Increasingly, I’m starting to think that there not being a right way, merely a bunch of wrong ways, is why we need humans. How a teacher explains something is also linked to their own personality and preferences in ways that are non-trivial!
On the negative numbers example, I am still trying to work out how helpful the double sided counters are. They are a good model… but to what extent are they useful in introducing the concept and to what extent does it add unhelpful complexity?
The tension described here between comprehensibility and generalisability is one that curriculum designers live with constantly, and it rarely gets named as precisely as this. What sits behind so many curriculum debates is exactly this: not one side being wrong, but both sides holding a principle that is genuinely true, with the real question being which matters most for this concept, with these learners, at this point in the sequence. Is this perhaps why curriculum design remains more art than science, and why the most experienced practitioners are often the hardest to convince?
I think it all depends on the learner, not the teacher.
Coming in with a set routine independent of who you are teaching is always going to be hit and miss. But if you start from where the student's understanding is and build on it, it's more effective.
Novices aren't blank slates. They come with a worldview, and introducing something that contradicts it will confuse them. If you help them refine and expand their existing model rather than replace it with yours, there's less friction.
As more often a physics teacher filling in gaps or lapses of say vectors or my most recwnt bug-bears N/m^2 wt al I am both fearful and resentful if this be the case:" It matters how the first representation of something is presented, because it becomes the object through which much of what follows is interpreted."
Increasingly, I’m starting to think that there not being a right way, merely a bunch of wrong ways, is why we need humans. How a teacher explains something is also linked to their own personality and preferences in ways that are non-trivial!
On the negative numbers example, I am still trying to work out how helpful the double sided counters are. They are a good model… but to what extent are they useful in introducing the concept and to what extent does it add unhelpful complexity?
The tension described here between comprehensibility and generalisability is one that curriculum designers live with constantly, and it rarely gets named as precisely as this. What sits behind so many curriculum debates is exactly this: not one side being wrong, but both sides holding a principle that is genuinely true, with the real question being which matters most for this concept, with these learners, at this point in the sequence. Is this perhaps why curriculum design remains more art than science, and why the most experienced practitioners are often the hardest to convince?
I think it all depends on the learner, not the teacher.
Coming in with a set routine independent of who you are teaching is always going to be hit and miss. But if you start from where the student's understanding is and build on it, it's more effective.
Novices aren't blank slates. They come with a worldview, and introducing something that contradicts it will confuse them. If you help them refine and expand their existing model rather than replace it with yours, there's less friction.
As more often a physics teacher filling in gaps or lapses of say vectors or my most recwnt bug-bears N/m^2 wt al I am both fearful and resentful if this be the case:" It matters how the first representation of something is presented, because it becomes the object through which much of what follows is interpreted."