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Mathematics 1

Running head: CURRICULUM GUIDE AND CRITIQUE FOR MATHEMATICS

Course Project: Curriculum Guide and Critique for a High School Mathematics Program

J. Michael Dillon

Walden University

Curriculum Theory and Design EDUC-8807-001

Dr. Linda Crawford, Professor

11-23-08

Week 12 Mathematics 2

Course Project: Curriculum Guide and Critique for a High School Mathematics Program

Introduction

The Iowa Core Curriculum is a curricular document developed by a team of educators, professionals, and content experts in direct response to legislation in the state of Iowa. The document is an attempt to provide guidance to schools across the state as they develop content for the core curricular areas of mathematics, literacy, and science. In addition, the actions taken at the state level to develop guidelines for educators reflect emerging social, economic, and political issues facing the United States. However, the Iowa Core Curriculum is intended to act only as a guide for schools in the development process and requires each district to uniquely design curriculum that reflects the local educational needs as well as the general aspects of the statewide model. In the area of mathematics, those responsible for curriculum design must find ways to update the existing mathematics curricula to meet the standards of a “world class” mathematics curriculum (Iowa Department of Education, 2006a, p. 16).

Social Demands and Purpose

Despite the general feeling that most students in the state of Iowa receive a high level education, the authors of the Iowa Core Curriculum emphasized that the demands of an ever- changing, technology-based society requires educators to rethink curriculum in the core educational areas (Iowa Department of Education, 2006a). The authors also stressed that economic and political threats face the United States as students from other nations experience educational opportunities that prepare them to meet 21st-century demands. Leaders in the

Department of Education cautioned that “Iowa schools cannot afford to stand still and expect to graduate students who are adequately prepared. We must jettison the status quo just to stay competitive…” (Iowa Department of Education, 2006a, p. 6). Although the warnings issued by Mathematics 3 the department of education apply to all educational areas, math is one content area receiving special attention.

Members of the work team responsible for developing curriculum in mathematics stated that the country is facing a “crisis in mathematics education” (Iowa Department of Education,

2006b, p. 1). The work team noted that not only are American students performing well behind students from other countries, but the typical mathematics curriculum also lacks the depth necessary to engage those students and to promote critical thinking and the development of problem-solving skills. The urgency behind updating existing mathematics curricula was delineated in the description of several issues currently being faced: a broad mathematics curriculum that lacks both depth and a focus on problem-solving; a lack of rigor and relevance within the curriculum; and global economic pressures from other countries excelling in their own educational programs (Iowa Department of Education, 2005, 2006a, 2006b). Educators at the local level have been given the challenge and mandate to reorganize mathematics curricula to ensure that today’s students will be well-prepared for whatever the future demands.

Content Elements of the Mathematics Curriculum

Although specific learning objectives were not included in the Iowa Core Curriculum, the authors described the important aspects of effective mathematical content and provided examples for educators to work from. Areas of the content that were identified included: algebra, with an emphasis on functions, equations, simplification of expressions, rate, and iteration; geometry, with an emphasis on coordinates, basic properties, transformations, trigonometry, and graph theory; statistics, with an emphasis on descriptive and inferential statistics and probability; and quantitative literacy (Iowa Department of Education, 2006b). The Iowa Core Curriculum also describes the importance of utilizing an integrated approach to content delivery. An integrated Mathematics 4 curriculum does not present content through distinct classes that center around specific topics, but, instead, it presents material in a manner that ties the various content strands together through common ideas, themes, and/or units (Iowa Department of Education, 2006b). Since the model curriculum is intended to guide learning in all math classrooms in all high schools across the state, educators are responsible for converting the learning goals into specific learning objectives.

Guidelines for Instruction

In order for students to acquire the desired skills, administrators and educators must address how information is presented. The authors of the curriculum delineated what must be done in order to implement a high-quality mathematics curriculum. The characteristics of effective mathematics instruction are: teaching for understanding; the use of activities rooted in problem-solving; meaningful opportunities for student practice; the incorporation of mathematical modeling; a focus on depth; the incorporation of rigor and relevance; the effective and appropriate use of relevant technology; and the use of learning contexts that integrate mathematics with other areas of the curriculum (Iowa Department of Education, 2006b). A curriculum that mirrors the expectations developing in the Iowa Core Curriculum not only addresses the necessary content but also accommodates the needs of teachers as they implement classroom instruction.

