New Maths and Dyscalculia Assessment!

A new assessment for identifying and supporting difficulties in mathematics learning was launched in late July that has the potential to be of significant help in strengthening school’s and parent’s ability to appropriately plan for learners struggling with key aspects of maths study.

The assessment has been designed by the co-founder of the Dyscalculia Network and experienced specialist teacher, Rob Jennings, alongside Jane Emerson, the Director of Emerson House – a centre for dyscalculia, dyslexia and dyspraxia.

Rob Jennings and Jane Emerson (used with permission)

Split into 19 sections, the assessment provides for a comprehensive examination of learner’s abilities regarding early number concepts such as number sense and counting, different mental and written calculation strategies, interpreting word problems, working with and converting between fractions and decimals, as well as basic length measure. The sections have a mix of verbal and written questions, with some of the earlier ones requiring the use of counters. The authors suggest that the assessment should take roughly one hour (based on trials that have taken place to help refine the test questions and assessment approach), however are at pains to point out that the assessment should not be limited by time, either overall or for any one of the sections, as this could lead to anxiety for the pupil that could skew the results.

What sets this assessment apart, for me, from other assessments and on-line screeners for dyscalculia and/or other maths difficulties, is the level of detail that the assessor (which could be a parent, teacher, or TA – not necessarily a qualified assessor) is encouraged to capture about the pupil. As well as simply getting an answer correct or incorrect, the assessor is encouraged to note down (through the use of a provided assessor’s booklet) how long each section took, the strategies that pupils used (to help capture whether these are efficient or immature strategies), and any comments or questions that the pupil makes – either to themselves or to the assessor. This sort of information is potentially crucial in formulating a proper plan for addressing the difficulties that the assessed pupil is facing with mathematics. In addition, there are actually two assessments, an A assessment and a B assessment, which means that they could be used as a pre- and post-test for an intervention specifically designed to support the pupil.

Access to the assessment, assessor’s booklet, and answers comes through the purchase of the companion guide called (straightforwardly), “The Maths and Dyscalculia Assessment”, with a link and redemption code for the online materials included in the guide.

The companion guide comes with much more than simply a step-by-step guide on administering the test itself. Included is also guidance for the assessor in preparing for the test, including how to make sure the necessary things are organised in advance, how to create a good environment for conducting the test, what to be on the lookout for and to record during the assessment, and even making sure that the learner is at ease during the assessment. There is also a chapter on interpreting  the results of the assessment, including what issues may have been highlighted by the assessment, what might then be included in a teaching plan if these issues have arisen, and suitable specific interventions that might be required, which I would definitely recommend reading this before  administering the assessment – I feel like it would sharpen my focus on certain aspects of the test and approaches that a child takes to the test beyond what is given in the step-by-step guide.

In addition to the guidance provided for before, during and after the assessment, the companion guide also contains a host of other information and support for working with pupils that have difficulties with mathematics, including different checklists or screeners that could be used in advance of the assessment or to support its findings, a template summary report along with a completed exemplar to help capture the results of the assessment and plan for future teaching with the pupil, and an in-depth family questionnaire that can be used to provide extra detail and context to be used in the summary report. Both the templates for the summery report and the family questionnaire are both included in the online materials accessed through the guide, which means that the complete assessment package can be used with as many pupils as is needed. The guide also contains lots of information about how dyscalculia and other maths specific and non-specific learning difficulties (such as maths anxiety, dyslexia and the like) might impact maths learning and attainment, as well as some interesting statistics about occurrence of maths difficulties and co-occurrence with other difficulties, and a host of sources of further information and resources that could be useful for parents and educators.

As I went through the book and the online assessments, I reflected heavily about the numerous pupils I had encountered that displayed some or many of these difficulties. In the last department that I led we benefited from the assistance of a part-time numeracy intervention tutor and I can see how this sort of assessment would be invaluable in supporting her work alongside those of the main class teachers, as well as contributing hugely to the work of our SENDCo and inclusion team in pinpointing the difficulties that pupils were having and initiating conversations with parents and other agencies about the diagnoses and support that these pupils might benefit from. Despite my limited experience in the field, I have never seen anything that is designed to capture how pupils approach maths problems alongside their accuracy and time taken in completing them, or the depth of insight to guide future planning that this test provides and for that reason alone I think this book/assessment is well worth a look for any school or parent that wants to get a real handle on the maths difficulties that their children are facing.

From GCSE Sequences to Calculus

As part of the recent mathsconf39 I ran a session on using structure to promote algebraic thinking, and as part of that I looked at using a couple of different structures that can be used to help with the teaching of sequences. This got me reflecting on a link that I taught and highlighted to my Level 2 Further Maths and A-Level students about the links between the sequences they learn at GCSE/pre-GCSE and the calculus with polynomial functions that they first learned at Level 2 Further Maths or A-Level.

Part of the first work that pupils do with sequences is learn about linear sequences, and their defining characteristic being that of constant difference. Later they learn about quadratic sequences, and their defining characteristic being constant second difference. I always highlight that this means that the differences in the terms for a quadratic sequence form their own linear sequence. This establishes a progression from quadratic to linear to constant, or, in reverse, constant to linear to quadratic. I then ask them to predict what type of sequence would have a quadratic difference – which pupils quickly identify as cubic and can then extrapolate further from there.

This of course perfectly mirrors the progression that different polynomials go through when they are differentiated or integrated. Cubics differentiate to quadratics, which differentiate to linears, which differentiate to constants. So, when I first introduce differentiation to pupils, I ask them about where they have seen this progression before. Someone almost invariably mentions sequences, and if they don’t, I might start by writing out a quadratic sequence to prompt them.

A great question to ask students then is, of course, why? Why this connection between sequences and differentiation? This allows me to reinforce the idea of differentiation as about the rate of change, as when we are identifying the type of sequence we are examining the change between the terms.

It is worth then examining a particular sequence and its nth term, and the links to differentiation. I will typically pick one that has a coefficient of n2, such as the sequence 3n2 + 4n – 1:

n = 1

 

= 2

 

= 3

 

= 4

6

 

19

 

38

 

63

 

+13

 

+19

 

+25

 

 

 

+6

 

+6

 

 

 

If we differentiate the nth term of the sequence, we of course get 6n + 4, whereas the nth term of the linear 1st differences is 6n + 7. At first this appears to be a discrepancy, until you see that the 1st differences are in the 1.5th, 2.5th and 3.5th position. We can account for this by shifting each value back by 3 (half of 6):

n = 1

 

= 2

 

= 3

 

= 4

6

 

19

 

38

 

63

10

 

16

 

22

 

28

 

+6

 

+6

 

+6

 

 

Or by substituting n = 1.5 and Tn = 13 into 6n + a = Tn:

6 × 1.5 + a = 13

9 + a = 13

a = 4

Either way the nth term of the first differences can be shown to actually be 6n + 4, which is precisely what we get when we differentiate the nth term.

This provides a nice way of interleaving other algebraic manipulation and structure into the study of calculus, as well as reinforcing the idea that differentiation and sequences are two sides of the same mathematical idea – sequences being an introduction the study of changes and growth of discrete functions whilst differentiation (and integration) being the study of changes and growth of continuous functions. It also just adds to the impression for pupils that the landscape of mathematics is connected in unexpected but beautiful ways.