The Fallacy in Mathematical Education

Note : This is a significantly updated version of the previous post (since removed) under the name ‘Assessing the Adequacy the Mathematics Curriculum at KS4 in Preparing Students for Post-16 Pathways‘ on September 25th. This post now expands on the entire curriculum instead of only focusing on Key Stage 4. The original thesis is under editing to expand on this and will be published at a later date, whether here or elsewhere (August 2024 – Edit : the thesis is now found here). I am not affiliated with any of the groups mentioned below, and opinions mentioned are solely mine, unless mentioned otherwise.

Prologue

“Education is not the filling of a pail, but the lighting of a fire.” 

William Butler Yeats

January 2020

‘So to simplify, you move everything to the left hand side and subtract 7 from 8…’

This is simple, I thought. It’s just apples and oranges. You group apples with apples, oranges with oranges. Why are we still stuck here when this is so obvious? 

Bored out of my mind, I decided to venture onto the internet. Surely there must be something much better I could do in the midst of this stupid pandemic. 

This, dear readers, was the start of my true mathematical voyage. In a short 6 months time, I had learnt more than what I could have ever imagined had I stuck with the school curriculum, and was more motivated to learn mathematics than I had ever been. By the end of the summer, I left with a more concrete understanding of algebra, calculus and trigonometry than what I could have been taught in school. 

January 2023

Whilst searching for topics for my EPQ (Extended Project Qualification), I found myself reminiscing how I got to my current mathematical standing. I had admittedly not paid any attention in maths – this was however to no one’s surprise, given that I had somehow taken AS Maths, scoring an A before even attempting GCSE. I had instead taken the liberty to work on other topics of mathematics I had come to take a liking to – one such that caught my attention was  fractional calculus. Given that my teachers were on the topic of how to take derivatives for polynomials, I thought I might as well ask. 

‘Sir, what happens if you take a half derivative for these functions?’

Stunned silence.

Apparently no one had an idea of what would happen, despite it being a natural question – if we could have a first-ordered derivative and a second-ordered derivative, why couldn’t we have a half-ordered one? It stood out to me that this thought had never even occurred to the better of us, despite being very well documented. Obviously it was way more complicated than what the course could cover, but the fact a mention of such was never even raised was still odd (for more information on this please see my post “Fractional Calculus – A Study of Differintegrals with Fractional Orders”). This thus prompted me to write about how the Mathematics curriculum at Key Stage 4 should be reformed, and this formed the basis of my EPQ project. 

However, I was never very satisfied with this. It was only 15 pages long, not officially graded and never really touched  on why I think the current Mathematics curriculum is unsatisfactory. More specifically, I wish to dive deeper into criteria of what makes a mathematics course good and how shortcomings of the current methodology should be tackled. At the end of this, I hope to establish an intuitive course for students, specifically with emphasis on how the ability to research and using technology should be stressed as a requirement for further development in the field. 

Either way, I hope you enjoy this article and may I present to you my vision for the Mathematics Revolution of this decade – from a student’s perspective. 

Part I – The Current Mathematics Curriculum

“Mathematics is, in its own way, the poetry of logical ideas.”

— Albert Einstein

The current curriculum features 5 Key Stages for mathematics, with Key Stage 4 featuring a mandatory examination for Mathematics at GCSE and Key Stage 5 being optional. With 12.7 million students under 16 studying in UK as of 2019, and conjoined with  the education reform proposed by current Prime Minister Rishi Sunak extending mandatory mathematics education until 18, I find it more crucial than ever to evaluate how successful our current curriculum is at achieving the goal of both preparing students for further studies and applying such skills in real life, and it remains more relevant than ever that we outline our goals for enhancing our quality of education in terms of revamping our education system.

In Key Stage 1, the focus is on ensuring ‘pupils develop confidence and mental fluency’ with numbers and basic operations. Students learn to work with whole numbers, counting, and place value using various tools and resources. They also start recognizing, describing, and comparing shapes, as well as using measures like length, mass, and time. By the end of Year 2, students should know number bonds to 20 and understand place value accurately, and should also practice reading and spelling mathematical vocabulary to enhance their word reading skills.

Moving to Lower Key Stage 2 (Years 3 and 4), the emphasis is on developing fluency with whole numbers and operations, including fractions and decimals. Students learn efficient calculation methods and problem-solving skills. They also improve their drawing accuracy and mathematical reasoning, while mastering multiplication tables up to 12, which, quoting the official document ‘should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers’, and most importantly ‘develop their ability to solve a range of problems’.

