Waxing eloquently on the basic importance of Mathematics in human life, Roger Bacon (1214-1294), an English Franciscan friar, philosopher, scientist and scholar of the 13th century, once stated: "Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world." And the ingenuity of his statement is there before us to see, in this Internet era.
However, mathematics, in itself, has all the ingredients that make it a universal language shared by all human beings irrespective of culture, religion, or gender. Pi is always 3.14159 regardless of where we are. Similarly, the elemental mathematical processes (like addition, subtraction, etc.) never get changed due to a change in the location or for any other reason whatsoever. These all say about the close intertwining of mathematics in our daily life.
At a time when even a common man is being increasingly dependent upon the application of science and technology in the day-to-day activities of life, the role of mathematics has undoubtedly been redefined. Right from getting up in early hours of the day to the ringing of an alarm, to wait for the counts of whistles of the cooker, to exchange currency at a ticket outlet while availing a public conveyance, almost every next moment we do the simple calculations at the back of our mind. Of course, these are all done pretty unconsciously without a thought being spared for the use of mathematics on all such occasions.
Reading time on a watch, rounding a date on a calendar, checking up the mileage of your car, halting at the filling station, attending to a roll call at school, getting scores in the class exams, scoring in a game, betting on a horse race, preparing a recipe in the kitchen, - the list is just endless if one goes on to note down the situations when our computational skill, or more specifically, simple mathematics comes to play a role. It scares us to certain extent to think of a life without any knowledge of calculation or computation, or in other words mathematics. In many a case, lack of a formal education hardly affects gaining a relative mastery in these computational skills which are so essential in our daily life.
At a psychological level, exposure to mathematics helps in developing an analytic mind and assists in better organization of ideas and accurate expression of thoughts. At a more general level, far away from dealing with the higher mathematical concepts, the importance of mathematics for a common man is underpinned whenever he visits banks, shopping malls, railways, post offices, insurance companies, or deals with transport, business transactions, imports and exports, trade and commerce, and the ilk. Even when we think of role of mathematics in our recreational activities, we surprisingly have a list that runs quite long: video games, computer games, puzzles, riddles, and so on.
It ensues from the above discussion that a modern life style seems completely handicapped and at times, highly improbable, in the absence of mathematics. For, unless we are well versed in the language of numbers, we would find it difficult to reach at important decisions and perform everyday tasks. Be it to shop wisely, or buy the right insurance, or refashion a home within a budget, knowledge of mathematics holds the key, and hence, barely necessary.
from http://www.articlesbase.com/k-12-education-articles/mathematics-in-daily-life-390556.html
Monday, September 27, 2010
Learning Mathematics Through Mastery of Concepts
There are many ways to learn a subject. Different people has different learning style. However, regardless of learning style, there are two particular ways to learn knowledge within any style. The first way is memory method. This way of learning focuses on remembering facts and details, and recalling them when required. The second method is to understand the underlying concepts of the facts. It is the "what" versus the "how" and "why". Either way has their merits and demerits. It depends on the education level and complexity of the subject matters. Sometimes, a hybrid of both are necessary to reap the best possible results, especially in learning mathematics.
For elementary mathematics level, the amount of mathematical facts and concepts to be learned are marginal, and only serves to lay foundation for further advancement into higher mathematics learning. At the basic level, memory method of learning may be acceptable and manageable. But how about maintaining this way when one progresses into higher mathematics studies?
At a higher level of mathematics learning, the learning taxonomy moves into the application level and beyond. Mastery of concepts becomes an important factor in analyzing and solving more complex mathematics questions. Mathematical equations and expressions get more integrated with detailed concepts. Pure memory will not be able to extract out the true meaning of these equations and expressions. A few mathematical tools may be required to solve a mathematics problem. This combination of solving methods and concepts rendered pure memory way of learning mathematics unacceptable. The scope to cover all possible combinations of solving tools and questions is far too wide to manage. Staying firm with this facts-remembering method will only cause the performance and outcome to dwindle low. This will reduce the motivation to study and may decline towards the fearful mathematics anxiety situation.
Studying mathematics is different from other subjects in that a unique language is used to present mathematical meaning. Symbols and variables are used to form relationships and conditions. Many in-depth concepts are embedded into these few symbols and variables, including mathematical operators. True mastery of the mathematics concepts is therefore needed. However, it should not be said that memorising facts are completely useless at the higher level of mathematics education. It still serves its purpose in that they are the basic ingredients for the brain to link up with the solving methods and conceptual approach. It is these linkages or bondings that retain and strengthen the acquired knowledge longer. This is real mastery of the subject and its topics. A hybid of both learning methods is therefore necessary to achieve a level of acceptable comprehension with more emphasis going towards mastery of concepts.
In conclusion, time causes facts to fade off. Memorising facts will therefore not last long. A better learning platform is to focus on relationship and concept mastery. Linking facts to solving strategy becomes a useful problem-solving skill and has the benefit that it will last longer. Understanding the "why" and "how" is better when they complement the "what". Resistance to studying advance mathematics will then be lowered when the correct method of learning mathematics through mastery of concepts is applied.
from http://www.articlesbase.com/science-articles/learning-mathematics-through-mastery-of-concepts-246250.html
For elementary mathematics level, the amount of mathematical facts and concepts to be learned are marginal, and only serves to lay foundation for further advancement into higher mathematics learning. At the basic level, memory method of learning may be acceptable and manageable. But how about maintaining this way when one progresses into higher mathematics studies?
At a higher level of mathematics learning, the learning taxonomy moves into the application level and beyond. Mastery of concepts becomes an important factor in analyzing and solving more complex mathematics questions. Mathematical equations and expressions get more integrated with detailed concepts. Pure memory will not be able to extract out the true meaning of these equations and expressions. A few mathematical tools may be required to solve a mathematics problem. This combination of solving methods and concepts rendered pure memory way of learning mathematics unacceptable. The scope to cover all possible combinations of solving tools and questions is far too wide to manage. Staying firm with this facts-remembering method will only cause the performance and outcome to dwindle low. This will reduce the motivation to study and may decline towards the fearful mathematics anxiety situation.
Studying mathematics is different from other subjects in that a unique language is used to present mathematical meaning. Symbols and variables are used to form relationships and conditions. Many in-depth concepts are embedded into these few symbols and variables, including mathematical operators. True mastery of the mathematics concepts is therefore needed. However, it should not be said that memorising facts are completely useless at the higher level of mathematics education. It still serves its purpose in that they are the basic ingredients for the brain to link up with the solving methods and conceptual approach. It is these linkages or bondings that retain and strengthen the acquired knowledge longer. This is real mastery of the subject and its topics. A hybid of both learning methods is therefore necessary to achieve a level of acceptable comprehension with more emphasis going towards mastery of concepts.
In conclusion, time causes facts to fade off. Memorising facts will therefore not last long. A better learning platform is to focus on relationship and concept mastery. Linking facts to solving strategy becomes a useful problem-solving skill and has the benefit that it will last longer. Understanding the "why" and "how" is better when they complement the "what". Resistance to studying advance mathematics will then be lowered when the correct method of learning mathematics through mastery of concepts is applied.
from http://www.articlesbase.com/science-articles/learning-mathematics-through-mastery-of-concepts-246250.html
Thursday, September 16, 2010
Manipulatives: The Missing Link in High School Math
by Marilyn Curtain- Phillips, M. Ed.
