During 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. To give students this kind of experience, Connected Mathematics is organized so students continually solve problems that contain important mathematical concepts and skills. This kind of organization is quite different from the way most of us learned.
Because the purpose of studying mathematics is to understand and apply concepts and procedures and be able to solve a variety of problems, students should spend most of their time in mathematics solving problems. When time is spent solving problems, reflecting on solution methods, examining why the methods work, comparing methods, and relating methods to those used in previous situations, students are likely to build more robust understandings and strategies and to view mathematics as a valuable tool for making sense of and solving interesting problems.
Selecting Good Problems
In a problem-centered curriculum, the types of problems students are asked to solve is important. First and foremost, problems must embody critical mathematical concepts and skills while engaging students in making sense of the mathematics. Good problems also help students build understanding through reflection and communication. Problems in Connected Mathematics were selected using the following criteria:
Source: Connected Mathematics Web site (http://www.math.msu.edu/cmp)
The Connected Mathematics Project received National Science Foundation funding between 1991 and 1998 to develop this mathematics curriculum for grades 6, 7, and 8. Each Connected Mathematics unit has been field tested, evaluated, and revised during a three- to four-year period. Approximately 160 teachers and 45,000 students in diverse school settings across the nation participated in the curriculum development. Research shows consistently that Connected Mathematics students outperform other students on tests of problem-solving ability, conceptual understanding, and proportional reasoning. In addition, Connected Mathematics students do as well as, or better than, other students on tests of basic skills. Connected Mathematics was the only middle school mathematics curriculum awarded “exemplary” status by the U.S. Department of Education’s Mathematics and Science Education Expert Panel (1999). Of the 61 elementary, middle, and high school curricula submitted for review, only five reached this status. The American Association for Advancement of Science (1999) reviewed 12 nationally available middle school mathematics curricula and ranked Connected Mathematics highest, stating that it “contains both in-depth mathematics content and excellent instructional support.” Connected Mathematics is now used in all 50 states and many foreign countries.
The research-based features of Connected Mathematics include:
In Connected Mathematics, units are aligned so students build on their knowledge; each year, key concepts or big ideas are explored more deeply. Students study number and operations, algebra, geometry, measurement, and probability and statistics in each year of Connected Mathematics. For example, students learn number and operations in “Bits and Pieces” in sixth grade and build on that knowledge in seventh grade in the unit, “Comparing and Scaling.” In eighth grade, students extend their proficiency with number, operation, and geometry in “Looking for Pythagoras.” Student knowledge in each strand spirals throughout the years until they have gained the proficiency or mastery expected at the end of each school year.
Connected Mathematics is committed to developing mathematical skills, skills that include more than quickness with paper-and-pencil algorithms. Skills also encompass the ability to use mathematical tools, resources, procedures, knowledge, and ways of thinking to make sense of new situations. For example, students need to recognize when an exact answer is required and when an approximate answer is sufficient, and they need to have a variety of methods for finding an answer. If an approximate answer is called for, a paper-and-pencil algorithm may not be the most efficient method.
Students need to know how and when to use paper-and-pencil algorithms, mental computation, calculator procedures, and estimation strategies, and it is critical that students acquire these methods with understanding. For example, students should be able to add two simple fractions quickly by finding a common denominator, but they should also understand why this algorithm works. Understanding equivalent fractions is the foundation for developing algorithms for fractions.
In Connected Mathematics, students develop understanding of algorithms and strategies in a variety of ways. As they work on investigations, they use and refine their skills. They learn to recognize when an algorithm or strategy applies to a new context and when they can build on the skills and strategies they know to develop new strategies. In these processes, students practice skills as an ongoing activity throughout the curriculum.
Source: Connected Mathematics Web site (http://www.math.msu.edu/cmp)
A calculator is a valuable tool for people to use when thinking mathematically. Calculators are not meant to replace the need for students to learn their basic facts or compute with a paper and pencil using mental mathematics. When students solve problems, calculators serve as a resource for them, aiding in their attempts to find a reasonable answer and make sense of the mathematics. In some instances, calculators will be chosen and used wisely to deepen understandings, where in other circumstances, calculators will prove to be ineffective. Discovering what tools to use and when to use them is part of the learning process for students.
Learning with understanding is critical in the area of mathematics. Group work actively engages middle school students in learning; they are expected to participate, contribute, and be reflective thinkers. Teachers strive to create learning environments where serious mathematical thinking and collaboration between all learners is the norm. When students work in collaborative groups, they are engaged in opportunities to clarify and justify their thinking. Group work provides challenges for students at all levels of understanding. Students learn from one another, as well as from the teacher.
Classroom activities should provide students the opportunity to work both individually and in small- and large-group arrangements. The arrangement should be determined by the instructional goals as well as the nature of the activity. Individual work can help students develop confidence in their own ability to solve problems but should constitute only a portion of the middle school experience. Working in small groups provides students with opportunities to talk about ideas and listen to their peers, enables teachers to interact more closely with students, takes positive advantage of the social characteristics of the middle school student, and provides opportunities for students to exchange ideas and hence develops their ability to communicate and reason.
NCTM Standards, page 67
Writing is a form of communication that requires complex thinking and the ability to articulate through the written word so someone else can understand the thinking. Writing gives students an opportunity to explain, clarify, and solidify their thinking so they can share it with others. When a teacher reads a student’s writing, she or he is given the opportunity to see into the child’s mind and to see what kinds of reasoning are taking place. Through reading a student’s writing, the teacher can get to know the learner better, can clarify misunderstandings, and enrich mathematical thinking. When students share their writing with each another, it leads to discussions, which honor students thinking, allow them to clarify understandings, and allow them to become more effective mathematicians.
