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Core-Plus Mathematics Project

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Core-Plus Mathematics is a comprehensive high school mathematics program consisting of a four-year series of print and digital student textbooks and supporting materials for teachers, developed by the Core-Plus Mathematics Project at Western Michigan University, with funding from the National Science Foundation. Each text consists of a significant common core of mathematics and statistics for all students, plus extensions of core content to challenge the most motivated and able students. The Core-Plus development team consists of mathematics educators, mathematicians, and statisticians from several U.S. universities.

From 1992 to 2015, several updated revisions and editions were developed with advice and contributions from curriculum developers, mathematics teachers, mathematicians, statisticians, and mathematics education researchers, informed by formative evaluation findings from field testing nationally in urban, suburban, and rural schools. The first edition, entitled Contemporary Mathematics in Context: A Unified Approach, was completed in 1999. The third edition, entitled Core-Plus Mathematics: Contemporary Mathematics in Context, was published by McGraw-Hill Education in 2015.

Key Features

The first edition of Core-Plus Mathematics was designed to meet the curriculum, teaching, and assessment standards from the National Council of Teachers of Mathematics[1][2][3][4] and the broad goals outlined in the National Research Council report, Everybody Counts: A Report to the Nation on the Future of Mathematics Education.[5] Later editions were designed to also meet the American Statistical Association Guidelines for Assessment and Instruction in Statistics Education (GAISE)[6] and most recently the standards for mathematical content and practice in the Common Core State Standards for Mathematics (CCSSM).[7]

The program is a problem-based, technology-rich, four-year college-preparatory program with an emphasis on teaching and learning mathematics through mathematical modeling and mathematical inquiry. Each year, students learn mathematics in four interconnected strands: algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematical modeling.[8][9]

Evaluations and Research

Project and independent evaluations and many research studies have been conducted on Core-Plus Mathematics, including content analyses, case studies, surveys, small- and large-scale comparison studies, research reviews, and a longitudinal study. A general trend is that Core-Plus Mathematics students performed significantly better than comparison students on assessments of conceptual understanding, problem solving, and applications, and results were mixed for performance on assessments of by-hand calculation skills.[10][11][12][13][14][15][16][17][18][19][20]

For example, in 2013, three large-scale comparison studies of Core-Plus Mathematics and more conventional curricula were reported by independent researchers at the University of Missouri–Columbia.[10][11][12] The research was reported in the March and July 2013 issues of the Journal for Research in Mathematics Education and in the December 2013 issue of the International Journal of Science and Mathematics Education. The three studies examined student achievement in schools in 5 geographically dispersed states. The first study involved 2,161 students in 10 schools in first-year high school mathematics courses, the second study involved 3,258 students in 11 schools in second-year mathematics courses, and the third study involved 2,242 students in 10 schools in third-year mathematics courses. Results in the first study showed that Core-Plus Mathematics students scored significantly higher on all three end-of-year outcome measures: a test of common objectives, a problem solving and reasoning test, and a standardized achievement test. Results in the second study showed that Core-Plus Mathematics students scored significantly higher on a standardized achievement test, with no differences on the other measures. Results in the third study showed that Core-Plus Mathematics students scored significantly higher on a test of common objectives, with no differences on the other measure.

Older comparative studies in peer-reviewed journals report that students using early versions of Core-Plus Mathematics did as well as or better than those in traditional single-subject curricula on all measures except paper-and-pencil algebra skills;[13] students using the first field-test versions of Core-Plus Mathematics scored significantly better on tests of conceptual understanding and problem solving, while Algebra II students in conventional programs scored significantly better on a test of paper-and-pencil procedures;[18] and Core-Plus Mathematics students displayed qualities such as engagement, eagerness, communication, flexibility, and curiosity to a much higher degree than did students who studied from more conventional programs.[16] A review of research in 2008 concluded that there were modest effects for Core-Plus Mathematics on mostly standardized tests of mathematics.[19]

With regard to achievement of students in minority groups, an early peer-reviewed paper documenting the performance of students from under-represented groups using Core-Plus Mathematics reported that at the end of each of Course 1, Course 2, and Course 3, the posttest means on standardized mathematics achievement tests of Core-Plus Mathematics students in all minority groups (African Americans, Asian Americans, Hispanics, and Native/Alaskan Americans) were greater than those of the national norm group at the same pretest levels. Hispanics made the greatest pretest to posttest gains at the end of each course.[20] A later comparative study reported that Hispanic high school students using Core-Plus Mathematics made modest gains compared to the performance of students with other demographic backgrounds.[14]