The thrill of achievement comes from overcoming adversity in the accomplishment of an important goal ~ UnknownAlgebra, Geometry and Trigonometry can be challenging and complex subjects for many high school students in the United States. These subjects are usually taught using textbooks, workbooks and examinations. While these resources are essential in developing learning skills in math, it doesn’t encourage problem-solving skills and retention. It is not surprising to learn many students view math as boring, difficult and irrelevant, rather than fun and interesting. The scope of what it means to be successful in mathematics expands beyond procedural knowledge and skill-acquisition to include sense-making and conceptual understanding (Buckley 2004). An integrated approach is needed to motivate students to learn math in a more relaxed environment, thus eliminating persistent math anxiety, the lack of math application skills and poor standardized test performance. The Third International Mathematics and Science Study (TIMSS) tested the students of 41 nations. Children in the United States were among the leaders in the fourth grade assessments, but by high school graduation, they were almost last. High school math should be taught in a way in which students can grasp and understand skills. Manipulatives are visual objects that help illustrate mathematical relationships and applications. These resources are used primarily in elementary schools and somewhat in middle schools. Manipulatives are valuable resources for accelerating and deepening students understanding of math, yet its use is almost non-existent in high school. Marilyn Burns, Creator of Math Solutions has "used manipulative materials at all levels for 30 years." In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels (Bellonio), yet many high school teachers are reluctant to use this type of resource.Almost all mathematics-teaching activities take place at the abstract level (Sharma, 1997). According to Sharma, students have a tendency to forget when taught only at the abstract stage. Thus students become frustrated because mastery was never fully attained. Students will begin to have difficulty in learning mathematics. The results in failure will cause many students to develop a fear of mathematics (Sharma, 1997). Students’ attitudes towards any curriculum area can be related to their achievement in ways that reinforce higher or lower performance (timss.org). Full 88% of Bill and Melinda Gates Foundations’ survey respondents said they had passing grades in high school. Asked to name the reasons they had left school, more respondents named boredom than struggles with coursework. Over 1 million students drop out of school each year. That includes nearly half of all African Americans, Hispanics, and Native Americans who fail to graduate from public high school with their class. Leaving many of them with a host of poor outcomes to follow, from low lifetime earnings to high incarceration rates to a high likelihood that their children will drop out of high school thus eliminating the cycle (Thornburg, 2006).
Sharma (1997) feels there are six levels of mastery of mathematical concepts: intuitive, concrete, representation (pictorial), abstract, applications and communication. According to Sharma, ideally, each mathematics concept should be introduced beginning at the communication level. Manipulatives will teach concrete understanding to the abstract math process, especially when the student may not understand the concept behind the skill. When students develop a concrete understanding of math skills, then they are more likely to perform that math skill and understand math concepts at the abstract level. Manipulatives can make math concepts come alive. According to Spikell (1993), most learners whether adults or children, will master mathematical concepts and skills more readily, if they are presented first in concrete, pictorial and symbols. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for learners to understand the abstractions they represent (Spikell, 1993). Students will be able to practice and demonstrate mastery using concrete objects. Manipulatives appeal to the learning style of kinesthetic learners because they actually touch the objects. Pictures appeal visually for visual/spatial learners. "Visualization is the natural way one begins to think. Before words, images emerge" (Sharma, 1987). "Almost every mathematics idea, except simple arithmetic facts, consists of three components: linguistic, conceptual and skill/procedural" (Sharma, 1987). Therefore manipulatives offer benefits to a variety of learning styles. They also provide a change from the textbook for mathematical/logical learners.
Games with manipulatives are also valuable with helping students to apply what they learned to the real world, as well as provide a means to improve their math skills interactively. Using board games and card games along with cooperative learning are ways that students can become involved in a positive mathematical environment. Games are highly motivational to students and can be used effectively to practice specific skills. "Using games in the classroom and at home will maximize students' problem-solving competence, ability to communicate and reason mathematically, perception of the value of mathematics, and self-confidence in their ability to apply mathematical knowledge to new situations." Cooperative groups provide students a chance to exchange ideas, to ask questions freely, to explain to one another, to clarify ideas in meaningful ways and to express feelings about their learning. These skills acquired at an early age will be greatly beneficial throughout their adult working life.
Many teachers feel as though they do not know how to teach using manipulatives and therefore, hesitate to use them in the classroom. Many math teachers, who attended college more than ten years ago, were usually taught on the abstract level, textbook, pencil and paper, all through their school days. There are classes and workshops for teachers to learn how to teach using manipulatives. The companies that make the manipulatives also provide books and pamphlets on ways the material can be used. There are articles on using manipulatives in mathematics teaching journal such as the National Council of Teachers of Mathematics’ Arithmetic Teacher magazine. Manipulatives help relieve boredom in children allowing them to explore and use their imagination.
Many manipulatives are inexpensive and can be everyday objects. Money, 2 –color counters, calculators, rulers, dominoes, playing cards, button and number cubes are a few of the commonly available manipulatives that can successfully be used in the classroom. These manipulatives can be used to teach such concepts such as angles, area, decimals, factoring, estimation, fractions, measurement, counting, percent, prime numbers, probability, geometry and whole numbers.
There are companies, which specialize in manipulatives that can be ordered from a catalog or online. No matter where a school is located, materials can be made available through the mail. These companies also make available to teachers such manipulatives as tangrams, pattern blocks, fraction towers, geoboards, algebra tiles, Cuisenaire rods, miras and polyhedral models.
When I initially introduced pattern blocks to my Geometry students, I was met with opposition and disbelief. It was a consensus that these students had not used pattern blocks since elementary school. However, many students actually began to explore and build figures with these blocks. I gave them this little bit of play time before taking on the task for that particular lesson, which was angle measurement of polygons. We must remember, everyone likes to play games even adults, look at our various hobbies, such as bowling, tennis, golf, art, race cars, chess, playing cards, etc. This holds true for high school students, they still enjoy playing games. Manipulatives require a great deal of prior planning and organization. But considering the many benefits that manipulatives offer, it is well worth the effort. Students will find math more attainable, fun and exciting, hopefully returning them to the prior successes of their early math experiences.
REFERENCES:
Buckley, L. A. (2004, Oct) Course Taking and Equity: The Efforts of One High School Mathematics Department Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF> Retrieved 2008-01-12 from <http://www.allacademic.com/meta/p117666_index.html>Burns, M. How to make the most of math manipulatives – A fresh look at getting students’ heads — and hands! — around math concepts. Retrieved 2008-01-1-10 from <http://teacher.scholastic.com/lessonrepro/lessonplans/instructor/burns.htm>
Sharma, M. (1987). How to take a child from concrete to abstract. Math Note book, 5, 8-10
Sharma, M. (1997, July). Improving mathematics instruction for all. Fourth Lecture in Colloquium "Improving Schools from Within: Your Role. Pp. 2-12
Spikell, M. (1993). Teaching mathematics with manipulatives: A resource of activities for the K-12 teacher. New York: Allyn and Bacon.
