- Why is it not a good idea just to tell my child my way of doing the homework?
- My child says the teacher rarely demonstrates methods. When my child is stuck the teacher often asks more questions. Why is this?
- Is there some general way of helping my child no matter which unit we're in?
- What if I have forgotten much of my school mathematics? How can I help my child?
- If there are few worked examples in the student text what do students refer to when they are doing homework or studying for tests?
- Why do students work in groups in CMP? How does the teacher know who is doing the work?
- How much Algebra is there in CMP? What high school class will my child be ready for after CMP 8 th grade?
- How much Geometry is there in CMP? How well does CMP prepare my child for High School Geometry?
- Does CMP pay enough attention to developing skills? There don't seem to be the pages and pages of short skills questions that are in traditional texts.
- Why does my child do so much writing in CMP?
- What is the research basis for CMP's design?
- How does CMP accommodate gifted and talented students?
- How does CMP accommodate English as a Second Language learners?
- How does CMP accommodate students with special needs?
- Why are there so few worked examples in CMP texts?
Why is it not a good idea just to tell my child my way of doing the homework? ^ Back to Top ^
When your child seems to need help with mathematics homework, as a caring parent/guardian, your first wish and impulse is to try to tell your child how to do the assignment or show them a method you recall. Teachers are care-givers too and they are similarly tempted to show and tell, but if we want our children to be independent learners then we have to resist this impulse and develop ways to have mathematical conversations in which the child is an active participant.
CMP is problem-centered. This means that important mathematical concepts are embedded in engaging problems. Students develop understanding, skill, and confidence in themselves as they explore problems, individually and in groups, or with the class. Parents and guardians can support this approach by the way that they interact with their children over homework. See General Suggestions on this website for help. See Problem-Centered on this website for more about this approach and for the research base that supports this approach.
My child says the teacher rarely demonstrates methods. When my child is stuck the teacher often asks more questions. Why is this? ^ Back to Top ^
CMP was designed to be compatible with research from the Cognitive Sciences about how knowledge is developed. Specifically, the materials support student-student and student-teacher dialogues about mathematics. The single mathematical standard that has been a guide for all the CMP curriculum development is: All students should be able to reason and communicate proficiently in mathematics. If students are to reason and communicate proficiently then these are the skills they must spend time practicing in the classroom. Sometimes it is appropriate for students to work individually, and to make written explanations, but often the materials require that students discuss problems, argue with and convince each other, make conjectures and draw conclusions, and make summaries. The teacher's role is to aid and guide students as they work with problems to develop understanding of the embedded ideas, and abstract powerful mathematical principles, strategies and algorithms. Of course, there will be occasions when the teacher will demonstrate some mathematical idea, but generally the teacher will try to help students draw as much sense as they can from a problem before intervening. For more on these ideas, see Communication as a Learning Tool on this website.
Is there some general way of helping my child no matter which unit we're in? ^ Back to Top ^
One of the best ways to help your child is to make it a habit to find out first what the child does and does not understand, and what resources he or she has to draw on. A set of questions has been developed to assist you in starting up a helpful dialogue. You will want to personalize these to fit the relationship you have with your child and also the particular circumstances of the classroom your child is in. These are always useful, whether, as a parent/guardian, you feel competent with the mathematics or not. See General Suggestions on this website.
What if I have forgotten much of my school mathematics? How can I help my child? ^ Back to Top ^
You may need a reminder before any real mathematical conversation with your child can take place. In that case you probably went to the student text and looked for an example. You may be disappointed, then, to find there are almost no examples shown, nor methods told. Instead there are questions for students to discuss, followed by more questions and problems for them to solve. Without being part of the classroom discussion it is hard for you to reconstruct the mathematics embedded in these problems. To help parents and guardians bridge this gap, an on-line summary of goals, a glossary of vocabulary words and concepts (with examples!), and sample solutions for homework problems are available, for each unit.
