Cover Design with Name, Class, Title ……… +5 _______

Cover Design with Name, Class, Title............……… +5 _______
Grading Criteria Cover Design with Name, Class, Title ……… _______ Binder/PowerPoint/ or other Electronic Presentation _______ Organized/Neat/Easy to Read … +10 _______ Table of Contents/Cover Letter with Summary _______ Includes all Sections Mentioned Above _______ Math Auto. and Philosophy Bullets … +10 _______ Research Articles on Math Ed./Websites…………… +10 _______ Professional Section with Resume, etc … +10 _______ Quality Presentation/Demonstration of Effort _______ Above and beyond minimum(Tech., Org., Qual., & Pres.) _______ Possible Portfolio Grade = 100 ______

Electronic Portfolio MAE 4360 November, 2006

Table of Contents Summary……………………………………………. 4
Sunshine State Standards…………………………. 5 FAU lesson plan…………………………………… 19 Website/Articles………………………………… Journal/Readings…………………………………. 41 Math Problems ……………………………………. 66 Research …………………………………………... 87 Creative Projects …………………………………100 Clinical Experience ……………………………… 130 Math Autobiography…………………………… My Math Education Philosophy………………… 151 My resume………………………………………

Summary This class over exceeded my expectations. What I thought as another math-psychology oriented class ended up as an incredible vast of resources any math teacher could just not teach without. Entire sections were devoted to understanding the system, the students, parents, and even the community and how interact and affect mathematics education. The use of updated technology (e.g. virtual manipulatives) really amazed me. I also tried to organize the multiple activities that can be used in the future as worksheets, extra credit work, or even fun quizzes. I truly believe this entire textbook should be preserve for years to come. In regards to class activities, I really enjoyed the problems of the weeks for the challenge they all presented. I also liked the research part that complimented the lessons. The history of pi, algebra, magic squares, etc, were just fascinating. I enjoyed doing all the class projects. I probably learned the most with the FAU lesson plan, although I consider it has a flaw: it does not consider ESE, or even the many ESOL students needs that live in this particular state where I am planning on teaching. I devoted long hours in its research and was very proud of the outcome. Nevertheless, as an experienced substitute teacher I also know that usually these nice plans cannot be implemented for lack of time and sometimes resources. Yet, it is good to be able to develop them. I especially appreciated learning about standard-based education, as I was allowed to relate it to the controversial “No Child Left behind” philosophy and organize my own thoughts. One topic that I thought was not covered enough was classroom management. It is well known that middle and high school students are going through physical and emotional changes that makes classroom management a real ordeal. In my opinion, the author underestimated this important fact. The grade I expect for this course? Humbly, a big “A”, not because all I was to prove to my teacher, but for all I learned.

Sunshine State Standards (SSS)
Mathematics 9-12

General Objective To provide senior high school mathematics teachers with the content, concepts, teaching strategies, and assessment needed to incorporate the Sunshine State Standards into the mathematics curriculum. The five strands are included in the specific objectives as indicated: NUMBER SENSES 1.0 – 5.0 MEASUREMENT 6.0 – 9.0 GEOMETRY AND SPACIAL SENSE 10.0 – 12.0 ALGEBRAIC THINKING 13.0 – 14.0 DATA ANALYSIS AND PROBABILITY 15.0 – 17.0

Specific Objectives

1) Understand the different ways numbers are represented and used in the real world
Associate verbal names, written word names, and standard numerals with integers, rational numbers, irrational numbers, real numbers, and complex numbers. Determine relative sizes of integers, rational numbers, irrational numbers, and real numbers. Use concrete and symbolic representations of real and complex numbers in real-world situations. 4. Represent numbers in a variety of equivalent forms, including integers, fractions, decimals, percents, scientific notation, exponents, radicals, absolute values, and logarithms.

S.S.S 2) Understand number systems Use the basic concepts of limits and infinity. Use the real number system. 3. Understand the structure of the complex number system.

S.S.S. 3) Understand the effects of operations on numbers and the relationships among these operations, select appropriate operations, and compute for problem solving Explain the effects of addition, subtraction, multiplication, and division on real numbers, including square roots, exponents, and appropriate inverse relationships. Select and justify alternative strategies, such as using properties of numbers, including inverse, identity, distributive, associative, transitive, that allow operational shortcuts for computational procedures in real-world or mathematical problems. 3. Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator.

4) Use estimation in problem solving and computation
S.S.S. 4) Use estimation in problem solving and computation 1. Use estimation strategies in complex situations to predict results and to check the reasonableness of results. 5) Apply theories related to numbers. 1. Apply special number relationships such as sequences and series to real-world problems.

S.S.S. 6) Measure quantities in the real world and use the measures to solve problems. Use concrete and graphic models to derive formulas for finding perimeter, area, surface area, circumference, and volume of two-and three-dimensional shapes, including rectangular solids, cylinders, cones, and pyramids. Use concrete and graphic models to derive formulas for finding rate, distance, time, angle measures, and arc lengths. 3. Relate the concepts of measurement to similarity and proportionality in real-world situations.

S.S.S. Compare, contract, and covert within systems of measurement 1. Select and use direct or indirect methods of measurement as appropriate. Solve real-world problems involving rated measures (miles per hour, feet per second). 9) Select and use appropriate units and instruments for measurement to achieve the degree of precision and accuracy required in real-world situations. Determine the level of accuracy and prevision, including absolute and relative errors or tolerance, required in real-world measurement situations. 2. Use appropriate instruments, technology, and techniques to measure quantities in order to achieve specified degrees of accuracy in a problem situation.

S.S.S. 10) Describe, draw, identify, and analyze two- and three-dimensional shapes 1. Use properties and relationships of geometric shapes to construct formal and informal proofs. 11) Illustrate ways in which shapes can be combined, subdivided, and changed Use geometric concepts such as perpendicularity, parallelism, tangency, congruency, similarity, reflections, symmetry, and transformations including flips, slides, turns, enlargements, rotations, and fractals. 2. Analyze and apply geometric relationships involving planar cross-sections (the intersection of a place and a three-dimensional figure).

S.S.S. Use coordinate geometry to locate objects in both two and three dimensions and to describe objects algebraically. Represent and apply geometric properties and relationships to solve real-world and mathematical problems including ratio, proportion, and properties of right triangle trigonometry. 2. Use a rectangular coordinate system (graph), apply and algebraically verify properties of two- and three-dimensional figures, including distance, midpoint, slope, parallelism, and perpendicularity.

S.S.S. 13) Describe, analyze, and generalize a wide variety of patterns, relations, and functions. Describe, analyze, and generalize relationships, patterns, and functions using words, symbols, variables, tables, and graphs. Determine the impact when changing parameters of given functions. 14) Use expressions, equations, inequalities, graphs, and formulas to represent and interpret situations. Represent real-world problem situations using finite graphs, matrices, sequences, series, and recursive relations. 2. Use systems of equations and inequalities to solve real-world problems graphically, algebraically, and with matrices.

S.S.S. 15) Use the tools of data analysis for managing information Interpret data that has been collected, organized, and displayed in charts, tables, and plots. 2. Calculate measures of central tendency (mean, median, and mode) and dispersion (range, standard deviation, and variance) for complex sets of data and determine the most meaningful measure to describe the data. 3. Analyze real-world data and make predictions of larger populations by applying formulas to calculate measures of central tendency and dispersion using the sample population data, and using appropriate technology, including calculators and computers.

S.S.S. 16) Identify patterns and make predictions from an orderly display of data using concepts of probability and statistics. 1. Determine probabilities using counting procedures, tables, tree diagrams, and formulas for permutations and combinations. Determine the probability for simple and compound events as well as independent and dependent events. 17) Use statistical methods to make inferences and valid arguments about real-world situations. Design and perform real-world statistical experiments that involve more than one variable, then analyze results and report findings. 2. Explain the limitations of using statistical techniques and data in making inferences and valid arguments.

Lesson / Unit Plans

FAU Lesson Plan Understanding the Graph of a Function

Teacher: Subject: Algebra I Length of Lesson: 120 minutes Grade Level: 9-12 Date: November Instructional Objective(s)/Outcomes: Lesson Objectives: Students will be expected to identify the basic elements of a graph. Students will define, find, and state verbally the domain and range using the graph of a function Students will create a graph in a given domain and range. Students should use responsive listening skills, including paraphrasing, and asking short and simple questions for elaboration and clarification. Students will demonstrate mastery of these subject areas through a closure assessment during the last day of the lesson

Florida Sunshine State Standards:
MA.D Describes a wide variety of patterns, relationships, and functions through models, such as manipulatives, tables, graphs, expressions, equations, and inequalities. MA.D Creates and interprets tables, graphs, equations, and verbal descriptions to explain cause and effect relationships.MA.B.2.4.2, Measurement: The student selects and uses direct (measured) or indirect (not measured) methods of measurement as appropriate. MA.C.1.4, Geometry and Spatial Sense: The student describes draws, identifies and analyzes two- and three-dimensional shapes.

