In terms of getting the desired outcomes, this homework set had mixed success. The first two problems did get the students to do the sorts of work I had hoped for. Problems 3 and 4, however, did not for the most part. For example, Neal and Don did the problem from the derived formula, which was not my intent. John on the other hand solved the problem the way I wanted students to. When grading the set, I reread the questions, and I realized that questions 3 and 4 as written did not actually ask the students to work through the derivation the way John did. Consequently, I graded the students on the method they chose, and next time I teach the course, I will change the problem set to require the students to work through the derivation.
The fifth question seemed to serve the desired purpose. Don, showed a weakness in his understanding of the property of closure under addition. When he states
Since e + p is algebraic, both e and p are algebraic,
he shows a common student fallacy of the closure laws. This response then enables a discussion of the law, and the need for understanding contrapositives correctly. (This statement arises from applying a basic logical fallacy to the closure law for algebraic numbers, namely that the converse of a statement is equivalent to the statement.) Neal, on the other hand, shows a growth in proof by this point of the course and is able to provide a valid proof (this is discussed further in the section a student's journey to mathematics). John's proof, shows a great deal of understanding, as he synthesizes the four parts of the problem into one well written proof.
The sixth question is one of the most interesting on this assignment, and begs for a case by case analysis. John's response is quite insightful, showing a fairly deep understanding of the number systems for a student at his level. He notes that:
The algebraic numbers, roots of polynomials with integer coefficients, is a fairly intuitive structure for most high school students.
He justifies this statement by pointing out that finding roots of polynomials is a process students are familiar with by late algebra or precalculus. He then makes the statement that
Furthermore, algebraic numbers, out of their nature, have for the most part simple algebraic representations ... using integers and simple symbols.
While his statement is not exactly true, he gets one important difference between the numbers, namely that students can get a handle on most of the algebraic numbers that they are introduced to in high school and college, thus they are less abstract. Turning to the transcendental numbers, John notes:
Like the irrationals, transcendentals are defined negatively: that is, they are numbers that are the roots of no polynomial with integer or rational coefficients. Transcendentals cannot be easily represented in terms of integers the way that algebraic numbers can. Furthermore, they cannot be constructed with any finite set of tools.
John is getting at the key point that most transcendental numbers cannot be presented in a tangible way, and thus a generic transcendental number can only be defined abstractly. I should comment briefly that we had in class been using finite tools to me tools that allowed only for solving of algebraic functions, much the way that Descartes' uses the idea in his Geometry text. I did not discuss in class how you could construct all roots using the circatrix.
Jumping ahead in John's entry, we look for how this would affect his teaching of e and p. Here he shows a deep understanding of the importance of how we define e and p when he says:
The difficulty of characterizing p and e algebraically would lead me to focus not on p and e's decimal representation or other interpretations, but rather p and e do, how they function with respect to other operations, such as trigonometric and logarithmic most obviously. I think we have to think of p and e as numbers which satisfy certain geometric and analytic properties rather than interpreting them algebraically.
Here we see that John has made a conceptual leap by recognizing the importance of interpreting number in a broader sense. This is a huge conceptual move, that I don't expect students in this course to get, but I hope to move them in this direction.
We can compare this to Neal's answer, which is good, but evidences an understanding that is not quite as deep. Neal is focusing in on the division of the real numbers (and later on a slightly larger realm of number). We see this when he states:
... in each parallel, we're building a system of numbers. We are exhausting a list of every single number in different ways that we can think of them.
Later he differentiates between the “form” a number takes and the application of this form in terms of polynomials. This is showing an understanding that the transcendental numbers for the most part have a different form from the algebraic numbers. This appears to be a step just before the step that John is taking, namely focusing in on the tangible versus intangible nature of the types of numbers we are looking at. Neal, however, also realizes that he doesn't understand the transcendental numbers fully when he says:
I don't completely understand these “harder” numbers. We can understand a few examples, such as e and p, but what about the rest?
He then discusses briefly the stunning (to them) idea that the transcendental numbers “outweigh” the algebraic numbers by far, getting to one of the fundamental mysteries of mathematics. What I find particularly impressive on this answer is how he then takes his insecurity about his understanding transcendental numbers, and decides that he can use it to communicate with students. In the end, he concludes that
In this way, we need to motivate why we look at e and p and then we can teach our students more exactly what they are (i.e., transcendental, irrational) for a much better understanding of the whole system of numbers.
In contrast to John, Neal suggests that p and e should be presented as points on the number line. Again, this is deeper than presenting them as infinite decimals, and it shows an emerging understanding of the role they play. The other contrast to John is that Neal suggests that part of the reason for discussing the role of e and p is to effect a better understanding of the whole number system. Pedagogically, this shows a burgeoning understanding of the idea that one role for studying higher mathematics is that this study will enhance the understanding of lower level mathematics.
The attentive reader, will have noted that Neal's response received 12 points out of 10, more than John's. There were two basic reasons for this. First, I see an important piece of this class to stretch students as far as I can, and that includes using small grading irregularities to encourage students. Neal's response showed a great leap for him, which was less true of John's, since Neal started the class with weaker understandings.
Don's response, while perfectly acceptable, does not show a particularly deep understanding of the issues inherent in applying these ideas to the high school classroom. His interpretation of the question is narrow and does not think about the larger implications of the distinguishing characteristics between transcendental and algebraic numbers. Curiously, none of the students used the idea of graphing polynomials and identifying crossings on the axis. I was surprised by this because to me a key idea behind algebraic and transcendental has to do with the ability to place the numbers on the number line via the graph, and the ability to make this placement in an algorithmic fashion for all algebraic numbers using something like Newton's method.