To participate
fully in the 21st century, all citizens must be comfortable
and fluent with mathematics. In
addition to dealing with quantitative ideas in virtually any job, today’s
students will need mathematics to handle their finances, evaluate medical
risks, and understand public policy issues – from the future of Social Security
to the risks of genetically modified foods. (Foundations for
Success, 2001)
All young Americans must learn to think mathematically, and they must think mathematically to learn. (Kilpatrick, 2001)
In July 2001, Montgomery County Public Schools (MCPS), a major school district in the suburbs of Washington DC, completed a major project, the release of a new, revised, mathematics curriculum for all students. There were many reasons for this revision, but the major purposes were: elimination of repeated topics, alignment with Principles and Standards for School Mathematics, the standards created by the National Council of Teachers of Mathematics, and development of more coherent vertical articulation across the grades. (Montgomery County Public Schools, 2001)
These goals raise several questions, but one of the most important is does this new curriculum meet the needs of the students? Will students who follow this curriculum be ready to enter college and/or find and keep a job? This question can be answered by looking at the level of rigor contained within. Middle school is a very important time for students as it is the continuation of the learning that took place in elementary school and it sets the stage for high school, college, and adult life. According to the National Council of Teachers of Mathematics, “during this time, many students will solidify conceptions about themselves as learners of mathematics – about their competence, their attitude, and their interest and motivation. These conceptions will influence how they approach the study of mathematics in later years, which will in turn influence their life opportunities.” (Principles and Standards for School Mathematics, 2000) Another organization, Achieve, Inc. states, “the middle grades are the crossroads where students ready themselves to acquire advanced mathematical skills and knowledge.” (Foundations for Success, 2001) By focusing, in-depth, on the curriculum for sixth through eighth grade, it is possible to get a good idea of the entire curriculum and its level of rigor.
One of the best ways to evaluate the level of rigor contained in the MCPS curriculum is to compare it to well known standards. Two different sets of standards seem to be the benchmark in mathematics education. The National Council of Teachers of Mathematics has been involved in developing standards for mathematics curricula for decades. Their most recent publication, the Principles and Standards for School Mathematics, is actually the basis for new curriculum. These standards are fully developed for grades Pre-Kindergarten through twelve. Achieve, Inc. has also released a set of standards in draft form. These standards were developed at the request of business executives and state governors to improve the standing of U. S. students in comparison to other students internationally. This need to “catch up” is based on the results of the Third International Mathematics and Science Study, which shows American students ahead in fourth-grade but falling behind several other countries by the end of eighth grade. (Mullis, 2001) Eventually, these standards will cover all grades, but they have only been developed fully for grades six through eight. A list of elementary school skills that are pre-requisites for the middle school standards and a list of possible high school topics to follow the standards have also been included in this initial draft.
According to MCPS, the National Council of Teachers of Mathematics Standards were the basis for the curriculum revisions. (MCPS, 2001) For this reason, the most natural standards with which to begin the comparison are those listed in the Principles and Standards for School Mathematics. The National Council of Teachers of Mathematics has five content standards and five process standards. The focus will be on the content standards because they are consistent for MCPS, the National Council of Teachers of Mathematics, and Achieve, Inc. However, it is important to note that Montgomery County has included a section in the new curriculum overview to address each of the process standards listed in the Principles and Standards. (MCPS, 2001) These standards are: problem solving, reasoning and proof, communication, connections, and representation. (Principles and Standards for School Mathematics, 2000) Each of the content standards is discussed, in detail below.