An important pillar of the Iowa Core Curriculum is the rigor and relevance framework developed by W.R. Daggett. The rigor and relevance framework involves a combination of

Bloom’s Taxonomy with a continuum for application of ideas from specific to general situations

(Iowa Department of Education, 2005). According to the Iowa Department of Education, the framework involves four quadrants into which instruction and learning can be categorized. At the basic end of the spectrum, students acquire basic knowledge and apply that knowledge within Mathematics 5 the classroom context. At the other end of the continuum, students adapt information in order to evaluate and synthesize information into wider settings. The framework provides basic guidelines for classifying various instructional and learning situations; and it is essential for curriculum designers to accurately and purposefully balance learning activities throughout the continuum outlined by Daggett. The framework is intended to ensure that students learn content in a manner that provides a solid educational foundation in conjunction with the skills necessary to successfully apply knowledge to real-world situations.

Curriculum Document

The following curriculum document is geared toward high school mathematics and, specifically, the content for Algebra I. The future of mathematics instruction and curriculum in the state of Iowa is based on the guidelines delineated in the Iowa Core Curriculum. Therefore, the design of this document attempts to reflect the key ideas outlined in the mathematics portion of the Iowa Core Curriculum.

Part I: Program Level

The subject area being addressed in this curriculum document is mathematics at the high school level (grades 9 through 12). The following statements are intended to guide learning and instruction throughout the entire mathematics program:

 Students will accurately execute the procedures for solving various rote and

application-based mathematics problems. They will also effectively communicate the

reasoning behind the chosen process in order to evaluate the accuracy and

applicability of their solutions.

 Students will effectively chose and apply various technology-based tools and

applications to aid in the problem-solving process. Mathematics 6

 Students will develop connections between mathematical concepts in various areas

such as algebra, geometry, statistics, and so; and they will apply their understanding

of those connections to problem-solving situations or applications outside of the

mathematics classroom.

The following scope and sequence chart outlines how mathematics content will be organized and presented: Mathematics 7

COURSES → Remedial Algebra I Algebra II Geometry Trigonometry Statistics Pre-Calculus Calculus CONTENT ↓ c H T d s D c

o  Basic properties of  Review properties of  Applications of algebra  Applications of algebra  Applications of algebra  Applications of algebra e h o Algebra u h m t e w e e

algebraic expressions: algebraic expressions: including expressions and including expressions and including expressions and including expressions and e

r

p d t t e m y o l

order of operations, order of operations, equations, inequalities, equations, inequalities, equations, inequalities, equations, inequalities, u v e p

i l e t l n e i

e simplifying, and simplifying, and functions, rates, and functions, rates, and functions, rates, and functions, rates, and o r c d e ,

g a A d evaluating. evaluating. iteration will be addressed iteration will be addressed iteration will be addressed iteration will be addressed s i l t s o o l b i t g

as necessary in the as necessary in the as necessary in the as necessary in the m n  Rules for solving multi-  Graphing and interpreting i y p e c c

a e b delivery of geometric delivery of trigonometric delivery of statistical delivery of advanced a o step linear equations. inequalities and absolute t r h l r m s t

a e i

concepts and in the and geometric concepts concepts and in the mathematical concepts c t  Rules for solving multi- value c

u

i o c o n i

d problem-solving process. and in the problem- problem-solving process. and in the problem- n p o step linear inequalities,  Slope, rate of change, r g e s g a u

n

solving process. solving process. t t

f conjunctions, and graphs of linear functions. a r r e t r s n a s e

e disjunctions.  Review basic properties of  Introduction to graph i i  Review basic function i

s s n n z w h

a t series and introduce power theory and the use of a  Introduction to functions, notation, ways to represent i a s m t l s i , n l

o and trigonometric series vertex-edge graphs a

notation, evaluation, and functions, and evaluating s w r n o n t e u

e  Review of concepts representation. functions. n

q t w d l l h u l i

associated with series and e

 Slope, rate of change,  Analyzing quadratic a i n i a l n r t e l s

e sequences t graphs of linear functions. functions c

s b

p m o r

e r w e  Solving non-linear (i.e. o g e o m r h e i g

polynomial) equations d n t o e r

i

a d

g n h

 Introduction to basic a m i r a i r a a s t e .

