This week, I read "New Questions for Old" by Prestage and Perks (2001). The authors start out the chapter with stating "take an old question, change it a bit, and hey presto, a new question appears." This was an instant red flag for me and may have influenced my opinion of the chapter. I'll summarize before I state my opinion of the chapter.
The authors recommend four strategies to alter an existing questions: (1) change a bit of an existing question, (2) give the answer, rather than the question, (3) change the resources, and (4) change the format. The authors provide explicit examples of altered questions in each of the categories, why the changes are significant, and how they may have a positive influence on student learning.
Generally, I did not like the examples that the authors provided. I found that many of their examples were arbitrary and would not necessarily help students understand a particular concept better, since they didn't necessarily provide anything meaningful for students to hold onto in the future. For example, the problem "What right-angled triangles can you find with an hypotenuse of 17cm" is an extremely general question. What are students supposed to get out of this exercise? The authors claim that since students have to apply the algorithm many times and make decisions about the number and types of solutions. But do they have a more rich understanding about right triangles or do they just know how to compute better?
Ok....enough of a rant for now. Onto my altered problems! I've chosen a problem from differential calculus, integral calculus, and
For integral calculus:
It is a very common problem for students to find the volume of a cone using the disk method for solids of rotation. As a change, have students extend this to a tetrahedral of side length a, and then to a cone with an arbitrary polygonal cross-section. This could lead students to thinking about some topology, if we want to take it that far. :P This would be changing the question a bit, according to Prestage and Parks.
For differential calculus:
Original problem: Show that 2x^2 + x -2 has a zero on the interval (-1,1) using the intermediate value theorem. As an extension, we could ask students to consider any odd degree polynomial and have them prove that the function has at least one zero using the intermediate value theorem.This changes the format, since students cannot do the normal trick of finding explicit x values to evaluate the curve at.
For linear algebra:
Original problem: Given u = [5,2] and v = [-2,4], and the linear transformation T(x) = [.5 0 | 0 .5] [x_1, x_2] (that's supposed to be a 2x2 matrix and a vector in R^2). Find the images of u and v under the transformation T.
As an extension, have students plot this into a computer program and discuss geometrically what T does to each vector x in R^2. As a further extension, transformations in R^3 and do the same in a computer program. This could allow students to better visualize linear transformations in three dimensions. This change would be an example of changing the resources.
Hi Vanessa,
ReplyDeleteI must admit I was skeptical that the authors actually used the word 'presto'... But lo and behold! Upon only a brief skim of this chapter, I can certainly relate to your issues with it. The goal in some of the examples seems to be: keep the students on this problem longer. In others, the original problem has been transformed to such an extent it seems to be a whole new question - I think this might be the case with your second example :) Still others seem to be such obvious extensions I wonder about the use of mentioning them...I can understand your lack of enthusiasm. That said, each of the questions you mention seem pretty reasonable of what might be asked in these classes. The first, with enough repetitions might drive home the point about the discs stacking up better, the second is much trickier than the original (but still good for them!) while I know something like the third is often demonstrated, but it is better to have students actually execute themselves.
The 'hey presto!' certainly sounds like a magician's trick...but I'm sorry it immediately put you off. There is the bigger question that you and Sophie both raise: the question of pedagogical purpose. Why change these problems? What is it meant to achieve in terms of students' mathematics learning? We'll talk more about this today in class.
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