See Mathematical Help on this website. (For more on why there are so few worked problems see Learning in a Problem-Centered Curriculum on this website.)
If there are few worked examples in the student text what do students refer to when they are doing homework or studying for tests? ^ Back to Top ^
Students should be keeping notes of how they solved in-class problems, noting new vocabulary, summarizing each Problem with the teacher's help (typically 3 per Investigation), and completing a Mathematical Reflection on the entire Investigation (typically 4 per Unit). Different teachers will have different ways of handling this organization. A comprehensive student notebook is a record of what each student understands and can do. The creation of the notebook is crucial for each student's success.
Why do students work in groups in CMP? How does the teacher know who is doing the work? ^ Back to Top ^
There is a consistent and substantial body of research indicating that when students engage in cooperative work on challenging problem solving tasks, their mathematical learning will be enhanced. Therefore, CMP materials have been designed to be suitable for use in cooperative learning instructional formats, as well as individual learning formats. The mathematical task determines the format of the problem. Teachers are assisted and advised in the teacher materials in planning for and executing both kinds of lesson designs. In fact, in any given lesson there is likely to be a blend of individual, group, and whole class activity.
When students are working in groups the classroom looks very different from when they are working individually; and the teacher's role is very different. However, group work is not the opposite of individual responsibility and accountability. Teachers can and do observe closely what individuals are doing and learning, while they monitor group progress. Each student will be responsible at minimum for taking notes, asking questions, and summarizing. Depending on the particular problem, groups or individual students may be responsible for sharing ideas or leading class summaries.
How much Algebra is there in CMP? What high school class will my child be ready for after CMP 8 th grade? ^ Back to Top ^
The Connected Mathematics Project was funded by the National Science Foundation and designed by its authors with one of its goals being to provide more algebra before high school. The NCTM Principles and Standards guided the development of the algebra strand in the Connected Mathematics Project.
Algebra is developed during all three grades of Connected Mathematics. By the end of grade 8, most students should be able to:
Identify and use variables to describe relationships between quantitative variables in order to solve problems or make decisions
Recognize and distinguish among patterns of change associated with linear, inverse, exponential and quadratic functions
Construct tables, graphs, symbolic expressions and verbal descriptions and use them to describe and predict patterns of change in variables
Move easily among tables, graphs, symbolic expressions and verbal descriptions
Describe the advantages and disadvantages of each representation and use these descriptions to make choices when solving problems
Use linear and inverse equations and inequalities as mathematical models of situations involving variables
Connect equations to problem situations
Connect solving equations in one variable to finding specific values of functions
Solve linear equations and inequalities and simple quadratic equations using symbolic methods.
Find equivalent forms of many kinds of equations, including factoring simple quadratic equations.
Use the distributive and commutative properties to write equivalent expressions and equations
Solve systems of linear equations
Mathematics Courses in High School
Depending on what program of studies is being offered in the high school, it may well be the case that students who have been successful in CMP in 8 th grade can skip the first year of the high school program. Obviously this decision can only be made based on knowledge of both programs; the best guides are the teachers involved.
If the high school in the district is still offering a traditional Algebra 1, Geometry, Algebra 2 sequence, then, based on what courses are available at 9 th grade, and on how successful a particular student has been in 8 th grade CMP, there are several options for the district to consider. Two options are outlined below. In neither of these options is it necessary for a student who has been successful in the algebra units in CMP to spend a valuable year of high school in a traditional Algebra 1 class.
Students who have been successful in the CMP algebra units will have met and mastered most of the ideas and skills that are part of a traditional Algebra 1. But, they also will have done very much more than this in their study of algebra in CMP. Their experience will have been a coherent functions approach to important mathematical relationships, especially linear, exponential, inverse proportion and quadratic, including solving linear, exponential and quadratic equations, and inverse and direct proportions. Therefore, CMP algebra units are an excellent preparation for a traditional functions-based approach in Algebra 2. Because of this extensive and thorough study of algebraic ideas in CMP, many students entering a high school with a traditional curriculum in place may successfully proceed to Algebra 2.