Third International Mathematics and Science Study (TIMSS), Institute of Educational Sciences, United States Department of Education
Thornburg, N. (2006), April 7), Dropout Nation, Time Magazine
TIMSS 1999 International Mathematics Report. Findings from IEA's Repeat of the Third International Mathematics and Science Study at the Eighth Grade. Ina V.S. Mullis, Michael O. Martin, Eugenio J. Gonzalez, Kelvin D. Gregory, Robert A. Garden, Kathleen M. O'Connor, Steven J. Chrostowski, Teresa A. Smith. Retrieved 2008-01-12 from <http://timss.bc.edu/timss1999i/pdf/T99i_Math_TOC.pdf>
Yale-New Haven Teachers Institute Home Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete by Judith L. Bellonio Contents of Curriculum Unit 01.06.12: Narrative Math Anxiety Types of ... Retrieved 2008-01-12 from <http://www.yale.edu/ynhti/curriculum/units/2001/6/01.06.12.x.html>
Marilyn Curtain-Phillips teaches high school mathematics and is the author of Math Attack: How to Reduce Math Anxiety in the Classroom, at Work and in Everyday Personal Use,(1999), $15.00. You can purchase her book through our Online Bookstore.
Algebra, Present and Past
What is Algebra? Algebra is one of the many branches of mathematics. This branch of math utilizes mathematical equations to describe the relationships between any two things that may change over a period of time. Whenever a math equation has to represent something that fluctuates and does not stay the same, letters or symbols are often used to represent the varying quantity. This letter or symbol is called a variable because it varies.
The very start of modern algebra began with the ancient Babylonians. They were the very first people who used an advanced math system for their time. This math used an algebraic system to do calculations, and it enabled the Babylonians to utilize formulas and calculate the answer using unknown values for a problem class that would be solved in the modern day by quadratic equations, linear equations, and indeterminate linear equations. During this same time period in the first millennium BC, most of the mathematicians in India, China, Greece, and Egypt were using geometric methods to solve such equations. The word algebra comes from the Arabic word al-jabar, which means reunion of the broken parts. A Persian Muslim
mathematician named Muhammad ibn Masa al-khwarizmi wrote a book in 820 whose title is translated to mean The book of Summary Concerning Calculating by Transposition and Reduction. There is debate in math circles on whether the mathematician Diophantes, who has always been known as the father of Algebra, or Al-Khwarizmi should actually be named the father of Algebra. There is support for Al-Khwarizmi instead of Diophantes because a lot of his work on reduction is still currently in use today. He also explained how to solve a quadratic very extensively.
In the fourteenth century the word algeber referred to bone setting, which is quite similar to the original meaning of the word. The mathematical form of the word that is spelled algebra had made its debut by the sixteenth century. A man by the name of Robert Recorde was the very first person to use the term algebra in its mathematical meaning. He was also the inventor of the equality symbol(=). The original meaning of the word Algebra would refer today to the sub class of elementary algebra. There are numerous different fields of algebra in the modern definition of the word.
Some of these other mathematic fields include elementary algebra, s-algebra, abstract algebra, geometric algebra, boolean algebra, linear algebra, and many more. The mastery of elementary algebra must be accomplished before calculus or any of the other advanced mathematics can be learned. Many careers today require that at least elementary algebra has been mastered. The symbolism that is first introduced in elementary algebra extends throughout all the fields of mathematics. This is because the symbolism is the letters in the language of math. Modern fields of algebra are vastly more advanced than the first equations and formulas that were developed by the Babylonians. Numerous jobs and careers in the modern work environment require that algebra is mastered as a legitimate requirement for employment.
The very start of modern algebra began with the ancient Babylonians. They were the very first people who used an advanced math system for their time. This math used an algebraic system to do calculations, and it enabled the Babylonians to utilize formulas and calculate the answer using unknown values for a problem class that would be solved in the modern day by quadratic equations, linear equations, and indeterminate linear equations. During this same time period in the first millennium BC, most of the mathematicians in India, China, Greece, and Egypt were using geometric methods to solve such equations. The word algebra comes from the Arabic word al-jabar, which means reunion of the broken parts. A Persian Muslim
mathematician named Muhammad ibn Masa al-khwarizmi wrote a book in 820 whose title is translated to mean The book of Summary Concerning Calculating by Transposition and Reduction. There is debate in math circles on whether the mathematician Diophantes, who has always been known as the father of Algebra, or Al-Khwarizmi should actually be named the father of Algebra. There is support for Al-Khwarizmi instead of Diophantes because a lot of his work on reduction is still currently in use today. He also explained how to solve a quadratic very extensively.
In the fourteenth century the word algeber referred to bone setting, which is quite similar to the original meaning of the word. The mathematical form of the word that is spelled algebra had made its debut by the sixteenth century. A man by the name of Robert Recorde was the very first person to use the term algebra in its mathematical meaning. He was also the inventor of the equality symbol(=). The original meaning of the word Algebra would refer today to the sub class of elementary algebra. There are numerous different fields of algebra in the modern definition of the word.
Some of these other mathematic fields include elementary algebra, s-algebra, abstract algebra, geometric algebra, boolean algebra, linear algebra, and many more. The mastery of elementary algebra must be accomplished before calculus or any of the other advanced mathematics can be learned. Many careers today require that at least elementary algebra has been mastered. The symbolism that is first introduced in elementary algebra extends throughout all the fields of mathematics. This is because the symbolism is the letters in the language of math. Modern fields of algebra are vastly more advanced than the first equations and formulas that were developed by the Babylonians. Numerous jobs and careers in the modern work environment require that algebra is mastered as a legitimate requirement for employment.
(ArticlesBase SC #499774)
Wednesday, September 15, 2010
Thursday, September 9, 2010
Math and Social Injustice
by Jack Ucifferri
When you walk into a typical math class on a typical day in almost any school, you'll notice that most of the students are bored and distracted. That, believes Jonathan Osler, founder of RadicalMath, is a social justice issue."Math classes should give students the tools to better understand their reality. Who cares if 'Train A goes x+4 times faster than train B' when your community isn't adequately served by public transportation?"
Traditional math curricula don't teach students how to compare the density of check-cashers to banks in low-income communities, evaluate college loan plans to determine which offer the most favorable rates, or analyze data on rates of diabetes and asthma in communities of color. Lesson plans for addressing all of these issues can be found at RadicalMath.org, a free website for educators interested in integrating issues of economic and social justice into their math classes.
"I believe in engaging and empowing students to learn about issues that are relevant to their lives and communities," says RadicalMath founder, Jonathan Osler, who taught in a public high school in Brooklyn, New York for six years and now coaches math teachers in a public high school in Los Angeles. "But there were no sources of information for how I could integrate social justice issues into my math classes, so I began writing my own curricula and posting it online." Two years later, RadicalMath contains over 800 lesson plans, data sets, and articles, has received over 1,000,000 page views, and has drawn visitors from all over the world.
Osler explains that it is critical for students to graduate from high school with strong math skills, prepared for math-based college majors and careers. But equally strong is his belief that in order to address our country's most pressing problems, young people need to become agents for change in their lives and communities, and math is a tool that can help them do so.
RadicalMath.org contains information on dozens of issues including racial profiling, immigration, global warming, and the criminal justice system. There are also numerous financial education resources and lesson plans on economic topics such as minimum vs. living wage, predatory lending, the mathematics of the lottery, and home ownership.
