Yesterday I was discussing how I didn't include any graphs on my Geometry final because I wanted it to double as a possible computerized test, and graphs are hard to draw on the computer. This is ironic because the very same day, Dan Meyer's post is all about drawing graphs on the computer:
Meyer begins by linking to a New York Times article that discusses the relationship between parental income and children's college attendance. Here is the article:
But Meyer's post has nothing to do with poverty or the achievement gap. Instead, he discusses how technology can be used to improve how math is taught. First of all, the New York Times article doesn't merely give the relationship. Instead, readers have to graph what they think the relationship is before they can proceed with the article.
Dan Meyer writes about why he likes the way this New York Times article can be made into a lesson:
You can always ask a student to move higher but it’s difficult to ask a student to move lower, forgetting what they’ve already seen. You can always ask for precisely plotted points of a model on a coordinate plane. But once you ask for them you can’t unask for them. You can’t then ask the question, “What might the model look like?” Because they’re looking at what the model looks like. So the Times asks you to sketch the relationship before showing you the precise graph.
That isn’t just their intuition about learning. It’s Lisa Kasmer’s research. And it won’t happen in a print textbook. We eventually need students to see the answer graph and whereas the Times webpage can progressively disclose the answer graph, putting up a wall until you commit to a sketch, a paper textbook lacks a mechanism for preventing you from moving ahead and seeing the answer.
This isn’t just great digital pedagogy, it’s great pedagogy. You can and should ask students to sketch relationships without any technology at all. But the digital sketch offers some incredible advantages over the same sketch in pencil.The lesson is the same but it is presented differently and responsively from student to student. I watched an adult experience this lesson yesterday, and while she read the personalized paragraph with interest, she only skimmed the later prefabricated paragraphs. It should go without saying that print textbooks are entirely prefabricated.
Since the relationship turns out to be linear, this lesson is geared more towards an Algebra I class than a Geometry class. But one of Meyer's commenters mentions how this can apply to Geometry:
right off the bat, multiple transformations with either
1 mapping rules or
2 dynamic geometry commands.
1 mapping rules or
2 dynamic geometry commands.
Is it in the correct quadrant? proximate in size? orientation?
So both Dan Meyer and the commenter Travis seem to think that having the students draw graphs on the computer is a good idea. Yesterday, I argued against graphing because I knew that many Common Core opponents don't like how students have to draw graphs on the computer.
Here's a link to a webpage from Mathforum (no, it's not Dr. John Conway, but a Dr. Steven Rasmussen), where the author criticizes the SBAC computerized math exam:
In this article, we see that Rasmussen isn't opposed to computerized testing per se, but only the way that SBAC does it. He begins by reviewing some of the tenth grade questions -- which is interesting, since SBAC is only given in grades 3-8 and 11, not to sophomores. Anyway, the first question requires students to plot a fraction on a 1D number line:
Then, as I went to indicate my answers, I was simply confused by the technology I was seeing. The “innovative” technology here is a dynamic number line—a digital manipulative familiar to me for which I’ve developed many compelling applications while working with The Geometer’s Sketchpad, and which I know has a deep research pedigree going back at least to Paul Goldenberg’s Dynagraphs project. But Question 1 has nothing to do with elementary number theory, proportion, continuity, the real numbers, or any of the other mathematical concepts that dynamic number lines are productively used for. The problem simply tests a procedural skill—division of mixed numbers—and the dynamic number line is used only as a mechanism for filling in a blank with a specific value, and a whole number value at that. This was supposed to be an innovative use of mathematics technology? The technology “enhancement” has nothing whatsoever to do with the actual problem. A multiple-choice response would serve this question perfectly well.
So Rasmussen questions why a graph is even needed at all for a simple division problem. The second equation involves circles that are tangent to each other -- and there is no graph at all for this one:
Give me a grid and a circle-drawing tool if I am to show circles on a coordinate plane! Let me use a dynamic-dragging representation—like the one they misused in Question 1—to drag a dynamic circle to its correct position and size on a given coordinate system. Asking students to communicate their thinking should be done in fair and appropriate contexts. This one is neither.
So whereas Question 1 should not have had a graph but does anyway, Question 2 should have had a graph yet doesn't! And this goes on and on.
Of course, we can't quite compare Meyer's use of computers in a lesson to the SBAC's use of computers on a test. But the point has been made -- using computer technology in education is good when it is done correctly. We should not be surprised, then, when Common Core opponents say that they'd rather just give the test with good ol' pencil and paper than with the SBAC interface.
Here is a replacement page for the test that I posted yesterday. I changed the last three questions so that they are now Representations questions, one each from Chapters 1, 4, and 6. They are written in multiple-choice format for a final, but these can be conceivably converted to computer format in the manner described by Meyer and Rasmussen. The answer key is the same as the originally posted test.