Primary Mathematics/General Overview of the Teaching of Mathematics
The face of the teaching of mathematics is in constant flux. This course, therefore, is an attempt to stay in step with the current pedagogy(s) used in the teaching of mathematics to the primary grades. While educators should find this book helpful in learning or reacquainting themselves with some of the methodologies currently used, this course attempts to use as much lay language as possible to also be helpful to parents who, in looking over the shoulders of their children, are sometimes baffled by the "New Math". Those who perhaps might find the most value from this course would be those who wish to home-school their child(ren). It is hoped that teachers of mathematics will continually make updates to this course as these changes inevitably occur.
What This Book is Not
This book assumes that the reader has reached some level of competency with the topics it covers. As such, this book is not designed to teach math to the reader. Rather, it is meant to give the reader insight into how and why mathematical core understandings and skills are taught the way they are. Activities will be suggested that will present teachers and prospective home-schoolers with ideas for their own course content. A basic mathematics background should be all that is needed to find value in this course. Often times parents (and even teachers) only understand a topic in the way they were taught it, but modern math curriculums are often constructed in such a way that topics are covered in many different ways in order to accommodate the different learning styles of students. By way of example, this course will not teach the reader in step by step fashion how to multiply two large numbers, but the various building blocks and algorithms that students are introduced to in the process of learning this skill will be explained.
One of the overarching ideas that is highly valued in the teaching of mathematics in the primary grades involves the connections students make in their learning. All of our mathematical understandings are intertwined. For this reason, different skills such as multiplication and understandings like those learned from the study of geometry are not taught separately. Rather, they are taught together, in such a way as to reinforce each other. Students should be encouraged to make connections by looking for patterns, exploring extremes, and forming and testing conjectures. Teachers and parents should refrain from simply "telling them how to do it." Letting student "guess at" and then check their work, learning from their mistakes as they go, leads to much stronger mathematical understandings. It also leads them to form mathematical habits that make future mathematically based explorations more efficient.
Primary and even upper school teachers recognize that understanding mathematics in the abstract is not the goal of most students. They need to see connections to the real world that inspire their learning. For this reason, teachers prefer to use real world problems that require the need for mathematical models (see below). Teachers prefer not to "teach". They prefer to "guide". Modern Educators realize that students gain true "ownership" of their understandings through, inasmuch as it is possible, making connections on their own - by way of their own work and explorations. Internal links in this course, when found in this course will purposefully be placed there to emphasize what should be an ever-present concept of the connectedness of mathematical understandings.
Manipulatives and Models
One very important component of the contemporary methods used to teach math is the use of manipulatives (such as toys) and models (visual representations) to give added dimension to students' understandings. In each chapter of this course, the reader will find various examples of models used to teach different mathematical understandings. Often, they will find that these models serve to make connections to material covered in other sections. Keep in mind that teachers in the classroom tend to be very creative and resourceful. Often, the models found in this course have many possible permutations, and can come in various shapes, sizes, and guizes. They are purposely presented here in simple forms to facilitate for the reader their identification.