Improving maths attainment for students from lower income households

1. The mathematical journey to Foundation GCSE Maths

2. Conceptual difficulties with numbers and the role of metaphors

3. Treating low-attaining KS4 students as ‘delayed’ not written-off

4. Real life experiences and learning maths for students from lower income families

5. Why do many lower income students under-perform at GCSE compared to SATs?

6. Informal peer support for low attaining maths students

7. Compound nature of a falling rate of mastering maths skills in secondary schools

8. Secondary school strategies for teaching students from low income households

9. A new way forward for the lowest attaining KS4 maths students

10. The joy of a broad education and then focused maths revision

References

Appendix: Output from ChatGPT:

• ‘Average score in maths sats at Year 6 to get 7 8 or 9 grade at gcse’

• ‘Convert sats scores to approximate percentages’

• ‘Common Techniques to Describe Negative Numbers’

The mathematical journey to Foundation GCSE Maths

The journey to understanding proportion starts, before school even, with one-to-one correspondence in the form of counting. When a teacher models and talks about natural numbers for example, students will attempt to connect their knowledge with classroom teaching. Arithmetic practised in primary school, therefore, may not be sufficiently embedded, when overall cognitive understanding is developing at a slower rate than expected.

Number two, could be represented in many ways including:

• labels (door numbers, football shirt number); or

• discrete number counting (natural numbers); or

• continuous measures combined with a unit descriptor e.g. 1.9 metres (at 1 significant figure); or

• product of measures e.g. area 2m2; or

• constants, differences, ratios or coefficients e.g. on average for x people, the number of legs in a room 2x.

Addition and subtraction methods and algorithms (Years 1 and 2), involve a level of abstraction that teachers may gloss-over, intentionally or otherwise, that can make learning further foundation maths problematic and nudge students into misconceptions that reach KS4 and thereafter if no further mathematics training is acquired after Year 11.

Conceptual difficulties with numbers and the role of metaphors

Grouping numbers, essential for multiplication and division, can be cognitively problematic. Units, tens, hundreds, essential for base 10 manipulatives can be easily represented by counting physical or picture blocks, but by the time students reach multiples and factors in Years 3 and 4, rote learning of ‘times-tables’ cannot assist students who struggle with number line positioning once a number is deemed too big or small to visualise. Negative numbers and parts of numbers also require more than a focus on a number line or counting, and are cognitively tricky, without contextual meaning.

Large absolute numbers

To think of a number between 20 and 100, most people can structure and create multiple imagery. Between 100 and 1,000 this task becomes more difficult. That is why metaphors are often used with numbers to create imagery: 1,000 - students in a school; 10,000 to 50,000 - fans in a football stadium; 1 million - 6 million - population in cities. When metaphors are no longer useful e.g. 1 million to 1 billion, the science world uses standard form with the focus on the power of ten.

Small absolute numbers

When analogue scale measurements are so small as not to be observable to the human eye, they also become difficult to visualise. The science community uses negative powers in standard form to denote how small these numbers really are. Such notation is confusing to KS4 students who barely understand negative numbers in many of their discrete/continuous/binary forms (Appendix 1 Chat GPT 2025).

Conceptual difficulties in learning proportion

Proportion provides the focus for approximately 40% of the Foundation GCSE. Proportion relies on a strong multiplicative understanding, comprehension of parts and what a ‘whole’ means in a variety of contexts. We cannot scale without multiplication or find unit rates without division (the inverse of multiplication).

Just as there are for natural numbers a variety of meanings, the word ‘fraction’ too has similar issues (Lesh 1983), which confuse many Foundation students.

The fraction ⅔ could mean:

• ‘inclusive’ reference to a whole/unit, ⅔ of a pizza (which if measured would be a continuous unit); or

• 2 out of 3 sweets (where the discrete units relating what is consumed are sometimes the expressed fraction); or

• ‘exclusive’ comparisons e.g. John has ⅔ of the points that Sarah has. Note that unlike in the pizza/ sweets example there is an inverse fraction that can be derived from the above fact Sarah has ³⁄₂ of the points John has.

