Example of weak quantitative reasoning
- These are students who are relatively weak in understanding relationships between numbers, in seeing patterns and order in numbers, and in their flexibility in combining and recombining quantitative elements in meaningful ways.
- If the absolute quantitative score is very low (stanines 1-3), and if in class work the student’s difficulties appear to be confined to numbers, then it is possible that a specific arithmetical difficulty (or dyscalculia) is involved.
Aelwyn Probert has verbal and non-verbal scores that are at or above the national average, but her quantitative score is 85, placing her in stanine 3 and below average.
What does this look like in the classroom?
As noted previously, if a student’s quantitative score is very low in relation to other scores then this can indicate some specific arithmetical difficulty or dyscalculia, or mathematical anxiety.
- Research has suggested that dyscalculic students may have problems understanding that number words and numerals refer to the size (or numerosity) of the sets they denote.
- Such students may not understand that collections of things have a numerosity and that manipulations (for example, combining collections or taking sub-collections away) affect this.
- Additionally, these students may also not understand that collections need not be of visible things – they can equally be audible things, tactile things or abstract things (like wishes). These indications may alert the teacher to the need for a more detailed investigation of the student’s capabilities and, as with dyslexia, an appropriate test will be beneficial.
GL Assessment provides a Dyscalculia Screener, an assessment tool which identifies dyscalculic tendencies in learners aged 6–14+ years. It provides recommendations for intervention strategies that will support students and help them to achieve their potential. This 30-minute test can be delivered to a whole class or to individual students and will play an important role in helping both specialist and non-specialist teachers distinguish between those individuals who have poor maths attainment and those whose difficulties are associated with dyscalculia.
Separately, the Dyscalculia Guidance will tailor the general advice to students who have specific problems with numerosity. Like the Dyslexia Guidance, it contains resources that are useful for students with or without a formally recognised diagnosis.
Dyscalculics do not have reduced working-memory span for words, unlike dyslexics. Problems with numerosity will be expected to impact CAT4’s Number Series, while those students with poor workingmemory span for words might also be expected to do worse at QR, as with other batteries. There is some evidence that developmental dyscalculia is related to problems with working memory for visuospatial material (Szucs et al., 2013), so it is possible that these students will also exhibit lower scores in spatial ability. CAT4 is not designed to be indicative of dyscalculia, and has not been validated as such, so these areas are merely suggestions for consideration.
Readers will notice here that it has been recognised that working memory might be split down the lines of verbal/spatial rather than being monolithic, as in the introduction to memory earlier in the chapter (see page 10). This was not introduced earlier for the sake of simplifying the explanation of working and long-term memory, but cognitive psychology indeed suggests that there are distinct aspects of working memory (Baddeley and Hitch, 1974) and offers refinements to the simpler Atkinson and Shiffrin (1968) model by positing separate sub-systems, called the ‘visuospatial sketchpad’ and ‘phonological loop’, to accommodate evidence of dissociations observed between young people’s working-memory abilities. Ultimately, these dissociations are mirrored in the CAT4 profiles by contrasting spatial ability with verbal reasoning, which, although correlated to a high degree, can show important divergences.
Examples of strategies for weak quantitative reasoning
Strategies for supporting weak quantitative reasoning learners include activities to support and improve working memory.
1. Reduce burden on working memory
One way to improve the learning experience for students with weak quantitative reasoning or a below average no bias is to reduce the number of things that must be held simultaneously in working memory. As identified in the guidance for Bisset Billy (see page 31), some students will have difficulty coordinating what they hear with what they see, or what is on the board with what is on the paper in front of them. Eliminating the need to remember ideas, even temporarily, can greatly assist these students. Working-memory burdens can also be reduced by relating new material to familiar concepts, for example, using a 12-inch ruler as a physical analogy to illustrate the measurement of mental qualities such as attitude scales or market research.
This can also work at a more complex level by making concrete analogies to familiar physical systems, for example, thermostats; mechanical systems such as levers, balances and scales; hydraulic systems such as drip feeds and overflows. Finally, ‘overlearning’ (and therefore making automatic) basic processes such as writing or number facts will free up the student to spend greater time on the more demanding aspects of the learning activity or task.
All learners (including teachers) gain more when they are actively involved in the learning process. Developing and then sustaining engagement is a powerful motivation, and teachers should start by ensuring that the learning environment they create encourages active involvement.