According to the Principles and Standards for School Mathematics, the overall focus of the number and operations standard is to:
·
Understand numbers, ways of representing numbers,
relationships among numbers and number systems
·
Understand meanings of operations and how they
relate to one another
· Compute fluently and make reasonable estimates (2000)
The National Council of Teachers of Mathematics has also provided specific expectations for grades six through eight. These standards seem to match, almost completely with the sixth grade objectives for Montgomery County. The focus on fractions, decimals, and percents is the same focus as the Number Unit of the Math A (sixth-grade on level) curriculum. (MCPS, 2001) The only objective from the National Council of Teachers of Mathematics that appears to be left out of the specific goals listed in the revised curriculum is the development of a meaning for percents greater than one-hundred and less than one. However, many teachers do cover this topic. An interview of teachers at one middle school in Montgomery County found that all of the sixth grade teachers include percents over one-hundred and less than one in their discussion of percents in general. Additionally, all of the eighth grade classes of Functional Skills Improvement (FSI), a remedial math class for struggling students, include a review of the topic when they discuss converting among percents, decimals, and fractions. (Parkland Middle School, 2001)
It is also interesting to note that many middle school students who follow the new curriculum will learn topics in the number and operations domain that the National Council of Teachers of Mathematics has suggested for high school. The Math C (on level eighth grade) curriculum includes combinations and permutations without using that terminology, and the Math Investigations curriculum includes this topic and several others. Math Investigations, the honors pre-algebra course offered to some seventh grade students, also covers the properties of number systems, matrices as a means to display data, and matrix operations (addition, subtraction, and multiplication by a scalar). (MCPS, 2001) Students who complete the Math Investigations curriculum will actually complete about half of the high school objectives in addition to those assigned to middle school.
According the Principles and Standards for School Mathematics, the overall focus of the algebra strand is to:
·
Understand patterns, relations, and functions
·
Represent and analyze mathematical situations and
structures using algebraic symbols
·
Use mathematical models to represent and understand
quantitative relationships
·
Analyze change in various contexts (2000)
These general standards are also broken down into specific middle school expectations. Montgomery County has done a very good job of covering all of these specific goals in the new curriculum. Every one of the goals is included in the official curriculum that the vast majority of students will encounter. The Math C course covers a majority of the topics, which makes a great deal of sense because Math C is the pre-algebra course for most MCPS students. The honors students take Math Investigations in seventh-grade instead of Math C. This course covers all of the objectives of Math C as well as several additional, more advanced, topics. One of the topics, developing an initial understanding of the different uses of variables, is covered in fourth-grade, well before the prescribed middle school placement of the National Council of Teachers of Mathematics. (MCPS, 2001)
Unlike with the number and operations standard, none of the high-school algebra topics is taught in the MCPS middle-school curriculum. However, the curriculum assumes that at least forty-percent of all students in Montgomery County Public Schools will take Algebra I in middle school. The MCPS Algebra I course covers all of the high school algebra topics with the exception of periodic, iterative, and recursive functions, which are covered in the Algebra II course. (MCPS, 2001) This goal is not yet a reality, but it must be taken into account when evaluating the match of the curriculum to the National Council of Teachers of Mathematics standards. At the current time, all of the seventh grade Math Investigations students take Algebra I in eighth-grade as well as several students who skip from Math B (seventh grade on level) due to exceptional performance.
According to the Principles and Standards for School Mathematics, the overall focus of the geometry standard is to:
·
Analyze characteristics and properties of two- and
three-dimensional geometric shapes and develop mathematical arguments about geometric
relationships
·
Specify locations and describe spatial relationships
using coordinate geometry and other representational systems
·
Apply transformations and use symmetry to analyze
mathematical situations
· Use visualization, spatial reasoning, and geometric modeling to solve problems (2000)
Each of the above standards is broken into specific expectations for grades six through eight. The Montgomery County curriculum for middle school includes almost all of these goals. There are, however, parts that are not covered. The relationships among the volumes of similar figures is not taught, or even introduced, until geometry, a course that most students do not encounter until tenth grade. The curriculum also does not include parts of the geometric modeling piece of the strand. The use of networks to solve problems is never covered in the curriculum, and the use of geometric models to represent and explain numerical and algebraic relationships is not officially a part of the curriculum. Some teachers use geometric models to help students develop a deeper understanding of material, but they are not required to do so. (MCPS, 2001)
Montgomery County did include one topic in the middle school curriculum that the Principles and Standards for School Mathematics does not list until high school. The Math C curriculum includes a unit on geometric constructions with a compass and straightedge. (MCPS, 2001) While this unit does not cover the three-dimensional construction and drawing part of the goal, it does go further than required by the National Council of Teachers of Mathematics by introducing constructions at all. The topic is reviewed in geometry, but the bulk of instruction on constructions is left for the eighth-grade course.
