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Prior work exploring preschool-aged children’s reasoning with repeating patterns has shown that patterning ability is an important predictor of math achievement; however, there is limited research exploring older children’s growing pattern task performance. The current study tested whether presentation format impacts performance on growing pattern problems, and whether the effects of presentation format extend to transfer word problems for which no patterns are provided. Sixth grade students were randomly assigned to complete several growing pattern tasks in one of three presentation formats (figures, sequences of values, or tables of values), and later completed transfer story problems with no figures, sequences, or tables provided. Findings suggest that presenting growing patterns as figures can benefit performance, although these benefits may depend on both pattern type and task. No differences were observed in performance on transfer problems, likely because students rarely spontaneously generated figures. Additional exploratory analyses suggest that performance on growing pattern problems may be related to both standardized math ability and fraction task performance, whereas inhibitory control may only be related to performance for specific patterning tasks. These findings have implications for educators because describing/expressing patterns is critical to algebra and higher-level mathematics.

Mathematics ability is related to several important life outcomes, including better-informed health and financial choices (

Two main types of patterns have been the focus of existing research: repeating patterns and growing patterns. In repeating patterns, a set of elements, the unit of repeat, is repeated indefinitely. For instance, in the repeating pattern ABCABCABC, the unit of repeat, ABC, consists of the elements A, B, and C. Growing patterns increase or decrease in well-defined, predictable ways, and can be expressed recursively (e.g.,

A growing body of work supports a connection between patterning ability and math outcomes; most of this research has focused on preschoolers completing repeating patterns (

In the remainder of this introduction, we provide a brief review of the literature motivating our primary research question: how does the presentation format of growing pattern tasks (e.g., visual versus numerical) impact performance and transfer? Then, we review relevant research motivating our exploratory analyses for our second research question: what are the relations between growing pattern task performance and individual differences including inhibitory control, fraction task performance, and math ability? In our study, 72 sixth-grade participants were randomly assigned to complete growing pattern tasks in one of three experimental conditions (numerical sequences, tables, or figures). Next, all participants completed transfer problems, in the form of algebra story problems with no sequences, tables, or figures provided, and individual difference measures. Our findings suggest that the way patterning tasks are presented may impact performance, but these effects may vary depending on the type of pattern and the type of task.

Growing patterns can be presented to learners in different formats, such as tables of input and output values, sequences of values, or figures (see

Presenting mathematical problems or tasks in different ways can impact performance. When solving a math problem, a learner must first form an accurate internal representation by encoding relevant problem elements before selecting and executing a solution strategy. Thus, the nature of a learner’s internal representation can impact strategy selection or which prior knowledge is applied during problem solving (

Although children can successfully reason with visually presented patterns (usually in the form of a sequence of shapes and colors), there is also evidence that visual format could hinder patterning performance relative to other formats. Preschoolers are less successful at extracting the underlying structure for repeating patterns presented with concrete descriptions referencing the shape or color of the elements in the pattern rather than abstract descriptions (

We argue that visual format (see

It is also an open question whether different presentation formats impact performance on transfer problems for which no figure, sequence, or table is provided. Teaching students a mathematical rule using concrete symbols (e.g.,

Presentation format may impact students’ choice of strategies to solve patterning tasks. When children and adults reason about growing patterns presented with figures, use of two specific strategies may help them attend to both the constant properties of a pattern as well as the parts that change (

Although there is evidence that students may rely on both recursive and functional strategies when reasoning about growing patterns presented with figures (

To address our secondary research question, we consider the role of individual differences in patterning performance based on evidence supporting a link between patterning and math achievement (

Fractions are a specific relational mathematical concept because they entail a relation between parts of a whole and the fraction’s whole (

