Piloting and Initial Considerations
should be given to many issues as you study the Core-Plus Mathematics program
and plan for an effective implementation of the program. Excerpted
below are a selection of entries from Implementing Core-Plus Mathematics. You may wish to access
resources related to improving mathematics education from organizations
such as the Mathematics Assessment Resource Service (MARS) "Resources
for Leaders in Mathematics" (toolkitforchange.org).
When considering the adoption of mathematics textbooks, districts often decide to pilot two or more programs to feed into their program review. If your district mathematics goals for student learning align with the problem-based, inquiry-oriented Core-Plus Mathematics program, you may wish to pilot a course or selected units to inform decisions. If so, we recommend that pilot teachers receive professional development based on the content they will be teaching. If your district is shifting to inquiry-based instruction, when evaluating the pilot, keep in mind that it takes time to change classroom expectations and norms. There may be some push-back initially from students who have been successful in
mathematics courses that have required memorization and imitation rather than learning through problem-solving, reasoning by sense-making, and by discussing mathematical ideas with classmates. Some students may find mathematics learning more difficult when the expectations are shifted. Other students who have been less successful in earlier mathematics courses may find that they are learning more with the approach of the Core-Plus Mathematics program.
If you are piloting within your current mathematics course offerings, you might consider piloting Course 1, Unit 2, Patterns in Data. This unit may provide students the opportunity to learn statistics-related standards that your current course does not and make use of the affordances of the CPMP-Tools built-in data sets and statistical software. If you wish to pilot material from the algebra and functions strand, consider teaching the CCSS Pathway outlined in the Unit Planning Guide for Course 1, Unit 1, Patterns of Change, and Unit 3, Linear Functions. As you select pilot material, keep in mind the trade-offs, particularly within mathematical strands, of teaching a later unit without teaching earlier units in the strand. Try to retain the coherence and connectedness purposefully developed into the curriculum.
Building a Strong Foundation
Careful consideration should be given to many issues as you plan for an effective implementation of the Core-Plus Mathematics program. If you are a considering a pilot to inform adoption, suggestions from schools that have implemented the program may be helpful as you develop an implementation plan.
Begin adoption with Course 1 and add a course level each year. Encourage teachers to progress from Course 1 to Course 4 in stages, so they can develop a thorough understanding of the curriculum and the growth of mathematical ideas across the courses.
Schedule classes to allow for common planning periods for teachers teaching the same course.
Produce a Frequently Asked Questions document containing your district's responses to community questions so that there is consistent message from district administrators, mathematics teachers, counselors, teachers of other subject areas, and school office staff.
Consider how district and individual teacher decisions can affect the amount of material taught each year. (See Pacing Considerations, page 40, and the CCSS Pathways in the Teacher's Guide Unit Planning Guides.)
Assess classroom organization needs, such as tables and chairs rather than desks for students.
Assess district technology needs. Handheld technology (such as the TI-84) and CPMP-Tools should be available for each student. Some districts provide access to handheld technology for each student to use at home, as well as a set of handheld technology for each mathematics classroom. (See the CPMP-Tools overview on pages 14-18.)
Consider ways to align your district curriculum with your district goals and state mathematics standards. (See Curriculum Adoption: Advice and Tools on pages 18-20.)
Develop an ongoing professional development plan that will provide continuous support for high school, middle school, and special education teachers, and paraprofessionals. High quality professional development will weave together mathematics, pedagogy, and assessment and allow time for teachers to become familiar with new technology.
Develop a professional development plan for newly hired teachers, such as a common planning period with mentor teachers and additional summer or academic year professional development opportunities.
Consider providing collaborative learning, technology, literacy, and alternative assessment workshop opportunities for mathematics teachers before they begin teaching the curriculum.
Provide opportunities for instructional leaders, particularly building principals, to understand the goals of your mathematics program and discuss ways to promote effective classroom instruction. (One professional development option for leadership teams is workshops based on the three Lenses on Learning modules: www2.edc.org/CDT/cdt/cdt_lol1.html.)