l exponential and

e d

o

a s c u T r

logarithmic functions b o a

t h h h l u t e i e

e  Arithmetic and geometric e r o

s r

t c n p

e sequences o c u

s r

r o r t o

e a

i  Series, sums of series, and r u g n q i k r c r

u

infinite series e s o a u e i

m r l C r w u d  Basic properties of  Review basic properties of  Address the basic  Advanced review of trig  Applications of geometry Geometry e . a m e m o l

r coordinate plane and coordinate plane geometric properties functions and including basic r c

e k f t u n o graphing linear and including: points, lines, Pythagorean’s Theorem trigonometry, o

 Applying the distance and l i t

r u n s s quadratic equations. and planes; parallel and transformations, circular midpoint formulas  Solving triangles using s . t u

h A

c w T  Graphing, interpreting, perpendicular lines; trig functions, functions, and identities i c l s i g w e

l

triangles; polygons; Pythagorean’s Theorem, will be addressed as c

e and applying conic l s o

o s b b

similarity; circles; and so necessary in the delivery f sections and the laws of sines and u r r e u a r e

l s on cosines of advanced mathematical m  Introduction to the basic l I e y .

e  Basic review of trig concepts and in the

 Analyzing graphs of i

trig functions d s

i  Solving triangles using functions and circular functions problem-solving process. a l

trig functions, Pythagorean’s Theorem  Review of transformations Pythagorean’s Theorem,  Review basic properties of including translations and Mathematics 8

courses, and laws of sines/cosines the coordinate plane rotations General  Introduction to circular  Transformations including functions reflections, translations, Math  Introduction to basic trig rotations, and dilations and identities (scaling) Statistics and Introduc  Use of bar and line graphs  Representation and  Frequency distributions  Applications of statistics Probability tion to to represent functions. analysis of data using and graphs including probability and Algebra graphs, functions, etc.  Review measures of combinatorics including  Introduction to the normal central tendency and will be addressed as , will be curve introduce measures of necessary in the delivery offered.  Permutations and variation of advanced mathematical These combinations  Combinatorics concepts and in the courses  Introduction to probability  Discrete and continuous problem-solving process. distributions will  Review the normal introduc distribution e the  Introduce hypothesis same testing topics  Introduce correlation and regression outlined  Introduce concepts in the associated with Algebra nonparametric statistics Quantitative I and  Basic properties of real  Review basic properties of  Properties of real numbers  Properties of real numbers  Properties of real numbers  Properties of real numbers numbers: algebraic real numbers: algebraic will be reviewed as will be reviewed as will be reviewed as will be reviewed as Literacy Geomet properties, number properties; sums, necessary with regard to necessary with regard to necessary with regard to necessary with regard to ry families, fractions, etc. differences, products, and the presentation of the the presentation of the the presentation of the the presentation of the courses. quotients; fractions; etc. content content content content Howeve **Problem-solving will be an integral part of every math course offered in the program. Problems associated with decision-making, information processing, and systematic counting will be incorporated into every class. The statistics course will offer the most in-depth look at these topics. However, each course will utilize a unique approach to tying these situations to the r, the content being presented. Additional content  Properties of real numbers  Properties of real numbers  Deductive and inductive  Deductive and inductive  Introduction to parametric Content will be  Rules of exponents and  Rules of exponents reasoning reasoning equations more simplifying polynomials  Expanding and factoring  Direct and indirect proof  Direct and indirect proof  Introduction to fractals basic  Factoring quadratic polynomial expressions  Constructions of trigonometric identities  Introduction to basic polynomials and simplifying  Perimeter and area of  Vectors calculus ideas including and will  Introduction to solving polynomials plane figures  Polar graphing limits, derivatives, and be quadratic equations  Working with fractions  Surface area and volume  Graphs of complex integration presente  Working with fractions and simplifying rational of 3-D objects numbers d at a and simplifying rational expressions expressions  Simplifying radical Mathematics 9

