys do remedial work to keep one son on top of the game and the other son can strengthen his skills. You can purchase some educational software that the boys can use over the summer to give both boys extra practice in an enjoyable atmosphere. Answered by : Question: I have been in business for years and now I am seriously considering going into teaching. I would like to teach middle school mathematics. I already have a bachelor's degree and several mathematics courses completed while I was in college. What do I need to become a teacher? Asked by: Expert's keyword(s) : career change, teaching requirements, teaching certification Answer : You should start by contacting your state education office. You will probably be able to find some of the information on the Internet. You should also contact a local college offering education courses to inquire about their application process and to find out how they can help you fulfill the requirements for your teaching credentials. Some states require a master's degree but will allow you to begin teaching while attending school. You will also need to go through a state certification process that includes course work, workshops, examinations and possibly a video.You change of career will take some time but I am sure that you will find the switch to be quite rewarding. Many of our best teachers are second career professionals. These people bring their own wealth of experience with them to the classroom and are able to explain to their students how they used mathematics in their previous occupations. Good luck! Answered by : Question: With the new grading system using rubrics and conversion charts on standardized tests, it seems increasingly more difficult for students to move up to a satisfactory level. Why can't we use a system with stablized and equitable conversion charts? Asked by: Expert's keyword(s) : standardized test scores, rubrics, conversion tables, grading Answer :The scoring of standardized testing has become very complicated. Instructions on mathematics tests will state that the problems are worth a certain amount of points. Then there is the rubric on how these points should be allotted. Finally after the scores are tallied, there is the conversion scale which gives the actual score or placement of the student. I have seen tests where the students need an extraordinary amount of points to move up. Once they move up, the number of points the following year also increases, so they have a good chance of sliding back in the standings. On top of that, the conversion scales change from year to year so neither teacher nor student can predict what grade is needed to "pass." Sometimes the conversion scales arrive days after the test is administered and graded.
Answered by : Question: We would like to help our children with their homework but we sometimes are unable to do the work! Is it possible to get solution manuals so we can work through the problems and then help our children with the work? What else can we do? Asked by: Expert's keyword(s) : Problem solutions, homework help Answer : I had several similar requests just this past week. It is heartening that so many parents are so interested in helping their children succeed. Unfortunately, solution manuals are usually not available until students reach very advanced mathematics areas. Many textbooks today provide selected answers in the back of the books as a help. Unfortunately, as one parent explained that the solution was useless without a working knowledge of the difference between a permutation and a combination as an example.I suggest that you look at the sample problems in the chapter of the particular textbook that your child is using. Another source would be to utilize the numerous homework help sites on the Internet. You might also write a note or email to your child's teacher. Possibly the teacher might be able to provide you with some material that further explains the topic. Answered by : Question: We are trying to get a curriculum integration committee started for the various grades in our school. Do you have any suggestions on where we should start? Asked by: Expert's keyword(s) : Integration, curriculum, projects, exteneded tasks, field trips Answer : Congratulations on your efforts to integrate curriculum in the middle school where students can benefit from practical and hands-on learning while seeing how the material they study in school is from related subject areas. It would be a good idea to begin with an outline of topics that would be covered during the school year. The committee should look for overlaps or places where topics can be connected to various disciplines. You might need to realign the order in which the topics are presented so there can be a smooth link up.Once the course outlines are completed, discuss projects, extended tasks, or class outings that can be assessed by teachers in two or more disciplines. The teachers involved should hone the assignment to make it doable in the allotted time for both the students and teachers involved while not detracting from the interest of the theme. A time line of projects or extended tasks should be developed so students are only working on one at a time and have some breathing space in between. It would be good if your committee can meet regularly throughout the year during the school day so that integration does not become another burden on your busy schedules. During these meetings, assess how the integration projects are working and adjust the scheduling accordingly. At the end of the year, see what worked and what needs to be improved or deleted. Good luck with your new committee! Answered by : Question: I have been going over the review sheets with my daughter to help her prepare for her final examinations. I noticed that she needs to know a number of conversion tables (linear, weight, liquid, etc.). She also needs to know how to convert metric measures. However, when I asked her to approximate the number of centimeters there are in foot, she was totally confused. She stated