The average percent correct on the first in-class homework was 72.4 and 78.3 for Conditions T and GC, respectively, F (1,32)   =   0.40, p   =   .531, ns, showing that the two conditions were closely matched for solving area and circumference computation-only problems. The average percent correct on the second in-class homework was 47.1 and 54.1 for Conditions T and GC, respectively, F (1,32)   =   0.72, p   =   .402, ns, demonstrating that the two conditions were also closely matched in their abilities to solve area and circumference story problems. In contrast, on the unexpected 2-day delayed quiz, Condition GC students outperformed Condition T students by 14.7 percentage points. Percent-correct means were 75.5 and 60.8 for Conditions GC and T, respectively, F(1,32)   =   4.75, p   =   .037.

The average percent correct on the first in-class homework was 88.6 and 83.9 for the T and GC Conditions, respectively, F (1,18)   =   0.40, p   =   .536, ns, indicating that the two groups were well matched in their ability to solve computation-only integer problems. The average percent correct on the second in-class homework was 71.8 and 80.0 for the T and GC Conditions, respectively, F (1,18)   =   0.59, p   =   .454, ns, showing no significant difference in solving integer story problems. In contrast, on the unexpected 3-day delayed quiz, which reflected learning from the in-class homework, Condition GC students outperformed Condition T students by 15.6 percentage points. Percent-correct means were 86.1 and 70.5 for Conditions GC and T, respectively, F (1,18)   =   7.63, p   =   .013.

It is illuminating to compare the results of these two experiments as shown in Figure 4.1. Whether studying geometry (Experiment 12) or integer addition (Experiment 13), students whose homework included the Guided Cognition cognitive events performed reliably and similarly better (about 15 percentage points on an unexpected 2- or 3-day delayed quiz) than did students who had Traditional homework problems. The relative gains were also very similar across the two experiments. If we divide Condition GC quiz performance by Condition T quiz performance, we find that Condition GC students performed 24% better than Condition T students in Experiment 12, and 22% better than Condition T students in Experiment 13.

In the next nine mathematics-content experiments, we explore further what is learned from Guided Cognition homework, the long-term benefits of Guided Cognition homework, the effectiveness of Guided Cognition homework for students of various ability levels, and some practical design issues.

Analysis of review activity (quiz) performance, without regard to problem type, confirmed an overall Guided Cognition effect, GC   >   T, t (94)   =   −2.67, p   =   .009. This analysis gives more weight to the story problems because according to the grading rubric, each story problem was worth five points, and each numerical problem was worth two points. This scoring method is analogous to how an actual quiz might be graded with more points based on more complex problems: From that viewpoint, there is nearly a letter grade improvement due to Guided Cognition, assuming that letter grades are 10 percentage points apart. Means are shown in the top row of Table 4.9.

To equalize the weighting of story and numerical problems, point totals for each problem type were converted to percent-correct values for each student. Analysis of in-class homework condition (T vs. GC) by review activity problem type (numerical vs. story) revealed a main effect of homework condition, GC   >   T, F (1,94)   =   8.02, p   =   .006; a main effect of problem type, numerical   >   story, F (1,94)   =   234.71, p   <   .000, and no interaction, F (1,94)   =   0.15, p   =   .697, ns. The means for performance on numerical problems and story problems, when the two types of problems are equally weighted, are shown in Table 4.1, below.

Mean pre-experiment grade averages of the 71 students who participated in all parts of Experiments 14 and 15 were found to be nearly identical across conditions (Condition T, n   =   34, GPA   =   86.1; Condition GC, n   =   37, GPA   =   87.9, t (69)   =   −1.08, p   =   .286, ns).

Analysis without regard to problem type of in-class homework condition (T vs. GC) by review activity delay (3–6   days vs. 6   months) confirmed a main effect of condition, GC   >   T, F (1,69)   =   7.40, p   =   .008, a main effect of review activity delay, 3–6   days   >   6   months, F (1,69)   =   25.60, p   <   .000, and no interaction, F (1,69)   =   0.03, p   =   .867, ns. This analysis gives more weight to the story problems because according to the grading rubric, each story problem was worth five points, and each numerical problem was worth two points. This unequally weighted analysis is analogous to the way tests are actually graded, with more points from more complex problems, and in that sense, it is a valid analysis to estimate the grade improvement that can be attributed to Guided Cognition homework. Whether after 3–6   days or 6   months, students who had the Guided Cognition homework performed about one grade better, assuming that letter grades are 10 percentage points apart. Means are shown in Table 4.2.

