教授猫头鹰代数：在Duolingo上发展数学

You've heard of Spanish or vanish, but what about divide or hide? Last year, Duolingo launched a new Math course in an effort to expand our mission beyond language, into subjects we believe have the power to create new opportunities for our learners. Along the way, our team has discovered that teaching math comes with its own unique challenges – and we've been developing a formula that we're excited to share.

你听说过西班牙语或消失，但除法或隐藏呢？去年，Duolingo推出了一个新的数学课程，以努力将我们的使命扩展到语言之外，进入我们相信能够为学习者创造新机会的学科。在此过程中，我们的团队发现教授数学有其独特的挑战——我们一直在开发一个公式，我们很高兴与大家分享。

How do we teach language?

我们如何教授语言？

In our language courses, each lesson is constructed from knowledge components – you can think of these as words, characters, or phrases. While the prompt and input of a given exercise may change, the knowledge component itself will always be represented as text.

在我们的语言课程中，每节课都是由知识组件构成的——你可以将这些看作是单词、字符或短语。虽然给定练习的提示和输入可能会改变，但知识组件本身将始终以文本形式呈现。

What makes math different?

数学有什么不同？

For math, this gets more complicated. We can't represent all content in the same way, and how we teach depends on the context.

对于数学来说，这变得更加复杂。我们不能以同样的方式表示所有内容，教学方式取决于具体的情境。

For example, addition lessons may represent the equation using countable blocks, integers, and word problems. Rectangular perimeter lessons – while technically rooted in addition – look significantly different because the context has changed.

例如，加法课程可能会使用可数的积木、整数和文字问题来表示方程式。矩形周长课程——虽然技术上根植于加法——看起来显著不同，因为情境已经改变。

Geometry adds a whole other layer of complexity. Addition and perimeter are based on numbers and equations – but what about classifying a trapezoid? We need new visuals and exercises to explain these concepts!

几何增加了另一层复杂性。加法和周长是基于数字和方程的——但如何分类梯形呢？我们需要新的视觉效果和练习来解释这些概念！

So, "knowledge components" in math don't have a consistent form like they do in language. Instead, visuals and exercises need to change depending on the lesson's difficulty and context. 

因此，数学中的“知识组件”不像语言中那样具有一致的形式。相反，视觉效果和练习需要根据课程的难度和情境进行变化。 

This introduces some interesting engineering challenges, particularly in how we can develop m...