that they never learned those types of conversion. This seems important in today's world. Why don't the students learn to approximate measures in different systems? Asked by: Expert's keyword(s) : conversion. curriculum, test prep Answer : I would first like to commend you for helping your daughter with her review for her final examinations. Too many parents leave the education of the child entirely up to the school. When it comes time for reviewing for big tests, often students feel overwhelmed and give up even before they begin.You are also an astute parent to recognize that important concepts are missing from the curriculum. Unfortunately, there are lots of holes in our curricula. When scope and sequences are composed, educators often refer to state guidelines, textbook resources, and standardized testing criteria. As a result, we have high school students graduating with honors in mathematics that can find the foci of the various conics but cannot explain the difference between a debit and a credit. Your reference to estimating the number of centimeter in a foot is another excellent example of concepts that students should be taught. With all the international traveling people do today, students should also be taught how to convert money quickly and efficiently. If you are involved with the parent-teacher association, there might be a committee that could bring these missing pieces of learning to the forefront. If not, you might want to bring this up at the next parent-teacher meeting you attend. Good luck! Answered by : Question: We are moving to another state, and we just registered our son in his new school. It seems that he is almost a full year behind the math they are doing. The testing also suggested that he is behind in another subject. The principal suggested we leave him back, but my son became upset with the thought of grade retention. What are your feelings? Asked by: Expert's keyword(s) : retention, changing schools, moving Answer :Changing schools and making new friends are often difficult situations for a middle school student. When these circumstances are compounded by the fact that the student is behind in work through no fault of his or her own, the situation may appear unacceptable to the student. By retaining the student to fill an academic gap, the result may be a social and physical mismatch with the student's new classmates. This could affect the student's self-esteem and motivation.
Answered by : Question: This is the time of year when we need to review textbooks and put in orders for the upcoming school year. What are your recommendations? Asked by: Expert's keyword(s) : textbooks, book selection Answer : If you are planning to change textbooks, you should start well in advance of the needed deadline to purchase books. You do not want to purchase just any book from an over zealous publishing representative. You also need to check which books are available from your state?s approved book list.In making your selection, revisit your scope and sequence to see which topics should be sufficiently covered in the text. Ask the publishers for popular copies of textbooks for review; they are usually sent to schools for free. Next, make a checklist of what is important in a textbook that would serve the appropriate students in your class. Some considerations include the types of problems in each section. Will the same text be used for all levels of students; and if so, are there enough problems for each level? Are the explanations written in language that students can easily understand? Is the text filled with too many distracting pictures? Does the text provide a proper transitioning between grade levels? (If you are purchasing books for more than one grade level, it might be better to choose the same series.) What kind of supplemental material is available? What else is important to the teachers using the textbook? The best assessment of a text comes after you have used the text to teach. Since you want to be sure that you make a good decision based on your own school?s parameters, ask opinions of teachers who are presently using the textbook in other schools. Answered by : Question: Some of my weakest students don't even attempt to solve the word problems on their tests. I'm under the impression that they are having a harder time with the reading than with the mathematics. What should I do? Asked by: Expert's keyword(s) : reading, word problems, problem-solving techniques Answer : Students need to be taught the techniques involved in solving word problems. Before that, they must learn how to read and select the proper information to set up an operation or equation. There are several things that could be done to help students with word problems.To start, try composing small paragraphs with both pertinent and irrelevant information and include some numbers written in numeral form and others spelled out in words. Ask the students what are they actually supposed to find in the problem. Then let them guess at some plausible answers. Next, have them circle the information (numbers, units of measure, etc.) that is relevant to the question. Have the students decide what method they would use to arrive at the solution. Then let them set up the operation or equation and solve. Before they move to the next problem, have the students decide if their answer makes sense. Once, the students are able to extract the right information, have the students try a variety of plans to tackle word problems. For example, they can draw a picture or chart to clarify the information. They can work similar problems or problems with easier numbers. They can examine the problem to see if a pattern exists. The problem-solving books by George Polya might be good resources for middle school teachers interested in teaching problem-solving techniques. When new concepts are taught, students should be encouraged to put a model problem on an index card. They can refer to these cards while doing their assignments and later on, these cards can become flashcards for