- Connected Mathematics
Connected Mathematics is a comprehensive, problem-centered curriculum designed for all students in grades 6-8 based on the NCTM standards. The curriculum was developed by the Connected Mathematics Project (CMP) at Michigan State University and funded by the National Science Foundation.
Each grade level curriculum is a full-year program, and in each of the three grade levels, topics of number, algebra, geometry/measurement, probability and statistics are covered in an increasingly sophisticated manner. The program seeks to make connections within mathematics, between mathematics and other subject areas, and to the real world. The curriculum is divided into units, each of which contains investigations with major problems that the teacher and students explore in class. Extensive problem sets are included for each investigation to help students practice, apply, connect, and extend these understandings.
Connected Mathematics addresses both the content and the process standards of the NCTM. The process standards are: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. For example, in Moving Straight Ahead students construct and interpret concrete, symbolic, graphic, verbal and algorithmic models of quantitative and algebraic relationships, translating information from one model to another.
Like other curricula implementing the NCTM standards, Connected Math has been criticized by supporters of traditional mathematics for not directly teaching standard arithmetic methods.
One 2003 study compared the mathematics achievement of eighth graders in the first three school districts in Missouri to adopt NSF-funded Standards-based middle grades mathematics curriculum materials (MATH Thematics or Connected Mathematics Project) with students who had similar prior mathematics achievement and family income levels from other districts. Significant differences in achievement were identified between students using Standards-based curriculum materials for at least 2 years and students from comparison districts using other curriculum materials. All of the significant differences reflected higher achievement of students using Standards-based materials. Students in each of the three districts using Standards-based materials scored higher in two content areas (data analysis and algebra), and these differences were significant.
Another study compared statewide standardized test scores of fourth-grade students using Everyday Mathematics and eighth-grade students using Connected Mathematics to test scores of demographically similar students using a mix of traditional curricula. Results indicate that students in schools using either of these standards-based programs as their primary mathematics curriculum performed significantly better on the 1999 statewide mathematics test than did students in traditional programs attending matched comparison schools. With minor exceptions, differences in favor of the standards-based programs remained consistent across mathematical strands, question types, and student sub-populations.
As one of many widely adopted curricula developed around the NCTM standards, Connected Mathematics has been criticized by advocates of traditional mathematics as being particularly ineffective and incomplete and praised by various researchers who have noted its benefits in promoting deep understanding of mathematical concepts among students. In a review by critic James Milgram, "the program seems to be very incomplete... it is aimed at underachieving students." He observes that "the students should entirely construct their own knowledge.. standard algorithms are never introduced, not even for adding, subtracting, multiplying and dividing fractions." However, studies have shown that students who have used the curriculum have "develop[ed] sophisticated ways of comparing and analyzing data sets, . . . refine[d] problem-solving skills and the ability to distinguish between reasonable and unreasonable solutions to problems involving fractions, . . . exhibit[ed] a deep understanding of how to generalize functions symbolically from patterns of data, . . . [and] exhibited a strong understanding of algebraic concepts and procedures," among other benefits.
Districts in states such as Texas were awarded NSF grants for teacher training to support curricula such as CM. Austin ISD received a $5 million NSF grant for teacher training in 1997. NSF awarded $10 million for "Rural Systemic Initiatives" through West Texas A&M. At the state level, the SSI (Statewide Systemic Initiative), was a federally-funded program developed by the Dana Center at the University of Texas. Its most important work was directing the implementation of CM in schools across the state. But in 1999, Connected Mathematics was rejected by California's revised standards because it was judged at least two years below grade level and it contained numerous errors. After the 2000-2001 academic year, state monies can no longer be used to buy Connected Mathematics
The Christian Science Monitor noted parents in Plano Texas who demanded that their schools drop use of CM, while the New York Times reported parents there rebelled against folding fraction strips rather than using common denominators to add fractions. For the improved second edition, it is stated that "Students should be able to add two fractions quickly by finding a common denominator". The letter to parents states that students are also expected to multiply and divide fractions by standard methods.
What parents often do not understand is that students begin with exploratory methods in order to gain a solid conceptual understanding, but finish by learning the standard procedures, sometimes by discovering them under teacher guidance. Large-scale studies of reform curricula such as Connected Mathematics have shown that students in such programs learn procedural skills to the same level as those in traditional programs, as measured by traditional standardized tests. Students in standards-based programs gain conceptual understanding and problem-solving skills at a higher level than those in traditional programs.
Despite disbelief on the part of parents whose textbooks always contained instruction in mathematical methods, it is claimed that the pedagogical benefits of this approach find strong support in the research: "Over the past three to four decades, a growing body of knowledge from the cognitive sciences has supported the notion that students develop their own understanding from their experiences with mathematics."
