K-12 MATHEMATICS CURRICULA, STANDARDS, AND ASSESSMENTS:

IMPLICATIONS FOR STUDENT

MATHEMATICS SUCCESS AT

THE CITY UNIVERSITY

OF NEW YORK

 

 

A Report of

the CUNY Council of Math Chairs

 

 

 

 

Prof. Joseph Bertorelli, Queensborough Community College

Prof. Robert Feinerman, Lehman College

Prof. Warren Gordon, Baruch College

Prof. Ed Grossman, City College

Prof. Stanley Ocken, City College

Prof. George Shapiro, Brooklyn College

 

 

Writers:

Prof. Stanley Ocken, City College

Prof. Ed Grossman, City College


Executive Summary

 

This report is the response of the CUNY Council of Math Chairs to NYCDOE Chancellor Joel Klein’s request for recommendations regarding a critical shared goal: ensuring that as many high school graduates as possible are prepared for success in college-level mathematics courses. These courses require both symbolic and abstract mathematical competency, both of which can only be built on a sound foundation. Although the need of our citizenry for mathematical proficiency has never been greater, that foundation seems to be crumbling.

 

Far too many CUNY students, whether they are pursuing a technical career or are fulfilling a mandatory mathematics requirement, arrive on campus with neither the conceptual framework nor the symbolic skills required by CUNY mathematics courses.

Indeed, the mathematics preparation of CUNY’s incoming students is so weak that large proportions of students cannot compensate for their inadequate preparation, fail mathematics courses repeatedly, and, in many cases, drop out of school. Evidence of weak preparation includes both placement test performance and survey quizzes administered to beginning mathematics students. These weaknesses have been apparent for decades but seem to be worsening, despite well-meaning but often ineffective attempts to address the problem. The goal of this report is to make constructive suggestions for improvement. 

 

If change is to be effected, it is crucial to deal with the sources of weak student preparation. While there are undoubtedly many factors at play, some beyond the reach of academic intervention, we believe that systematic flaws in New York State’s assessments and standards are a significant source of the poor mathematics preparation we see in students entering CUNY. As we will document below, the State has mandated a system of mathematics instruction that is superficial, unfocused, and, above all, lacking in critical mathematics content. Large numbers of high school graduates, including many in CUNY mathematics courses, are at substantial risk for failure. In our view, K-12 students and teachers alike have been victimized by content-poor mathematics instruction resulting from New York State’s defective mathematics standards and assessments.

 

We hope that this report will mark the beginning of a constructive collaboration between the K-12 educational leadership in the city and the CUNY mathematics faculty in addressing these issues. In practice, we realize that the needed systemic change in New York State mathematics instruction will take considerable effort and time. However, neither the NYCDOE nor the CUNY chairs can afford to wait for that to happen. Our more realistic hope for the short term is that as quickly as possible, and within existing constraints imposed by the NYSDOE, mathematics instruction in New York City classrooms will be modified to better serve the needs of both NYCDOE and CUNY mathematics students. 

 

It is important to state at the outset that we are advocating a more focused K-12 mathematics curriculum, not a harder one. The current New York State standards and assessments include a number of topics that are introduced prematurely and treated superficially. The principal consequence of the State’s approach to mathematics instruction is that it is difficult or impossible to cover in sufficient depth the core topics in algebra and analytic geometry that are critical for future success in college mathematics.

 

Nevertheless, until and if New York State revises its mathematics curriculum, CUNY and the NYCDOE will have to work together to address partially inconsistent goals for New York State K-12 students:

  • achieving high scores on New York State mandated assessments, and
  • offering more focused instruction in the core pre-algebra, algebra, and analytic geometry material needed to prepare students for college mathematics courses.

 

We are fully committed to cooperating with NYCDOE stakeholders, including teachers, administrators and math specialists, to achieve both goals. We can offer at least the following levels of assistance:

 

  • With support from NYCDOE and CUNY, we will provide online review material keyed to existing New York State assessments. All students in computer-equipped classrooms will be able to take practice online assessments covering a wide range of materials in current New York State standards and assessments.

 

  • We will work with NYCDOE math specialists to prepare a grade-by-grade online assessment question bank with solutions, together with support materials for teachers, that would represent what students should be able to do in a college-preparatory mathematics curriculum.

 

Our recommendations, immediately following, are consistent with those contained in the recently released report of the National Mathematics Advisory Panel (NMAP), established in April, 2006 by Executive Order 13998. That report is available at http://www.ed.gov/about/bdscomm/list/mathpanel/index.html. The panel’s charge was to advise the President and the Secretary of Education on how “…to foster greater knowledge of and improved performance in mathematics among American students.” We anticipate that states will refocus their mathematics standards and assessments in accord with that report, and we stand on record as supporting its perspectives and major conclusions.

 

In particular, we support the NMAP report’s critique of the mathematics content of the National Assessment of Educational Progress (NAEP), which inadequately emphasizes many important K-12 pre-algebra and algebra topics. We choose instead as a gold standard California’s standards and assessments, whose mathematics content reflects our shared academic judgment as to which topics must be emphasized (and, of particular importance in New York, which should be de-emphasized or omitted) if high school graduates are to succeed in college mathematics courses. We are unsurprised that California’s implementation of standards and assessments with rigorous mathematics content has been accompanied by a sharp decrease in the proportion of students in the California State University system who require mathematics remediation; see http://www.asd.calstate.edu/performance/proficiency.shtml.

 

 

Our immediate goals for CUNY mathematics students are to see significant reductions in the need for mathematics remediation in the CUNY system, as well as an increase in the number of students who enroll in, and who successfully complete, mathematics-related programs and majors. In that spirit, we hope that our suggestions for revisions in New York State standards and assessments will be brought to the attention of the New York State Department of Education as a matter of utmost urgency. We also believe that curricula currently used in New York City should undergo careful scrutiny, with the goal of eliminating those whose use is inconsistent with the recommendations of the NMAP report. In addition, we recommend field tests of promising curricula, such as Singapore Mathematics, a program praised by the NMAP Report, that would better serve our students.

 

 

 

Recommendations

 

1. The NYCDOE and CUNY should establish a joint working committee to establish coherence between K-12 and college mathematics content.

 

The goal of the committee will be to ensure that K-12 curricula adequately prepare high school graduates for college-level mathematics courses, while addressing simultaneously the need to prepare students for existing New York State assessments. Since coherence of mathematics content will be the focus of the committee, its membership should consist primarily of CUNY mathematics faculty and NYCDOE middle and high school teachers. An immediate charge will be to provide materials and support structures for teachers in pursuit of these goals. Materials should include

·        online review material keyed to existing New York State assessments;

·        a grade-by-grade online compilation of mathematics problem sets and solutions that K-12 students need to master in order to be properly prepared for CUNY mathematics courses; and

·        a teachers’ handbook containing grade-by-grade discussions of content and pedagogy relevant to these problem sets.

 

A good model for the teachers’ handbook would be Chapter 3: Grade Level Considerations in the Mathematics Framework for California Public Schools [ http://www.cde.ca.gov/ci/ma/cf/ ].