review. Finally, make word problems a part of almost every lesson. Students will then be able to connect concept and theory with application, and word problems themselves will become less intimidating. This response has been cross-posted in Adolescent Literacy in the Content Areas. Answered by : Question: What are your thoughts on separating the best students and giving them accelerated math courses in middle school? Asked by: Expert's keyword(s) : accelerated students, honors classes, course pacing, skipped grades Answer : It is a wonderful idea to challenge students with more difficult problems or enrich their learning by giving them a broader view of the theory, concept, or application being taught. You never want a student being so bored with the material presented in class that he or she begins to shed the natural curiosity to learn. If it is possible, these accelerated students could be taken out of class and be given projects or more difficult material to tackle. If this cannot be done, these accelerated students can be given more difficult assignments and assessments to work on in their regular class.In my opinion, these accelerated students may be placed in the next higher level mathematics class by increasing the pace of their courses. However, courses should not be skipped as these students will have gaps in their learning. Additionally, it is not wise to pace the courses too quickly. Part of the process of learning mathematics is developmental, and one cannot rush Mother Nature. If pushed too quickly ahead, these students, who were once promising mathematics students, could become frustrated and lose their self-confidence. Answered by : Question: The schools' report card results were just published in the newspaper. My child unfortunately is enrolled in a district with low math and verbal scores. We can't afford private schools. What can we do? Asked by: Expert's keyword(s) : school report cards, standardized test results, academic success Answer :School report cards indicate the results of the schools' pass rate on a variety of standardized tests. This report card does not indicate the individual scores of students. Consequently, you do not know how your child did on the standardized tests for his or her grade level.
Answered by : Question: I would like to use manipulatives to help the students in my resource room better understand the topics that are taught in their mathematics class. However, we are on a tight budget. Can you recommend anything that I can use? Asked by: Expert's keyword(s) : interactive activities, Internet activities, manipulatives Answer :Having students see why something works helps them to better understand the underlying concepts. Of course, mathematics manipulatives have been around a long time for this purpose. There is also great interactive software, but if you are on a budget, this may also be prohibitive.
Answered by : Question: Should students lose points for omitting the units of measure for their answers? Asked by: Expert's keyword(s) : units of measures, solutions, credit Answer : Having the students label their answers with the proper units of measure seems to have taken the backseat in recent years. Even on most standardized tests, students are only required to put down the correct numerical answer for the short answer sections. However, units of measure are very important. Students need to understand when they are working with linear measures, such as perimeters, that their answers are also linear. When they are finding areas, their answers are in square measures; when finding volumes, their answers are in cubic measures. Additionally, students seem to have lost the skill of converting units of measure. Many do not know how many feet are in a mile or how many pints in a gallon.Teachers should stress the importance of putting the units of measure along with their numerical answers. If the students omit the units, the teacher can fill them in on the tests in red. When students, answer questions orally or go to the blackboard, their answers should only be accepted if labeled correctly. I believe reinforcement such as this works better than taking points off on a test. Another way to stress the importance of the units of measure is to give the students word problems where proper conversion is necessary in order to attain the correct solution. This will sharpen their thinking process during problem solving. Answered by : Question: What is a webquest and how can it be used in the middle school classroom? Asked by: Expert's keyword(s) : webquest, math project, cooperative learning activity Answer : A webquest is an activity in which students seek some or all of the information needed to complete a particular activity from resources found on the Internet. The webquest includes an introduction detailing the problem and the real-world application of a concept or theory taught in class. The task is then defined. Students are next given a list of sources or sites to assist them in obtaining the necessary information, although they are not limited to accessing these sites. The students are then instructed on the media that may be used to present their findings. Finally, the webquest conclusion informs the students what they will be learning in such a way to pique their interest.For the technology novice, creating a webquest may require a lot of time and patience. However, once made, the webquest may be used each succeeding year. The sharing of webquest projects is highly recommended among faculty. In addition, ready-made webquests can be found on the Internet and can be customized accordingly. Webquests are ideal as projects that are used to reinforce a unit in middle school mathematics. For example, students may be asked to construct their ideal house involving area and perimeter formulas. The downside is that a webquest takes several class periods to complete, and if students are absent during any of these sessions, the project becomes very fragmented. Answered by : Question: In New York, students are required to take