A colleague of mine had the following problem for an algebra midterm exam:
Simplify:
Firstly, it can be argued that it already is as simple as possible; if a student wrote this down I would certainly award full marks. Secondly, the question leaves a number of mistake paths a student could take to answer, some of which raise some interesting questions about how people understand exponents and mathematics in general. In my own teaching, I make sure to emphasize one of those paths:
Surely exponents "distribute" like this? I use a question mark over the equal sign to denote this unsureness.
This is known as the "freshman's dream". It seems the name fits as it is "freshman" in the sense that it should be an easily clarified mistake and it is a "dream" in the sense of a misuse of some potential patterns students would have seen relatively close to the above problem. Here are a few that I think can contribute to this:
1. Distribution of a single multiplicative factor:
2. Unfamiliarity with writing number 1 but from the right:
3. Exponents over multiplication:
It is my humble opinion that some twisted and illogical combination of the above mathematical facts leads to the freshman's dream. In other words, students try applying past patterns on something not calling for it; while this is generally a passable strategy for proving results or thinking about problems, it does require a bit of the latter.
“The Freshman’s Dream is an algebra instructor’s nightmare ”
Falling for the freshman's dream at first is something I can understand coming from my guesses 1-3 above, but repeating the freshman's dream quickly turns me into a sort of mathematical Freddy Kruger. It indicates that poorly minted pattern memorization takes precedence in their learning over critical thought, something that I am not sure what to do about other than present the freshman's dream as a true/false question repeatedly. Some methods to reinforce against having the dream:
- PEMDAS! Clearly addition must come after exponentiation, since exponentiation is just repeated multiplication, which itself is repeated addition.
- Evidence-based instruction on what exponents and logarithms are: This paper from 2002 explores that concept in detail.
- Playing devils' advocate. This article from Forbes details a what if scenario on the nature of linear functions.
- Advanced instruction and development in the Binomial Theorem.
“The Freshman’s Dream is indicative of students’ misusing pattern memorization; a grave mistake when trying to gather understanding from mathematics!”