Understanding Primary Math – For Effective Teaching

The full title of the book is “Understanding Primary Math - for Effective Teaching”.

This section is the introduction. The latter chapters of this book form the content sections of this book.

Its objective is to help teachers understand the basic concepts & concepts underlying procedures in primary (K-8) math so that they can, in turn, help students understand these concepts better and enjoy learning math.

Two Levels of at which Math can be Understood

We saw briefly that Primary math curriculum can be understood at two levels, both by teachers & students.

First is the level which can be called doing math or the instrumental level.

At this level several procedures and algorithms for performing calculations and symbol manipulations are learnt and remembered. Examples are addition or multiplication of multi-digit numbers or fractions. At this level many concepts are turned into ‘rules of operation” and understood only at a shallow level. An example is that of dividing with fractions where most children are able say “invert and multiply”. This level may also be called as “how to do math” level.

Math textbooks deal with the math curriculum at this level. Most schools teach math at this level.

At the second or next higher level, which can be called relational level, many of these concepts and their relation to other concepts would be understood. The logic behind procedures also would be understood.

Some examples are the role of the place value system, its relation to the algorithm for multi-digit multiplication and the logic of the procedures for multiplying or dividing fractions. In the example quoted in the above paragraph, students need to understand why “inverting and multiplying is equivalent to dividing”.

If students understand math at this level, they will find the increasing abstractness of math concepts easier to understand. They can understand the logical flow behind many of the algorithms, rather than memorising them. Students who understand at this level will also start enjoying math and find it easy.

Teachers also need to understand math at this level so that many difficult concepts can be demonstrated to students in easy-to-understand ways.

What is Understanding in Math

We also give below an extract from the opening paragraph to Teaching Student-Centered Mathematics by John Van de Walle, LouAnn Lovin, Karen Karp, and Jennifer Bay-Williams, which beautifully summarises what is understanding in math.

“What is understanding? Understanding is being able to think and act flexibly with a topic or concept. It goes beyond knowing; it is more than a collection of information, facts, or data. It is more than being able to follow steps in a procedure. One hallmark of mathematical understanding is a student’s ability to justify why a given mathematical claim or answer is true or why a mathematical rule makes sense (Council of Chief State School Officers, 2010). Although children might know their basic multiplication facts and be able to give you quick answers to questions about these facts, they might not understand multiplication. They might not be able to justify how they know an answer is correct or provide an example of when it would make sense to use this basic fact. These tasks go beyond simply knowing mathematical facts and procedures. Understanding must be a primary goal for all of the mathematics you teach.”

About This Book

This book will deal with transacting the primary math curriculum at this second level. A Math class which focusses only on definitions and computations is like an English class focusing only on grammar and spelling, leaving out literature, creativity and poetry.

The book is divided into 52 sections and a Bibliography. First section covers the introduction & the background. The next 28 sections cover the K-8 math curriculum. There are additional sections dealing with "Logic & Math"; and about issues in the Primary School Curriculum itself. Two sections are in the nature of "enrichment topics", which, though are not part of the formal curriculum, are easily accessible to primary school students. The next chapter 2.8 gives a graphic framework of the topics & sections.

Section 37 & 38 would be an elaboration of the theme of this chapter & other aspects of the discipline of math. This would be for teachers to get a deeper understanding of math, including primary school math.

I have read several books which broadened my understanding of primary math concepts and their role in the development of higher-level concepts. I have also drawn from memory of my experiences in the schools and training workshops. The idea has been to present the material in as simple a language as possible. I had not kept any notes on the source of most of these ideas. Hence I have not been able to give specific references to other sources. In the last Bibliography section, I have just included a list of books, that I read or mostly browsed through. Teachers will certainly benefit by reading any of them.

Aspects of Mathematical Thinking

Through the content of the math topics, we will be constantly touching upon the following aspects of mathematical thinking.

Representation of math concepts, connecting concepts in various math topics and focusing on reasoning & proving using the logic inherent ion math ideas. We will also touch upon the aspect of the use of correct vocabulary and syntax to communicate math ideas. The fifth aspect of problem-solving is beyond the scope of this book.

The concept of mathematical thinking itself would be dealt with in detail in one of the sections.

Explaining Activities

Explaining the various activities has been a difficult part of writing these chapters. I have tried to keep the explanations readable and tried to avoid dwelling too much on the theory. Hence I would not be dealing with definitions & procedures already explained in standard textbooks. I would also not be dealing much with actual math procedures in the classroom, except to explain concepts which explain why particular procedures work.

Keeping in mind the complexity of the concepts, the fact that teachers may be reading about them for the first time and the difficulty of explaining them through language, I have tried to ensure that each chapter does not exceed 2 pages. Wherever necessary, I have provided additional information in separate annexures and diagrams where necessary.

Many math concepts & processes which arise from different life situations have no universally accepted names. I have used terms which I feel reflect the concepts closely and are also easy for teachers to understand. It is possible that some math educators may not agree with the terms but I am sure they will agree with the intent behind my explanations.

Another of my target groups is parents. Most parents remember the "negative feelings"& they used to have with Math, when they were in school and readily pass on the same to their children. In fact Math is the only subject which adults are almost "proud" to admit that they were weak in. When schools try out different approaches, parents usually confuse children by explaining the procedures to their children in the same way they (parents) had learnt it. I hope these articles would give parents a clearer understanding so that they would be able to support the efforts of the school & their children.

Feedback from readers will be most welcome. It would help me make this book, easier too understand and use with children.