the Regents A math examination. Some schools begin teaching the topics as early as seventh grade, while others wait until the ninth. In a school where new students transfer in throughout the entire school year, what is the best way to make sure that students have the proper background? Asked by: Expert's keyword(s) : Math A, exit exam, scope and sequence Answer : The Regents A examination combines topics from algebra and geometry that are generally taught in high school with topics including measurement, sets, and percentage that are generally covered in middle school. A student may take this examination in any grade from eighth grade on but needs to pass the test at some point in order to fulfill one of the New York State graduation requirements.Many schools have developed a scope and sequence to insure that students will be taught all the material before sitting for the examination. However, there is no uniform curriculum, and consequently as students transfer between schools, they may have missed topics, resulting in learning gaps. It would be beneficial to both the teacher and the student if a checklist could be created that includes all of the topics under the key ideas covered on the test. As the topics are taught in class, the checklist would reflect the material that was presented. This checklist could be included with the student's transcript when the student changes schools. Answered by : Question: We have been advised not to use multiple choice questions on the mathematics test that we make up. What is your opinion of these tests? Asked by: Expert's keyword(s) : multiple-choice, test questions Answer : Multiple-choice tests are often used by teachers because of their ease in grading. However, any student can easily select the incorrect answer just because of a minor mechanical error or because they did not carry out their solution to completion. There is no partial credit on these tests; the answers are completely right or completely wrong. Additionally, some students just select an answer randomly rather than work through the problem.On the other hand, most standardized tests have multiple-choice questions, and students should have the opportunity to practice answering these question types. They need to be able to recognize the distractor and become skilled at eliminating the impractical choices. If multiple-choice questions are left only for a portion of the final examination, students do not get sufficient experience working with these problem types. It might be better to intersperse some multiple choice questions on several of your class tests if this format is acceptable with your supervisor. Answered by : Question: Should parents help middle school students with their homework? Asked by: Expert's keyword(s) : homework, parental help Answer : Parents should help their middle school children with their homework. This can be done in a number of ways. First, they should provide a regular time slot and quiet space where the homework should be done. They should provide the proper supplies: pencils, pens, loose leaf, etc. Then depending on the age and maturity of the child, the parent should spend time with the student and answer questions the child may have with the assignment. The supervision should include making sure that the assignment is completed.If a parent sees that the child is doing the homework incorrectly, the parent should help the child do the work correctly. Otherwise the incorrect mathematical procedure might be reinforced during the entire assignment. The teacher may not have the time to correct all of the homework each day. Possibly, only the correct answers may be given the following day in class. The student might mark the problems as incorrect, but never be given the explanation as to what was wrong. Answered by : Question: What is the best way for schools to provide remediation for their students? Asked by: Expert's keyword(s) : math help, labs, remediation Answer : It is common to have students need extra help in mathematics. Some might not understand the concepts when they are first presented. Other students might be absent and need to fill in the gaps. Still others need to go over their mistakes on tests so that the mistakes are not repeated in the future.Under the ideal situation, a math lab is built into the students' schedules where they can get extra help during the school day without missing class time from one of the other subjects. However, these labs add to an already packed schedule, and consequently math help labs are often offered after school. Some schools even have math help labs on Saturdays. Whenever the math labs are held, it is vital that an experienced mathematics teacher is the instructor. Answered by : Question: To what extent should mathematics examinations (midterms and finals) be given in the middle school? Asked by: Expert's keyword(s) : examinations, midterms, finals Answer : Schools administer midterm and final examinations to assess student achievement level on the material that was taught during the prior months. There are other benefits to administering these tests. Middle school students do not realize that the mathematics concepts that they are learning serve as building blocks for more advanced courses. Examinations give them an immediate purpose for learning. They understand that their score on the examination may be one of the determining factors for promotion to the next grade level. Consequently, these examinations encourage the students to review and integrate material to insure their success. These examinations also teach students how to prepare and take cumulative examinations. This learning experience will become useful during their senior high school and college years.Middle school examinations should not be stressful situations for students. Teachers should review topics that will be tested, and students should practice the different types of questions that will be included on the examination. The level of difficulty should be established so that all of students are able