Errors and Misconceptions
Identifying and Understanding Gaps in Understanding
At the start of my inquiry, I focused much of my data collection on gathering student work with evidence of misconceptions. These artifacts have been helpful to me in recognizing the various types of misunderstandings that can arise and in being able to closely analyze what in particular my students were having trouble understanding. Solely looking at an incorrect answer did not provide me with much information; however, looking at their written work showed me more of their thoughts. Although their reasoning was flawed, it demonstrated to me that my students were making efforts to make sense of the material.
The deeper that I reflected on the work that my students produced, the more I realized how informative looking closely at incorrect student work could be in addition to how many questions that they got correct. Both Titus (2010) and Kelley (1993) reason that deciphering why students are making their mistakes can help teachers guide their students to more accurate understanding. In this spirit, I tried to get inside my students’ heads and their thinking as much as possible to understand their rationalizations and interpretations of the material. I encountered several different types of misconceptions from my students, including computational, procedural, and conceptual ones.
Computational Misunderstandings
I attribute instances of computational errors to a lack of a strong foundation in basic math operations and content. Adding and subtracting integers and working with fractions are skills that ideally students should come to high school already equipped with and be able to do with ease. Some of my students were not the strongest with the fundamental skills which made it difficult for them to be successful with the Algebra I material that built upon such skills. Furthermore, as the school year went on, my students who had not mastered prior material began to face challenges with the new content that related to it (i.e. solving systems of equations from solving one equation).
In particular, one of my students seemed to have most of the procedural understandings of linear equations but had difficulty with solving equations. At first glance, one would think that she did not really understand any of the material because she got so many of the questions on the quiz incorrect (she received 11/30 points on the quiz). Looking more closely, I saw that she understood many of the concepts about linear equations in order to set up the problem but was having trouble with solving the equations and completing the basic calculations.
Computational misunderstandings seem basic and minuscule, but because these fundamental foundations are necessary to accurately complete calculations, they can stand as barriers to students gaining conceptual understanding. As such, I saw the need to find ways to address and respond to misconceptions without having to completely re-teach fundamental concepts, like solving equations.
Procedural Misunderstandings
Similarly for errors in performing mathematical process, I looked closely at which particular steps my students were having trouble with and how they were interpreting those steps. A student’s solution and work may initially appear completely incorrect because the steps depend on each other (i.e. doing an incorrect calculation at one point during the process affects all other calculations thereafter). However, with careful observation and thoughtful analysis a teacher can justify their students’ incorrect, but logical, ways of thinking.
When I began taking over the grading for my students’ work, I found myself pausing a lot to look at their work in detail because I wanted to see what they were struggling with and why they were getting the problems wrong. While grading a daily quiz for graphing linear equations, I came across many graphs that were executing certain portions of the graphing process correctly but not in the correct order or in a fully proper fashion.
One student seemed to understand how to use slope to find other points on a line but was having sign issues and not addressing y-intercepts. This student’s work shows evidence that he was making some sense of the material. He correctly memorized that slope is rise over run and how to utilize that to find points on the coordinate plane. However, his work also reveals gaps in his conceptual understanding of what slope and the graph of a line really are. With a good enough memory, a student can gain procedural understanding, but to truly learn and possess a deep understanding, procedural knowledge should be tied to a conceptual understanding of the content and steps of the process.
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Equipping for Anticipating and Responding
As I brought more of my attention to identifying the problems that my students were getting wrong, I was better able to figure out where they were struggling and the reasons for why they were thinking and solving the problems in certain ways. In more cases than not, the reasoning and logic behind my students’ misconceptions were valid and understandable. I collected several pieces of student work that documented the various mistakes that my students were making. Then, I also began to take note of how I was adjusting my instruction in reaction to these misconceptions. In this fashion, the focus of my inquiry shifted away from the student errors themselves and more towards how a teacher takes those errors and adapts their teaching.
In education, student attempts to demonstrate their understanding and makes mistakes that often reveal gaps in this understanding. Santagata (2002) points out that “Students’ mistakes are an unavoidable and necessary part of the learning process” (p. 12-13). To assist in this furthering of student learning, the teacher can utilize the incorrect work that their students produce to inform their instructional decisions. Anticipations and responses to these misconceptions are proactive measures on the part of the teacher to help guide their students in correcting their understandings.