Some students arrive in KS4 classes still unable to comprehend addition and subtraction of numbers with decimal places, because the algorithms for carrying and borrowing belie their current cognitive understanding of real numbers and place value. Therefore comparison of numbers with regards to statements such as: ‘more’, ‘less’ and ‘equal’ are particularly troublesome to such students, and fractions, decimals, percentages and ratio can therefore make little sense.

Treating low-attaining KS4 students as ‘delayed’ not written-off

Note in a mainstream class in KS4 we still treat students in the lowest attaining sets assuming that they are subject to ‘delay’ in cognitive and therefore mathematical learning (Hodgen et al 2020). This is an important point. Often students arrive at the start of KS4 maths learning in a disheartened state. The students are still only 14 or 15 years old and a lot can change for them cognitively and therefore in their maths education. Even students who have been ‘written-off’ by themselves or their KS3 teachers can see surprising leaps in mathematical understanding in a supportive environment where appropriate catch-up maths is taught in the classroom.

Out of the twelve different strategies to intervention strategies, with a large sample size, traditional explicit teaching had the largest moderate cohen effect size (Hodgen et al 2020) only surpassed by ‘prompts for self-instruction’ which had a very strong effect but had 24% of the sample size used for calculating the effects of explicit teaching. Teaching catch-up material in a structured emotionally sensitive way is the most important methodology of the Positive Foundation Maths Project for this reason.

Real life experiences and learning maths for students from lower income families

Attempts in schools over the last decade to keep disadvantaged students from ‘sliding down’ the maths sets from Year 7 to Year 11 by actively focusing on pupil premium students, for example, with extra resources and ensuring they are a priority in the classroom. However by KS4, the lower attaining sets have proportionally more disadvantaged students that have underperformed compared to their end of KS2 scores.

The Positive Foundation Maths Project was set-up in 2020 to plug the gaps in KS4 of the students that slide into the two lowest attaining sets. Many (but not all) of these students had potential, as demonstrated by KS2 results. Regardless of the cause, we provide an in-class catch-up system that is emotionally appropriate and enables students to fill in the gaps at their own pace, in class, with high work-ethic expectations.

Why do many lower income students under-perform at GCSE compared to SATs?

Cognitive understanding of the world facilitates literate and verbal fluency, which is positively correlated with learning mathematics. To assist students on a one-to-one basis with difficulties in KS3 or KS4 maths, comparisons to real-life situations and language expressing the concept and mathematics usually are modified by teaching staff.

When observing students from low-income backgrounds, compared to other students in the same classroom, regardless of attainment setting in maths, lower income students appear to have less real-life experiences from which to draw on, to connect to the dots mathematically speaking. This has to be worthy of future study.

Measurements of absence and behaviour incidents are poor proxies for the reasons for falling mathematical performance, sometimes leading to inappropriate remarks in academic studies from low-income households without looking at their complex lived reality for the causality of poor relative mathematical performance at GCSE,

Studies often use a version of ‘pupil premium’ and or ‘free school meals’ as low income indicators, which again do not directly explain why many students from poorer backgrounds appear to not benefit from secondary education as much as their richer cohort. From classroom observation, student language skills and outside world experience seem to be connect and correlate with access to the following resources:

• adults with higher levels of education; and/or

• parental and student time; and/or

• health of parent and student; and/or

• learning ‘tools’ such as technology and also analogue resources such as books; and/or

• access to extra-curricular and outside school experiences

Clearly, no one household will be the same, and each student will have different outcomes for similar inputs (not that a study could control for the myriad of environmental and inherited characteristics) but it is clear that students from poorer backgrounds are over-represented in lower attaining sets (Jerrim 2024), and these students are seen with fewer resources.

