Responding to Misconceptions
Acknowledging and Addressing
When considering what responses to student misconceptions look like, I believe that teachers should try to go beyond telling students that “they are wrong” and acknowledge the errors in more productive ways. Mistakes are an indication that students are attempting to make sense of the material and actually serve as good starting points to begin working towards understanding.
Especially in the beginning of my student teaching, I was unsure about how to respond to my students making mistakes when answering questions during class. Smith and Stein (2011) articulate my own experiences well in “When teachers feel overwhelmed by the needs and frustrations of their students, it is easy for them to revert to just telling students what to do when an alternative course of action does not immediately come to mind” (p. 36). I did not want my students to feel embarrassed or ashamed when they did a problem wrong, so I would not push them any further and instead ask another student or say the correct thing myself.
Teacher-Centered Responses
Case #1: An instance where I took over completely was when a student asked about determining which numbers are greater or less than each other from a number line. One student made an observation that the bigger number was closer to zero, and another student questioned if that was only true when comparing negative numbers. I was so intent on answering the question myself that I denied my students a good opportunity to correct their perceptions and to reach that point of understanding for themselves. Although I actively responded to the misconception with a clear, understandable explanation, a more valuable and useful response would have been one where I had stepped back more and given more agency and student voice. |
Recording Transcript:
Me: Do we want to include the number 2? Class: No. Me: How do we show that we want values greater than 2? Class: Arrow to the right Student (raises hand to ask this question): Wouldn’t you start at 3? Me: So we don’t want to include 2, but we want to include numbers that are greater than 2. If we think of decimals, a decimal between 2 and 3 is like 2.5, right? Is that greater than 2? This is a really good question. We don’t want to include 2 but we want to include values between 2 and 3. The notation in math is that we circle the 2 and we want to include everything after the 2, so we use the arrow to show that. Student: So, start the line at 2? Me: Yeah, so start the line at the 2. |
Case #2:
Another lesson that was very teacher-centered was a re-teach that I did for graphing and writing linear equations. When I noticed that my students were struggling with internalizing the multi-step processes, I designed the lesson to be an extremely explicit breakdown of the steps. During the lesson, I also mentioned the common mistakes that I had seen them making. Many of my students were able to follow the algorithm given in the examples, but others were still struggling because the steps seemed arbitrary. A disconnect may have remained for some because I had not addressed their misconceptions with a logical or reason-based approach. Starting the graph of a linear equation by going up some amount on the y-axis had no meaning unless students understood that this was the graphical representation of the y-intercept, where the graph of the equation crosses the y-axis and the x-value is zero. |
More Student-Centered Approaches
Questioning
In the two teacher-centered cases, I could have used questioning as an instructional technique to respond to my students’ misconceptions and still promote student-centered and inquiry-based learning. Questioning allows the teacher to guide students to understanding, but the onus is largely on the students to discover and articulate that understanding independently (Sahin, 2007; Smith & Stein 2011).
For Case #1, the other student had already posed the probing question of if it only works for negative numbers. I should have just stepped back and opened the floor for my students to explain and pick apart the observations— thinking of their own examples and counterexamples. For Case #2, I could still keep the design of the lesson relatively the same and even possibly have it still be direct instruction. However, the addition of some guiding questions that push my students to explain the “why’s” behind each step would be beneficial. Just as when trying to convince someone to change their mind or way of thinking, one has to provide good reasons and justifications to do so. “How” and “why” questions can serve this function of giving students logic to ground and modify their understanding in.
In the two teacher-centered cases, I could have used questioning as an instructional technique to respond to my students’ misconceptions and still promote student-centered and inquiry-based learning. Questioning allows the teacher to guide students to understanding, but the onus is largely on the students to discover and articulate that understanding independently (Sahin, 2007; Smith & Stein 2011).
For Case #1, the other student had already posed the probing question of if it only works for negative numbers. I should have just stepped back and opened the floor for my students to explain and pick apart the observations— thinking of their own examples and counterexamples. For Case #2, I could still keep the design of the lesson relatively the same and even possibly have it still be direct instruction. However, the addition of some guiding questions that push my students to explain the “why’s” behind each step would be beneficial. Just as when trying to convince someone to change their mind or way of thinking, one has to provide good reasons and justifications to do so. “How” and “why” questions can serve this function of giving students logic to ground and modify their understanding in.
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Projects
Like questioning, projects when designed properly can be highly student-centered and helpful in addressing misconceptions. Compared to a traditional test or quiz with straight-forward problems to solve, projects can be more performance-based in nature and ask to students to rationalize, explain, and justify their thinking. Projects can be more powerful in tackling misconceptions than any direct-instruction re-teach or quiz because students have the reins. A project that I assigned to my students addressed misconceptions that they were having with distinguishing between the graphs of equations and inequalities. The major misconception that my students were having was that they were shading the graphs of both equations and inequalities. My students were admirably trying to synthesize their knowledge by applying their new knowledge about inequalities to their previous understanding of equations. Alternatively, this confusion could also point to my students just memorizing the material (i.e. all graphs need to be shaded) and not rationalizing through their understanding. The project pushed students to develop these rationalizations for themselves and to explain why inequality graphs are shaded and equation graphs are not. Examples of successful and unsuccessful student work can be found here. |
Value of Responding
Not only is it important and valuable for teachers to respond to their students’ misconceptions, but they should also do so in a way that aims towards equipping their students with long-term understanding. Deep and lasting understanding cannot emerge if teachers simply tell their students what is wrong and exactly how to correct it. I would like to aim to structure my responses such that my students have ownership over the knowledge because they are discovering it mostly on their own.
Anticipation takes responding to student misconceptions to another level. Here, teachers are predicting the approaches their students will utilize and preparing for how they will respond to them. Doing this preparation helps alleviate situations when teachers must think on their feet and respond in the moment to a student mistake.