For review copies
of Courses 1,
2, 3, and 4 (Preparation for Calculus) student and teacher
materials, contact your McGraw-Hill representative. You may wish
to request the CCSS Guide to Core-Plus Mathematics and the Implementing
Core-Plus Mathematics guide to aid in your review and/or implementation
of the program.
As noted previously, differentiation to challenge students is designed into the Core-Plus Mathematics program. But if your district has a history of accelerating some students, you may wish to continue this approach.
There is a trend in most states to include in middle school expectations some algebra and geometry content that historically has been taught in high school. The increased middle school expectations makes it more challenging to develop accelerated courses and to select students into these courses. If this is the case in your state, it may not be advisable for students to skip a middle school course in order to accelerate. One method of acceleration suggested in the Common Core State Standards, Appendix A is to compact middle school courses.
Some methods for acceleration to allow some students to take Advanced Placement Calculus or Advanced Placement Statistics their senior year include:
Provide an 8th-grade course that includes middle school expectations and some of the content from Core-Plus Mathematics Course 1. (High expectations in this 8th-grade course is one way to determine whether a student has the ability and work ethic to maintain accelerated classes through high school. If not, the student could enroll in Course 1 in 9th grade.) Then in 9th grade, develop a course that includes the remainder of Course 1 and Course 2 content.
Compacting courses may occur by teaching Courses 1–3, and Course 4: Preparation for Calculus or Transition to College Mathematics and Statistics in grades 9 through 11.
Provide a summer Course 1 for 8th-grade students so they may enroll in Course 2 as 9th-graders.
In schools with semester block scheduling at the high school level, a student may enroll in two courses in a given year.
In schools with academic-year schedules, two mathematics classes may be scheduled back-to-back to allow a class of students to study one course in the first semester and the next course in the second semester.
Students who have not accelerated earlier may choose as seniors to double up on classes by enrolling in both Course 4: Preparation for Calculus or Transition to College Mathematics and Statistics and AP Statistics.
Equity, and Differentiation
One question frequently
asked by districts adopting Core-Plus Mathematics is related to
equity and approaches to accommodate the program for underrepresented students. Mixed-ability classes, a focus on problem-solving, high expectations for all students, attention to a broad array of mathematical topics, and allowing students to restate problems in their own words also appear to help students from different racial, ethnic, and linguistic groups be more successful in mathematics. In addition, several research studies have provided
evidence that introducing activities through class discussion, teaching
students to explain and justify, and making real-world contexts accessible
to students promote greater access and equity in mathematics classrooms.
(Boaler, J. "Learning from Teaching: Exploring the Relationship Between
Reform Curriculum and Equity," Journal for Research in Mathematics
Education, 2002, Vol. 33, No. 4, 239-258, and Brown, C.A., Stein,
M.K., and Forman, E. A. "Assisting Teachers and Students to Reform Their
Mathematics Classroom," Educational Studies in Mathematics, 1996,
31-93). Practices that help promote equity are briefly discussed below.
Through Class Discussions Group and class discussions regarding the aim of investigations, the meaning of contexts, the challenging points within problems, and possible problem access points to which students might turn make tasks more evenly accessible to students. In cases where students use informal or non-mathematical language to explain their reasoning, the teacher may consider rephrasing or re-voicing the student's explanation, using more formal mathematics language.
Teaching Students to Explain and Justify their Thinking Giving explicit attention to explaining thinking and evaluating what makes a good piece of work helps students improve their work. Explicit attention should include having students think about and discuss how a given solution differs from others' solutions, make connections between the methods by indicating points of agreement and disagreement, and explain why they selected their particular methods.
Making Real-World Contexts Accessible Considering the constraints that real situations involve and connecting these situations with issues and topics in their own lives helps students view mathematics as something that will help them interpret their world. The focus of class discussion should aim at supporting students' understanding of the culturally relevant suppositions intrinsic to a problem context and the development of imagery of key mathematical relationships described in a task.
