Effects of Figural and Numerical Presentation Formats on Growing Pattern Performance

Tasks for Session 1

Tasks for Session 2

Accuracy

For the near extension task, participants were asked to provide outputs for three stages (see Appendix A) and received a point for each correct response for a maximum of 3 points per near extension task. For all other tasks, participants received one point for a correct response. For both patterning and transfer problems, accuracy was scored as either correct or incorrect, based on whether the correct numerical answer was present somewhere on the page. If a participant indicated a correct answer and then crossed it out and provided a different answer, it was not counted as correct. If a participant crossed out a correct answer, but did not provide an incorrect answer, it was counted as correct. If both a correct and an incorrect answer were provided and neither was crossed out, it was counted as correct. For matching task responses, if a participant indicated that neither alternative rule was correct, their response was scored as incorrect even if they indicated that one of the alternatives was “close” or “almost right” (e.g., “neither but Matt is close” and “technically none are correct but Matt is correct”). For the reversal task on the last transfer problem, participants were given a point for a correct response that featured a remainder (indicating 32.5 or 32 remainder 12). However, they were not given a point if they rounded up (33 instead of 32) because this indicates they did not recognize that the remainder is meaningful (i.e., indicates a starting value).

Because the prompt for the figure condition did not specify that participants should respond with a single numerical answer, several possible responses were coded as correct in order to account for the drawing ability demands that were only present in the figure condition. In the absence of a numerical response, participants’ drawings of the figures or verbal descriptions of the figures were coded as correct or incorrect. If participants drew the correct number of squares, their response was considered correct. For the proportional near extension task, if participants indicated an understanding of the pattern of grey and white squares (e.g., “the pattern is 4 gray and 3 white”) but did not accurately draw each box, they were also given a point for a correct response. For the non-proportional near extension task, if participants drew a “+1” next to elements of the previous figure instead of drawing a whole figure as a response or indicated how far the squares would extend (e.g., “+27 on each side”) this was counted as correct. Vaguely extended figures (e.g., indicating “and so on”) were not counted as correct for the far extension tasks.

Accurate written descriptions of the correct figure were also scored as correct (e.g., “3 blocks empty at the top and 43 pairs of 2 shaded at the bottom” or “there would be 140 shaded blocks and 105 white blocks”), but vague descriptions were not (e.g., “equivalent of the second figure with it’s appropriate proportions” or “it would be bigger”). Explicit mention of the starting values for the far extension task in problems 2 and 3 were required in order to receive a point (e.g., “would have 86 shaded squares beneath it” did not receive a point because this response did not indicate there were 3 squares above the 86 squares).

Two raters independently coded the whole data set for accuracy based on these criteria. Percent agreement was calculated by taking the average agreement across each item (17 total across patterning and transfer tasks) and each participant. Percent agreement was 96%, so one rater’s coding was used for the following analyses.

Target Strategies

Using the coding scheme described below, two independent raters coded all responses (regardless of response accuracy) for target strategies. Examples of recursive and functional strategies can be found in Figure 3, and the percentage of students who used each strategy can be found in Table 3 for proportional problems and Table 4 for non-proportional problems.

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Figure 3

Examples of Recursive and Functional Strategies

Note. Children's writing in the bottom two panels of the figure read: "because you would multiply 2 times 43" and "I know because we are counting by two and 43 is the 43rd number so you do it times two and we started at five so you add 5".

Student responses to near extension, far extension, and reversal (transfer problems only) tasks were coded based on the strategies students reported, and the results of this coding were summed separately for each task. The matching task was treated as a recognition task (that is, did students recognize a possible recursive rule), and was only coded for accuracy as described above. However, if the student did not indicate a rule or strategy on any other task within a problem, the student’s response to the matching task was taken into consideration when coding the other tasks. Broadly, strategies were coded into two main categories: recursive and functional (see Figure 3 for examples). The recursive and functional codes were not mutually exclusive, and it was possible for a student to demonstrate both recursive and functional strategies within a single task, for instance if the student indicated a rate (e.g., she reads 7 pages a day) but relied on repeated addition instead of multiplying.

Recursive strategies relied primarily on output values to generate a response and could be either full recursive or part recursive. Full Recursive strategies entailed repeated addition of a constant to all intervening output values in order to obtain an answer for one input value (e.g., adding a constant 43 times to find the 43^rd output value in a pattern). Part Recursive strategies were only coded for far extension tasks, and entailed addition to, or multiplication of, known output values without repeated addition to all intervening output values (e.g., multiplying the 5^th output value in a pattern by seven to get the 35^th value in a pattern). These strategies still rely primarily on output values and do not reflect thinking about patterns as a relation between an input and an output but are more efficient than Full Recursive strategies and less prone to error.