slower  Systems of linear expressions and working pace. equations and inequalities with complex numbers  Properties of radical  Advanced systems of expressions linear and non-linear  Evaluating, solving, and equations and inequalities graphing quadratic  Properties of radical functions expressions.  Evaluating, solving, and graphing quadratic functions.  Vectors  Polar graphing  Graphs of complex numbers  Applications of matrices **Note that the content choices are based on the Iowa Core Curriculum model (Iowa Department of Education, 2006b). **The following textbooks will be utilized in the implementation of the program outlines above: Bluman, A. G. (2007). Elementary statistics: A step by step approach. Boston: McGraw Hill Higher Education. Brown, R. G., Dolciani, M. P., Sorgenfrey, R. H., & Cole, W. L. (2000). Algebra: Structure and method: Book 2. Evanston, Il: McDougal Littel. Brown, R. G., Dolciani, M. P., Sorgenfrey, R. H., & Kane, R. B. (2000). Algebra and trigonometry: Structure and method: Book 2. Evanston, Il: McDougal Littel. Giarrusso, J. A., Gunsallus, Jr., B. E., Keeney, J. R., Molina, D., & Nicholson, P.O. (2000). Geometry. , Il: McDougal Littel. Gordon-Holliday, B. W., Yunker, L. E., Crosswhite, F. J., & Vannatta, G. D. (1999). Advanced mathematical concepts: Precalculus with applications. New York: Glencoe McGraw-Hill. Graham, J.A., & Sorgenfrey, R. H. (1986). Trigonometry with applications. Atlanta, GA: Houghton Mifflin. Mathematics 10

The typical basic coursework for most students involves the following the courses: one full year

(three trimesters) of Algebra I, two trimesters of Algebra II, and two trimesters of Geometry.

This sequence of courses not only allows students to meet local graduation requirements, but it also meets the basic requirements for most post-secondary institutions. For students who requirement remediation, there are two available courses: General Math and Introduction to

Algebra. These courses provide surveys of basic concepts and are intended to prepare a student to successfully complete Algebra I. Although these courses may be utilized to meet local graduations requirements, they do not qualify for the basic entrance requirements to colleges and universities. The program is also set up so that eighth grade students who meet designated requirements may take Algebra I prior to entering high school. The Algebra I, Algebra II, and

Geometry sequence is structured to meet the basic requirements outlined in the Iowa Core

Curriculum.

Part II: Course Level

A critical course in the mathematics program is Algebra I. This course is taken by every student in high school at some point in his or her mathematics coursework. For students who require remediation, Algebra I serves as a capstone to their learning. For those students who plan to take advanced courses in mathematics, Algebra I serves as the springboard to higher level learning. Therefore, the curriculum for Algebra I will be described in greater detail.

The following statements are intended to guide the learning and instructional activities during Algebra I:

 Students will develop the necessary skills in order to think in an abstract manner and

to apply general concepts to specific situations. Mathematics 11

 Students will practice and implement a wide range of techniques to solve

mathematical problems and problems found in real-world situations. The students

will learn to use the “Problem Solving Plan” to guide the problem-solving process.

 Students will accurately execute the procedures for solving various rote and

application-based algebra problems. They will also communicate the reasoning

behind the chosen process in order to evaluate the accuracy and applicability of their

solutions.

 Students will develop logical connections between algebra concepts and other content

areas in the high school curriculum as well as real-world situations.

The following table outlines the topics to be addressed, instructional materials, and suggested instructional activities. Mathematics 12

LEARNING/INSTRUCTIONAL SUGGESTED LEARNING TOPICS MATERIALS ACTIVITIES UNIT I Variables,  Textbook.  Large group instruction led by teacher. Expressions, and  Written notes for each section of the unit.  Use of PowerPoint presentations (either Simplifying  Overhead transparencies for example led by teacher or used as supplemental problems. activities on an individual basis).  Textbook assignments and supplemental  Review games to check for worksheets. understanding and to review key  Copies of the Problem Solving Plan. concepts.  Additional materials include:  Small group activities. Introduction to  Journaling and vocabulary exercises (i.e. Solving Equations  Materials for instructional activities (i.e. puzzles, word finds, matching LINCS tables) to develop in-depth activities, etc.) understanding of key ideas.  Journal questions, vocabulary  Student led discussions or problem- exercises, additional word problems, solving exercises at the board or in small mini-quizzes, and so on. groups.  Review Game materials  Puzzle activities for connecting various Translating Words  Additional technology-based materials forms (i.e. words, symbols, etc.) forms of into Expressions and include: mathematical communication. Written Equations  Overhead projector notes for each section of the unit.  Teacher Desktop and LCD projector  Computer-based assessment activities.  PowerPoint presentations for each of  Internet WebQuests to learn terminology, the sections. practice algorithms and procedures, and  Computer-based curriculum used to check for understanding. supplement instruction (i.e. PLATO,  Paper-and-pencils assessments. Introduction to ALEKS, etc.) Problem Solving  Websites with supplemental materials  UNIT I Activity/Application –  Smartboard hardware/software for Developing algebraic expressions to transferring board examples to digital calculate gross and net pay. form  Classroom website for sharing information and posting journal responses, vocabulary, etc. UNIT II Properties of Real  See instructional materials for Unit I.  See instructional activities for Unit I. Numbers (+/ –/×/÷) Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Additional activities include:  Manipulatives for developing an  Using a “human” number line to help understanding of arithmetic operations students understand the relationship Mathematics 13

