Research School Network: My mind on algebra tiles and algebra tiles on my mind Using algebraic tiles to introduce the abstract world of algebra to students in a more concrete sense.

Please excuse the Snoop Dogg (or 2Pac for the OGs amongst us) inspired title but I’ve been considering algebra tiles a lot this year as a way of introducing the abstract world of algebra to students in a more concrete sense. I’m relatively inexperienced and new to teaching but already have picked up on and grown uneasy with some students’ difficulties and frustrations in ​‘getting algebra’ which can quickly develop into a terminal phobia of the topic.

For those unaware, algebra tiles are provided in three sizes and different colours: small yellow squares representing a unit, green rectangular strips representing and large blue squares representing. They are all double sided with the red reverse sides representing the negative of the aforementioned quantities. They can be used for, but not limited to: adding and subtracting directed numbers, adding and subtracting algebraic expressions, solving linear equations and manipulating both linear and quadratic expressions. The longest side of a tile is equal to the smallest side of the next tile, so that they fit together snuggly when demonstrating a problem, essential when you look at the practices of expanding brackets and factorising expressions to name but a few.

I won’t explain the above different ways of how to use the tiles in any detail as it would simply take too long and be too tricky to do justice in a blog but here is an image of how I would use tiles to expand (2x + 1)(x + 2).

The centre tiles represent the space enclosed by the two sides i.e. the answer of 2x² + 5x + 2.

So how have we arrived at using these seemingly fiddly pieces of card/​paper as a tool in teaching what we can surely agree is a fundamental and hugely important part of our curriculum? (Algebra is one of the most heavily assessed topic area on the reformed GCSE syllabus – an unrivalled 30 to 33% of marks on the Higher Tier examination). It wouldn’t be fitting for me to publish this post through the Blackpool Research School and not start to view the use of algebra tiles from an evidenced informed perspective. At the back end of 2017, the EEF published the eagerly anticipated ​“Improving Mathematics in Key Stages Two and Three” guidance report which lists eight recommendations to improve outcomes in mathematics. Recommendation Two of these concerns the use of manipulatives and representations in the classroom, with algebra tiles being physical items which students can use move come under the manipulative heading rather than pictorial aids such as bar modelling which would be classed as representations. The report can be found in full online here:

https://educationendowmentfoundation.org.uk/tools/guidance-reports/maths-ks‑2 – 3/

The report encourages the use of these manipulatives as a way of strengthening students’ understanding of new concepts and I would argue that algebra tiles addresses this in an area which classically has been difficult to represent in a physical sense.

In my view secondary maths departments can be guilty as seeing manipulatives as ​‘too primary school’ and reserve them only for ​‘nurture group’ type classes, but I would argue that even top set KS4 classes could benefit from using tools to consolidate their understanding. Algebra tiles for example can be used to illustrate completing the square for quadratic expressions and hence, where the method coins its name. Jonathan Hall (@studymaths) has an interactive version of tiles which can be used for this on his excellent resource page ​‘MathsBot’:

https://mathsbot.com/manipulatives/tiles

His page also has a wealth of other representations that I encourage you to explore, but returning to my previous point, the guidance report states that ​“in general the use of multiple representations appears to have a positive impact on attainment” and mentions how this arises from strengthening students’ conceptual understanding. It does also warn that too many methods can cause confusion so it is worth treading carefully with this.

I am fortunate enough to work in an area which has recently rolled out a KS2 to KS3 transition project whereby a number of local primary schools have agreed to teach the basics of using tiles as their unit of work after KS2 SATs, with the idea being that they arrive to Secondary with a basic understanding of their uses and nomenclature. This ideally then means less time introducing them in Year 7, allowing more time to be spent on their applications. I’m not convinced that tiles are used widely nationally yet and whether they have a future in being used so remains to be seen. From my personal experience and that of my department, I would best describe them as ​‘promising’, with a key focus on careful explanation at the introductory stage needed. CEO of La Salle Education Mark McCourt (EmathsUK) produced a really helpful video showing how tiles can be used to explore the concept of adding and subtracting directed numbers and focuses in on a concept called zero pairs, which has important uses when later looking at linear algebra. You can see Mark explain it better than me here:

https://www.youtube.com/watch?v=15urVcS7XcQ

There’s also another video from Mark on solving linear equations which is equally as good:

https://www.youtube.com/watch?v=SGIrxqjrMeQ

I’m aware that this post barely scratches the surface of algebra tiles and their uses, but my takeaways from having used them are:

1) Have a clear rationale of why it is you want to use them i.e. not as a gimmick or a way to engage

2) Be sure students record their methods alongside their use of tiles so that they will eventually be able to work independently of them

3) Question the students as to why they are using the tiles in such a way on questions, in order to get them to see the structure behind the problem and not just mindlessly learn routines when there are too many minimally different questions for this to be feasible

4) Do not underestimate the importance of spending time introducing the tiles and their meanings (although in our case this was completed via the transition project)

5) Consider using the tiles for directed number practice as well as algebra (see above mentioned video)

6) Ensure students have a solid grounding in directed numbers before using the tiles for algebra purposes – think cognitive load!

7) Logistics – have clear rules and routines for their use and collection (think of the nightmare of using multilink only magnified!)

To conclude, the main reason I looked to use algebra tiles more was due to my worry that students struggled with the underlying structure behind algebraic problems. If we want our students to go on to excel in this subject, it is crucial they build their understanding of basic algebra on solid foundations. If you feel a similar way, I encourage you to look into them a little more and consider trialling them in your classroom. They are surprisingly expensive things but you can easily find templates online to make your own if you have the patience to cut them out!