Informal peer support for low attaining maths students

Around school, my colleagues and I anecdotally observed how older, more impoverished students are more likely to socialise and nurture, at break, lunch, arriving/leaving school, younger students with similar backgrounds to themselves. In the Maths Department we identified that spending resources improving confidence in maths of students in low-income families in the highest attaining sets had a positive effect on students’ learning in the lower attaining sets. Therefore we surmised there was a larger effect of providing resources to students who would pass the GCSE without help, but without further school input would definitely not achieve their potential of the highest grades. It appears that higher attaining students can provide some support which would be probably provided in richer households by parents. This is a subject worthy of deeper study.

Compound nature of a falling rate of mastering maths skills in secondary schools

When language and conceptual understanding in maths progresses at a slower rate than the average cohort student, the loss in mastery of skills compounds for every year of subsequent education. This overall effect can be simply demonstrated by a series of compound interest calculations.

When secondary maths classes are harder to navigate than at primary school due to a combination of social, emotional or physical reasons, each year the student finds the work more difficult compared to the cohort average, and even if all other variables remain constant, and will probably find it even more difficult to keep up in the next and so on.

The KS2 high achiever

Imagine a student achieves 90% in their Year 6 maths SATS, most students with this score will be expected to get a Grade 7 to Grade 9 in their maths GCSE (Chat GPT (2025)..

In a very simplified model, all other variables remaining constant, should they find the new material in year 7 difficult they may see a 5% (conservative estimate) fall in their end of year exam score. If this pattern repeats each year until the GCSE exams, the compound effect means that the student scores 70% in Year 11. This number is a little difficult to interpret because of the huge step between Foundation and Higher material (usually around Year 9) which would probably give a bigger drop in grade if Foundation material is not fully understood. What we do often see in top sets is ‘Pupil Premium’ students gaining grades 5 or 6 in Year 11 , well below their expected scores based on their SATs.

KS2 Moderately high achiever

Keeping with the same model, if another student keeps dropping sets and moves from a Higher to a Foundation class, the KS4 years will inevitably see a larger fall in performance, due to not studying any higher content. Taking a student who gains 80% at KS2 (on average with an expectation of grades 6 or 7 in Year 11) a 5% fall per annum in Year 7 to Year 9 and 10% fall in Year 10 and 11, because they were put in a Foundation set means they now achieve 53%, relatively speaking. In layperson's terms the middle student may get a grade 3 or 4, so again all students underachieve and some actually fail the GCSE.

KS2 Average achiever

Consider now a student who, at KS2, attains the average scaled score of 100. In average circumstances they should pass a GCSE with a Grade 4. Therefore using 65% (an upper boundary estimate) in the SATS examination, and assuming this student remains in foundation sets all the way through, if they experience a 5% (modest) fall per annum from Year 7 through to Year 11, this would equate to 50% in the Foundation GCSE examination which would in all probabilities equate to a grade 3 (a fail).

The large step between Foundation maths (grades 1 to 5) and Higher maths (grades 4 to 9) make analysis by percentage a little difficult to parse: Higher papers give grade 4s with approximately 25% mark, at Foundation a grade 4 is approximately 60-65%. However it cannot be denied that the cumulative effect for students in poorer households experiencing a reduction in the effectiveness of their studies in secondary school due to their socio-economic status cannot be overemphasised (Jerrim 2024 and Mazenod 2018).

Secondary school strategies for teaching students from low income households

Traditional policies maths departments to mitigate loss of maths performance include:

• extra intervention lessons;

• keeping students in higher sets than their test scores suggest they should be;

• seating students at the front of the class;

• school policy for such students to get priority in one to one assistance;

• school policy to follow one scheme of work for all KS3 classes and in KS4 one scheme of work for all Foundation sets regardless of attainment.

Mazenod et al (2018) describe very well the effect of putting ‘at risk’ students (which include students from low income households) into a lower attaining set. It might seem reasonable to do so on the basis of test scores, as such classes have the best adult to student ratios and are sold to students, parents and teachers as a place for ‘nurture’ and ‘protection’. The reality on the ground is that they can often feel like battlegrounds with disruptive elements and disengaged students mixed with SEND students. Teachers can be exhausted trying to keep the peace between the school and parents’ high expectations and adapting work for students that lack ‘confidence and resilience’ and act out as a consequence. It is no surprise students and staff feel the work becomes ‘repetitive and dull’ and does not take students learning ‘forward’ (Connoly 2019).