Distributive Property with positive and negative numbers. between positive and negative  Word problem template for working numbers. with the Problem Solving Plan.  Inspiration software for building  UNIT II Activity/Application – Students interactive quizzes about algebraic must choose a real-life situation where properties. positive and negative numbers are used. Develop a mini-presentation that Reciprocals illustrates how the numbers applied in the given setting.

 UNIT II Activity/Application – Students will describe the “series” nature of consecutive integer problems and will explain in their own words the procedure used for solving these types of problems. UNIT III Solving Multi-Step  See instructional materials for Unit I.  See instructional activities for Unit I. Equations Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include: Solving Word  Copies of the “Golden Rules” for  UNIT III Activity/Application – Students Problems by Writing solving equations. will develop word problems based on Equations  Pre-developed charts for students to be personal experiences. Students will used while solving word problems. develop equations associated with their Solving Word  Word problem template for working situations and develop “quizzes” for Problems Using with the Problem Solving Plan. other students to take. Organizational Charts UNIT IV Properties of  See instructional materials for Unit I.  See instructional activities for Unit I. Exponents Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Additional activities include:  Review materials for continually  Play matching game to related “FOIL- comparing the various rules for ed” and their “un-FOIL-ed” Simplifying adding, multiplying, and taking the counterparts. Polynomials power of polynomial expressions.  Compare and contrast the process for  Puzzle manipulatives for modeling the simplifying polynomials that involve process of algebraically transforming adding, multiplying, and powers. formulas. Students will create their own Mathematics 14

Transforming  Word problem template for working problems, solutions, and explanations Formulas with the Problem Solving Plan. that illustrate the understanding of the various sets of rules and when they are used.

 UNIT IV Activity/Application – Students Solving Rate and will be given various parameters with Area Word Problems regard to materials, budget, measurements, etc. and will required to design a backyard landscape (i.e. patio/deck, green space, areas of mulching, etc.) that addresses specifications for area, perimeter, etc. UNIT V Factoring  See instructional materials for Unit I.  See instructional activities for Unit I. Monomials Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include: FOIL-ing  Word problem template for working  UNIT V Activity/Application – Students with the Problem Solving Plan. will gather data for various types of situations. Using excel (or some other graphing program) they will need to Factoring Quadratic determine the nature of the data (i.e. Polynomials linear, quadratic, etc.) (Note that this activity serves to connect why quadratics are important and serves as an introduction to future graphing topics.) UNIT VI Simplifying  See instructional materials for Unit I.  See instructional activities for Unit I. Fractions Comparable materials will be used for Comparable activities will be used for Adding/Subtracting this unit. this unit. Rational Expressions Multiplying/Dividing  UNIT VI Activity/Application – See Rational Expressions activity for UNIT VII. UNIT VII Ratios and  See instructional materials for Unit I.  See instructional activities for Unit I. Proportions Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Additional activities include: Solving Fractional  Word problem template for working  Students will work with manipulatives Equations with the Problem Solving Plan. to explore the relationships between  Demonstration models for illustrating various mixtures and the results of Mathematics 15

