I raved for a minute on Twitter last week about this New York Times article. You should read it (play it? experience it?) and then come back so I can explain why it’s what math curriculum could and should become.
The lesson asks for an imprecise sketch rather than a precise graph.
This is so rare. More often than not, our curricula rushes past lower, imprecise, informal, concrete rungs on the ladder of abstraction straight for the highest, most precise, most formal, most abstract ones. That’s a disservice to our learners and the process of learning.
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.
Their reason is exactly right:
We asked you to take the trouble to draw a line because we think doing so makes you think carefully about the relationship, which, in turn, makes the realization that it’s a line all the more astonishing.
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 builds your thinking into its instruction.
Once it has your guess – a sketch representing your best thinking about the relationship between income and college participation – it tailors its instruction to that sketch. (See the highlighted sentences.)
The lesson is the same but it is presented differently and responsively from student to student. All the highlighted material is tailored to my graph. 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.
It makes your classmates’ thinking visible.
The lesson makes my classmates’ thinking visible in ways that print textbooks and flesh-and-blood teachers cannot. At the time of this posting, 70,000 people have sketched a graph. It’s interesting for me to know how much more accurate my sketch is than my classmates. It’s interesting to see the heatmap of their sketches. And it’s interesting to see the heatmap converge around the point that the lesson gave us for free, a point where there is much less doubt.
In a version of this article designed for the classroom, students would sketch their graphs and the textbook would adaptively pair one group of students up with another when their graph indicated disagreement. Debate it.
I’m not saying any of this is easy. (“Sure! Do that for factoring trinomials!”) But we aren’t exactly drowning in great examples of instruction enhanced by technology. Take a second and appreciate this one. Then let me know where else you think this kind of technology would be helpful to you in your teaching.
And as far as I know, even with Apple proclaiming “Textbooks that go beyond the printed page” since 2012?, there isn’t a single digital math textbook doing this yet.