Report to:        Daniel S. Papp, Senior Vice Chancellor

for Academics and Fiscal Affairs,

                    Board of Regents of the University System of Georgia

Submitted to: Dorothy Zinsmeister, Senior Faculty Associate,

                          Academics and Fiscal Affairs

 

Submitted by: Advisory Committee on Mathematical Subjects (ACMS)

Catherine C. Aust, Chair

 

Date:                March 5, 2004                     

 

In response to your request, the ACMS has reviewed the proposed Georgia Performance Standards in Mathematics. We appreciate the opportunity to express our views and contribute to the Chancellor’s coordinated USG response to the Department of Education.

 

While we applaud the intention to replace a curriculum that is “a mile wide and an inch deep” with one that actively engages students in the development of mathematical understanding and emphasizes the NCTM process skills, we have serious concerns about the Georgia Performance Standards in mathematics as presented at http://www.glc.k12.ga.us/spotlight/gps2.htm (currently located at the new site http://www.georgiastandards.org/). We doubt that adoption of these curriculum standards will lead to the desired improvement in the mathematical performance of Georgia’s public school students because of problems in the content and difficulties with the implementation plan.

 

The ACMS met on Feb. 26-27, 2004, at South Georgia College and developed consensus on the following issues:

·      Statements of some standards are too generic, and the sample tasks do not provide needed specificity.

·      Important topics have been omitted.

·      Planned teacher training is insufficient to support such a drastic revision of the curriculum.

 

In addition, the document seems to have been written without careful consideration of the following issues:

·      Many topics are classified under the wrong content strand, and a number of mathematical terms are misused or used ambiguously.

·      The implementation schedule is unrealistic.

·      The standards have been taken from the Japanese mathematics standards without adequate consideration of cultural differences.

 

Your request for input asked that we address five questions. The answers to these questions cut across several issues, so responding to them one by one would lead to unnecessary repetition. Thus, we have chosen to provide detailed discussions of each of the above issues; collectively these discussions address the questions. In our discussions, numerous specific examples from the standards are cited, but these are not intended as an exhaustive list. They are used to illustrate and clarify our concerns.

 

The first issue is the generic language used in the statement of some of the standards; for example, in grade 3, section M3N5b, the standard indicates that students will “know that addition and subtraction can also be applied to decimals and fractions with like denominators in pictures and problems.” The range of skills and understanding to which this statement can legitimately be applied is wide. One class can know that the picture of two halves of a sandwich is equivalent to the picture of a whole sandwich (when the slice that would create the two halves is shown in the picture) and also perform the subtraction “7.5 – 2.5.” Another class might have to “take away” ¾ of a pie from the picture of a pie and a half by cutting the one pie into fourths and the half pie again in half and, separately, later perform the subtraction “3.5 – 6.2347,” a subtraction in which the decimal expressions for the two numbers differ in the number of decimal places shown and in which the result is a negative number. Both sets of students could be considered as having satisfactorily achieved the specified standard. Students who pass forward from these two classes would all enter the same grade 4 curriculum. Needless to say, having two such disparate groups in one classroom would have major pedagogical implications for the grade 4 curriculum in that classroom.

In the Foundations of Mathematics II, sections MFM2A1a and b, the meaning of the standard is not clear, and there are no examples in the tasks for the statements “generate ordered pairs from a quadratic relationship” and “create and interpret graphs of quadratic equations from data.” Also, in section MFM2A3 of the same course, the standard states that “the student will investigate the rate of change of .” This statement could mean that the student should be in some way doing a “derivative,” a calculus topic. Other possibilities are that the student could be looking at a table of values in which the “x-values” are uniformly distributed and comparing second differences, or that the student could be looking at average rates of change in the function. Since no tasks are given, the specific nature of the desired investigation remains unclear.

 

The Pre-Calculus standards include the statements “The student will explore parametric representations of plane curves” (section MPCA10) and “The student will use polar representations” by converting from rectangular to polar coordinates and graphing various equations( section MPCA11). Again, the range of skills and understanding to which such statements could apply is vast, and there are no tasks to clarify the intent.

 

Additional examples occur in the eighth grade standards. Section M8G1 states “ The student will find geometric properties of a figure in a plane and confirm them by using properties of parallel and perpendicular lines including the right triangle.” Since a right triangle is not a property of parallel and perpendicular lines, this phrasing makes no sense. Section M8N3c states that students will “multiply and divide simple monomials and polynomials.” This standard could be satisfied in one instance by being able to perform the indicated multiplication and division, respectively, in and, yet in another instance, be satisfied by computing and . With no language to clarify the range and level of algebraic concepts that should be introduced, and no sample tasks addressing multiplication and division, the standard fails to specify a particular level of performance.