Goal 3 Standards: Standard 3, Numeric Problem Solvers: Students use numeric operations and concepts to describe, analyze, desegregate, communicate and synthesize numeric data, and to identify and solve problems. Standard 4, Creative and Critical Thinkers: Students use creative thinking skills to generate new ideas, make the best decision, recognize and solve problems through reasoning, interpret symbolic data, and develop efficient techniques for lifelong learning. Standard 5, Responsible Workers: Students display responsibility, self-esteem, self-management, integrity and honesty. Standard 8, Cooperative Workers: Students work cooperatively to successfully complete a project or activity. Standard 10, Multiculturally Sensitive Citizens

Subject Matter Content:
Using this lesson plan, the instructor will impart concept, skill and knowledge in the general areas of: Graphing functions and identifying its important features (x and y axis, grids, lines, and plotted points. Describing the real world activity and explaining how the characteristics of this event relate to the general form of the equation and graph (e.g., the unknown x represents time). Naming and recognizing the domain and range of a function Given a linear equation, define the general form of this equation and the General pattern the graph takes.

Instructional Procedures:
Lesson Initiating Activities: This lesson should be taught in the middle to the end the unit on functions and understand the properties of graphs, and after the students have been taught the skills necessary. Using the overhead projector and laminates, students should be asked to identify some of the defining characteristics and properties of graphs of equations (e.g., slope, x-intercept) (10 min. total Day 1) Kinesthetic Activity: “Creating a human graph” Have students make a 12 X 12 grid on the floor using masking tape (min. 30 cm. wide). Draw a large dot on an index card (origin). Indicate axis using tape. Write a linear equation y = 2x + 3 on the board. Have student stand at intersection of the cells to indicate points of the line. At this point, students should recognize their positions are creating a line. Repeat other equations with different students. (30 min total –Day 1)

Core Activities: Students and teacher will review previous day activity creating a human graph by plotting “human” points. (5 min. total –Day 2) Students will be put into groups of four (chosen by the instructor) prior to completing the lesson. Each team will design and draw what they believe to be a graph” using grid paper (provided) before the lesson begins. (15 minutes total – Day 2) As a class, compare and contrast each group’s drawings and explanations. At the conclusion of the discussion, introduce the definitions of a domain and range, the difference between x and Y intercepts. (20 minutes total – Day 2)

Closure Activities: Students will be put into groups of four (chosen by the instructor) at the beginning of the class, and have them complete a group worksheet on domain and range (10 min. total - Day 3) Share answers in a class discussion format, and allow student groups to explain their individual work to peers (10 minutes – Day 3) Assessment: 30 minute quiz on graphing linear functions using paired coordinates, vocabulary, and identifying domain and range provided a linear graph. (20 minutes total –Day 3)

ESOL Activities: The lesson plan includes additional visual material and worksheets for ESOL students (See appendix A)

ESE Students can also be paired up with a buddy to assist in completing the assignment. Students should follow their IEP plan.

Classroom Management:
Cooperative learning requires self discipline and self control. It is a very rewarding methodology that makes learning easy and fun. In day 1 (human graph) students are asked to listen to directions carefully. This is a team work where everybody will have a job, (grid designers, measures, players, and clean-up team). If a student on this team fails to follow his or her duties, no fun activity can be implemented; therefore, to create and maintain discipline during this lesson, the teacher will announce expected behavior and consequences before starting the lesson. List of rewards: fun learning, working with their peers. Consequences: losing their assigned jobs, moving their cards, teacher conference after class.

Materials and Equipment:
Masking tape Grid paper Overhead projector Assessment sheets Practice worksheets Flash cards Whiteboard or screen for viewing purposes Extension chord and necessary connections to Internet, USB plugs and electrical outlets

Assessment/Evaluation:
After conducting the entire lesson, each student must take a multiple-choice graphing quiz (attached – Appendix B) which also contains graphing questions and real life applications. At least 80% of the class should receive a mastery grade for the lesson concepts; otherwise, re-teaching is necessary as these ideas are an essential piece of the FCAT. An alternate quiz for ESOL students with a higher visual context and limited language is also proposed (Appendix A) ESE students assessment should be discussed with ESE specialist taking into consideration the child’s IEP.

Follow-up Activities:
Students will be asked to investigate for extra credit the history of the Cartesian plane. This site will be suggested to complete the assignment:

ESOL Assessment (Appendix A)
This is a plane of coordinates x and y We can draw a graph using coordinates x and y.

ESOL assessment (cont)
We can draw a graph using coordinates x and y.

Look at the domain of this function. All numbers are on the x-axis. The domain starts on -1 and ends on +4. We write it like this: Dom = {-1, 4} Now look at the range of this function. All the numbers on the y-axis The range starts on -5 and ends on +4. We write it like this: Range = {-5, 4}

Now… answer these questions: True or False. Circle your answer. Only positive numbers can be a domain T F x and z are a pair of coordinates T F The range is in the x axis T F The x-axis goes in direction right-left T F The number zero is part of the x axis T F

2. Look at the graph Write the domain of the function Dom: __________ Write the range of the function Range: _________

3. True or False: The domain of this function is {-1, 1 } True ( ) False ( )

Articles/Websites

Useful Websites Sunshine State Standards: Standard-based education: National Council of Teaching Mathematics: Assessing Special Education: CREDE (Center for Research on Education, Diversity and Excellence): Educator’s Reference Desk: National Library of Virtual Manipulatives: Purple Math: Tour Algebra Resource: Math Websites links: Teaching Mathematics for the 21 st Century: Florida Department of Education: Principles and Standards for School of Mathematics: Mathematical Association of America:

Journal/Readings

Chapter History and Introduction to Reform
Summary: Chapter one deals with the history and evolution of different concepts and education strategies during the last fifty years that lead to the current reform to the teaching of mathematics. Reforms started back in 1927 when math instruction was based in the sole memorizing of concepts. The first teaching reform was the “New Match”, in the 1950’s, which considered social pressures towards increase equity and the development of student-centered classrooms. In the 1960’s and 1970’s Jean Piaget and George Polya proposed the “back to the basics” concept in response to the failed achievements of students attending open classes. Mathematics as a problem solver was again reviewed. But it was not until the publication of A Nation at Risk (1983) that a new approach towards mathematical instruction took a last shift towards a new trend on equal education available to all students, The Core Curriculum, and the call for continuous research on teaching and use of technology. Learning theory also evolved, ranging from behavioral models to cognitive models. All these considerations led to the publication of a final document, in 1995, the Assessment Standards for School Mathematics. This latest reform includes the practical applications of mathematical tools. The chapter finalizes with a summary for both teachers and students, of cooperative learning in secondary mathematics.

Chapter (cont) Reflections
Reading the chapter taught me to appreciate the dynamic nature of the teaching of mathematics at the secondary level. I appreciated extreme positions dominating different decades according to the times. Upon reflecting in this thought I could not stop visualizing the mathematics instruction of the next decade. Will children carry advanced technology gadgets in their backpacks instead of pencils and notebooks? How would a 100% virtual class would be? What would be the role of the math teacher? Would educators go back once again to the basics in instruction? How would time really impact The Standards? I guess millions of questions jumped into my mind and left me with the frustration of not really knowing a thing about tomorrow’s educational trends. Technology advances in such a fast pace that, with my limited knowledge in technology, I just cannot picture an accurate image of tomorrows mathematics instruction. It was also very interesting reading about the cooperative learning approach since I firmly believe in it. I think that once students enter the workforce, they will find out that most pivotal decisions are taken by a group of people working together towards a common goal. Instruction, though, should also make emphasis in this cooperative approach

Chapter Learning, Motivation, and Basic Management Skills
Summary Chapter two gives as a short but concise description of diverse aspects of learning theory as main sources of learning motivation. The chapter starts with a brief description of behaviorism, a theory that recognizes learning as a result of observable behaviors and related environment. It describes Pavlov’s classical and operational conditioning theories and its applications to education. Other theories briefly discussed were the social cognitive, which states that much can be learned just by observing others, the information processing theory, which defines learning as a significant changer in mental processes by the means of storage, encoding, retrieval, and control processes. The biggest weight on this chapter was placed on Piaget’s constructivist theory, for much of The Standards rely on principles acknowledged by this theory, such as cognitive learning, where learning should comprise processes of active individual construction and a process of enculturation into a wider society. Curiosity, inventions and research are perfect ingredients to a positive learning experience. The chapter also acknowledges Vygotsky, another important constructivist, and his theories about how though and language are interrelated for the young child and, to this matter, the importance of guided instruction. Because there are alternate ways to view learning such as Howard Gardner’s multiple intelligence theory and different learning styles, there is a reference to motivating a student through learning tasks, teachers’ beliefs, and students’ beliefs. Also mentioned in the chapter were managing techniques such as classroom arrangements, teachers’ expectations and daily organization.