At Key Stage 3, students further develop fluency in the number system, calculation strategies, and algebraic concepts. They deepen their understanding through various representations and learn to reason mathematically, apply their knowledge to solve problems, and make connections between different mathematical concepts. Additionally, the curriculum emphasizes the importance of reasoning mathematically, wherein students extend their understanding of the number system, identify and express mathematical relationships algebraically and graphically, and begin to reason deductively in geometry, number, and algebra. Moreover, students are encouraged to solve problems by applying their mathematical knowledge to both routine and non-routine situations, modeling mathematical situations, and selecting appropriate concepts, methods, and techniques to tackle unfamiliar problems.

Transitioning to Key Stage 4, students encounter compulsory examinations in Mathematics in the form of the General Certificate of Secondary Education (GCSE) assessments. This requirement was established following the enactment of the “Raising the Participation Age” (RPA) legislation in 2013. Under this legislation, it became obligatory for students to undertake and achieve a minimum grade 4 in the mathematics exam. Hence, it is crucial that we delve into this further as it is of utmost importance that we understand whether it can properly arm students with the abilities needed to progress into future studies.

The objectives of the curriculum at KS4 consists of 3 main aims :

  • Use and apply standard techniques (Assessment Objective 1);
  • Reason, interpret and communicate mathematically (AO2); and
  • Solve problems within mathematics and in other contexts (AO3).

A noteworthy observation lies in the considerable overlap evident across the objectives spanning from Key Stage 1 to 4, implying that the learning goals remain largely consistent throughout these stages. However, this uniformity can potentially lead to a sense of monotony among students as they progress through the curriculum, which is one of the key problems I wish to solve through this proposal. We must however, also mention the addition of Further Mathematics and Additional Mathematics (OCR) at the KS4 level, which has considerably improved upon this aspect. Despite this, only a handful of schools and colleges have chosen to offer such courses, whether because of a lack of interest in the subject or continuously diminished funding.

For post-16 options mathematics is no longer mandatory, with numerous different courses provided such as Business and Technology Education Council (BTECs) and International Baccalaureate (IBs) still building on knowledge accumulated at GCSE level. These options often feature more real-life applications of Mathematics than A-Levels – heavily employing one’s implementation of skills learnt. Other A-Levels, such as Physics and Economics to name just a few, also require students’ knowledge of the GCSE Mathematics syllabus to be capable of calculating in different contexts. The overlap between different qualifications and subjects accentuates the importance of applying mathematical concepts to real-life situations, and this will have to be reflected in the GCSE curriculum.

Part II – Why are we failing?

“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.”

— William Paul Thurston

Without a doubt, the administration of the current curriculum requires desperate improvement. From my own experience of having to communicate with school authorities, it has become quite apparent that there are no policies for ‘outliers’ who have taken separate examinations, both in Mathematics and other subjects alike. To illustrate this, I had taken an AS grade qualification in Mathematics before I had started Year 11. Now, logically that would probably imply that I could go straight into Year 13 content – however I was stuck in a GCSE Further Maths class for a year before going on to study the same content at Year 12. Case in point : a better system must be put in place to accommodate those who are talented (or more familiar, in my case) at mathematics. Even though I was able to dedicate my time to other more useful ventures, plenty don’t have that liberty and in such cases, one might simply lose interest in mathematics overall. 

I would also like to take this time to outline some of the different criticisms of the current curriculum, of which some are my own opinions and others I have obtained from different sources such as social media. In particular, I drew my attention to Reddit for this post titled “r/math, what do you think is the limit for ‘mandatory maths’?” in r/math. We will draw from some of these advice provided.

With more mature students we fact the same problem as I have personally suffered : the current curriculum was way too rigid to allow for such students to truly shine, and the combination of lack of time and resources most certainly did not help. However, the current situation does allow for one exception : Olympiads. With a plethora of mathematical olympiads such as BMO (British Mathematics Olympiad) and SMC (Senior Mathematics Challenge), it comes as a huge surprise that these are the only choices for students wishing to pursue further mathematics studies outside of school.

However, Mathematics Olympiads may not provide students with opportunities to engage in research methodologies such as literature review, experimentation, and collaboration with peers and mentors. These skills are vital for success in academic research. With such limited exposure to research methods, the lack of availability for such projects are deeply concerning.