A fraction comparison task and a number line estimation task were used in our study. In addition to fractions being inherently relational, fraction tasks may encourage relational thinking in various ways. For instance, in the number line estimation task a child is asked to place the fraction 5/6 on a number line with endpoints of 0 and 1. The child may consider the relation between the numerator and denominator to estimate the magnitude of the fraction. The child might think about how half of 6 would be 3, so 5/6 should be more than halfway between 0 and 1. The child may try to relate her prior experience with whole numbers on number lines when deciding where to place the fraction on the number line (

Patterning ability is thought to be related to relational reasoning ability (

Another individual difference that may be related to patterning ability is inhibitory control, or the ability to attend to task-relevant information while ignoring irrelevant information. Inhibitory control predicts overall math ability (

Lastly, given the correlational evidence supporting a link between patterning and general math outcomes (

The primary goal of the current study was to test whether sixth graders approach patterning tasks differently depending on whether the patterns are presented numerically or with figures. By sixth grade, students have exposure to the Common Core standards in the “Operations and Algebraic Thinking” domain which deal with generating numerical sequences using rules that relate input to output (CCSSM 6.EE.C.9;

A secondary goal of the current study was to explore the potential relations between patterning and individual differences in inhibitory control, relational reasoning, and math ability. Based on prior work with repeating patterns (

Participants were recruited from a single intermediate school which reported 24% of students were eligible for free or reduced lunch. The school provided standardized reading and math scores from the prior school year for each participant. Seventy-two sixth grade students (38% female; 79% White) were randomly assigned to one of three conditions (24 per condition): sequence, table, or figure. Eleven participants were excluded^{1}

Nine of these participants ran out of time, and 2 expressed that they did not wish to continue. These participants did not differ from the participants with complete data on any of the individual difference measures: standardized math and reading scores, relational reasoning (fraction magnitude comparison and number line estimation performance), or inhibitory control (numerical Stroop performance).

from the analyses of patterning and transfer task performance for not completing all of the problems (sequence condition:Measure | Sequence |
Table |
Figure |
|||
---|---|---|---|---|---|---|

Standardized Math Score | 717.00 | 71.33 | 721.11 | 24.74 | 722.50 | 35.43 |

Standardized Reading Score | 708.33 | 72.79 | 722.68 | 33.62 | 718.35 | 50.60 |

Fraction Comparison Proportion Correct | .84 | .13 | .81 | .17 | .86 | .13 |

Number Line Estimation (PAE) | 26% | 6% | 25% | 4% | 26% | 4% |

Stroop Task Inhibition RT (ms) | 960 | 150 | 890 | 210 | 900 | 190 |

Participants were taken out of class and run individually in two sessions in a quiet location in their intermediate school. In the first experimental session, children completed growing pattern problems (see

Children were randomly assigned to complete the growing pattern problems in a figural format, sequence format, or a table format. No figures, tables, or sequences were provided for the transfer problems. Initial pilot work indicated that this number of problems was the maximum that could be completed in a single, hour-long session. In the second session, children completed the fraction tasks (fraction comparison and number line estimation), and the inhibitory control task (numerical Stroop) in a randomized order on a laptop using Qualtrics software.

Children completed the same multi-component growing pattern problems (see ^{rd} number/figure be?). None of the far extension tasks featured “seductive” stage numbers such as 50 or 100, which may encourage incorrect proportional multiplication (

Participants completed the same multi-component transfer problems (see

Participants completed the same 32 fraction magnitude comparison problems as in

Participants completed the same 20 number line estimation problems as in

Participants completed a numerical Stroop task as a measure of their inhibitory control. As in _{dif} = 149ms, _{dif} = 141 ms,

The mean time between the two experimental sessions was 19.20 days (

For the near extension task, participants were asked to provide outputs for three stages (see

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.

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

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

Recursive strategies relied primarily on output values to generate a response and could be either full recursive or part recursive. ^{rd} output value in a pattern). ^{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

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.

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

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

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.,

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% (

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,

For the matching task, children's performance did not reliably differ by presentation type, χ^{2}(2, 61) = 4.01,

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, χ^{2}(2, 61) = 13.23,

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.

