# A common misconception in the application of exponents

Watching the video provided this week and seeing some of the common misconceptions that even the most educated students have about the solar system reminded me of a recent misconception that was shared among my grade 8 students about the concept of repeated multiplication.  In the video, Heather’s misconceptions of the videos stem from misinterpretations of common illustrations used in textbooks, and a misconstrued view of the solar system and its planetary bodies. I find my own student’s misunderstanding to have similar causes.

The problem that my students struggled with this year had to do with an application of exponents, specifically with the number of bacteria in a population after it has undergone a period of exponential growth. The problem I pose to students is as follows:

“In the beginning of an experiment, there is 50 bacteria cells. The bacteria population grows when each bacteria splits in half. How many bacteria would there be after 4 divisions?”

A common solution given by my students is as follows: “50 * 50 * 50 * 50 = 6250000, so there are 62500000 bacteria after 4 divisions”

This calculation is incorrect because the concept of splitting in half cannot be captured directly by multiplying the initial population repeatedly. The correct solution would be that “50 * 2 * 2 * 2 * 2 = 800 bacteria”

One may classify this type of error as an error due to “rigidity of thinking leading to inadequate flexibility in decoding and encoding new information” (Comfrey, 1990) When students are first introduced to exponents, they are usually taught the “fact” that for a given number n, n^m = n * n * n * n m times, usually without any accompanying illustrations or physical models, thus it could become very difficult to come up with different ways of using that rule. As constructivists would have it, the individual mental construction of the concept is largely incomplete. (Cobb, 2004)

In order to better integrate this knowledge and apply it to bacteria growth, students need time to reflect on their solution in different ways (Davis, 2000). The students should be encouraged to ask themselves, “Is it plausible for 50 bacteria to turn into 62500000 bacteria just after 4 divisions?”, “If the same pattern was consistent, would 3125000000 bacteria after 5 divisions make sense?” Students should be asked not only to reflect on this through thinking, but also through illustration. How would one draw a bacteria population of 62500000? How does this drawing compare to that of 50 bacteria?

In order to solve the given problem, different strategies should be taught in addition to the rule. Some of which include drawing pictures depicting the number of bacteria after each split, or constructing a table to record the number of bacteria after each split. Other strategies involving digital technology would be showing videos or animations of bacteria growth in order to further help students in developing their understanding of exponential growth. I believe these are all strategies that will assist in helping students develop a more accurate model of knowledge.

Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational researcher23(7), 13-20.

Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. Review of research in education, 16, 3-56.

Davis, E. A. (2000). Scaffolding students’ knowledge integration: Prompts for reflection in KIE. International Journal of Science Education22(8), 819-837.

### 2 comments

1. AndrewYeung says:

Thanks for sharing the example Gary. I like the fact you have students reflect whether it is likely ’50 bacteria would turn into 62500000′ to appreciate the real-world relevance of examples posed in class. I wonder however when often numbers are given on the fly during lessons and are way out there, when can students rely on this intuition to solve problems?

Andrew

2. Gary Ma says:

Having a good number sense is a very important skill to have and is vital in helping students realize their mistakes while working on problems. I have had students calculate the length of a airplane to be over a km long, which clearly doesn’t make sense. Number sense comes in especially handy whenever we are working with problems that involve calculating lengths, and money, but in practically, the intuition is useful across any mathematical topic.