A. Common Features of NSF–sponsored Curricula

1. The curricula are organized using multiple strands of algebraic, geometric, statistical, probabilistic, numerical and discrete mathematical ideas, which build upon each other throughout each grade level;

2. Core mathematical ideas within each strand are carefully sequenced and articulated with each other through more advanced grades;

3. These core ideas are conceptual integrated and presented in the form of thematic units or content strands designed to intrigue and engage students at different levels of depth and abstraction;

4. The curricula use modeling, group data collection, simulations and predictions;

5. Students work individually and in collaborative learning groups to actively investigate non-routine problems over an extended period of time;

6. Graphics and scientific calculators are used as an integral component of the lessons;

7. The curricula are college-preparatory material accessible to all students.

These design features of the above NSF-sponsored curricula, accompanied by student-centered teaching methods, are supported by a substantial body of cognitive science research (Bruer, 1993; Caine, R., & Caine, G. 1991; Piaget, 1971). These curricula presuppose students are inherently "active learners" who interpret and construct meaning from their engagement with interesting mathematical questions and concrete materials. In these curricula, a series of carefully sequenced activities lead students to discover relationships and, therefore, acquire deeper conceptual understanding of important mathematical ideas. Students work in small groups and collaborate on developing strategies to solve open-ended problems. Teachers guide the groups through "organized discovery" whereby the teacher asks a probing question provoking students to think rather than memorize. Students generate a variety of algorithms and are assessed using a variety of measures.

The internal organization of these new curricula is in sharp contrast to pre-standards texts, which organize and present mathematics formally, topic-by-topic, emphasizing algorithmic manipulations and computational tasks. A mathematics curriculum organized linearly by topic encourages an instructional method whereby the teacher stands and tells concepts and procedures to students, interspersed with teacher-led whole class questioning. Students in these classes typically sit passively in rows watching and listening as the teacher shows them a procedure on how to solve a particular problem. Students are then assigned homework problems to practice the day's new procedure and are later tested for mastery of the algorithms. The teacher then moves on to the next topic in an effort to "cover" the material. As a result, according to the Third International Mathematics & Science Study (TIMSS) (1997), the U.S. curriculum has become "a mile wide and an inch deep."

Probability and statistics: Students will be able to devise an experiment to estimate the probability of success using a specific strategy, and they will understand that accuracy of their estimate usually depends upon the number of times the experiment is repeated. They "I recognize that the probability of an event is a number between 0 and I inclusive and they will be able to determine the probabilities for events involving equally likely outcomes. They will be familiar with the concept of independent events and ascertain whether previous events affect future events in given situations. They will be able to find probabilities (including the probability of a sequence of events) by using and creating an area model that can be subdivided into regions which can represent equally likely events, by developing a tree diagram, or by listing and counting all possible outcomes. They will be able to distinguish between and compare experimental probabilities and theoretical probabilities. They will be able to calculate the expected value for various situations, including multi-stage (2 or 3) events, using the "large number of trials" method. They will be able to determine how to adjust payoffs so that a game is considered fair. Students will investigate a variety of strategies in the pursuit of a "best strategy" to play a game or make a decision in real-life situations. They will be able to analyze and compare these strategies by drawing an area model and finding the expected value for each.

Geometry: Students will utilize geoboards to investigate the concept of area, including unit of measurement. They will be able to approximate the area of a figure using a nonstandard unit of measure. They will be able create figures (including triangles and quadrilaterals) on a geoboard and find their areas. They will develop formulas for the area of a triangle, parallelogram, and trapezoid. They will realize that figures with the same perimeter can have different areas. By examining relationships among the areas of squares made from sides of a triangle, students will "discover" the Pythagorean Theorem, explore a geometric proof of it, and be able to apply the theorem to problem situations. They will also investigate the triangle inequality. Given the lengths of the sides of any triangle, they "I be able to find the area. They will ascertain that a square has maximum area of all rectangles with a fixed perimeter. They will explore and compute areas of regular polygons with fixed perimeters and conclude that the greater the number of sides, the larger the area. They will recognize that areas of similar polygons are proportional to the squares of their perimeters. They will develop a formula to find the area of an n-gon with fixed perimeter. Students will be able to create a two-dimensional net for a three-dimensional rectangular solid. They will develop formulas for the surface area and volume of a rectangular solid, and recognize that figures with the same volume can have different surface areas. They will develop and use formulas for the lateral surface area and volume of a night prism, and recognize that figures with the same lateral surface area can have different volumes. They will generalize the Pythagorean Theorem to three dimensions. They will establish the general principles that volumes of similar solids are proportional to the cubes of their corresponding linear dimensions, and that surface areas are proportional to the squares of their perimeters. Students will investigate tesselations of regular polygons and discover that only equilateral triangles, squares, and regular hexagons tesselate.