Last April, Osler, along with several other RadicalMath contributors, organized a national conference to discuss teaching math through a social justice lens. This first annual "Creating Balance in an Unjust World" conference drew over 500 educators, activists, parents and students from around the country to Brooklyn, NY. Osler and the other organizers expect to draw twice as many participants to this year's conference.
This article was provided to us by Jonathan Osler. Osler is the founder of RadicalMath, a resource for educators interested in integrating issues of social and economic justice in their math classes and curriculum.
(ArticlesBase SC #499871)
Manipulatives: The Missing Link in High School Math
by Marilyn Curtain- Phillips, M. Ed.
The thrill of achievement comes from overcoming adversity in the accomplishment of an important goal ~ UnknownAlgebra, Geometry and Trigonometry can be challenging and complex subjects for many high school students in the United States. These subjects are usually taught using textbooks, workbooks and examinations. While these resources are essential in developing learning skills in math, it doesn’t encourage problem-solving skills and retention. It is not surprising to learn many students view math as boring, difficult and irrelevant, rather than fun and interesting. The scope of what it means to be successful in mathematics expands beyond procedural knowledge and skill-acquisition to include sense-making and conceptual understanding (Buckley 2004). An integrated approach is needed to motivate students to learn math in a more relaxed environment, thus eliminating persistent math anxiety, the lack of math application skills and poor standardized test performance. The Third International Mathematics and Science Study (TIMSS) tested the students of 41 nations. Children in the United States were among the leaders in the fourth grade assessments, but by high school graduation, they were almost last. High school math should be taught in a way in which students can grasp and understand skills. Manipulatives are visual objects that help illustrate mathematical relationships and applications. These resources are used primarily in elementary schools and somewhat in middle schools. Manipulatives are valuable resources for accelerating and deepening students understanding of math, yet its use is almost non-existent in high school. Marilyn Burns, Creator of Math Solutions has "used manipulative materials at all levels for 30 years." In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels (Bellonio), yet many high school teachers are reluctant to use this type of resource.
Almost all mathematics-teaching activities take place at the abstract level (Sharma, 1997). According to Sharma, students have a tendency to forget when taught only at the abstract stage. Thus students become frustrated because mastery was never fully attained. Students will begin to have difficulty in learning mathematics. The results in failure will cause many students to develop a fear of mathematics (Sharma, 1997). Students’ attitudes towards any curriculum area can be related to their achievement in ways that reinforce higher or lower performance (timss.org). Full 88% of Bill and Melinda Gates Foundations’ survey respondents said they had passing grades in high school. Asked to name the reasons they had left school, more respondents named boredom than struggles with coursework. Over 1 million students drop out of school each year. That includes nearly half of all African Americans, Hispanics, and Native Americans who fail to graduate from public high school with their class. Leaving many of them with a host of poor outcomes to follow, from low lifetime earnings to high incarceration rates to a high likelihood that their children will drop out of high school thus eliminating the cycle (Thornburg, 2006).
Sharma (1997) feels there are six levels of mastery of mathematical concepts: intuitive, concrete, representation (pictorial), abstract, applications and communication. According to Sharma, ideally, each mathematics concept should be introduced beginning at the communication level. Manipulatives will teach concrete understanding to the abstract math process, especially when the student may not understand the concept behind the skill. When students develop a concrete understanding of math skills, then they are more likely to perform that math skill and understand math concepts at the abstract level. Manipulatives can make math concepts come alive. According to Spikell (1993), most learners whether adults or children, will master mathematical concepts and skills more readily, if they are presented first in concrete, pictorial and symbols. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for learners to understand the abstractions they represent (Spikell, 1993). Students will be able to practice and demonstrate mastery using concrete objects. Manipulatives appeal to the learning style of kinesthetic learners because they actually touch the objects. Pictures appeal visually for visual/spatial learners. "Visualization is the natural way one begins to think. Before words, images emerge" (Sharma, 1987). "Almost every mathematics idea, except simple arithmetic facts, consists of three components: linguistic, conceptual and skill/procedural" (Sharma, 1987). Therefore manipulatives offer benefits to a variety of learning styles. They also provide a change from the textbook for mathematical/logical learners.
Games with manipulatives are also valuable with helping students to apply what they learned to the real world, as well as provide a means to improve their math skills interactively. Using board games and card games along with cooperative learning are ways that students can become involved in a positive mathematical environment. Games are highly motivational to students and can be used effectively to practice specific skills. "Using games in the classroom and at home will maximize students' problem-solving competence, ability to communicate and reason mathematically, perception of the value of mathematics, and self-confidence in their ability to apply mathematical knowledge to new situations." Cooperative groups provide students a chance to exchange ideas, to ask questions freely, to explain to one another, to clarify ideas in meaningful ways and to express feelings about their learning. These skills acquired at an early age will be greatly beneficial throughout their adult working life.
Many teachers feel as though they do not know how to teach using manipulatives and therefore, hesitate to use them in the classroom. Many math teachers, who attended college more than ten years ago, were usually taught on the abstract level, textbook, pencil and paper, all through their school days. There are classes and workshops for teachers to learn how to teach using manipulatives. The companies that make the manipulatives also provide books and pamphlets on ways the material can be used. There are articles on using manipulatives in mathematics teaching journal such as the National Council of Teachers of Mathematics’ Arithmetic Teacher magazine. Manipulatives help relieve boredom in children allowing them to explore and use their imagination.
Many manipulatives are inexpensive and can be everyday objects. Money, 2 –color counters, calculators, rulers, dominoes, playing cards, button and number cubes are a few of the commonly available manipulatives that can successfully be used in the classroom. These manipulatives can be used to teach such concepts such as angles, area, decimals, factoring, estimation, fractions, measurement, counting, percent, prime numbers, probability, geometry and whole numbers.
There are companies, which specialize in manipulatives that can be ordered from a catalog or online. No matter where a school is located, materials can be made available through the mail. These companies also make available to teachers such manipulatives as tangrams, pattern blocks, fraction towers, geoboards, algebra tiles, Cuisenaire rods, miras and polyhedral models.
When I initially introduced pattern blocks to my Geometry students, I was met with opposition and disbelief. It was a consensus that these students had not used pattern blocks since elementary school. However, many students actually began to explore and build figures with these blocks. I gave them this little bit of play time before taking on the task for that particular lesson, which was angle measurement of polygons. We must remember, everyone likes to play games even adults, look at our various hobbies, such as bowling, tennis, golf, art, race cars, chess, playing cards, etc. This holds true for high school students, they still enjoy playing games. Manipulatives require a great deal of prior planning and organization. But considering the many benefits that manipulatives offer, it is well worth the effort. Students will find math more attainable, fun and exciting, hopefully returning them to the prior successes of their early math experiences.