 

2. Mathematics curricula used in New York City schools should be examined for compliance with the National Math Panel Report.

 

K-8 curricula used in New York City should be critically re-examined to verify that they comply with the recommendations of the NMAP report. Those that do not should be replaced. We recommend that the Singapore Mathematics Curriculum, highly praised by the NMAP Report, be examined carefully and piloted by the NYCDOE.

 

3. All New York City high school graduates should be offered, and should be encouraged to complete, at least three years of Regents Mathematics courses in high school.

 

Current New York State degree requirements call for students entering after fall 2008 to complete three units of math for graduation. In line with this requirement, we strongly urge the NYCDOE to provide all students with the opportunity to take Regents examinations in Integrated Algebra, Geometry, and Algebra 2/Trigonometry as those courses are phased in. Furthermore, all students should be urged to take mathematics during their senior year in high school if they hope to succeed in a mathematics-related major when they get to college.

 

4. The NYCDOE should investigate the expanding base of mathematics tutorial and drill-and-practice software.

 

In particular, the DOE should initiate a cooperative effort with interested CUNY STEM faculty who are establishing web sites for K-12 mathematics instruction. Despite the paucity of quality research studies that met the rigorous review criteria of the NMAP, there was substantial and solid evidence for the effectiveness of drill-and-practice software used by learning disabled students. Indeed, some studies indicated that software alone achieved a better result than a combination of software and traditional teaching. We suggest as well exploring the potential benefits of having general and gifted NYC students use both tutorial and drill-and-practice software.

 

 5. New York State mathematics standards should be rewritten to comply with the recommendations of the National Math Advisory Panel (NMAP) Report.

 

Teachers of college mathematics courses, especially STEM oriented courses, should be well represented on the committee that revises the standards. Those individuals should be responsible for ensuring that the standards have the focus, rigor, and coherence demanded by the NMAP Report. As California has already done in its Mathematics Framework for California Public Schools [ http://www.cde.ca.gov/ci/ma/cf/ ], the revised mathematics standards should be embedded in a comprehensive framework document that provides guidance for teachers, administrators, and publishers as to how the standards should be implemented in practice.

 

 

 

 

6. New York State mathematics assessments should be rewritten to comply with the recommendations of the NMAP report.

 

The California assessments should serve as a model that can be improved even further. The process for devising new State assessments should include substantial input from mathematicians who will ensure that the focus, rigor, and coherence of these assessments reflect NMAP recommendations, and who will also check the mathematical correctness of assessment questions, since past New York State assessments have contained numerous ambiguities and some outright errors.  

 

References

 

New York State Standards http://www.emsc.nysed.gov/3-8/MathCore.pdf

New York State Assessments (K-8) http://www.emsc.nysed.gov/osa/elintmath.html

New York State Assessments (HS) http://www.emsc.nysed.gov/osa/hsmath.html

 

California Mathematics Framework http://www.cde.ca.gov/ci/ma/cf/

California Standards http://www.cde.ca.gov/ci/ma/cf/ Pages 14-106

California guide to Instructional Materials http://www.cde.ca.gov/ci/ma/im/

California Released Test Questions http://www.cde.ca.gov/ta/tg/sr/css05rtq.asp

 

California State University data on student proficiency: http://www.asd.calstate.edu/performance/proficiency.shtml

 

National Mathematics Panel Final Report: http://www.ed.gov/about/bdscomm/list/mathpanel/index.html


 

1. K-16 Mathematics Education in New York City:

Challenges and Solutions

 

College mathematics instruction at CUNY addresses four student audiences, namely those enrolled in

  • remedial mathematics courses;
  • liberal arts mathematics courses that satisfy a core requirement;
  • advanced courses that are part of a major in mathematics;
  • service mathematics courses that are required by programs and majors in science, engineering, technology, finance, business, and other mathematics-based disciplines.

 

All four cohorts need to have acquired appropriate levels of mathematics competency by the time they graduate high school if they are to succeed in their respective courses.  It follows that an important goal of K-12 mathematics instruction should be to prepare high school graduates for success in college-level mathematics.

 

That is indeed the case in countries with high-performing STEM students. However, international comparisons show that the United States is not among these countries. Part of the problem is that there are fifty independent educational systems in this country, one per state, and failure to establish a seamless K-16 curriculum that connects high school mathematics content with the requirements of college mathematics is the rule rather than the exception. New York, in our view, ranks high among those states in which the transition from high school to college mathematics is anything but seamless.

 

Throughout CUNY, we find that all student audiences listed above are badly prepared for their respective mathematics courses:

  • Thousands of students entering our community colleges are unable to pass a remedial course in arithmetic.
  • Large numbers of students fail liberal arts core courses in mathematics, including those that  require as background at most a high school course in ninth grade algebra.
  • At senior colleges, the number of mathematics majors is far lower than in the past, despite the fact that the demand for graduates with strong backgrounds in mathematics has never been greater.
  • In pre-calculus service courses students discover that three years of mathematics in New York high schools did not prepare them for college level work and are forced to change their majors.

 

The basic reason for presenting this report is that the mathematics preparation of CUNY’s incoming students is so weak that large proportions of students cannot compensate for their inadequate preparation, fail mathematics courses repeatedly, and in many cases drop out of school.  Evidence of weak preparation includes both placement test performance and survey quizzes administered at the beginning of the semester.

 

If change is to be effected, it is crucial to deal with the sources of weak student preparation. While there are undoubtedly many factors at play, some beyond the reach of academic intervention, we believe that systematic flaws in New York State’s assessments and standards are a significant source of the poor mathematics preparation we see in students entering CUNY.   As we will document below, the State has mandated a system of mathematics instruction that is superficial, unfocused, and above all lacking in critical mathematics content.  Large numbers of high school graduates, and in particular those in CUNY mathematics courses, are at substantial risk for failure. In our view, K-12 students and teachers alike have been victimized by content-poor mathematics instruction resulting from New York State’s defective mathematics standards and assessments.

 

2. The Goal of this Report

 

The constructive intent of this report is to explain our view of content-rich mathematics instruction and to suggest steps that the NYCDOE can take to properly prepare its students for success in college mathematics courses.  To do so, we will review relevant findings of the recently released report of the National Mathematics Advisory Panel (NMAP), a report commissioned by the President as part of an effort to get a “broken system of U.S. instruction” in American K-12 mathematics back on track.  We will discuss the current instructional focus in mathematics in K-12 in NY State, and why this is not providing the proper background, as described by NMAP, for students entering CUNY. Finally, we will discuss an existing model of mathematics education, namely California’s standards and assessments, that has been in place for a decade and that has in large part anticipated the NMAP recommendations. We suggest that those standards and assessments (good but by no means perfect) can and should be adapted to provide a revised model for K-12 instruction in New York State that would properly prepare students for success in CUNY mathematics courses.

 

We are concerned that only one member of the 24-person committee that drew up the current (2005) New York State mathematics standards holds a research mathematics Ph.D., despite the fact that four persons were identified in the committee report as professors of mathematics. The revision process should address what is in our view a critical shortcoming of the current standards, namely that they fail to emphasize formal and algebraic mathematical competency as crucial outcomes of K-12 instruction.

