Classroom Environment
Jump to:
Organizing the Classroom
Homework in CMP
Student Notebooks
Vocabulary
Pacing
Absenteeism
Organizing the Classroom
Helping students become independent learners is an important goal for Connected Mathematics. The Teacher's Guides point out opportunities for helping students reach this goal.
Classroom Setup
The way your classroom is set up can have a significant impact on learning. Here are some suggestions for creating an effective learning environment for students:
Post the Mathematical and Problem-Solving Goals for the unit given in the Teacher's Guide and student books and check them off as the class meets them.
Post a vocabulary list in the room so students know what words should be in their glossaries. Add words to the list for each unit as you proceed through the unit.
Keep a list of assignments in the room for students who have missed class.
Post upcoming assessments and due dates so students can anticipate your expectations.
Make tools (rulers, grid paper, angle rulers, and so on) accessible so students can decide which tools are appropriate for solving a problem.
On the board, keep a list of unresolved questions for future discussion.
Have textbooks, a mathematics dictionary, and other reference materials (almanacs, atlases, and so on) available for students to use.
Have materials on hand that allow students to display their work and share their results with the class. Save student work for future reference by the class or for parent meetings. Record student work on overhead transparencies for class discussions and then make copies for absent students or save for future reference by the class.
Homework in CMP
In Connected Mathematics, homework takes a role different from that in other curricula. Homework in Connected Mathematics is intended as an opportunity for students to think further about the ideas in a lesson. The lesson, rather than the homework assignment, is the primary unit of instruction, with homework as a vehicle for teachers to help students to process, practice, connect, and extend the ideas from the lesson. On a typical day in a Connected Mathematics classroom, far less time is spent assigning, doing and checking homework than may have been the case with other programs. The following sections contain some approaches Connected Mathematics teachers have taken in order to maximize the effectiveness of their time and students' time spent on homework.
Assigning Homework
You can use the Assignment Guide feature in the Teacher's Guide to help you assign homework. This feature appears on the At a Glance page for each Problem and indicates the ACE exercises which students should be able to answer after completing the Problem. The Assignment Guide for a Problem typically includes questions from each of the three ACE sections.
Teachers have generally found that the ACE exercises in Connected Mathematics are more substantial than the homework assigned in other curricula. So they often think differently about homework assignments. In particular, teachers begin to make more careful choices about which questions to assign and how to assign and grade them.
In general, the Assignment Guide in the Teacher's Guide for an Investigation will include all ACE exercises in that Investigation. In the spirit of Connected Mathematics materials, many more ACE exercises are provided than can reasonably be assigned as homework. However, this gives the teacher choices, so the materials can be tailored to the needs of a particular classroom of students. In addition, different communities have different expectations about homework, and classes meet for different lengths of time. These and other factors influence the pace and amount of homework assigned.
By answering the ACE exercises yourself before you assign them, you can anticipate difficulties and estimate the time it will take students to complete the assignment. Some teachers read and briefly discuss the ACE exercises in class before assigning them so that students understand what they are to do.
Students should attempt to answer all the assigned ACE exercises, but they may struggle with some. You might suggest that, if a student cannot solve a problem, he or she write a question about it, such as, "What are 'increments of 5 campers'?" or "Which variable should go on the x-axis?" Questions such as these focus the student on the area of difficulty, let you know the student's thoughts about the problem, and give you insight into the difficulty the student may be having. Some Connected Mathematics teachers begin class by allowing students to ask questions about the previous night's homework. The students are then given the opportunity to revise their work before turning it in.
Responding to and Grading Homework
How you respond to student work will depend on the reason you assigned the work. Connected Mathematics teachers have listed, among many others, the following reasons for assigning homework:
To provide additional explanation and practice of the key mathematical ideas in the lesson
To grade students' work
To assess what students do and do not know in order to plan instruction
To connect learning experiences on two consecutive days
To instill good study habits
To accomplish more mathematical study outside the time limits of the classroom
Because ACE exercises are rich, they may elicit a variety of answers and strategies from students. Dealing with this variety of responses can be time-consuming for teachers. Many Connected Mathematics teachers adapt how they respond to student homework based on the reason for a particular assignment. Clearly, if an assignment is given to assess understanding to plan instruction, the work will need a different kind of attention from the teacher than if it is assigned to instill good study habits.
Some methods used by Connected Mathematics teachers to respond to students' homework are listed here.
Prepare an answer key that covers the main elements of each exercise. (Detailed answers for all Problems and ACE exercises are in the Teacher's Guides.) Assign points to each Problem, and have students correct their work and total the points for correct answers.
Write the exercise numbers for the previous night's assignment on the board. As students come into class, have them make a checkmark next to the numbers for which they have questions. Discuss only those problems.
Choose a few exercises to read carefully and grade; discuss the rest in class.
Go over the answers in class and have students check and revise their work. Then, each Friday choose a small set of exercises from the week's homework for students to turn in (this is sometimes referred to as a "homework quiz") for a grade.