According to the Principles and Standards for School Mathematics, the overall focus of the measurement standard is to:
· Understand measurable attributes of objects and the units, systems, and processes of measurement
· Apply appropriate techniques, tools, and formulas to determine measurements (2000)
The document breaks these standards into expectations for middle school students. Montgomery County did a very good job including these goals in the revised curriculum. In fact, only one expectation is not specifically listed in the MCPS curriculum for grades six through eight. The use of common benchmarks to estimate measurements is not included in the list of objectives. (MCPS, 2001) However, a survey of middle school teachers at one school found that ninety-percent of the teachers (nine of the ten interviewed) present this material in their classrooms. (Parkland Middle School, 2001)
Students in Montgomery County learn almost all of the high school expectations for this standard while in middle school as well. The National Council of Teachers of Mathematics lists four high school expectations for the second standard in measurement. Two of these expectations are covered in the Math C course. These eighth-grade students learn to analyze precision and accuracy in measurement, and they are expected to understand and use surface-area and volume formulas for geometric figures. In fact, students are introduced to surface-area and volume in Math B. Students who take Math Investigations learn one more of the high school objectives. They are expected to become quite proficient with unit analysis as a means to check measurement computations. (MCPS, 2001) For these students, there is only one expectation for the second standard that is left for high school, the use of successive approximation, upper and lower bounds, and limit in measurement.
According to the Principles and Standards for School Mathematics, the overall focus of the data analysis and probability standard is to:
· Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
· Select and use appropriate statistical methods to analyze data
· Develop and evaluate inferences and predictions that are based on data
· Understand and apply basic concepts of probability (2000)
The National Council of Teachers of Mathematics has also included specific expectations for different grade bands. Montgomery County included almost all of the middle school expectations in the revised middle school curriculum. There are only two expectations that are not specifically included. Understanding and using the appropriate terminology to describe complementary and mutually exclusive events is only covered in the Math Investigations curriculum. The use of conjectures to formulate new questions and plan new studies is not specifically listed anywhere in the curriculum. (MCPS, 2001) An interview of several middle school teachers found that some teachers do cover these topics. One of the Math B teachers introduced the above terminology, and all of the Math C teachers talked to the students about using conjectures. (Parkland Middle School, 2001)
In June 2001, a draft form of another set of standards for middle school mathematics was released in draft form. Achieve, Inc.’s Mathematics Advisory Panel provided another view of what students need to know before entering high school in Foundations for Success. This document proposes that the National Council of Teachers of Mathematics “set the stage” for higher standards in mathematics education and proposes that Achieve, Inc. has the solution. According to their document, “MAP recognizes that [their vision] encompasses more mathematics than most U. S. students are currently learning by the end of the eighth grade. In fact, it covers more than some students now learn by the time they finish high school.” (Foundations for Success, 2001) Since Maryland is listed as one of the states that will be administering an end-of-eighth-grade test based on the Achieve, Inc. standards in the spring of 2002, it makes sense to compare the new curriculum to these standards. The standards are broken into four areas that are discussed, in detail, below. Because only middle school standards have been developed at this point, it is not possible to look for areas where MCPS has included high school level objectives.
According to the Achieve, Inc. Mathematics Advisory Panel’s 2001 report, “conceptual depth and computational fluency in the arithmetic of rational numbers is the backbone of mathematics in daily life and the foundation of quantitative thinking in science and business.” This standard focuses on the following three concepts:
· Whole Numbers
· Rational Numbers
· Real Numbers
Each of these topics is broken down into more specific expectations and sample problems are included to assess these expectations. The authors also state, “students will need to accomplish much of the work in the number strand early in their middle school years.” (Foundations for Success, 2001) In the whole numbers strand, MCPS has followed this recommendation very well. All of the expectations are covered in Math A, the sixth grade course. With the exception of one expectation, the same can be said for the rational numbers strand. The definition of rational numbers as the quotient of integers (with a non-zero denominator) is only taught, officially, in the Math Investigations curriculum. However, half of the Math C teachers at one middle school in Montgomery County stated that they do present the material. (Parkland Middle School, 2001)
Unfortunately, almost none of the expectations in the real numbers strand are part of the MCPS curriculum. There are four expectations dealing with rational and irrational numbers. These topics are only part of the Math Investigations curriculum, but the same teachers who included the definition of rational numbers also included this material. (Parkland Middle School, 2001) Only three expectations in this area are actually included in the MCPS curriculum, but they are in Math A. (MCPS, 2001)