 

In practice, we realize that the needed systemic change in New York State mathematics instruction will take considerable effort and time. However, neither NYCDOE nor CUNY chairs can afford to wait for that to happen. Our more realistic hope for the short term is that as quickly as possible and within existing constraints imposed by the NYSDOE , mathematics instruction in New York City classrooms will be modified to better mesh with our vision of a college preparatory mathematics curriculum. To aid such efforts we would encourage the Chancellor to appoint NYCDOE math specialists to work with CUNY math faculty in compiling a grade-by-grade assessment question bank, together with support materials for teachers, that would represent what students should be able to do in a truly college-preparatory mathematics curriculum.

 

3. The Report of the National Mathematics Panel http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

 

The National Mathematics Advisory Panel (NMAP) was established in April, 2006 by Executive Order 13998. Its charge was to advise the President and the Secretary of Education on how “…to foster greater knowledge of and improved performance in mathematics among American students…with respect to the conduct, evaluation, and effective use of the results of research relating to proven-effective and evidence-based mathematics instruction.”  The Executive Order further calls for recommendations that are “based on the best available scientific evidence.”  Components of the charge most relevant to the current report included addressing

 

  • the critical skills and skill progressions for students to acquire competence in algebra and readiness for higher levels of mathematics;
  • the role and appropriate design of standards and assessment in promoting mathematical competence;
  • instructional practices, programs, and materials that are effective for improving mathematics learning; and
  • the role and appropriate design of systems for delivering instruction in mathematics that combine the different elements of learning processes, curricula, instruction, teacher training and support, and standards, assessments, and accountability.

 

The NMAP membership included mathematics education specialist from schools of education, mathematicians from university mathematics departments, statisticians, cognitive and educational psychologists, and educators in the field.

 

A critical part of the panel’s charge was to rely upon the “best available scientific evidence” to reach its conclusions.  It is unfortunate but unsurprising that only a tiny percentage of the sixteen thousand research papers reviewed met the criteria for rigor and scientific method required by the panel. Based in part on the small number of qualifying studies, the Panel’s summary of conclusions included the following items, all directly relevant to our report, since each bullet’s recommendation is to some extent (and in some cases, entirely) lacking in New York State’s standards and assessments.

 

·        The mathematics curriculum in Grades Pre-K–8 should be streamlined and should emphasize a well-defined set of the most critical topics in the early grades.

·        Use should be made of what is clearly known from rigorous research about how children learn, especially by recognizing the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic (i.e., quick and effortless) recall of facts.

·        A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.

·        To encourage the development of students in Grades Pre-K–8 at an effective pace, the Panel recommends a set of Benchmarks for the Critical Foundations, included in the current report as Appendix V. They should be used to guide classroom curricula, mathematics instruction, textbook development, and state assessments.

 

·        All school districts should ensure that all prepared students have access to an authentic algebra course—and should prepare more students than at present to enroll in such a course by Grade 8.  

 

·        A major goal for Pre-K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions), for such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped. Proficiency with whole numbers is a necessary precursor for the study of fractions, as are aspects of measurement and      geometry. These three areas   whole numbers, fractions, and particular aspects of geometry and measurement are the Critical Foundations of Algebra.

 

·        The Major Topics of School Algebra (Appendix IV) should be the focus for school algebra standards in curriculum frameworks, algebra courses, textbooks for algebra, and in end-of-course assessments.

 

This report will offer our perspective on what it would mean in practice for New York City and State to implement the recommendations of the NMAP report in an effort to best serve the mutual interests of the NYCDOE and the CUNY mathematics departments and colleges.

 

In many cases, problematic visions of mathematics have found strong expression in New York State standards and assessments, as well as in certain mathematics curricula that are in wide use locally and nationally. Fixing what the NMAP Report describes as a ‘broken’ system of mathematics education in the United States will require substantial alterations in the content and emphasis of

  • New York State standards, which delineate the mathematics content that is to be taught in the schools;
  • New York State  assessments, which in principle test student knowledge of material in the State standards and which are used as indicators of  compliance with the No Child Left Behind Act;
  • programmatic materials stand-alone K-12 textbooks as well as more extensive materials, often referred to as ‘curricula,’ provided by publishers as a complete support system for both teachers and learners that determine in practice how mathematics is conveyed to students.

 

 

4. Mathematical Preparation of Entering CUNY Students

 

As is well-known and widely documented, vast numbers of students enter CUNY completely lacking facility in basic arithmetic. An extensive educational enterprise exists in the community colleges to attempt to repair these problems, but the long-term prognosis of dealing with these issues at this stage is bleak.  For example, it is the experience at one community college that only 8% of students placed in arithmetic have received a 2- or 4- year degree six years after admission.

 

Yet it is not only at the two-year colleges where we find evidence of prior mathematical mis-education.  Non-mathematics   senior college faculty routinely report in conversation that students cannot perform simple manipulations with decimals (multiplying and dividing by 10), or lack an understanding of percents.  Even students who have apparently taken and passed a fair amount of high school-level mathematics are often dismayingly incapable of answering the simplest questions on that material.

 

 For example, the following first-day survey quiz was administered in Spring 2007 to students enrolled in the first mathematics course, College Algebra,   at one of the CUNY senior colleges.  To the right of each question are the percentages of students who answered the indicated question completely correctly For all students tested,, the course was the first post-high-school math course they had taken, and so the results  represent an upper bound for what students retain as they make the transition from high school to college. It is deeply troubling that only one student was able to determine an equation of the line through two given points.  This should be a focus topic for high school mathematics.  Some of the other listed outcomes are scarcely more encouraging.

 

 

          Math Survey Quiz      Time: 25 minutes 

                                                                                       

                                                                                                                 % of students

                                                                                                           answering correctly

Q1:  Write 3/8 as a decimal

34%

Q2:  How much is 0.2 divided by 5.0?

44%

Q3:  Find the value of  5 + 3 ( 4 + 6 )

64%

Q4:  Solve the equation x^2 = 9

10%

Q5: Multiply out and collect terms: 

19%

Q6: Reduce completely: (2x+4y) / (2xz)

19%

Q7: Solve the equation x^2 = x

7%

Q8: Find an equation of the line through points (2,3) & (3, -3)

1%

Q9: Solve x (x + 1) = 2

24%

Q10: Reduce completely (x+2) / (4 - x^2)

25%

Q11: Write 1/3 + 1/4 +1/8 as a reduced fraction

63%

The members of this panel, with over 400 person-years’ experience teaching college mathematics, certainly agree that   students in both courses ought to be able to answer these questions in the allotted time.  Their extensive failure to do so indicates that the students have missed something important in their preparation. We will address this below, but one consequence is  that CUNY mathematics instructors face grave difficulties in running coherent courses with acceptable pass rates and honest grading policies.

