You’ve never seen the GCSE Maths curriculum like this before…

Whole network no labels low res

What you’re looking at is the GCSE Mathematics curriculum. Each node represents a topic, e.g. transformations, ordering decimal numbers, frequency polygons etc. There are 164 nodes in the diagram representing all topics on both the foundation and higher tier curriculum. The nodes are connected by 935 links. Each link represents a connection between two topics whereby one is the prior learning required to be able to access the other. For example, equivalent fractions is linked to adding fractions because you need to be able to do the former before you can learn how to do the later.

I created this diagram using Gephi to investigate which topics are the ‘essential skills’ required to access as many topics on the GCSE as possible. If pupils need to completely master certain topics in KS3 in order to be able to learn as much of the GCSE syllabus as possible, what are those topics? I have scaled the node size based on how many links they have, i.e. how many topics they are prior learning for. The larger the node, the more topics it is prior learning for. The largest nodes are the essential skills needed to be able to access the full GCSE.

The nodes have also been colour-coded based on the part of the curriculum they relate to:

Number and calculating- red

Shape space and measure- purple,

Algebra- turquoise

Data handling and probability- light green

Click here to download a high-resolution pdf of the network diagram.

Before I started the project my experience led me to predict that number was going to be the most connected topic in the diagram and it certainly is. The largest nodes are red indicating that if pupils have not mastered the important number topics you will struggle to teach them much else. My teacher’s instinct knew this already but the diagram certainly confirms it in a visually powerful way. If you removed all the red nodes, and those they are connected to, you would have little left that you could teach.

What are the topics specifically? Here is the diagram again with the node labels shown. These are also scaled proportionally like the nodes.

Whole network with labels low res

For a high resolution pdf of the image, click here.

In rank order, the most important topics for students to master, based on the number of topics they are prior knowledge for are as follows:

Multiply and divide whole numbers (90 topics this is prior knowledge for)

Add and subtract whole numbers (73)

BIDMAS (50)

Multiply and divide decimal numbers (43)

Understand place value and identify the value of digits in a number (38)

Add and subtract decimal numbers (34)

Multiply and divide negative numbers (34)

Write a fraction in its simplest form (29)

Round to decimal places (29)

Substitute into an expression or formula (26)

Add and subtract negative numbers (24)

Put a number on and read a number off a number line (23)

Round to significant figures (23)

Two letter notation for a line and three letter angle notation (21)

Use a calculator to evaluate complex calculations (20)

Plot and identify coordinates (19)

Use and calculate with index notation including squares, cubes and powers of 10 (19)

Equivalent fractions (18)

Express one number as a fraction of another (18)

Extract data from lists and tables (16)

16 out of the top 20 topics are number and could be summarised as four operations, BIDMAS, place value, rounding, negative numbers and basic fractions. I’m sure this comes as no surprise to the experienced maths teachers out there. It has reinforced my own belief in the importance of pupils mastering number skills in KS3, in particular mental and written techniques for the four operations.

There are of course limitations to this analysis which I should highlight. I grouped topics together to keep the analysis manageable within the timeframe I was willing to put into it. For example, I haven’t differentiated between mental, written or calculator based four operations techniques. This may have accentuated the importance of the top two skills to a certain amount. There were many assumptions built into my decisions of which topics were required prior knowledge for the others. Too many to explain and we could debate endlessly about particular links.

However, all I wanted from this analysis were the key trends and I think in that regard it is a success.  It confirmed what my experience has told me. Number skills must be mastered early in secondary education if we want to keep the other topics accessible to students at GCSE. In addition, seeing the complexity of the connectivity of the topics also reinforces how difficult it is to come up with a sequential syllabus. It shows the importance of good AFL in lessons to establish whether pupils are secure with all the necessary prior learning before teaching them a new topic. It shows what a skilful job being a good maths teacher is! We as teachers have our own versions of this network in our minds, but seeing the complexity of it in the diagram really brings home why students find it so hard to make links between topics and ‘see the big picture’!

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39 Responses

  1. Niki Di Palma says:

    Great stuff!
    You mention importance of mastering the skills at KS3 – just as, if not more important is developing and mastering those skills at KS2 (and even 1). As a primary school teacher, the topics look oh so familiar . . .

  2. Bruno Reddy says:

    Excellent analysis Will, thanks for taking the time to make it and share. It adds support to the idea of slowing things down in KS3 and focusing deeply on a shorter list of essentials. At KSA, the weightings we give to various maths topics in our KS3 curriculum matches your findings above.

  3. @TSLAMaths says:

    This is one of best pieces of work I’ve ever seen when it comes to maths teaching! Thank you for the ingenuity and sheer effort it must have took to put this together!

