Connected Mathematics

Connected Mathematics is a comprehensive, problem-centered curriculum designed for all students in grades 6-8 based on the NCTM standards. Since its introduction in the 1990s, this curriculum and its latest revision have been challenged for purported ineffectiveness in its treatment of basic mathematics skills.

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. CM follows the TERC investigations into junior high and middle school.

Pearson Education has published the revised Connected Mathematics 2 with a full color design. It was again funded by the National Science Foundation, revising and fixing some of the problems which sparked controversy with the original version. The units were field-tested and evaluated in over 30 classrooms nationwide, with feedback from nearly 15,000 students and 300 teachers.

Controversy

As one of many widely adopted, but controversial curricula developed around the NCTM standards such as TERC, CM 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 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." It has been reported that previously successful students have been "reduced to tears" from CM's teaching approach" and that other 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. One parent remarked "There were virtually no example solutions or general equations for anything." Unlike traditional textbooks, no complete method to add, subtract, multiply or divide fractions is actually presented in the textbook. The student textbooks contains an investigation for deriving the formula of a circle, but the actual formula is not stated in the book. This is because "The student role of formulating, representing, clarifying, communicating, and reflecting on ideas leads to an increase in learning. If the format of the texts included many worked examples, the student role would then become merely reproducing these examples with small modifications." 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

Connected treatment of some topics either are padded with subjective exercises which have nothing to do with the mathematical concept such as primes, or significantly omit standard methods such as the formula for average, or how to compare fractions with different denominators.

Average

In the first edition, an entire booklet takes the student from computing medians up to figuring out an average by moving blocks from different piles together. The standard method of taking the sum of the data divided by the number of items was omitted entirely, though later included in the revised version.

Comparing Fractions

In the 6th grade unit on fractions, students are instructed to use benchmark fractions, fraction strips, or other strategies to compare fractions with different denominators. 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. However, common denominators are not only utilized, but are a central concept in the revised edition (CMP2), which has been in use since at least the 2003-2004 school year. Students are taught how to use common denominators in adding fractions, and they are taught through multiple models and methods why common denominators are needed and why their use makes sense in the process of adding fractions. As stated in the "Concept with Explanation" page for Bits & Pieces II, from the parent support website, "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." Thus, not only are students taught to use common denominators; they are also taught to understand why they use common denominators, and to understand and be able to articulate the mathematical concepts underlying this procedure.

Prime Numbers

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 is a typical illustation of a non-traditional teaching approach. The student is asked to write numbers which a student "likes" more than other numbers, there are no "correct" answers.

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.

"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. "

External links and reviews

• http://www.math.msu.edu/cmp
• 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