 

The above comments point to a frequent complaint. Even when students have taken a mathematics course, their mastery level is often very shallow and quickly erodes.  This problem is by no means restricted to New York State or New York City. Nevertheless, an examination of New York State assessments and standards, which we will now undertake, reveals that their lack of focus and coherence, with insufficient attention to development of mathematical proficiency, can reasonably be viewed as a major contributor to weak student performance. The NMAP report defines these terms as follows:

 

  • Focus: the curriculum must include and engage with adequate depth the most important topics underlying success in school algebra.
  • Coherence: the curriculum is marked by effective, logical progressions from earlier, less sophisticated topics into later, more sophisticated ones.
  • Proficiency:  students should understand key concepts, achieve automaticity as appropriate, develop flexible, accurate, and automatic execution of the standard algorithms, and use these competencies to solve problems.

 

We will document deficiencies in New York standards and assessments. By offering as a contrast the corresponding California materials, we will suggest changes in mathematics instruction in New York City and State that would be needed to improve the performance of entering college students. 

 

5. New York Standards and Assessments

 

It is not possible within the body of this report to completely document the deficiencies in NY standards and assessments. Rather, we will examine typical weaknesses that would eventually undermine our own students’ mathematical competency. Our discussion is based on the following web documents: 

 

New York State Standards: http://www.emsc.nysed.gov/3-8/MathCore.pdf

New York State Assessments (K-8):  http://www.emsc.nysed.gov/osa/elintmath.html

New York State Assessments (HS):  http://www.emsc.nysed.gov/osa/hsmath.html

 

We believe that mathematics standards should give a clear sense of the content, difficulty, and emphasis of both the material that is to be taught and the questions that will appear on assessments.  New York’s standards fall far short of these goals.

 

New York’s standards are divided at each grade into process strands that indicate pedagogical focus and content strands that describe what should be taught. The process strands, comprising more than half of the 125-page MST-3 standards document, are devoted to pedagogical abstractions divorced from content, and in our view send a questionable message to teachers as to where their instructional efforts should be directed.

 

When we turn to the critical issue of content we find, in contrast to the recommendations of the NMAP, that content coverage at each grade is diffuse and lacking coherence. At each grade, the content standards are presented simply as a list of topics devoid of context that would provide the following critical guidance to teachers:

  • how each year’s work builds on earlier instruction;
  • how student competencies should increase from year to year; and
  • what are the key content standards that  should be given priority if teachers find it difficult to cover  a very long list of content goals, e.g., 67 goals in grade 5 and 64 in grade 7.  

 

In addition, we are troubled by an overemphasis of specific content areas. We believe that material on transformations of geometric figures is largely out of place in the K-12 curriculum. More disturbing is that over 25% of the 83 content goals listed for Integrated Algebra deal with statistics and probability, topics that are at best tenuously related to learning algebra. In our view, the consequent necessary de-emphasis on teaching core instructional material contributes to the weak arithmetic and algebra skills that we observe in students entering CUNY.  In contrast, the NMAP report’s outline of the major topics of school algebra devotes just one of 25 bullets to statistics and probability.

 

In terms of establishing procedural fluency as recommended by NMAP, the content standards are particularly evasive and lay the groundwork for student failures in understanding how to manipulate decimals and fractions. The operative phrase in the standards from grades 3 through 5 concerning the teaching of basic arithmetic is "use a variety of strategies to…", in contrast to "learn how to…".  Students are encouraged to create their own methods for multiplication and division, rather than learn the standard methods upon which their understanding of more advanced mathematics will ultimately depend. In contrast, the California standards, anticipating the NMAP report, consistently and wisely omit reference to any algorithms other than the standard ones.

 

The “use a variety of strategies” perspective of the New York standards is implemented in many curricula, notably those that present multiple alternate versions of the standard algorithms. These curricula encourage the use of inefficient methods to solve straightforward problems and should not be adopted,   If they are being used, as is the case in some New York City schools, they should be replaced. [ SOFTEN????]

 

As noted above, by default, assessments drive the entire instructional process in K-12 and largely define what students will know, and the levels of proficiency they will have, when they enter college. If, as we believe, the goal of the assessments is to establish and evaluate mathematics proficiency appropriate ultimately for college-level study, then assessments that fail to do so should be judged as deficient. For example, while most of the topics on elementary arithmetic covered in a community college basic math course are examined in the NY assessments from grades 4 through 6, the level at which those topics are tested is simply too low to provide students with the necessary skills.   The table below presents some typical contrasts:

 

                NY Assessment

Community College Basic Math     Assessment (without calculator)

(5th grade) Compare ¼ to 1/5.

(expressed as a word problem)

Compare 11/13 to 7/9.

(6th grade) Find 25% of 80

 (expressed as a word problem)

Find 4.7% of 80.

(5th grade) Convert 60% to a decimal. (expressed as a word problem)

Convert 21.6% to an equivalent decimal.

(5th grade) Find the perimeter of a rectangle with integer side lengths.  Formula given.

Similar problems, but area and perimeter formulas are assumed memorized.

(6th grade) Add 2/6 and 3/8

(expressed as a word problem)

Compute.

 

The examples in the above table are simply an illustration that even elementary arithmetic is assessed by questions whose level of difficulty is unreasonably low.  Furthermore, the New York assessments persistently and seemingly dogmatically avoid problems that test computational skills outside the context of a word problem.  This practice places an unfair burden on students with weak English skills, who might otherwise perform at superior levels.  Furthermore, it confounds both the learning and evaluation process for these skills to be constantly intertwined.

 

As the examples above make clear, the achievement level required in the NY assessments is far below the expectations of College faculty who teach this material in remedial classes.  Given the level of the assessments it is hardly surprising that students in these classes have only the weakest grasp of basic arithmetic. 

 

Similar problems arise when we look at assessments in the higher grades.   We oppose the pervasive pattern of minimizing the number and difficulty of questions testing numerical and algebraic skills. Furthermore, the sequencing of some topics is flawed. For example, pre-algebra assessments in 7th and 8th grades lack any requirement that students demonstrate proficiency in the arithmetic of signed numbers, although this is an essential skill needed to do algebra. A detailed comparison of grade 7 standards and assessments is given in Appendix II.

 

Of principal concern to us is the quality of New York’s high school assessments, exams that are linked most directly to the mathematical competence of students who enter college.  At a minimum we would hope that the Integrated Algebra Assessment strongly emphasize the critical basic algebra material needed for success in college mathematics. Precisely the opposite is the case. Indeed, the sample assessment provided by New York State, presumably a model for the Integrated Algebra Regents that will first be given in June, is so deficient in testing algebraic technique that the lack of skill development in the preceding years will scarcely matter. The deficiencies of that sample assessment are detailed in the next section and in Appendix III, which document the sharp contrasts between the content and quality of assessments and standards in New York and California.

 

In summary, we believe the content of the New York assessments described above to be so weak that they cannot be used as a floor on student achievement. Simply put, mastery of basic mathematics at the level specified in these exams does not prepare a student for mathematics study at the college level, nor is it likely to prepare students to deal with the mathematical issues they will face as adults.  For a more complete picture of just how weak is New York assessments’ coverage of critical content areas in grades 5-11, see Appendix VI.