The primary purpose of Experiments 14 and 15 was to determine the effects of Guided Cognition homework on each type of problem. To equalize the weighting of story and numerical problems, point totals for each problem type were converted to percent-correct values for each student. Analysis of in-class homework condition (T vs. GC) by review activity problem type (numerical vs. story) by review activity delay (3–6   days vs. 6   months) revealed a main effect of condition, GC   >   T, F (1,69)   =   7.81, p   =   .007; a main effect of delay, 3–6   days   >   6   months, F (1,69)   =   206.78, p   <   .000; and a main effect of problem type, numerical   >   story, F (1,69)   =   33.14, p   <   .000. There was no interaction of delay and homework condition, F (1,69)   =   0.23, p   =   .634, ns; of problem type and homework condition, F (1,69)   =   0.23, p   =   .635, ns; or of delay by problem type by homework condition, F (1,69)   =   1.25, p   =   .268, ns. There was an interaction of delay and problem type, F (1,69)   =   5.59, p   =   .021, indicating that over the 6-month interval, the drop in performance is greater for numerical problems than for story problems. This interaction may be partly due to the fact that the possible range of decreased performance is much greater for the numerical problems than for the story problems. The relative decrease in performance for each type of problem is, however, nearly identical. For numerical problems, performance on the 6-month delayed review activity was 77.9% of the students' performance on the 3- to 6-day delayed review activity. For story problems, performance on the 6-month delayed review activity was 78.7% of the students' performance on the 3- to 6-day delayed review activity. The average percent correct for each type of problem at near-term and very long-term review activity delays is shown in Figure 4.2.

As in Experiment 14, the lack of interactions of homework condition (T or GC) with review activity problem type (story or numerical) means we cannot say that Guided Cognition helps more with one type of problem than the other. After a 6-month interval, the Guided Cognition advantage was found for story problems that required interpretation and set up, and calculations; this advantage was also found for numerical problems that required only calculations. The former result was expected because it was hypothesized that thinking about how to solve problems, even in the absence of calculation practice, should help students interpret and set up similar problems at a later time. That the Guided Cognition tasks resulted in better execution of calculation-only problems seems less likely and is especially interesting because the cognitive events did not include additional calculation practice—students in both conditions performed exactly the same calculations. This finding implies that even without specific calculation practice, Guided Cognition resulted in more elaborate or more stable representations of the calculations. That the effect lost none of its strength after 6   months suggests that Guided Cognition can provide a highly desirable very long-term performance advantage, which makes Guided Cognition a valuable tool for designing effective mathematics homework.

All students received the same four story problems in Part A for each set of homework, so performance on Part A serves as another comparison of a priori problem-solving ability across conditions. Analysis of Part A homework for multiplying fractions and mixed numbers showed the groups were well matched, with percent correct for T   =   46.2 and GC   =   51.0, t (92)   =   −0.80, p   =   .427, ns. Similarly, analysis of Part A homework for dividing fractions and mixed numbers showed the groups were well matched, with percent correct for T   =   43.4 and GC   =   41.8, t (92) = 0.35, p   =   .729, ns. Because the review activity included half multiplication and half division problems, the Part A homework scores were combined for the two topics and not surprisingly, showed that overall, the students in the two conditions were well matched for Part A problem solving, T   =   44.8 and GC   =   46.4, t (92)   =   −   0.37, p   =   .710, ns.

The three parts of Condition T homework were analyzed to determine whether or not students were improving with problem-solving practice. The mean performance for each 12-minute time period is shown in Table 4.4.

Multiplication homework performance increased significantly over the three time periods, as confirmed by a significant linear trend, F (1,47)   =   14.06, p   <   .000. Division homework performance also increased over the three time periods, and this was confirmed by a significant linear trend, F (1,47)   =   49.73, p   <   .000.

Performance on the review activity showed that performing the cognitive events was as helpful and efficient for learning as was more practice solving problems, with percent correct for T   =   56.6 and GC   =   52.1, t (92)   =   1.15, p   =   .252, ns.

From these data, we cannot predict the optimal ratio of problem-solving practice to conceptual practice for effective long-term problem-solving performance. The data do show, however, that Guided Cognition tasks can be as efficient as additional computational practice for learning mathematics: Holding homework time strictly constant, students may learn as well from working fewer problems if they spend the additional time performing non-calculation cognitive events.

