# Nine Principles For Great Mathematics Content

## 1. Learning Should Inspire

Mathematics should inspire and empower students, not scare or confuse them. We should show the surprising beauty and great power of mathematics – and that *everyone* can “do maths”.

## 2. Tell a Story

Storytelling can motivate students, make the content more memorable, and justify why what you’re learning is important – including real-life applications, curious puzzles, or historical background. More…

## 2. Tell a Story

Rather than presenting mathematics as an abstract collection of results, we should introduce every new topic with an interesting narrative that shows students why what they are about to learn is useful and worthwhile doing. This is not just more interesting and motivating, but it also makes the content much more memorable.

Stories could be based on real-life applications (“predicting the weather”), historical events (“measuring the height of Mount Everest”), a mathematical puzzle (“which shapes tessellate”) or even fictional characters. There could even be some suspense and plot twists, where students don’t initially know where the story might lead, and are later surprised with an unexpected mathematical result.

## 3. Exploration and Creativity

Allow students to explore, be creative, make mistakes, practise critical thinking, and discover new ideas – rather than just telling them the final results and procedures to memorise. More…

## 3. Exploration and Creativity

In mathematics, the *process* of learning is often more important than the actual results and knowledge: it teaches students problem-solving, logical reasoning, critical thinking, abstraction and generalisation. These skills are transferable to many other parts of life, even if the specific mathematical topics have no real-life applications.

To make this most effective, students should be able to freely explore and be creative. With the guidance of a teacher or tutor, they should be able to discover new patterns and ideas on their own, and not just be the consumers of pre-packaged results and procedures.

## 4. Mathematics is Everywhere

We are always surrounded by mathematical patterns and relationships. Students should be able to recognise these, and harness the power of maths to solve problems in everyday life.

## 5. Not *Useful*, but *Meaningful*

Not all topics in the curriculum have to be *useful* in everyday life (neither are Mozart or Shakespeare), but every topic should be *meaningful* – because of its applications or mathematical significance. More…

## 5. Not *Useful*, but *Meaningful*

A large part of the mathematics that students learn at school won’t be **useful** in everyday life, even if they end up working as a scientist or software engineer. And that’s ok – as we’ve seen in principle 3, one of the reasons to study mathematics is to learn transferable skills like problem-solving and critical thinking. There are many other subjects in school that are also not “useful”, from Shakespeare’s Sonnets to Mozart’s Symphonies, and even Newton’s laws of motion. Instead, these subjects tell us about culture and history, or they help us understand and make sense of the world around us.

However, a lot of the *existing* mathematics curriculum is also not **meaningful** – and that *is* a problem. Rather than teaching about boring and essentially meaningless topics like long division, rationalising denominators or two-column geometry proofs, we should teach about networks, chaos, data science, cryptography, or game theory: topics which are exciting and beautiful, and which have a direct impact on all our lives. They help students better understand the world we live in, even if they are not directly useful in everyday life.

## 6. Mathematics is Visual

Equations are useful, but there are often much better representations of mathematical concepts and relationships. The content should be as visual and colourful as possible.

## 7. Intuition over Rigor or Fluency

Rigor is an important part of mathematics, and there is also a place for practising fluency – but the main goal should be to develop intuition, deep understanding, and general numeracy. More…

## 7. Intuition over Rigor or Fluency

When mathematicians think about their subject, they might primarily associate the rigor and formality of proofs. When high-school students think about mathematics, they might associate fluency problems when preparing for exams. In reality, neither of these two approaches are what we need from school mathematics.

The focus should be much more on *mathematical intuition and understanding*: estimating the answer to problems or verifying an existing answer, recognising and generalising patterns, deriving procedures and equations which you don’t remember exactly, and being aware of common mistakes and misconceptions (especially in topics like probability and statistics).

Many concepts in mathematics have a wide range of different representations (e.g. fractions as shaded areas, decimals, percentages, groups or rates). Students should be familiar with as many representations as possible, understand their relationships, and be able to decide which one is most suitable for a specific problem.

## 8. Discussion and Teamwork

Mathematics is rarely a solitary pursuit, and many real problems don’t just have a single, correct answer. Discussions, collaboration and teamwork should be a key part of every curriculum.

## 9. Mathematics is Alive

To make mathematics more relevant, it is important to portray its history, recent discoveries, and current research – as well as the diverse groups of mathematicians and scientists doing this work.