Recently I have been teaching linear equation solving with Year 7. We have explored various interpretations; function machines and their inverses, balancing, inverse operations, blank box etc. But one thing I did before any of this was explicitly define the role that x plays.
It is well known that pupils can struggle with algebra when it is introduced immediately as an abstract concept. Much has been made about the use of representations to support the teaching of algebra - indeed I dedicate a good proportion of my upcoming book to exploring different ways of representing algebraic expressions and equations, and how these representations can aid pupils in understanding how algebra is manipulated. However, one thing the representations struggle with is communicating the nature of the letter itself.
I once saw an excellent use of Geogebra to create a dynamic visual representation of completing the square that would allow the value of x to be varied, the squares shrinking and growing really hammered home the idea of x being a variable in the expression. However the static representations that we often use cannot convey the same meaning. So before I started working with them I decided to introduce some of the basic interpretations of letters in mathematics; in particular viewing them as variables, as parameters and unknowns.
To be fair, I didn't stress parameter too much, just a vague definition about them having a particular meaning for a value that doesn't change, like b for the base of a triangle or A for the area of a particular shape. But we did talk a lot about the difference between an unknown and a variable, and then we revisited the ideas as we went through the different lessons. At the beginning of each lesson I would ask the class what role x was playing in an equation, and that meant I went through the entire topic without once being asked what x is - which could be a first for the introductory teaching of algebra!
It seems like it really helped the pupils to understand something about the different roles that the letters can play, and in particular the role that they played in the equations we were working with. I would certainly recommend teachers introducing algebra by giving clarity over the roles that letters play in mathematics so that pupils have a sense of what they are working with before they are asked to manipulate them.
It is well known that pupils can struggle with algebra when it is introduced immediately as an abstract concept. Much has been made about the use of representations to support the teaching of algebra - indeed I dedicate a good proportion of my upcoming book to exploring different ways of representing algebraic expressions and equations, and how these representations can aid pupils in understanding how algebra is manipulated. However, one thing the representations struggle with is communicating the nature of the letter itself.
I once saw an excellent use of Geogebra to create a dynamic visual representation of completing the square that would allow the value of x to be varied, the squares shrinking and growing really hammered home the idea of x being a variable in the expression. However the static representations that we often use cannot convey the same meaning. So before I started working with them I decided to introduce some of the basic interpretations of letters in mathematics; in particular viewing them as variables, as parameters and unknowns.
To be fair, I didn't stress parameter too much, just a vague definition about them having a particular meaning for a value that doesn't change, like b for the base of a triangle or A for the area of a particular shape. But we did talk a lot about the difference between an unknown and a variable, and then we revisited the ideas as we went through the different lessons. At the beginning of each lesson I would ask the class what role x was playing in an equation, and that meant I went through the entire topic without once being asked what x is - which could be a first for the introductory teaching of algebra!
It seems like it really helped the pupils to understand something about the different roles that the letters can play, and in particular the role that they played in the equations we were working with. I would certainly recommend teachers introducing algebra by giving clarity over the roles that letters play in mathematics so that pupils have a sense of what they are working with before they are asked to manipulate them.