Mean pre-experiment grade averages of the 88 students who participated in all parts of Experiments 16 and 17 were found to be nearly identical across conditions (Condition T, n   =   45, GPA   =   86.0; Condition GC, n   =   43, GPA   =   87.0; t (86)   =   −0.64, p   =   .525, ns).

Recall that in Experiment 16, all students received the same four story problems in Part A for the multiplication homework, and all students received the same four story problems in Part A for the division homework. Performance on Part A of the multiplication homework and on Part A of the division homework serves as another comparison of a priori problem-solving ability across conditions for the 88 students who completed both experiments. Analysis of the Experiment 16 Part A homework for multiplying fractions and mixed numbers showed the groups were well matched, with percent correct for T   =   47.7 and GC   =   51.4, t (86)   =   −0.60, p   =   .550, ns. Similarly, analysis of the Experiment 16 Part A homework for dividing fractions and mixed numbers showed the groups were well matched, with percent correct for T   =   43.9 and GC   =   42.6, t (86)   =   0.28, p   =   .782, ns. Because the review activity included half multiplication and half division problems, the Part A homework scores were combined for the two topics, and overall, the students in the two conditions were well matched for the Experiment 16 Part A problem solving, T   =   45.8 and GC   =   47.0, t (86)   =   −0.27, p   =   .788, ns.

The three parts of Condition T homework were analyzed to confirm that the 45 Condition T students who had completed both Experiments 16 and 17 showed improvement with problem-solving practice. The mean performance for each 12-minute time period is shown in Table 4.5.

Multiplication homework performance increased significantly over the three time periods as confirmed by a significant linear trend, F (1,44)   =   11.32, p   =   .002. Division homework performance also increased over the three time periods, and this was confirmed by a significant linear trend, F (1,44)   =   43.09, p   <   .000.

Table 4.6 shows the review activity means for students in Conditions T and GC after 1–3   days (Experiment 16 results) and after 14   weeks (Experiment 17 results). A delay (1–3   days vs. 14   weeks) by condition (T vs. GC) analysis of variance yielded a significant main effect of delay, F (1,86)   =   59.57, p   <   .000. The analysis confirmed that there was no main effect of condition, F (1,86)   =   0.28, p   =   .597, ns, and no interaction of delay by condition F (1,86)   =   1.50, p   =   .225, ns.

But what are students learning from performing the cognitive events? Logically, performing these cognitive events may lead to improved understanding of the concepts underlying the problems or may lead to better long-term memory of the procedures, or to both. The results of Experiments 14 and 15 suggest that performing cognitive events promotes both concept comprehension and procedural learning: On an unexpected quiz, students whose homework had consisted of working story problems and performing cognitive events scored better on story problems and on numerical problems, than did students whose homework had consisted of just working story problems.

Taken together, the results of Experiments 14 and 15, and the results of Experiments 16 and 17, indicate that performing cognitive events is both effective and efficient for learning. Experiments 14 and 15 illustrate the effectiveness of Guided Cognition-designed homework by showing that holding the amount of calculation practice constant while adding conceptual thinking tasks (cognitive events) leads to better near-term and also to better very long-term problem-solving performance. Experiments 16 and 17 illustrate the efficiency of Guided Cognition-designed homework by showing that holding study time constant while replacing some of the repetitive calculation practice with cognitive events can result in equivalent performance.

Analysis of review activity (quiz) performance, without regard to problem type, found no Guided Cognition effect, with T and GC Condition means   =   55.9 and 60.4 percent correct, respectively, t (90)   =   −1.16, p   =   .249, ns. Further analysis by problem type confirmed no Guided Cognition effect for numerical problems, with T and GC Condition means   =   81.4 and 83.6 percent correct, respectively, t (90)   =   −0.49, p   =   .626, ns, and also confirmed no Guided Cognition effect for story problems with T and GC Condition means   =   45.7 and 51.1 percent correct, respectively, t (90)   =   −1.20, p   =   .234, ns.

Having found no benefit of merely reading completed cognitive events among the 92 students who participated in the experiment, we decided to look further by examining the performance of only those students who had performed reasonably well on their in-class homework. Data from students who scored 50 percent or better on the multiplication homework problems and on the division homework problems were included in these new analyses. We first determined that students in Condition T and Condition GC were well matched for ability by confirming that the pre-experiment grade averages were nearly identical across conditions (Condition T, n   =   28, GPA   =   89.4, Condition GC, n   =   27, GPA   =   88.4, t (53)   =   0.46, p   =   .649, ns).