Given a real-world situation, students will be able to identify an appropriate population and a representative sample of the population. They will recognize that observed differences in samples could be due to sampling fluctuations. They will be able to construct a double bar graph for an applicable set of data. They will be able to formulate hypotheses based on actual data. When comparing a population with a theoretical model or another "real" population, students will be able to devise a hypothesis and a null hypothesis, design and carry out a plan for gathering data (if necessary), and analyze results by calculating the - ' statistic and interpreting it using the - ' probability chart and distribution curve. Based on this analysis, they will be able to decide whether or not the two populations are really different, i.e. whether or not to reject the null hypothesis. They will use proportional reasoning to find expected numbers when assuming a true null hypothesis in comparing two "real" populations. They will become familiar with the distinction between correlation and causation, i.e. a correlation between two variables does not necessarily indicate that one caused the other. Students will review normal distribution and standard deviation, use the standard deviation of a normal distribution to draw conclusions about hypotheses made, and compare the statistic with standard deviation.

Logic/reasoning: Students will continue to develop their reasoning skills; in particular, the POWs are more generally phrased and provide a framework for forrnwating "completely convincing arguments." They will be expected to provide full explanations of solutions and, when appropriate, a proof that a solution is correct. Students will be familiar with analyzing a situation on a case by case basis in order to conclude that there is a unique solution. They will use deductive reasoning to draw valid conclusions from two or more statements assumed to be true.

vertex form is the lead coefficient of the quadratic function and that the sign of this number determines whether the graph of the function opens up or opens down. They will be exposed to finding roots of a quadratic using the vertex form in specific cases, but will probably not know the quadratic formula at this time. They will be able to use quadratic functions to create mathematical models and do mathematical analyses in a variety of settings. For example, using the algebraic properties of the vertex representation of a quadratic they will be able to prove that of all rectangles with a perimeter of 200 feet, a square is the rectangle with the largest area. In addition, they will be able to fit a quadratic to three "data" points in the coordinate plane, where the graph contains each of the points. Students will know and be able to use such standard terminology as quadratic, parabola, polynomial, trinomial, coefficient, term, maximum (minimum) of a function.

Geometry: Students will understand the concept of slope of a line geometrically as the rise over the run as one moves toward the right. They will be able to associate the sign of the slope with the direction that the line tilts. They will know that horizontal lines have zero slope and they "' I be able to explain why slope is undefined for vertical lines. They will also know the slope as the ratio of the difference of the y-coordinates to the x-coordinates. They will be able to determine the slope of a line given two points on the line and understand that the computation is independent of the points chosen as a result of the geometric similarity of triangles determined by the line and the two points. They will be able to connect the geometric properties of y-intercept and slope of a line to the coefficients in the algebraic representation of the line written as y=a+bx. They will be able to determine the formula for a linear function given two points on the graph or one point on the graph and the slope. They will also be able to represent the slope of a line as a rate of growth in many contexts. Students will understand the basic shape of an exponential function y=(kb)^cx, for O<b. They will be able to compare linear and exponential functions. For example, in the expressions above, changes in the linear parameter a and the exponential parameter k both change the y-intercept. Changes in the linear parameter b and the exponential parameter c both change the rate of growth of the function. Students will understand the derivative as the instantaneous rate of change, or growth, of the function at a point (x-value) and that it can, except in the case of a linear function, be depicted geometrically as the slope of the line tangent to a curve determined by the graph of a function over that point. They "I understand that a positive derivative indicates that the function is increasing at that point, a negative derivative indicates that the function is decreasing at that point and that at "turning points" the derivative is zero, if it exists. They will also have explored at least one case where a function does not have a derivative at a point. They will know that the magnitude of the absolute value of the derivative of a function at a point determines its steepness. They will be able to interpret the derivative of a function as a new function in some instances. Students will be able to write (and will have rigorously developed) the algebraic equation for the equation of a circle with specified center and radius in the coordinate (Cartesian) plane. Moreover, they will be able to transform the general equation x²+bx+y²+dy+e = 0 into the form (x-a)² +(y-c)² = r², when b, d, and e are specific numbers. (There is even some exposure to imaginary numbers when r is negative.) They will be able to use (and will have rigorously developed) the distance formula between two points. They will be able to find the distance between a point and a line (not containing the point) in coordinate geometry, although no general formula is developed. They Will know and be able to use (and at least experimentally developed) the formula for the midpoint of a line segment in coordinate geometry. Using results and techniques in synthetic geometry, coordinate geometry and trigonometry, students will develop strategies to prove or establish (at least with a partial proof) many results from Euclidean geometry, such as (but not limited to): a point equidistant from two points is on the perpendicular bisector of the line segment joining the two points; a line through the midpoint of a segment is equidistant from the two end points of the segment; there is a point equidistant from three if and only if the three points are not collinear; every triangle has an inscribed circle; vertical angles are equal; the bisectors of each pair of vertical angles formed by two intersecting lines are perpendicular; the tangent to a circle is perpendicular to the radius of the circle. (Their strategies for proof should extend beyond the particular results they prove.) Students will have deepened their insight (or developed insight) into formulas for the area and circumference of circles by developing an outline for a proof (some details are omitted) that the circumference of a circle of radius r is 2rπ and the area is πr². They will have strategies for estimating the value of pi. They will have developed and be able to use the formula for the volume and surface area of a cylinder given the radius of the base and its height. They will know and be able to use such terms as cross-sectional area, equidistant, circumscribed/inscribed circle/polygon, angle bisector, perpendicular bisector of a segment, line, tangent to a circle, (integer) lattice point in the coordinate plane, and unit circle. Students will deepen their ability to solve two-dimensional linear programming problems graphically and be able to describe geometric properties of the feasible region.