Collect the homework papers and check each exercise.
To prevent losing some class time each day while checking homework, assign exercises over the course of the week and grade them all on one day.
Assign a set of exercises at the beginning of the Investigation, informing students which of these exercises they should attempt each day. Spend a few minutes each day taking student questions about the previous night's work. Collect all of the exercises at the end of the Investigation.
Give complete credit for satisfactory completion of the assignment. Give partial credit, as warranted, based on the number of exercises completed satisfactorily.
Student Notebooks
It is helpful for students to keep their work in an organized notebook. The notebook can include notes, vocabulary, solutions to investigation problems, homework, and responses to mathematical reflections. By reviewing your students' notebooks, you can get a clearer picture of their mathematical development.
Because the Connected Mathematics units are three-hole punched, students can keep their books, along with their important work, in a three- ring binder. The binder can easily be divided into sections for a journal or notes, homework, vocabulary lists, quizzes, and tests. The binder can include work and notes written on loose-leaf paper or in spiral notebooks, which can be removed when the teacher wants to check some part of the notebook.
Some teachers have students designate a section of the notebook as a journal. In their journals, students record solutions to the Investigation problems, responses to Mathematical Reflections, and respond to queries generated by the class discussion, the teacher, or other students. Journals should be seen as an aid for students as they try out their thinking and develop complete responses and thoughtful conjectures.
Some teachers combine the journal and notes. Students record all journal entries on the left-hand side and the notes on the right-hand side. With this arrangement, students and teachers can separate the experimentation ideas from the summary of classroom ideas.
The following example shows the notebook guidelines one Connected Mathematics teacher gave to her students.
An Example of a Notebook Organization
(These suggestions are adapted from Jan Palkowski, a middle school teacher in Traverse City, Michigan. This teacher also has the students keep daily logs of their participation in classroom activities and dialogue.)
Section 1: Forms
In this section, keep assignment sheets, participation logs, and classroom rules and procedures.
Section 2: Journal
This section should include:
Any and all work you do for in-class problems; this includes your work on Investigation Problems and any handouts. Include words, charts, pictures, or anything else to show your thinking.
Any notes you take; write anything that will help you remember your thinking. You should also record notes about the class summary of the ideas in each Investigation. These notes are for your reference as you solve in-class problems, answer homework questions, work on quizzes, and prepare for tests.
Your Mathematical Reflections from each Investigation.
Section 3: Vocabulary
In this section, you will create mathematical descriptions with examples of words you need to know. Use lined loose-leaf notebook paper for this section.
Section 4: Homework Assignments
This section should include your work on the ACE assignments. Your homework should be written on lined loose-leaf notebook paper or graph paper and clearly identified.
Section 5: Assessment
This section will include all check-ups, partner quizzes, tests, projects, and self-assessment.
Section 6: Your Book
Keep your unit inside your binder at all times. Please do the following to help you organize your work and to make it easier for me to review your notebook:
Date every entry and identify problems with problem numbers and the unit name.
Always revise what you have written by crossing it out, rather than by erasing. This saves you time and helps me to follow your thinking. It does not count against you to cross out your old work.
I will check your notebooks at unannounced times, and homework grades will be given. After you complete a unit, clean out all the sections of your notebook except the "Forms" and "Vocabulary" sections. I will file your work for future reference and portfolio selections.
It is recommended that you check notebooks often during the first few weeks of the semester. It is important to give students feedback early to make sure notebooks are being used correctly and to address any problems. Many teachers walk around the room while students are working and give comments or suggestions on maintaining notebooks.
Since keeping notes in mathematics class is new for many students, it is helpful to keep models of outstanding notes. This helps students understand your expectations. You can photocopy good examples to share with students. It might be helpful to have students evaluate their notes, journal entries, or vocabulary according to the models. At the end of the year, ask a student if you may keep his or her notes for the next year to have a complete example of how a notebook should look.
The Notebook Checklist can be used to evaluate students' notes periodically throughout a unit or at the end of a unit. In the Homework Assignments, list the items you would like to assess. Having students assess their notebooks before turning them in allows them to critically review their entries and organization.
Many teachers grade the journal, notes, or vocabulary sections of students' notebooks as well as the overall organization. Rubrics lend themselves nicely to the grading of notebooks, as you are generally looking for the completeness of ideas, notes, and vocabulary descriptions and records of the discussions from class. Some teachers give "Credit," "Partial Credit," or "No Credit" as a grade for notebooks.
There are a variety of methods for checking student notebooks. Here are some ideas you might try:
Read and respond to a few students' journal entries each day.
Collect papers from students at the end of each Investigation. Grade or respond to student work and then return the papers for students to replace in their notebooks.
Collect notes at the end of a unit and grade them.
Spot-check notebooks while students work on an assessment resource
Check notebooks at random
Give notebook quizzes. That is, periodically have students copy information from their notes on a sheet of paper; then grade just that information. (For example, What were the three strategies we discussed for solving Problem 3.1? What was the answer to Problem 3.3 Part B?)
