In defense of Algebra

Recently I have started reading the book Unknown Quantity: A real and imagined history of algebra by John Derbyshire. For any history (or mathematics) students, the book is a fascinating read, detailing the development of the number system, algebraic notation and use from over four thousand years ago to as we know it today.

From time to time I have shared with my students snippets of what I have read. For example, the ancient Babylonians didn’t have a conceptual understanding of the numbers 1 to 10. Instead they worked with a base 60 notation, which explains why this same notation appears in various instances of our numerate world today. For my students, who were studying trigonometry and the breakdown of degrees into minutes and seconds, this explanation proved helpful.

Derbyshire’s book has highlighted for me how complex the development of algebraic thinking, language and writing has been. Mathematicians struggled over the course of generations trying to figure out how to represent the unknown number (i.e. a pronumeral or variable, often written as ‘x’). Each step of the way, they didn’t know how their findings would be of use to secondary students today, nor did they often know what had immediately preceded them or was being simultaneously discovered somewhere else in the world.

The study of Algebra today gives many students grief. They struggle with its abstractness and the ideas of simplifying an expression or working with a balanced equation.

Alongside these students, however, there are others who enjoy it and who get a sense of achievement when they find that they do understand this new language they are learning.

In response to recent debate over the worthiness of algebra in the classroom, why limit what our students are learning because it is difficult for some? Learning is not intended to be easy. It is intended to set students up to navigate and make sense of our world. And who knows, maybe there is another Diophantus or al-Khwarizmi growing up today, who will in the years to come make some amazing leap forward in mathematical thinking.

A great contribution to this debate can be found here in an article by Judy Bolton-Fasman.

Stepping through the looking glass

It is very tempting to sit here this evening and write a post about a terrible class I had today, where particular students hijacked the lesson, ruining it for those who came to class wanting to learn. BUT this will only push me further into ruminations about the not-so-enjoyable 48 minutes of my day, when instead I could be focusing on another 48 minutes that went breathtakingly well.

Last week I began teaching my Year 7 class algebra. This is a topic that I have been eagerly awaiting, and have told the students that, almost as a matter of principle, they will enjoy it as much as I do.  Following the advice of my university tutor, the sequence of learning activities is taking place as follows:

  1. Number patterns: introducing the concept of equations e.g.
    Triangles Matches
    1 3
    2 5
    etc
    M
    [find no. of matches i.e. rule]
    [find no. of triangles i.e. rule]
    N
  2. Number machines: spitting out numbers e.g. 3 is input, then x2, output is 6
  3. Number sentences: e.g. think of a number (3), add 2 (5), multiply by 8 (40).  Put this in equation form
  4. Flow-charting/tracking
    S  x3   F                                            a  +2      x4  b
    [] –> []     i.e. F = 3 x S                 [] –> []  –> []           i.e. b = (a+2) x 4
  5. Backtracking: looking at inverse operations
    +2              x4
    [3] –> [5] –> [20]
    <–             <–
    -2               /4

Today’s session was somewhere in between 2 and 3: the introduction of the pronumeral.  Now this might not seem like such a big deal, but up until now everything we have been looking at has involved concrete numbers, such as multiplying decimals, finding averages and reading column graphs about the number of M&Ms in a packet.

So today, we took a leap into the unknown, abstract world of algebraic expressions. Big thanks must go to a former teacher of mine, Tal Greengard, who has helped to make the divide between the concrete and the abstract somewhat less formidable through cups and counters. By modelling physical but unknown amounts of counters in a cup, students were quickly able to grasp the concept of a letter in algebra representing an unknown number.

With a group of students whose ability levels range from Year 3 to Year 9, I had a deeply held concern that those students perhaps classed at the Piagetian developmental stage of concrete operations would not be able to cope.  Interestingly, these students have so far proved my concerns unfounded.