Manifesto on the Teaching of Mathematics
[We] must not introduce any topic for which we cannot first convince the students that they should want to pursue it.
We learn when we are challenged, when we push ourselves. If you’re not stuck you’re not learning. If it’s not a struggle you’re not doing it right.
[The] goal of teaching is independent thought. We want students to be able to think and reason and apply what they know in new situations.
Concrete before abstract.
You must understand the layers of abstraction to the concept you’re trying to teach. You must teach in terms of the lower levels of abstraction before the higher. For instance, if you’re teaching fractions, you should teach in terms of fractional quantities of a thing, like a pie, before teaching in terms of the numbers. Teach the abstract, then the concrete. 3b1b shows a pie chart next to the equations which represent it. It is also important to teach the reader
- how the abstract relates to the concrete
- how the rules in the abstract make since in terms of the concrete
- why they should work in the abstract space (because it requires less thinking)
Try very hard to structure your explanation to go from the concrete, to the abstract.
Topic choice over production quality.
You spend 1% of your time choosing your topic and 99% of your time writing about it, but it’s the topic that drives interest. Spend more time than you otherwise would choosing a topic.
When you’re just starting out, choose something more esoteric and specific than you might be inclined to. It builds a more loyal audience. Also, it might not be as niche as you think, especially if you’re passionate about it.
Know a genre.
There are different types of explanitory writing. There is
- the well-researched explainer, where you document a subject for which you are an expert
- the journaling learner, where you document your own learning process
- the walk-through guide, where you walk the reader through an explicit example
- the demo presenter, where you demo a concept for the purposes of inspiration Be aware of your own experties, and follow a format for which you are qualified.
Definitions are not the beginning.
Make sure the definition is well motivated. Explain why that’s the definition and what else it could have been.
Production quality > 0
Sound quality matters. Try to use a good microphone.
On the macro scale, you need to hook someone into the video as a whole. This is the hook. Ask an interesting question, or quickly relate it to a real-world problem. You can also ask a “nerd sniping question”—one to which the audience has to know the answer. You can also show the historical significance of a topic. Finally, you can make the learner feel like they’re taking part in the learning. This last option likely only works in interactive media.
Motivation is critical, but it shouldn’t take long. Getting to the core lesson can often capture your audience better than a long-winded pitch. If you can, motivate using clear examples, not sweeping statements of promises of things to come.
On a micro statement, you need to motivate every new lesson in the context of the overall explanation. Before you introduce any new topic, try outlining the main idea and intuition of the topic. Build up to any esoteric, abstract concepts and motivate their existence along the way. Try starting with a naive but flawed solution, and then progressively refine it.
If Motivation determines how much focus a student is willing to give you, clarity determines how quickly you expend that focus. A clear lesson will retain the students focus throughout the lesson.
Keep music quite and decidedly in the background.
Try to be unique. Find your own voice and asthetic. But, most importantly, try to be original in your content. You should either have a unique topic, or a unique perspective.
This is subjective, but try to be memorable. Ask a question that’s fun to think about, or provide an answer so satisfying that it sticks with you long after the lesson.