Rachel Dunsmore was teaching at Manor C of E Academy in York, part of the Hope Learning Trust when they started to roll out iPads so that every single child had their own device for maths lessons. She was delighted with the technology but felt their potential significantly exceeded their use.

‘I was becoming increasingly aware of the educational discussions around memory and in particular the impact of retrieval practice on long-term memory,’ she said. ‘This was something that the iPads could administer far more effectively than I could. Furthermore, all the results could be recorded allowing spaced practice to be incorporated on a personalised basis. It struck me that the potential impact on progress was phenomenal.’

One incident set her on track to create her own app. She was revisiting the topic of straight line graphs with all year groups and found that most pupils could not remember what ‘gradient’ meant, let alone find one. This meant going back to basics all over again to reinforce both concepts and the language of maths.

### Creating a secondary maths app from scratch

Rachel had a clear concept for the app. It was to make sure learners retained their prior knowledge of maths so they could move on without constantly going over old ground.

It has taken three years so far, has 15000 questions and has been produced with support from technology experts, cognitive scientists and experienced maths teachers.

With stalwart support from such a band of professionals, bringing the app to market should have been a straightforward process, but for a long time it was impossible to find an app designer who shared her views.

Many were intent on producing something which was enjoyable and fun, rather than a serious educational tool but as Rachel pointed out, ‘If children want fun then they are going to play FIFA or Fortnite. We were never going to be able to make an educational App that could compete with computer games and there is a real danger of dumbing down the educational content if one tries. Also, it is important from a learning perspective that children are able to attend to the maths and that cannot happen if there are busy screen animations such as exploding chickens.’

The app has 10 Levels covering all the topics for KS3 and KS4. Each session has 12 questions, is a finite task and should take no more than 10 minutes.

The simple dashboard means that pupils can track both their progress using an attainment measure and also their effort using a daily streak counter and XP (experience points – 1 for every correct answer).

### Exploiting the power of digital technology

While many apps focus on gamification, assessment and rewards, these were not key elements for Rachel. Her starting point was the intelligence of the software. ‘I had always started lessons with 10 quick questions to gets students tuned into maths thinking and refresh their memory. Many teachers do. The trouble is that often we focus on recently taught topics. Pupils have not had time to forget these and they are in the forefront of our minds too so the questions are not necessarily recapping what is needed, and of course, what each pupil needs will be different.’

Arc Maths tracks pupils’ knowledge and personalises the content it offers them, choosing different items from its question bank.

This means it is providing in-built differentiation which saves a considerable amount to teacher time; pupils don’t waste time practising what they already know and only skills or knowledge that has been forgotten is revisited on a systematic basis, with increasing time lapses until it passes into long-term memory.

Computers and phone screens are not ideally suited to maths: indices and standard form require superscript, differentiating between the multiple sign and the variable *x* is tricky and then there are symbols such as pi π and square root signs that do not feature on a standard keyboard at all.

Arc Maths uses powerful handwriting technology and now children can use a stylus or their finger to write formulae and fractions in a way that they can’t using a computer or phone keyboard.

‘It is amazing,’ said Rachel. ‘It can decipher even the worst handwriting and can recognise complex algebraic expressions. Thanks to this software, we have been able to move beyond multiple-choice questions and students can input answers in a way that more closely resembles their schoolwork.’

A year 10 pupil said: ‘This app is ten times better, it’s so much easier to use. The graphics are clear and I like being able to write my answers with my finger, it’s so much easier not having a keyboard. The app has levels you can work towards and I find the space for working out my answer really helpful.’

### Focusing on memory and mathematics

While there is a considerable body of research about memory but the theory is often divorced from classroom practice but this was fundamental to Arc Maths and Rachel has benefited from advice from the Department of Psychology at the University of York

Silke Goebel, a senior lecturer in the department who leads the Numerical Cognition Lab, said:

‘Right from the start Rachel was interested in designing the app based on scientific evidence – the app is based on the principle of retrieval practice. In general, retrieval practice produces better retention than restudying the same information (Roediger & Butler, 2011). There is solid evidence for the benefits of retrieval practice for many school-related subjects (e.g. language learning, Butler, 2010). However, surprisingly, there is little research on the effectiveness of retrieval practice for mathematical learning despite many studies showing that retrieval of facts and learned procedures plays an important part in mathematical learning (Gilmore, Inglis & Goebel, 2018). One particular advantage of this app is that it is not only adaptive to the level of the individual student, it also takes the learning history into account’

