This week, we have been working on subtracting, using tables and graphing. Although these topics seem unrelated, a homework assignment asked the student to analyze the graph and answer questions. While grading papers, I noticed a common theme among the homework assignments and the questions the students had a misunderstanding about. The question says, "Looking at the graph, how many more students like chocolate milk than white milk?" Due to our current work on subtraction, the students were supposed to subtract the amount of students who like chocolate milk (11) from the students who like white milk (4). However, many of the students answered the question by simply saying the answer was 11. These answers showed me that either the student did not understand the question (because the student simply wrote chocolate milk had 11 more students), or the student does not know what operation to use between the two different types of drinks to either count up from 4 or count back from 11. I think this artifact reveals gaps in the student's current mathematical understanding. Since my third graders have been reviewing subtraction facts, this may reveal that the student does not recognize familiar language when dealing with subtraction problems.
Due to this misunderstanding, there are two ways to advance this student's mathematical thinking. I would ask the student to explain why he/she approached the problem the way that they did. From there, I would be able to better understand the student's thinking and work. However, from my own assumptions, I would push the student to understand what more means. How many more students like chocolate milk than white milk? Or, framing the question a different way: What is the difference between the amount of students who like chocolate milk and white milk? In order for the students to better understand this concept, future assignments such as analyzing graphs and word problems will help this student grow in this area.
This is a good example; indeed, this is a common misunderstanding and it's excellent that you have been able to produce an experience / artifact that reveals this misunderstanding so clearly. To advance students' thinking, you might 1) present a similar problem, 2) circulate to see which students and saying what, and 3) lead a whole class discussion where you can scaffold student understanding. That is, note who says 11 and who has 7. You can have the students who say 11 defend their thinking, and then ultimately lead to the students who say 7 and have them defend their thinking. This is a great opportunity for a whole class discussion where students talk to each other, defending their ideas, as opposed to a teacher-centered classroom where the teacher says, "No, the answer is 7."
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