 

Fortunately, alternatives exist.  In the next section we describe the California standards and assessments and explain why they suggest an instructional model that should be adapted as quickly as possible for use in New York.

 

6.  The California Model

 

California’s standards and assessments were designed with substantial input from mathematicians whose goals included providing rich content that prepares students for college level mathematics. To a large degree the California model of instruction anticipated the NMAP report’s recommendations by over a decade.

 

We make no claims about the success of California’s implementation of mathematics instruction. Such an analysis is far beyond the scope of this report, although the sharp drop in the proportion of California State University students who require remediation

[http://www.asd.calstate.edu/performance/proficiency.shtml] would seem to be an encouraging sign. Rather we assert vigorously that the California standards and assessments are good reflections of what students should know in order to be prepared for the rigors of college mathematics.

 

California’s standards hold to a sensible minimum the New York focus on process and instead clearly specify at each grade level what students are expected to know, as well as the level of difficulty of problems they should be able to solve. Each year’s content goals are summarized in a carefully crafted paragraph. A two page document concatenating those paragraphs reads well as a narrative of coherent development of mathematical skills and knowledge from kindergarten through high school.

 

While the standards are ambitious, they recognize that not all classes will be able to complete all of the material. To deal with this reality, a subset of key standards is singled out for emphasis.  As a result, teachers are provided with a clear map that specifies how to prioritize mathematics instruction. The reduced set of key standards, while not as rich as the full set of standards, nevertheless offers an achievable blueprint for coherent mathematics instruction. For example, while New York lists 63 content goals in grade 7, California specifies but 41, of which 17 are identified as key standards.

 

The California distribution of topics aligns closely with the NMAP report’s Major Topics of School Algebra (Appendix IV) and the Benchmarks for Critical Foundations (Appendix V). Topics presented superficially in the New York Standards are absent from California’s.  In line with the NMAP outline of major topics in algebra, only one key standard in California’s Grade 7 standards deals with probability and statistics.

 

The superior mathematical vision of the California standards is implemented as a sequence of world-class assessments. There are few if any of the giveaway questions that litter New York’s exams. The assessments contain challenging yet grade-appropriate questions, both computational and conceptual. Each assessment contains a limited number of challenging word problems, in contrast to large numbers of very simple computational questions that are embedded in overly verbose word problems on New York exams. 

 

Unlike New York, which presents its standards as a largely unsupported stand-alone document, California embeds its standards in the California Mathematics Framework

[ http://www.cde.ca.gov/ci/ma/cf/ ],  a comprehensive 400-page resource manual including chapters on

 

  • sample problems keyed to the standards;
  • mathematical and pedagogical overview of each grade level’s content;
  • (for publishers) criteria for evaluating mathematics instructional materials;
  • professional development;
  • instructional strategies;
  • the use of technology;

 

and much more.   We recommend that document as a model for a correspondingly rich resource that should be implemented in New York as quickly as possible.

 

Next we outline here and detail in Appendix III the stark differences between New York and California assessments administered at the critical transition from middle school to high school. In New York State, the grade 8 assessment is administered in March of each year. The Integrated Algebra assessment will be offered for the first time in June 2008 to students completing ninth grade. In California, there is no grade 8 assessment labeled as such. The Algebra I assessment is offered to students completing eighth grade, but more typically is taken by ninth or tenth graders.  

 

For our analysis, we consider the two New York assessments as one and compare their combined content with that of the California Algebra I exam.  We recommend that high school content experts peruse Appendix III, which contains side-by-side comparisons of corresponding questions from the two assessments.  

 

Nearly without exception, New York examination questions that test formal algebra skills require a minimal level of algebraic competence and provide very superficial measures of student knowledge. In contrast, California’s questions are challenging and parallel closely the difficulty of corresponding questions in CUNY math exams, for example, the sample final exam in  College Algebra (the prerequisite to pre-calculus) in Appendix I.  Furthermore, the California assessment includes difficult multi-step word problems involving mixtures, rates, fluid flow, and more. The New York exams include only very simple word problems, most of which can be solved by word-for-word translation of an English sentence into a very simple algebra equation whose solution again requires only minimal algebraic skill.  

 

The bottom line is this. The content of the sample New York Integrated Algebra assessment is so weak that students who achieve a near-perfect score may nevertheless be ill-prepared for the standard College Algebra course that is offered at CUNY campuses. It is questionable even whether simply passing this exam with the current CUNY math competency standard of a scaled score of 75 indicates an achievement level suited for taking any college-level mathematics course. Simple fairness requires upgrading the quality of that assessment, and indeed of all New York mathematics standards and assessments in grades K-12.


 

 

Appendix I.  A Sample CUNY College Algebra Final Exam

 

Instructions: Show all work. You may use a scientific calculator.

There are 2 parts. In each part choose 10 questions. Each question is worth 5 points.

 

Time: 2 hours and 20 minutes

 

 

 

Part I. Choose any 10 questions from this part.

 

1) Rewrite  as a simplified fraction.

2) Solve the system of equations:   

 

3) Given   simplify

 

 4) In triangle ABC, angle C measures 70 degrees, the length of side AB is 5, and the length of side AC is 4.  Find angle A to the nearest tenth of a degree.

 

6) Use long division to simplify the fraction.

7) Rewrite  as a simplified fraction.

8) Find the center and radius of the circle whose equation is .

9)    Simplify  so that only positive powers appear in your answer.

10) Use Cramer’s rule to solve for x.   Do not solve for y or z:

 

11) Find an equation of the circle with center (-4,–3) that passes through the point (1,1).

 

12) Find an equation of the line through point (4, 1) that is perpendicular to the line.

 

 

Part II . Choose any 10 questions from this part.

 

 

13) Solve:    for x.

 

 

14) Solve the equation  for x and check your answer(s).

 

15)  A is an acute angle with .     Find the exact value of .

 

16) In triangle ABC, angle A measures 35 degrees, side AC measures 10 centimeters, and the length of side AB is 8 centimeters. Find the length of side BC, correct to two decimal places.
 

17)  Sketch the graph of the parabola  by first finding and labeling the co-ordinates of its vertex and intercept(s).

 

 18) The sum of three consecutive odd integers is 24 more than the largest of the three integers.   Find the integers.

 

19) Simplify  completely.

 

20) Given the function, find and simplify the expression

 

21) Simplify     completely.

22) Rewrite  as an expression that doesn’t involve a fraction.

 

23)  It takes a boat 2 hours to travel 36 miles downstream(with the current) and 3 hours to travel 27 miles upstream (against the current). What is the speed of the boat in still water?

 

 

24) Simplify .

 

25) A ball is thrown upward.  You are told that the height of the ball above the ground at time t is given by the formula  feet. When the ball reaches its maximum height, how many feet is it above the ground?

 

 

End of exam

 

Appendix II. Analysis of 7th Grade

Mathematics Standards and Assessments

 

Comparison of Standards: NY, California, Massachusetts

 

Here is some preliminary data that may serve to shape the discussion of the standards.  The first table separates the content goals of each standard into four strands. Geometry and Measurement have been combined for MA and NY in order to conform to the CA classification scheme.  The point here is to delineate the unusually large number of goals in the NY curriculum, as well as the very different distribution of topics. This information is readily obtainable from the published standard in each state.

