1 Preamble for Instructors and Self-Learners

This is a course intended as an introduction to numbers and arithmetic. It attempts to make the concepts understandable to persons of young age and/or little background in numeracy. The course was motivated partly by our work with our grandson and partly by the appalling sparseness of the elementary school mathematics curriculum in Ontario.

1.1 Course Organization

The course is organized into major topics with subsections identified by a scheme which numbers the topics in order of appearance in the Table of Contents. Subsections are numbered in sequence by placing a period after the section number followed by the subsection number. This process can be followed to create as deep a division in a topic as desirable. All such (sub)sections have an active link anchored to them making navigation easy.

The major topics are ordered in a way that each builds on what has gone before. Subsections are used to contain material that develops some aspect of the parent topic.

We try and give simple examples and use familiar concepts when developing a topic. For example, pies are very good for discussing fractions, rulers for the number line, and thermometers for introducing negative numbers.

Generally we do not provide work exercises. The instructor should be able to develop simple but comprehensive examples for each topic. The material found at the Kahn Academy (see 1.4 Other Resources below) is limited but pertinent.

1.2 Writing Conventions

We have tried to provide explanations of concepts in the simplest terms possible for students. We use everyday objects where appropriate and in a manner consistent with their everyday use.

When we introduce an important word, we mark its first occurrence by using emphasis and italics. In addition, we add it to a list of new words at the end of the section. We develop a vocabulary that is consistent with mathematics in general.

Some words and discussions go beyond what a beginning student needs to know or remember. We use them because we feel that it is important that the student be exposed to them. These we mark in red text. Some students will appreciate the exposure. Other students may return at a later date to review the text in red. Other students simply won’t care and their progress will not be impeded by ignoring the marked text.

The wide range of symbols used in mathematics cannot be produced directly from the computer keyboard. Special software and symbol look-up tables have to be used. Even in basic arithmetic we run into this problem. For example, for multiplication, the common representation is with an “x” as in “2 x 3”. In other branches of mathematics such as algebra, we wish to use the “x” for other purposes.

More formal math writing (including the Kahn Academy) uses the “dot” symbol, “•” which is not to be confused with the period “.”. Because we had to use a symbol table to get this symbol, a laborious process, we wish to have a symbol on the standard keyboard instead. Such a symbol is the asterisk,  “*”. This is used by computer programmers and student who advances in this area will appreciate the early exposure to its use. We will use it throughout.

The same problem is even more complex for division. The common symbol is “÷” and the structure for long division  and the division property of fractions simply cannot be generated except by very special software. We will use the next best thing and that is the forward slash or “/”. Hence, “1/2” represents the fraction one half or one divided by two.

We have used a formal vocabulary in this course. The instructor may choose to simplify the language to some extent, but we would urge the preservation of key words (emphasized and italicized). In a sense, the student is learning to speak a new language – the language of mathematics – with its own symbols and structures.

1.3 Notes on Instruction

As the instructor you are encouraged to be creative with, and attentive to the student. Some ides (in no particular order):

  • Create written problems. If the student can read, assign them and evaluate the results. If the student can’t read, help her to whatever extent she requires.
  • Use common everyday objects, especially any that you know the student is strongly interested in. Perhaps you will use cars or lipsticks as numerical quantifiers.
  • Use simple numbers and expressions such as “1 + 2” or “3 – 2” and gradually increase complexity at the students pace. This is our approach and seeing how we do it may give you ideas.
  • Create games as part of the learning process. I ask you to name the biggest number you can think of. Then I give you one bigger. You in turn must give me one still bigger. If the game gets silly – into zillions, say – then you are both having fun. As a side note, there is no reason why a number of some size can’t be called a “zillion”,
  • Be sensitive to the state of the student. If he is over tired or his brain is fried after a hard day at school, or she is upset over a schoolyard incident, you won’t get any new learning from the student. Simply talk or go for an ice-cream cone or whatever the student will enjoy. There’s always tomorrow.
  • You can learn too. Even though my background includes a number of math courses both at the undergraduate and graduate levels, my understanding of numbers was sharpened by creating this course. Math has an inherent simplicity and beauty. May you discover some of it.
  • Be silly at times and above all, have fun!

1.4 Other Resources

An excellent online resource with a graphical interface is that of the Kahn Academy. It is widely used as a free (donations accepted) source of video instruction on a wide variety of subjects. We have consulted much of their material on arithmetic, numbers and other areas of elementary mathematics while preparing this course.

1.5 Key Points

Every major section in this course will conclude with a summary subsection of key points such as this one. The intent is to provide the student with the key points extracted for quick review.

In the list below, any text in red is for informational purposes only and the student need not remember it for this course.

1.5.1 New Words
  • arithmetic: the field of mathematics that deals with numbers and the operations that can performed on them such as addition (to be introduced in section );
  • number: a symbolic representation of a quantity of something;
  • numeracy: the ability to understand and work with numbers.
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