Regarding preparation for college, studies of SAT and ACT test results reported that Core-Plus Mathematics students performed significantly better than comparison students on the SAT and performed as well on the ACT.[21] Several studies examined the subsequent college mathematics performance of students who used different high school textbook series. These studies did not detect any differential effect of high school curriculum on placement in college mathematics courses, in subsequent performance, or in course-taking patterns.[22][23][24][25][26][27][28]

In terms of core content development, a study comparing the development of quadratic equations in the Korean national curriculum and Core-Plus Mathematics found that some quadratic equation topics are developed earlier in Korean textbooks, while Core-Plus Mathematics includes more problems requiring explanations, various representations, and higher cognitive demand.[29]

Several studies have analyzed the teacher’s role in Core-Plus Mathematics.[17][30][31]

Review by Prof. Harel

In 2009 professor of mathematics at the University of California in San Diego, Guershon Harel reviewed four high-school mathematics programs. The examined programs included Core-Plus Courses 1, 2, and 3. The examination focused on two topics in algebra and one topic in geometry, deemed by Prof. Harel central to the high school curriculum. The examination was intended "to ensure these topics are coherently developed, completely covered, mathematically correct, and provide students a solid foundation for further study in mathematics".[32]

From the outset, Prof. Harel noted that the content presentation in Core-Plus program is unusual in that its instructional units, from the start to the end, are made of word problems involving "real-life" situations. This structure is reflected in the subtitle of the Core-Plus series: Contemporary Mathematics in Context. To review the program, it was necessary to go through all the problems in the core units and their corresponding materials in the Teacher’s Edition. Despite the unconventional textbook structure, the language used by the Core-Plus program was found mathematically sound.

In the algebra section, fundamental theorems on linear functions and quadratic functions were found not justified, except for the quadratic formula. Theorems are often presented without proof.

Like in the algebra texts, the geometry text does not lead to a clear logical structure of the material taught. Because theoretical material is concealed within the text of the problems, "a teacher must identify all the critical problems and know in advance the intended structure to establish the essential mathematical progression. This task is further complicated by the fact that many critical problems appear in the homework sections. Important theorems in geometry are not justified. Moreover, with the way the material is sequenced, some of these theorems cannot be justified".[32]

According to Prof. Harel, the Core-Plus program "excels in providing ample experience in solving application problems and in ensuring that students understand the meanings of the different parts of the modeling functions. The program also excels in its mission to contextualize the mathematics taught". However, it fails "to convey critical mathematical concepts and ideas that should and can be within reach for high school students".[32]

Review by Prof. Wilson

Professor W. Stephen Wilson from Johns Hopkins University evaluated the mathematical  development  and  coherence  of  the Core-Plus program in 2009. In particular, he examined "the  algebraic  concepts  and  skills  associated with linear functions because they are a critical foundation for the further study of algebra", and evaluated how the program presents the theorem that the sum of the angles of a triangle is 180 degrees, "which is a fundamental theorem of Euclidean geometry and it connects many of  the  basics in  geometry  to  each  other".[33] 

Prof. Wilson noted that the major theme of the algebra portion of the program seems to involve creating a table from data, graphing the points from the table; given the table students are asked to find a corresponding function. In case of linear function, "at no point is there an attempt to show that the equation's graph really is a line. Likewise, there is never an attempt to show that a line graph comes from the usual form of a linear equation". Prof. Wilson considered this approach to be "a significant flaw in the mathematical foundation".[33]

Quoting the textbook, "Linear functions relating two variables x and y can be represented using tables, graphs, symbolic rules, or verbal descriptions", Prof. Wilson laments that although this statement is true, "the essence of algebra involves abstraction using symbols".[33]

Prof. Wilson says that the Core-Plus program "has a multitude of good problems, but never develops the core of the mathematics of linear functions. The problems are set in contexts and mathematics itself is rarely considered as a legitimate enterprise to investigate". The program lacks attention to algebraic manipulation" to the point that "symbolic algebra is minimized".[33]

In regards to geometry portion, Prof. Wilson concludes that the program fails to build geometry up from foundations in a mathematically sound and coherent way". He stresses out that "one significant goal of a geometry course is to teach logic, and this program fails on that account".[33]

Overall, the "unacceptable nature of geometry" and the fashion in which the program downplays "algebraic structure and skills" make the Core-Plus program unacceptable.

