The problem that I referenced in my previous post is turning out to be an interesting case study in just how much variety you can get in a single exercise, and how very similar-looking responses can get a wide range of grades. The problem (#18 from section 1.2 of Stewart Early Transcendentals) gives a table showing the rate at which crickets chirp at various temperatures. The students are asked to (a) make a scatter plot; (b) find and graph the regression line; and (c) use the linear model to estimate the chirping rate at 100 degrees.
Part (c) is where the grading gets interesting. Recall that on these 10-point assignments, if a student fails to make significant progress on every question in each part of each exercise, the entire assignment gets a 1 out of 10. In particular, giving an answer without a justification counts as not having made significant progress and earns the student a 1 out of 10 on the assignment regardless of the rest of the work.
The best way to answer this question is to have Excel generate the equation for the trendline, plug in 100 for x, and do the math. A few students did that… but not everyone.
1. A student makes the scatterplot with the trendline and equation and writes: "I put 100 degrees F into the equation to solve for the chirping rates", and then writes down the answer. I gave this full credit — because the student is clearly indicating the answer and how s/he got it. The only math that was skipped was a multiplication and an addition step, nothing complicated. If it were something more complicated — like doing the same problem with an exponential or logistic trendline — I would probably take off points for skipping all the math in the middle. But not for two arithmetic steps.
2. A student makes the scatterplot with the trendline and equation and writes: "I used the equation to solve for the chirping rates", and then writes down the answer. This student loses the credit (one point out of 10) for that exercise, but does not get a 1 out of 10 — because the student is telling me how that answer came to be, but it’s not explicit or precise. Just saying "I used the equation" isn’t very helpful. The exercise tells you explicitly to "Use the model." HOW did you use the equation?
3. A student makes the scatterplot with the trendline, and then writes: "By looking at the graph, I estimate the chirping rate at 100 to be about…" This student loses the credit (one point out of 10) for that exercise, but does not get a 1 out of 10 — because the student is telling me how they got their answer, but the explanation isn’t precise or detailed. The exercise tells you explicitly to "Use the model." HOW did you use the graph?
4. A student — well, several over a dozen students — make the scatterplot with the trendline, and then writes down the answer in another cell of the spreadsheet (without using an Excel formula, ie. just a plain transcription of an answer) or on their main writeup, separate from the scatterplot. These students get 1 out of 10 — because they have given an answer with no discernible justification. The presence of a graph on the page in itself tells the reader nothing about how the answer came to be.
5. A student makes the scatterplot with the trendline, and then types in the answer as a text annotation to the scatterplot, on top of the trendline roughly at x = 100. this student gets full credit — because s/he has indicated the answer and visually shown me how they got it. Remember that the exercise does not say, "Use the equation for the trendline…" but just "Use the linear model…".
6. A student makes the scatterplot with the trendline in the Excel file. Then the student puts a copy of the plot in his main writeup, and hand-draws a little extension of the trendline out to around x = 100 (which is the very edge of the plot box). Then, in a separate location, he puts the answer. This student got a 1 out of 10 — because again, the existence of a plot, however modified, does not justify the answer if there is no explicit connection made between it and the answer. The hand-drawn extension doesn’t constitute an explanation of the answer. How do I know that has anything to do with the answer, and it’s not just he dropped his pencil and that’s where it landed and sort of skidded around?
It’s strangely difficult to convince students that detailed, precise justifications of mathematical results are important and even necessary — and harder still to teach them to make such justifications.
One of these days I’ll actually finish this assignment.