According to Foundations for Success, “gathering, measuring, counting, representing, summarizing, transforming, projecting, and interpreting data are essential mathematical skills. Students’ exposure to the complexity of measurement is their first step toward becoming skeptical and critical users of data.” (2001) The expectations in this domain focus on the following areas:
· Measurement and Approximation
· Data Analysis
· Probability
Each of the above areas is broken down into several specific expectations. In the area of measurement and approximation, the majority of the concepts listed are covered before students leave eighth grade. Three of the expectations are part of Math A, two are part of Math B, and four are part of Math C. There are, however, six expectations that are not listed in the MCPS curriculum. Converting between different systems of units, such as from meters per second to miles per hour, and solving problems using unit rates are only in Math Investigations. Algebra I, the on level ninth grade course integrates two more expectations, the use of derived quantities and the calculation of weighted averages. However, two objectives are not taught in the middle school at all. Students do not learn about the role of significant figures or how the precision of measurement affects the accuracy of the derived quantities until high school physics! (MCPS, 2001)
Almost all of the expectations in the data analysis and probability strands are covered in the middle school curriculum. Two of the data analysis expectations are in Math A, three are in Math B, and one is in Math C. Three of the probability expectations are in Math A, and two are in Math C. Students who are in Math Investigations also learn to recognize and avoid common misconceptions about probability associated with dependent and independent events. One of the Math C teachers at a MCPS middle school also presents this information. (Parkland Middle School, 2001). It is interesting to note that four of the data analysis expectations, and one of the probability expectations, are not presented in middle school at all. Relative and cumulative frequency, the difference between causation and correlation, normal distribution, estimating correlation, and the relation of probability to relative frequency are not part of any courses other than statistics or AP statistics. These courses are not required for any students, and are offered as electives in late high school. (MCPS, 2001)
According to Foundations for Success, “geometry provides the foundation for many common uses of mathematics, from architecture and manufacturing to computer graphics and telecommunications.” (2001) The expectation in this domain focus on the following areas:
· Geometric Figures
· Measurement
· Transformations
Each of the above areas is broken down into specific expectations. In the geometric figures strand, most of the expectations are part of the MCPS curriculum. Two expectations are in Math A, six are in Math B, and three are in Math C. There are even four of the objectives in the elementary school curriculum (three of the four are in fifth grade)! There are eight concepts in this domain that are not part of the middle school curriculum. The sum of the exterior angles of a polygon, the characteristics of polygons inscribed in circles, tangents, cross sections of spheres, and the intersections of planes and solids are part of the geometry curriculum and not taught until high school. The vocabulary of two-dimensional representations of three-dimensional solids (orthogonal, projective view, and isometric view is never discussed, and the Pythagorean theorem is used but never proven. (MCPS, 2001)
The measurement and transformations domains are covered more completely by the MCPS curriculum. One of the transformations expectations is part of the Math B curriculum, and all of the others are in Math C. Two of the measurement expectations are in Math A, six are in Math B, and two are in Math C. One expectation, solving problems using the area of triangles, quadrilaterals, and circles, is even listed ahead of schedule. It is a fourth grade objective in the new MCPS curriculum. There are, however, three expectations that are not covered in the middle school curriculum. Students learn to find the length of an arc in geometry and the distance between two points in Algebra I. One expectation, finding the volume of a hemisphere, is not included at all. (MCPS, 2001)
According to Foundations for Success, “the middle grades provide a crucial and challenging transition from the concrete mathematics of elementary school to the more abstract and conceptual mathematics of high school. Nowhere is this challenge more striking or more important than in the transition from arithmetic to algebra.” (2001) The expectations in this domain focus on the following areas:
· Symbols and Operations
· Functions
· Equations
Achieve, Inc. breaks the symbols and operations domain into seven expectations. Almost all of these expectations are part of the MCPS curriculum by the end of eighth grade. There are two in fourth grade, two in fifth grade, one in Math A, and one in Math B. The coverage of one of these topics is not complete because Achieve, Inc. expects students to “understand the meanings of constant, variable, and parameter, and the relations among them.” (Foundations for Success, 2001) The fifth grade curriculum includes the first two words, but parameters are not included until Algebra II. One objective is not part of the middle school curriculum at all. The following formulas are not included until Algebra I.