If, on examining what is expected of students coming out of 8 th grade CMP, teachers in a high school offering the traditional curriculum see skills which they believe are integral to Algebra 1 and which CMP students have not met, then they may create a short "patch" which can be added to the 8 th grade CMP units. However, Algebra 2 textbooks typically include a lot of review of Algebra 1, and, therefore, would review and supplement what students know from CMP.
For more information on Algebra in CMP see Content by Strand on this website.
How much Geometry is there in CMP? How well does CMP prepare my child for High School Geometry? ^ Back to Top ^
Two Units per grade level (6 out of a total of 24 units in 3 grade levels) are primarily Geometry/ Measurement. Geometric ideas are connected and reviewed in all other units. See Content by Strand on this website for a list of topics and goals. The CMP authors were influenced by research from Mathematics Education indicating a shift from a focus on shape and form to the related ideas of congruence, similarity, and symmetry transformations.
A traditional high school Geometry class will spend a semester on shape and measurement, two topics thoroughly explored in CMP, and a semester on Proof. All CMP units require students to reason and communicate their reasoning. In CMP Geometry units students are asked to reason about ideas which are part of the high school curriculum, such as similarity and congruence. At first the reasoning will be informal, but as students mature both their ability to reason and their ability to communicate their reasoning develops.
Does CMP pay enough attention to developing skills? There don't seem to be the pages and pages of short skills questions that are in traditional texts. ^ Back to Top ^
The authors of CMP have been influenced by theory and research indicating that mathematical understanding is fundamentally a web of logical and psychological connections among ideas. They have interpreted research on the interplay between conceptual and procedural knowledge to say that sound conceptual understanding is an important foundation for procedural skill, not an incidental and delayed consequence of repeated rote procedural practice. Therefore, CMP Problems concentrate on conceptual development, but often draw on previous conceptual and procedural knowledge so that there is repeated procedural practice embedded in new problems. Homework questions, particularly Connections questions, repeatedly connect new ideas to prior procedural knowledge. For example, if the unit under study is Covering and Surrounding then area is one of the concepts under study; this gives ample opportunity to review and practice all the rational number (fraction) work done prior to this. The authors have paid deliberate attention to this kind of connectivity and distributed practice. For procedures or algorithms for which students need to reach fluency (for example, fractions operations) substantial numbers of practice problems are distributed throughout all three grades.
As a parent you might be wondering what the difference is between a skill and a concept, between procedural knowledge and conceptual knowledge, and how one can assess student knowledge. Suppose we want to find out if a student understands decimal multiplication. We might assign 4.56 x 2.35 to be done by hand. When the student produces the answer 10.716, which is correct, we can not be sure that the student actually understands anything about the concept of place value and how that concept is used in calculating the correct answer. We can be pleased that the student has apparently mastered that skill, but if we want to find out what the student understands about place value we need to ask a different question.
If, instead of assigning 4.56 x 2.35, we start by giving the student the fact that 4.56 x 2.35 = 10.716 and then ask for the result of 4.56 x 0.235, without access to paper and pencil or a calculator, then the student will have to reason that since one of the factors is a tenth of its original value the answer should also be a tenth of its original value, or 1.0716. Now the student is using the place values for the digits in 2.35 and 0.235 to think about the calculation. Similarly, giving the student the fact that 4.56 x 2.35 = 10.716, and asking for the result of 456 x 235, without access to paper and pencil or calculator, forces students to consider place value.
The other procedure that students often learn as a shortcut is to "count the decimal places in the two factors, and make the answer have the same number of decimal places." Suppose we start by telling the student that a broken calculator has produced the answer 4.56 x 2.35 = 10716, and ask the student to place the decimal point. If the student applies the shortcut, without thinking about place value, then he/she might state that the answer is 1.0716, which has 4 digits after the decimal point. But the student who understands multiplication and estimation can reason that the answer must be more than 4 x 2, and, therefore, the correct answer is 10.716.