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% (^{2}

For the matching and far extension tasks, the results do not change when each matching item and each far extension item are analyzed separately using non-parametric analyses, so the continuous analyses are reported here for ease of comparison across tasks. Item-level accuracy for each task can be found in

Task by Problem Type | Sequence |
Table |
Figure |
|||
---|---|---|---|---|---|---|

Pattern | ||||||

Near Extension | 94% | 18% | 74% | 34% | 84% | 36% |

Matching | 88% | 27% | 76% | 35% | 88% | 28% |

Far Extension | 5% | 15% | 11% | 28% | 45% | 46% |

Transfer | ||||||

Near Extension | 68% | 36% | 68% | 30% | 55% | 43% |

Matching | 77% | 34% | 84% | 24% | 80% | 30% |

Far Extension | 30% | 33% | 16% | 29% | 15% | 29% |

Reversal | 43% | 44% | 37% | 37% | 23% | 34% |

The analysis revealed main effects of task, ^{3}

When the degrees of freedom are not whole numbers, this indicates corrections for sphericity assumption violations.

,In contrast, there was no simple two-way interaction of condition by task for transfer problems,

We used Chi-square tests to determine whether children used different strategies in each condition for near and far extension tasks (see ^{2}(2, 61) = 11.53, ^{2}(2, 61) = 31.10,

Strategy by Task | Sequence | Table | Figure |
---|---|---|---|

Near Extension | |||

Full Recursive | 68% | 16% | 40% |

Full Functional | 18% | 84% | 5% |

Far Extension | |||

Full Recursive | 0% | 5% | 0% |

Part Recursive | 9% | 5% | 40% |

Full Functional | 86% | 84% | 30% |

For the far extension task, few children in any condition relied on ^{2}(2, 61) = 2.25 ^{2}(2, 61) = 9.82, ^{2}(2, 61) = 18.72

Near and far extension tasks were analyzed separately because of slightly different coding for each task (far extension tasks were coded for

For near extension tasks, we conducted a 3 (condition: sequences, tables, figures) x 3 (strategy:

For the far extension tasks, a 3 (condition: sequences, tables, figures) x 3 (strategy:

Once again, there was no simple two-way interaction of condition and strategy at transfer,

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.

Strategy by Task | Pattern |
Transfer |
||||
---|---|---|---|---|---|---|

Sequence | Table | Figure | Sequence | Table | Figure | |

Near Extension | ||||||

Full Recursive | 89% | 74% | 55% | 73% | 87% | 85% |

Full Functional | 0% | 8% | 20% | 5% | 5% | 3% |

Part Functional – No Intercept | 0% | 5% | 8% | 34% | 24% | 45% |

Part Functional – Wrong Intercept | 0% | 3% | 0% | 0% | 0% | 3% |

Far Extension | ||||||

Full Recursive | 0% | 11% | 3% | 0% | 5% | 0% |

Part Recursive | 19% | 13% | 0% | 41% | 16% | 28% |

Full Functional | 2% | 11% | 68% | 23% | 24% | 23% |

Part Functional – No Intercept | 61% | 37% | 10% | 30% | 45% | 25% |

Part Functional – Wrong Intercept | 14% | 24% | 0% | 27% | 11% | 30% |

Reversal | ||||||

Part Recursive | – | – | – | 53% | 39% | 48% |

Full Functional | – | – | – | 28% | 37% | 20% |

Part Functional – No Intercept | – | – | – | 19% | 24% | 25% |

Part Functional – Wrong Intercept | – | – | – | 21% | 3% | 20% |

Although this study lacked the sample size to adequately explore how these individual difference measures might predict performance on each of the patterning tasks with regression analyses^{4}

Using G*Power 3.1.9.2 (^{2} = .15) with four predictors.

