# 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.