Functional strategies, in contrast, rely primarily on multiplicative thinking and demonstrate an understanding of the relation between input values and output values. There were two sub-codes within the functional strategy code, full functional and part functional. Full Functional indicated consideration of both the rate of change and the starting value of a pattern. Part Functional indicated either consideration of only the rate of change (ignoring the starting value completely) or indicated consideration of an incorrect starting value (e.g., x = 1 not x = 0). These two strategies were coded as Part Functional No Intercept and Part Functional Wrong Intercept, respectively. These two codes for functional strategies were only used for non-proportional problems (2 and 3, not 1). Problem 1, which had a starting value of 0, only had a Functional code instead of two codes for part and full. Also, near extension tasks did not have a Part Recursive code because the input values were too small for students to rely on this strategy. For reversal tasks in the transfer problems, no Full Recursive code was used because no students relied on this strategy.

For the transfer problems, three additional codes were added in order to track whether participants spontaneously generated a presentation format (sequence, table, figure) while attempting to solve the near and far extension tasks in the transfer problems. To differentiate between a table and simply indicating which response was associated with which input value (e.g., How many pages did she read on the 2^nd, 4^th, and 6^th day?), we only coded a response as having relied on a table if consecutive input values beyond those associated with the response were included. For instance, if the near extension task prompt provided the output values for the first, third, and fifth input values, and the child was required to find output values for the second, fourth, and sixth input values (see Appendix B), a response was only coded as a table if it included input/output pairs for input values x = 1 through x = 6. In contrast, a response that only included input/output pairs for input values x = 2, x = 4, and x = 6 was not coded as a table.

Non-Target Strategies

Additional codes were included in the coding scheme for responses that could not be interpreted as meaningful strategies. Some strategy codes captured errors that students made when executing a solution procedure. For instance, if a student demonstrated understanding of the rule (either in a recursive or function sense), but incorrectly executed an operation or used the wrong operation (e.g., divided instead of multiplied), this was coded as a Computation Error. If a student demonstrated understanding of the rule but did not answer all parts of the question or missed a step in their calculations (e.g., not dividing the difference between non-consecutive input values in order to get the slope), this was coded as a Missed Step Error. If the student only submitted a response with no justification or calculations provided, this was coded as No Rule. An Other code was used for instances in which the student misunderstood the question, or their response did not fall into any of the above code categories. These non-target codes were analyzed separately (see Appendix C). No condition differences were expected for any of these analyses. However, condition differences did emerge for No Rule codes in patterning problems, F(2, 60) = 5.60, p = .006, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .16. Post-hoc pairwise comparisons with Bonferroni corrections indicated that the figure condition led to more instances of No Rule codes than either of the numerical format conditions, which did not differ. The additional drawing demands may have contributed to slightly more No Rule Errors on pattern problems for the figure condition compared to the other conditions.

We coded for up to 13 possible codes across the non-proportional and proportional problems, though not all codes were relevant for each task. Percent agreement was calculated by taking the average agreement across relevant codes for each task (e.g., Full Recursive, Full Functional, Part Recursive, Part Functional, No Intercept, Wrong Intercept, Computation Error, Missed Step Error, No Rule, Other, and presence of tables, sequences, or figures on the transfer tasks; see Table 4 and C5Table C5) across participants. Percent agreement was 91%, and all disagreements were resolved through discussion, which resulted in the final coding data used for analyses.

Proportional Patterning Problem

Across conditions, we calculated mean percentage correct on the near extension task for each participant by summing the number of points and dividing by three possible points and converting to percentages. Mean percentage correct on the near extension task was 79% (SD = 38%). For the matching task, 86.9% of participants selected the recursive solution that yielded a correct response across conditions, and for the far extension task, 62.3% of participants indicated a correct response.

To test whether presentation type impacted accuracy on the proportional near extension task, we conducted a one-way ANOVA comparing the percentage correct responses among the three conditions. There was a significant effect of condition, F(2, 58) = 14.62, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .34. Follow-up pairwise comparisons with Bonferroni corrections revealed that the figure condition had lower accuracy on the task (M = 48%, SD = 44%) than both numerical conditions, which did not differ (sequence: M = 94%, SD = 22%; table: M = 95%, SD = 23%).

For the matching task, children's performance did not reliably differ by presentation type, χ²(2, 61) = 4.01, p = .14, Cramer’s V = .26. A similar number of children selected the correct recursive rule in the sequence condition (95.5%), table condition (89.5%) and figure condition (75.0%).

For the far extension task, 77.3% of children in the sequence condition responded correctly, compared to 78.9% in the table condition and 30.0% in the figure condition, and the effect of presentation condition was significant, χ²(2, 61) = 13.23, p = .001, Cramer’s V = .47. Post-hoc pairwise comparisons with Bonferroni corrections showed that the numerical presentation conditions did not differ from each other, but the figure condition had lower performance than both the sequence condition (p < .01), and table condition (p < .01).