Performance on the number line estimation task, fraction comparison task, and the math ability measure were related to performance for each of the three patterning tasks. The strength of the correlations among fraction task performance, math ability, and patterning performance was similar for both the near extension and matching tasks. For the far extension task, however, these correlations were slightly weaker. This could be due to a higher likelihood of computation errors for far extension tasks, or children’s reliance on different strategies across the patterning tasks. Performance on the near extension and matching tasks was highly related, and children tended to rely heavily on recursive strategies for both of these tasks. In contrast, the relation between near and far extension performance was weaker, and there was no relation between matching and far extension performance. It is possible that performance on near extension and matching tasks, for which children tended to use recursive strategies, relies more on magnitude understanding, but also less on inhibitory control, compared to performance on far extension tasks. Overall, these results suggest that for growing pattern tasks, math ability and fraction task performance have similar relations to patterning performance.

Measure | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1. Numerical Stroop | – | -.45** | .31* | -.35** | -.22 | -.15 | -.26* |

2. Fraction Comparison | – | -.53** | .52** | .47** | .38** | .31* | |

3. Number Line Estimation | – | -.75** | -.41** | -.51** | -.29* | ||

4. Standardized Math | – | .47** | .56** | .29* | |||

5. Near Extension | – | .81** | .28* | ||||

6. Matching | – | .21 | |||||

7. Far Extension | – |

*

The primary goal of the current study was to test the effect of pattern presentation format on performance and strategy choice for patterning tasks presented in either figural or numerical format and on subsequent transfer problems with no accompanying figural or numerical patterns. Consistent with prior work in other domains of mathematics (

Analysis of the strategy coding for participants’ open-ended explanations for their solutions revealed that presentation format also impacted students’ choice of strategies for solving growing pattern problems. This effect was most apparent in the far extension tasks for the non-proportional patterns. For these tasks, figural presentation format facilitated the use of function-based strategies, which are more efficient and more aligned with expert thinking compared to recursive strategies. Further, presenting non-proportional far extension tasks with figures seemed to help students attend to the correct starting values for these problems. Taken together, the differences in strategy use and performance suggest an advantage of presenting patterning tasks in figure format, specifically for far extension tasks with non-proportional underlying functions. However, children’s lower performance in the figure condition on the near and far extension tasks for the proportional problem relative to the other two conditions suggests one of two possibilities. Either the presence of figures themselves hurt performance, or some specific features of the figures hurt performance. Because the findings of the current study are based on participant responses for three open-ended questions for a single proportional pattern with a single associated figure, these two possibilities require further exploration.

The differential effects of figures for proportional and non-proportional growing patterns suggest that the specific surface-level features of the figures may be important. In the case of non-proportional patterns, these features may have enhanced encoding of relevant pattern features, but for the proportional pattern these features may have impeded encoding of the underlying relation between stages of the pattern and the number of elements in the figure. In the proportional pattern used in the current study, the figure was shaded in such a way as to potentially obscure the most efficient expression of the underlying function (i.e.,

Although the shading of the figures may have hurt performance on the proportional pattern tasks, presenting the non-proportional patterns with figures helped performance and encouraged more frequent use of functional strategies on the near and far extension tasks. Again, there may be two possible explanations for this effect. The mere presence of a figure may be beneficial, or figures with certain visual features that specifically guide attention to important aspects of the pattern may be driving the benefit. For the non-proportional problems in the current study, the shading of the squares in the figures may have helped draw students’ attention to the starting value of the pattern, which was always shaded differently than the part of the pattern that varied. It is not clear whether students would have been more likely to attend to the starting values in these problems if the figures had not been shaded in such a way as to highlight the starting values.