Students will know and be able to use the standard coordinatization and associated visual representation of three-space. They will know the connection between linear equations in three variables and their graphical representation as planes in three dimensions. They will understand the geometry of lines and planes in three dimensions. For example, they will be able to describe the possibilities for intersections of two, three, and four planes in three-space (resulting in a plane, a line, a point, or the empty set.) They will deepen their understanding of parallel lines in three dimensions as more than lines that do not meet. That is they will understand the difference between parallel lines (which must be in a plane) and skew lines (which don't meet, but are not in a single plane).

Probability and statistics: Students will enhance their ability to use area and tree diagrams (some with probabilities as branch weights) and simulation to analyze sequences of events. They "I focus on three phases in the mathematical process: knowing how to compute, knowing when to apply, and knowing why a method works. They mill be able determine the number of outcomes for a multi-stage event using several strategies: organized lists (when the number is small), the multiplication principle, tree diagrams, and identifying situations that involve permutations or combinations. They will understand the multiplication principle as the number of ways one can chose one object from each of two disjoint sets and its generalization to choosing one object from each of n disjoint sets. They will be able to justify why one multiplies the probability weights along the branches to an outcome in order to establish the probability of the outcome. They will recognize that the concept of permutations may be utilized in situations where multi-stage outcomes are different if the sequencing of the individual stages, and their respective outcomes, is changed. They will recognize that combinations are useful in situations where outcomes are not considered different if only the sequencing of the stages is changed. They will know and be able to use factorials. They will know, be able to use, and have developed symbolic formulas for nPr, the number of permutations of r objects selected from n objects (n,r,0). They will know, be able to use and have developed symbolic formulas for nCr, the number of combinations of r objects selected form n objects (n,r,0). They will be able to calculate both nPr, and nCr, using pencil and paper if r is small and using technology otherwise. They will be able to use these concepts in situations involving equally likely probabilities. They will also be able to use combinatorial coefficients in situations where outcomes are not equally likely. For instance, they will understand the term binomial experiment as a situation involving just two outcomes, success and failure, repeated a number of times. They will be able to determine when binomial probabilities are appropriate in situations and be able to utilize and calculate binomial probabilities or a cumulative distribution of binomial probabilities in situations that prescribe both the number of trials and the probability of success. They will recognize the general shape of a binomial distribution displayed in a bar graph and know, for instance, that the distribution is symmetric when the probability of success is 0.5. They will have strategies for determining the "most probable" outcome in some situations. They will be able to recognize Pascal's Triangle and know several patterns in it. For example, they will know that each row (other than the first row, which only has one entry in it) begins and ends with a I and that each internal entry has the property that its value is the sum of the two entries in the previous row closest to the given entry. They will also know that the sum of entries in a row is a power of 2 (and be able to justify this with a combinatorial proof). They will know and understand the relation of the entries of Pascal's Triangle to the combinatorial coefficients, _nC_r , and the coefficients in the expansion of (a + b)^n. They will know ₁C_r and be able to justify several identities involving the combinatorial coefficients such as _nC_r = _n-1C_r-1,_n-1C_r ; _nC_r = _nC_n-r ; _nC_{o = n}C_n = 1; and _nC_r = _nP_r /r!. Students will know and be able to use the strategy of determining the probability of the complementary event as a way calculating the probability of an event. They will enhance their ability to use probability to evaluate null hypotheses and make decisions. They will also be able to use the pigeon hole principle in counting arguments.

