As part of our professional growth and school improvement process, several members of my department and I have formed a focus group to study formative assessment; testing which focuses on testing to make instructional decisions rather than for grading. One of our goals is to try and develop a deeper understanding of how to write better questions in order to measure student learning.
Recently I imposed on several members of my staff to give a brief assessment to their students that included two different forms of a question on slope. The questions, shown below, were of the lines y=3/4 x -3 and y=3/4 x - 2. I assumed their would be a slight difference in the number of correct responses since one had integer intercepts on both the x and y axis, and one did not.
What I didn't expect was the degree of difference. The students ranged from Algebra I students who have only recently been introduce to slope, through geometry students who had reviewed the topic twice this year according to their teacher, my own sections of Alg II students whom I assumed were much stronger on the topic than my results made me know, and a section of statistics students; many of whom are taking an additional pre-calc or calculus class. In all there were 47 students who took each form. Of the 47 who had the question with integers on both intercepts, a total of 26, or 55%, got the correct answer. Of the 47 who had the line translated one unit up the y-axis, only 10, or 21% gave the correct response.
Four more of the students who had the "easier" question had the expected types of mistakes; they either gave the reciprocal of the correct response, or the negative of it. I was not prepared for how strongly the students were bound to the x and y intercepts. Thirteen of the students using the second form had given slopes of .8 or 2/2.5 using the values they interpreted as the intercepts. Of the 22 students in Alg II who took the questions, 8 of 11 gave the correct answer for the "easier" problem, but only three of them were correct (that is, they answered 3/4) on the second form. Five more of them, however, gave the answer of 2/2.5 or .8 to the second form. There is no question in my mind that they understood the idea of slope. I have no quarrel with the people who would give full credit to the answer of .8 or 4/5. There is however, I think, some difference between the student who focuses solely on the intercepts and the ones who see the whole line, coordinate plane relationship. I think a question which differentiates between them tells me something about how they see and treat slope.
I guess I am more concerned than ever about learning how to write good assessment items that will help me, and my students, understand what they know, and don't know, about the mathematical ideas we are covering.