REFERENCES:
Buckley, L. A. (2004, Oct) Course Taking and Equity: The Efforts of One High School Mathematics Department Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF> Retrieved 2008-01-12 from <http://www.allacademic.com/meta/p117666_index.html>Burns, M. How to make the most of math manipulatives – A fresh look at getting students’ heads — and hands! — around math concepts. Retrieved 2008-01-1-10 from <http://teacher.scholastic.com/lessonrepro/lessonplans/instructor/burns.htm>
Sharma, M. (1987). How to take a child from concrete to abstract. Math Note book, 5, 8-10
Sharma, M. (1997, July). Improving mathematics instruction for all. Fourth Lecture in Colloquium "Improving Schools from Within: Your Role. Pp. 2-12
Spikell, M. (1993). Teaching mathematics with manipulatives: A resource of activities for the K-12 teacher. New York: Allyn and Bacon.
Third International Mathematics and Science Study (TIMSS), Institute of Educational Sciences, United States Department of Education
Thornburg, N. (2006), April 7), Dropout Nation, Time Magazine
TIMSS 1999 International Mathematics Report. Findings from IEA's Repeat of the Third International Mathematics and Science Study at the Eighth Grade. Ina V.S. Mullis, Michael O. Martin, Eugenio J. Gonzalez, Kelvin D. Gregory, Robert A. Garden, Kathleen M. O'Connor, Steven J. Chrostowski, Teresa A. Smith. Retrieved 2008-01-12 from <http://timss.bc.edu/timss1999i/pdf/T99i_Math_TOC.pdf>
Yale-New Haven Teachers Institute Home Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete by Judith L. Bellonio Contents of Curriculum Unit 01.06.12: Narrative Math Anxiety Types of ... Retrieved 2008-01-12 from <http://www.yale.edu/ynhti/curriculum/units/2001/6/01.06.12.x.html>
Marilyn Curtain-Phillips teaches high school mathematics and is the author of Math Attack: How to Reduce Math Anxiety in the Classroom, at Work and in Everyday Personal Use,(1999), $15.00. You can purchase her book through our Online Bookstore.
by Marilyn Curtain- Phillips, M. Ed.
The thrill of achievement comes from overcoming adversity in the accomplishment of an important goal ~ UnknownAlgebra, Geometry and Trigonometry can be challenging and complex subjects for many high school students in the United States. These subjects are usually taught using textbooks, workbooks and examinations. While these resources are essential in developing learning skills in math, it doesn’t encourage problem-solving skills and retention. It is not surprising to learn many students view math as boring, difficult and irrelevant, rather than fun and interesting. The scope of what it means to be successful in mathematics expands beyond procedural knowledge and skill-acquisition to include sense-making and conceptual understanding (Buckley 2004). An integrated approach is needed to motivate students to learn math in a more relaxed environment, thus eliminating persistent math anxiety, the lack of math application skills and poor standardized test performance. The Third International Mathematics and Science Study (TIMSS) tested the students of 41 nations. Children in the United States were among the leaders in the fourth grade assessments, but by high school graduation, they were almost last. High school math should be taught in a way in which students can grasp and understand skills. Manipulatives are visual objects that help illustrate mathematical relationships and applications. These resources are used primarily in elementary schools and somewhat in middle schools. Manipulatives are valuable resources for accelerating and deepening students understanding of math, yet its use is almost non-existent in high school. Marilyn Burns, Creator of Math Solutions has "used manipulative materials at all levels for 30 years." In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels (Bellonio), yet many high school teachers are reluctant to use this type of resource.
Almost all mathematics-teaching activities take place at the abstract level (Sharma, 1997). According to Sharma, students have a tendency to forget when taught only at the abstract stage. Thus students become frustrated because mastery was never fully attained. Students will begin to have difficulty in learning mathematics. The results in failure will cause many students to develop a fear of mathematics (Sharma, 1997). Students’ attitudes towards any curriculum area can be related to their achievement in ways that reinforce higher or lower performance (timss.org). Full 88% of Bill and Melinda Gates Foundations’ survey respondents said they had passing grades in high school. Asked to name the reasons they had left school, more respondents named boredom than struggles with coursework. Over 1 million students drop out of school each year. That includes nearly half of all African Americans, Hispanics, and Native Americans who fail to graduate from public high school with their class. Leaving many of them with a host of poor outcomes to follow, from low lifetime earnings to high incarceration rates to a high likelihood that their children will drop out of high school thus eliminating the cycle (Thornburg, 2006).
Sharma (1997) feels there are six levels of mastery of mathematical concepts: intuitive, concrete, representation (pictorial), abstract, applications and communication. According to Sharma, ideally, each mathematics concept should be introduced beginning at the communication level. Manipulatives will teach concrete understanding to the abstract math process, especially when the student may not understand the concept behind the skill. When students develop a concrete understanding of math skills, then they are more likely to perform that math skill and understand math concepts at the abstract level. Manipulatives can make math concepts come alive. According to Spikell (1993), most learners whether adults or children, will master mathematical concepts and skills more readily, if they are presented first in concrete, pictorial and symbols. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for learners to understand the abstractions they represent (Spikell, 1993). Students will be able to practice and demonstrate mastery using concrete objects. Manipulatives appeal to the learning style of kinesthetic learners because they actually touch the objects. Pictures appeal visually for visual/spatial learners. "Visualization is the natural way one begins to think. Before words, images emerge" (Sharma, 1987). "Almost every mathematics idea, except simple arithmetic facts, consists of three components: linguistic, conceptual and skill/procedural" (Sharma, 1987). Therefore manipulatives offer benefits to a variety of learning styles. They also provide a change from the textbook for mathematical/logical learners.
Games with manipulatives are also valuable with helping students to apply what they learned to the real world, as well as provide a means to improve their math skills interactively. Using board games and card games along with cooperative learning are ways that students can become involved in a positive mathematical environment. Games are highly motivational to students and can be used effectively to practice specific skills. "Using games in the classroom and at home will maximize students' problem-solving competence, ability to communicate and reason mathematically, perception of the value of mathematics, and self-confidence in their ability to apply mathematical knowledge to new situations." Cooperative groups provide students a chance to exchange ideas, to ask questions freely, to explain to one another, to clarify ideas in meaningful ways and to express feelings about their learning. These skills acquired at an early age will be greatly beneficial throughout their adult working life.
Many teachers feel as though they do not know how to teach using manipulatives and therefore, hesitate to use them in the classroom. Many math teachers, who attended college more than ten years ago, were usually taught on the abstract level, textbook, pencil and paper, all through their school days. There are classes and workshops for teachers to learn how to teach using manipulatives. The companies that make the manipulatives also provide books and pamphlets on ways the material can be used. There are articles on using manipulatives in mathematics teaching journal such as the National Council of Teachers of Mathematics’ Arithmetic Teacher magazine. Manipulatives help relieve boredom in children allowing them to explore and use their imagination.
Many manipulatives are inexpensive and can be everyday objects. Money, 2 –color counters, calculators, rulers, dominoes, playing cards, button and number cubes are a few of the commonly available manipulatives that can successfully be used in the classroom. These manipulatives can be used to teach such concepts such as angles, area, decimals, factoring, estimation, fractions, measurement, counting, percent, prime numbers, probability, geometry and whole numbers.
There are companies, which specialize in manipulatives that can be ordered from a catalog or online. No matter where a school is located, materials can be made available through the mail. These companies also make available to teachers such manipulatives as tangrams, pattern blocks, fraction towers, geoboards, algebra tiles, Cuisenaire rods, miras and polyhedral models.