to pass the examination. Students should not be put under undo time pressure and shoudl be allowed sufficient time to complete the examination. Students should also have the opportunity to review the corrected examination so they can learn from their mistakes. Answered by : Question: Many of my colleagues say that their students have math tutors to help them with their homework or to teach them the new lesson before it is presented in class. My students are unable to afford private tutors. How do you feel about middle school students being tutored privately? Asked by: Expert's keyword(s) : tutor, resource room Answer : Many parents are under the misconception that their child needs to get an A in mathematics in middle school so that they will be guaranteed a seat in the Advanced Placement courses in high school. As soon as their child's marks begin to slip, they feel it is time to hire a tutor. Other parents take the preventive approach. They believe if their tutor stays one step ahead of the classroom teacher, their child will certainly understand the material when it is presented in class and consequently score high on assessments. Unfortunately, these parents seem to be more concerned with the actual grade than the importance of learning mathematics as the future building blocks for senior high school courses. They also do not realize that learning mathematics has a developmental component that is integral to learning the subject.Tutors would be helpful in situations where students have missed a number of classes, and it would be impractical for the teacher to instruct the child on all of the missing topics. Tutors are also helpful in situations where the child is consistently performing poorly on all assessments. By sitting down one on one with the student, the tutor may discover a learning disability (for example, dyslexia ) that would go unnoticed by a teacher with a large class. The student could then seek the help that is needed from a learning specialist. Ideally, the school's resource room should be a place to go for remediation. However, the resource teachers are expected to be masters of all subject areas, and often cannot pinpoint the problem that is causing the student to have difficulty in mathematics. Answered by : Question: Is it better to group students heterogeneously or homogenously in middle school mathematics classes? Asked by: Expert's keyword(s) : homogenous; heterogeneous; tracking Answer : Many schools begin tracking their mathematics classes around sixth or seventh grades. The students are sorted into the high-achieving, middle-achieving, and low-achieving sections. The teacher can then vary both the intensity and the pace of the course according to the track of students. As a result, the high-achievers can really be challenged and compete with each other as they grow in their ability. The low-achieving students are placed in a situation where they feel they can finally succeed in mathematics.However, the problem with tracking is that it is rare that the students in the lower tracks can ever catch up and take the more demanding mathematics courses. Since learning mathematics has a developmental factor, these students may have had a slow start but can later assume more challenging work. Also, as the work becomes increasingly more difficult more and more students are relegated to the lower tracks. Consequently the original tracks take on the form of a pyramid. If it is possible, it might be best to keep students in heterogeneous classes in middle school and provide special resource work for both the remedial and high-achieving students. Answered by : Question: Many students are required to take city or state examinations in mathematics at the end of the school year. How much classroom time should the teacher use to prepare the students for these tests? Asked by: Expert's keyword(s) : end of year exams, finals, teaching to test, standardized tests Answer : It would be ideal if end of year examinations were based on the curriculum that was taught throughout that year. The teacher would then present new topics, review old ones, and integrate all the material whenever possible during the school year. Several weeks before the test, the material should be reviewed in entirety, and the students should be given questions in the same format that will appear on the examination; for example, multiple choice, short answer, and qualitative comparison. In addition, the teacher should become very familiar with the test itself and the accompanying rubric and explain the expectations to the students. I have seen some peculiar rules lately. On one test where the guess-and-check method is permitted, the student must show work leading to at least two incorrect answers before arriving at the correct solution in order to receive full credit. It is important that students are made aware of other factors that can deflect from their true mathematics score. Will the test be answered in pencil or pen? Are students penalized more for incorrect answers than answers left blank? Are calculators permitted? Is partial credit allowed?Unfortunately, end of year examinations are not always based on the material taught during the year. School districts may adopt their own curriculum for the different grades, but the city or state includes different concepts on the test. It is still expected that the students do well, and teachers are held accountable for the scores. This becomes a large problem when the educrats are out of touch with the practitioners. In the end, it is the students that suffer. Answered by : Question: While I believe in guiding students through explorations to discover mathematics, I find it difficult to include exploration, clarification of concepts, guided practice, going over homework, informal and formal assessment, and reteaching concepts in the alloted time. I would rather increase the depth of a student's understanding, rather than the breadth of their exposure, but state testing covers so many topics, I find myself doing primarily direct