After reading this chapter I believe that instruction cannot set a unique, defined method of instruction because no two children are alike; they all have different needs and grow up in different environments; therefore, learning styles also differ. I also believe in Gardner’s multiple intelligence theory and that not to children learn the same way. For this reason I believe that concerned teachers should balance in their teaching strategies different activities to accommodate every single child learning potential. In terms of motivation, I also think that teachers should be creative enough to develop different activities using both, classical and operational theories, since they both prove to be efficient and some kids respond to motivation/gratification differently.

Chapter Concrete to Abstract
Summary Chapter three makes emphasis in manipulatives and also small electronics as means of motivational instruments in today’s classrooms. Manipulatives are designed to help student understand mathematical concepts by moving from concrete experiences (manipulatives) to abstract thinking. It is important to consider the levels of association of each age group to fully take advantage of them. Hence, cognitive development is important here. Another key factor when using manipulatives in the classroom is the quality of teacher-conducted instruction/behavior. Different manipulatives can used to aid algebraic, geometric, number operations, measurement, and probability concepts, but is essential that they are appropriate to the age of the children. Some manipulatives described in the chapter were fraction stacks, pattern block, fraction bars, Cuisenaire blocks, colored chips, base-ten blocks, mirrors, geoboards, and algebra tiles. The author included sample activities and instructions on their use. The chapter also mentions the use of technology in the classrooms, such as the use of computer software emulating manipulatives and the use of calculators in the classroom.

Chapter (Cont) Reflections
The use of manipulatives in the classroom can be one of the essential motivators a mathematics teacher should take advantage of. They will help students understand concepts through hands-on experiences easy to remember instead of easy to forget lecture. The use of colors, shapes, group participation and exploration are great assets to mathematical instructions. Manipulatives will accommodate needs of students with different types of intelligence other than the “traditional" verbal-linguistic, such as spatial, visual, and kinesthetic. It caught my attention the author’s point of view on the issue of use of calculators in the classroom as a study aid. I really do not think its use is appropriate to daily use in certain grades such as middle school, for I think that at this age, according to Piaget’s learning stages, is when the most meaningful learning takes place, and the use of a calculator at this point, may lag the student to learn basic operational concepts. For example, it might be easier for them find the x-intercepts through a graphic calculator, than through manual computations. I think they can be very motivating, eliciting curiosity and love-for-math at the early school years and that they definitely enhance learning in high school grades. I am still not convinced about middle school.

Chapter Standard-base Curricula
Summary Chapter four focuses in new developments in secondary mathematics curricula. It introduces the term instructional sequence to distinguish between the standards-based models described in this chapter, such as The Core-Plus Mathematics Project and Contemporary Mathematics in Context, developed by the University of Michigan, or MathScape, and the traditional daily lesson format of most traditional textbooks. These middle and high school models depict standard-based curricula intended for all students The chapter also outlines segments of institutional sequences from prominent middle school and high school programs. The models gear towards a better understanding of mathematical concepts and its applications to daily life activities. Research on these new approaches has shown a great deal of success. The variety of activities and techniques used make mathematics a motivating subject. The chapter also makes emphasis on activities that incorporate methods of data collection, including probeware, and ways to analyze mathematics.

This week I had the opportunity to look in retrospect to my secondary-years math instruction twenty years ago and compare it to my most recent experiences at BCC and FAU. As a math student in the past two years I have truly enjoyed the paths of the standards in mathematics education. My recent experiences include working cooperatively, analyzing data, daily problems solving, and case presentations, among some of them. I have seen how math works in today’s world. Thanks to this new teaching approach I have come to love math so much that I even changed my career twice, from accounting to elementary education, and finally, to math secondary education. My passion for math developed, without a doubt, to the new teaching practices, like the ones depicted in this chapter. Definitely, I would like to emulate my experience with my future students. I used to see mathematics as a bored, multi-task, senseless, repeated activity suited only for “higher minds”. Today I view it as an integrated part of this world. There is math all around me, and math is the origin of many things. Its exploration is a fun activity, and it is definitely good for the “common” person just like me!

Chapter Geometry and Algebra Redefined
Summary New strategies on today’s algebra and geometry instruction is the main focus of this chapter. It explores current developments in each area, making emphasis in the way they integrate to each other. According to the text, the “old” model of teaching algebra and geometry as independent disciplines, reported very low passing grades (only between 40-50%). The low return in academics forced teachers and researchers to explore new methods for teaching them. The result of this efforts are the use of an integrated curricula, use of inductive and deductive geometric reasoning, use of three-dimensional geometry, formal proof on its three forms: justification of conclusions, analytic geometry, and transformational geometry. The new models also include the use of manipulatives, visuals, hand-on activities, and a big stress on electronics such as computers and graphic calculators. All of the developments are adapted to the NCTA standards.

In this chapter we can have a “good felling” about the importance of research in the area of academics. Once a flaw is devised in the system, mathematicians and educators do commit to working on a better quality of education. They have concluded in a more effective way of in teaching mathematics by adapting techniques and developing new models that takes into account the evolution of society and the introduction on new technologies. The use of the standards is also important for it directs the efforts to every single child in a classroom. For this reason, the use of assisted technology, manipulatives, models, realia, among others, makes the concepts easier to understand to a larger group of students with different intellectual capabilities. On the same token, the idea of integrating algebra and geometry also facilitates understanding of both related subjects.

Chapter Planning Instruction
Summary This chapter deals with lesson planning and methods of instructions. In order to be an effective teacher it is important to come to class prepared an organized to deliver instruction. This can be attained through detailed lesson planning that motivate students to learning and satisfy curricula requirements. Before starting the actual planning, three aspects must be considered: colleagues’ expectations, abilities of your students, and available resources for instruction. Long-range planning includes a semester-plan, which assists in setting goals for instruction, and a unit planning. It is recommended to start a unit plan with a concept map detailing the main concepts and their interconnections. Unit planning also requires balanced decisions in pacing the lesson, sequence, depths, breaths, skills and processes. Daily planning is crucial for beginning teachers. This one-page document must instate the day’s goal, activities, timing, a beginning-middle-end structure, and reflections notes. The author also acknowledges different types of effective instruction, such as explicit, or teacher oriented, problem solving and inquiry-discovery.

As a beginner teacher, I realized the importance to come prepared to class with a motivating, fluid lesson plan every day. I learned that class management problems occur more frequently when the teacher has an incomplete lesson plan. Students tend to misbehave when they have nothing interesting to do in the class. Regaining control of the class can be very time consuming. It then becomes a struggle between managing behavior and assessing consequences, or teaching under unfavorable conditions (if possible) and cover at least some the activities of the lesson plan; a position very unfavorable to any teacher, especially, a non-experienced one. In order to avoid this circumstance as much as possible I will look for advice from experienced teachers and try to do my best to come up with varied lesson plans every day. I also think that knowing your students is crucial, for this knowledge is a determining factor is when doing your planning and the way you deliver instruction. For example, an active, very well behaved group will make it easy to cooperative learning and try the problem solving or the inquiry-discovery approach to learning, while a hard to focus group will benefit more of a teacher- guided instruction (explicit instruction.) I also think, that no matter how diverse your group is, a teacher should try different types of instructions to make classes fun, varied and interesting every day.

Chapter Promoting Communication in the Classroom
Summary This chapter` emphasizes the importance of interpersonal communication, both written and verbal, to enhance the environment in the classroom. The author talks about the importance of class participation, the art of questioning and listening, both, in whole class discussions and cooperative learning, as well as a list of do’s and don’ts. Writing is also described as an important form of communication. Journals, learning logs, and student-writing problems are some of examples of creative writing that can enhance communication between teachers, students, and even their parents. The chapter ends with the mention of maintaining an appropriate class atmosphere where this interpersonal communication can take place smoothly. Some class misbehavior- preventive techniques are described from excerpts of Kounin's research, such as the “ripple effect”, whithitness (teachers awareness of what is happening everywhere I in the class at all times), momentum, and group accountability. The author also suggests a list of preventing class disruptions, ranging from small misdemeanors to more serious, chronic situations. Disciplinary sanctions depend much on the relationship student- teacher and school policies.