Fractions, Decimals, the nature of mixture and work combining mixtures together. and Percents problems (i.e. solutions with different percentages that can be mixed to  UNIT VII Activity/Application – illustrate the new amounts with new Students will be given data regarding the Solving Mixture and percentages). voting characteristics of various Work Problems  Real life examples to illustrate the demographic groups. The students will “size” of quantities that require be running for office, and will have to scientific notation determine how the percentages of the Negative Exponents various groups will influence how they will campaign based on the issues and voting patterns associated with each group. UNIT VIII Graphs of Lines  See instructional materials for Unit I.  See instructional activities for Unit I. Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Additional activities include:  Graph paper for sketching graphs.  Graphing activities using graphing Introduction to  Rulers calculators. Functions  Individual whiteboards for sketching  Using a “human-size” coordinate graphs in small groups. plane, students will explore the nature  Word problem template for working of slope. with the Problem Solving Plan. Graphing and  Additional technology-based materials  UNIT VIII Activity/Application – Interpreting Linear include: Students will gather data to generate and Quadratic  Graphing calculators for generating the linear and quadratic graphs of the data. Functions graphs of functions. The students will then be responsible for  Excel software for analyzing the linear determining how the characteristics of Direct and Inverse nature of data. the graphs relate to the nature of the data Variation  Smartboard technology and/or they collected. Small groups of students Internet-based graphing tools for will gather their own types of data and developing the connection between will develop a presentation of their graphs of lines and their equations. findings to the rest of the class. UNIT IX Solving Systems of  See instructional materials for Unit I.  See instructional activities for Unit I. Linear Equations Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Graph paper for sketching graphs.  UNIT IX Activity/Application – Students  Rulers will be given recent data about current  Individual whiteboards for sketching economic issues in the country and will graphs in small groups. be asked to utilize their knowledge of  Word problem template for working systems to assess how the various data with the Problem Solving Plan. sets compare and contrast. Mathematics 16

Solving Word  Additional technology-based materials Problems Using include: Systems of  Graphing calculators for generating the Equations graphs of functions.  Excel software for analyzing the linear nature of data.  Smartboard technology and/or Internet-based graphing tools for developing the connection between graphs of lines and their equations and for locating solutions to systems of equations. UNIT X Solving and  See instructional materials for Unit I.  See instructional activities for Unit I. Graphing Comparable materials will be used for Comparable activities will be used for Inequalities this unit. this unit.  Additional materials include:  Graph paper for sketching graphs.  UNIT X Activity/Application – Students Conjunctions and  Rulers will learn the basics of linear Disjunctions  Individual whiteboards for sketching programming to model various situations graphs in small groups. with inequalities in order to  Word problem template for working maximize/minimize certain variables. with the Problem Solving Plan. (i.e. students could sell two types of  Additional technology-based materials products and determine how many to sell Solving Equations include: to maximize profit). with Absolute Value  Graphing calculators for generating the graphs of linear inequalities.  Excel software for analyzing the linear nature of data. Graphing Systems of  Smartboard technology and/or Linear Inequalities Internet-based graphing tools for developing the connection between graphs of lines and their equations and for locating solutions to systems of inequalities. UNIT XI Properties of  See instructional materials for Unit I.  See instructional activities for Unit I. Irrational Numbers Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Additional activities include: Mathematics 17

Pythagorean’s  Manipulatives for developing a  Students will work with manipulatives Theorem concrete understanding of the to explore the nature of right triangles relationship between the sides of a and the relationship between the sides right triangle. of the triangle. Adding, Subtracting,  UNIT XI Activity/Application – Students Multiplying, and will be given a limited set of materials Dividing Radicals (i.e. straws, tape, etc.) will need to design and build the tallest possible structure. Working with Before building, though, they will have Conjugates to develop a schematic of the structure complete with measurements, dimensions, etc. The structure will have to include triangles in order to maximize strength. UNIT XII Review of Quadratic  See instructional materials for Unit I.  See instructional activities for Unit I. Functions Comparable materials will be used for Comparable activities will be used for this unit. this unit.  Additional materials include:  Graphing materials for visually  UNIT XII Activity/Application – determining the solutions to quadratic Students will be given various types of Completing the functions for determining how the quadratic functions based on real-world Square and the types of solutions compare the applications and will have to use their Quadratic Formula characteristics of the graph. knowledge of quadratics functions in  Word problem template for working order to interpret the meaning of the with the Problem Solving Plan. functions.