Some additional differences between the figure condition and the two numerical conditions may warrant additional exploration in future research. The figures used in this study may have been more variable than sequences or tables in terms of visual features since there was a different arrangement of squares for each pattern. Although shape is a salient perceptual feature (e.g.,

Presenting patterns as figures led to more variability in patterning task performance compared to presenting patterns numerically, and there are a number of possible reasons for this. The figure condition did impose drawing demands that were not present in the other conditions, and this may have contributed to greater variability in performance when patterns were presented with figures as opposed to tables or sequences of values. Related to these additional demands, the figure condition may have introduced some ambiguity about how children should indicate a response for the tasks, as evidenced by slightly more instances of

In the current study, differences in performance did not emerge between the two numerical presentation formats on any of the tasks. This may be surprising given that tables included both input and output values which could have helped students identify the underlying functional relation. However, this benefit would depend on whether students attended to the provided input values in the tables. Although there were no differences in performance between the two numerical presentation formats, there was a difference in strategy use, though only for the proportional near extension task. For this task, the sequence presentation format may have led children to rely on recursive strategies and the table presentation format may have led children to rely on functional strategies. Given that this result is based on a single pattern and task, this should be interpreted with caution, but it suggests that for some types of problems it might be advantageous to provide students with tables instead of sequences. Future research could test whether drawing students’ attention to the input values might strengthen potential benefits of presenting problems with tables relative to sequences.

Although presentation format had an effect on both performance and strategy use for growing pattern problems, these effects did not transfer to problems presented without accompanying figures, tables, or sequences. No figure condition advantage emerged for near extension or far extension tasks for transfer problems, suggesting that students did not spontaneously leverage benefits of figures when figures were not explicitly provided. Students rarely generated their own presentation format when solving the transfer problems, which is in line with prior work (

Given that the findings of the current study suggest an advantage for presenting growing non-proportional patterns with figures and that children are unlikely to spontaneously generate figures when solving transfer problems, how can we get children to generate figures when they are not present? One possible explanation for why students were unlikely to transfer strategies they had used with pattern problems presented with figures to transfer problems may be that they needed to practice with many more patterns before transfer occurred. However, students in the study conducted by

A secondary goal of the current study was to explore how individual differences in math ability, fraction task performance, and inhibitory control relate to performance on growing pattern tasks. These analyses were exploratory, but they provide valuable preliminary insights into how these individual differences may relate to growing pattern performance specifically. Although prior work involving young children who completed repeating pattern tasks has not consistently demonstrated a relation between inhibitory control and patterning performance (

Prior work (

Although we found several significant relations among patterning performance, fraction task performance, and a measure of math ability in our exploratory analyses, future studies should endeavor to include a larger sample size, as is customary with individual differences studies, to test the unique contributions of math ability and inhibitory control on growing pattern performance. Understanding the role of individual differences that predict patterning ability may inform potential interventions to improve this ability.

Another important contribution of the current work is the extension of patterning research to older students' performance on growing patterns. The bulk of the existing research on patterning has relied on preschool-aged children’s repeating pattern performance (

The presentation format of growing pattern problems can impact performance and strategy use on certain patterning tasks. Presenting non-proportional growing patterns with figures may improve performance on far extension tasks by making elements of the pattern more salient, however children are unlikely to spontaneously generate figures when solving transfer problems. The current study also demonstrated preliminary evidence of relations between several individual difference measures--performance on fraction tasks, inhibitory control, and math ability--and patterning performance. These findings have implications for educators because describing/expressing patterns is critical to algebra and higher-level mathematics.

Children’s understanding of patterns has received much international attention due to the intriguing connection between patterning and math achievement. This study addresses an important gap in the existing patterning literature and represents important first steps towards better understanding how older students reason with growing patterns. This work also provides a valuable contribution to existing research on how problem presentation format impacts learning and transfer in mathematics. Furthermore, the findings from the current study provide a springboard for future work testing the impact of surface-level features of visual patterns on performance, which may help provide specific guidance to educators seeking to support children’s understanding of growing patterns.

Note: Participants were shown the problems with one of the three presentations below (depending on condition).

