Annin, S., and Lai, K. (13 Mar 2010) Common Errors in Counting Problems. Mathematics Teacher, 103(6), 403-409.
In their article, Annin and Lai address the issue of the difficulty that most students have with counting problems, and in particular with permutation and combination problems. There are four main categories of counting problems and problems are fairly simple if it falls directly under one of the four categories. However, most problems will not and thus a broader understanding of how to do counting problems is necessary. Students must understand the principles behind permutations and combinations; they must recognize that how to do the problem depends on whether the order of the objects matters or not. Using examples of different problems, the authors present common mistakes that are made and how teachers can overcome these misconceptions. Annin and Lai suggest having a variety of problems for students to solve so they are exposed to the different wording and set-up that problems may have.
Annin and Lai make compelling arguments in hteir article that I believe are relevant to teaching mathematics. When I was first learning permutations and combinations, my classmates and I really struggled to understand the concepts. My teacher gave us more simple problems to do during class and for homework. By the time the test came around, most students had a difficult time doing the problems that were not straight forward, the problems that did not fall directly under one of the four main types. At the end of the section, most people had an instrumental understanding but not a relational understanding. Students need to understand all the concepts of counting to be able to do any problem that they may encounter.
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This sounds like a very helpful article. I was hoping that you would say more about what the four categories are. I also wondered if there wasn't a fifth category, one for the problems that didn't fit into the first four categories.
ReplyDeleteWhat did the authors think were the main concepts associated with permutations and combinations? What were the most common error types? Is there a way that you could have worked brief descriptions of these into your paper?
As I was reading your critique of the main idea, I couldn't identify the three main points you were making to support your stance toward the main idea. It seemed like the argument was that you had a bad learning experience with combinations and permutations because you had only developed instrumental understanding. What other points could you have added that would have made your argument stronger.
Also, please note that your citation is not in APA format. Next time, consider checking out the website I linked on our course blog that shows you how to use APA formatting.
You did a really nice job of sounding scholarly. Your topic sentence was also referred back to throughout your entire blog.
ReplyDeleteI did wonder if it talked about how we can help students get a better understanding. Also what were the basic four categories?
While I agree that this is a relevant topic for math teachers and classrooms today, I am curious as to why these types of problems are difficult for students. Also, it would be helpful to see how else teachers can better explain this mathematical concept to students besides through example problems. Sometimes it's helpful to explain the structure of the problem and how it relates to real world problems or even apply it to something for better understanding. Good job though.
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