Examples of criticism
Connected Mathematics treatment of some topics include exercises which some have criticized as being either "subjective" or "having nothing to do with the mathematical concept" or "omit standard methods such as the" formula for arithmetic mean. (See above for discussion of reasons for initial suppression of formulas.) The following examples are from the student textbooks, which is all the parents see. (See discussion below.)
In the first edition, one booklet focuses on a conceptual understanding of median and mean, using manipulatives. The standard algorithm was not presented. Later editions included the algorithm.
In the 6th grade unit on fractions, students develop a conceptual understanding of comparing fractions with different denominators by using benchmark fractions, fraction strips, and other strategies. The standard method, which is to convert to fractions using the least common denominator, may not have appeared in the first edition, according to some critics. Even in the revised edition (CMP2), which has been in use since at least the 2003-2004 school year, the standard method is not listed in the index, though it later appears in decimal arithmetic units. Parents are told that students do learn how to use common denominators in adding fractions, but some have expressed concern because a direct explanation does not appear in the student textbook. In the "Concept with Explanation" page for Bits & Pieces II, from the parent support website, parents are told "The goal is to make sense of the strategy of renaming with common denominators, so that this becomes an efficient and sensible algorithm, which can be used without the supporting models."
Area of a circle
Students learn the standard formula that the area of a circle is pi multiplied by the square of the radius, but this formula does not occur in the 6th grade textbook, and is only mentioned as "one possible" method in the teacher's guide. Rather than a conventional derivation in which a rectangle is constructed of wedges cut out of a circle, students are guided to cut up a circle into many small pieces, and conclude that they take up slightly more than 3 radius squares, which does not really explain why the standard formula works.
The following exercise is from the first of the sixth grade booklets, which is named "Prime Time", after the prime factorization of whole numbers. It represents one type of non-traditional teaching approach. The student is asked to select a number he or she "likes" and to analyze that number. There are no unique correct answers, of necessity, since whether an answer is correct or not depends on the number the child chose to analyze.
My Special Number: Choose a whole number between 10 and 100 that you especially like. In your Journal:
- Record your number
- Explain why you chose that number
- List three or four mathematical things about your number
- List three or four connections you can make between your number and your world.
The third item above is where a student could state whether or not the number is prime, or the number of different primes in the chosen number's prime factorization, for example. "As you work through the investigations in Prime Time, you will learn lots of things about numbers. Think about how these new ideas apply to your special number, and add any new information about your number to your journal. You may want to designate one or two "special number" pages in your journal, where you can record this information. At the end of the unit, your teacher will ask you to find an interesting way to report to the class about your special number."
In the second edition of the program, a 6th grade book "Bits and pieces 2" teaches how to add, subtract, multiply, and divide fractions. When multiplying, instead of just multiplying the numerator and the denominator, Students are asked to diagram. Most students who already know how to do this operation, dislike the unnecessary steps. The use of cross cancelation is not mentioned anywhere in the program.
Context of above examples
The intended use of such materials is that the teacher provide mathematical "scaffolding" (background material needed to successfully negotiate the exercises, correct student errors, facilitate mathematically accurate answers and classroom discussion, provide closure and summary, and so forth). In fact, an extensive Teacher's Guide book exists in parallel to the student text. The Teacher's Guide includes segments on how to introduce a unit or section; how to "Launch" the activity so students are given a mathematical orientation; and a "Summarize" section in which the teacher is expected to check for the mathematical correctness of answers shared during discussion of different methods students came up with individually or in their groups.
- ^ http://connectedmath.msu.edu/mathcontent/goals.shtml#process
- ^ Assessing the Impact of Standards-Based Middle Grades Mathematics Curriculum Materials on Student Achievement
- ^ The Impact of Two Standards-Based Mathematics Curricula on Student Achievement in Massachusetts
- ^ NYCHold
- ^ a b Pearson review of research
- ^ Educational controversies: toward a discourse of reconciliation by Pamela LePage, Hugh Sockett. p. 23.
- ^ Becker, J., & Jacob, B. (2000). The Politics of California School Mathematics. Phi Delta Kappan, 81(7), 529. p531 http://search.ebscohost.com
- ^ Connected Mathematics
- ^ Boston Spends $4M to Avoid Math Basics
- ^ CMP website
- ^ Selecting the Right Curriculum
- ^ Learning in a Problem Centered Curriculum
- ^ a b CMP2 Parent Website - Bits & Pieces II
- NYCHold several reviews of CMP, mostly strongly critical
- http://www.ridgewood.k12.nj.us Ridgewood recently implemented CMP2 in both of its junior high schools
Standards-based mathematics controversy Traditional mathematics Reform mathematics
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