Historical Controversy

Mathematics programs initially developed in the 1990s that were based on the NCTM’s Curriculum and Evaluation Standards for School Mathematics, like Core-Plus Mathematics, have been the subject of controversy due to their differences from more conventional mathematics programs. In the case of Core-Plus Mathematics, there has been debate about (a) the international-like integrated nature of the curriculum, whereby each year students learn algebra, geometry, statistics, probability, and discrete mathematical modeling, as opposed to conventional U.S. curricula in which just a single subject is studied each year, (b) a concern that students may not adequately develop conventional algebraic skills, (c) a concern that students may not be adequately prepared for college, and (d) a mode of instruction that relies less on teacher lecture and demonstration and more on inquiry, problem solving in contextualized settings, and collaborative work by students.

While discussion of these issues continues, a growing body of research (referenced above) has provided reassuring results about learning algebraic skills and college preparation and also suggests that Core-Plus Mathematics students are advantaged in terms of conceptual understanding, problem solving, and reasoning. An integrated high school mathematics curriculum is now increasingly recognized as the most common curriculum organization outside the U.S. and also acknowledged as a valid curriculum organizational structure in the Common Core State Standards for Mathematics in the U.S. (e.g., CCSSI-Courses and Transitions). And classroom instruction that is more inquiry- and problem-solving-oriented is recognized as a viable instructional methodology.