Schools often try to counterbalance this with stating ‘high standards’ for all by institutionally insisting on all Foundation maths sets following the same programme. Students who are from lower income households are more likely to have time off from school due to their (or a family member’s) illness and often have caring responsibilities which make taking on catch-up classes or doing homework difficult. In addition homelife may mean sharing bedrooms making sleep and additional study more difficult; concentration in class can also therefore be negatively affected. Therefore such students need a catch-up session which is more joined-up in the main classroom than the traditional Rosenshine review methods and unconnected ‘Interleaving’ starters and plenaries

For time purposes, this paper will not step into the mixed ability setting debate, which one could argue would be a logical way of teaching lower income students in a more equitable way through better quality teaching, improved behaviour, student collaboration and independence. Most schools reject this method of teaching and even Mazenod describes this as ‘risky and untested’.

A new way forward for the lowest attaining KS4 maths students

The Positive Foundation Maths Project advocates therefore a simple approach which suits most students who have not made expected progress by KS4, regardless of the cause, because students learn the most when they are in the classroom. Key take-aways discussed in more details in other zines include:

• provide full equipment for all students in the classroom;

• reduce distractions in the presented teaching material to keep students attention;

• foster success by re-teaching two scaffolded connected 10 minute topics per lesson that are carefully planned to ensure there is no bias in the choice of revision topics over the course of a year.

• repeat the two scaffolded connected 10 minute topics per lesson over the course of the week (same questions different numbers) to improve recall and fluency; this gives students confidence to enter the classroom calmly and work more independently giving easy lesson structure and pace by starting and ending with predictable tasks.

• do interleaving and exam question practise after Year 11 February half-term holiday for the lowest attaining class, and the next set up, after the Year 11 October half-term holiday; the students that have gaps in basic mathematical learning will learn less from pop-quizzed unconnected topic questions;

• modelling all examples from the main scheme of work, for the middle of the lesson, in the Rosenshine method, but write and talk as you explain and leave up on a whiteboard, rather than presenting on the screen.

• using the screen for repetitive practice questions for the majority to practise the modelled main part of the lesson questions

• using easily accessible good textbooks or worksheets to differentiate for those who need to move faster or slower in the lesson.

• having sympathetic teaching assistants and all teaching staff engage with students around the building, encouraging students to emulate respectful professional communication.

The joy of a broad education and then focused maths revision

When students have more time and resources to widen their horizons, although they are unaware they are also forming memories which provide language and cognitive hooks for mathematical abstraction. Activities such as: sports, arts, music, holidays in locations that differ from home (even inland/seaside, rural/city, as well as obviously overseas), Scouts, Guides, cultural trips to museums and historical sites, books, learning materials including access to the internet and technology, play-dates with people outside your usual social group or generation, social gatherings, alternative food experiences, movies, theatre trips. An example of context through experience, could be travelling to many different places close and far away gives cognitive resonance to geometry and measures for example.

The stronger the real-life experience and the wider language and vocabulary a student has, the easier it is to conceptualise mathematical measures. In fact problem solving in mathematics is not possible without some contextualisation which allows abstraction and connection to mathematical tools.

It is a strongly held view of the author, based on 20 years of observation in a wide variety of schools of students of varied maths attainment levels between Year 7 and Year 13, that life experiences that the school can offer in KS3, are essential and there should be some time allocated for more free-form discussions and experimentations by students. How much would be worthy of future study. Should a student be ‘underperforming’ in maths by KS4 then in-class catch-up learning should be provided in a structured emotionally appropriate way for their age. Lessons in KS4 have to look and feel different to the KS2 and KS2 to make a difference; they must demystify maths and be straightforward with many chances for students to feel success in relation to the GCSE they are aiming to pass or do better in.