It really caught my attention the author’s mention that as a new teacher, maybe I should be more concerned about having a disciplined class than in knowing the content area I am teaching. Content I would acquire, without a doubt, from my own schooling; but class management is an art that can best be executed through experience. A luxury no new teacher has. I know there is a chapter in this book that deals with class management, and I know the semester I will spend in a clinical placement will give me a better understanding in this topic. I really liked the ideas the author reveals in classroom communication. Some of them, although obvious, really made me think about them twice; like waiting 3-5 seconds after posting a question. Five seconds do seem an “eternity” when you, as a teacher, want to assess your students understanding on material previously taught. Also, I think time is a big issue in today’s teaching system. Even though there is an interesting rationale for the extended waiting time, I do have the concern about being able to warm-up the class, create empathy and motivation towards the day’s lesson, class lecture, activities, assessment and wrap-up /conclusion in just one block of class. I believe in quality education rather than quantity. Quality education requires time; time to communicate and create a positive rapport with your students, time to assess the adequately (3-5 minute wait to elicit as many responses from diverse students.), and even time to enjoy and have fun with them. Learning time-managing techniques, then, is going to be an important issue for me.

Chapter : Assessing Performance
Summary This chapter gives an insight on individual student performance. It describes formative assessment as a way to evaluate students between major or end-of-unit tests. Student academic process should take into account not only content goals, but also computational skills, abilities, and to use language and symbols of each mathematical domain. Formative assessment can take place in the form of homework, an important instrument that, if used appropriately, could measure in depth reasoning, understanding, and application of mathematical processes. Some guidelines for effective homework are as follows: 1) all students should receive the same assignment, 2) Do not use it to teach essential skills, and 3) Never give it as punishment. Individual self-assessment in form of a journal is another way of formative assessment. Summative assessment relates to teacher-produced tests and standardized tests in a standards-based classroom. Even though summative assessments are necessary to measure students’ knowledge, according with standards, it should not be used as a sole way of information. Conceptual understanding and problem-solving skills, for example, are not considered in this type of testing. Test corrections, test timings, design, and test reviews are also considered in the chapter. Chapter eight ends with a discussion on grade determination. Scholars like Kohl and Guskey oppose grading as a sole mean to determine mathematical understanding. Moreover, Guskey even thinks that it may be very harmful upon placing the teacher in two incompatible roles: advocate and judge. Assignment sheets are described as a powerful tool to allow students participate in the grading process.

Chapter eight left me with an “uneasy feeling” inside…. Testing,… cheating,…. Does grading really measures how much children are learning in our schools? I would need to agree with Kohl and Guskey on the harm and unfairness concern. Cheating homework and testing is a common issue every teacher acknowledges. So how can anyone determine learning under this condition? To add up against testing I also consider that tests are biased and not every child is a good test-taker. Timed tests also bother me. How many times a timed-test restricts a student ability to demonstrate knowledge? Also, how do you measure effort? How do I comply with school grading policies and what I believe asses best my students’ performance? Maybe a complete rubric that includes a grade for content knowledge, comprehension, oral presentations, and a portfolio would be a good choice. I would not include homework for a grade because I consider homework to be very useful for class review and extra practice. . For this reason, I would provide my students with answer sheets and spend a portion of class time to go over questions. Yet, I would never grade them. To encourage students to work on them, I would include homework questions on written tests.

Chapter “Equity” Summary
Chapter 9 “Equity” addresses multiple issues that deal with the many efforts to make mathematics education available to most students, taking in consideration their different abilities and other circumstances such as genre, socio-economic factors, and cultures. Being aligned with standards based education is an important cue here. The chapter addresses the frequent use of tracking students according to their abilities, mainly in middle and secondary schools. It also addresses strategies for “detracking” students by encouraging them to take advanced mathematics course at early grades, such as fourth or seven grades and also by eliminating low-track levels in some other schools. The chapter also introduces the QUASAR project, an educational reform project to promote the study, development and implementation of enhanced mathematics instructional programs for middle schools students in low socioeconomic areas. The chapter goes one wit the description of multiple techniques to make math accessible to every students, especially those linguistically diverse. Most of the techniques mentioned do comply with those associated with NCATE standards for math education, such as cooperative learning, scaffolding instruction, use of manipulatives, and challenging all students. Being sensible to their differences is also pivotal to multicultural education. Gender differences in mathematics are present after fifth grade, when females start to lag behind boys. Use of calculators seems to be broadening this gap. The chapter ends with the special education section, which focuses on instruction for children with disabilities, both, the group with mild learning disabilities and the gifted group needs special assistance to fulfill their capabilities.

As a teacher I just cannot imagine how to address a single lesson plan to students with special needs (gifted or mentally retarded), or linguistically disadvantaged. . I just do not think it would be easy to give a mentally challenged student and a gifted one the same lesson plan, especially, because of time constraints that would not allow me to “teach” two versions of the same content, and because they both use the same textbook. Assessments would also be a challenge. If I give the “smarter” kid a more challenged assignment than the rest of the class, how do I assess that student if his or her assignments are different from that of the rest of the class. Thinking it over, I just think that this extra material should not be graded at all but given only as motivation and support for the child’s special condition. In the case I have a mentally challenged student mainstreamed to my classroom I just hope I have enough support from the ESE department in providing me with those extra activities especially designed for these students. As a good teaching strategy, I could have them work together, though. Unfortunately, In the ESOL strategies course that I took at FAU the curriculum never addressed assessments for these children. It is also very disappointing to see that al the great theories approached by the SIOP model cannot be implemented in class due to lack of time and FCAT demands. Experience and peer communication will guide me through, I hope.

Chapter “More on Assessment”
Summary Chapter 10 focuses on the use of performance tasks and rubric scoring as alternate ways of assessments to the previously discussed in chapter 8. They are both a very important factor of standard-based teaching, for students are expected to show their reasoning and justify their approaches to the given problem situations, it is a two-way communication tool between educators and students, and demonstrates applications of mathematical concepts. The performance assessment is also considered part of the curriculum, and it provides valuable information about students’ progress to both the teacher and the student as an ongoing process so that weaknesses can be corrected along the way. Some of the multiple forms of performance assessment are mathematical tasks, ongoing projects, investigations, and portfolios. Scoring rubrics are usually provided to facilitate assessment. It is important to note that in this type of approach students are not being evaluated by comparing them to other students, but to specific performance standards. The chapter also touches on standardized or “high stakes “tests. The author supports the abolition of these tests providing ideas that support the statement that these biased tests just do not measure educational effectiveness.

At the end of the chapter there is a phrase that really caught my attention: the author states “It is our belief that the job of the teacher is primarily to deliver curriculum to students… However, you will be doing a disservice to your students and profession if tests objectives become your curriculum” What a powerful statement! Unfortunately, nowadays we see the majority of teachers –at all levels- teaching just for a test (FCAT, SAT). This practice leaves behind a greater opportunity to standards teaching. There is just no time for discussion work, group activities, hand-on experiments, or even devoting time to ESL students. The SIOP model, also, becomes just a utopia, never a reality. Use of mathematical cooperative learning, use of manipulatives, portfolios and research are just not possible when teaching for a test. Sometimes I just think on how many great teaching strategies I have learned these pasts semesters , and I just wonder if I will ever be able to use them in my classrooms if I am also “forced” by the system to just teach for a test, as many good intended teachers are.

Chapter Communicating with Parents and Community
Summary: Chapter eleven is about effective ways to establish a positive communication and interaction between parents and the school community. Teachers not only have the responsibility to interact with others beyond the classroom and model their students how to integrate themselves to society. There are many different ways to establish this positive interaction such as community service hours and educational resources such as libraries and museums. The chapter is very helpful in providing new teacher tips on how to approach parents during school nights and parents conferences. Items to be discussed in these meetings were not only school performance but also personality related, or even sudden changes of behavior. By working as a team, Parents and educators could more easily address any present or future concerns. Some of the school-based events described activities were back-to school nights, open houses, and parent-teacher conferences. Mathematics contests and clubs were also depicted as an effective way to motivate students into math learning.

Chapter (cont) Reflections:
The low return data on parents’ involvement in school related activities such as parents- teachers conferences really caught my attention. Then I remembered a day a high school teacher told me that she was tired of calling parents to conferences related to student’s behavior. She told me on that occasion that “parents just didn’t want to know”. She commented that her feeling was just that for those parents was easier to send them to school where they will be looked after, than having them at home in gang activities. The conference, them, was just a waste of time for them. So the question arises on how to motivate these parents to get involved and motivate their kids themselves? School cannot just be an escape goat to gang problems. Schooling is just more than that, it is a treasure; the only way out in life. And this message has to be passed out to parents. It is also important to make the time for them. Flexible hours for the appointments, web communication, folder communication, and phone accessibility are just a few to mention. Fun activities like Family math nights, play nights, family fun days, are also good examples on how to attract parents to school.