When I initially introduced pattern blocks to my Geometry students, I was met with opposition and disbelief. It was a consensus that these students had not used pattern blocks since elementary school. However, many students actually began to explore and build figures with these blocks. I gave them this little bit of play time before taking on the task for that particular lesson, which was angle measurement of polygons. We must remember, everyone likes to play games even adults, look at our various hobbies, such as bowling, tennis, golf, art, race cars, chess, playing cards, etc. This holds true for high school students, they still enjoy playing games. Manipulatives require a great deal of prior planning and organization. But considering the many benefits that manipulatives offer, it is well worth the effort. Students will find math more attainable, fun and exciting, hopefully returning them to the prior successes of their early math experiences.
REFERENCES:
Buckley, L. A. (2004, Oct) Course Taking and Equity: The Efforts of One High School Mathematics Department Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF> Retrieved 2008-01-12 from <http://www.allacademic.com/meta/p117666_index.html>Burns, M. How to make the most of math manipulatives – A fresh look at getting students’ heads — and hands! — around math concepts. Retrieved 2008-01-1-10 from <http://teacher.scholastic.com/lessonrepro/lessonplans/instructor/burns.htm>
Sharma, M. (1987). How to take a child from concrete to abstract. Math Note book, 5, 8-10
Sharma, M. (1997, July). Improving mathematics instruction for all. Fourth Lecture in Colloquium "Improving Schools from Within: Your Role. Pp. 2-12
Spikell, M. (1993). Teaching mathematics with manipulatives: A resource of activities for the K-12 teacher. New York: Allyn and Bacon.
Third International Mathematics and Science Study (TIMSS), Institute of Educational Sciences, United States Department of Education
Thornburg, N. (2006), April 7), Dropout Nation, Time Magazine
TIMSS 1999 International Mathematics Report. Findings from IEA's Repeat of the Third International Mathematics and Science Study at the Eighth Grade. Ina V.S. Mullis, Michael O. Martin, Eugenio J. Gonzalez, Kelvin D. Gregory, Robert A. Garden, Kathleen M. O'Connor, Steven J. Chrostowski, Teresa A. Smith. Retrieved 2008-01-12 from <http://timss.bc.edu/timss1999i/pdf/T99i_Math_TOC.pdf>
Yale-New Haven Teachers Institute Home Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete by Judith L. Bellonio Contents of Curriculum Unit 01.06.12: Narrative Math Anxiety Types of ... Retrieved 2008-01-12 from <http://www.yale.edu/ynhti/curriculum/units/2001/6/01.06.12.x.html>
Marilyn Curtain-Phillips teaches high school mathematics and is the author of Math Attack: How to Reduce Math Anxiety in the Classroom, at Work and in Everyday Personal Use,(1999), $15.00. You can purchase her book through our Online Bookstore.
The Causes and Prevention of Math Anxiety
by Marilyn Curtain-Phillips, M. Ed.
| Mathematics anxiety has been defined as feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations Math anxiety can cause one to forget and lose one’s self-confidence (Tobias, S., 1993). | |||||
| Research confirms that pressure of timed tests and risk of public embarrassment have long been recognized as sources of unproductive tension among many students. Three practices that are a regular part of the traditional mathematics classroom and cause great anxiety in many students are imposed authority, public exposure and time deadlines. Although these are a regular part of the traditional mathematics classroom cause great deal of anxiety. Therefore, teaching methods must be re-examined. Consequently, there should be more emphasis on teaching methods which include less lecture, more student directed classes and more discussion. | |||||
| Given the fact that many students experience math anxiety in the traditional classroom, teachers should design classrooms that will make children feel more successful . Students must have a high level of success or a level of failure that they can tolerate. Therefore, incorrect responses must be handled in a positive way to encourage student participation and enhance student confidence. | |||||
| Studies have shown students learn best when they are active rather than passive learners (Spikell, 1993). The theory of multiple intelligences addresses the different learning styles. Lessons are presented for visual/spatial, logical/mathematics, musical, body/kinesthetic, interpersonal and intrapersonal and verbal/linguistic. Everyone is capable of learning, but may learn in different ways. Therefore, lessons must be presented in a variety of ways. For example, different ways to teach a new concept can be through play acting, cooperative groups, visual aids, hands on activities and technology. Learners are different than they were forty years ago. These learners today ask questions why something is done this way or that way and why not this way? Whereas years ago learners did not question the why of math concepts; they simply memorized and mechanically performed the operations needed. | |||||
| Students today have a need for practical math. Therefore, math needs to be relevant to their everyday lives. Students enjoy experimenting. To learn mathematics, students must be engaged in exploring, conjecturing, and thinking rather than, engaged only in rote learning of rules and procedures. | |||||
| Students’ prior negative experiences in math class and at home when learning math are often transferred and cause a lack of understanding of mathematics. According to Sheila Tobias, millions of adults are blocked from professional and personal opportunities because they fear or perform poorly in mathematics for many, these negative experiences remain throughout their adult lives. | |||||
| Math is often associated with pain and frustration. For instance, unpaid bills, unforeseen debts, unbalanced checkbooks, IRS forms are a few of the negative experiences associated with numbers. Parents should show their children how numbers are successfully used by them in positive pleasant ways, such as in cooking, sewing, sports, problem solving in hobbies and home repairs. | |||||
| Math must be looked upon in a positive light to reduce anxiety. A person’s state of mind has a great influence on his/her success. Many games are based on math concepts. Some games that are beneficial to learners and are enjoyed are cards playing, Life, Yahtzee, Battleship and Tangrams. | |||||
| With all the tension and anxiety, math humor is greatly needed. Young children enjoy cartoons and jokes. Cartoons may be used to introduce a concept or for class discussion. Most children will master mathematical concepts and skills more readily if they are presented first in concrete, pictorial and symbols. For example manipulatives are concrete objects used to teach a concept. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for young learners to understand the abstractions they represent. Students enjoy the change from lecture and books and they are more inclined to explore with manipulatives and show greater interest in classwork. | |||||
| Cooperative groups provide students a chance to exchange ideas, to ask questions freely, to explain to one another, to clarify ideas in meaningful ways and to express feelings about their learning. These skills acquired at an early age will be greatly beneficial throughout their adult working life. | |||||
In conclusion, math anxiety is very real and occurs among thousands of people. Much of this anxiety happens in the classroom due to the lack of consideration of different learning styles of students. Today, the needs of society require a greater need for mathematics. Math must be looked upon in a positive light to reduce math anxiety. Therefore, teachers must re-examine traditional teaching methods which often do not match students’ learning styles and skills needed in society. Lessons must be presented in a variety of ways. For instance, a new concept can be taught through play acting, cooperative groups, visual aids, hands on activities and technology. As a result once young children see math as fun, they will enjoy it, and, the joy of mathematics could remain with them throughout the rest of their lives. References
Marilyn Curtain-Phillips teaches high school mathematics and is the author of Math Attack: How to Reduce Math Anxiety in the Classroom, at Work and in Everyday Personal Use,(1999), $15.00. You can purchase her book through our Online Bookstore. |
Why Study Calculus? - Related Rates
By Joe Pagano 
One of the more interesting applications of the calculus is in related rates problems. Problems such as these demonstrate the sheer power of this branch of mathematics to answer questions that would seem unanswerable. Here we examine a specific problem in related rates and show how the calculus allows us to come up with the solution quite easily.