 

New York

California

Massachusetts

Strand

# of Goals

%

# of Goals

%

# of Goals

%

Number Sense

19

30%

12

29%

9

32%

Algebra

10

16%

13

32%

6

21%

Geom. & Meas.

23

36%

13

32%

10

36%

Stats. & Prob.

12

19%

3

7%

3

11%

Total # of Goals

64

 

41

 

28

 

Table 1: 7th Grade Content Goals by State

The New York standard has by far the largest number of instructional topics. Assuming instructional time is uniformly distributed, in a school year of 180 days the New York curriculum would allow less than 3 days for each of the 64 topics.  

 

Given the large number of topics in the NY curriculum, one would expect some description of priorities. Unfortunately, there is none.  In contrast, the California standards explicitly label key standards.  From the total of 41 goals cited in Table 1, 17 are listed as "key" standards.  In Massachusetts, standards which will be taken up again in 8th grade are marked, so that by implication those need not be taught to the same level of mastery.  Fully half of the 28 Massachusetts standards are designated in this way, implying that the remaining 14 are essential for the students to know by the end of 7th grade.  The lack of clarity in the NY standards means that the teaching content can not be reliably determined from the standards themselves.  Rather, this must be inferred from the limited scope of the published assessments, which at this date consists of one exam of 38 questions, which of necessity must omit almost 40% of the prescribed topics.  Whether or not omitted topics will be tested in the future and the extent of such testing appears to be an open question.

 

We have examined certain broad features of the state standards that can be summarized by numeric measures.  They demonstrate in measurable ways that the NY standards are less focused and provide a different instructional emphasis than those of California or Massachusetts.  At this point we examine that emphasis in more detail.

 

Based on the deficiencies we see in students entering CUNY, the primary goal in middle school mathematics should be deepening the student's understanding and facility with arithmetic principles, with an eye towards their generalization in the study of algebra.  This should include intensive work with fractions – including decimal and percentage representations – as well as a thorough introduction to the extension of these methods to the integers and certain types of irrational numbers.  All of these can be thoroughly embedded within the context of many applications to geometry, measurement, and statistics.

 

Given that the 6th grade NY standards introduce students to percents and manipulations with fractions, the 7th grade standards are conspicuously deficient in extending these concepts. (We note, moreover, that the released 6th grade assessment asks exactly one — mathematically ambiguous — question (NY-gr6Bk1.11) requiring students to arithmetically combine fractions in any way.  One wonders, therefore, what emphasis these methods actually receive in the 6th grade classroom.) The computational focus in 7th grade is on integer manipulations (NY-7.N.11 to NY-7.N.13).  Rational numbers and percents appear simply as different representations of a fraction and a number of goals (often repeating material from the 6th grade) address the conversion between different forms.  However, the total neglect of fraction manipulations means that whatever skills students may have developed in 6th grade in this area will be vitiated in the 7th grade.  In fact, it is not clear where these skills are ever again considered, since they are not explicitly mentioned in the 8th grade (where in fact the discussion of polynomial arithmetic specifically precludes polynomials with fractional coefficients).

 

In contrast, the California and Massachusetts standards, in spite of their relative brevity, explicitly recognize the importance of fraction computations. To wit, CA-7NS1.2, a key standard, says students must be able to

 

"Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers."

 

Additionally, key standards CA-7NS2.2 and CA-7NS2.3 require students to

 

"Add and subtract fractions by using factoring to find common denominators."

"Multiply, divide, and simplify rational numbers by using exponent rules."

 

Similarly, the Massachusetts standards prescribe (MA-7.N.9) that students

"Select and use appropriate operations – addition, subtraction, multiplication, division, and positive integer exponents – to solve problems with rational numbers (including negatives)."

 

This is a standard that is repeated in 8th grade, as well it might, since the topic is so crucial in developing students' understanding of number systems.

 

In applications of rational numbers the NY standards do not effectively lead students beyond the level of the previous grade.  Whereas the 6th grade standard introduces students in some way to the concept of rate (NY-6.N.6), this is not explicitly dealt with at all in grade 7.  The latter grade focuses almost exclusively on conversions between units, virtually duplicating the work assessed in the 6th grade.  In contrast, the California standards (CA-7MG1.3 – a key standard) require that students

"Use measures expressed as rates (e.g. speed, density) and measures expressed as products (e.g. person-days) to solve problems; check the units of solutions; and use dimensional analysis to check the reasonableness of the answer.

 

The NY standards in algebra on the whole seem ambitious as written, but in practice as reflected on the released assessment, the expectations are lower than expressed in the standard. For example, though the standards (NY-7.A.4) speak of students being able to solve equations such as , the more realistic goal appears to be for students to solve (and formulate) simple linear equations such as , with positive integer solutions.  The latter target seems to be an appropriate goal for 7th grade (though one might wish that positive rational solutions would be allowed as well), so it is perplexing that the standards contain a stated goal that is so unrealistic (in fact, solving general linear equations does not currently appear until the first algebra course).

 

Some of the weaknesses in the New York standards could be addressed by sharply reducing the number of content goals dealing with statistics and probability. Doing so would permit redirecting  scarce instructional time to arithmetic and algebra.

 

The statistics material in the standards and assessments focuses on bar graphs and pictographs, both of which use categorical (e.g., non-numerical) data as the independent variable. These useful representations are better treated in science and social science curricula.  Mathematics curricula should address only graphical representations with real mathematical content, such as  scatter plots, which use the co-ordinate plane, and  pie charts, which can provide useful practice with fractions and percentages.

 

Similarly, we fail to see the benefit students derive from recipe-like probability problems prescribed by the standards.  As is well known, it is extremely difficult to apply combinatorial methods when problems are posed with even small variations from template descriptions.  Moreover, these methods, while of importance in some advanced contexts, are far less important to the development of basic skills than is reinforcing the fundamental methods of algebra and geometry.

 

The Assessments

 

Each state has published an assessment.  The published assessments contain a breakdown of the questions according to the content strand to which each is related.  The following table gives the relevant information:

 

 

 

 

 

 

New York

California

Massachusetts

Strand

Questions (#)

%

Questions (#)

%

Questions (#)

%

Number Sense

12

32%

22

34%

11

28%

Algebra

5

13%

25

38%

12

31%

Geom. & Meas.

10

26%

13

20%

8

21%

Stats. & Prob.

11

29%

5

8%

8

20%

Total # of questions

38

 

65

 

39

 

Table 2: Grade 7 Math Assessments by Content Strand

In effect, the assessment exam serves as a notice to teachers and students as to what is important in the syllabus.  The striking point from Table 2 is that while California and Massachusetts devote 60% or more of their assessments to Number Sense and Algebra, both of which what might well be classified as pre-algebra, the NY assessment devotes only 45% to those topics.   Moreover, the NY assessment overweights Statistics and Probability even beyond the already heavy emphasis allotted in the standards.

