Spacing and The Practice Meter

Without a doubt, students need to practice mathematics thoughtfully. Classroom instruction of any kind is not enough. Practicing not only helps to consolidate learning, but it can be a source of good extended instruction on a topic. And in recent years, research has uncovered—or rather re-uncovered—a very potent way to make that practice effective for long-term learning: spacing.

Dr. Robert Bjork here briefly describes the very long history and robustness of the research on the effectiveness of spacing practice:

It seems that not only is spaced practice more effective than so-called “massed” practice, but spaced learning is more effective than massed learning. A recent study by Chen, Castro-Alonso, Paas, and John Sweller, for example, provides some evidence that spaced learning is more effective than massed learning for long-term retention because spaced learning does not deplete working memory resources to the same extent as massed learning.

In one experiment, Sweller, et al. provided massed and spaced instruction on operations with negative numbers and solving equations with fractions to counterbalanced groups of 82 fourth grade students (from a primary school in Chengdu, China) in regular classroom settings. In both conditions, students were instructed using three worked example–problem-solving pairs. A worked example was studied and then a problem was attempted—for a total of three pairs (they were not presented together). In the massed condition, these pairs were given back to back, for 15 minutes. In the spaced condition, this same 15 minutes was spread out over 3 days.

In both conditions, a working memory test was administered immediately after the final worked example–problem-solving pair. And immediately following the working memory test, students were given a post-test on the material covered in the instruction. In the massed condition, this post-test occurred at the end of Day 1. In the spaced condition, the post-test occurred at the end of Day 4.

Students in the spaced condition scored significantly higher on the post-test than students in the massed condition. And there were some indications that working memory resource depletion had something to do with these results.

In the absence of…stored, previously acquired information, it was assumed that for any given individual, working memory capacity was essentially fixed. Based on the current data, that assumption is untenable. Working memory capacity can be variable depending not just on previous information stored via the information store, the borrowing and reorganizing, and the randomness as genesis principles, but also on working memory resource depletion due to cognitive effort.

Shorter, Smaller Chunks

spacing and practice meter

Taken together, the research on the spacing effect for both practice and instruction suggests that both instruction and practice should happen in shorter, smaller chunks over time rather than packed all together in one session.

As an example of this, here is a video of a module from the lesson app Add and Subtract Negatives. The user runs through this very quickly (and correctly), skipping the video and worked examples on the left side and the student Notes—and a lot of other things that accompany the instructional tool—to demonstrate how the work of this module flows from beginning to end. The Practice Meter is shown in the center of the modules (and instructor notes) on the homepage as a circle with the Guzinta Math logo. If you want to skip most of the video, just forward to the end (2:11) to see how the Practice Meter on the homepage changes after completing a module.

You can see that the Practice Meter fills up to represent the percent of the lesson app a student has worked through (approximately 55% in the video). Although not shown in the video above, hovering over the logo on the homepage reveals this percent. The green color represents a percent between 25 and 80. Under 25%, the color is red, and above or at 80%, the color is blue.

Whether or not the lesson is used in initial instruction, the Practice Meter fades over time. Specifically, the decay function \(\mathtt{M(t) = C \cdot 0.75^t}\) is used in the first week since either initial instruction or initial practice to calculate the Practice Meter level, where \(\mathtt{C}\) represents the current level and \(\mathtt{t}\) represents the time since the student last completed a module.

In our example above, during the first week after initial instruction or practice, the student’s Practice Meter level of 55 will drop into the red in about 3 days. If she returns to the app in 15 minutes to see a Practice Meter level of 54 and then raises that up to an 80 by completing the same module again or a different module (100 is max score at any time), then her Practice Meter level will drop to below 25 in about 4 days. If she raises it up to 100, then that will decay to below 25 in a little less than 5 days.

This fairly rapid decay rate applies only to the first week. After Day 7, and up until Day 28, the decay rate changes to \(\mathtt{M(t) = C \cdot 0.825^t}\), whether the student practiced during that time or not. This provides some incentive for spacing out practice a little more over time. Mapping this onto our example above, an initial Practice Meter level of 55 would decay to below 25 in a little over 4 days. A level of 80 would decay to below 25 in a little over 6 days, and a level of 100 would take about 7 and a half days to go red.