References

  1. ^ National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  2. ^ National Council of Teachers of Mathematics. (1991). Mathematics professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.
  3. ^ National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  4. ^ National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  5. ^ National Research Council; Mathematical Sciences Education Board; Board on Mathematical Sciences and Their Applications. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: The National Academies Press.
  6. ^ Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., & Scheaffer, R. (2007). Guidelines for assessment and instruction in statistics education. Alexandria, VA: American Statistical Association.
  7. ^ Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.
  8. ^ Fey, J., & Hirsch, C. (2007). The case of Core-Plus Mathematics. In C. Hirsch (Ed.), Perspectives on the design and development of school mathematics curricula (pp. 129–142). Reston, VA: National Council of Teachers of Mathematics.
  9. ^ Maurer, S.; McCallum, W. (2006). "Advising a precollege curriculum project". Notices of the AMS. 53 (9): 1018–1020.
  10. ^ a b Grouws, D. A.; Tarr, J. E.; Chávez, Ó.; Sears, R.; Soria, V. M.; Taylan, R. D. (2013). "Curriculum and implementation effects on high school students' mathematics learning from curricula representing subject-specific and integrated content organizations". Journal for Research in Mathematics Education. 44 (2): 416–463. doi:10.5951/jresematheduc.44.2.0416.
  11. ^ a b Tarr, J. E.; Grouws, D. A.; Chávez, Ó.; Soria, V. M. (2013). "The effects of content organization and curriculum implementation on students' mathematics learning in second-year high school courses". Journal for Research in Mathematics Education. 44 (4): 683–729. doi:10.5951/jresematheduc.44.4.0683.
  12. ^ a b Chávez, Ó.; Tarr, J. E.; Grouws, D. A.; Soria, V. M. (2013). "Third-year high school mathematics curriculum: Effects of content organization and curriculum implementation". International Journal of Science and Mathematics Education. 13: 97–120. doi:10.1007/s10763-013-9443-7.
  13. ^ a b Schoen, H. L., & Hirsch, C. R. (2003). The Core-Plus Mathematics Project: Perspectives and student achievement. In S. Senk & D. Thompson (Eds.), Standards-based school mathematics curricula: What are they? What do students learn? (pp. 311–344). Hillsdale, NJ: Lawrence Erlbaum Associates.
  14. ^ a b Capraro, M. M.; Capraro, R. M.; Yetkiner, Z. E.; Rangel-Chavez, A. F.; Lewis, C. W. (2010). "Examining Hispanic student mathematics performance on high-stakes tests: An examination of one urban school district in Colorado". Urban Review: Issues and Ideas in Public Education. 42 (3): 193–209. doi:10.1007/s11256-009-0127-0.
  15. ^ Harwell, M.; Post, T. R.; Maeda, Y.; Davis, J.; Cutler, A.; Anderson, E.; Kahan, J. A. (2007). "Standards-based mathematics curricula and secondary students' performance on standardized achievement tests". Journal for Research in Mathematics Education. 38 (1): 71–101.
  16. ^ a b Latterell, C. M. (2003). "Testing the problem-solving skills of students in an NCTM-oriented curriculum". The Mathematics Educator. 13 (1): 5–14.
  17. ^ a b Schoen, H. L.; Finn, K. F.; Cebulla, K. J.; Fi, C. (2003). "Teacher variables that relate to student achievement when using a standards-based curriculum". Journal for Research in Mathematics Education. 34 (3): 228–259. doi:10.2307/30034779.
  18. ^ a b Huntley, M. A.; Rasmussen, C. L.; Villarubi, R. S.; Sangtong, J.; Fey, J. T. (2000). "Effects of Standards-based mathematics education: A study of the Core-Plus Mathematics Project algebra and functions strand". Journal for Research in Mathematics Education. 31 (3): 328–361. doi:10.2307/749810.
  19. ^ a b Slavin, R.; Lake, C.; Groff, C. (2007). "Effective programs in middle and high school mathematics: A best-evidence synthesis". Review of Educational Research. 79 (2): 839–911. doi:10.3102/0034654308330968.
  20. ^ a b Schoen, H. L., Hirsch, C. R., & Ziebarth, S. W. (1998). An emerging profile of the mathematical achievement of students in the Core-Plus Mathematics Project. Paper presented at the 1998 Annual Meeting of the American Educational Research Association. San Diego, CA.
  21. ^ Schoen, H. L., Ziebarth, S. W., Hirsch, C. R., & BrckaLorenz, A. (2010). A five-year study of the first edition of the Core-Plus Mathematics curriculum. Charlotte, NC: Information Age Publishing, Inc.
  22. ^ Schoen, H. L., & Hirsch, C. R. (2003, February). Responding to calls for change in high school mathematics: Implications for collegiate mathematics. American Mathematical Monthly, pp. 109–123.
  23. ^ Norman, K. W.; Medhanie, A. G.; Harwell, M. R.; Anderson, E.; Post, T. R. (2011). "High school mathematics curricula, university mathematics placement recommendations, and student university mathematics performance". PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies. 21 (5): 434–455. doi:10.1080/10511970903261902.
  24. ^ Dupuis, D. N.; Medhanie, A. G.; Harwell, M. R.; Lebeau, B.; Monson, D. (2012). "A multi-institutional study of the relationship between high school mathematics achievement and performance in introductory college statistics". Statistics Education Research Journal. 11 (1): 4–20.
  25. ^ Harwell, M. R.; Medhanie, A. G.; Post, T. R.; Norman, K. W.; Dupuis, D. N. (2012). "Preparation of students completing a Core-Plus or commercially developed high school mathematics curriculum for intense college mathematics coursework". Journal of Experimental Education. 80 (1): 96–112. doi:10.1080/00220973.2011.567311.
  26. ^ Post, T. R.; Monson, D. S.; Anderson, E.; Harwell, M. R. (2012). "Integrated curricula and preparation for college mathematics". The Mathematics Teacher. 106 (2): 138–143. doi:10.5951/mathteacher.106.2.0138.
  27. ^ Post, T. R.; Medhanie, A.; Harwell, M.; Norman, K. W.; Dupuis, D. N.; Muchlinski, T.; Anderson, E.; Monson, D. (2010). "The impact of prior mathematics achievement on the relationship between high school mathematics curricula and post-secondary mathematics performance, course-taking, and persistence". Journal for Research in Mathematics Education. 41 (3): 274–308.
  28. ^ Teuscher, D.; Reys, R. E. (2012). "Rate of change: AP calculus students' understandings and misconceptions after completing different curricular paths". School Science and Mathematics. 112 (6): 359–376. doi:10.1111/j.1949-8594.2012.00150.x.
  29. ^ Hong, D. S.; Choi, K. M. (2014). "A comparison of Korean and American secondary school textbooks: The case of quadratic equations". Educational Studies in Mathematics. 85 (2): 241–263. doi:10.1007/s10649-013-9512-4.
  30. ^ Herbel-Eisenmann, B.; Lubienski, S.; Id-Deen, L. (2006). "Reconsidering the study of mathematics instructional practices: The importance of curricular context in understanding local and global teacher change". Journal of Mathematics Teacher Education. 9: 313–345. doi:10.1007/s10857-006-9012-x.
  31. ^ Ziebarth, S. W., Hart, E., Marcus, R., Ritsema B., Schoen, H. L., & Walker, R. (2008). High school teachers as negotiators between curriculum intentions and enactment. In J. Remillard, G. Lloyd, & B. Herbel-Eisenmann (Eds.), Mathematics teachers at work. (pp. 171–189). New York: Routledge Falmer.
  32. ^ a b c Harel, Guershon (2009). "A Review of Four High-School Mathematics Programs" (PDF).
  33. ^ a b c d e Wilson, W. Stephen (2009). "Washington State high school math text review" (PDF). {{cite web}}: no-break space character in |first= at position 3 (help); no-break space character in |title= at position 11 (help)
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