This is the INTRODUCTION to the book:
This book is based on a blog, Algebradabra! The blog’s aim was to provide tasks that would help
lower secondary school students get a better feel for algebra - and to give
teachers a fresh perspective on familiar areas of the school algebra
curriculum.
The format of the blog was to post a
new task at the beginning of each week, followed by a variant of the task on
each of the next four weekdays, together with comments and guidance. This was a
device aimed at maintaining readers’ interest in the blog which also imposed a
discipline to keep writing on the author - me! The result was 20 sets of tasks,
making 100 tasks in all, which were released one at a time on the blog between
February and June 2018.
Each of these 100 tasks appears in
this book, and we have adopted the same format as the blog, of having a
‘weekly’ set of 5 ‘daily’ tasks. However, this format should not be interpreted
too literally. Not every class will have a maths lesson every day, and even if
it did, it is unlikely that a teacher would want to devote at least part of
each maths lesson to algebra. And many of the tasks in this book could easily
be developed into a whole lesson, and many of the sets of tasks into a sequence
of lessons. It is therefore up to you to determine which tasks you wish to use
in the classroom (and adapt to your liking) and how much lesson time to devote
to any one of them.
However, whether or not you
eventually use a task or set of tasks in the classroom, we hope you will work
through them yourself and that some at least will give you the pleasure and the
satisfaction that can come from being challenged - and of gaining insights and
a wider understanding of the nature and demands of school algebra.
Some of the ideas that have gone
into this book can be traced back to my experience on the CSMS project in the
1970s, where we looked at students’ responses to a range of carefully designed
written tests. The Algebra test looked at the uses of letters in generalised
arithmetic, which we placed into these six categories: Letter evaluated, Letter
ignored, Letter as object, Letter as specific unknown, as generalised number,
as variable.
These categories have proved useful
for devising the current tasks. I have also hugely benefited from my work since
2008 on the ICCAMS project directed by Jeremy Hodgen, though I have tried hard
not to duplicate activities that I helped develop there.
The current tasks are wide-ranging,
but there is perhaps a particular emphasis on the notion of variable (as
opposed to specific unknown) and on generalising. This should
come as no surprise in an algebra book, though these ideas are often poorly
represented in school textbooks. Thus many of the tasks involve dynamic
contexts and make use of graphs or focus on general structure by, for example,
analysing generic patterns. I have also tried to devise tasks that can help
students see the purpose and utility of algebra (to use a phrase of
Janet Ainley and Dave Pratt), though this can be notoriously difficult to
achieve! However, I make no claims that the content covered by the book is
comprehensive. [If you sense any glaring ommissions, please let me know - if we
can get to 20, there might be scope for another book!]
The tasks vary in cognitive demand,
though most should be accessible (or could be made accessible) to students at
Key Stages 3 and 4. Most, it is hoped, will also engage the teacher.
Note 1: Each of the tasks in this book is also available as
a slide in a pdf file (so that it can be displayed on a screen), along with a
small number of QuickTime movies that have been created for some of the tasks.
Note 2: Many tasks are in several parts which, because of
the nature of the original blog, appear on the same slide. On the principle
that less is more (or more haste, less speed), we recommend that
where practicable, these parts are displayed in the classroom one at a time, so
that sufficient time is devoted to each and to avoid students wanting to rush
through the parts in order to ‘complete’ the task.
Dietmar Küchemann