In addition, achieving success in Mathematics Olympiads requires a particular set of problem-solving skills and techniques, which are undoubtedly valuable. However, these skills alone may not adequately prepare students for the diverse challenges encountered in mathematical research. Unlike Olympiads, which emphasize quick and elegant solutions to pre-defined problems, mathematical research demands a broader skill set, including creativity, critical thinking, and the ability to adapt to new ideas and approaches. Engaging in research methodologies such as literature review, experimentation, and collaboration with peers and mentors provides students with essential skills that are vital for success in academic research. Therefore, the restricted availability of opportunities for students to engage in such projects is a matter of great concern, as it impedes their development as mathematicians and researchers.

Terence Tao on Mathematics Competitions

Another concern we raise is the importance of such mathematics taught in schools in the real world. With the aid of Chat-GPT, we have established a ranking score from 1 to 10 for all of the professions listed in ISCO-08 in terms of mathematics, which I have included for reference in 1. The established baseline is as follows :

  1. Knowledge of numbers: This score indicates a basic understanding of numbers and their significance. Individuals with this level of mathematical knowledge can recognize and work with numbers but may not have advanced skills in mathematical operations or concepts.
  2. Knowledge of addition, subtraction, multiplication, and division: This level of mathematical proficiency encompasses basic arithmetic operations, including addition, subtraction, multiplication, and division. Individuals at this level can perform these operations with whole numbers, fractions, and decimals, enabling them to handle everyday tasks such as budgeting, measuring, and calculating simple quantities.
  3. Understanding of basic mathematical concepts: At this level, individuals have a grasp of fundamental mathematical concepts beyond arithmetic, such as place value, fractions, decimals, percentages, and basic geometry. They can apply these concepts to solve simple problems in various contexts, including budgeting, measurement, and basic calculations.
  4. Proficiency in algebraic concepts: Individuals with this level of mathematical ability understand and can manipulate algebraic expressions, solve equations, and work with basic functions and graphs. They can apply algebraic principles to solve problems in real-world scenarios, such as analyzing patterns, making predictions, and solving basic engineering or scientific problems.
  5. Competency in geometry and trigonometry: This score indicates proficiency in geometric concepts, including shapes, angles, areas, and volumes, as well as trigonometric functions and identities. Individuals at this level can apply geometry and trigonometry to solve problems involving measurement, spatial reasoning, and trigonometric calculations, such as calculating distances or angles in real-world applications.
  6. Understanding of calculus and its applications: At this level, individuals have a solid understanding of differential and integral calculus, including concepts such as limits, derivatives, and integrals. They can apply calculus techniques to analyze rates of change, optimize functions, and solve advanced problems in physics, engineering, economics, and other fields.
  7. Proficiency in statistics and probability: This score indicates competence in statistical analysis, including data collection, organization, analysis, and interpretation. Individuals at this level can apply statistical methods to analyze data sets, make predictions, test hypotheses, and draw conclusions in various fields, such as economics, social sciences, and research.
  8. Mastery of advanced mathematical topics: Individuals with this level of mathematical proficiency have a deep understanding of advanced mathematical topics, such as linear algebra, differential equations, discrete mathematics, and complex analysis. They can apply advanced mathematical techniques to solve complex problems in fields such as physics, engineering, computer science, and theoretical mathematics.
  9. Expertise in specialized mathematical fields: At this level, individuals have specialized expertise in specific branches of mathematics, such as number theory, topology, mathematical logic, or applied mathematics. They can conduct original research, develop new mathematical theories or techniques, and contribute significantly to advancements in their respective fields.
  10. World-class mastery and contribution to mathematics: This score represents the highest level of mathematical achievement, reserved for individuals who have made groundbreaking contributions to the field of mathematics, such as Fields Medalists, Abel Prize winners, and other distinguished mathematicians. They have made significant discoveries, solved long-standing mathematical problems, and significantly advanced the frontiers of mathematical knowledge and understanding.

To assess these scores, we may cite directly from Chat-GPT, which stated that “these scores provide a framework for evaluating the level of mathematical knowledge and proficiency required or demonstrated in various professions, with higher scores indicating more advanced mathematical skills and expertise”.

To analyze, the results above have shown a huge disparity between the mathematics level required for most professions. Above is the bar chart for the data, with a mean of 3.6 and a standard deviation of 1.6. Notably, the majority of professions listed in the ISCO-08 framework demand only a modest level of mathematical skill, typically around a score of 2. This stands in stark contrast to the mathematical content covered in secondary education, where students are often taught up to a level equivalent to a score of 5. This raises questions about the connection between the mathematical skills taught in academic institutions and those necessary for the majority of occupations in real-world settings, implying a potential necessity for a more customized approach to mathematics education. Such an approach should align more closely with the varied mathematical demands of different professions and adequately equip students with the appropriate skills for their chosen career paths.

