This article inspires me from the following points. I will rethink how to question my students, how to explain curriculum content, and how to plan my lesson.
1. "However, these are stories invented by teachers mainly as memory aids rather than justifications." (p.3)
If students feel it is hard to memorize names, I think inventing stories as memory aids is a great idea. This is what I am always doing to help my kids and myself memorize arbitrary things. When I was teaching my children Chinese characters and ten digits (0, 1, 2…9), I often invented some stories related to their structures or shapes. These stories work as labels that are attached to those arbitrary things. When we want to recall some arbitrary things, our brain could find them by their labels. However, I think most labels and symbols are not arbitrary. Most of them make sense, and it is easy to explain their meanings. For example, the symbols of "greater than >" and " less than< ", "perpendicular ⊥" and "parallel ∥", "angle ∠ " and "triangle Δ", and so on. For those arbitrary labels and symbols in the mathematics curriculum, such as "division sign ÷", I will encourage students to invent labels by themselves to help memorize. I personally don't suggest teachers invent stories for students, because the brains of different people work in different ways. Students, especially high school students, need methods more than to be informed of some things.
2. " Those things which are necessary can be worked out: it is only a matter of whether particular students have the awareness required to do so. If not, then maybe it is not the best choice of topic to be taught at that moment in time." (p.4)
This explains what an invalid lecture looks like. Invalid means that the content of the lecture doesn't match for students' level of understanding. For example, if I taught five laws of exponents, I asked students to calculate (3/n^2)^4. To solve this expression, students needed to use two laws to solve this single problem. The majority of students should have the awareness required to do so, otherwise, I think I need to add extra lessons to teach them some basic mathematical knowledge before teaching laws of exponents. If I did not notice this circumstance and not add any extra lesson, students would feel confused when they doing practice.
3. "Inviting students to 'think about it' is appropriate for what is necessary, but not for what is arbitrary." (p.5)
Questioning is an important assessment method. If we ask students to think about something that is arbitrary, we are just asking students to recall in their memories for something they cannot remember. When we ask students questions, we need to make sure that "it is possible for students to use their awareness to try to work out for themselves" (p.5). These questions are necessary and in the realm of awareness. For example, if I ask "what are the four basic arithmetic operations," this is a question for something is arbitrary. If I ask " which one of the four arithmetic operations is the inverse operation of addition," this is a question for something is necessary. I think the second question is the one that students need to work out.
Good! I'm glad that you are thinking deeply about this article, and I think it still has more to offer. Think about giving students a great deal more responsibility (within constraints, naturally) for making sense of new ideas that are logically necessary within the topic you are studying. This involves more than simply giving them exercises or questions at an appropriate level -- it also means that some of the big concepts can actually be worked out collaboratively by the students, with their peers, and with way less 'feeding' of information and explicit scaffolding from you. You can do this! It does take some courage to let go of the role of classroom authority and expert, and you'll want to start on a small scale, but you'll be surprised at how much kids can work out for themselves, if they have good open-ended questions, sufficient background (but not too much), and a motivation to try! I'd love to hear how this goes as you try it out on practicum.
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