Access to Technology The use of graphing calculators and computer software is beneficial for many special needs students. Using technology allows students to make more extensive use of multiple representations without needing to construct them by hand. It may be the case that some students do not have access to CPMP-Tools software or to handheld technology for homework tasks. Approaches taken by schools using the CPMP program to reduce this inequity include providing multiple locations for students to complete homework during the school day. CPMP-Tools should be available in classrooms used by special education teachers or other professionals who are assisting students with homework. For homes that do not have Internet access, but have computers, schools provide the software on a USB drive for students to download to their home computer. Technology Tips are available in the Unit Resource Masters to assist students who have difficulty retaining methods.
Core-Plus Mathematics offers many
opportunities for teachers to incorporate these practices into daily
routines. One such built-in opportunity is the Think About This Situations
(TATS) used to introduce lessons through discussions. Although no TATS
questions are in the student text for individual investigations there
are often suggestions in the Teacher's Guide for class launches
of investigations. Since much of the mathematical content is based on
real contexts, it is important that all students understand the contexts
and draw on their own or a classmates background knowledge. Opportunities
for students to explain and justify their thinking are built into all
curriculum features. Look for opportunities to encourage the habit of
mind of justifying their thinking, individually and in small group or
In addition, in the Teacher's Guide periodically,
notes provide specific ideas for differentiation. Look for DIFFERENTIATION margin
notes and student masters.
It is important to recognize that implementing
the Core-Plus Mathematics curriculum with classes consisting only of
students previously unsuccessful in mathematics will create additional
implementation challenges. These classes should not be expected to complete
all units from a CPMP course.
Some schools provide special-needs students
with a second hour of class devoted to support. This class typically
follows the regular mathematics class. During this class time, students
are assisted with their homework and sometimes pre-read material for
the next day's class period. This support for special-needs students
increases the access to the mathematics content during the regular class
As one special education teacher indicated:
"I have been teaching the CPMP in a resource setting for the past
2 years and have been teaching mathematics in the resource setting
for over 8 years. I have to admit that initially I was adamant
that my students (the majority have specific learning disabilities
either with respect to math or reading) would be incapable of using
this program. After 2 years, I have found that the opposite is true.
My students are more engaged and achieving
higher-level concepts because of CPMP. We do move at a slower pace,
but they are learning the concepts of Algebra and Geometry thanks to
the contextual component and the guided discovery approach. The program
also provides ample opportunities to differentiate instruction. I am
fortunate to work in a district where the staff received excellent
training and we have access to technology. Overall, the students are
doing well. I am looking forward to the training for Course 3.
Mathematics is definitively a program that works well for students
with learning disabilities."
Other Practices that Promote Equity
Some instructional strategies that are helpful for making mathematics more accessible for a diverse population of students, including students with special needs and English language learners, are listed below:
Use engaging and meaningful contexts.
Use multiple representations.
Sequence instruction to move from concrete to representational to abstract (from specific to the general).
Provide examples and nonexamples.
Offer templates and graphic organizers.
Use cooperative group work.
Teach metacognitive and problem-solving strategies.
Provide opportunities for students to build on their prior knowledge and experiences.
Immerse students in the language of mathematics.
Provide opportunities for guided and independent practice.
Use frequent assessments.
Provide timely and constructive feedback.
Have students create their own resources.
Use organizational systems for notebooks/binders.
Reduce amount of copying for students.
Adjust time for tasks and pacing.
Adjust amount of work.
Help students to become independent learners.
Core-Plus Mathematics field-test materials were reviewed by the Educational Development Center, Inc. (EDC) through an accessibility lens in order to identify strengths and potential barriers for students with special needs. EDC found that many of the above strategies were already an integral part of the materials. In particular, the following features of Core-Plus Mathematics improve access for all students.