Now we head onto some of the criticisms received regarding the current curriculum. The first and arguably most important aspect that we are heavily lacking in currently in our curriculum is the limited relevance of mathematical concepts to real-world applications, resulting in disengagement. AO3 makes a valid attempt at tackling this problem at the Mathematics GCSE level – requiring pupils to ‘solve problems within mathematics and in other contexts’. However, it only takes up 25% of the total marks across all exam boards – problematic when considered that this may be the hardest mathematics exam one will ever sit. Again, tackling this problem is of utmost priority as the exam must be able to prepare all students for their future post-16 choices.

I would also like to cite a comment here from u/banana_grandmaster regarding this issue. This sentiment was echoed in my previous rendition of this post, where I proposed the creation of a new qualification called “Applicational Mathematics” where it focuses on real-life usages of mathematics. This will be elaborated further into this post.

“For example, and this is probably quite controversial and it’s very possible I will change my mind on this, but I actually think solving quadratics is not that important for most people. I agree on learning algebra up to (linear) simultaneous equations in two variables (including graphical perspective), but if there something to cut out of GCSE algebra it is all the time that is spent towards solving quadratic equations.”

u/banana_grandmaster’s comment on my Reddit post linked above

In stark contrast, another definitive lack in the curriculum we now observe is the sheer lack of ‘proof-oriented’ questions. Many students are not exposed to enough proof-oriented questions, which are crucial for developing logical reasoning skills and understanding the underlying principles of mathematics. In order to answer proof-oriented questions, students must provide justifications using logical arguments and mathematical reasoning, rather than relying on memorized algorithms or formulas. Through working with proofs, students develop skills in systematic problem analysis, pattern recognition, and constructing coherent arguments to support their solutions. At A-Level this is much more emphasized on, though more emphasis on this at lower levels would be beneficial for students in their understanding of logical relationships.

We must also discuss teaching quality. The inability for teachers to explain topics in a way intuitively has also been one of the main reflected concerns for the current curriculum. Teaching math effectively requires more than just presenting information or demonstrating procedures. It requires the ability to clearly communicate abstract concepts, connect new ideas to prior knowledge, and engage students in meaningful learning experiences.

Yet, not all teachers have the necessary pedagogical skills, subject knowledge, or resources to meet these expectations. Educators may face difficulties in simplifying complex concepts, which can result in students feeling confused or disengaged. Some may focus too much on rote memorization or procedural methods, disregarding the importance of developing conceptual understanding and problem-solving abilities. As a result of this recurring issue, students are often left with a lack of understanding and comprehension of the topic at hand.

For students who may face difficulties with mathematics, the limited time allotted for comprehending topics in the curriculum, which may prioritize breadth over depth, can impede their ability to fully grasp each concept. It is possible that they may require additional time and support to fully comprehend each concept and establish a strong understanding. Nevertheless, the constraints of the curriculum often allow for limited opportunities for extensive examination and improvement.

Consequently, students who encounter difficulties with mathematics may experience falling behind or feeling overwhelmed by the pace of instruction. It is possible that they may experience difficulty keeping pace with their classmates, resulting in emotions of frustration, anxiety, and disengagement from the subject. Moreover, the emphasis on covering a broad range of topics within a limited timeframe can lead to a superficial treatment of concepts. In the current educational climate, teachers may feel immense pressure to prioritize surface-level understanding and procedural fluency, often at the expense of fostering deeper conceptual understanding and honing problem-solving skills among their students.

As a result, students may perceive mathematics as a series of disconnected rules and procedures to be memorized rather than a coherent and meaningful subject. This shallow understanding can lead to disengagement and frustration, as students struggle to make sense of mathematical concepts and apply them in real-world contexts.

The repetitiveness in the curriculum for certain topics once again emphasizes this point. Within the current mathematics curriculum in the UK, specific topics are repeatedly revisited throughout various grade levels or key stages. The frequent repetition of tasks may result in a tendency to prioritize procedural fluency and superficial comprehension, rather than promoting a deeper grasp of concepts and problem-solving abilities.

For instance, subjects such as fundamental arithmetic operations (including addition, subtraction, multiplication, and division), fractions, and algebraic equations are initially introduced during primary education and revisited in subsequent years. While this repetition can be beneficial for reinforcing foundational skills, the curriculum quickly become repetitive or stagnant, particularly those who grasp the concepts quickly, resulting in boredom and disinterest among students.