Chapter “Professional Growth”
Summary The second and the third sections of the Professional Standards for Teaching Mathematics deals with teaching evaluation and professional development. Both aspects will be the main concern for this last chapter. Also, there is a section on the national board certification and professional teaching standards. Evaluation is a continuing process that bases in gathering data, reflecting on it, and acting accordingly to improve instruction. It should be used as an important tool throughout the whole life of teaching, regardless the years of experience, with the aim to improving their teaching. Evaluation can be done by an evaluation supervisor as a planned or impromptu visit. There is also individual, self-reflecting evaluation, and this happens when the teacher reflects on his or her experience at the end of the day. Another option is to ask students about the lesson or even videotape a class for further reflection or analysis. Professional development includes participation in projects, conferences and workshops, as well as subscription to professional organizations. This type of involvement adds both to content and pedagogical knowledge. The chapter ends with a brief description and advantages of the National Board Certification, as well as on the process to obtain it.

I would like to comment on the tenure process since I had never heard about it before. I believe that a tree-year probation period benefits both, the teacher and the school. For the teachers, evaluation supervisors frequent visits or portfolio reviews do lead to a continuous self-observation and analysis. Then teachers have the opportunity to seek further resources, like workshops, or seeking advise from experienced teachers to improve their instruction abilities. The school also receives a positive benefit from the probation period just by knowing at the end of it, that the tenured teacher is reliable to do a good job. It gives the administrative officials peace of mind. Even though being observed may be very intimidating, I think only a positive outcome can result from it.

Math Problems

Problem 1: “Japanese Swords on Trains”
On Japanese trains, there is a rule that forbids passengers bringing onto the train objects longer than 36 inches. How did a passenger travel with a ceremonial sword that was 42 inches long? Solution: A possible solution is that the passenger was a respected foot soldier samurai. As a military, could not be dispossessed of his sword, no matter how long it was. Strategy: To solve this problem, firs I read it several times (ESOL strategy) in order to fully understand the problem. My plan to solve it was to investigate about Japanese culture. First, I investigated about ceremonial swords, and learned that they were usually rounded. This approach did not solve my problem. Then I looked at a different approach: the passenger. And this is where I found out about the noble military class of samurais. When checking the result, it made sense to me. It had logic.

Problem 2: Juan’s Spending
A. Juan spent 3/5 of his money and then had $12 left. How much money did Juan originally have? Let T= total money I had T-3/5T = 12 Solving the equation T= $ 30

Problem 3: “The Census Bureau”
1.In 1995 the U.S. Census Bureau reported that the average lifetime earnings of a person who holds a high school diploma is $821,000 and that the average lifetime earnings of a person who holds an associate's degree is $1 million. Draw a bar graph that appears to show: a. a small increase b. a large increase Data: U.S. Census Bureau (1995)

Problem 4 “Dancers” What is the least number of dancers that a producer must hire if the choreographer wants to arrange them in groups of 4, 5, or 6 with none left over? It all depends in the number of groups he wants to form. For example, if he is going to work with a minimum group of just four dancers, the answer would be 4. For five dancers, the answer would be 5, and for six dancers answer is 6. There would be none left in any of the cases.

Problem 5 “Holly” .Holly's mean average on six exams is 87. Find the sum of her scores. Test1+Test2+Test3+Test4+Test5+Test6 / 6 = 87 87x 6 = 522 is the sum of the six tests

Problem 6 “Phone companies”
In a college dormitory, each student has a choice of two phone companies. Company A charges $7.46 per month plus 13 cents a call; Company B charges $6.17 per month plus 17 cents per call. A) About how many calls do you make per month? B) For each company, write an equation which represents the cost in a given month in terms of phone calls. C) Graph each equation you wrote in part B. Be sure to label the graph. D) Which plan is best for different types of users? I make approximate 100 calls per month Company A: f(x)= x Where x represents number of calls per month Company B: f(x) = x

Problem 7 “ Firefighters”
A firefighter stood on the middle rung of a ladder spraying water on a burning building. As the smoke cleared, she stepped up 3 rungs. But a sudden flare-up forced her to go down 5 rungs. Later she climbed 7 rungs where she worked until the fire was out. Then she climbed the remaining 9 rungs to the top of the ladder and stood on the roof. How many rungs did the ladder have? Explain your thinking throughout. I drew a ladder and marked the middle of the ladder as point (1), from there, I drew and counted 5 more steps down and marked this step (2). From there I counted and drew 7 steps up and labeled this step (3). From (3) I drew 9 more steps up that made the top of the ladder. Finally I counted the steps from the top to the middle one (where I started counting up) , and ended with 29 steps without counting the top one (where you finally stand).

Problem 8 “Logical Buying”
What could you be buying? (A logical purchase) If you buy: 1 The cost is: $1.00 $1 item If you buy: 86 The cost is: $ cents/ object If you buy: 923 The cost is: $3.00 0.003 cents /object I could not find an algebraic answer that made any sense. Of course I cold buy a grain of rice for $0.003 (923 items for $3), but this is not logical. No one counts the grains of rice. So I went ahead and think on “something else” and came up with a reasonable answer: I could buy 3 numbers (maybe plastic ones) , each number with a total value of $1.00

Problem 9 “Half-a Job” Mr. Muddle's house was a disaster. "Everything needs to be fixed, everything needs to be painted, " Mr. Muddle groaned. He felt there was so much to do and he didn't know how to get started. "I'll tell you how I handle problems like that," said Ms. Eng. "When a job is so big that I don't feel like starting it, I just do half the job each day. That way it isn't so hard." How long does it take Ms. Eng to do the job? "Half the job each day," said Mr. Muddle. "That sounds like an excellent idea. I'm going to try it. I think I'll start today with painting the four walls in this room." Ms. Eng said, "Good luck, I'll be back in a week and I am sure you will have your house in much better shape if you follow my advice and do half a job everyday." Mr. Muddle got out the paint, brushes, and rollers, and he started painting the walls in the room. How many walls should he paint the first day if he is going to do half the job? Two walls

Problem 9 (cont) Mr. Muddle figured that half of 4 is 2. And so the first day he painted two walls. The next day Mr. Muddle got out his painting materials again and was ready to start work. "Now, what was it Ms. Eng told me to do?" he asked himself. "Ah, yes, I remember. Do half the job each day. There are two walls that need painting, so if I do half the job today, that means I paint one wall." Mr. Muddle painted one of the walls. Do you think that is what Ms. Eng meant? What should Mr. Muddle have done instead? "Mr. Eng said “When a job is so big that I don't feel like starting it, I just do half the job each day.” By this statement, probably Mr. Eng probably meant that he should do one half of the job on one day and the other second half the second day. So if he needs to paint 4 walls, he would do “half of the job each day”: two walls on one day, two other in the second. The complete job would be finish in just two days!

Problem 9 (cont) The next day Mr. Muddle continued with painting only half of the job. Now how much is left to be painted? How much will Mr. Muddle paint tomorrow? How much painting is still left to do? Do you think Mr. Muddle will finish the job soon? Why or why not? Do you think Mr. Muddle will ever finish painting the wall following what he believed was what Ms. Eng suggested him to do in order to get the job done? He will always be infinite times one half away to finish his job. He needs to do half of the remaining job each day. In this pattern, he will never finish the job. No, he will not. He will always be one half away from finishing (infinite times)

Problem 9 (cont) Why will there continue to be a little strip of the wall that isn't painted yet? Would it probably take Mr. Muddle forever to finish this job? How does this problem relate to Calculus in some way? It is always half of the half, and so on… He will never finish Because it has to do with the concept of infinity over a change of a period of time.

Problem 10 “Coins” 1. I have two coins for a total of 35 cents. One is not a dime. What coin values do I have? This problem cannot be solved using American coins. The Euro coins do not have a 25 cent denomination. So I decided to use Netherlands coins. A 25 cent coin plus a 10 cents coin=35 cents.