Any quantity which increases or decreases with respect to time is a candidate for a related rates problem. It should be noted that all functions in related rates problems are dependent on time. Since we are trying to find an instantaneous rate of change with respect to time, the process of differentiation (taking derivatives) comes into play and this is done with respect to time. Once we map out the problem, we can isolate the rate of change we are looking for, and then solve using differentiation. A specific example will make this procedure clear. (Please note I have taken this problem from Protter/Morrey, "College Calculus," Third Edition, and have expanded upon the solution and application of such.)
Let us take the following problem: Water is flowing into a conical tank at the rate of 5 cubic meters per minute. The cone has altitude 20 meters and base radius 10 meters (the vertex of the cone is facing down). How fast is the water level rising when the water is 8 meters deep? Before we solve this problem, let us ask why we might even need to address such a problem. Well suppose the tank serves as part of an overflow system for a dam. When the dam is overcapacity because of flooding as a result of, let us say, excessive rain or river drainage, the conical tanks serve as outlets to release pressure on the dam walls, preventing damage to the overall dam structure.
This whole system has been designed so that there is an emergency procedure which kicks in when the water levels of the conical tanks reach a certain level. Before this procedure is implemented a certain amount of preparation is necessary. The workers have taken a measurement of the depth of the water and find that it is 8 meters deep. The question becomes how long do the emergency workers have before the conical tanks reach capacity?
To answer this question, related rates come into play. By knowing how fast the water level is rising at any point in time, we can determine how long we have until the tank is going to overflow. To solve this problem, we let h be the depth, r the radius of the surface of the water, and V the volume of the water at an arbitrary time t. We want to find the rate at which the height of the water is changing when h = 8. This is another way of saying we wish to know the derivative dh/dt.
We are given that the water is flowing in at 5 cubic meters per minute. This is expressed as
dV/dt = 5. Since we are dealing with a cone, the volume for the water is given by
V = (1/3)(pi)(r^2)h, such that all quantities depend on time t. We see that this volume formula depends on both variables r and h. We wish to find dh/dt, which only depends on h. Thus we need to somehow eliminate r in the volume formula.
We can do this by drawing a picture of the situation. We see that we have a conical tank of altitude 20 meters, with a base radius of 10 meters. We can eliminate r if we use similar triangles in the diagram. (Try to draw this out to see this.) We have 10/20 = r/h, where r and h represent the constantly changing quantities based on the flow of water into the tank. We can solve for r to get r = 1/2h. If we plug this value of r into the formula for the volume of the cone, we have V = (1/3)(pi)(.5h^2)h. (We have replaced r^2 by 0.5h^2). We simplify to get
V = (1/3)(pi)(h^2/4)h or (1/12)(pi)h^3.
Since we want to know dh/dt, we take differentials to get dV = (1/4)(pi)(h^2)dh. Since we want to know these quantities with respect to time, we divide by dt to get
(1) dV/dt = (1/4)(pi)(h^2)dh/dt.
We know that dV/dt is equal to 5 from the original statement of the problem. We want to find dh/dt when h = 8. Thus we can solve equation (1) for dh/dt by letting h = 8 and dV/dt = 5. Inputting we get dh/dt = (5/16pi)meters/minute, or 0.099 meters/minute. Thus the height is changing at a rate of less than 1/10 of a meter every minute when the water level is 8 meters high. The emergency dam workers now have a better assessment of the situation at hand.
For those who have some understanding of the calculus, I know you will agree that problems such as these demonstrate the awesome power of this discipline. Before calculus, there would never have been a way to solve such a problem, and if this were a real world impending disaster, no way to avert such a tragedy. This is the power of mathematics.
See more of my articles here at Articles Page and see a list of my ebooks here at Order Ebooks.
One of the more interesting applications of the calculus is in related rates problems. Problems such as these demonstrate the sheer power of this branch of mathematics to answer questions that would seem unanswerable. Here we examine a specific problem in related rates and show how the calculus allows us to come up with the solution quite easily.
Any quantity which increases or decreases with respect to time is a candidate for a related rates problem. It should be noted that all functions in related rates problems are dependent on time. Since we are trying to find an instantaneous rate of change with respect to time, the process of differentiation (taking derivatives) comes into play and this is done with respect to time. Once we map out the problem, we can isolate the rate of change we are looking for, and then solve using differentiation. A specific example will make this procedure clear. (Please note I have taken this problem from Protter/Morrey, "College Calculus," Third Edition, and have expanded upon the solution and application of such.)
Let us take the following problem: Water is flowing into a conical tank at the rate of 5 cubic meters per minute. The cone has altitude 20 meters and base radius 10 meters (the vertex of the cone is facing down). How fast is the water level rising when the water is 8 meters deep? Before we solve this problem, let us ask why we might even need to address such a problem. Well suppose the tank serves as part of an overflow system for a dam. When the dam is overcapacity because of flooding as a result of, let us say, excessive rain or river drainage, the conical tanks serve as outlets to release pressure on the dam walls, preventing damage to the overall dam structure.
This whole system has been designed so that there is an emergency procedure which kicks in when the water levels of the conical tanks reach a certain level. Before this procedure is implemented a certain amount of preparation is necessary. The workers have taken a measurement of the depth of the water and find that it is 8 meters deep. The question becomes how long do the emergency workers have before the conical tanks reach capacity?
To answer this question, related rates come into play. By knowing how fast the water level is rising at any point in time, we can determine how long we have until the tank is going to overflow. To solve this problem, we let h be the depth, r the radius of the surface of the water, and V the volume of the water at an arbitrary time t. We want to find the rate at which the height of the water is changing when h = 8. This is another way of saying we wish to know the derivative dh/dt.
We are given that the water is flowing in at 5 cubic meters per minute. This is expressed as
dV/dt = 5. Since we are dealing with a cone, the volume for the water is given by
V = (1/3)(pi)(r^2)h, such that all quantities depend on time t. We see that this volume formula depends on both variables r and h. We wish to find dh/dt, which only depends on h. Thus we need to somehow eliminate r in the volume formula.
We can do this by drawing a picture of the situation. We see that we have a conical tank of altitude 20 meters, with a base radius of 10 meters. We can eliminate r if we use similar triangles in the diagram. (Try to draw this out to see this.) We have 10/20 = r/h, where r and h represent the constantly changing quantities based on the flow of water into the tank. We can solve for r to get r = 1/2h. If we plug this value of r into the formula for the volume of the cone, we have V = (1/3)(pi)(.5h^2)h. (We have replaced r^2 by 0.5h^2). We simplify to get
V = (1/3)(pi)(h^2/4)h or (1/12)(pi)h^3.
Since we want to know dh/dt, we take differentials to get dV = (1/4)(pi)(h^2)dh. Since we want to know these quantities with respect to time, we divide by dt to get
(1) dV/dt = (1/4)(pi)(h^2)dh/dt.
We know that dV/dt is equal to 5 from the original statement of the problem. We want to find dh/dt when h = 8. Thus we can solve equation (1) for dh/dt by letting h = 8 and dV/dt = 5. Inputting we get dh/dt = (5/16pi)meters/minute, or 0.099 meters/minute. Thus the height is changing at a rate of less than 1/10 of a meter every minute when the water level is 8 meters high. The emergency dam workers now have a better assessment of the situation at hand.
