I’ve spent a significant amount of time over the last four or five years thinking about integration of mathematics and computer science. My efforts have been driven, in particular, by a general dissatisfaction with a lot of the available “integrated” materials for elementary school. Without calling out any programs by name (because criticism of any program is not at all my point), I’ll say that I see an awful lot of what I’d call “slapping a math standard on a CS activity.” The integration was superficial at best — in many cases, kids could (and probably would) complete the programming activities without really engaging with the mathematics. And that makes sense, because the math is not what the activities were claiming to support.
I just felt like there had to be a better way.
I looked as several models describing different ways to think about integrating mathematics and science to help push my thinking about what that better way might be.
While working on a paper with some colleagues, I came across something called the balance model for integration (Kiray, 2012). The article was about integration of math and science, so science was swapped for CS, but otherwise the diagram of the model looked something like this:
Kiray (2012) doesn’t use exactly those words (“Math with opportunistic CS” is “Math-centred science-assisted integration” and “Math with coherent CS” is “Math-intensive science-connected integration”), but I think my version captures the differences he is trying to get at. Integration is opportunistic when activities take advantage of any available connection, without worrying about whether the content that is connected is a part of the expected curriculum or related to anything else students have explored in the second subject. This is what I feel like a lot of the math connections in elementary CS materials look like. Yes, kids could count the number of blocks in a program that moves a robot around a maze, and technically that’s a math connection. But it’s opportunistic, and therefore likely not all that meaningful in terms of developing math learning.
And it works the other way, too. Full-time CS ed folks (I’m only a part-timer, really) feel the same way when I say kids could think about drawing a mathematical model in terms of CS abstraction. It took me a while to understand the skepticism coming from some corners of the CS ed world when I would talk about such things, but now I think I get it. It’s as hard for them to see how abstracting numbers from a word problem is connected to meaningful CS learning as it is for me to see how counting blocks is connected to meaningful math learning.
The types of integration that are closer to the middle, math with coherent CS and CS with coherent math, still have one discipline or the other in the driver’s seat. However, the connections to the other discipline are made with careful attention to the development of both disciplines. The way each discipline builds on itself is considered, and how the two might be made to build on each other. Still, in all the cases when a choice has to be made for better development and opportunities for learning, the driving discipline wins. So, one discipline progresses at a typical, grade-level appropriate rate, and the other lags behind what might be possible otherwise.
Finding a way to plan for coherent progressions through CS/CT content as it gets integrated into mathematics was a big part of the goal of the LTEC project. By developing learning trajectories for CT concepts to use as a guide while working on a math+CT curriculum, we hoped to help make the learning in both disciplines meaningful.
It seems to be working (results coming soon!). Still, I think one of the biggest lessons I’ve learned as I’ve watched the work progress (I want to be clear that my colleagues are doing most of the work) is the delay in one discipline is unsatisfying, even for the people (like me) who claim to care more about one discipline over the other. My inner dialogues about these issues tend to go like this:
Well, we could introduce the idea of looping here. The connection seems really solid.
Yes, but the kids haven’t really been examining repetition anywhere else, so it’s going to take some build up to that.
But… the fit is so good! We can’t give up that opportunity! Maybe we do just a straight-up CT activity in here to get them ready?
We said we weren’t going to do that.
I know. Blargh.
We either choose to give up the good connection opportunity and let the CS lag behind, or put in a CS-only or CS-with-opportunistic-math activity. Then the curriculum feels like it’s got “extra” stuff in it, and it feels kind of jerky. So even though math with coherent CS is better than math with opportunistic CS, it still just feels like there’s got to be a better way.
That leaves us with the balance point: total integration. Here’s Kiray’s description of that:
In total integration, the target is to devote an equal share of the integrated curriculum to science and mathematics. Neither of the courses is regarded as the core of the curriculum. This course can be called either science-mathematics or mathematics-science. Separate science or mathematics outcomes do not exist. An independent person observing the course cannot recognise the course as a purely mathematical or a purely scientific course. One of the aims of the course is to help the students to acquire all of the outcomes of the science course as well as the mathematics course. (p. 1186)
Sounds great, right? I’ve conceptualized much of my work over the last five years to be a pursuit for what total integration looks like. Unsurprisingly, I have not figured it out. I can think of one or two example activities, maybe, but not a whole curriculum. Plus, no school is going to be happy with no recognizable math or CS outcomes. Total integration was supposed to be the solution I was searching for, yet sometimes it felt unachievable or inappropriate for real schools, teachers, and kids.
A week or two ago, though, I was doing revisions on another paper. We got some thoughtful comments that prompted me to step back and try to articulate some of the assumptions underlying the analysis in that paper. One of them that I included in my response to the editor was this:
We believe that curriculum development has to take a longer view than one activity at a time.
It’s true. This belief has underscored all of my mathematics curriculum development work for the last 10 years. (My colleague — shout out to Carla — left a comment on that sentence saying, “Actual snaps for this!”) Yet I had not really said it to myself for a while. And when I did, I realized I’d been thinking about this balance framework in the wrong way.
What if it isn’t about picking a kind of integration and staying there?
What if, instead, development of integrated curricula is about moving across the balance?
We start with math. (I mean, it’s already happening anyway!) We take some opportunities to dabble with CS in math (which mostly involve use of CT practices, I think). We find places to build reasonable, coherent activity sequences, still letting the math drive. We work on identifying those few, golden opportunities for total integration.
And here’s the part I thought I’d never say:
Then we keep developing the CS, and start being ok with letting the strong connections to math fade and maybe even disappear.
Integration is hard to achieve even one activity at a time, and it’s really, really, hard to maintain. Maybe a continual push for fuller integration isn’t really what we want. Maybe we just want to drive toward an example or two of really rich, total integration activities and make sure they’re functioning as good transitions into learning CS without the mathematics.
How do we find and develop those golden, fully integrated, strong transitionary activities?
I’ll talk about one idea I have — next week.
Kiray, S. A. (2012). A new model for the integration of science and mathematics: The balance model. Energy Education Science and Technology Part B: Social and Educational Studies, 4, 1181-1196.