Context: The ability to find the volume of a cylinder would allow you to determine the capacity of many different types of containers (e.g. cup, water bottle, rainwater tank). Furthermore, because you have already learned how to find the volume of a cube and a rectangular prism in the last lesson, this will extend your general ability to work with real world objects. Note: “Because your teacher wants you to”, is not a context! Neither is, “because it’s the next lesson of the chapter/book”.
Some exemplary SuperMemo items for reinforcing the context may look like this:
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Q: In order to find out how much water these objects can hold you must be able to find the volume of a [...]
A: cylinder
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Q: In order to find out how much water these objects can hold you must be able to find the [...] of a cylinder
A: volume
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Procedure:
The example in the book says:
There is a rainwater tank of height 3m and diameter 4m. What is its volume?
The general formula for finding the volume of an object with a uniform cross-section is: volume = (area of base) x (height)
The base is a circle, so:
(area of base) = πr2
--> (area of base) = π*(4/2)2
--> (area of base) = 12.6m2
volume = (area of base) x (height)
--> volume = 12.6 x 3
--> volume = 37.8m3
After you read through this, cover it up and re-write the 3 steps on a sheet of paper. Once you get the answer, check it against the example. If you get it correct, great! If not, check which step you missed and try again.
The SuperMemo item can simply be:
Q: A cylindrical rainwater tank is 3m tall and 4m wide. What is its volume?
A: 37.8m3 (pg440)
Answer this item by writing out the entire solution. The page reference is there in case you forget the procedure and need to re-learn the procedure. Alternatively, you can simply write the entire solution in the answer section of your SuperMemo item. Although there’s more clutter that way, you can get rid of your books, which is a big plus.
If you consistently have trouble with the second step, you can formulate it as a separate item. For example:
Q: A circle has radius 2m, what is its area?
A: 12.6m2
However, this does not mean that you should learn the primary procedure as a group of smaller ones. Instead, you should learn both the primary procedure (i.e. volume of a cylinder) and the secondary one (i.e. area of a circle) as separate items. This increases redundancy and makes it less likely that you will ever have to go back to the book, even if you must relearn the primary procedure.
Rationale:
Step 1 is not much more than a definition. It can be formulated as follows:
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Q: How can the volume of this shape be calculated? Hint: it has a uniform cross-section
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Since the height is already known, step 2 simply involves determining the area of the base of the cylinder. Hence:
Q: What shape is the base of a cylinder?
A: a circle
Q: If a circle has radius r, what is its area?
A: πr2
Q: If a circle has diameter D, what is its radius r?
A: r = D/2
The last step simply involves substitution of the height and area into the initial formula. If you wish, you could formulate this as:
Q: Given a cylinder with a base area of 12.6m2 and a height of 3m, what should you do to find the volume?
A: multiply them
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Hopefully you were able to get through this example and get a feeling for the way procedural knowledge can be acquired, formulated and reviewed blindly and independently, and yet intimately combined with a declarative overview.
Please leave any comments below.
What about artistic skills such as sketching and perspective practice?
ReplyDeleteThanks for your question.
ReplyDeleteI sense that it comes from a feeling that artistic skills are less well-defined than mathematical skills, so you would like to know how to apply the technique to something more fluid...? If this interpretation is correct, I would say that although this idea is somewhat true, it is also not as big of a deal as it seems at first.
In many cases, the hardest thing about formulating procedural knowledge is that it seems so fluid - there seem to be no hard edges. This applies to maths as much as anything else. After all, there are so many ways to solve the same mathematical problem (the geniuses are the ones who find the shortcuts!), as there are to draw the same visual concept, or play the same musical theme on a piano, or encode the same solution as a software algorithm, or... on and on. This is why I said in the "procedural knowledge" post that "a procedural item is characterised by the purpose it serves". This statement means that it is sufficient to know at least one way of getting things done. Of course, the more important the result is, the more ways you may want to learn in order to achieve the same result (e.g. several perspective drawing techniques).
Having said that, I will try to answer your actual question more directly as my next post