Teaching and learning with algebra tiles
Written by Bernie Westacott Sunday, 09 June 2019
Edited and compiled by Robert J Smith @RJS2212
'Teaching and learning with algebra tiles: no tricks, no gimmicks, just correct mathematics’ ' is a blog preview of Bernie Westacott's #MathsConf19 session/workshop being run at #MathsConf19.
The correct use of algebra tiles is based on an understanding of the field axioms, particularly the existence of an additive inverse (zero pairs). We will work through the learning trajectories for aspects of algebra from UKS2 to KS4, using algebra tiles in online apps. Please bring your laptop/iPad/tablet so that you can play along – wi-fi access will be available.
Some further thoughts from Bernie:
Teaching and learning with algebra tiles: no tricks, no gimmicks, just correct mathematics
Many of the problems my pupils experienced in algebra were actually due to them not having a sound grasp of dealing with positive and negative integers. Over the 39-year period where I was a full-time Maths teacher, I think I tried every which way to get my pupils to understand how we deal with integers - I was never truly happy with any of these ways.
I then discovered how this was being taught in Singapore when I came across a paper that described how the Singapore Ministry of Education and the National Institute of Education developed the ©Algetools programme for use in secondary schools. I came to realise that this approach was based on the Field Axioms and was for me an accurate representation of those underpinning principles. The transition from integers to algebra was seamless – I wish I had discovered this approach decades before!
After reading further, and spending months getting to understand the app (no manual available), I slightly adapted the introduction to be accessible to our Year 6 pupils. This ran successfully for a few years in our school until the app disappeared from the internet. In the UK, teachers and pupils are using algebra tiles/discs but I think that some of the strengths of the Algetools app have been lost. I also worry that there may be some misunderstandings surrounding the methods explained in Singapore textbooks (or UK textbooks based on these) currently being used in the UK - some of these issues revolve around the use of the ‘-‘ sign. The diagrams in these textbooks are based on the Algetools app (the use of which is a requirement of the Singapore curriculum and its use is stated clearly in the Singapore textbooks), but UK teachers do not have access to this app. We will be using an algebra tiles app, but I will briefly refer to the original Algetools app.
I intend to cover as much ground as I can in the 50-minute slot, hopefully from integers up to some aspects of KS4 algebra. However, as is true for other areas of maths, it is the beginnings that need to be carefully and thoroughly presented if the later elements are to make sense.
The extracts below, taken from a paper I was recently reading, reflect my own experiences and are a good introduction to the challenges of using manipulatives (be they real or virtual) to teach integers and certain aspects of algebra. My experience has been that once the integers are fully understood, then the related aspects of algebra are grasped more quickly and with fewer errors occurring.
Making Sense of Integer Arithmetic: The Effect of Using Virtual Manipulatives on Students’ Representational Fluency
Many mathematics educators argue that students over-learn “take away” as an interpretation of subtraction in whole number arithmetic (Moses, Kamii, Swap, & Howard, 1989)…When students expand into arithmetic situations with negative integers, the complexity increases and “take away” does not adequately model subtraction with positive and negative integers. Thus, a more flexible interpretation of subtraction (i.e., one that includes comparison, difference, and other contexts) that allows for both quantity and direction features of integers to be made explicit is needed (Moses, Kamii, Swap, & Howard, 1989).
You can see Bernie Westacott speak about "Teaching and learning with algebra tiles: no tricks, no gimmicks, just correct mathematics" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June
Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.
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