· (a+b)2 = a2 + 2ab + b2
· (a-b)2 = a2 - 2ab + b2
· (a+b)(a-b) = a2 – b2 (MCPS, 2001)
The functions domain is split into two separate pieces, linear functions and nonlinear functions. Montgomery County includes all of the linear functions objectives in the middle school curriculum. One is in Math A, four are in Math B, and four are in Math C. On the other hand, nonlinear functions are not covered in the curriculum at all. Achieve, Inc. suggests that students learn to recognize the graphs of quadratic, rational, cubic, square root, and exponential functions. They also want students to recognize and represent simple nonlinear functions in problem contexts, factor quadratic functions, and work fluently with nonlinear functions that represent relationships. All of these concepts are part of the Algebra I curriculum. One of the expectations, factoring quadratic functions by completing the square, is not covered until Algebra II. (MCPS, 2001)
Even fewer of the expectations in the equations domain are part of the Montgomery County math curriculum at the middle school level. Two of the expectations are in the Math B curriculum, and four are in Math C. Seven of the expectations are not covered in middle school. Understanding that the solution to a linear equation is the point where the graph crosses the x axis, solving a system of linear equations in two variables (graphically, by substitution, and by linear combination), recognizing and expressing the relationship between equations and functions, using graphs to check algebraic solutions, and solving quadratic equations (by factoring and by finding the intercept) are all part of the Algebra I curriculum. Solving quadratic equations by completing the square is not part of the curriculum until Algebra II, a class that most students do not take until eleventh grade. (MCPS, 2001)
When Montgomery County released a new curriculum in the summer of 2001, they claimed that it was based on the expectations set forth by the National Council of Teachers of Mathematics in their 2000 publication Principles and Standards for School Mathematics. In fact, copies of these standards were distributed with the new curriculum, and teachers were encouraged to refer to it when questions arose during the year. It seems that MCPS was successful in aligning their curriculum with the National Council for Teachers of Mathematics. Almost every one of the objectives in the grades six through eight band are incorporated into the MCPS middle school curriculum, and there are more objectives taught ahead of the prescribed schedule than behind. Additionally, the results so far have shown that the new curriculum is sufficiently challenging for the students without placing the bar so high as to ensure failure. (Parkland Middle School, 2001) When the honors objectives of Math Investigations and Algebra I are considered as well, MCPS goes well beyond the goals listed in the Principles and Standards for middle school students, providing a challenging mathematics education for the above grade level students as well.
On the other hand, it appears that the Montgomery County curriculum does not meet the standard set by the Mathematics Advisory Panel of Achieve, Inc in Foundations for Success. The number domain is fully covered, but not at the prescribed time. Achieve, Inc. suggests that students should learn these concepts early in middle school, but MCPS students do not complete the domain until the end of eighth grade. Falling behind in this area sets students up to fall even further behind in the other domains. The problem becomes most evident as students reach the algebra domain. In this area, only students who take Algebra I in middle school come close to meeting the standard. A few of the topics are not covered until Algebra II, which even honors students do not take until tenth grade. It seems that the students in Montgomery County will not be able to pass the proposed end-of-eighth-grade test based on the new curriculum. In fact, the only students who can pass are the ones who take Algebra I in eighth grade, a trend that is becoming less popular in many areas. (Kilpatrick, 2001) However, instead of pushing more students into Algebra I, MCPS should stick to the new curriculum because it better meets the needs of the diverse student population.
Foundations for Success: Mathematics for the Middle Grades (Draft). Washington, DC: Achieve, Inc. Mathematics Achievement Partnership, July 2001.
Keys to Math Success: A Report from the Maryland Mathematics Commission. Baltimore, MD: Maryland State Department of Education, June 2001.
Kilpatrick, J., J. Swafford, and B. Findell, Eds. Adding it Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press, 2001.
Montgomery County Public Schools. MCPS Mathematics Curriculum Framework. Rockville, MD: Montgomery County Public Schools, July 2001.
Mullis, Ina V. S. et al. Mathematics Benchmarking Report TIMSS 1999 – Eighth Grade. Boston, MA: International Study Center, Boston College, International Association for the Evaluation of Educational Achievement, April 2001.
Parkland Middle School Math Department. Personal Interview. Rockville, MD December 2001.
Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2000.