Logic/reasoning: Students will know and be able to formulate the converse of an if…then (conditional) statement; particularly in geometry. They will also know the meaning of if and only if and be able to write two conditional statements given an if and only if statement in geometry. Students will have developed strategies for direct proof of many theorems in Euclidean geometry. They will develop some conversational proofs for some facts in linear algebra, such as: if a square coefficient matrix has an inverse, then the corresponding system of linear equations has a unique solution. The will also work on mathematical reasoning in terms of situations such as winning strategies for games or solutions to card tricks. They will enhance their ability to make simplifying assumptions in modeling situations. They will develop the strategy of making analogies, such as relating counting situations to either ice cream cone possibilities (permutations) or bowls of ice cream (combinations). They 'II justify several combinatorial identities on the abstract level such as _nC_{r = n}C_n-r or _nC_{i =} 2^n .

Discrete Mathematics: Students will deepen their understanding of linear programming problems (Ipp's). They will review and extend their knowledge of two dimensional lpp's. They will be able to describe the general (mathematical) form of an lpp in n variables using constraints that contain both equalities and inequalities. (However, terms such as profit function or cost function are used instead of objective function.) They will understand the geometry of the three dimensional case. They will be able to model verbal (written) situations with Ipp's in many cases and using up to six variables. They will be able to use matrices to represent many types of information. In particular, they will be able to use a matrix equation of the form [Al [X]=[B] to describe a given system of linear equations in n unknowns where [Al is the matrix of coefficients and [B] is the constant matrix. They will have actually worked with as many as six unknowns. They will have developed the operations of addition and multiplication of matrices (and know when multiplication of matrices is possible) and apply these operations in many contexts. They will have developed several properties of matrix algebra. They will know that multiplication of matrices is associative (when it is defined) and is not commutative, in general. They will have developed the identity matrix for multiplication of square matrices and know that some matrices have inverses. They will be able to find the inverse of some two dimensional square matrices using strategies for solving systems of equations in two unknowns and be able to use technology (graphing calculators) to find inverses of matrices (when they exist) for square matrices of larger dimensions (where dimension limitations depend upon the technology). They will know the relationship between being able to solve a system n linear equations in n unknowns and the existence of the inverse of the coefficient matrix, [Al. They will be able to use the inverse of the coefficient matrix (when it exists) to a system of n linear equations in n unknowns They will have done this cases when n=6.

Students will know the formal definition of functions as a set of ordered pair in which no two ordered pair have the same first coordinate. They will also understand usual functional notation (such as f(x)=2x). They will be able to represent functions using tables, graphs, symbols (algebra or formulas), and verbal descriptions. They will be able to move between these representations in many instances. They will have considerable familiarity with several abstract families of functions such as: linear (y=ax+b), quadratic (y=ax²+bx+c), cubic (y=ax³+bx²+cx+d) polynomial (y=ax^_n+bx^_n-1+...+c), exponential (y=ab^x), sine (y=Asin(Bx)), reciprocal (y=K/x), rational (quotient of two polynomials), and, to a lesser degree, power functions (y=ax^p). Less formally, they will understand step functions and absolute value functions. They will know the basic shapes of graphs of functions in each of these families, except, probably, the general polynomial, general rational, and general power families. They will know that the linear family and the exponential family each as "two parameter" families and have strategies to "fit" unique functions from each o these two families to two data points. Similarly, they will know that the quadratic family is a three-parameter family and have strategies to fit quadratic function to three data points. They will be able to recognize patterns in tables that suggest linear, quadratic, exponential, and sine. For example, they will know and be able to prove that linear functions have constant first differences, quadratics have constant second differences, exponential have constant ratios, and sine function tables portray periodic behavior. They will be able to determine by reasoning or using data which family of functions to use and which member of the family is most appropriate in many modeling situations. They will understand the concepts of domain of a function as the set of inputs or, implicitly, as the set of numbers for which a formula representing a function makes sense.

