17 February 2018

Elementary Algebra for Schools

I've really enjoyed looking through a fully digitised version of Elementary Algebra for Schools, a maths textbook which was first published 1885. I wish I had more time to do so. Unlike A Classbook of Algebra, a set of algebra exercises that I blogged about a couple of months ago, Elementary Algebra for Schools is a proper textbook featuring explanations, definitions and worked examples (plus over 3,500 questions for students to practise). I read the explanations with interest - it's fascinating to see some of the things that changed in the teaching of algebra over the course of the 20th century. For example algebraic division was previously taught very early on and is now not taught until A level.
In their introduction, the authors tell us that "The examples are very numerous, and have been compiled with great care". Pause for a minute and think about whether we can say the same about modern textbooks, in a world of ever changing curricula and pedagogical fads. Things aren't built to last anymore, including textbooks.

It's interesting to see the order in which topics are tackled. It is clear that the authors put a lot of thought into this. For example, in the preface they explain that the skill of factorising is purposely left until after "the student has acquired some freedom and readiness in the use of symbols". They go on to explain that leaving factorisation until students have developed algebraic fluency allows them to deal with it in greater depth, which is of course preferable to factors being "introduced and disposed of in one short early chapter".

The book also covers common misconceptions, which are rarely mentioned in modern textbooks. The authors say, "under the belief that prevalent mistakes are not sufficiently guarded against, we have given occasional notes to caution the reader against the blunders which experience shews to be almost universal amongst beginners". 

There is a lot I want to share from Elementary Algebra for Schools - in this post I'll feature a few things that may be of interest from the first four chapters. You can read the whole textbook yourself here, if you wish to do so.

Chapter I - Definitions. Substitutions.
We start with a set of very clear definitions and examples (extract below).
We're told that "the beginner must be careful to distinguish between coefficient and index" (eg know the difference between 2a and a2). It looks like a x a = 2a was just as a common a misconception 130 years ago as it is now. The first exercise in the book is specifically designed so that students can develop fluency in making the distinction between a coefficient and an index. This is done through substitution.
Modern textbooks have comparable explanations and exercises, though there's considerably less rigour.

To give a direct comparison, this is what the Victorian textbook says about writing the factors of a product in alphabetical order, compared to a modern day GCSE textbook.
Extract from 'Elementary Algebra for Beginners', 1885

Extract from 'Edexcel GCSE Maths Higher', OUP, 2015

I suppose we could argue that as long as teachers' verbal explanations are sufficiently clear and detailed, then it doesn't matter what the textbook says. These days, textbooks and worksheets are used primarily for practice rather explanations. So I suppose the brevity in textbooks explanations is acceptable, as long as the teaching itself is thorough and the exercises are well written.

In Elementary Algebra for Schools, it's interesting to see that guidance is given regarding the setting out of working. I expect we all wholeheartedly agree with the following points:
Chapter II - Negative Quantities. Addition of Like Terms.
The first chapter covered algebraic definitions and notation, and these fundamentals were practised through substitution exercises. The second chapter covers collecting like terms, focusing on how to deal with positive and negative quantities. Modern day teaching of algebra often takes these two concepts in reverse order, starting with simplification before later tackling substitution.
The first exercise on collecting like terms has a higher level of challenge and granularity than equivalent modern day exercises on the same topic. Here we have thirty questions where students are not required to identify whether the terms are 'like' or not - they are only required to decide whether to add or subtract each term.
In a GCSE textbook, we only have ten questions to practise this particular skill, before students move onto the next skill where like and unlike terms are mixed.

Chapter III - Simple Brackets. Addition.
In the third chapter of Elementary Algebra for Schools, we move onto simplifying expressions with a mixture of like and unlike terms. It is interesting that like terms are collected using a column addition method.
Note the explicit mention of descending powers - this convention regarding the ordering of terms is something I don't normally mention until I teach Binomial Expansions in Year 12.

Chapter IV - Subtraction.
There is an entire chapter on subtracting one expression from another. In my experience students these days don't spend any time on this skill at all, hence they often stumble in Year 12 when they are required to simplify 2x2 + 5x + 8 - (x2 + 6x - 7) for example, to find the area between two curves.
I wonder when, and why, questions of this form disappeared from most classrooms.

So that's the first four chapters: definitions, substitution, and collecting like terms. Forming algebraic expressions doesn't come until Chapter 9, but often comes earlier in modern day algebra teaching. The order in which skills are taught is really interesting, as is the isolation of particular skills. I don't know what the 'best' order is to introduce algebra, but it's certainly something I want to research further.

You may have noticed that I'm very geekily addicted to this old textbook thing now! If you're interested in this stuff too, look out for another post about Elementary Algebra for Schools in the near future.