 

On the surface, the topics included in the courses make it appear that in the future students should be better prepared to take college-level mathematics classes. However, lack of specificity makes some of the standards unusable as guides to the classroom teacher. This lack of specificity, often accompanied by a failure to provide any appropriate tasks for the standard, makes it difficult to judge, for a number of topics, both the intended depth of coverage and whether students will be prepared to take college-level mathematics.

 

A second issue concerns the omission of topics from some of the course sequences – topics that, in the opinion of this committee, should have been included. In the Accelerated Mathematics – Pre-Calculus course sequence, which leads to AP Calculus, piecewise-defined functions are omitted. These functions play an important role in calculus and have many real-world applications, such as postal rates, cell-phone rates, state and federal income tax rates, and costs of purchases where different quantities purchased have a different price per item. Students taking this sequence should also study absolute value and root functions since calculus assumes understanding of these functions. Exponential functions are omitted from the Foundations of Math courses, even though they have many real world applications, including compound interest, population growth, and radioactive decay. The course descriptions for Foundations of Mathematics I-IV do not include statements about using manipulatives, appropriate technology, and multiple representations. Such statements are included in the Mathematics and Accelerated Mathematics course sequences and should be included in Foundations of Mathematics.

 

In all three of the high school course sequences, the development of the concept of “function” needs attention. The term is used in all high school courses except Mathematics I (a curious omission), and the concept is fundamental to higher mathematics, but the prerequisite courses in the eighth grade and below do not properly address the development of the general concept. These courses do extensively address the idea of a function in the context of studying linear relationships between two quantities. However, linear functions are among the simplest examples of functions. An appropriate development of the concept of function requires more than an extensive discussion of one type of function, just as the development of the concept of numbers requires more than an extensive study of the counting numbers. Development of the use of functional notation is also missing. There is no reference to the notation in the list of terms/symbols for the eighth grade and below, yet this notation is used repeatedly in the statement of standards and tasks for the high school strands. Understanding the notation does not come automatically, but introducing it is not included as a topic of study.

 

There are other topics, in addition to the development of the concept of function, that have been omitted and need to permeate the high school curriculum . All of the high school course sequences lack sufficient content on factoring, operations on rational expressions, and manipulations expressions involving radicals and fractional exponents for students intending to study calculus (whether in high school or college). Additional content on factoring, rational expressions, fractional exponents, and expressions involving radicals should be included in all of the high school course sequences, and these topics should be studied in depth in the Accelerated Mathematics – Pre-Calculus sequence. In addition, while the process standard of “Reasoning and Proof” applies to all of the high school courses and while improving students’ abilities to reason mathematically should be an important part of every high school course, the actual standards and corresponding tasks provide very little evidence that this process standard will receive adequate attention in any of the high school course sequences.

There is also a content omission associated with the flow chart of possible course sequences for high school students. According to the flow chart, a student who takes Foundations I and II can proceed to Mathematics II. A close comparison of the content of Foundations I and II with the content of Mathematics I shows that the Foundations student will be missing algebraic topics of cubing binomials and solving quadratic equations by completing the square, and will have studied a very different set of data analysis topics. Very few of the geometry topics in Mathematics I appear in either of the Foundations courses so that any student who takes this path will have not seen these topics also. Presumably, students who move from Fundamentals I-II to Mathematics II will be college bound. These missing topics, especially those involving geometry, will put these students at a disadvantage when they take the SAT.

 

A third content issue involves the misuse or ambiguous use of mathematical terms and misclassification of topics. Several instances of misuse of terminology are found in the eighth grade tasks. In task M8N4B the statement “…you subtracted A and B” makes no sense because subtraction is not commutative. If you subtract $50 from your $20 bank account you will get a very different reaction than if you subtract $20 from your $50 account. You must subtract A from B or subtract B from A. In task M8A1B the student is asked to calculate (which has a value of 4,194,304) and “write it in scientific notation.” This is a misuse of scientific notation. It would be much better to say "provide an approximation of 2²² correct to the nearest thousand and express the result in scientific notation." Task M8A2G asks students to "name" some equations, when it should ask students to "write" these equations.