Geometry: Using their knowledge of similar triangles, students will develop and be able to use formulas that give the coordinates of a point a given fraction of the distance along a line segment in terms of the endpoints of the segment in two and three dimensions. They will understand the basic geometric transformations (isometrics) of translation, rotation and reflection in two and three- dimensional coordinate geometry. (In this material, 3-space is often represented as an "extension" of the standard x y - plane with the z-axis pointing "out.") For example, they will understand how the coordinates of a point a-re affected by a translation in two and three dimensions, a reflection about an axis in two and three dimensions, a rotation about the origin in 2-space and a rotation about an axis in 3-space. (E.g. (x,y)•(x+ay+b) for a translation in 2-space). They will also have developed and be able to use matrix representations and matrix multiplication for rotations and reflections in order to move a single point at a time or a finite set of points at a time ' two and three dimensions. They 'II be able to develop and use coordinate transformations to project (from "center of projection" or "viewpoint") points onto a plane in 3-space and use this technique to project a three dimensional cube onto its two dimensional image on a plane. Using their knowledge of coordinate geometry, geometric transformations and their matrix representations, they will develop computer algorithms and coded programs to animate the movement of three dimensional objects on a calculator (computer) screen. They will also know and be able to use the fact that the area of a triangle is one-half the product of the length of two of its sides and the sine of the angle of the triangle formed by the two sides. They will know and be able to use the algebraic expression for an ellipse in coordinate geometry.

Trigonometry: Students will be able to describe sine and cosine (developed previously in terms of right triangles) in terms of circular functions and any (positive or negative) angle degree measure. They 'II know the graphs of y=Asin(Bx) and y=Acos(Bx), for different values of A and B, and understand how the change in value of the parameters A and B effect the graphs. They will recognize the periodic behavior of these functions. They will be able to use sine and cosine to describe failing objects with an initial velocity in a given direction using the vertical and horizontal components of velocity. They will understand the inverse functions sin-'x and cos-'x (in terms of angle values) and understand to domain and range restrictions on these functions. They will have developed and be able to use several trigonometric identities including the Pythagorean identity sin²x + cos²x = 1; as well as sin(-°) = -sin(°), cos(-°) = cos(°); cos (°) = sin(90°- ° ); and sin(°) = sin(180°-°). In addition, they will be able to use the Law of Sines and the Law of Cosines. For example, given the measure of two angles and a side of a triangle, they will be able to determine the measure of the remaining sides and angles using the Law of Sines. They will know how to convert rectangular coordinates to polar coordinates. Students will also understand the meaning of radian measure of an angle and be familiar with the procedures for converting from one angle measure to another.

Students will have strategies to determine the mean and standard deviation of a probability distribution. They will have seen a relation between the probabilistic concepts of mean (expected value) and standard deviation and the corresponding statistical measures. Moreover, they will know and have used patterns to determine general formulas for the mean (expected value) and standard deviation of binomial distributions. They will also have such formulas for the percentage binomial distribution (when the x-axis represents the percentage of success in N trials rather than the number of trials). Students will also understand the term variance. They will be able to analyze reliability of samples from population where the true percentage and standard deviation are known. Conversely, for sample percentages which are assumed to be normally distributed, they will be able to use a single sample percentage, the standard deviation of the sample, and a strategy to determine an upper bound for the standard deviation of the distribution of sample percentages to analyze the reliability of the sample and to estimate the true percentage of the population characteristic. In particular, they will be able to construct confidence intervals and determine margin of error. They will also understand the terms confidence level and confidence limits. Moreover, they will understand the "popular" use of the term margin of error. In addition, given some information about confidence intervals (e.g. confidence level and margin of error) they will be able to determine sample size. Students "I also be able to use tables for the normal curve to derive conclusions from or about sample or population statistics. They will also have seen the general analytic function whose graph is the normal curve. They will know that the overall size of the population might not affect the reliability of the sample, provided the population is large enough for the sample size and is also large enough so that distinctions between sampling, with and without replacement can be ignored.

The promulgation of new mathematics content, teaching and assessment standards has coincided with the need to change the "system" at the state and school district levels. These systemic change efforts include:

- Establishing statewide curricular standards as a matter of policy;

- Reformulating student assessments in light of these new content standards;

- Aligning school and state administrative policies to support these standards;

- Arranging for intensive and extended whole-staff teacher professional development;

- Adopting standards-based curriculum.

