I chose this problem and this topic because order of operations is found in mathematics courses from middle school through college. However, some students still have many misconceptions about it and how it is solved.
Here's what I learned about student understanding of order of operations:
- Other grouping symbols are often not recognized as forms of parentheses in PEMDAS.
- When students are just taught to memorize the acronym PEMDAS, they may fail to recognize the connection between other grouping symbols and parentheses. In this example, the brackets caused some confusion because of lack of immediate recognition.
- Students taught to memorize PEMDAS may always simplify parentheses first, even when they don't need to be simplified.
- In the example above, -5 and -2 are in parentheses, but they cannot be simplified further because they are only a number in parentheses rather than an expression. However, if students just memorize an acronym and do not have a deeper understanding, they will immediately jump to anywhere that they see parentheses rather than knowing that it really involves simplifying within the parentheses.
- Students can have difficulty associating the exponent with the entire grouping symbol, rather than the last number of the group.
- Students in the example above may want to square 5 or -5 rather than the entire group because it is in closest proximity. Students who are able to immediately recognize other grouping symbols as parentheses in PEMDAS in Misconception #1 are better able to avoid this misconception.
Other things that I learned from this interview:
1. I noticed that I frequently interchanged the words SOLVE and SIMPLIFY when asking the student about the problem. This can be so confusing for students because they are not synonymous! I need to check my use of mathematical terms if I want my students to understand the true meaning of the vocabulary.
2. I need to ask questions about the right answers as much as I ask questions about the wrong ones. Students can become conditioned when they hear questions to know that they are wrong and immediately correct their response, rather than develop deep thinking about the material. They will search for what they think you want to hear instead of thinking critically about whether their answer was actually correct or incorrect. If I question their correct and incorrect thinking, they become accountable all the time, not just when they are mistaken.
This was a huge learning opportunity for me to check my own difficulties with questioning and learn order of operations misconceptions from a student perspective.