Moreover, the quality of the questions in the NY assessment is considerably below those in the California and Massachusetts assessments.  For example, here is a NY problem that tests a student's ability to substitute in a formula: (NY-19)

The expression below represents the total cost in dollars, including shipping, for a certain number of music CDs, m.

Based on the expression above, what is the total cost for 4 music CDs?

 

Unfortunately, this questions tests two skills, an interpretive skill and a very modest algebraic proficiency.  The interpretive skill is adequately assessed in many problems – in fact there are almost no problems that do not require substantial reading proficiency.  If a student answers this question incorrectly, what have we learned?  Conversely, if the student answers correctly, does he or she really have a firm grasp of the mechanics of algebraic substitution, since the algebra involved is so simple (moreover, the distracter answers do not even contain the response generated by the most likely type of algebraic error).

In contrast, the California assessment asks a straightforward question that any mathematician would pose to see if a student understood how to do substitution (CA – 25).

If  and , then

The choices include answers that are generated from common errors, most importantly evaluating the expression from left to right without using the precedence rules of operations.  In this way, even wrong answers provide feedback to instructors or the institution.

The Massachusetts assessment also tests substitution.  Again, the wording is straightforward (MA – 18):

 

What is the value of the expression below when  and ?

We have examined one particular question, but it illustrates a persistent flaw in NY math assessments — an almost dogmatic avoidance of assessing computational skill.  For example, the NY standard includes (NY-7.N.11 and 7.N.12) the requirement that students simplify expressions including absolute values, and learn how to add integers.  How is this actually assessed? Here is NY-31, the only question on the test that deals with negative numbers.

The temperature in St. Cloud, Minnesota, was F (Fahrenheit) on January 27 and F on January 28.

A)    On the number line below, plot the temperatures for January 27 and January 28.  Be sure to label both points with the appropriate date.

B)     How many degrees warmer was it on January 28 than on January 27?

This question requires no understanding of integer arithmetic. It is possible that the principles involved in the latter topic are taught at the end of 7th grade, but then one might hope to see some explicit testing of this material in the 8th grade.  Although the 8th grade assessment includes manipulations with negative integers as coefficients of polynomials, there are no questions that simply assess students' skills in integer arithmetic.  In contrast, the California 7th grade assessment includes three computations dealing with negative numbers, as well as four computations dealing with simplification of fractions involving exponents.

 

Our final comment concerns the use of reference tables and calculators. The NY assessment has absolutely the lowest standard of mathematical knowledge required of students.  The included 7th grade reference sheet contains formulas for the volume of a rectangular box, area and circumference formulas for a circle, as well as conversion equivalents for the units in the metric system. What is the point of teaching metric units if students are not expected to know the meaning of the prefixes and their relationship to the base 10 numeration system?

 

Part B of the NY exam is calculator active. It would appear that students can bring any four-function calculator, including non-graphing scientific calculators.  As a consequence, much of the computational value in this part is severely diminished.  For example NY-38 requires that students compute  and express the answer in standard form. This is a perfectly reasonable computation for students to do by hand.  Doing so would test their understanding of scientific notation and reinforce their arithmetic skills.  However, using a calculator to solve this problem destroys whatever computational value it has.

 

Summary

 

The NY 7th grade standards are too broad and not sufficiently prioritized as to important goals.  From the released assessment, we can infer that approximately 30% of instructional time should be devoted to preparing students in statistics and probability, goals that have little bearing on developing numeric and algebraic skills that will be needed in 8th grade and beyond. (Note that the 8th grade standard includes no statistics or probability.)  The curriculum provides a very weak foundation for moving the students into mainstream algebra.  Pass rates in the latter will not improve until the middle schools provide more rigorous and focused mathematics instruction.

 

Appendix III. Comparison of Algebra I Assessments

 

The California questions listed on the following page are extracted from a set of 80 Released Test Question, 65 of which would appear on a typical Algebra assessment, an untimed multiple choice test. Each question counts for approximately 1.5 points on a 100 point grading scale.

 

New York questions are taken from the 2007 8th grade assessment and the Sample Integrated Algebra Regents, both of which consist primarily of multiple choice or open response questions counting approximately 2.2 points each on a 100 point grading scale.

 

                        Rewriting or simplifying algebra expressions.

 

                 California        

 

Algebra I: 11 questions, 16.5 points                                         

               New York

 

Grade 8: 6 questions, 13.2 points

Integrated Algebra : 4 questions,  8.8 points

 

             

                

 

 

 

 The differences between the two sets of questions are remarkable:

 

1.  The California questions test the full range of math operations with rational polynomials (fractions of polynomials), whereas rational polynomial division and multiplication are absent from the New York assessments.

 

2.   Answering the California questions involves factoring 16 nonlinear (degree 2 or higher) polynomials, while only 2 such factorizations (both of which are very easy to carry out) are required for the New York questions.   

 

3.  The California questions include factoring two non-monic quadratic polynomials and two quadratic polynomials in two variables, whereas New York’s assessments omit all such examples.

 

4.  Perhaps most striking, the listed California questions appear on a single assessment, whereas the New York questions are split between two assessments that will be taken two years apart, since it is anticipated that the majority of New York students will take the new Integrated Algebra assessment at the end of tenth grade.

 

  Additional critical differences in other content areas are as follows.

  • The California assessment, following the California standards, requires answering many questions involving the solution of quadratic equations, including questions that test student understanding of the theory behind completing the square. In contrast, the New York standards postpone study of quadratic equations until the 11th or 12th grade Algebra II assessment, which has yet to be given. Only the easiest kinds of questions on this topic are included on the current Math B advanced algebra assessment.
  • The California assessment includes difficult multi-step word problems involving mixtures, rates, fluid flow, and more. The New York exams include only very simple word problems, most of which can be solved by word-for-word translation of an English sentence into a linear polynomial. See the questions listed below.

 

California Word problems from Algebra I Exam: (1.5 points each)

 

63. The height of a triangle is 4 inches greater than twice its base. The area of the triangle is 168 square inches. What is the base of the triangle?  Note: similar problem (#26) for 2.2 points on NY Grade 9 exam asks only for the equation that is needed to solve the problem.

 

72. A pharmacist mixed some 10%-saline solution with some 15%-saline solution to obtain 100 mL of a 12%-saline solution. How much of the 10%-saline solution did the pharmacist use in the mixture?

 

73. Andy’s average driving speed for a 4-hour trip was 45 miles per hour. During the first 3 hours he drove 40 miles per hour. What was his average speed for the last hour of his trip?

 

74. One pipe can fill a tank in 20 minutes, while another takes 30 minutes to fill the same tank. How long would it take the two pipes together to fill the tank?

 

75. Two airplanes left the same airport traveling in opposite directions. If one airplane averages 400 miles per hour and the other airplane averages 250 miles per hour, in how many hours will the distance between the two planes be 1625 miles?

 

76. Lisa will make punch that is 25% fruit juice by adding pure fruit juice to a 2-liter mixture that is 10% pure fruit juice. How many liters of pure fruit juice does she need to add?

