The end of the semester is approaching, and I’m a bit crunched for time this week. But I do have one more brief thought about integration to share.

A few years before I left my full-time work as a curriculum developer, a freelance journalist interviewed me via email about mathematics word problems and my theories on why students often say they hate them. I told her that my years of curriculum development work really opened my eyes to just how inauthentic word problems can be. Is it possible to write word problems that target a particular mathematics concept and also are meaningful to children? Yes, definitely. Is it possible to write 50+ such problems targeting that same mathematics concept? I can tell you from experience that it gets really tough, really fast.

And that’s before applying the list of constraints that comes along with the task in large-scale curriculum development. During our latest round of development, we were not allowed to reference any junk food in our problems. We also had to stick to round objects when we talked about fractions, because our chosen manipulatives for fractions were circles. How many round items can you think of that aren’t considered junk food, but are round and make sense to divide into parts? It starts out easy: oranges, tortillas, cucumber slices. But then come the descriptors that help make semi-junky food sound ok: veggie pizzas, whole-wheat pita. By the tenth problem or so, I promise you’ll be grasping at straws. I’m pretty sure we wrote some fraction problems about cans of cat food.

My point is simply this: Starting with some form of disciplinary content and back-tracking to a reasonably authentic task is difficult after the first few times. And when tasks start to lose authenticity, kids notice. The activities they complete start to feel like busy-work (because they are).

The issue I’ve been thinking about this week is whether the task of contextualizing content becomes easier or harder when you’re thinking about two disciplines, as in integrated curricula. On one hand, it seems like finding a task authentic to both disciplines might be more difficult. But on the other hand, I think part of the difficulty of generating authentic tasks is that usually authentic tasks require multiple kinds of component skills. Finding one that gives kids exposure or practice to one particular thing, but does not require any other skills they don’t yet have, is a challenge. So I think it is possible that considering two disciplines might actually open up some space to move in task development.

Take the number grid activity I discussed last week. I’ve written other activities before asking kids to map out paths on a number grid. And I’ve asked them to limit their movements to moving in rows or columns — adding or subtracting 10s or 1s. But I never had a great reason for that restriction, other than a desire to focus on place value, so often the task felt inauthentic. But when I added the element of programming a robot, suddenly the restriction in movements had new meaning: Programming languages are made up of a limited set of commands. So a very similar activity became more authentic — along one dimension, at least — through integration.

I’m hoping to find more of these happy compatibilities as I continue to think about integrated curricula.

Thanks, Katie, for your series on integration. You nicely articulated the challenges of developing an integrated program. A couple of comments:

I like Kiray’s balance model you cite and the idea that as you move towards the center, it gets closer to a true, fully integrated program. However, the model does not capture how the difficulty increases the closer you get to the center. The level of increased difficulty is not linear as you get closer the center. I think of the difficulty level increasing exponentially with each step. Think of a graph with the x axis representing increasing levels of integration—developing a good activity is a modest challenge; developing a good, authentic activity is more difficult; developing a coherent curriculum is significantly more difficult than good activities; developing a coherent, curriculum with strong connections to another discipline is harder yet; and developing the truly integrated curriculum is hardest. The y axis shows the level of difficulty. The exponential increases in difficulty as one moves along the x axis create a visual “explanation” of why the holy grail of the truly integrated curriculum is rarely, if ever, done. Perhaps the idea of a curriculum with strong connections to another discipline, e.g., your “math with coherent CS” category, is the best that mere mortals who have limited time and resources can reasonably accomplish.

Your number grid example reminded me of David Page’s fantastic problems with number lattices. Prof. Page was a colleague of Max Beberman during the “new math” era. (He later spent 3 decades in the math department at UIC.) Page focused on elementary mathematics and his work was viewed as among the most creative work done during the new math era. Among his body of work is a set of problems that he called Maneuvers on Lattices. They included grid problems like the one you showed. But he also included lattices of different shapes, with barriers, with “sticky points” and other neat ideas. They were hardly “authentic” but they were super fun and super motivating. The obstacles and shapes weren’t random; he included them to demonstrate some concept or idea, or sometimes just to add some fun to the problem. But they were great problems. And as I think about them in relation to your goal of math-CS integration, they also create interesting programming possibilities — a nice way to enhance both the math and the CS. I can try to track some down if you are interested.

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Yes, I would love to see them! Thanks for reading and commenting. I totally agree about the level of difficulty.

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