Arc Maths is based on ideas drawn from evidence-based research into how the memory work.:

**Retrieval practice**– often children revise by reading their text books or looking at exercises they have done but it is the retrieval of knowledge with high-frequency, low-stakes testing that fixes it in the memory.**Spaced practice**– research shows that short study sessions, spaced out over a period of time, work better than massed practice in which the same skill is practised repeatedly in one sitting. The increased effort required to retrieve the information after a period of forgetting appears to help strengthen the memory. If a pupil forgets a skill or topic, the algorithm within Arc Maths ensures a similar question is presented to the pupil the next day and then with increasing time lapses until the knowledge has been retained over a 3-month interval, thereby helping to secure this information in long-term memory.**Interleaving**– In Arc Maths, the question topics are mixed up. This is known as interleaving and requires pupils to switch between topics. Whilst this approach can feel slower than massed practice, the evidence is clear that it has a greater impact on long-term retention. When using Arc Maths, a question on rounding may be followed by one on substitution so the pupil needs to identify what maths is required. This is a more demanding task than repeat practice and the increased effort needed to retrieve the knowledge or skill has a positive impact on the strength of retention.**Reducing the cognitive load**– The user interface of Arc Maths has been created in a way that minimises the extraneous cognitive load. Research has shown that adding unnecessary complexity to a task inhibits learning by overloading the working memory. The questions in Arc Maths are deliberately presented with minimal wording and without real-life context where possible. The interface is uncluttered and consistent so that the user can give their full attention to the maths.

According to dyscalculia expert Steve Chinn, maths anxiety seems to be a factor in blocking memory:

‘Mental block anxiety may be initiated, for example, by a mathematics symbol or a concept that creates a barrier for the person learning mathematics. This could be the introduction in algebra of letters for numbers or the seemingly inexplicable procedure for long division or failing to memorise the ‘basic’ multiplication facts. This type of mathematics anxiety may be addressed by appropriate teaching, if the mathematics curriculum allows for this, both in time and in structure. Ashcraft et al. (1998) have shown that, under certain circumstances, anxiety can adversely affect the working memory that is used for mathematical tasks. Working memory has been shown to be a significant predictor of mathematical achievement. Thus anxiety does not just block the willingness to learn, it can also reduce the capacity of one of the key sub-skills needed to succeed.’

Arc Maths also works well for SEND pupils. Partly this is because the app offers a consistent approach but also, they can see the progress they are making which is motivating. ‘It makes maths look straightforward,’ said one teacher, ‘so it overcomes some of the difficulties of getting these students engaged who have not done well in maths before.’

One unlooked for
advantage is that during lockdown there has been an increased uptake from girls
who lack confidence in their maths skills and are getting good results. One pupil
said: ‘It’s very easy to fit the sessions in to each day. The streaks part is a
really good way of making sure you do the app every day. My streaks are up
to 60 now and I don’t think I have ever done maths for 60 consecutive days
before! I don’t want to lose my streaks now. It’s also a good way of
proving to Mum that I have done it. I also like the fact that there are so
many levels to work through and you can see how you are progressing as you
move up a level.’

Rachel Dunsmore has created an app that saves hours of teacher time. Arc Maths is an annual school subscription service. The price is on a per pupil basis. With regular term-time use, a session of Arc Maths is comparable to the cost of a single black-and-white photocopy. However, instead of a simple worksheet, this provides questions for each pupil that are differentiated, chosen, set, marked, recorded, followed-up and, most importantly, that have a measurable impact.

References

Roediger, H.L., & Butler, A.C. (2011). The critical role of retrieval practice in long-term retention. Trends in Cognitive Sciences, 15, 20-27.

Rohrer, D., Dedrick, R. F., & Stershic, S. (2015). Interleaved practice improves mathematics learning. Journal of Educational Psychology, 107, 900-908.

Rohrer, D., & Taylor, K. (2006). The effects of overlearning and distributed practice on the retention of mathematics knowledge. Applied Cognitive Psychology, 20, 1209–1224. Soderstrom, N.C.,

Kerr, T., & Bjork, R. (2016). The Critical Impor tance of Retrieval – and Spacing – for Learning. Psychological science, 27 2, 223-30.

Sweller, J. (1988). Cognitive Load During Problem Solving: Ef fects on Learning. Cognitive Science, 12, 257-285

Chinn,S. (2018) Spotlight on social and emotional wellbeing in school A Routledge Free Book produced with nasen