 

 New York word problems from Grade 8 Assessment (2.2 points each)

 

8. Sarah earned a 4% commission on all of her sales in March. Her total sales were

$80,000 in March.  How much money did she earn from commissions?

 

14. In order to purchase a new CD player, Rosa must save at least $85.00. What inequality represents the amount of money,  m, Rosa must save?

 

22. A 20-ounce bag of popcorn costs $2.80. If the unit price stays the same, how much

does a 35-ounce bag of popcorn cost?

 

26. Ms. Snyder wants to buy a television at an electronics store. All televisions at the store are 3/4 of the original price, p . She has a $40 discount coupon she will use during the sale. Which [of the following equations] should Ms. Snyder use to find the final price, f, of a television?

 

31. The transportation department recommends increasing highway tolls by 20%

to raise money for road repairs. The current highway toll is $1.50. What will be the new toll after the 20% increase?

 

 

 

 

New York word problems from Sample Integrated Algebra Assessment (2.2 points each unless otherwise indicated)

 

15. An electronics store sells DVD players and cordless telephones. The store makes a $75 profit on the sale of each DVD player (d) and a $30 profit on the sale of each cordless telephone (c). The store wants to make a profit of at least $255.00 from its sales of DVD players and cordless phones. Which inequality [from among those given below] describes this situation?

 

26. The length of a rectangular window is 5 feet more than its width, w. The area of the window is 36 square feet. Which [of the equations given below] could be used to find the dimensions of the window?

 

34 (3.3 points) [Compare the complex wording with the simple mathematics]

            a) Hannah took a trip to visit her cousin. She drove 120 miles to reach her cousin’s house and the same distance back home. It took her 1.2 hours to get halfway to her cousin’s house. What was her average speed, in miles per hour, for the first 1.2 hours of the trip?

            b) Hannah’s average speed for the remainder of the trip to her cousin’s house was 40 miles per hour. How long, in hours, did it take her to drive the remaining distance?

            c) Traveling home along the same route, Hannah drove at an average rate of 55 miles per hour. After 2 hours her car broke down. How many miles was she from home?

 

35. A prom ticket at Smith High School is $120. Tom is going to save money for the ticket by walking his neighbor’s dog for $15 per week. If Tom already has saved $22, what is the minimum number of weeks Tom must walk the dog to earn enough to pay for the prom ticket?

Appendix IV: The Major Topics of School Algebra

 

These are based on a table in the NMAP report. However, for clarity we have added the topics listed under the heading Algebra of Rational Expressions. It is clear that the writers of the NMAP report understood these topics as implied by the topic ‘rational expressions’ under the first heading ‘Symbols and Expressions.’

 

The Major Topics of School Algebra

 

Symbols and Expressions

Polynomial expressions

Rational expressions

Arithmetic and finite geometric series

Linear Equations

Real numbers as points on the number line

Linear equations and their graphs

Solving problems with linear equations

Linear inequalities and their graphs

Graphing and solving systems of simultaneous linear equations

Quadratic Equations

Factors and factoring of quadratic polynomials with integer coefficients

Completing the square in quadratic expressions

Quadratic formula and factoring of general quadratic polynomials

Using the quadratic formula to solve equations

Functions

Linear functions

Quadratic functionsword problems involving quadratic functions

Graphs of quadratic functions and completing the square

Polynomial functions (including graphs of basic functions)

Simple nonlinear functions (e.g., square and cube root functions; absolute value;

   rational functions; step functions)

Rational exponents, radical expressions, and exponential functions

Logarithmic functions

Trigonometric functions

Fitting simple mathematical models to data

Algebra of Polynomials

Roots and factorization of polynomials

Complex numbers and operations

Fundamental theorem of algebra

Binomial coefficients (and Pascal’s Triangle)

Mathematical induction and the binomial theorem

Algebra of Rational Expressions

Factoring rational expressions with linear and quadratic denominators

Operations on rational expressions

Solving equations involving rational expressions.

Combinatorics and Finite Probability

Combinations and permutations, as applications of the binomial theorem and

   Pascal’s Triangle

 

Appendix V:  Benchmarks for the Critical Foundations

 

Source: National Mathematics Advisory Panel, 2008.

 

Fluency With Whole Numbers

1) By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers.

2) By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.

Fluency With Fractions

1) By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.

2) By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals.

3) By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.

4) By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.

5) By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.

6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.

Geometry and Measurement

1) By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).

2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three dimensional shapes and solve problems involving surface area and volume.

3) By the end of Grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.

 

Appendix VI: Content Analysis of New York State Mathematics Assessments: Grades 5 through 11

 

The following table lists all questions in the following categories:

·        simplifying expressions;

·        solving equations;

·        equations of graphs in the x,y-plane.

 

The symbol (WP) means that the stated algebra problem is phrased (usually transparently) as a word problem.

 

In the six year 2006 examinations analyzed:

·        Only two questions require adding fractions with different denominators:  and .

  • No question requires finding the product or quotient of fractions.
  • No question requires finding the sum, difference, product, or quotient of signed numbers.
  • Three questions require simplifying a numerical expressions with parentheses;
  • There is one question about vertical lines and one question on horizontal lines.
  • There is no question on slanted lines, their slopes, or their graphs.

 

 

 

 

 

Exam

Simplify or evaluate expression

Solve equation

Graphs of equations

 

NYS Grade 5,

March 2006

 

17. (WP) Evaluate  2L + 2W when L=14; W = 10

 

18. 2/7 + 3/7 = ?

 

10. 2.5 feet + 1 foot = ?inches

27. (WP) 216   54 = ?

31.a) 37+451+50= T = ?

     b)    ?

32. 180 -100-50 = ? degrees

 

 

 

 

None

 

Locate points in plane (1 question)

 

NYS Grade 6,

March 2006

 

 

 

 

 

 

None

Exam

Simplify or evaluate  expression

Solve Equation

Graphs of Equations

 

NYS Grade 7

March 2006

 

 

 

 

None

None

 

 

 

 

NYS Grade 8

March 2006

Calculator permitted

 

 

 

 

 

 

None

 

 

 

 

NYS Math A

June 2006

Calculator permitted

 

 

 

 

 

 

 

Multiple choice questions with answers:

 

Graph of x = 4 is parallel to ?

 

Slope of a horizontal line is ?

 

NYS Math B

August 2006

Calculator permitted

 

 

 

The vertex of the parabola with equation  is ?

 

 

 

 

There is only a modest improvement in the 2007 Integrated Algebra Sampler, as shown on the following page. A detailed comparison of that exam with the corresponding California exam was given in Appendix III.

  

 

Exam

Simplify or evaluate expression

Solve equation

Graphs of equations

 

NYS Integrated Algebra Sampler

Fall 2007

 

(Multi-year depreciation)

 

Find perimeter of Norman window

 

 

 

#34) Basic D = RT word problem

 

#38 Solve linear/quadratic system graphically

 

#39  Solve

 

Eqn. of line through (2,0) and (0,3)

 

Slope of line through (3,4) and

(-6,10)

 

Identify quadratic graph

 

Identify inequality graph

 

Shift absolute value graph

 

Plug-in to verify a solution of

  -2x+5>17