During the second semester, I only assigned homework using Delta Math. This worked well for several reasons, with the most important one being that it gave me better insight into what my students actually understood than pencil-and-paper homework ever did.

The Delta Math homework I assigned surprised me right off the bat. The image below displays one student’s results on the first assignment of the second semester. Many students had similar results.

This assignment consisted entirely of review material. We had assessed linear functions at least once in December, and for the most part, students did fine. I knew that students would be a bit rusty after Winter Break. That’s why we completed this assignment. The surprise came from the misconceptions I saw in student responses.

Although I could only see their answers, I found it relatively easy to determine their misconceptions. Unsurprisingly, I saw all of the classic slope mistakes.

- Writing slope as an ordered pair instead of a ratio
- Subtraction errors, especially with negative numbers
- Putting the change in in the numerator instead of the denominator
- Mismatching and
- Forgetting to simplify (not exactly a mistake, but Delta Math marks this wrong)
- Misinterpreting as either or as

These aren’t new mistakes. I’ve seen them many times, and we’ve corrected them together many times. We said “vertical divided by horizontal” at least ten times each class period every day for a month. And the students *got* this! Like I said, students did fine when we assessed linear functions in December.

So why did so many students continue to display these misconceptions? I’m not naive. I realize that students master the material at different rates. But how could so many of them who *had* mastered the material make these mistakes? Why did they return to misconceptions that they had overcome a month earlier?

One explanation is that students managed to know the material well enough to pass an assessment, but they did not develop the robust understanding necessary to maintain their skills even a month later. That’s the explanation I originally subscribed to, but I think it lacks something important. Many of my students *did* understand slope. Throughout the fall, they worked hard to connect proportional relationships, steepness, and lines. They built a solid conceptual foundation. They thought mathematically. They solved problems. I think what they lacked – and this is on me – was a proper emphasis on procedural fluency. Sure, we had plenty of opportunities to practice and to develop that fluency, but rarely did they have the “Uh oh, I’m wrong!” moments that Delta Math gave them. My Delta Math assignments required students to get a certain number of problems correct to get credit. You just can’t fake a right answer. That’s what I like about Delta Math – it holds them accountable.

The other major advantage of Delta Math over pencil-and-paper homework? I can look through as many assignments as I want to in a relatively short time period without having to carry papers around. I have a bit of a problem with keeping papers organized, so moving homework online and avoiding paper altogether saves me some serious time and energy. The obvious drawback is that I cannot see the work that the students did. In the examples above, it’s relatively straightforward to identify misconceptions without seeing the student’s work, but that certainly won’t always be the case. I think, though, that simply being able to see that a student struggled with a problem type may be enough, especially given that some sort of intervention would need to take place anyway.

I’m not sure yet what homework will look like in my classes this year, but it seems like I’ll want to place more emphasis on procedural fluency. Perhaps such an emphasis earlier in the year will lead to better understanding all year long. I hope 2018-19 is the year I finally figure out how to make homework work for me and for my students!

I think you hit the nail on the head with the procedural fluency. Right now, the pendulum has moved to conceptual fluency > procedural fluency. And even though in many cases this is probably a better way to default to as teachers, brain science tells us that our brain needs to practice procedures in order to memorize and be able to apply them in more complex (conceptual) situations. I believe that we need to look at it as conceptual fluency procedural fluency. They go hand in hand.

So, we can get frustrated when students can’t break down more complex problems into smaller, more accessible pieces; however, we’ve sometimes set them up tomorrow this by assuming that all of them can master these procedures with one or two or three really good examples in class. For some, yes, they can probably do this. But, definitely not all.

I liken it to a foreign language. You’d never expect a student to be able to learn a language by just practicing three or four really good examples. No, you’d use those examples as a framework for the conceptual learning, but then expect them to practice vocabulary, speaking, writing, etc. to solidify those concepts. It’s why immersing yourself in a place that only speaks the language is the most effective way to learn a new language.

Why would any other discipline work any differently?

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