  4. Bob Jackson says:

    Absolutely brilliant and your timing couldn’t be better as I’ve been trying to get across to SLT how students lack of numeracy skills has a major impact on subsequent progress. How can I get a large scale poster of this?

    • Hi Bob. Thanks for your comments. If you download the pdf from the link in the post that will be a start. Then I think the Posterazor software which is free to download from the net can cut it up into sections that print on individual pages to make a large poster. Hope that helps.

  5. John Peterson says:

    Hi William,

    Than you for taking the time to put this together. I think KS3 and 2 should aim to cover/master/achieve the biggest nodes. I will definitely be sharing this article with my department at school.

    Great work!

  6. Justin Dodd says:

    I love this! As the comments above, this is exactly this issue with current secondary mathematics. Students are not comfortable nor confident with the basic understanding of number. This is a key research theme of the forthcoming China visit by NCTL.

  7. Dan File says:

    This is great and amazingly similar to what I am trying to put together for the new Primary and Secondary curriculum – just not been able to find the right software. Thank you!

    Some thoughts:

    1) The theme of numeracy mastery particularly when re-deigning the curriculum for KS3 to take in to account new GCSE keeps recurring whenever I am planning for next year and beyond. I still don’t have it straight in my head but I am convinced that getting it right will be extremely beneficial.

    2) What I think would be really amazingly useful for teachers at Primary/Secondary is to be able click on a particular topic node and then all linked nodes (prior and subsequent) would be highlighted and the rest of the graph disappears. This would become an amaingly powerful tool for planning, teaching, AfL, curriculum design…Is the software able to do this?? Also the ability to dynamically zoom and navigate the map. Sorry this sounds ungrateful, but just my mind wandering.

    • Hi, thanks for your comments and suggestion. I think there is a plugin that gives the map the functionality you requested. I shall look into it when I get a minute and get back to you.

  8. Daniel Stevenson says:

    Impresive stuff!!
    Which version of the curriculum has this been created for? (We are all, I guess, looking forward to the 2013/2014+ version).
    Are you willing to share the source GEPHI file? (if so can you send me a copy)

  9. Raj Kumar says:

    Cooltastic! An excellent way of showing the importance of interconnections and how we must promote conceptual understanding in order to develop deep learning for our future mathematicians.

  10. Jill todd says:

    It’s great! I’ve taught my lessons, over the past 2 years, by aiming to get the students to see/make the connections. It would by great to show them – if only I was still teaching kids!

  11. Nicki P says:

    This is amazing and so interesting. Thank you!

  12. Gary Owen says:

    This is an amazing piece of work. If possible could you upload a non pdf version of the high res images. As posterazor will not open pdf files.

  13. Joe Cooper says:

    This is absolutely fantastic! Is there any chance you could provide a link to the gephi file of the network? That would be amazing.

  14. Julia Upton says:

    Hi Will

    I think this diagram is great – so powerful to see how things build on each other in Maths. I would like, with your permission and obviously appropriate referencing, to use your diagram in a book I have written aimed at trainee maths teachers. Could you drop me a line to have a chat about this.

    Julia

  15. Claire Hobbs says:

    Thank you – what am amazing way of seeing what we know to be true. The power is in sharing with the students as well I think. We are now (like many others) doing a root and branch curriculum rethink – it would be really handy to have a list of all those topics starting from the maximum number of links and going down to the least. I would use it to underpin my planning. Thanks.

  16. Monika says:

    Hi Will

    What a Fab piece of work!! Thank you so much for sharing ….. I think this is where I shall be starting my departments next CPD session from…..cann’t appreciate enough!!

  17. Monika says:

    And ….Will….is it possible to have a link to the Gephi file please?

    Thanks

  18. nicky says:

    Hi, I have looked at your work with great interest, I was sent a link by a colleague I was chatting to about this very thing!

    The bit that interests me the most is the number of links, is a list of all topics and number of links available?

    Thanks for putting in so much hard work, it really is a great piece of creativity.

    Nicky

  19. Hi Will,

    Is this the same for the new GCSE curriculum also?

    • Hi, not done it for the new curriculum yet. Am certain the underlying foundations won’t have changed… The big message will be the same. The top 20ish nodes need to be learned to mastery…

  20. IndieP says:

    I believe the analysis undervalues the significance of fractions.

    True, they’re not normally a prerequisite for topcs such as shape/space but that’s because of ‘teaching to the test’ – assuming that ‘angle facts’ Qs will always involve intergers because ‘they always do’.

    Lack of confidence with fractions is very limiting at A level and we should be including fractions (&decimals, &surds) everywhere and not just as that last special ‘challenging’ question for extension.

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