Solving Quadratic  UNIT XII Activity/Application – Equations Students will work with projectiles. They will gather data about the flight of a projectile and will have to derive a quadratic function that describes the nature of the motion of the projectile.

In order to accurately assess the achievement of students in the learning process, a variety

of different evaluation instruments should be utilized before, during, and after instruction.

Evaluation instruments will include paper-and-pencil tests midway through each chapter and at

the conclusion of each chapter. Students will also be asked to complete mini-assessments on an

individual basis in a one-on-one setting with the teacher. These mini-assessments are intended to Mathematics 18 allow the students to demonstrate their knowledge of procedures and to allow for direct feedback from the teacher. Daily homework will also be utilized as an activity for students to practice their skills for both teachers and students to gauge understanding of various concepts. In accordance with the guidelines in the Iowa Core Curriculum (2006b), the development of homework activities will need to center on purposeful practice that goes beyond rote learning.

Additional assessment tools will also include: teacher observation, in-class question and answer sessions, small group activities, and so on. Technological activities will be incorporated into the learning process. Therefore, the same technological applications should be used during assessment in order to truly reflect how and what the students have learned. Finally, mini- projects will also be incorporated into each chapter. These projects will reflect: the problem- solving activities introduced in each chapter, the algebraic skills focused on during the chapter, and the quantitative literacy goals outlined in the Iowa Core Curriculum.

Learning algebraic concepts can be a difficult process for students due to the abstract nature of the content. In addition, mathematical concepts tend to develop in a hierarchical manner that requires students to build from one concept to the next. Therefore, continual, consistent, and reliable evaluation techniques are necessary in order to ensure that students master the material. The use of varied evaluation activities that reflect how the students learn is essential to accurately gauge student learning.

Part III: Critique

English (2000) defined curriculum as some type of document that guides the actions of teachers in the classroom. Although a curricular document is primarily based on the content that is anticipated to be presented in the classroom setting, there are many factors that contribute to the actual translation of curriculum into instruction. For example, teachers must strike a balance Mathematics 19 in the classroom that addresses the amount of material that is covered in class, the level of mastery attained by the students, the management of the classroom environment, the attitudes of students regarding the learning of content, and the assessment of student learning (Posner, 2004).

Posner noted that there are many constraints that can affect the implementation of instruction. In addition, there are many additional influences that affect the learning process including: political and social influences; student motivation and parental support; individual teacher interpretation of the curriculum; availability of resources such as money, materials, and time; and so on. Due to the fact that a curriculum is influenced by many different factors, the preceding curricular document will be critiqued for its strengths and weakness.

Strengths of the Curriculum

Flexibility. The curriculum, specifically the outline for Algebra I, provides a significant amount of detail so that individual teachers could easily determine what is to be taught.

However, the curriculum remains flexible enough to allow educators to incorporate their unique teaching styles into the educational process. The guide provides details about the topics to be covered and provides suggestions for the types of learning materials and activities to be utilized.

However, specific details are purposefully avoided to allow teachers to place their own spin on the various topics. The nature of the flexibility provides enough structure to delineate what needs to be taught, but it also give individual teachers the professional freedom to develop instruction that aligns with their teaching style and the learning needs of each unique group of students.

Alignment to state expectations. The Iowa Core Curriculum is a document that will guide mathematics instruction in the state of Iowa for years to come. Therefore, it is essential that mathematics curricula intended for use in Iowa classrooms align with the model accepted at Mathematics 20 the state level. Although the original curriculum was derived from a “pre-core curriculum” artifact, the concepts addressed in the document were purposefully aligned with the expectations set forth in the state model. The mathematics curriculum developed here serves as an example of how existing curricula can be adapted to meet the changing requirements of the state as well as the changing educational demands of the society. Although educators must continually adapt instruction and curriculum to reflect the dynamic nature of the world, they do not necessarily have to reinvent the wheel with each change that occurs.

Problem-solving. A key characteristic of mathematics learning is the ability to solve problems. The authors of the Iowa Core Curriculum (2006b) emphasize the need for students to learn how to effectively solve real-world problems and to utilize various forms of technology in this process. The Algebra I curriculum places a significant amount of emphasis on the problem- solving process. In virtually every unit, students are expected to utilize various techniques to solve problems. In addition, emphasis is also placed on the need for students to demonstrate their ability through communication instead of simply providing answers to the problems.