Taken together, these results suggest that for patterns based on proportional functions, numerical presentation may lead to better performance than figural presentation for both near and far extension tasks. However, there was only a single proportional problem, and the task demands may have been greater in the figural condition than in the numerical condition for two reasons. First, the figure condition entailed drawing demands that were not present in the other conditions. Second, the presence of two different colors of squares in the figure representing the pattern may have impeded children from finding the rate of change by encouraging them to think about the white and shaded squares as separate quantities.

Non-Proportional Problems

We calculated the percentage of correct responses for each participant for all of the tasks in the non-proportional problems presented with patterns (either figures, tables, or sequences depending on condition) and as story problems without patterns at transfer. For patterning problems, overall mean percentage correct was 84% (SD = 31%) for the near extension tasks, 84% (SD = 30%) for the matching tasks, and 20% (SD = 36%) for the far extension tasks. For transfer problems, overall mean percentage correct was 64% (SD = 36%) for the near extension tasks, 80% (SD = 29%) for the matching tasks, 21% (SD = 31%) for the far extension tasks, and 34% (SD = 39%) for the reversal tasks, which were only present at transfer. To test whether presentation type impacted performance on the three tasks for non-proportional problems, we conducted a 3 (condition: sequences, tables, figures) x 3 (task: near extension, far extension, matching) x 2 (problem type: patterning, transfer) mixed ANOVA with accuracy as the dependent variable² (see Table 2).

The analysis revealed main effects of task, F(2, 116) = 232.93, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .80, and problem type, F(1, 58) = 5.11, p = .03, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .08, and no main effect of condition, F(2, 58) = 0.59, p = .56. The main effect of problem type indicates that overall accuracy was lower for transfer problems than patterning problems, which suggests that transfer problems are more difficult and may be tapping separate constructs (e.g., reading comprehension) than patterning problems. All two-way interactions were significant (condition by task: F(4, 116) = 2.92, p = .02, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .09, condition by type: F(2, 58) = 4.25, p = .02, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .13, task by type: F(2, 116) = 6.35, p < .01, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .10). These effects were qualified by a three-way interaction, F(4, 116) = 5.10, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .15. Follow-up analyses revealed a significant simple two-way interaction of condition and task for patterning problems³, F(2.98, 86.40) = 7.46, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .09. When problems were presented with patterns as either figures, tables, or sequences, children performed similarly on the near extension task, F(2, 58) = 2.32, p = .11, and on the matching task, F(2, 58) = 1.04, p = .36. However, pattern presentation format did impact children’s performance on the far extension task, F(2, 58) = 9.91, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .26. Pairwise comparisons with Bonferroni corrections revealed that this effect was driven by the figure condition outperforming both numerical presentation conditions, which did not differ from each other. These results suggest that for patterns with non-proportional underlying functions, there may be an advantage for figural presentation, specifically for far extension tasks.

In contrast, there was no simple two-way interaction of condition by task for transfer problems, F(4, 116) = 1.26, p = .29. An additional one-way ANOVA comparing reversal task performance across conditions also revealed no differences in accuracy, F(2, 58) = 1.53, p = .23, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .05. Taken together, these results suggest that the figural presentation advantage observed for non-proportional far extension pattern problems does not transfer to new problems in which figural presentation is not provided.

Proportional Patterning Problem

We used Chi-square tests to determine whether children used different strategies in each condition for near and far extension tasks (see Table 3). All follow-up pairwise comparisons are reported with Bonferroni corrections. For the near extension task, condition differences emerged for Full Recursive strategies, χ²(2, 61) = 11.53, p = .003, Cramer’s V = .44, and Full Functional strategies, χ²(2, 61) = 31.10, p < .001, Cramer’s V = .71. Children in the sequence condition were more likely to rely on Full Recursive strategies than children in the table condition, and children in the table condition were more likely than children in either of the other two conditions to rely on Full Functional strategies.

For the far extension task, few children in any condition relied on Full Recursive strategies, and there were no differences across conditions, χ²(2, 61) = 2.25 p = .33. However, there were condition differences for Part Recursive, χ²(2, 61) = 9.82, p < .01, Cramer’s V = .40, and Full Functional strategies, χ²(2, 61) = 18.72 p < .001, Cramer’s V = .55. Children in the figure condition were more likely to rely on Part Recursive strategies than children in the table condition, and children in the figure condition were also less likely to rely on Full Functional strategies than children in either of the two numerical format conditions. As with the difference in performance on the proportional near and far extension tasks for the figure condition relative to the other two conditions, the difference in Full Functional strategy use may be due to the shading of the figure in the proportional problem. Several students in the figure condition referred to elements in the pattern based on whether they were shaded or unshaded squares, so the shading may have obscured the overall pattern (i.e., y = 7x).

