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Symbolic number knowledge is strongly related to mathematical performance for both children and adults. We present a model of symbolic number relations in which increasing skill is a function of hierarchical integration of symbolic associations. We tested the model by contrasting the performance of two groups of adults. One group was educated in China (n = 71) and had substantially higher levels of mathematical skill compared to the other group who was educated in Canada (n = 68). Both groups completed a variety of symbolic number tasks, including measures of cardinal number knowledge (number comparisons), ordinal number knowledge (ordinal judgments) and arithmetic fluency, as well as other mathematical measures, including number line estimation, fraction/algebra arithmetic and word problem solving. We hypothesized that Chinese-educated individuals, whose mathematical experiences include a strong emphasis on acquiring fluent access to symbolic associations among numbers, would show more integrated number symbol knowledge compared to Canadian-educated individuals. Multi-group path analysis supported the hierarchical symbol integration hypothesis. We discuss the implications of these results for understanding why performance on simple number processing tasks is persistently related to measures of mathematical performance that also involve more complex and varied numerical skills.

Symbolic representations are central to mathematical processing (

The underlying assumption of the HSI model is that various symbolic number associations become progressively more

Symbolic number comparison (i.e., which is larger, 3 or 7;

Individual differences in ordinal knowledge have also been linked to individual differences in mathematical skill (

In studies with young children, basic arithmetic facts (e.g., 3 + 4, 7 – 4, 5 x 8) are used as an index of developing mathematical skill (e.g.,

In older children and adults, fluency (i.e., speed and accuracy) of solving multi-digit arithmetic problems (e.g., 34 + 57, 23 x 7) is typically used to index arithmetic skill (

Among adults, skilled knowledge of cardinal, ordinal, and arithmetic associations reflects the development of fast and accurate performance in the criterion tasks described above. Because the same number symbols are used in cardinal, ordinal, and arithmetic tasks, ability to inhibit, suppress, or control the activation of specific associations must exist (

In this paper, we propose a model of the relations among number comparisons, ordinal judgments, and arithmetic performance in which these tasks all reflect individual differences in the fluency of access to associations among symbolic numbers. Furthermore, we propose that these various tasks represent stages in the ongoing integration of the numerical associations into a coherent network. Accordingly, the HSI model captures patterns of correlations that reflect shared individual differences among symbolic number tasks (

Research using number-matching tasks supports the view that activation of the various associations among symbolic numbers change over time and vary with arithmetic skill (

Learning arithmetic is complicated because numbers have multiple associations that need to be differentially activated depending on the context.

Individuals who can manage competing associations and thus respond quickly in tasks such as ordinal judgments may also develop fast and direct access to arithmetic associations. According to the HSI model, the underlying associations with cardinal and ordinal relations do not disappear but they are no longer the main determinant of individual differences in tasks which include access to number associations as part of the solution process (e.g., fraction and algebra arithmetic). Instead, arithmetic performance will be the best predictor of these higher-level mathematical tasks both because arithmetic associations are often relevant in task performance and because individual differences in arithmetic performance will be representative of the individual’s ability to form and access integrated associations. This model of symbolic number integration implies that number comparisons and ordinal judgments are correlated with arithmetic because all three tasks depend on accessing an integrated network of symbolic number associations (

Another possible factor in the hierarchical relations among arithmetic, ordinal, and cardinal associations is the extent to which each task relies on or activates non-symbolic information.

The HSI model that we have proposed differs from other accounts of the strong association between ordinal judgments and arithmetic performance. For example,

Furthermore, the specific pattern of observed correlations depends on the extent to which individuals have developed an integrated associative network. We stress that integrated, in this context, means that there is a single representation that captures all of the learned associations, rather than one in which different networks are accessed depending on the required task. A fully integrated network allows successful performance of many related tasks and yet is tuned for specific instances or activities. Just as skilled readers show Stroop effects in color-word naming (

In summary, the HSI model shares with other theories the view that symbol-symbol associations are fundamental to numerical processing tasks (

The goal of the present research was to test certain predictions of the HSI model in two different groups of university students; one group had completed elementary and secondary school in Canada and the other in China. Both groups learned cardinal, ordinal, and arithmetic associations in the course of acquiring mathematical skills and, although the learning experiences may have been very different, both groups can competently perform number comparison, ordinal judgment, and arithmetic tasks. We tested three general hypotheses. First, we assessed specific predictions of the symbol integration model by exploring the relations among prototypical number comparisons, ordinal judgments, and arithmetic tasks. We expected strong correlations among these measures in both groups, replicating other work (

Second, we extended the analyses to three other mathematical tasks that are plausibly linked with these fundamental skills through access to symbolic number skills. Number line performance (i.e., placing a number on a line with endpoints marked) is consistently correlated with other mathematical skills and involves knowledge of the interrelations among symbolic numbers (

Third, we tested the HSI model in a group of adults who were educated in China and compared them to a Canadian-educated sample. Why is this an interesting comparison? In various studies, Chinese students studying at Canadian universities have been shown to have much better arithmetic skills than Canadian-educated students (

We tested the HSI model shown in

Proposed Hierarchical Symbol Integration (HSI) model. A mediating effect of ordinal judgments on the relations between number comparisons and all other outcomes were predicted for Canadians (dotted lines and solid lines). The relations between ordinal judgments and number line, fraction/algebra arithmetic, and word problem solving were expected to be mediated by arithmetic fluency for the Chinese (solid lines).