Although answers to problems are important, reasoning, interpretation, and the communication of ideas is often more important in real-world settings. Throughout the curriculum, students are asked to practice these skills in order to develop connections between concepts and their applications to settings outside of the mathematics classroom.

Weaknesses of the Curriculum

Role of technology. As noted in the Iowa Core Curriculum (2006b), technology is an important part of today’s society and students must learn how to effectively utilize technology- based applications in the problem-solving process. Although the use of technology is emphasized throughout the curricular document, a statement of use and the actual Mathematics 21 implementation of technology as part of instruction are two different things. One weakness of the curriculum is that specific activities and/or pre-developed technology-based activities are not included in the curriculum guide. Therefore, there is a risk that teachers may not be able to successfully utilize technology in the delivery of instruction. In order for technology to become a staple in the mathematics classroom, teachers need proper training, professional development activities, and general support in order to use technology to truly enhance the learning process.

A curriculum document may state that technology should be used, but this is only realistic if educators have the support they need.

Time constraints. Posner (2004) described a variety of factors that act as constraints to the translation of curriculum into instruction. One of the significant “frame factors” (p. 193) that affects classroom instruction is time. The curriculum document presented here outlines a variety of topics that encompass a quality mathematics program. The curriculum meets all of the essential aspects of a high quality mathematics curriculum (Iowa Department of Education,

2006b). The curriculum also emphasizes application-based learning activities and problem solving. However, a question that must be addressed involves the amount of time that the teacher will need to cover every component of the curriculum. Is there enough time in the schedule to allow teachers to sufficiently cover all of the indicated topics? Although ideal conditions would allow for teachers to present every concept, the reality is that educators must manage other factors that tend to limit instructional time.

Diverse learning needs of students. At the high school level, students generally tend to follow diverse learning paths that are rooted in scheduling issues, personal interests and needs, and previous experiences. Taken as a whole, the mathematics curriculum covers a wide range of topics that would sufficiently prepare students for their future experiences. However, an Mathematics 22 assumption that underlies the curriculum is that students will take a large majority of the courses offered? What happens to students who require remediation? Will students who are ahead of the game have sufficient learning opportunities? How do local graduation requirements align with the expectations for mathematics achievement? Although the curriculum is comprehensive, educators must find a way to ensure that every student meets the same minimum requirements in order to be successful after high school. Since students have diverse learning needs, this can be a challenge that is not easily addressed within the framework of a curriculum document.

Summary

Although the basic definition of curriculum is simple, the nature of curriculum in the practical settings is very complex. There are many different factors that influence both the development and implementation of curriculum. Although a particular curriculum document may be developed with best intentions and with the careful consideration of external influences, no curricular document is perfect. It is always subject to changing demands of a dynamic learning setting. The mathematics curriculum presented here is no expectation. Although the document is comprehensive, incorporates the ideals of the Iowa Core Curriculum, and provides a flexible structure for educators, it has certain weaknesses that must be addressed. Ultimately, curriculum is more than a piece of paper that guides instruction. Curriculum is an entity that embodies the ever-changing nature of education and the inherently human characteristics of learning. Therefore, the curriculum design process is an ongoing process that requires continual reflection, experimentation, and redesign. The document presented here is an example of that process and demonstrates the general nature of curriculum development. Mathematics 23

References

English, F. W. (2000). Deciding what to teach and test: Developing, aligning, and auditing the curriculum. (Millennium ed.). Newbury Park, CA: Corwin Press.

Iowa Department of Education. (2005, September). Improving rigor and relevance in the high school curriculum. Retrieved September 13, 2008, from http://www.iowa.gov/educate/content/view/673/1024/.

Iowa Department of Education. (2006a, May). Model core curriculum for Iowa high schools: Final report to the state board of education. Retrieved September 13, 2008, from http://www.iowa.gov/educate/content/view/674/1023/.

Iowa Department of Education. (2006b). Iowa high school mathematics model core curriculum. Retrieved September 13, 2008, from http://www.iowamodelcore.org/docs/Mathematics %20Model%20Core%20Curriculum.pdf.

Posner, G. J. (2004). Analyzing the curriculum (3rd ed.). New York: McGraw-Hill.

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