 

Several misuses of mathematical terminology occur in those high school standards that require students to solve equations. In the following cases a mathematical expression is given, but the choice of expression is inappropriate for the context. In Foundations of Mathematics II, standard MFM2A1c reads, “solve quadratic equations of the form ”; standard MFM2A1d reads, “solve quadratic equations of the form by….” In Mathematics I and II, standard MM1N3c reads, “Solve quadratic equations in the form by…”; standard MM2A2 reads, “Solve quadratic equations and inequalities in the form, ”; and standard MM1N3b reads, “Create and interpret graphs of quadratic equations from data.” Standard MAA3 in Accelerated Mathematics I reads, “Solve quadratic equations and inequalities in the form .” Some of these problems could be the result of editing errors, and are easily corrected, while others need to be examined for mathematical accuracy. Throughout the document there seems to be a lack of distinction between two very distinct concepts: solving a quadratic equation of the form and graphing a quadratic function of the form . These incorrect uses of mathematical terminology will be embarrassing for Georgia if they are not corrected.

 

The Georgia Performance Standards adapt the content strands of the NCTM curriculum; these are: Number and Operations, Algebra, Geometry, Measurement, Data Analysis and Probability. Many standards are classified in the wrong content strand. In the Foundations of Mathematics courses, standards MFM2N1d, MFM2N2, MFM4N1, and MFM4N2 in the Numbers and Operations section all deal with algebraic topics and thus belong in the Algebra section. In Mathematics I, standards MM1N2 and MM1N3 in the Numbers and Operations section deal with algebraic topics and, again, should be in the Algebra section. In the middle grades curriculum, standards M6A3 and M6A4 in the Algebra section deal with data analysis and should be included in the Data Analysis and Probability section. Standards M7D1 and M7D2 in the Data Analysis and Probability section deal with algebraic topics, which belong in the Algebra section. Examples in the eighth grade include standards M8N1, M8N2, M8N3, and M8N4 in the Numbers and Operations section; all deal with algebraic topics and belong in the Algebra section. Misclassification of standards displays a disregard for mathematical accuracy, contradicting the goal of increasing the mathematical rigor of the curriculum.

 

The remaining discussion addresses the committee’s concerns about issues other than content. We begin with concerns about the implementation schedule for the standards. Based on the information we have been able to find, we understand that this plan calls for teacher training in year 1 of implementation and teaching and testing of the new curriculum in year 2, with the actual calendar year corresponding to years 1 and 2 dependent on grade level. Since year 2 will be the year that students experience the new curriculum, as far as students are concerned, the grade 6 standards will be implemented in 2005-2006; grades K-2 and 7, in 2006-2007; grades 3-5 and 8, in 2007-2008; grades 9-12, in 2008-2009 (see Appendix A). Students entering high school next year will not be affected by the new curriculum, but those entering middle school in Fall 2004 will. Next fall’s sixth, seventh, and eighth graders will not see the new curriculum until their sophomore, junior, and senior years in high school, respectively, at which time they will be placed in the second, third, or fourth course of one of the three high school sequences without the benefit of any experience with the curriculum at any of the earlier levels. It is possible that this is not the intent, that full high school implementation in the fourth year is intended to apply only to the ninth grade and move forward each successive year. We hope that this is the case; if not, we anticipate major problems for the students and teachers of grades 10-12 in the year 2008-2009.

 

If the plan is to allow current middle school students to complete high school in transitional courses that presume only that prerequisite content which students have actually had an opportunity to learn, there is still a serious concern for all of the students who will be asked to enter the new curriculum in grade 6 (these are students who will enter grades 3 and 4 in Fall 2004 – see Appendix A). According to the standards, these students will be asked to evaluate their assertions and conjectures, use inductive and deductive reasoning to formulate mathematical arguments, and present their thinking orally and in writing; however, they will not have experienced the rigorous and complex content proposed for grades K-5 and will not have spent their elementary school years focusing on developmental levels of the Reasoning and Proof process standard. Grade 6 is the beginning of middle school and a socially challenging time for many students. The section on process standards for grades 6-8 notes issues of social norms in middle-grades classrooms and urges teachers to “build a sense of community in middle-grades classrooms so students feel free to express their ideas honestly and openly, without fear of ridicule.” In many grade 6 classrooms, building such a climate with students who are being introduced to the new standards for the first time will prove not to be possible, and the strains of attempting to do so will be very stressful for teachers and will create an extremely hostile environment for some students, especially exceptional students at both ends of the achievement spectrum. Thus, we believe that this implementation plan places too great a burden on current third and fourth grade students and the teachers who will instruct them in grades 6-8. To implement this radical change in content and pedagogical approach, we believe that the plan with the best chance for success is one which begins with kindergarten students and progresses through the grades one by one in successive years.