The National Science Foundation has been the lead agency in supporting large-scale mathematics and science systemic reform. Since 1990 the NSF has launched various systemic programs: the Statewide Systemic Initiative (SSIs), the Rural Systemic Initiatives (RSIs), the Urban Systemic Initiatives (USIs), and most recently the Local Systemic Initiatives (LSCs) programs.

Many schools have realized they must re-train their mathematics faculty if these standards are to have any impact in the classroom. Teachers will need to learn new inquiry-based curricula and student-centered pedagogical techniques. They will need to infuse their classrooms with more statistics and probability, more algebraic and geometric problem solving, and more real-world applications. Such re-training requires facilitating a major paradigm shift in the habits of mind and behavior of traditionally schooled teachers. Even eager and willing teachers have difficulty making this transition. This is not an easy or quick task.

The Greater Philadelphia Secondary Mathematics Project (GPSMP) is a "Local Systemic Change" program funded by the National Science Foundation. It builds upon nearly five years of work in Philadelphia implementing the Interactive Mathematics Program. Its five-year mission is to include more NSF reform curricula for both middle and high schools, and extend its service to include schools and school districts in southeastern Pennsylvania and southern New Jersey. From a total of possible 1,100 schools in both states, this project will eventually work with approximately 20 school districts that have demonstrated readiness to engage in standards-based mathematics reform. The goal is to train 600 teachers over five years in this local area to who are able to sustain the reform process after this grant ends.

GPSMP Professional Services

The Greater Philadelphia Secondary Mathematics Project provides the necessary professional development, in-classroom follow-up mentoring, and administrative technical assistance to schools that have made a commitment to align the curriculum and instructional practices of their secondary mathematics classrooms with the new mathematics standards. A school district’s professional development plan is developed in joint consultations with the GPSMP project directors and the district’s administrators and teachers. All plans have the following components:

1) a multi-year, intensive in-service schedule,

2) student-centered instructional methods,

3) inquiry-based, integrated curriculum,

3) teacher leadership development,

4) provisions for classroom mentoring, and

5) administrative technical assistance.

1. In-service Curriculum: The professional development plan for each districts' math staff is designed to meet the needs of a range of teachers who are at different stages in their careers, who have different teaching assignments in their schools, and who have had different amounts of previous professional development. The professional in-service curriculum consists of two types of training. Each teacher will have a somewhat different mix and schedule of training depending on the particular needs of the school and the teaching assignments and experience of its teachers. The first type is centered on one of several NSF-sponsored middle and/or high school curricula, which have been selected. This type of training ranges from 180 to 240 hours depending on whether the curriculum spans 3 or 4 years.

2. In-service Instructional Methods: The in-service instructional methods for the above training model an inquiry-based, student-centered classroom. For example, in IMP in-service sessions, teachers are seated in groups of four, like their students. The in-service presenters guide teachers through the units as they would their own students. Teachers work on selected problems in each unit. In addition to new math content, the training also focuses on student-centered instructional strategies and various forms of assessment, including portfolios, long-term problem-based essays, and group presentations.

An integral part of training is for teachers to actually teach all levels of a full-replacement curriculum. The in-services focus on practical issues of classroom implementation. Teacher manuals, videotapes and electronic networking supplement the in-services.

3. Promoting Teacher Leadership: Teacher leaders can be important agents of systemic change are within schools (Day, Goertz, & Floden, 1995). Teacher leaders are important because they: 1) act as agents of change within their building, 2) persuade their colleagues to take risks involved with change, 3) provide programmatic stability in the face of administrative turnover, 4) persuade parents of the need for change, and 5) become better classroom teachers. We promote teacher leadership by providing teachers with the opportunity to co-present in-service sessions with more experienced presenters, including the co-directors. Teachers leading in-services encourages the development of teacher-to-teacher networking and collegial support. Therefore, one goal of this project is to train more veteran teachers and mathematics department heads to lead or co-present in-service sessions. For example, over two dozen veteran Philadelphia IMP teachers have been involved in presenting or co-presenting IMP in-service sessions to new teachers in Philadelphia, New York and Boston. Experienced teacher leaders are utilized as much as possible to help schedule and conduct in-services for newer teachers.