In short, CMP values both skills and concepts, and the authors have created ways for students to build concepts and practice skills, and for teachers to assess both.
See Research Results (Prentice Hall) for results of studies favorably comparing CMP students' performance on tests of basic skills.
What is the research basis for CMP's design? ^ Back to Top ^
See Learning in a Problem Centered Curriculum and Communication, both on this website, for a summary of the research ideas from the Cognitive Sciences that influenced the authors. See Content by Strand, on this website for a summary of research from Mathematics Education that was influential for the authors.
How does CMP accommodate gifted and talented students? ^ Back to Top ^
The Implementation Guide that comes with CMP materials when a school adopts CMP pays detailed attention to the needs of Gifted and Talented students in the Mathematics classroom. A National Council of Teachers of Mathematics publication, Providing Opportunities for the Mathematically Gifted K-12, proposed several essential components, which relate directly to the mathematics curriculum. CMP possesses all the components described by NCTM's publication, notably:
nurturing higher-order thinking processes in open-ended investigations,
prompting students to communicate effectively,
having problem-solving as a major focus,
including applications of mathematics to real-life situations,
encouraging experimentation, connecting mathematics to other subject areas.
Cognitive Science Researchers have suggested that organizing the curriculum around key concepts and complex problems, promoting use rather than acquisition of knowledge, and pushing for abstraction are all appropriate modifications for G&T students. All of these are already present in the organization of CMP. In particular the Extension questions in the ACE ask for levels of abstraction that may not be available to all students. But even within the in-class Problem there are thought-provoking questions that can be answered at different levels of abstraction. (See Learning in a Problem-Centered Curriculum on this website.)
Interestingly researchers stress opportunities for divergent thinking, and group interaction to allow G&T students the opportunity to explain their ways of thinking to others, and to assume leadership roles. The focus should be on explaining why and how strategies work, and on classroom patterns of interaction that make learning, not the teacher's approval, the focus.
How does CMP accommodate English as a Second Language learners? ^ Back to Top ^
The Implementation Guide that comes with CMP materials when a school adopts CMP pays detailed attention to the needs of English Language Learners in the Mathematics classroom. Several strategies are outlined, and illustrated, to help teachers with the challenge of including ELL students. Each of these strategies in some way addresses instructional delivery. None of these strategies is based on modifying the cognitive demand of the mathematics in CMP.
In addition there are two versions of parent/guardian letters, English and Spanish, made available to teachers to help them keep parents informed.
How does CMP accommodate students with special needs? ^ Back to Top ^
The Implementation Guide that comes with CMP materials when a school adopts CMP pays detailed attention to the students with Special Needs in the Mathematics classroom. Several strategies are outlined, and illustrated, to help teachers with the challenge of including all students. Each of these strategies in some way addresses instructional delivery. None of these strategies is based on modifying the cognitive demand of the mathematics in CMP.
In addition sample lessons, one per unit, have been developed to help teachers see how these modifications can be made, without substantively altering the cognitive demands on students.
Strategies that reach out to students with Special Needs are also strategies that help all children learn. Indeed the blending of group and individual work, of student conversation and silent reflection, and of conceptual development and procedural skills practice, characteristics that are integral to CMP's format, offer more ways for more students to be successful.
Why are there so few worked examples in CMP texts? ^ Back to Top ^
The overarching goal in CMP is for students to make sense of and take ownership of mathematical concepts. 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. This would have the effect of having students become fluent at a particular skill, or with a particular strategy, but would work against having students develop the independent reasoning skills that can be applied flexibly in new situations. This is not to say that worked examples are not valued at CMP; but the "work" in these examples is done by students, independently, in groups, or with the help of the teacher, and is captured in notebooks, to be referred to whenever needed.