Analysis of review activity (quiz) performance, without regard to problem type, confirmed an overall Guided Cognition effect for these students, GC   >   T, t (53)   =   −2.07, p   =   .044. This analysis gives more weight to the story problems because according to the grading rubric, each story problem was worth five points, and each numerical problem was worth two points. As was mentioned in Experiment 14, this way of grading performance is analogous to how an actual quiz might be graded, with more points based on more complex problems: From that viewpoint, there is nearly a letter grade improvement because of Guided Cognition, assuming that letter grades are 10 percentage points apart. Means are shown in the top row of Table 4.7.

To equalize the weighting of story and numerical problems, point totals for each problem type were converted to percent-correct values for each student. Analysis of in-class homework condition (T vs. GC) by review activity problem type (numerical vs. story) revealed a borderline main effect of condition, GC   >   T, F (1,53)   =   3.86, p   =   .055; a main effect of problem type, numerical   >   story, F (1,53)   =   96.85, p   <   .000, and no interaction, F (1,53)   =   0.44, p   =   .510, ns. The means for performance on numerical problems and story problems, when the two types of problems are equally weighted, are shown in Table 4.7.

A practical implication of these results is that embedding completed examples of Guided Cognition tasks into assigned story problems may facilitate subsequent performance in interpreting and solving story problems, and in performing calculations for plain numerical problems. Typically, homework problems do not contain such explanatory and exemplary content: This is usually confined to the introduction of each topic in mathematics textbooks. The results of Experiment 18 suggest that such content, if embedded with assigned practice problems, may boost the value of practice, but only if students do reasonably well on their homework.

The next experiment was designed to determine whether the advantage of merely reading completed examples of cognitive events lasts for a very long time.

Analysis without regard to problem type of in-class homework condition (T vs. GC) by review activity delay (3–6   days vs. 5   months) found no main effect of condition, GC vs. T, F (1, 53) = 0.01, p   =   .926, ns, a main effect of review activity delay, 3–6   days   >   5   months, F (1, 53)   =   17.98, p   <   .000, and a significant condition by delay interaction, F (1, 53)   =   5.27, p   =   .026. The interpretation of this interaction is made clear by the means shown in Table 4.8.

Performance on the 3- to 6-day delayed review activity was 7.9 percentage points better for the students who had the Guided Cognition homework when compared to students who had the Traditional homework, but after 5   months this effect was reversed such that students who had Traditional homework performed 7.1 percentage points better than the students who had the Guided Cognition homework.

This analysis gives more weight to the story problems because according to the grading rubric, each story problem was worth five points, and each numerical problem was worth two points. To equalize the weighting of story and numerical problems, point totals for each problem type were converted to percent-correct values for each student. Analysis of homework condition (T vs. GC) by review activity problem type (numerical vs. story) by review activity delay (3–6   days vs. 5   months) revealed no main effect of condition, F (1, 53)   =   0.00, p   =   .997, ns; a main effect of delay, 3–6   days   >   5   months, F (1,53)   =   22.15, p   <   .000; and a main effect of problem type, numerical   > story, F (1,53)   =   105.24, p   <   .000. There was a borderline interaction of delay with condition, F (1,53)   =   3.69, p   =   .060, reflecting the findings, illustrated in Figure 4.3, that GC   >   T on a 3- to 6-day delayed review activity, but T   >   GC on a 5-month delayed review activity. There was no interaction of problem type with condition, F (1,53)   =   0.16, p   =   .688, ns; or of delay by problem type by condition, F (1,53)   =   0.31, p   =   .582, ns. There was an interaction of delay and problem type, F (1,53)   =   12.06, p   =   .001, indicating that over the 5-month interval, the drop in performance is greater for numerical problems than for story problems.

In Experiment 18, we analyzed the results of students who scored 50 percent or better on each of the multiplication and division homework assignments. Students in the Guided Cognition condition, who merely read completed examples of cognitive events, performed somewhat better on a 3- to 6-day delayed review activity compared to students who had Traditional homework. In sharp contrast, when retested 5   months later, in Experiment 19, we obtained the opposite results—the students who had Traditional homework performed better than the students who had read the completed cognitive events as part of their homework.