4. Classroom Teacher Mentoring: As a follow-up to the in-service sessions, teachers are regularly visited in their classrooms to receive one-to-one mentoring. Secondary math teachers who were traditionally schooled must not only shift their thinking about the way students learn, but must also adopt different approaches to classroom management and student assessment. At the same time, many teachers must re-learn a large volume of mathematics content that is not usually taught in high school, such as probability, statistics, matrix algebra, and linear programming. Teachers also need to master the use of graphing calculators and must work through many unfamiliar, non-routine problems. New IMP teachers, for example, are regularly visited in their classroom, by a veteran IMP teacher-mentor or IMP director. During these visits, a mentor may team-teach part of a lesson or help lead classroom discussions. After each class, the mentor provides feedback to the math teacher.

5. Administrative Technical Assistance: The project directors provide technical assistance to principals, mathematics department heads, roster persons and other administrative personnel concerning:

1) Budgeting support for program implementation in their school;

2) Classroom materials, book, and calculator requisitions;

3) Teacher and student recruitment and classroom rostering issues;

4) Student transfer, absentee, retention and readiness issues;

5) Intra and inter-school and grade articulation issues;

6) Student test performance and program evaluation;

7) School-based implementation issues, such as ESL, special education inclusion, college

admissions, NCAA, and AP Calculus and AP Statistic courses.

1. Schedule: One half of the training takes place during the months of June, July and August, and occurs in 5-day blocks, 6 hours a day. (Lunch is an additional 45 minutes). Approximately 10 weeks is available for summer training. The academic-year training typically occurs on Saturdays or on other in-service days for 6 hours each day. Approximately twenty-two (22) Saturdays per academic year are available for training. (In-service sessions may run concurrently during the same summer week and on Saturdays.) All training dates are scheduled at the most convenient times for teachers. Teachers are also permitted to attend in-service training scheduled at other times with other participating school districts.

2. Location: Depending on the number of teachers per course, the training occurs either at La Salle University, or on-site in a school district, or another nearby school district. We strive for a class size of about 20 participants. A calendar of the dates, times and locations of summer and academic year training is distributed to all teachers by the end of the month of March preceding the in-services.

3. Classroom Mentoring: Teachers who have undergone training during the summer are provided with classroom mentoring during the following academic year. Teachers are visited an average of 8 times during the first year of their training; 4 times during the second year and 2 times during the third year. Each mentor has a set of teachers and schools as his/her responsibility. It is expected that each mentor will visit three different teachers per day, usually within the same school. Each visit entails a pre- and post-conference with the teacher. Each mentor submits a brief report of every visit. Periodically the mentors meet to discuss the progress of their teachers.

4. Administrative Technical Assistance: The co-directors and other project specialists will provide the technical assistance to schools. The average number of project days per year devoted to providing technical assistance for schools is approximately 8 director-days per school district, which includes meetings and telephone conferences. The number of consulting days diminishes each year until there is sufficient implementation expertise at each school site.

All participating GPSMP schools are selected based on three criteria:

1. Administrative Support: The administrative staff of a school district--superintendent, curriculum supervisors, school principals, roster chairs and mathematics department heads--have to be willing and prepared to commit school-based and district resources to support sustained multi-year teacher enhancement. In particular, they have to commit to a local cost share in the form of books, classroom materials, audiovisual equipment, graphics calculators and adequate classroom space, including desks for students to work in groups.

Principals must be visible supporters of change in their schools. This means teachers must have stable teaching assignments as they undergo extensive staff development over time. Teachers-in-training must also be provided the time during the school day to plan the use of new curriculum materials and discuss their classroom experiences with other teachers. For example, in Philadelphia high schools, IMP teachers-in-training have typically received a reduced course load or compensation for an extra preparation period while in training.

2) School District Policy: Perhaps the greatest single challenge to institutionalizing local systemic reform is maintaining the continuity of the reform process in the context of administrative personnel changes. For this reason, it is important that school districts adopt policies and practices, which will continue the change process with different administrators. The following policy indicators are used to select schools:

a) Adoption of standards-based curriculum and assessment frameworks;

b) Previous professional development on the NCTM Standards;

c) Willingness to align other professional development resources with this project (such as Title 1, Eisenhower, and Goals 2000 fund);

d) Adoption of administrative policies to support classroom reform, such as ensuring teachers have stable classroom and building assignments from one year to the next;

e) Provision for all students from diverse racial and economic backgrounds to have equal access to an inquiry-based, student-centered classroom;

f) Development of parent and community relations outreach and information program;

g) Alignment of new math teacher hiring criteria with the NCTM Teaching Standards.