Problem 11 “Equation” Add one line and one sign to make a true equation: 99 = 8 9X9=81 1 is a straight line X is a sign

Problem 12 “Magic Squares”
Make a "Magic Square" by placing the digits 1 through 9 in a 3x3 grid so that the sum of the three numbers in any vertical, horizontal, or diagonal line equals the same number, which is called the magic number. What is the magic number? How did you go about solving the problem? *This is a Critical Assignment Problem* I started from the Lo Shu Magic Square 3 5 7 6 8 1 And rotated “clockwise” each digit 2 positions to obtain Maribel’s Magic Square: 8 3 4 6 7 2 The magic number is 15

Problem 13 “A Famous Goat Problem”
. A square barn measures 14 meters on each side. A goat is tethered outside by a rope that is attached to one corner of the barn: a. Suppose that the length of the rope is 8 meters. On how many square meters of land is the goat able to graze? What shape is formed? The shape formed is a circle, which I divided into four quadrants to “see” the section of the barn that was not accessible to the goat due to the fence. A= πr² A= (8)² A≈201 meters ¼ of the total circle area≈ 51 meters (area inside the circle/fence) 201-51≈ 151 meters of land available for the goat to gaze

Problem 13 (Cont) Suppose that the rope is made twice as long as in part a. On how many square meters of land is the goat now able to graze? A= πr² A=3.1416(16) ² A≈804 meters ¼ of the total circle area≈201 meters ≈603 meters available for grazing

Problem 13 (Cont) Lastly, an addition to the barn was built making it rectangular with dimensions 14 meters by 20 meters and extended the length of the rope to 24 meters, on how many square meters of land is the goat now able to graze? The radius of the new circle would be 24 mts. The shape inside the fence not allowed for gazing is now a rectangle 14x20 mts, plus the area of the two triangles formed in the circle where the 24 meter-rope does not allow the goat to gaze. Area for gazing: Area of a circle with radius 24 minus ¼ of this area, plus the two small sections of the circle (radius 4 and 10) beyond the fence reachable by the 24 meter rope. Total area of the circle A= πr² A= (24)² A≈1810mts ¼ of the total circle area≈453 mts ≈1357 meters Small triangle 1: (radius 4) A≈ 50 meters Small triangle 2 (radius 10) A≈314 meters Total area available for gazing≈ ≈1721 meters

Problem 14 “Mackenzie's favorite clothes”
Mackenzie's favorite clothes include four T-shirts, three pairs of designer jeans, and two pairs of sandals. How many days in a row could she wear a different outfit using her favorite clothes? 4X3X2 = 24 days

Research

What is Date Analysis? It is a method for correlating times and dates contained within a file to the modified, accessed, and created/change of status. The purpose of date analysis is to bring together all date-related elements and normalize their usages. A resource may have several dates associated with it, including: creation date, copyright date, revision date, edition date, modification date, etc. Each element has a specific application. The date element is associated with the life cycle of the resource itself (when it was created, when it was digitized, when it was published, etc.). The coverage element is associated with the 'temporal period' of the content of the resource (the era that a photograph is from -- World War II, for example). The recommended best practice for the date element is to use the YYYY-MM-DD format. The recommended best practice for the temporal aspect of the coverage element is to use a controlled vocabulary with named time periods rather than date ranges.

"Quantitative Literacy"
(Also known as “numeracy”. It refers to an ability to handle numbers and other mathematical concepts. In the United States, it is somewhat better known as Quantitative Literacy, and Quantitative analysis has two attributes. The first of these is an "at homeness" with numbers and an ability to make use of mathematical skills which enables an individual to cope with the practical demands of everyday life. The second is an ability to have some appreciation and understanding of information which is presented in mathematical terms. Quantitative literacy involves understanding the role of numbers in the world. It provides the ability to see below the surface and to demand enough information to get at the real issues. It requires logic, data analysis, and probability.... It enables individuals to analyze evidence, to read graphs, to understand logical arguments, to detect logical fallacies, to understand evidence, and to evaluate risks. Quantitative literacy means knowing how to reason and how to think. Some examples of quantitative analysis are the knowledge and skills required to apply arithmetic operations, either alone or sequentially, using numbers embedded in printed material (for example, balancing a checkbook or completing an order form). The Programme for International Assessment (Organization for Economic Cooperation and Development, 2000) defines mathematics literacy as an individual's capacity to identify and understand the role that mathematics plays in the world, to make well-founded mathematical judgments and to engage in mathematics in ways that meet the needs of that individual's current and future life as a constructive, concerned and reflective citizen.

What does the symbol for pi mean?
The mathematical constant π is a real number, approximately equal to , which is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant and as Ludolph's number. The constant π may be defined in many other ways, for example as the smallest positive x for which sin(x) = 0. The constant is named π because it is the first letter of the Greek words "περιφέρεια" (periphery) and "περίμετρον" (perimeter). The Swiss mathematician Leonhard Euler proposed that this number be given a particular name and suggested the use of π.

History of magic squares
Magic squares received their name because there are so many relationships between the sums of the numbers filling the squares. It has existed throughout history and in many different parts of the world. Magic squares have been studied for at least three thousand years, the earliest recorded appearance dating to about 2200 BC, in China. In the 9th century, Arab astrologers used them in calculating horoscopes, and by 1300 AD, magic squares had spread to the West. An engraving by the German artist Albrecht Dürer included a magic square in which the artist embedded the date, 1514, in the form of two consecutive numbers in the bottom row! Because the concept of a magic square is so easily understood, magic squares have been particularly attractive to puzzlers and amateur mathematicians. Even Benjamin Franklin dabbled with magic squares, and an magic square with some very interesting properties is attributed to him. Magic squares have fascinated humanity throughout the ages, and have been around for over 4,000 years. They are found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity and prevention of diseases.

History of magic squares (cont)
Lo Shu Magic Square (The Divine turtle): Chinese literature dating from as early as 2800 BC tells the legend of Lo Shu or "scroll of the river Lo". In ancient China, there was a huge flood. The people tried to offer some sacrifice to the river god of one of the flooding rivers, the Lo river, to calm his anger. Then, there emerged from the water a turtle with a curious figure/pattern on its shell; there were circular dots of numbers that were arranged in a three by three nine-grid pattern such that the sum of the numbers in each row, column and diagonal was the same: 15. This number is also equal to the number of days in each of the 24 cycles of the Chinese solar year. This pattern, in a certain way, was used by the people in controlling the river.

The Divine turtle Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.

Dürer Magic Square In 1514, Albrecht Dürer created an engraving named Melancholia that included a magic square. In the bottom row of his 4 X 4 magic square one can see that he placed the numbers "15" and "14" side by side to reveal the date of his engraving.

La Sagrada Familia Magic Square: The Passion façade of the Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4 magic square: The magic sum of the square is 33, the age of Jesus at the time of the Passion. Structurally, it is very similar to the Melancholia magic square, but it has had the numbers in four of the cells reduced by 1.

History of calculus The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations on vibrating chords; elastic membranes, elasticity of three-dimensional bodies; heat diffusion; light; electricity; spherical harmonics; acoustics; and physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics Calculus made it possible to compute the area and volume of regions and solids by breaking them up into recognizable shapes.

What is Trigonometry? Why is it important in life?
Trigonometry deals with the purely arithmetic relations between specific geometric characteristics of right angled triangles. It is also described as the branch of mathematics that deals with triangles, circles, oscillations and waves. Webster defines trigonometry as the study of the properties of triangles and trigonometric functions and of their properties. Though trigonometry refers to right angled triangles only, it naturally has much usefulness in all branches of geometry, since every polygon may effectively be described as a finite combination of such triangles. Dealing with purely arithmetic relations between the trigonometric functions - trigonometry has also a kin relationship to many branches of pure mathematics, as well as of applied mathematics (thus having many applications in science). Therefore, trigonometry is often considered to be a subtopic of many mathematical disciplines, not only of geometry. Among the scientific fields that make use of trigonometry are these: Acoustics, architecture, astronomy (and hence navigation, on the oceans, in aircraft, and in space; in this connection, see great circle distance), biology, cartography, chemistry, civil engineering, computer graphics, geophysics, crystallography, economics (in particular in analysis of financial markets), electrical engineering, electronics, land surveying and geodesy, many physical sciences, mechanical engineering, medical imaging (CAT scans and ultrasound), meteorology, music theory, number theory (and hence cryptography), oceanography, optics, pharmacology, phonetics, probability theory, psychology, seismology, statistics, and visual perception.

What is an Astrolabe? It is a medieval, historical instrument that was once used by astronomers and astrologers to determine the altitude of the sun orother celestial bodies. It was the chief navigational instrument until the invention of the sextant in the 18th century. Its many uses included locating and predicting the positions of the Sun, Moon, planets and stars; determining local time given local longitude and vice-versa; surveying; and triangulation. Astrologers of the Islamic world and European nations used astrolabes to construct horoscopes.

Why are geometry and measurement related
Why are geometry and measurement related? What is the metric system based on? Geometry is the branch of mathematics that is concerned with the properties and relationships of points, lines, angles, curves, surfaces, and solids. All these “objects” have a dimension, and to fully explain their existence, they must be measured. For example, a line is not a line unless it has a measurement (length). If it doesn’t have a measurement, does it really exist? How can we see it? Also, only by giving them a measurement we can connect them. The metric system is based on the metre and the gram.

