2、故事性

不要一开始就给学生讲一堆抽象的结论，应该将每一个话题讲得有趣味，而且要让学生知道，他们要学习的内容为什么是有用且有意义的。这么做不仅可以使课堂更有趣更能激发学生思考，也能使所学习的内容更容易被记住。

故事可以来源于实际生活（如预测天气）、历史事件（如珠穆拉玛峰的高度测量）、数学谜题（如哪种形状能够密铺空间）、甚至可以用小说里的人物。故事里甚至可以有一些悬疑和曲折，学生们一开始不知道故事的结局，最后的数学结论会使他们感到惊讶。

3、探索和创造

数学的学习过程往往比结果和知识更重要。学习过程可以培养学生解决问题的能力、逻辑推理能力、批判性思维、抽象和归纳的能力。这些能力往往可以迁移到生活的方方面面，即使具体的数学知识和解题技巧在生活中并无运用。

要获得这些能力，学生应该能自由地探索和创造。在老师的帮助下，学生能够成为数学规律的发现者、新想法的创造者，而不仅仅是流程和结果的被动消费者。

5、无用但有意义

学校里的数学，大多数在生活中都用不上，即使学生以后成为科学家或软件工程师。其实这也没关系，学习数学的一个重要原因是解决问题的能力和批判性思维的训练。学校里“无用的”科目其实很多，如莎士比亚的十四行诗、莫扎特的交响乐、牛顿的运动定律等。事实上，这些科目教给我们文化和历史，帮助我们理解所生活的世界，使我们的生活有意义。

然而，现在许多数学课毫无意义，这才是问题所在。与其教授乏味又无意义的数学，如长除法、分母有理化、两栏式几何证明，不如教一些令人激动又很美的数学，如图论、混沌、数据科学、密码学或博弈论，这些数学对我们的生活有更直接的影响。他们能帮助学生更好地理解我们生活的世界，即便在日常生活中他们并没有直接被应用。

7、直觉比严谨和熟练更重要

数学家们研究课题时，他们可能首先想到数学的严谨和数学证明的层次；高中生思考数学时，可能首先想到解题的熟练程度。事实上，我们不需要从学校的数学课上学到这两种技能。

关键点应该集中在数学直觉和理解力上：预估问题的答案、验证已经存在的答案、辨识并归纳出数学模式、推导出记不清处的过程和等式、意识到大众的错误和误解（尤其是概率和统计这样的数学分支）。

数学上的许多概念会有多种表达形式（如分数和阴影面积、小数、百分数、群、比率之间的关系），学生应该尽可能地熟悉各种数学表达形式，理解它们的关系，并能够知道，一个具体问题用哪一种数学形式更合适。

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

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.

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.

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.