|
|
 |
|
Foundations of Mathematics:
An Overview
The
foundational topics of mathematics—whole numbers, decimals,
fractions, ratios, proportions, percents, signed numbers,
exponents, and order of operations—are presented in six modules
of learning in Foundations of Mathematics. Though the
scope of this book does not differ from that of traditional
mathematics textbooks, both the goals and core philosophy of the
book do. While it is critical that students learn the content of
the course and how to perform calculations, it is equally
important that students learn how to learn and how to improve
their ability to learn as well as their performance. The
innovative and holistic nature of this philosophy entails that
the presentational style of Foundations of Math be
equally innovative. The following presents the parts of a sample
section, in the order and manner they are presented to students.
The book is designed to give
students not only the "hows" but also an understanding of the
"whys" of the basic math processes they may be reviewing and
those they are learning for the first time. The first
information the student encounters in each activity is a
real-life application of the skills and techniques covered by
that activity.
This portion of the activity clearly states the target
proficiencies that the student should gain over the course
of the activity.
Each module provides a terminology checkpoint, listing
previously used terms relevant to the new topic as well as the
new terms to be introduced, giving students a "heads up" on the
vocabulary they should have already learned before going on.
A Building Mathematical Language
section focuses on the specific and well-defined vocabulary of
mathematics as well as commonly used notation and translation.
This section does more than teach students to "speak math"; it
teaches them how to think through and talk about math.
|
Math processes that require more than
three steps are presented in table form as
Methodologies, which break the processes down
to their fundamental steps.
Techniques are presented for one or two-step processes,
translations, and conversions.
| |
|
Alongside a worked out example, students
are prompted to try a
second example
in an adjacent blank workspace. |
|
|
| |
|
Checking the
final answer (validation) and linking it back to the
original problem is demonstrated for every computation in
Modules 1 through 3, giving students confidence in the
accuracy of their answers. Once the students have taken on
the habit of validation for the basic whole number, decimal,
and fraction computational processes, topic-specific
validation techniques are offered in the remaining chapters.
|
 |
|
Models
for the special cases and shortcuts are presented in
step-by-step format. As students work through the models,
they are cued with the special case and shortcut
instructions. In chapters 4 through 6, specifically chosen
models also serve as opportunities for review by
incorporating computations with decimals and fractions. |
 |
When
appropriate for the section, estimation techniques
are presented with examples as additional (optional) tools
the student can use to verify the answer to a problem. |
 Gleaned
from the observations of Basic Mathematics instructors as to
what mathematical errors students are most likely to make, a
table-format section called Addressing Common Errors
lists and gives the student an example of each of those
problematic areas. Most importantly, however, it provides
resolution for each common pitfall—how to avoid this—and the
correct process for each example.
The Pre-Activity section ends with a Preparation Inventory
which provides students with an actual checklist of critical
skills and understandings they need to have before proceeding to
the Activity section.
In this section, students are given concrete guidelines as to
the criteria against which their performance is measured,
as well as guidelines for presentation of their responses to
questions and exercises.
This section presents a set of Critical Thinking Questions,
encouraging the student to articulate in writing what he or she
has learned thus far. The first few questions are direct,
followed by those convergent questions that require the students
to tie in previously learned knowledge with the topic at hand.
Finally, a question is posed to challenge the student to explore
the topic further. The convergent Critical Thinking Questions
are especially suitable for small group discussion, and may be
worth revisiting once the students have demonstrated that they
have mastered the processes.
This section provides important suggestions for successfully
performing the skills and solving the problems posed in the
Activity. Suggestions include the use of a horizontal bar in
fractions, as opposed to a diagonal bar, which can easily
interfere with correct alignment in the problem, careful use of
the original problem rather than an intermediate step when
validating a final answer, and so forth.
A carefully chosen set of exercises, including a few basic
problems, special cases, and those that might lead to the common
errors previously addressed are in a Demonstrate Your
Understanding section. Ample workspace is provided in the
book for both problem-solving and validation.
When an answer does not validate, many students first instinct
is to erase the entire problem and begin again. An Identify
and
Correct section presents actual student-written work for the
students to first validate for accuracy, and then analyze to
detect the error(s) made.
|
 Some
Activities also include Mental Math and Team
Exercises sections. In Mental Math, students are
challenged to apply what they have learned of the
problem-solving process to solve problems mentally, rather
than working them out in writing. The Team Exercises offer
students an opportunity to work in teams or groups to think
through meta-mathematical issues raised over the course of
the activity.
|
 |
At the end of each section there are Additional
Exercises to allow students the opportunity to
demonstrate their comprehension of the activity content.
|
 |
At the end of each chapter there are Application
Problems to allow students a further opportunity to
apply what they have learned over the course of the chapter. |
Foundations of
Math also provides additional student support tools
at the back of the book:
|
Reading Logs
Practice Tests
Self Assessment Tools |
Midterm Assessment Tool
Student Survey
Answer Keys |
|
|
|
|