Research shows that the most effective way to learn is to attach what you are trying to learn to what you already know. People should grapple with ideas until they create an understanding that makes sense to them. When students learn in this manner, rather than having someone show them the right answer, the learning is meaningful. The learner, in a sense, constructs his or her own knowledge. When this type of learning occurs, the knowledge is sustained over time and can be retrieved and reconstructed when needed.
It is important for students and their family members to understand the purposes of homework, the amount of homework assigned, and how family members can help. Homework is an opportunity for students to think further about what they have been learning in class. It is time to practice, process, and extend ideas and concepts learned in the classroom. Since homework is an extension of what is being learned in the classroom, the content is not typically new; homework is an opportunity to deepen the understandings of concepts being learned in class. If a student is struggling, family members can start by asking questions.
Helping Your Children with Homework
In helping children learn, one goal is to assist children in figuring out as much as they can for themselves. Family members can help by asking questions rather than telling what to do. Good questions and active listening will help children make sense of mathematics, build self-confidence, and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. Here are some questions family members might try; notice that none of them can be answered with a simple “yes” or “no.”
Getting Started
While Working
Reflecting about the Solution
Responding (helping your children clarify and extend their thinking)
Additionally, it is expected that middle school students know the basic addition, subtraction, multiplication, and division facts, as well as whole-number computation. If your children are not proficient with these skills, help them master the needed skills.
Helping Your Children Get Organized
Source: Connected Mathematics Web site (http://www.math.msu.edu/cmp)
Connected Mathematics provides a variety of tools that encompass the many dimensions and purposes of good assessment and reflect the philosophy of the curriculum. These assessments fall into three broad categories:
Source: Connected Mathematics Web site (http://www.math.msu.edu/cmp)
To answer CSAP test questions accurately, a student must be a complex thinker who is able to apply his or her knowledge of mathematics in a testing situation. She or he is required to communicate the reasonableness of an answer in a written response, answer multiple-choice questions that require the student to remember and apply mathematical concepts in new situations, and much more. Students who engage in quality teaching and learning using the Connected Mathematics curriculum should be well prepared for the CSAP tests. The curriculum allows students to internalize and understand what they are learning; students learn mathematics in a way that makes sense to them. When new learning is attached to what one already knows, the new learning is not forgotten. Students are then able to retrieve concepts and procedures they have learned and apply them in new situations.
Given the pivotal role of algebra in determining success both in high school and college, Denver Public Schools is committed to providing experiences in algebra that challenge and support success for all students. Because both the procedural and conceptual understandings are essential for student success in algebra, Denver Public School’s mathematics program is designed to ensure that all students have opportunities to develop these dual understandings by the end of ninth grade while also providing opportunities for students to demonstrate proficiency of these understandings during middle school.
However, “merely taking more prealgebra and algebra courses has not proved successful for many students” (Phillips and Lappan, 1998). Rather, students need algebra courses that support their making sense of the concepts. Algebraic concepts have been found to be best learned first when the concepts are embedded within a context or problem (NCTM, 1997). Research studies have shown that students who studied Connected Mathematics have a deep understanding of representing functions symbolically (Krebs, 1999), move successfully from “contextual language” to “official mathematical language” (Herbel-Eisenmann, 2002), and have the ability to connect problem contexts with symbolic expressions (Smith and Phillips, 2000).
Connected Mathematics was also written and designed to support all students in learning mathematical concepts and skills across the six content strands (algebra, geometry, probability and statistics, number sense, computation, and measurement) throughout sixth, seventh, and eighth grades. Together these experiences provide many students with enough understanding to earn algebra credit in middle school and to begin their freshman year in high school in a second-year mathematics course (geometry, for example).
Connected Mathematicsis a coherent and complete middle school curriculum that includes appropriate supports and challenges for diverse learners. Teachers differentiate instruction to meet the needs of individual learners through altering teaching and learning approaches, providing varied levels of support (scaffolding), spending more or less time on certain concepts, and asking a variety of questions.
In DPS, we understand some students may need more assistance to meet high expectations, while others may need enrichment to further excel. Use of assessment tools allows students to demonstrate what they do and do not understand, then teachers can plan the appropriate next steps to propel each student’s learning forward at the appropriate pace.
Good instruction in mathematics and English-language development share many teaching strategies. Connected Mathematics supports an effective learning environment in mathematics for English-language learners by incorporating group work into daily lessons. In small groups, students have opportunities to express ideas, ask questions, and clarify their thinking. Group work is complemented by whole-class instruction, in which the teacher models and uses mathematical language to reinforce skills and concepts addressed during group work. Connected Mathematics also supports teaching English through content that is relevant to students’ experiences and developing mathematical language proficiency through the use of manipulatives, models, demonstrations, and explicit vocabulary support.
The best place to get information about your child’s mathematics instruction is from the school. Attending mathematics class with your child is an invaluable experience that provides you with first-hand knowledge about the mathematics your child is learning. Scheduling a meeting with the teacher is another way to deepen your understanding of mathematics and how it is taught, as well as understand your child’s strengths and next steps. Attending mathematics events at your child’s school provides you with even more information about the mathematics curriculum, allows you to experience the teaching and learning process, and give you the opportunity to ask questions to clarify your understandings.
There are also a variety of Web sites for families who would like more information. The Denver Public Schools curriculum Web site (http://curriculum.dpsk12.org/math.htm) provides a plethora of information, as well as links to other mathematics Web sites. The Connected Mathematics Web site is also very helpful (http://www.math.msu.edu/cmp).
If you have technical questions about this Web site, contact Joel' Bradley-Hess at
joel_bradley-hess@dpsk12.org or 720-423-3723.
Page last updated:
Monday, October 24, 2005
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