 

We also have concerns about the implementation schedule relative to high school graduation testing. Once the high school standards are fully implemented in 2008-2009, the Georgia High School Graduation Test (GHSGT) for students in the eleventh grade will be based on the new standards, even if it is the student’s first or second year in the curriculum. Students entering grade 7 in Fall 2004 will have their first encounter with this new curriculum in the eleventh grade and will be tested on this curriculum; students entering grade 6 this fall will encounter the new curriculum in the tenth grade and be tested the next year. The plan to thoroughly implement the high school standards in 2008-2009 implies that the testing will not only assume the grade 11 standards but also presupposes (because of the strands) attainment of some level of competency in all of the new grade K-10 standards.

 

Another issue concerns the training of the teachers for this new curriculum. The plan for training states that teachers will receive at least eight days of professional learning; the schedule calls for a two-day workshop in the summer before year 1, two days during the school year in year 1, another two-day workshop in the summer before year 2, and two days during the school year in year 2. Based on the experiences of some of the members of the ACMS with other curriculum changes (such as the School Mathematics Study Group initiative in the 1960’s) and on mathematics education literature, it is inappropriate for teachers to have this minimal preparation for such a drastic revision of the curriculum. The third grade teacher needs to understand not only a mathematical concept as it relates to his/her class but also the impact of that concept on the students’ sixth and ninth grade curricula; analogous statements apply across grade levels. The content foundation of algebra and geometry for the high school mathematics courses is scheduled to be taught in middle school, yet one of the recurring topics in Georgia education concerns the number of teachers in middle school who are teaching mathematics “out-of-field.” As with other mathematics curriculum changes in the past, the best ideas on paper, implemented without adequate training of those who are to teach those ideas, could result in only a shadow of what was intended actually being done. It is the strongly held opinion of this committee that adequate training of the teachers must be an essential part of the implementation of a new mathematics curriculum.

 

Another concern relates to the eighth grade curriculum, which has been described earlier as ambiguous. It appears to contain an overabundance of topics and skills when compared to the sixth and seventh grade curricula, and gives the impression that it is trying to play “catch-up” on all of the mathematics necessary to succeed in any of the high school strands. According to the materials provided and studied, satisfactorily accomplishing the goals of the same grade 8 curriculum (although occasionally reaching this level in grade 7) prepares students for each of the three course sequences: Foundations of Mathematics, (regular) Mathematics, and Accelerated Mathematics – Pre-Calculus. Presumably, the student upon exiting the grade 8 curriculum can satisfactorily perform at each of these three levels; this is puzzling to the ACMS.

 

In addition, “all students will graduate with the mathematics requirements for college,” according to the video introduction and PowerPoint slides provided by the Department of Education. In the printed materials there is no such claim for the Foundations of Mathematics course sequence. It is the opinion of the ACMS that, in content areas such as understanding exponential functions and using technology, students who take the Foundations of Mathematics sequence will be minimally prepared for the Introduction to Mathematical Modeling college course and that such students will not be adequately prepared for the College Algebra course that includes more algebraic structures and methods than the modeling course.

 

The ACMS is also concerned that the new mathematics curriculum has been taken from the Japanese standards without sufficient consideration of cultural differences between Japan and Georgia. The amount of time spent with mathematics, in school as well as outside (both in tutoring and in homework), is vastly different in the Japanese culture from current practice in the American culture. Under the new standards, each student must read an entire book a week on the average (see “Reading Across The Curriculum” in the GPS Training media), spend time “investigating” and “exploring” mathematical topics to prepare for and supplement classroom activities, and do homework for more rigorous science and social science curricula. It is not clear that the students will have the time to develop themselves individually as we have come to expect, whether it be in afterschool activities, in organized sports like soccer, in independent investigation of topics of particular interest, or by just having enough free time to stimulate creativity and originality. Also, the Japanese environment for teachers is different; each school day allows time for mathematics teachers to meet and discuss the topics for the next day. In such a process, teachers talk about the various ways to present the material, try to anticipate the problems that students might have, and collaboratively generate ideas about the best way(s) to deal with them. We find nothing in the proposed standards indicating plans to develop a similar environment for Georgia’s mathematics teachers.