For those who have some understanding of the calculus, I know you will agree that problems such as these demonstrate the awesome power of this discipline. Before calculus, there would never have been a way to solve such a problem, and if this were a real world impending disaster, no way to avert such a tragedy. This is the power of mathematics.
See more of my articles here at Articles Page and see a list of my ebooks here at Order Ebooks.
Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Joe is the creator of the Wiz Kid series of math ebooks, Arithmetic Magic, the little classic on the ABC's of arithmetic, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.
Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com
Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com
Article Source: http://EzineArticles.com/?expert=Joe_Pagano
Mathematics - Beautiful For Many Reasons
By Joe Pagano
Mathematics is rife with beauty. Yes beauty. Oh, maybe not the type of physical beauty you are thinking of but a beauty that can only be perceived when you study this subject in all of its fabulous environs. Whether it be the elegance of a particular formula or the economy of expressing something very complicated in a concise way, mathematics trumps the field of aesthetics in a manner worthy of praise.
For one, mathematics has the special beauty of being without hypocrisy. Now you may wonder how being bereft of hypocrisy can be aesthetically appealing. You may ask: "Can you really compare such a situation with the beauty found in a majestic enclave or a sensuous piece of artwork?" Well maybe not exactly, but from an intangible perspective, a lack of hypocrisy is actually a very beautiful thing.
Double talk or double speak---hypocrisy's middle name---finds no place in mathematics. A formula gives the same result all the time and under any circumstance. Mathematical rigor does not succumb to pressure from without. You see, contradictions just do not abound in this wonderful discipline. For example, the quadratic formula always gives us the roots for any quadratic equation regardless of race, religion, or economic background. When a formula predicts an outcome, such as when the Pythagorean Theorem predicts the length of the hypotenuse, given the two side lengths, then this outcome will be obtained regardless of the dispositions of the sides.
When fractals can be condensed to a specific formula and then applied to describe a majestic coastline, we see evidence of beauty all around. When group theory can be applied to subparticle physics, thereby aiding the explanation of the origins of life, mathematics transcends reality and abuts upon divine territory.
In view of these contemplations, it is no wonder that the greatest minds of all time have plumbed as deeply as possible the caverns of this universe, and have tried so hard to get as close to this subject as possible. Maybe by getting closer to mathematics, we can all take a glimpse of that great beyond and behold a beauty unseen in this present world. Until then, we can only sample incomplete beauty in a very imperfect world.
Mathematics is rife with beauty. Yes beauty. Oh, maybe not the type of physical beauty you are thinking of but a beauty that can only be perceived when you study this subject in all of its fabulous environs. Whether it be the elegance of a particular formula or the economy of expressing something very complicated in a concise way, mathematics trumps the field of aesthetics in a manner worthy of praise.
For one, mathematics has the special beauty of being without hypocrisy. Now you may wonder how being bereft of hypocrisy can be aesthetically appealing. You may ask: "Can you really compare such a situation with the beauty found in a majestic enclave or a sensuous piece of artwork?" Well maybe not exactly, but from an intangible perspective, a lack of hypocrisy is actually a very beautiful thing.
Double talk or double speak---hypocrisy's middle name---finds no place in mathematics. A formula gives the same result all the time and under any circumstance. Mathematical rigor does not succumb to pressure from without. You see, contradictions just do not abound in this wonderful discipline. For example, the quadratic formula always gives us the roots for any quadratic equation regardless of race, religion, or economic background. When a formula predicts an outcome, such as when the Pythagorean Theorem predicts the length of the hypotenuse, given the two side lengths, then this outcome will be obtained regardless of the dispositions of the sides.
When fractals can be condensed to a specific formula and then applied to describe a majestic coastline, we see evidence of beauty all around. When group theory can be applied to subparticle physics, thereby aiding the explanation of the origins of life, mathematics transcends reality and abuts upon divine territory.
In view of these contemplations, it is no wonder that the greatest minds of all time have plumbed as deeply as possible the caverns of this universe, and have tried so hard to get as close to this subject as possible. Maybe by getting closer to mathematics, we can all take a glimpse of that great beyond and behold a beauty unseen in this present world. Until then, we can only sample incomplete beauty in a very imperfect world.
See a list of my ebooks here at Order Ebooks
Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Joe is the creator of the Wiz Kid series of math ebooks, Arithmetic Magic, the little classic on the ABC's of arithmetic, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.
Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com
Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Joe is the creator of the Wiz Kid series of math ebooks, Arithmetic Magic, the little classic on the ABC's of arithmetic, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.
Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com
Article Source: http://EzineArticles.com/?expert=Joe_Pagano
In Mathematics There is No Hypocrisy
Mathematics is rife with beauty. Yes beauty. Oh, maybe not the type of physical beauty you are thinking of but a beauty that can only be perceived when you study this subject in all of its fabulous environs. One particular beauty that hooks me to this discipline like no other is that of being without hypocrisy. That's right. In mathematics there is no hypocrisy.So what is beautiful about "lacking in hypocrisy?" Can you really compare such a situation with the beauty found in a majestic enclave or a sensuous piece of artwork? Well maybe not exactly, but from an intangible perspective, a lack of hypocrisy is actually a very beautiful thing. You see, hypocrisy-the art of saying one thing and doing another, or the deception that comes from presenting one front and actually supporting another-is one of the ugliest things on earth. Hypocrisy is an abstract noun copiously endowed with all kinds of negative attributes, subversive intentions, and evil chromosomes.
Whether we find the big "H," - hypocrisy - in politics, religion, or philosophy, we are no sooner duped by its subtle penetration into our being than we are left wondering how we ever could be fooled so easily. Thus the religious leader, who seems so pious, leaves us breathless at how moral and ethical he is, when only later we find out about some scurrilous act that he has committed; or the politician who pontificates on the need for Americans to cut wasteful energy consumption and yet is himself a wastrel in the prudent use of such resources.
Yet mathematics boasts of no such thing. In this discipline, we only find black and white. Indeed, one witnesses all the colors of the rainbow-and indeed every gradation of color in between as we run the gamut from red to violet-but we do not find that something of one color suddenly morphing into another shade. Thus when we have a mathematical law or equation, we can be certain that that equation will hold true today, tomorrow, and that it held true yesterday, even though it might not have been discovered yet.
Admittedly mathematics, because of its ability to explain the world around us so well, must by nature be complex; however, once we comprehend its beauty and penetrate into the depths of its soul and realize no hypocrisy is found lurking there, we immediately become a convert-a die-hard fan of epic proportions. Yes, mathematics might not have a beauty comparable to a majestic enclave, but it does have a beauty-unfathomable, consistent, and ever reliable.
So when you hear the word math and start to summon feelings of negativity because of past experience, stop and consider these words: math is not only the key to the universe but also one of the few things in life without hypocrisy. Just for that last statement, mathematics merits our profound respect. Here's to hypocrisy free living!
Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey. Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers." See his website http://www.mathbyjoe.com for more information and for testimonials, and try out one of his ebooks here http://www.mathbyjoe.com/page/page/2924777.htm to achieve better grades in math.
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