teaching so I can get through all the material. I think I was a better teacher, and my students better off before all the high stakes testing. How can we fit it all in? Asked by: Expert's keyword(s) : time management, high stakes testing, discovery learning, direct teaching Answer : I totally agree with you. The high stakes testing of today has taken the pleasure out of planning effective teaching. In an effort to leave no child behind, local and state governments have imposed a plethora of standardized tests on students of various grades levels. Testing companies, that are far removed from the classroom where their products are administered, create these tests. To teach to the test makes teaching mathematics a chore. Do you quicken the pace so that all the topics on the test are covered, or do you slow down to make certain that all students understand the topic? If you teach solely to the test and your students achieve good scores, does this make them better mathematics students? While limited standardized testing is a good measurement of students' growth in learning, we are now at a point where testing is out of control.Today's mathematics teacher must become a juggler. How can you make learning mathematics exciting through discovery and exploration and still teach, review, and prepare for tests? In addition, many teachers are asked to accomplish all these things in a 40-minute period each day. Since many teachers' jobs are based on their students' tests scores, it is critical that you follow the curriculum imposed upon you. However, when you can get a breath of fresh air (e. g., the day before vacations) it would be wonderful if you can create lessons that make teaching mathematics for you and learning mathematics for your students a memorable educational experience. There are some software programs that provide activities that allow students to discover and explore mathematical concepts and can easily fit into the structured school period. Answered by : Question: I am designing a workshop on how to use cooperative learning in the mathematics classroom. Have you used this strategy in the classroom and what would you recommend to help teachers get started with this strategy in their classroom? Asked by: Expert's keyword(s) : cooperative learning pointers; workshop suggestions,discovery learning Answer : Cooperative learning could be a wonderful opportunity to offer students a very different learning experience. If a student learns by doing, more senses are involved in the learning process. The student not only learns the concept, but the practical application and can transition the concept to other areas. In addition, students sharpen their social skills in group activities.I have used cooperative learning many times with a range of results. One problem occurs with the grouping of students. If left to the students to decide, friends will work with friends and often more socializing takes place than learning. If homogenous grouping is used, the bright students work diligently to complete the project, but the slower students seem to lose interest and consequently lose the lesson. If heterogeneous grouping is used, often the bright student will become the dominant person in the group, while the others become hitchhikers, sitting back while the work is completed for them. Small groups are best; however, you need to keep all the students on task while you are walking around and assisting the individual groups. Another problem comes if assessment is involved. If the group is assessed as a whole, again the bright student will take over so as not to sacrifice his or her grade. However, individual assessment defeats the purpose of cooperative learning. If students are given roles, such as facilitator and recorder, the roles become artificial, and the students spend more time trying to do their assigned jobs. Cooperative learning also requires a lot of time, a luxury that does not come with 40-minute periods. Most important, when using cooperative learning, teachers must be there to guide their students to the proper conclusion. Students can easily pick up misinformation that would have to be unlearned. Answered by : Question: How much mathematics homework should be given to middle school students? Asked by: Expert's keyword(s) : homework, assignments Answer : Students should be given homework in mathematics consistently to strengthen their skills. In the ideal situation, the teacher would give a few problems to the students to complete in class before assigning similar homework problems. Without classroom practice, the students could be making errors at home and consequently reinforcing the incorrect algorithm.The number of problems should be limited so that the students are able to complete the assignment in a reasonable length of time. Just as important, there should be a variety of problems that are meant to reinforce the class lesson. For example, when working with percentages, some problems could require the students to compute the rate, while others ask for the original amount. Problems should also have varying levels of difficulty that include those that can easily be completed and more challenging ones that require additional thought. It is also good to include some problems from past lessons for review. Homework should always be gone over in class to make the assignment meaningful and to assure that the students are doing the work properly. Answered by : Question: For several years, our district used the Math in Context curriculum, although the student scores never showed improvement. Now it is rumored that the curriculum will again be changed. Why is a set curriculum put in place in all schools, although the students come from diverse backgrounds? What happens to those students who just completed a curriculum that was found to be lacking? Asked by: Expert's keyword(s) : curriculum, math in context Answer : Math in Context is a method of teaching mathematics through application. Students are expected to learn theory through doing and then transition