3) Teacher Readiness: Many math teachers are simply not ready to undertake change. As of 1993, according to the National Science Board's Science and Engineering Indicators report (1996), 44% of high school math teachers and 72% of middle school math teachers were not "well aware" of the NCTM Curriculum and Evaluation Standards. An even greater percentage were not "well aware" of NCTM's Professional Teaching Standards. And, 44% of high school math teachers surveyed had 6 or less hours of in-service per year.

One major outcome of NSF's support for urban and statewide systemic change efforts has been to increase teachers' familiarity with the NCTM Standards and their implications for changes in their own classroom practice regarding curriculum, instruction and assessment. Nevertheless, teachers vary in respect to their own readiness to change. We look for schools with a sufficient critical mass of the school's teaching staff willing to participate in intensive, sustained professional development necessary to implement standards-based change.

The process of forming a partnership between the GPSMP and a participating school district usually involves the following steps:

1) An initial invitation to participate or a request from the school district for an initial discussion,

                    2) Meetings between the project co-directors and key administrative personnel

3) Meetings with the mathematics supervisor or department head and teacher leaders,

4) A 2 to 3 hour presentation before the mathematics teaching staff,

5) Teacher visits to other schools,

6) Four days (24 hours) of in-service featuring two replacement units teachers can use,

7) A series of meetings to plan the types of professional development in-services and to work out logistical and funding details,

8) Presentation and/or approval from the school board,

9) A letter-of-commitment from the school superintendent,

10) Further meeting to plan in-services and classroom implementation.

School District Obligations: Each school district is responsible for:

1) Providing incentives for teachers to attend the in-services,

2) Providing extra time for lesson planning during the academic year,

3) Purchasing books, graphics calculators, overhead projectors, overhead graphics calculator, computer software, and classroom durable and consumable materials.

Costs: The typical first year per teacher costs of these items is:

1) Teacher in-service stipends (or graduate credit) $1,000

2) Graphics calculators (1 classroom set = 35) 2,870

3) Books (2 classrooms sets) 2,520

4) Classroom materials 300

5) Audio Visual Equipment 700

$7,390

1. Project Directors: Mr. Joseph Merlino, Dr. Edward Wolff, and Dr. Alice Jordan are the directors and co-principal investigators of the project.

F. Joseph Merlino, M.A., Education, is the PI/PD for the Greater Philadelphia Secondary Mathematics Project and provides technical assistance to schools, supervisors all the mentors and presenters.

Edward Wolff, Ph.D., Mathematics, is a co-pi for the Greater Philadelphia Secondary Mathematics Project and has been a co-director of Philadelphia Regional IMP Center. He is also Chair of Mathematics and Computer Science Department at Arcadia University. Dr. Wolff provides training in IMP Level 4 training, Harvard Reform and Statistics in-services, mentors teachers and conducts statistical analyses of student outcomes.

Alice Jordan, Ed.D., Mathematics Education, has been a co-director of Philadelphia Regional IMP Center at La Salle University for 6 years while also being a Department Head at Strawberry Mansion High School where she taught IMP for three years. Now retired from high school teaching, Dr. Jordan constructs the in-service calendar and mentors teachers.

2. Additional Key Staff: Assisting the project directors are an experienced cadre of other in-service presenters and in-classroom teacher mentors familiar with IMP and Core-Plus and various NSF middle school curricula. Approximately 95 other teachers and university faculty are involved with providing in-services and mentoring to schoolteachers. A sample of these personnel are listed.

Barbara Stankus, an IMP 4 teacher on special assignment from Strath Haven High School in Wallingford/Swarthmore School District.

Anthony Campione is a recently retired IMP teacher from Furness High School. He has taught two levels of IMP and has co-presented numerous IMP workshops and has mentored dozens of new IMP teachers.

Richard Clancy is a recently retired IMP 3 teacher from Girls High school in Philadelphia. He has taught three levels of IMP, has co-presented numerous IMP workshops and has mentored dozens of new IMP teachers

3. Institutional Involvement: La Salle University is the fiscal agent for this project. Project partners include the Pennsylvania Department of Education Mathematics Office and the New Standards Project in Education-University of Pennsylvania. Fifteen schools from Delaware county, Montgomery county and Bucks county and the Center for a Greater Philadelphia have joined together to form the Southeastern Pennsylvania Standards Consortium. This consortium is based on the work of the New Standards Project at the University of Pittsburgh.

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