There are also decay rates for 28–90 days and after 90 days. For more information, see this Practice Meter Info page, which comes with the instructor notes in every lesson app.

(Lack of) Implementation Notes

The design of the Practice Meter is such that, if a student does not use a lesson for spaced practice, he or she will feel no interruption in their use of it. And it is important to implement it in a way that does not create extra responsibilities for the student if they aren’t required by their teacher. But if students and parents or students and their teachers do want to implement spaced practice, it can be easy to check in on the Practice Meter every so often, asking students to, say, keep their Practice Meter levels above 25 or above 80—perhaps differentiating for some students to start—at regular check-in intervals.

As always, though, implementing shorter and smaller in both instruction and practice is much more difficult than reading about it in research, especially when current practice or one’s institutional culture may be focused on “more” and more massed instruction and practice. But conclusions about spacing drawn from research are not regal edicts. We can keep them in mind as ideas for better practice and work to implement the ideas in the small ways we can—and then eventually in big ways.

Update: The Learning Scientists’ Podcast features a brief discussion of lagged homework, which definitely connects to what I discuss above. Henri wrote up something about it a few years ago.


Welcome to Guzinta Math’s Blog

Okay, all set. Welcome, welcome, welcome to the Guzinta Math blog! Here you’ll see things about what we’re developing and thinking about at Guzinta Math, along with content we can’t yet predict.

For the past two years, and likely for the next year or so, we have been and will continue to be mostly heads-down developing supplemental math lesson apps for middle school—6th, 7th, and 8th grades. As of this writing, we have fifteen Grade 6 lesson apps, seven Grade 7 lesson apps, and one Grade 8 lesson app. All of our math lesson apps are FREE Chrome apps, which can be found and installed on the Chrome Web Store or on our website. The Grade 7 and 8 lesson apps are compatible with Chrome OS (Chromebooks) only, and the Grade 6 apps are currently compatible with all operating systems (though that will stop being the case in early 2018). We have plans down the road to make sure that all of our apps work (in Chrome) with all operating systems. Stay tuned.

How Do the Apps Work?

Each lesson app contains 3 modules and instructor notes, which can be downloaded as a PDF. Each module contains video instruction, text or animated worked examples, and/or interactive instructional tools, along with short-response questions for students. Below, for example, is a short “in action” video from the first module of the Grade 6 Equations and Inequalities lesson app. It features Sal the Lizard, an interactive instructional game for teaching students about inequalities and graphing inequalities. Not shown here is that students are given audio directions for the interactive when they click on it.

The lesson app Equations and Inequalities is a bit of an outlier, because it is heavy on interactives. There is one in each module of that lesson app, actually. This is usually not the case. But the above layout is what you see in pretty much every module of every lesson app—interactives, videos, and worked examples on the left side and short-response questions on the right. Both sides are separately scrollable, and both sides also automatically scroll as students answer questions. You can see this in the “in action” video below, from the Grade 6 Quadrants in the Plane lesson app.

This is also a bit of an outlier, in that both sides scrolled together after just one question. That happens in the Quadrants in the Plane lesson app for a few questions, because we need the full right side to allow students to plot points to complete the problem. In most cases, a worked example, video, or interactive is used for a handful of questions on the right. These scroll automatically as students answer them, and then the left side scrolls automatically once the lesson is ready to move on to the next example. You can see that when you answer incorrectly, a red X is shown for a moment. When you answer correctly, a yellow star appears and a “ding” plays. No penalty applies to answering questions incorrectly. The red X is not accompanied by any sound at all.

We’ll come back to talk about features in a different post. But one thing that is visible in the video above is the drawing canvas, which is powered up by clicking on the left sidebar—the sidebar is present in all three modules of every lesson app. Turning on the drawing canvas allows teachers and students to draw all over the top of the screen—to highlight information, write notes, etc. You can choose four different colors: red, green, black, and blue.

When students complete all the items in a module correctly, an “applause” is played and a certificate of completion is displayed, which can be downloaded by a simple click. Below is an example of a certificate from the Grade 7 Proportion Equations lesson app.

certificate of completion welcome

More to Come

Okay, we’ve gone on with this welcome for far too long already. Thanks for sticking it out, and watch this space in the future for longer-form updates on our lesson apps, their features, and middle school math in general.