Creative Projects

My Math ABC Book

Algebra Definition: It is a branch of mathematics concerning the study of structure, relation, and quantity. The origins of algebra can be traced to the ancient Babylonians ( 1800 B.C.),who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion.

Bar graph Definition: A graph consisting of parallel, usually vertical bars or rectangles with lengths proportional to the frequency with which specified quantities occur in a set of data. A bar graph is a useful tool to organize and analyze information visually using attractive colors and fonts.

Circle Definition: A plane curve everywhere equidistant from a given fixed point, the center. The distance from the center is called the radius, and the point is called the center.

Direction Definition: It is the information contained in the relative position of one point with respect to another point without the distance information. Direction is often indicated manually by an extended index finger or written as an arrow.

Equation Definition: An equation is a mathematical statement, in symbols, that two things are the same. Equations are written with an equal sign, as in = 5. Laplace Equation

Function Definition: It is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted by f(x). The graph of a function f (x)

Geometry Definition: Area of mathematics dealing with solids, surfaces, points, lines, curves and angles, and their relationships in space. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers.

Hypotenuse Definition: The hypotenuse of a right triangle is the triangle's longest side, i.e., the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean Theorem.

Integer Definition: The integers consist of the positive natural numbers 1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. Fractions and decimals are not integer.

Jordan block Definition: A matrix, also called a canonical box matrix, having zeros everywhere except along the diagonal and superdiagonal, with each element of the diagonal consisting of a single number , and each element of the superdiagonal consisting of a 1. Any Jordan block is thus specified by its dimension n and its eigenvalue λ and is indicated as Jλ,n.

Knot Theory Definition: It is the mathematical study of knots.
A mathematical knot has no loose or dangling ends; the ends are joined to form a single twisted loop.

Limaçons Definition: They are heart-shaped mathematical curves. The cariotid is a special case, with a cusp. They arise in polar coordinates in the form r=a+ b sin δ, r=a+ b cos δ The word "limaçon" comes from the Latin limax, meaning "snail." limaçons (cariotids)

Monomial Definition: It is a product of positive integer powers of a fixed set of variables, for example, x, xy², or x²y³z. A monomial multiplied by some coefficient is called a term.

Natural number Definition: positive integer one to infinity.
Natural numbers are usually used for “counting”, and its symbol is Z+- Graph of natural numbers

Octagon Definition: An octagon is an eight-sided polygon.
The octagon with alternate sides parallel to the - or -axes is the shape used for traffic stop signs in the United States.

Pi (π) Definition: The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter. Pi is known to be an irrational number =

Quadratic equation Definition: It is a second-order polynomial equation in a single variable x. ax²+bx+c for a≠0 The Fundamental Theorem of Algebra guarantees that the quadratic equation has two solutions. The graph of a quadratic equation

Radian Definition: It is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle. Radians are the most useful angular measure in calculus because they allow derivative and integral identities to be written in simple terms 1 radian

Sine Definition: A ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. The sine function is one of the basic functions encountered in trigonometry. Graph of the sine function

Tangent Definition: In geometry, a straight lineis tangent to a curve at some point, if both line and curve pass through the point with the same direction; it is the best straight-line approximation to the curve at that point. The slope of a tangent line can be approximated by a secant line (a line that intersects two or more points on the curve.)

Uban numbers Definition: They are defined as numbers whose English names do not contain the letter "u" (i.e., "u" is banned). The first uban numbers are: 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, ...

Vector Definition: It is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction. A vector is often represented graphically by an arrow.

Wavelength Definition: It is the distance between two sequential crests (or troughs). The wavelength is denoted as λ and generally has the unit of metres.

X-axis Definition: It is the horizontal axis of a two-dimensional plot in Cartesian Coordinates that is conventionally oriented to point to the right . Physicists and astronomers sometimes call this axis the abscissa.

Y-intercepts Definition: The point at which a curve or function crosses the y-axis (i.e., when in two dimensions). For functions specified in form y = f(x), the y-intercept is easy to find by calculating f(0). Y=3; y-int. = (0,3)

Zero Definition: Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. Zero is the only integer that is neither negative nor positive.

Math About Me I am Mad about numbers 1, 4 and 5 and 9.
I was born on , and my daughter, my birthday present, was born on In month 11 (November) she will be 11 years old. I have 1 husband, 1 dog and 1 job as a substitute teacher (K-12) I am taking 4 classes this semester!, and I spent 4 years in BCC to gel my AA in elementary education last year. Today is august 22, which adds up 2+2=4 We were 5 of us in my family: three children and mom and dad. Now we are just three, my husband, my daughter and I. There are 5 numbers in my zip code and my house number. I am 5'2 and weigh 121 pounds, and 1+2+1= 5 (Trying to lose some, though) I guess number 9 is just my lucky number

Clinical Experience Teacher Observations

Nova Middle School Florida October, 2006

General Observations The math classrooms are located within a structural building, and once a person enters the area, he or she can visual math posters in all the hallways. The are varied in content. I did not see any children’s work in the hallways, but they were present inside Mr. A’s room. I was told that the only math family activity is “Family Math Night”, which is held every year towards the end of the year. Mr. A involves his kids in weekly competitions solving the “Problem of the Week”, which is posted in the board every Monday to be turned in by Friday. Children accumulate points for doing this that later can be traded for homework passes. Mr. A’s students are all very loud when they enter the room. It takes him too much of instruction time to noise control and task focus. He does threatens kids with calling their parents and detentions. Still, students do not respond accordingly. I think he should use some other type of classroom management different to the present one.

General Observations (cont)
Mr. A seems knowledgeable of the subject matter. He knows every single one of his students by their first name. When he poses a questions, he gives time for everybody to think about it and raise their hands. He asks a lot of questions, and a few kids are encouraged to work the questions on the board. Body language is accordingly. He moves around the room frequently to check on his student’s work. Mr. A mostly teaches through the overhead projector. This is useful because he faces the class at all times. My reflections: I believe that students have a motivating place to learn. Walls were covered with multiple colored posters and signs outside and inside the classrooms. This is definitely a “Mathematics Friendly” school.

Observing the Students
There is a lot of loud every morning as students enter the room. Approximately one student out of four are Hispanics and they talk only Spanish among themselves. I would estimate than only 60% of the class is listening to the teacher attentively and actively participating in the class. The rest are just vaguely listening or doing something else. Mr. A is on top of this students regularly. He stops the class frequently just to have the whole group immerse into learning. After they take a 20 minute quiz every morning, the teacher spends about 50 % of the remaining time either teaching (reviewing( or answering questions. The rest of the time is devoted to practice (multiple worksheets, which he also grades every day) There is a lot of grading going on in this class. Questions asked are mostly homework and quizzes questions. Very little questions are related to the teacher’s teaching.

Observing the Students (cont)
09/25/06 9: 30-9:35 Children enter the room. 9:34-9:40 Pledge of Allegiance, National Anthem and morning announcements 9:45-10:15 Daily quiz. 20 questions. 10:20-11:00 Students worked independently on a review worksheet. Collected. 11: 05-11:10 New group enters the room 11:10-11:40 Daily quiz. 20 questions. 11:45-12:15 Professor uses overhead to answer HW questions. Questions and answers period. Whiteboard and overhead. 12:15-12:55 Students worked independently on review worksheet. Collected

Interviewing the Teacher
What mathematics content is currently being studied in the classroom? What do the children already know about this topic when they come to this grade level? What are the key outcomes children of this grade level learn for this topic? Students are learning about converting fractions into mixed numbers and vice versa, and also about equivalent fractions Children know about simple fractions, time tables, multiplication, division, and decimals. Children learned about another way they will “see” fractions in the real world.

Interviewing the Teacher (Cont)
What will children learn about this topic next year? Are there particular materials you use to teach this topic at this grade level? What gives children at this grade level difficulty with this topic? They will learn mostly about problem applications. “ I have a video they saw at the beginning of the term, which covers this section. We also use worksheets to give them practice”, “I teach through the overhead projector” “Some kids still struggle with their time tables…”

Reflection: I had the chance to grade yesterday’s quiz on the subject, and the main problem I saw was, as Mr. A told me, that students just do not know their time tables. Most of them knew the procedure, but failed in the arithmetic. The solution I see is review their time tables often. I would use math multiplication flashcard games to motivate them into learning them, and I would also include a couple of multiplications in every quiz I give them. I think it is not acceptable that a student attending middle school still has trouble learning their math multiplication tables.

7. How do you believe children learn mathematics? 8. What are the characteristics of learning at this age? 9. What type of instructional activities best help children learn mathematics at this age? 7. By paying attention in class, doing their homework and much practice 8. At this age children are curious but also very easily distracted with other activities. It is hard to get them involved in homework every day. 9. These kids love to play math games and use manipulatives in the room. The only reason Mr. Adeshina limits their use is because children tend to get very loud with them. They also like the videos (especially the Spanish speaking students).

