Transition to College Mathematics and Statistics is designed
so that students engage in the modeling process. Students also
engage in the mathematical behaviors identified in the Common Core
State Standards for Mathematics (CCSS) Mathematical Practices and
other mathematical habits of mind as the primary vehicle for learning
the mathematics and statistics elaborated in the CCSS content standards.
Even prior to the era of the Common Core State Standards for Mathematics,
many mathematics educators have encouraged learning mathematics
through modeling of problems. Problems that engage students in
mathematical modeling have the following features and benefits
(Dewey, 1929, 195; Hiebert, Carpenter, Fennema, Fuson, Human, Murray,
Olivier, & Wearne, 1996):
- Problems are identified in context.
- Problems are studied through active engagement
- Conclusions are reached as problems are (at least partially)
- The benefits lie not only in the solutions to the problems,
but the new relationships that are discovered.
Process of Mathematical Modeling: Mathematical modeling
in the CCSS for high school mathematics is a conceptual category
and is also one of eight mathematical practices. The diagram below
describes the modeling process and the mathematical practices engaged
in during each phase of the process. Connecting Mathematical Practices
(MP) and Content Standards (CS) MP1 and MP4 are the overarching
focal points of the entire process.
The manner in which students encounter mathematical ideas can
contribute significantly to the quality of their learning and the
depth of their understanding. Transition to College Mathematics
and Statistics is designed so that students are engaged in
the mathematical behaviors identified in the Common Core State
Standards Mathematical Practices as the primary vehicle for learning
mathematics and statistics. Each unit includes multi-day lessons
centered on big ideas. Each lesson includes two to four focused
mathematical investigations that engage students in a four-phase
cycle of classroom activities—Launch, Explore, Share and
Summarize, and Self-Assessment. This cycle is designed to 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.
See the front matter of the TCMS Teacher's Guide for more
detail on the instructional model designed into the curriculum
materials. Also, see the available video
clips from a TCMS field-test
classroom of students working on a modular arithmetic investigation.
Transition to College Mathematics and Statistics is flexibly designed to meet students' present and future needs. In addition to the core sequence as presented in the text, selected units can be organized into courses that focus on either statistics or discrete mathematics while continuing to strengthen students' quantitative and algebraic fluency essential for college readiness or career apprenticeships. See recommended Statistics and Algebra Pathway and Discrete Mathematics and Algebra Pathway.
The TCMS Teacher's Guide has been developed to support
teachers in facilitating classrooms where students consistently
engage in authentic, cognitively demanding problems. The Teacher's
Guide includes each student page and a facing page containing
problem solutions and helpful teaching notes.
- unit and lesson overviews and objectives;
- unit planning guides with suggested assignments;
- notes related to instruction: possible student approaches,
possible incorrect or partial thinking, mathematical practices,
technology, and differentiation embedded at point of use;
- sample student and teacher discourse scenarios for whole-class
lesson launches and investigation summarizing discussions; and
- links to additional online printable resources such as student
activity and unit summary masters, technology tips, lesson quizzes,
unit tests, take-home assessments, projects, and college-readiness
See the front matter of the TCMS Teacher's Guide for an
explanation of the formative and summative assessment program.
College Readiness Assessment Sets: In addition to continuing
use of important competencies in the student text, you will find
College Readiness Assessment (CRA) sets in the unit resource masters
for each lesson in the course. The exercises will help students
build skills in strategic areas. The exercises are in multiple-choice
format as is commonly found on mathematics placement tests.
The 10 exercises in the CRA sets are drawn from 10 general areas.
The exercises are in the same order in each set. In the Unit
Resource Masters, you will find a record sheet that students
can use to keep track of their progress and performance on the
CRA sets. This record sheet shows the 10 categories. If students
are consistently missing the same numbered items in these sets,
you may wish to suggest extra work on similar exercises.
To help students develop test-taking strategies, discuss how items
are often written in novel ways, but still assess mathematical
understandings they have or should be able to reason about. TCMS
students should develop or refine the habit of reasoning numerically,
graphically, and algebraically. This skill, as well as general
test-taking strategies such as estimating and eliminator some multiple-choice
options, should be periodically discussed. Answers to the CRA sets
for each unit are found at the end of each lesson in the Teacher's
Guide. An online version of the CRA sets is available on ConnectED.
Student Help: Resources have
been developed to assist students with homework. Hints, scaffolding,
selected answers, and occasional full solutions are provided for
Professional Development Providers: For onsite professional
development facilitated by experienced TCMS facilitators, contact
your McGraw-Hill sales representative or any of the facilitators listed
Developing a Collaborative Classroom: To assist students
in learning how to collaborate effectively, you may wish to engage
them in some skits as outlined in Dysfunctional