I make it clear that the photo is optional. I had one student bring in a baby picture, which probably helped a lot more than he’d expected! I learn the names of the students who give me selfies and continue to struggle with the rest.

]]>In my blog I did not use the term “continuous”, and I used the term “discrete” just once: “Euclid’s treatment of proportionality is essentially that of discrete quantities: four magnitudes that have the same ratio are called proportional. (See Definition 6.)”

I realize that “discrete” is not an ideal term here. “Specific” is perhaps better. In any case, the distinction I was making is really quite natural. It is between two ways of expressing a quantity: using a specific instance of the quantity, or using a variable. In today’s school mathematics, a specific instance is generally a number. In Euclid it was a geometric quantity, such as a specific line segment in a figure.

The importance of the distinction is that it is at the core of two different approaches to problem situations: A traditional middle school approach uses specific instances of quantities (numbers) in cases where variables would be more natural. In more up to date approaches, variables are used.

We illustrate with a typical example problem: Suppose we are told that in an enlargement of a photo, a 6 inch line segment becomes 15 inches. We are asked what a 8 inch line segment becomes.

A traditional approach sets up a proportion 15/6 = x/8. The solution, x = 20 comes from solving this proportion equation. (The reasoning behind this approach is often left vague.)

A more modern approach reasons that, in an enlargement, each line segment is enlarged by the same dimensionless factor, call it e. If we let any length before the enlargement be “b” and the corresponding length after the enlargement be “a”, then we know that a = e b. The given information gives us a numerical instance of this relationship: 15 = 6 e. Solving this equation gives the numerical value e = 2.5. This allows us to express the general relationship: a = 2.5 b. The solution to the problem is then found by substituting b = 8 into this relationship: a = 2.5 x 6 = 20.

A central feature of the more modern approach is that it uses simple linear functions such as a = 2.5 b, where the quantities a and b are often called variables. A relationship expressed in a simple linear function such as a = 2.5 b is called a proportional relationship. The advantage of the more modern approach is that it makes explicit the general relationship underlying the problem situation. In the more traditional approach this underlying relational relationship is never found.

The blog illustrates the unfortunate consequences of the traditional approach, and urges the adoption of a more modern approach.

]]>Incidentally, the afore-mentioned book was published without an index, but the index can be found at this link: http://tinyurl.com/haho2v6 ]]>

But the difficulty discussed above came after this, revealing itself in the context of work on specific proofs. I’m speculating here, but perhaps one way to see it is that she was struggling with the idea of a uniform strategy; or else with the notion that a uniform strategy can be described in terms of a single (but generic) epsilon.

For what it’s worth, the analogy mentioned in note [14] did shift something for her, because she felt she understood how the algebraic proof worked.

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