2. The physical environment
Firstly, teachers should ensure that their classrooms look like the kind of place where learning is happening. It may be valuable to think about what the room feels like for students – so ask them! As Dillon notes:
I haven’t been in a classroom in the country that couldn’t remove 10 or 15 things … Every time a human being comes into a space, they visually process the entire room … by the time we actually ask [students] to intellectually engage, they’re visually exhausted.
There is an increasing amount of research that has endorsed the value of creating a physical learning environment that offers students choices and is characterised by an uncluttered, harmonious environment. This relates to the ideas of working memory again, such that the more focused the working memory on the task at hand, the better the retention will be.
Some teachers have redesigned their classrooms to look more like a contemporary coffee shop than a traditional classroom, as Delzer noted in a 2016 Edutopia blogpost:
I was working on my TEDx presentation at my local Starbucks and, looking around, I realized that everyone seemed to be happy, engaged in their work, and relaxed. Some people chose the traditional chairs and tables while I opted for a big, comfy chair with my MacBook on my lap. The quiet music, perfect lighting, and overall aesthetics of the coffee shop were favourable for a variety of learners. And if I wanted to switch up my seat during my stay, I was free to do just that. That’s when I decided that our classroom … was going to look radically different than anything I’d ever done before.
Although working-memory span is not readily changed with practice, reducing the ‘noise’ that might reach the mind and compete for mental resources with the ‘signal’ that a teacher wants a student to receive and think about, might effectively allow better cognition and metacognition.
3. Maths anxiety
CAT4 deals with quantitative reasoning that requires only basic levels of arithmetic skills, and this is deliberate to reduce the amount of prior education a student requires to access the demands of the items. However, one of the potential causes of particularly low quantitative reasoning scores relative to other batteries may be due to a modern phenomenon labelled ‘maths anxiety’: the emotional disturbance that some students feel when presented with challenges that look mathematical.
A new report, Building a numerate nation: confidence, belief and skills by National Numeracy and TP ICAP (2019), reveals that millions of adults still lack basic numeracy skills, leaving them unprepared for the workplace and everyday life. The report confirmed that there has been no progress in meeting targets set 20 years ago by the then UK government.
Brain-imaging studies have found that people with maths anxiety will experience activation in parts of the brain to do with threat detection and pain when they think about doing upcoming maths problems. For these students, maths hurts, and they will want to avoid anything to do with it. Students with maths anxiety experience less activation in areas associated with working memory and reasoning when working through problems. Worrying about maths will divert mental resources away from the task at hand, thereby decreasing the working memory that can be allocated to proper cognition and metacognition.
This may be related to the perception of maths being gendered. Mean SAS in QR are 1.5 points higher for boys than girls, despite the content having nothing inherently ‘masculine’ or ‘feminine’ about it. Adults can have stereotypes about children’s maths performance, such that they might expect a daughter to struggle more with maths than a son. Again, this might act as a self-fulfilling prophecy, since when females are made aware of such expectations their performance decreases. Gender differences in maths anxiety are more pronounced in secondary education than primary education, and more pronounced in adults than in secondary education, suggesting that girls are more likely to spiral faster than boys.
Cognitive psychology disagrees with this acquired and minority view. Evidence suggests that maths skills are like any other, and that targeted practice can and will improve such capabilities. There is no genetic coding for innate maths ability, and teachers who address this view might go some way to reducing students’ anxiety towards maths. Some evidence even suggests that teachers with low self-confidence that do not model mathematical reasoning properly can cause maths anxiety in their students by inadvertently suggesting that there is only right and wrong, so no pressure!
Instead, teachers should aim to encourage students to think about maths as a series of finding solutions to problems, and that the journey counts as much as the destination. Arguably, this is why maths more than any other subject asks students to ‘show their working’. Modelling these thought processes will show that experts apply a set of steps to approach new problems, and that these steps can be understandable and relatable to existing knowledge.
At the start of the different projects the teachers would introduce students to a problem or a theme that the students explored, using their own ideas and the mathematical methods that they were learning … Sometimes teachers taught the students mathematical content that could be useful to them before they started a new project. More typically though, the teachers would introduce methods to individuals or small groups when they encountered a need for them.
Teachers can explore these approaches further in Jo Boaler’s website (https://www.youcubed.org).
Understanding that students who suffer from maths anxiety might not have the same breadth or depth of learning from previous experience will also help establish the best starting point to rebuild any lost confidence or progress. This is a key issue to which we have returned throughout this chapter - small steps are effective when teachers start from the right place.