Instructional Design

The manner in which students encounter mathematical ideas can contribute significantly to the quality of their learning and the depth of their understanding. Core-Plus Mathematics units are designed around multi-day lessons centered on big ideas. Lessons are organized around cycles of instructional activities intended primarily for small-group work in the classroom and for individual work outside of the classroom.

In-Class Activities

The four-phase cycle of classroom activities,

is designed to actively engage students in investigating and making sense of problem situations, in constructing important mathematical concepts and methods, in generalizing and proving mathematical relationships, and in communicating, both orally and in writing, their thinking and the results of their efforts.

Most classroom activities are designed to be completed by students working collaboratively in groups of two to four students punctuated by whole-class discussions (that retain the cognitive demand of problems) once students have had some time to discuss ideas and have partial understanding of the mathematical ideas being developed.

  Students in a CPMP classroom

Short clips of classrooms are available.

LAUNCH class discussion

Think About This Situation
The lesson launch promotes a teacher-led discussion of a problem situation and of related questions to think about. This discussion sets the context for the student work to follow and helps to generate student interest; it also provides an opportunity for the teacher to assess student knowledge, to discuss cultural and mathematical context suppositions, and to clarify directions for the investigation to follow.

Sample launch

EXPLORE group investigation

Classroom activity then shifts to investigating focused problems and questions related to the launching situation by gathering data, looking for patterns, constructing models and meanings, and making and verifying conjectures. As students collaborate in pairs or small groups, the teacher circulates among students providing guidance and support, clarifying or asking questions, giving hints, providing encouragement, and drawing group members into the discussion to help groups collaborate more effectively. Periodically, it may be valuable to facilitate whole-class discussions, particularly at the close of the class period. The investigations and related questions posed by students drive the learning.

Sample exploration
Resource for developing collaboration skills—Dysfunctional Group Skit

SHARE AND SUMMARIZE class discussion

Summarize the Mathematics
This investigative work is followed by a teacher-led class discussion (referred to as Summarize the Mathematics) in which students summarize and explain the reasoning supporting mathematical ideas developed in their groups, providing an opportunity to construct a shared understanding of important concepts, methods, and approaches. This discussion leads to a class summary of important ideas or to further exploration of a topic if competing perspectives remain. Varying points of view and differing conclusions that can be justified should be encouraged. This discussion based on student thinking is crucial to building understanding of mathematical ideas for the procedural skill development to follow.

Sample Summarize the Mathematics

SELF-ASSESSMENT individual tasks

Check Your Understanding
Students are given an individual task to complete and reinforce their understanding of concepts and methods. Teachers should encourage students to develop the ability to determine for themselves whether their understanding is enough to secure progress to the tasks found in the On Your Own homework tasks.

Sample Check Your Understanding

On Your Own Homework Sets
In addition to the classroom investigations, Core-Plus Mathematics provides sets of On Your Own tasks, which are designed to engage students in applying, connecting, reflecting on, extending, and reviewing their evolving mathematical knowledge. On Your Own tasks are provided for each lesson in the materials and are central to the learning goals of each lesson. These tasks are intended primarily for individual work outside of class. Selection of homework tasks should be based on student performance and the availability of time and technology. Also, students should exercise some choice of tasks to pursue, and, at times should be given the opportunity to pose their own problems and questions to investigate. The chart below describes the types of tasks in a typical On Your Own set.

On Your Own: Homework Tasks
Applications These tasks provide opportunities for students to use and strengthen their understanding of the ideas they have learned in the lesson.
Connections These tasks help students to build links between mathematical topics they have studied in the lesson and to connect those topics with other mathematics that they know.
Reflections These tasks provide opportunities for students to re-examine their thinking about ideas in the lesson.
Extensions These tasks provide opportunities for students to explore further or more deeply the mathematics they are learning.
Review These tasks provide opportunities for just-in-time review and distributed practice of mathematical skills to maintain procedural fluency.

These tasks in the OYO are intended primarily for individual work outside of class. If a few students are identified as needing additional assistance with specific skills, they should be given additional assistance outside of class. Selection of homework tasks should be based on student performance, the availability of time, and technology access. Also, students should exercise some choice of tasks to pursue, and at times should be given the opportunity to pose their own problems and questions to investigate.

Practicing for Standardized Tests
Opportunities for additional review and practice are provided in the Practicing for Standardized Tests masters in each Unit Resources booklet. Each Practicing for Standardized Tests master presents 10 questions that draw on all content strands. The questions are presented in the form of test items similar to how they often appear in standardized tests such as state assessment tests, the Preliminary Scholastic Aptitude Test (PSAT), SAT, ACT PLAN, or ACT. We suggest using these practice sets following the unit assessment so students can become familiar with the formats of standardized tests and develop effective test-taking strategies for performing well on such tests.

Preparing for Undergraduate Mathematics Placement Tests
Most colleges and universities administer a mathematics placement test to incoming students. The results of this test together with grades in high school mathematics courses and intended undergraduate major are used to advise students regarding the first college mathematics course they study. About half of these post-secondary institutions allow students the use of technology (e.g., a graphing calculator) while completing the test. But you may wish to ensure that your students are prepared to take placement tests with and without technology.
     To perform well on these tests, there are a number of skills and reasoning techniques that need to be automatic. Developing that level of proficiency requires practice. Thus, you will find two Preparing for Undergraduate Mathematics Placement (PUMP) sets in the unit resources for each of the Course 4 units.

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