I found this article to be a great source of inspiration and advice for my lesson plans. My SA gave me a lot of similar advice.
For the subject of Math, there is a lot of knowledge that is arbitrary, that is not regular in terms of what needs to be helped for students to memorize. For example, the author uses the example at the beginning of the book to give examples of naming shapes, defining units, units of measurement, etc. This information is unfamiliar to all students and needs to be told. The teacher's role then is to help students find ways to remember these meanings and content that work for them.
However, the "necessary" parts are the ones that students can work out by their own exploration and thinking (for some able students), and these parts do not rely on memorization/ or partial memorization, but more on understanding. For example, the author gives conversions of formulas, solving triangles, etc.
"Necessary" does not mean memorizing facts, but rather logically deducing or intrinsically understanding from some given mathematical property. It would be helpful if students could explore certain aspects by reasoning from what they already know. The sense of achievement of successful pushback will bring great interest in maths to students.
When I was in high school, there were some so-called "formulas" that I had to memorize. My teachers didn't tell us that we needed to memorise all of them, such as the Pythagorean theorem, which students need to memorise in order to solve triangles. However, if this formula is memorized as "arbitrary", it will be easily forgotten. I think this can be deepened by showing the derivation process of the theorem, which not only makes students remember the theorem more deeply but also consolidates other related mathematical knowledge.
(e.g. using this proof of the area of a triangle to derive it)
As stated in the text, a careful explanation will improve students' ability to use their prior knowledge and experience to understand why this "accepted fact" is correct. This will shift the way they think about learning maths from memorizing and reciting mathematical formulas and facts to possibly trying to understand why these formulas and facts are correct. This will help them in their future learning of maths and, because they can understand, their interest in learning maths will be greatly enhanced.
Thanks Chernie. I think you understood part of this article: that some things are arbitrary (i.e., they can't be derived logically) and that we might as well just present them to kids. But Hewitt wasn't actually talking at all about whether or not to explain the source of formulae! And you haven't said anything about the 'necessary' -- those things that learners can actually figure out for themselves by using their mathematical thinking powers.
ReplyDelete** I would like you to revise this post by rereading the article and writing an additional paragraph or two about what Hewitt means by 'necessary', and how this might connect with your teaching!
OK, this is alright. There is something you have missed here from Hewitt -- and that is that it's NOT about the teacher explaining (however carefully and clearly), but about the teacher creating problems and activities where the students themselves can figure out the necessary knowledge using their own rational thought, logic and experimentation. In actual practice, we will probably choose some of the necessary knowledge for students to derive and work out on their own, and still explain a few things, to allow for enough time for the exploratory activities.
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