A correspondent writes:
We emphasize the idea that students should approach problems in multiple ways. This has caused me to wonder about patterns. For example:
- students might conclude that 3^0=1 because of the pattern 3^4=81, 3^3=27, 3^2=9, 3^1=1
- when the second difference is constant, students will conclude that the function is quadratic
- when a function is concave up, the second derivative is positive
However, I wonder if seeing patterns leads to understanding the reasons for the patterns. I worry that many students conclude that the reason that 3^0=1 is because of the pattern, or that the function is quadratic is because all quadratics have a second difference is constant, and so on.
What are the connections between seeing the patterns, memorizing them, and a deeper understanding?
This is a fantastic question, that gets to important issues in both pedagogy and mathematics. Here is my response.
Many students, parents, teachers, and administrators believe that learning math consists of memorizing huge lists of techniques, and deploying the right one when confronted with a familiar-looking problem. This method is seen as a good way for students to avoid having to think. If they are obedient and have photographic memory, they can handle anything we throw at them. As long, that is, as we only confront them with familiar-looking problems. Alas, that is not effective, as most students lack the required motivation or memory to learn that way. Moreover, that approach is not useful to people who will pursue mathematics at a higher level, or to people who will need to apply mathematics in the ever-changing complexities of the real world.
Teaching our students to look for patterns is a key component of moving away from that view of math, and towards conceptual understanding. Recognizing patterns involves students intellectually, and helps them take charge of their learning. When they look at a problem, they try to draw conclusions from what they see in the numbers, the symbols, or the figures, rather than looking for superficial similarity with a previously encountered problem.
Of course, like anything else, searching for patterns can be taught poorly: if the patterns are only introduced by the teacher, if no attempt to make a connection between the patterns and why they hold, if students are only asked to look for pre-learned patterns, we are trading one set of ill-understood memorizations for another.
* Students should be involved in discovering the patterns. In your first example they should be asked how one equation is related to the previous one and to the next one. They should be asked to extend the pattern in both directions and to justify their answers.
* This should be followed by a discussion about why the pattern holds. This is well within the reach of most students, who can see that it is not a coincidence: it has to do with the meaning of exponentiation when the exponent is a positive whole number.
* Finally, in exercises and assessments, students should be asked to analyze similar patterns. This would confirm that they have some grasp of what’s going on. For example, how does one extend this pattern: 3x4=12, 3x3=9, 3x2=6, 3x1=3, to find a meaning for 3x0 and 3x(-1)?
In this case, consistency with the pattern is a valid argument for the definition of 3^0. One could and should complement this with an application of the various laws of exponents, which would confirm that this makes sense. For example, 3^2 x 3^0 should equal 3^2, which confirms that 3^0 must equal 1. Again, in exercises and assessments, other laws of exponents can be brought into the discussion.
In other cases, such as your second example, the pattern is an insufficient argument. For example, the function f(x)=x^2·cos(2πx) yields constant second differences if the x’s are 1 unit apart. Yet, the function is not quadratic. Providing this example (at a precalculus level) would make for an exciting discussion of this pattern. As it turns out, quadratic functions always exhibit constant second differences, but the converse is not true. So the pattern tells us the function is not linear, and makes a quadratic function a likely candidate. What additional information do we need?
Finally, there are patterns that appear to hold, but in fact break down. For example, every odd number greater than 1 can be written as a prime plus two times a square. For example, 5 = 3 + 2x1^2. This works until you hit n = 5777, which cannot be written thus. You can find more examples at the start of this unit from my Infinity course, and even more in this article by Richard Guy. These examples offer a very strong argument about the fact that patterns do not constitute proof.
In short, do not give up on pattern-seeking. It is a very important mathematical habit of mind. But finding a pattern is merely the beginning. A necessary beginning, often, but only the beginning.