Taking students out of other subjects to do more maths, especially PE or creative arts is not helpful, as this increases the cognitive dissonance gap and is unhelpful in learning maths. It reduces the joy of school and can present more mental health challenges, which makes learning maths even harder. Students need the joy of a broad education, but then realistically, at least a pass in GCSE maths to move on with their lives.

Jane Hardy

June 2025

References

Adams, R. (Ed.) (19/9/2024) ‘Incredibly disheartening’ decline in special needs pupil attainment in England , The Guardian, https://www.theguardian.com/education/2024/sep/19/incredibly-disheartening-decline-in-special-needs-pupil-attainment-in-england?utm_source=chatgpt.com

Adams, R. (Ed.) (24/6/2024) ‘Poorer high-ability UK children fall behind peers at school from age of 11’

Connolly, P., Taylor, B., Francis, B., Archer, L., Hodgen, J., Mazenod, A., Tereshchenko, A. (2019). The misallocation of pupils to academic sets in maths: A study of secondary schools in England. British Educational Research Journal, 45(4), 873-897. https://pureadmin.qub.ac.uk/ws/portalfiles/portal/172083628/Connolly_et_al_2019_British_Educational_Research_Journal.pdf

Hodgen, J., Coe, R., Foster, C., Brown, M., Higgins, S., & Küchemann, D. (2020). Low attainment in mathematics: An investigation focusing on Year 9 students in England. Final Report. London: UCL Institute of Education. https://www.ucl.ac.uk/ioe/departments-and-centres/curriculum-pedagogy-and-assessment/research/low-attainment-mathematics-investigation-year-9-students-england

Jerrim, J, Palma Carvajal, M. (2024) ‘What happens to bright 5-year-olds from poor backgrounds? Longitudinal evidence from the Millennium Cohort Study.’ UCL Social Research Institute, https://johnjerrim.com/wp-content/uploads/2024/06/wp_mcs_high_ability_low_ses_v4_final.pdf

Lesh R., Landau, Vergnaud, G., M. Academic Press (Ed.) (1983) ‘Multiplicative structures In Acquisition of mathematics concepts and processes’ pp.127-174

https://www.gerard-vergnaud.org/texts/gvergnaud_1983_multiplicative-structures_acquisition-mathematics.pdf?utm_source=chatgpt.com

Mazenod, A., Francis, B., Archer, L., Hodgen, J., Taylor, B., Tereshchenko, A., & Pepper, D. (2018). ‘Nurturing learning or encouraging dependency? Teacher constructions of students in lower attainment groups in English secondary schools’. Cambridge Journal of Education, 49(1), 53–68. https://doi.org/10.1080/0305764X.2018.1441372

Appendix 1

Output from ChatGPT with GPT-4-turbo (as of June 2025):

‘Average score in maths sats at Year 6 to get 7 8 or 9 grade at gcse’

Estimated SATs to GCSE Progression: While there's no exact formula, here’s a rough guide based on national progression data:

Year 6 SATs Scaled Score Likely GCSE Outcome (if progress is maintained)

110–113 Grade 6–7

114–116 Grade 7–8

117–120 Grade 8–9

‘Convert sats scores to approximate percentages’

Approximate SATs Scaled Score to Percentage Correct in Maths:

Scaled Score Approximate % Correct (Raw Score)

120 ~98–100%

118 ~95–97%

115 ~90–93%

110 ~80–85%

105 ~70–75%

100 ~60–65% (National expected standard)

95 ~50–55%

90 ~40–45%

85 ~30–35%

80 ~<25%

Note: each year the raw score needed for a certain scaled score may vary slightly.

‘Common techniques to describe negative numbers’

Technique Example Context Strength

Number line −3 is left of 0 Visually intuitive

Temperature −5°C Familiar and real-world

Money/debt −$20 Emotionally resonant

Elevation/depth −100 metres Spatially understandable

Movement/direction −5 steps backward Physical embodiment

Integer chips/zero pairs Cancel out opposites Tactile and hands-on