 

Editing is one other issue that requires comment. Many of the previous examples show that careful editing is necessary to insure that the mathematics is accurately described and that the materials are useful for teachers attempting to locate resources. Since this is a drastic revision of the curriculum, teachers will be working with new instructional materials and seeking appropriate supplementary materials. Also, teachers with little or no training in mathematics may not even be aware that there is a problem with terminology or descriptions, so language becomes a problem for the student in the next course(s). The editing also needs to extend to any tasks that represent the content of a standard. For example, consider standard M8N4: “The student will understand the meaning of simultaneous linear equations, their solutions and applications.” One of the associated tasks is M8N4F: “Find the set of all integers which satisfy the following system of inequalities: , .” Either the standard needs to be edited to include systems of linear inequalities in one variable, or the task needs to be revised to contain a system of linear equations in two variables rather than the given system of inequalities. In the Terms and Symbols section of the grade 8 mathematics, “coordinate plane” is listed without mentioning “axes,” and “vertical line test” is listed without mentioning “graph of a function.” Grammar also needs to be checked. Repeated use of the plural personal pronoun “their” to refer to a single student is gender neutral but grammatically incorrect. The Mathematics Curriculum Revision – Executive Summary (obtained from the new Web site for the Georgia Performance Standards at http://www.georgiastandards.org/ ) indicates that the document is still a work in progress and states that additional student work and commentaries will be added. We hope that the developers will also see the need for editing as documented above; the ACMS strongly encourages this editing.

 

In closing, the committee notes that the current curriculum revision process has so far adopted an approach in which the Department of Education informs the colleges of the standards that will be met by students who complete high school. The committee recommends a more open and collaborative approach. Publishing information about the development process and identifying the authors of the current document would facilitate dialogue about ways to improve the proposal. There are many mathematics faculty members in University System of Georgia institutions who have both an interest in working with Georgia’s public school mathematics teachers to improve the performance of their mathematics students and the mathematical and pedagogical expertise to provide useful contributions. As work on the mathematics performance standards continues, closer cooperation among the Department of Education, the Board of Regents, and mathematics faculty from University System institutions would promote consistency with the University System of Georgia’s expectations for entering freshmen, provide coordination between teacher education programs and expectations for new teachers, and facilitate involvement of USG mathematics faculty in development activities for in-service teachers.

APPENDIX A

 

ACMS CHART DEPICTING IMPLEMENTATION OF TEACHING AND TESTING
THE PROPOSED GEORGIA PERFORMANCE STANDARDS IN MATHEMATICS
Age or Grade level in 2003-2004	Grade and curriculum in 04-05	Grade and curriculum in 05-06; GPS in grade 6 only	Grade and curriculum in 06-07: GPS in grades K-2, 6-7	Grade and curriculum in 07-08; GPS in grades K-8	Grade and curriculum in 08-09; GPS in grades K-12	Grade and curriculum in 09-10	Grade and curriculum in 10-11
2 yr. olds			K - GPS	1 - GPS	2 - GPS	3 - GPS	4 - GPS
3 yr. olds		K - QCC	1 - GPS	2 - GPS	3 - GPS	4 - GPS	5 - GPS
4 yr. olds	K - QCC	1 - QCC	2 - GPS	3 - GPS	4 - GPS	5 - GPS	6 - GPS
K	1 - QCC	2 - QCC	3 - QCC	4 - GPS	5 - GPS	6 - GPS	7 - GPS
1	2 - QCC	3 - QCC	4 - QCC	5 - GPS	6 - GPS	7 - GPS	8 - GPS
2	3 - QCC	4 - QCC	5 - QCC	6 - GPS	7 - GPS	8 - GPS	9 - GPS
3	4 - QCC	5 - QCC	6 - GPS	7 - GPS	8 - GPS	9 - GPS	10 - GPS
4	5 - QCC	6 - GPS	7 - GPS	8 - GPS	9 - GPS	10 - GPS	11 - GPS
5	6 - QCC	7 - QCC	8 - QCC	9 - QCC	10 - GPS	11 - GPS	12 - GPS
6	7 - QCC	8 - QCC	9 - QCC	10 - QCC	11 - GPS	12 - GPS
7	8 - QCC	9 - QCC	10 - QCC	11 - QCC	12 - GPS
8	9 - QCC	10 - QCC	11 - QCC	12 - QCC
9	10 - QCC	11 - QCC	12 - QCC
10	11 - QCC	12 - QCC
11	12 - QCC
Once the curriculum is implemented, the Criterion Referenced Competency Test (CRCT) for
	students in grades 1-8 will be based on the new standards as will the
	Georgia High School Graduation Test (GHSGT) in grade 11.
Yellow highlighting indicates that students will be taught using GPS and tested with the CRCT.
Green highlighting indicates that students will be taught using GPS but not scheduled for testing.
Pink highlighting indicates that students will be taught using GPS and tested with the GHSGT.