Furthermore, the prioritization of standardized testing and assessment structures, such as the Key Stage assessments and GCSE examinations, can perpetuate this cycle of repetition. Educators may feel obligated to give priority to teaching for the purpose of standardized testing, concentrating on particular subjects and methods that are expected to be included in evaluations, rather than delving into mathematics in a more comprehensive and investigative approach. To address this issue, there is an immediate need to rethink the structure and delivery of the mathematics curriculum to ensure a balance between repetition and exploration.

Part III – What do we do?

“Why do children dread mathematics? Because of the wrong approach. Because it is looked at as a subject.”

— Shakuntala Devi

In January 2023 MEI (Mathematics in Education and Industry) published its view on maths to 18, making the case that young people need confidence in using and applying maths, for example in dealing with their finances and being able to interpret data and statistical charts, where the ‘Core Maths’ qualifications were specifically designed for this purpose. For students choosing not to progress into academia, such qualifications should prove to be highly beneficial. The public’s knowledge of this however will need to be heavily bolstered, given that ‘Core Maths did not appear to be a priority for most survey respondents’ in 2.

I would also like to see the introduction for more ‘useful’ mathematics as some would deem, such as Fermi estimation, a technique used to make rough estimates of quantities or quantities that are difficult to measure directly. This involves breaking down complex problems into simpler components and making reasonable assumptions to arrive at an approximate solution, which is much required in daily life as opposed to tedious but accurate calculations. This is a valuable tool for developing intuition, problem-solving skills, and critical thinking abilities, something I find myself using way more in daily life than actual calculations. While these estimates may not be precise, they can provide useful insights and help guide decision-making in situations where exact solutions are impractical.

Additionally, we reached out to NCETM (The National Centre for Excellence in Mathematics) and Professor Sue Gifford at ECMG (The Early Childhood Maths Group) for their insight, for which I am sincerely grateful. The former suggested a ‘mastery curriculum’ in practice, which would feature “back and forth interaction, including questioning, short tasks, explanation, demonstration, and discussion, enabling pupils to think, reason and apply their knowledge to solve problems”3. Charlie Stripp, the Director for NCETM and MEI, has been a strong advocate for continuous mathematics for all, especially for post-16 studies – for those interested 4 and 5 are well worth a read.

Professor Gifford from the University of Roehampton on the other hand specialises in early mathematics education and was very kind to redirect me to some of the more recent works about a reformation in the mathematics curriculum, more specifically ACME (Advisory Committee on Mathematics Education) at the Royal Society, where a draft discussion paper was released in 20236, where a push for mathematics education up to 18 and a new need for pathways and qualifications for students with different needs was also mentioned. Linked below is Professor Gifford’s talk, where she highlights the priorities for younger learners in their mathematics journey.

Recent developments in education have also highlighted the integration of English and Mathematics, particularly for younger students. Helen Drury in 7 and 8 by Number Sense Maths have emphasized on this point by pointing out the similarities between the construction of phonics and numbers. By considering calculations as the construction of words, this methodology offers far greater clarity and consistency, leading students through a sequence of strategies that build upon each other to develop much-needed proficiency in mathematical operations.

We would also like to see more proof-based problems into the curriculum, starting from early stages and gradually increasing complexity as students progress. Encouraging students to explore and construct proofs not only fosters a deeper understanding of mathematical concepts but also prepares students for their further studies in STEM subjects and beyond where logical thinking is required.

To truly promote student’s skills at learning, schools will also need to provide more time for students to explore topics at their own pace, bolstering their own understand and promoting self-learning abilities, which are among the most crucial skills for academic success. By allowing for sufficient time for exploration, students have the opportunity to delve much deeper into mathematical ideas at their own pace, creating a more relaxed and in ways more productive atmosphere. By doing so, teachers can emphasize depth over breadth, ensuring that students are able to grasp fundamental concepts thoroughly before moving on to new topics.

Finally, we will also need to introduce innovative teaching methods and resources to keep lessons engaging and dynamic. By integrating a variety of learning opportunities, such as interactive simulations, group projects, and hands-on activities, student engagement and motivation can be sustained. Furthermore, through consistent review and updating of curriculum content, we can aim to maintain its relevance and level of difficulty. For example, solving quadratics, as u/banana_grandmaster mentioned, is not a crucial skill in our daily lives, particularly with the convenience of mobile phones. How we translate such nuances into a better curriculum will prove to be crucial in this modern age where the technological landscape is fast-changing.