Vocabulary
Vocabulary lists appear near the front of each Teacher's Guide. These lists are generally divided into three categories:
Essential terms developed in the unit
Terms developed in previous units
Useful terms developed in the unit
These lists indicate the mathematical terms developed in the unit. Based on your students' or school's needs, you may add to the lists or shift words from the useful to the essential column.
You may choose to hand out the list of vocabulary terms from the Teacher's Guide when you begin the unit, or you may prefer to have students generate their own list as they encounter the terms in the unit.
Although there is a glossary in the back of each student book, we suggest that you have your students develop their own lists of definitions and examples. For important mathematical vocabulary, students need to have descriptions that carry meaning at their level of verbal sophistication. Encourage students to view their lists as working glossaries that they can add to and refine as they gain new insight and encounter new examples. Revising and updating descriptions can help students improve their working knowledge of the vocabulary. You might find it helpful to have students occasionally work in a group or as a whole class to discuss the various descriptions they have written.
The vocabulary lists and the definitions students generate can become quite involved and personal. Many students like to keep their lists from each unit to use as reference tools in later units. In some schools, students are required to save the lists they generate during the year to use the following year. This helps them make sense of new ideas by giving them previous references on which to build. Some schools give extra credit to students who begin the year with their vocabulary from the previous year.
To keep the vocabulary section organized, students are directed by some teachers to begin the school year with 26 sheets of paper in the vocabulary section of their notebooks. Each piece of paper is then labeled with a different letter of the alphabet. Students write the words, descriptions, and examples under the appropriate letter. Although the words are not in alphabetical order on the page, there is enough organization for students to locate specific words.
Here is a replication of a hand-out that one teacher gave to her students as a guide to keeping their vocabulary section:
Student Guidelines for Vocabulary Section of Notebook
Vocabulary development is:
For personal and class development of mathematical vocabulary as the need arises.
In your vocabulary section you should have the following:
Any and all words that we as a class agree needs to be in everyone's notebook vocabulary section.
For each word recorded, a definition/description needs to be given by the end of study of the unit.
Definitions/descriptions can use words, charts, pictures, examples, and anything else to help make clear what the word means in our mathematics community.
To help you organize your journal for yourself and for me (the teacher) when I go through it, please do the following:
Underline each vocabulary word listed.
Leave room between words for a definition/description and space for revisions.
Always revise what you have written by "x-ing out" rather than erasing. This saves time and helps me to follow your thinking. It does not count against you to "x-out" work.
Pacing
When using Connected Mathematics, teachers should try to maintain a steady pace that will allow them to get through as much of the material as possible. Because ideas are developed over several problems, it is important for teachers not to spend too much time on any one problem. In some districts, district coordinators set timeline schedules to help teachers establish a sense of pacing. Each unit contains pacing schedules for 50-minute periods and block scheduling that were based on field testing. Depending on your district needs and schedule, it should be possible to do 6 to 8 units for each grade.
In the first year of implementation, some teachers may feel the need to supplement the materials with drill and practice. This will take time away from Connected Mathematics and slow the pace. Over time, teachers will learn the curriculum and understand that drill is incorporated into the lessons.
Although the primary focus of professional development is on the mathematics and pedagogy, teachers who are new to Connected Mathematics often have concerns about pacing, homework, grading, basic skills, and collaborative learning. These concerns may affect how a teacher sets the pacing of a unit. These issues should be addressed during professional development.
Absenteeism
Being absent in a CMP class is different from being absent in a traditional class. For example, students miss the experience of developing their understanding by working on a Problem and discussing key concepts and strategies. In Connected Mathematics, key concepts and skills are developed over several classroom Investigations. If students are absent for only a day or two, they have not missed the entire discussion on a key idea. The following suggestions come from CMP teachers.
When Students Are Absent
Keep assignments and activities posted in the classroom so students know what they missed.
Have group members collect any materials that are passed out for absent group members.
Establish note-taking buddies so students have someone to provide the notes.
Keep a master copy of the classroom notes for students' reference.
Have group or class members summarize what was done the previous day.
When Teachers Are Absent
Many teachers have found that, without some professional development, it is difficult for a substitute teacher to teach a Problem or lesson. Suggestions of activities that can be done when a substitute teaches the class are given here.
Partner Quizzes Pairs can work on quizzes fairly independently, using their notes and books as resources. When it is possible, assigning partners the day before can reduce confusion as the class begins.
Review Compile worksheets, using the additional practice problems from the units you have completed. If you teach seventh or eighth grade, you can use the problems from previous years.
ACE Exercises Assign a set of ACE exercises to be done in class. You can provide incentives for students. For example, if work is done diligently the first part of the period, you might allow students to work with a partner for the second half of the period.
Note that all three suggestions work well if the teacher knows in advance that he or she will be absent. The last two suggestions also work for unexpected absences.