their skills to other practical situations. Often middle school students are not able to grasp a mathematical concept or theory without a direct teaching approach. They are then better able to use the concept or theory in practical situations.Within school systems, curriculum specialists often develop a scope and sequence as a guide for a systematic presentation of material that is rich, rigorous, and recursive. After the scope and sequence is in place, someone finds textbooks and other resource material that support this orderly progression of learning. At this stage, it is very important to involve teachers since teachers are aware of the specific needs of their students. However, some curriculum programs are purchased by a central office, and all the schools in the district are mandated to follow the programs in a lock-step fashion. The program is then kept for several years in hope that standardized test scores go up as teachers become more familiar with the program. When scores fail to improve, the program is replaced. Unfortunately, during these transitions, students are continually moving through the educational system. Consequently, the program in place does not always best serve these students, and they may experience some difficulty with the subject matter in the future. Ideally, teachers should be able to use the methodology and the material that works best for their students when covering a standardized outline of curriculum topics. Answered by : Question: Do students in middle school learn mathematics better through cooperative learning or the traditional chalk and talk method? Asked by: Expert's keyword(s) : cooperative learning, discovery learning, methodologies Answer : Actually the chalk and talk method is extreme and should always be avoided. It would be difficult for students of any age to remain focused for an extended length of time if they were not expected to participate in their learning process.In middle school, mathematics is best taught through a variety of methodologies. Cooperative learning and discovery learning can precede a lesson on a new mathematical concept or theory as long as the students are guided so that they arrive at the proper solution. Often students are left to their own and come up with an incorrect or incomplete solution, which unfortunately is the one that stays with them. Learning projects are also good after a lesson is taught to reinforce the material through hands-on problem-solving exercises. On the other hand, having students copy rules into their notebooks can be a strong tool for future reference and review. The ideal learning environment balances a variety of teaching methods with even pacing of the curriculum while stimulating the natural curiosity of the student. Answered by : Question: Is the teaching of math and science in the middle school best done by one teacher or by two teachers, each specialized in one subject? What are the current practices in current middle schools around the world? Asked by: Expert's keyword(s) : mathematics, science, teacher training Answer : It would be ideal if a teacher with expertise in both mathematics and science were to teach students in middle schools. However, teacher education programs and state teacher certification requirements usually preclude teachers from crossing over disciplines. To do so, prospective teachers would be required to take extra courses and go through extra testing, which trickles down to extra expenses for the teacher-to-be. Teachers in middle schools usually major in education and minor in one discipline.Unfortunately, in the United States, we have a more pressing problem than trying to integrate disciplines on the middle school level. The problem is to get and retain qualified teachers in either the mathematics or science area. According to Ingersoll (Educational Researcher 28(2):26-37,1999), about one-half of the seventh grade mathematics teachers do not have the equivalent of a minor in mathematics and about one-third of middle school science teachers do not have the equivalent of a minor in any of the sciences. This problem seems to be growing internationally. In England, prospective teachers are offered incentives to enter the mathematics or science fields. In China, new teachers are given intensive induction and mentoring to keep new teachers from leaving the profession. In Japan and Germany, new teachers have a long-term structured apprenticeship. Answered by : Question: Parents sometimes express concerns when their middle school children are allowed to use calculators and/or computer software in their math classes. They worry that children can become dependent on these tools and fail to develop their own computation skills. How would you respond to parents who raise this concern? Asked by: Expert's keyword(s) : calculator, software, technology. computation skills Answer : Calculators and computer software bring excitement to today's middle school mathematics classroom. However, technology should not override the need for learning the basics. Students still need to learn their multiplication tables. They should be able to perform arithmetic operations involving fractions and decimals. On the other hand, software can aid in strengthening basic skills by challenging students to sharpen their speed and accuracy in fun-filled situations.Calculators and software are also wonderful tools for teaching problem-solving techniques. Calculators allow students to focus on the strategy of problem solving while simplifying the arithmetic operation process. Calculators also afford the student a quick check for their solutions. Software provides entertaining problem-solving situations utilizing multi-sensual stimulation to pique interest and enhance learning. It is important that a balance be achieved in order to keep technology usage at an appropriate level in the middle school classroom. One way is to give assessments that have calculator and non-calculator portions. Answered by : |
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