Non-Proportional Problems

Near and far extension tasks were analyzed separately because of slightly different coding for each task (far extension tasks were coded for Part Recursive strategies, whereas near extension tasks were not), and percentage use for each strategy can be found in Table 4. In order to reduce the number of statistical tests, the No Intercept and Wrong intercept codes were collapsed into a Partial Functional code for both near and far extension tasks. In addition, the Full Recursive and Part Recursive codes were collapsed into an Any Recursive code for far extension tasks.

For near extension tasks, we conducted a 3 (condition: sequences, tables, figures) x 3 (strategy: Full Recursive, Full Functional, Partial Functional) x 2 (problem type: patterning, transfer) mixed ANOVA to test whether condition impacted strategy use on non-proportional problems. The analysis revealed main effects of strategy, F(2, 116) = 199.07, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .74, and problem type, F(1, 58) = 31.59, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .35, and no main effect of condition, F(2, 58) = 0.30, p = .74. The two-way interaction of strategy by problem type was significant, F(2, 116) = 11.66, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .17, and the other two-way interactions were not (condition by problem type: F(2, 58) = 2.44, p = .10, condition by strategy: F(4, 116) = 2.18, p = .08). All of these effects were qualified by a three-way interaction, F(4, 116) = 4.70, p < .01, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .14. Similar to the accuracy findings, follow-up analyses revealed a significant simple two-way interaction of condition and strategy for patterning problems, F(3.24, 93.8) = 5.07, p < .01, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .15. When problems were presented with patterns as either figures, tables, or sequences, condition differences emerged in the use of Full Recursive, F(2, 58) = 4.53, p = .01, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .14, and Full Functional strategies, F(2, 58) = 4.33, p = .02, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .13, but not Partial Functional strategies, F(2, 58) = 1.37, p = .26. Follow-up pairwise comparisons with Bonferroni corrections revealed that children in the figure condition relied on Full Recursive strategies less frequently and relied on Full Functional strategies more frequently than children in the sequence condition. For transfer problems, the simple two-way interaction was not significant, F(3.44, 99.6) = 1.78, p = .15, suggesting that strategy use did not vary by condition when near extension problems were not presented with accompanying patterns.

For the far extension tasks, a 3 (condition: sequences, tables, figures) x 3 (strategy: Any Recursive, Full Functional, Partial Functional) x 2 (problem type: patterning, transfer) mixed ANOVA revealed main effects of strategy, F(2, 116) = 17.20, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .23, and problem type, F(1, 58) = 14.31, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .20, and no main effect of condition, F(2, 58) = 1.08, p = .35. The two-way interaction of condition by strategy: F(4, 116) = 6.91, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .19, was significant, but the condition by problem type, F(2, 58) = 1.85, p = .17, and strategy by problem type, F(2,116) = 2.27, p = .11, interactions were not. Once again, all of these effects were qualified by a three-way interaction, F(4, 116) = 10.61, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .27. Follow-up analyses again revealed a significant simple two-way interaction of condition and strategy for patterning problems, F(3.54, 103) = 16.90, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .37. When problems were presented with accompanying patterns, condition differences emerged in the use of Full Functional, F(2, 58) = 28.80, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .50, and Partial Functional strategies, F(2, 58) = 3.36, p = .04, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .10, but not Any Recursive strategies, F(2, 58) = 2.51, p = .09. Children in the figure condition used Full Functional strategies more frequently than children in the other two conditions, which did not differ. Moreover, the children in the figure condition were less likely to ignore the starting value or rely on a starting value other than x = 0 compared to children in the other conditions, which did not differ.

Once again, there was no simple two-way interaction of condition and strategy at transfer, F(4, 116) = 0.43, p = .79, suggesting that there were no condition differences in strategy use for far extension transfer problems. Additional one-way ANOVAs were conducted to determine whether strategy use varied by condition for reversal tasks (see Table 5). Condition differences did not emerge in strategy use on the reversal tasks (Recursive: F(4, 58) = 0.85, p = .43; Full Functional: F(2, 58) = 1.01, p = .37; Partial Functional: F(2, 58) = 1.54, p = .22).

Finally, we examined the use of spontaneously generated patterns (e.g., drawing a pattern or generating a table of values) at transfer. Students rarely spontaneously generated a pattern in any format when solving transfer problems, and this did not vary by condition. One student in the sequence condition generated a sequence for one of the near extension tasks. One student in the figure condition generated a figure for the reversal task in the second transfer problem. Otherwise, tables were the main type of format that students generated, and these were generated primarily for the near extension tasks. Three students (one in each condition) created a table for the first transfer problem, and six students (three in the sequence condition, two in the table condition, and one in the figure condition) created a table for the second transfer problem.