Part IV – Our Suggestions and Conclusions

“One should study Mathematics because it is only through Mathematics that nature can be conceived in harmonious form.” 

— George David Birkhoff

With all of the above considered, the mathematics curriculum cannot continue in its current state. One I would like schools to consider is grade acceleration for particular students. This provides intellectually gifted students with the opportunity to be challenged and engaged at a level that matches their abilities, preventing boredom and disengagement in the classroom. Accelerated students have the opportunity to receive instruction that is more closely aligned with their academic needs and interests, enabling them to advance at a pace that optimizes their learning potential. This option also proves to be cost-effective for the school, as it does not require significant additional resources, reducing the current need for specialized programs or interventions.

We would also like to propose two different qualifications, the first of which would be “Applicational Mathematics”, following the Core Maths9 and the Mathematics in Context curriculum offered by Pearson10. This will focus primarily on highlighting and strengthening the practical use of theorems and methods, covering a wide range of topics including but not limited to financial mathematics and statistics, and also introduce project-based work. This should effectively prepare students for the mathematical challenges they may face in their post-16 educational pursuits and be of much greater use for diplomas such as BTEC or apprenticeships.

For students wishing to approach mathematics from a more theoretical perspective, we would like a new qualification – “Pure Mathematics”, adapting the existing GCSE curriculum. Even though we appreciate the existence of the Further Maths and Additional Maths (OCR) qualification currently at KS4, most schools currently do not offer either. As someone who has taken all three current GCSE mathematics qualifications in one year, I would say that it is beneficial for those who would like to go on to do maths at a higher level. Thus, by merging these qualifications, we effectively streamline the pathway for students interested in pursuing mathematics at a higher level, placing a much stronger emphasis on theoretical rigor and preparing them for the demanding nature of post-16 mathematical studies and beyond.

However, the implementation of the above measures is highly constrained by current government funding limitations, further exacerbated by the additional mandate proposed by Prime Minister Rishi Sunak to make mathematics compulsory for all students in England up to the age of 18. In light of these challenges, it is imperative that the government prioritize increased funding to support these changes and ensure their successful implementation. As a closing note, I would like to recommend some resources available for teachers and students, while also urging the government to provide and back more funding for said changes.

To provide a better teaching and learning quality (at least for the short term), I would encourage teachers and students alike to leverage technology, utilizing services from government-funded initiatives such as the NCETM and the Millennium Mathematics Project (NRICH) based at the University of Cambridge. These online learning platforms enhance the accessibility of mathematics education while reducing delivery costs. Additionally, educators can explore professional development courses to enhance their teaching proficiency and deepen their understanding of mathematics education, for example “Teaching Secondary Mathematics” at the Open University which I have personally completed. These strategies can offer pragmatic solutions to navigate funding constraints and enhance the quality and accessibility of mathematics education.

In the rise of modern technology, mathematics has never been more crucial. Quoting Yuval Noah Harari in this interview with Stephen Colbert – “Today, nobody has any idea what to teach young people anymore that will still be relevant in 20 years.” If we truly want a better future for our younger generation, we must do more as a society and push for an education reform – and one that will still be relevant in 20 years, if not for life.

Thank you all for reading, and I will see you next month on April 15th!

  1. https://isaactengchithang.files.wordpress.com/2024/03/profession.zip ↩︎
  2. https://mei.org.uk/app/uploads/2021/08/Evaluation_of_the_AMSP_May_2020.pdf ↩︎
  3. https://www.ncetm.org.uk/teaching-for-mastery/mastery-explained/the-essence-of-mathematics-teaching-for-mastery/ ↩︎
  4. https://www.ncetm.org.uk/features/why-study-maths-at-university/ ↩︎
  5. https://www.ncetm.org.uk/features/maths-to-age-18-why-everyone-should-welcome-it-and-what-we-can-do-to-help-make-it-happen/ ↩︎
  6. https://royalsociety.org/topics-policy/projects/mathematical-futures ↩︎
  7. https://www.tes.com/magazine/teaching-learning/primary/primary-maths-needs-more-phonics ↩︎
  8. https://numbersensemaths.com/blog-and-research/why-number-fact-teaching-should-be-thought-of-as-phonics-for-maths ↩︎
  9. https://amsp.org.uk/teachers/core-maths/curriculum/ ↩︎
  10. https://qualifications.pearson.com/en/qualifications/edexcel-mathematics-in-context/mathematics-in-context.html ↩︎

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