Jenna just got a new book, and she starts reading the same amount every day. After the first day, she has 182 pages left to read. After the third day, she has 168 pages left to read. After the fifth day, she has 154 pages left to read.

Nathan already has some baseball cards, but he wants to start collecting more. When Nathan buys 1 pack of cards, he has 36 total cards in his collection. After buying his second pack of cards, he has 60 cards total. When Nathan buys his third pack of cards, he has 84 cards in all.

Problem Type / Underlying Function | Sequence |
Table |
Figure |
|||
---|---|---|---|---|---|---|

Session 1 (presented with patterns) | ||||||

y = 7x |
.94 | .22 | .95 | .23 | .48 | .44 |

y = 2x + 3 |
.92 | .23 | .74 | .41 | .85 | .37 |

y = 4x + 1 |
.95 | .95 | .74 | .42 | .83 | .37 |

Transfer (presented as story problems without patterns) | ||||||

y = 189 - 7x |
.70 | .42 | .54 | .49 | .57 | .47 |

y = 24x + 12 |
.67 | .41 | .81 | .34 | .53 | .49 |

Problem Type / Underlying Function | Sequence | Table | Figure |
---|---|---|---|

Session 1 (presented with patterns) | |||

y = 7x |
.96 | .90 | .75 |

y = 2x + 3 |
.91 | .74 | .85 |

y = 4x + 1 |
.86 | .79 | .90 |

Transfer (presented as story problems without patterns) | |||

y = 189 - 7x |
.68 | .74 | .70 |

y = 24x + 12 |
.86 | .95 | .90 |

Problem Type / Underlying Function | Sequence | Table | Figure |
---|---|---|---|

Session 1 (presented with patterns) | |||

y = 7x |
.77 | .79 | .30 |

y = 2x + 3 |
.05 | .11 | .40 |

y = 4x + 1 |
.05 | .11 | .50 |

Transfer (presented as story problems without patterns) | |||

y = 189 - 7x |
.27 | .16 | .15 |

y = 24x + 12 |
.32 | .16 | .15 |

Underlying Function | Sequence | Table | Figure |
---|---|---|---|

y = 189 - 7x |
.48 | .33 | .19 |

y = 24x + 12 |
.40 | .35 | .25 |

Strategy by Task | Sequence | Table | Figure |
---|---|---|---|

Proportional | |||

Near Extension | |||

No Rule | .14 | .00 | .45 |

Other | .00 | .05 | .15 |

Computation Error | .05 | .00 | .10 |

Missed Step Error | .00 | .00 | .10 |

Far Extension | |||

No Rule | .00 | .00 | .35 |

Other | .09 | .00 | .05 |

Computation Error | .14 | .16 | .15 |

Missed Step Error | .00 | .00 | .00 |

Non-Proportional | |||

Near Extension | |||

No Rule | .11 | .08 | .20 |

Other | .00 | .11 | .05 |

Computation Error | .02 | .11 | .00 |

Missed Step Error | .00 | .05 | .08 |

Far Extension | |||

No Rule | .05 | .00 | .18 |

Other | .14 | .11 | .18 |

Computation Error | .12 | .00 | .08 |

Missed Step Error | .00 | .00 | .08 |

Transfer | |||

Near Extension | |||

No Rule | .09 | .05 | .05 |

Other | .11 | .05 | .10 |

Computation Error | .16 | .11 | .20 |

Missed Step Error | .30 | .24 | .35 |

Far Extension | |||

No Rule | .02 | .05 | .05 |

Other | .14 | .13 | .18 |

Computation Error | .09 | .03 | .08 |

Missed Step Error | .07 | .00 | .05 |

Reversal | |||

No Rule | .05 | .16 | .18 |

Other | .16 | .11 | .08 |

Computation Error | .14 | .13 | .20 |

The instances of

This work was supported in part by the Institute of Education Sciences, U.S. Department of Education award R305A160295, to Kent State University.

The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.