What percent of the class time is devoted to conceptual development? Practice? How do you feel about mathematics and teaching mathematics? How do your attitudes and actions influence your student’s attitudes towards mathematics? About the same time (50/50) He has loved math since he was a young boy in Africa. He has taught math in the US for 10 years, an he still loves it. He says you have to come with a great attitude every morning. Some students tend to misbehave at this age, but they cannot be grounded forever. It is important to “let them be” once in a while as long as they are respectful. They like challenge, so including games and math mystery problems should be used frequently. It is important to relate to them every single them.

How do you support students that display mathematics anxiety? My Reflections: How did you answer to these questions match the teacher's responses? How did the interview give you better insight into what happens in the classroom? Mr. Adeshina always tells his class in advance when all his quizzes and chapter tests are going to be. My Reflections: I observed that it took 5-10 minutes every day to get his students in task every morning. I though it was a lot! I think this group age needs a lot of rewards (praise) for well-doing, instead of do many consequences (which I just do not think are effective with this instructor). Also, a big problem in his room were the non-English speakers, who got distracted very easily. Maybe some ESOL strategies need to be implemented.

What does the statement “all students can learn mathematics” mean to you? Do you hold the same high expectations for all your student? About assistance to meet expectations. What is one thing you do in the classroom to ensure all children learn. Is there school wide assistance? My Reflections: How did your answers to these questions match the teacher’s responses? Mathematics is for everybody. Everyone needs it. Nevertheless, not everybody learn at the same pace. He expects more from his “high learners”, but he really is rewarded when his “low learners” get good grades. 1He is available before and after class to answers any doubts. He is also reachable at home. School does support learning through a group of counselors and peer tutoring. My Reflections: I believe every child has a potential to learn; nevertheless, I have learned from my experience as a substitute teacher, that many times mainstreaming students to regular classes does not benefit the child, and the class is so distracted by these child, that their own learning is deeply affected.

My Teaching Experience
Unfortunately, I was not able to teach much during my visit to Mr. A’s class; first of all, there was an extensive chapter review, with children just asking questions on homework; also, he gave a quiz every morning, which limited teaching time, and he wanted me to grade his daily quizzes as well. Nevertheless, I was happy to teach one-on one some of his Spanish Speaker’s who had a lot of questions for him. I was able to solve all of their questions and the experience was worthwhile Due to the nature of my experience I did not use any lesson plan for its purpose. This was an “on the go” activity just to get them ready for next week’s unit test.

Classroom Manipulatives
All manipulatives were kept in armoires neatly organized. I was able to find the following items: Base ten blocks Cuisenaire rods Geoboards Meter sticks, Scales Pattern blocks Tangrams Fraction Models. Flashcards Math board games I did not see the teacher using class manipulatives during my four visits.

Classroom layout

The Textbook Mathematics Applications and Concepts Teacher’s Edition
Course 3 McGraw Hill, Glencoe editions. Analysis Students are working on Unit 5 Algebra Functions (Chapter 11) This is a very colorful edition with a large variety on fonts and images. There is a large section on higher thinking skills, a review of prior learning section, an introductory page on the lesson with a sample of useful applications, and even an ESOL corner with helpful ESOL resources for the teachers. There are also multiple boxes with tips and important facts Instruction goes by numbered steps with an illustration for each step. There is a solved activity for each stop and a “try on your own” one. Answers for these ones are not provided. At the end of the chapter there is a large section of application problems with answers for the odd ones at the end of the book.

Math Autobiography My first encounter with math started at age three in school. It really blows my mind that I can still remember the large blue room and the unit blocks the teacher was using to introduce the numbers. A different color for every block; brown for 8, yellow for 1, and so on. Then I remember Miss Eva, my first grade math teacher. Long black her and always with a smile. She was the best. My next recollection starts in middle school, when I first realized, and liked a lot, that I could finally give some use in physics and chemistry to all the math I had learned so far. Having a good math background was definitely a great asset to have in these two hard subjects. Thy upper instruction I will never forget. I entered in the Universidad Simon Bolivar, famous to be a “newer” modern university with the best teachers, technology, and curricula available on those days. There, the math program “Math 23”, it was called, was in an experimental phase.; and I was one of their guinea pigs. We reviewed and learned all the geometry, trigonometry and algebra through a self study course. We had tutors and a math lab available at all times, and we had a weekly test on Mondays for every unit. One could take the test as many times (sometimes up to 20) until you passed it. They were 50 multiple choice questions. Passing grade was 80, and every wrong answer took away a correct one. So guessing was not a choice. I went up to unit 13 (integrals), which I now do not remember a thing, but their existence. Our study group would spend the entire weekends in someone’s house having a lot of fun but studying as hard as one could imagine to pass the Monday test. Right after I changed my bachelor and went to study some place else, Math 23 was canceled. Passing rate was backing up the programs of study!. An experience it was!

Math Autobiography (cont)
The upper instruction I will never forget. I entered in the Universidad Simon Bolivar, famous to be a “newer” modern university with the best teachers, technology, and curricula available on those days. There, the math program “Math 23”, it was called, was in an experimental phase.; and I was one of their guinea pigs. We reviewed and learned all the geometry, trigonometry and algebra through a self study course. We had tutors and a math lab available at all times, and we had a weekly test on Mondays for every unit. One could take the test as many times (sometimes up to 20) until you passed it. They were 50 multiple choice questions. Passing grade was 80, and every wrong answer took away a correct one. So guessing was not a choice. I went up to unit 13 (integrals), which I now do not remember a thing, but their existence. Our study group would spend the entire weekends in someone’s house having a lot of fun but studying as hard as one could imagine to pass the Monday test. Right after I changed my bachelor and went to study some place else, Math 23 was canceled. Passing rate was backing up the programs of study!. An experience it was!

Math Autobiography (cont)
Then I remember BCC. This time I am in school because I want it, not because I have to, and I am taking my algebra classes without a problem. Twenty years after high school graduation and I am, getting all “A”s in all my math classes. I decided I am pretty good at them and decided to go for a bachelor degree on math education. I don’t expect to get an A in my calculus class this semester, but I will not fail my dreams!

My Math Education Philosophy
Every student deserves my committed attention Every student deserves my best subject knowledge. I will stay up-to-date in new math teaching strategies. Every student deserves to be respected as per its knowledge, aptitude and background. Starting my days with a positive attitude is crucial for a great teaching-learning experience. I will facilitate learning through the latest learning resources. Math is fun! Everyone can learn how to love it.

My resume NAME ADDRESS, PHONE Email Education:
2005-present Florida Atlantic University. Secondary Math Education. B.A expected by Summer 2008 Broward Community College A.A. in Elementary Education I.U.N.P. Caracas-Venezuela AA in Marketing and Advertising 1983 Aerotraffic, Caracas- Venezuela Reservations and Travel Agency Technician Colegio Mater Salvatoris, Caracas-Venezuela High School degree, Special mention in Science Other training: 2002 University of Augusta. Quickbooks Course. 1999 Atlantic Vocational Center, Florida: 8 months Medical Coding 1995 Broward Community College, Florida: 2 year Accounting

My resume (cont) Work Experience:
2002-present Broward County Schools. Substitute teacher K-12. Full time job. Weston’s Dollar, Inc. Responsible for purchases, inventory, payroll, cash register and other management related duties. “Things and That”. Retail Storeowner and manager. Retail and wholesale representative for SMC International. Responsible for activities dealing with customers, vendors, account receivables, account payable, distribution and inventory. Internet sales in Yahoo Store. Usana Health Science Independent Distributor Internet sore manager and retail home business with over 50 clients and distributors in downline. Internet sales in Yahoo Store and an independent website. “Watkins” Independent Distributor “Herbalife” Independent Distributor Leo Burnett Advertising Agency, Caracas/Venezuela Account Executive responsible for 5 mayor accounts: Heinz, Panasonic, Simmons, Continental Airlines and Buittoni. Mercedes Hercules&Associates, Marketing Research, Caracas. Marketing research analyst and programmer. JMC Young & Rubicam Advertising Agency, Caracas-Venezuela PR executive with a 12- client portfolio. Bates International, Caracas-Venezuela. Advertising Agency. Copywriter and Creative Department Supervisor.

My resume (cont) Languages: English and Spanish Merits and Awards:
FAU Member of Honor Society Phi-Theta Kappa Honor Society BCC Member of Honor Society Other Qualifications: Computer literate. Accounting (Quicken, Quickbooks, Peachtree), Word, Works, Excel, Access, Power Point, Outlook. Medical terminology

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