Guzinta Math

Although, this borders on the problematic (for me at least). Why should ‘underlying’ be a prerequisite for calling something conceptual knowledge as opposed to plain old knowledge? Even the straightforward addition and subtraction here presumably requires knowing what to do with the numbers and symbols presented in this (likely) novel problem and thus involves conceptual knowledge of some kind.

Still, it makes some sense to distinguish between knowing how to add and subtract numbers and knowing that adding and then subtracting (or subtracting, then adding) the same number is the same as adding zero (or doing nothing). But the following just a few paragraphs later doesn’t make much sense to me:

The biggest problem is that the concept of ‘conceptual knowledge’ of arithmetic laid out here relies on the fact that the “inversion shortcut” is not typically taught as a procedure. But it seems easily possible to train a group of students on the inversion shortcut and then sneak them into a research lab somewhere. After the experiment, the researcher would likely decide that all of the students had ‘conceptual knowledge’ of arithmetic, even though the subjects would be using the “rote application of a previously learned procedure”—something which contradicts the researcher’s own definition of ‘conceptual knowledge’. On a larger scale, instead of sneaking a group of trained kids into a lab, we could emphasize the concept of inversion in beginning arithmetic instruction in schools. If researchers were not ready for this, it would have the same contradictory effect as the smaller group of trained students. If the researchers were ready for it, then the inversion test would have to be thrown out, as they would be aware that inversion would be more or less learned and, thus (for some reason) not qualify as conceptual knowledge anymore.

Second, why should adding and subtracting the numbers from left to right count as an application of a rote procedure (which does not evidence conceptual knowledge) rather than as a transfer of a known procedure from a conceptually similar but superficially different problem (which does show evidence of conceptual knowledge)? The problem is novel and students would be transferring their knowledge of addition and subtraction procedures to a situation also involving addition and subtraction (conceptually similar) but with different numbers (superficially different).

I still see the value of knowing the concept of inversion, as described above. A person who notices the numbers above and can solve the problem without calculating (by just stating the first number given) is, most other things being equal, at an advantage compared to someone who can do nothing else but start number crunching (it’s also possible to not notice the equal numbers because you’re tired, not because you lack some as-yet undefined ‘critical thinking’ skill). What constantly perplexes me is why people insist on making something like knowing the inversion shortcut so damned mysterious and awe-inspiring.

You can know how to number crunch. That’s good to know. You can also know how to notice equal numbers and that adding and then subtracting the same value is the same as adding 0. That’s another good thing to know. The latter is probably rarer, but that fact alone doesn’t make it a fundamentally different kind of knowledge than the former. It is almost certainly rarer in instruction than calculation directions, so it should be no surprise that students are weaker on it generally. Let’s work to make it not as rare. A good place to start would be to acknowledge that inversion is not some deep or critical knowledge; it’s just ordinary knowledge that some people don’t know or apply well.

The section in question concludes:

But how does it show that ‘conceptual knowledge’ is applied when we test students on something they haven’t been taught (do not know)? Where is the knowledge in conceptual knowledge supposed to be coming from? As long as it’s not from the teacher, it must be ‘conceptual’?

Researchers conducted two experiments, asking participants in the first experiment to learn a word list and then a motor task and in the second experiment a motor task and then a word list. There were three conditions within each experiment: (1) the word recall and motor task had the same structure (see the supplemental material for how ‘same structure’ was operationalized here), (2) the two tasks had different structures, and (3) the tasks had no determined structure.

In the first experiment, participants first learned the word list and then their skill at the motor task was measured over three practice blocks. When the word list and motor task were of the same structure, participants did significantly better across the three motor-task practice blocks. Similarly, in the second experiment, after the motor skill was learned, participants who then practiced the word list with a similar structure to the motor task improved significantly more than participants in the other conditions. This improvement on an unrelated though similarly structured task was measured as transfer, and it occurred in both directions.

Somewhat surprisingly, however, this transfer of learning between word and motor tasks (or motor and word tasks) was correlated with a stronger decrease in performance on the original task, when participants were tested 12 hours later. That is, subjects who learned the word list and then successfully transferred that learning to the motor task (because the tasks were of similar structure) showed a sharper decline in their word list recall than subjects in other conditions. The same results appeared in the experiment where subjects first learned the motor task and then the word list.

At first blush, this seems obvious. The subjects who actually transferred their learning saw their learning on the original task displaced by the similarly structured and thus interfering second task. But when researchers inserted a 2-hour interval between the original task and the practice blocks, this decline disappeared—and the transfer learning was no longer present. Thus, it seems that not only the similar structure of the two tasks but also the weakness of the memory for the first task were both responsible for the effective transfer learning. The authors put it this way:

This is intriguing. In some sense, this reinforces results related to the spacing effect. Spacing causes forgetting which causes “unstable memories.” When learning is revisited after a period of forgetting, it finds this unstable memory in a “privileged state”: a state which allows it to strengthen the connections of the original learning.

But the above also suggests that extending learning for transfer to other situations or to other concepts may be done optimally in concert with spaced practice. In other words, the best time for transfer teaching might be after a space allowing for forgetting.

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.

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.

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.