It is likely that reading the completed cognitive events provided too much help such that students in Condition GC had to think less than students in Condition T to work the homework problems. By providing the completed cognitive events, we may have short-circuited students' essential thinking about how to solve the problems. In the near term (3–6   days), students may have recalled some of the guidance provided by the cognitive event examples, and this may have helped them solve the problems better than the Condition T students. If over time this specific guidance is forgotten, these students may remember less about how to solve the problems than the Condition T students who had to work out how to solve the homework problems without guidance.

These results join a long list of counterintuitive results that have been found in studies of human learning where making initial cognitive processing a bit more difficult results in better performance later (and conversely, making initial cognitive processing easier results in poorer performance later). For example, experiments have found that performance on initial recall tests of word pairs decreased as the initial test delay increased, but later, on a second test, recall of the word pairs increased as the initial test delay had increased. An explanation of this “spacing of tests effect” is that successful recall on an initial test requires more or different cognitive processing as the initial test is delayed, and this extra cognitive processing creates a more retrievable long-term memory and better recall on a later, second test (Whitten, 2011a; Whitten & Bjork, 1972, 1977).

Considering the results of Experiments 14, 15, 18, and 19, the recommendations are clear. For very long-term benefits, students need to perform the cognitive events. The results of these experiments suggest the following: Working through the steps of the cognitive events activates learning-effective cognitive processes. Merely reading cognitive events is a sort of crutch that may help moderately in the near term, probably by providing guidance that can be recalled for a few days. Ultimately, however, merely reading cognitive events is detrimental for very long-term performance. Once recall of the specific guidance fails, these students have less understanding of, and lower recall of, how to work the problems than Condition T students who had to work through the logic and mechanics on their own. So for the long term, working problems and performing corresponding cognitive events results in the best performance, working problems that are not enriched with cognitive events is next best, and working problems but merely reading completed examples of corresponding cognitive events is least effective.

It is important to contrast these results with seemingly contradictory results reported in the learning literature. Sweller and Cooper (1985), for example, have found that students benefit from seeing step-by-step completed examples of mathematics problems. There is little doubt that showing students procedures and patterns can help them solve similar problems later, assuming that the students recognize the problem type, recall the specific pattern, and then apply it. The cognitive events in our experiments differ from such examples in that they do not show all of the steps for working problems, but instead encourage specific sorts of cognitive processes. If performed, the cognitive events elicit these learning-effective processes, but if merely read, they apparently do not.

Analysis of review activity (quiz) performance, without regard to problem type, confirmed a main effect of homework condition, GC   >   T, F (1, 92)   =   7.93, p   =   .006; a main effect of ability level, average   >   low, F (1, 92)   =   12.58, p   =   .001; and no interaction, F (1, 92)   =   0.11, p   =   .743. Thus, Guided Cognition facilitated learning similarly for low- and average-ability students.

As in previous experiments, this analysis gives more weight to the story problems because according to the grading rubric, each story problem was again worth five points, and each numerical problem was again worth two points. As was mentioned in the results of Experiment 14, this way of grading the quiz is analogous to how an actual quiz might be graded, with more points based on more complex problems: This method of grading shows nearly a letter grade improvement due to Guided Cognition, assuming that letter grades are 10 percentage points apart. Means are shown in the top third of Table 4.9.

To equalize the weighting of story and numerical problems, point totals for each problem type were converted to percent-correct values for each student. Analysis of in-class homework condition (T vs. GC) by review activity problem type (numerical vs. story) by ability level (low vs. average) revealed a main effect of homework condition, GC   >   T, F (1,92)   =   8.80, p   =   .004; a main effect of problem type, numerical   >   story, F (1,92)   =   239.28, p   <   .000, a main effect of ability level, average   >   low, F (1,92)   =   11.12, p   =   .001, and no significant interactions. These results indicate that Guided Cognition facilitated learning similarly for low- and average-ability students for both types of problems. Means are shown in the lower two-thirds of Table 4.9.

It is also of interest to note that on the review activity, low-ability students who had Guided Cognition homework performed at levels similar to those of average-ability students who had Traditional homework. For numerical problems, low-ability students who had Guided Cognition homework scored 84.4 percent correct, and average-ability students who had Traditional homework scored 82.6 percent correct. For story problems, low-ability students who had Guided Cognition homework scored 48.3 percent correct, and average-ability students who had Traditional homework scored 52.2 percent correct. Thus, studying with Guided Cognition homework makes low-ability students look much like average-ability students.

The mean pre-experiment grade averages of 36 low-ability students were found to be very similar across conditions (Condition T, n   =   17, GPA   =   79.6; Condition GC, n   =   19, GPA   =   82.9), and the mean pre-experiment grade averages of 35 average-ability students were found to be nearly identical across conditions (Condition T, n   =   17, GPA   =   92.5; Condition GC, n   =   18, GPA   =   93.2). Analysis of variance confirmed a main effect of ability level, F (1, 67)   =   114.19, p   <   .000, no main effect of homework condition assignment, F (1, 67)   =   3.51, p   =   .065, ns, and no interaction of ability level with homework condition assignment, F (1, 67)   =   1.39, p   =   .243, ns.

Analysis of review activity (quiz) performance, without regard to problem type, confirmed a main effect of homework condition, GC   >   T, F (1,67)   =   8.96, p   =   .004, a main effect of review activity delay, 3–6   days   >   6   months, F (1,67)   =   26.43, p   <   .000, and a main effect of ability level, average   >   low, F (1,67)   =   13.64, p   <   .000. The only significant interaction was delay by ability level, F (1,67)   =   4.54, p   =   .037, indicating that lower-ability students forgot more over the 6-month interval.

As explained in previous experiments, this analysis of total points gives more weight to the story problems because according to the grading rubric, each story problem was worth five points, and each numerical problem was worth two points, but this is a valid method for estimating the grade improvement that can be attributed to Guided Cognition homework. Whether after 3–6   days or 6   months, low- and average-ability students who had the Guided Cognition homework performed about one grade better, assuming that letter grades are 10 percentage points apart. Means are shown in the top third of Table 4.10.

To equalize the weighting of story and numerical problems, point totals for each problem type were converted to percent-correct values for each student. Analysis of in-class homework condition (T vs. GC) by review activity problem type (numerical vs. story) by ability level (low vs. average) by review activity delay (3–6   days vs. 6   months) revealed a main effect of homework condition, GC   >   T, F (1,67)   =   9.60, p   =   .003; a main effect of problem type, numerical   >   story, F (1,67)   =   212.62, p   <   .000; a main effect of ability level, average   >   low, F (1,67)   =   14.30, p   <   .000; and a main effect of review activity delay, 3–6   days   >   6   months, F (1,67)   =   35.08, p   <   .000.

There were no interactions with homework condition, indicating that overall, Guided Cognition homework helped low- and average-ability students after short and long delays, and with both types of problems. There was an interaction of review activity delay and problem type, F (1,67)   =   5.55, p   =   .021, indicating that over the 6-month interval, the drop in performance is greater for numerical problems than for story problems. There was also a review activity delay by ability level interaction, F (1,67)   =   5.23, p   =   .024, indicating that the low-ability students forgot more over the longer delay. Means are shown in the lower two-thirds of Table 4.10.

Students in the Consolidator Condition performed better on the review activity than did students in the Advance Organizer Condition, t (92)   =   −2.08, p   =   .040, and also performed better on the in-class homework, t (92)   =   −2.01, p   =   .047. Means are shown in Table 4.11.

Because students in the Consolidator Condition performed better on the in-class homework than did students in the Advance Organizer Condition, we separated the homework scores into two parts—scores on the problems and scores on the cognitive events—to determine whether the performance difference was from one part or both parts of the homework. Analyses revealed that average problem-solving performance was not significantly different for the two conditions, with 73.9 and 76.8 percent correct for the Advance Organizer and the Consolidator Conditions, respectively, t (92)   =   −0.72, p   =   .473, ns. In contrast, the mean cognitive event performance was better for the Consolidator Condition (69.8 percent correct) than for the Advance Organizer Condition (61.7 percent correct), t (92)   =   −2.59, p   =   .011. These results indicate that better performance on the in-class homework was due to better performance on the cognitive events when they occurred after working some problems, as in the Consolidator Condition.

Our results suggest modifying standard teaching strategies. Typically, teachers instruct and demonstrate a topic, assign practice problems, and then proceed to the next topic. Our findings suggest that an instructional “sandwich” of teaching, problem-solving practice, and then reflecting on how to solve the problems by engaging in appropriate cognitive events, may significantly improve students' subsequent problem-solving performance.