One hundred and forty-two participants were recruited. All participants received a 2% bonus credit towards their introductory psychology or cognitive science courses or $20. Seventy-one Chinese-educated participants (49 females) had completed elementary and/or secondary school in China. Seventy-one Canadian-educated participants (43 females) had completed elementary and/or secondary school in Canada. Age was not significantly different between the Chinese-educated participants (

Participants were tested individually in a quiet room. Testing lasted for approximately two hours. Participants were given a short break after an hour of testing. To minimize differences in the task demands across groups, Canadian-educated participants were instructed in English whereas Chinese-educated participants were instructed in Chinese and each group responded to the verbal tasks (e.g., word problem solving) in their first language.

For the present analysis, we focus on the participants’ performance on number comparisons, ordinal judgments, arithmetic fluency, a 0-7000 number line task, word problem solving, and fraction/algebra arithmetic. This study was a part of a larger project. The full list of measures that participants completed is shown in the Supplementary material.

The number comparison task was used to measure participants’ ability to decide which of the two symbolic numbers (from 1 to 9) was numerically larger. Participants were presented with pairs of single-digit numbers (e.g., 3 4) and they were asked to cross out the larger digit as quickly and accurately as possible. They first completed three sample items for practice. Following the practice, participants completed two pages of stimuli (8 ½ x 11 inch white paper). Each page consisted of 30 stimuli presented in six rows. A mean score of correct items-per-second for each page was created using the formula: (number of correct items) / time in seconds. For example, if the participant completed one page of this task in 30 seconds and made two errors (out of 30), his or her score for that page would be [(30-2) / 30 = .93] items-per-second. The mean correct items-per-second score was the average across the two pages. Internal reliability based on performance on the two pages was high for both Chinese-educated (Cronbach’s α = .95) and Canadian-educated participants (Cronbach’s α = .96).

The ordinal judgment task was used to measure participants’ ability to judge whether three symbolic digits are in order or not. Participants were presented with three-digit number sequences and asked to put a “✓” beside the sequences that were in either ascending (e.g., 2 5 9 and 4 5 6) or descending order (e.g., 5 4 1 and 9 8 7) as quickly and accurately as possible. If the number sequences are not ordered (e.g., 2 1 7 and 3 1 2), they were asked to put a “✗” beside the sequences. Half of these were sequences with adjacent numbers (e.g., 1 2 3 and 4 6 5), and the other half were sequences with non-adjacent numbers (e.g., 4 7 9 and 3 4 8). Participants first completed six sample items for practice. Then, participants completed two pages of stimuli (8 ½ x 11 inch white paper). Each page consisted of 32 stimuli presented in eight rows. A correct items-per-second score was calculated the same way in the number comparison task. Internal reliability based on the two pages was high for both Chinese-educated (Cronbach’s α = .96) and Canadian-educated participants (Cronbach’s α = .95).

The arithmetic fluency test consisted of three pages of multi-digit arithmetic problems, one page each for addition (e.g., 34 + 56), subtraction (e.g., 45 - 19), and multiplication (e.g., 74 x 9). There were six rows of 10 questions on each page, and participants were given a one-minute time limit to write down the answer for each question from left to right as quickly and accurately as possible without skipping any items. The total correct score from the three subsets was used as the index of arithmetic fluency. Internal reliability based on performance on the three subsets was high for both Chinese-educated (Cronbach’s α = .86) and Canadian-educated participants (Cronbach’s α = .89).

A 0-7000 number line was used to assess participants’ number line estimation ability using an iPad application (

The percent of absolute error (PAE) was used as the index of how close (not considering direction) participants’ placement of each number was to the actual location of that number. In particular, the percent of absolute error was calculated as: PAE = [|(Estimate – Presented Number) / Scale of the Estimate| x 100]. For example, if a participant estimated the location of 231 at the position that corresponded to 350, the PAE would be 1.7% [|(231 – 350) / 7000 | x 100)]. Internal reliability was calculated based on the 29 trials was high for both Chinese-educated (Cronbach’s α = .91) and Canadian-educated participants (Cronbach’s α = .93).

Participants completed the problem-solving subtest from the KeyMath Numeration test (

Participants completed a Brief Math Assessment developed by

To evaluate whether participants showed the expected group differences in mathematical skills, comparisons for each task as a function of group are shown in

Measure^{a} |
Chinese-educated |
Canadian-educated |
Mean Difference | ||||||
---|---|---|---|---|---|---|---|---|---|

Number comparisons^{b} |
1.59 | 0.34 | 1.47 | 0.31 | +0.12 | 2.21 | 140 | .028 | 0.37 |

Ordinal judgments^{b} |
0.73 | 0.21 | 0.59 | 0.19 | +0.15 | 4.35 | 140 | < .001* | 0.70 |

Arithmetic fluency (180) | 58.10 | 14.50 | 33.72 | 15.90 | +24.38 | 9.43 | 140 | < .001* | 1.60 |

Word problem solving (15) | 12.82 | 1.88 | 11.01 | 2.61 | +1.80 | 4.70 | 123.15 | < .001* | 0.80 |

Fraction/algebra arithmetic (7) | 5.69 | 1.20 | 4.61 | 1.60 | +1.08 | 3.57 | 87 | .001* | 0.76 |

Number line^{c} |
5.28 | 2.77 | 6.33 | 3.81 | -1.06 | -1.89 | 127.69 | .061 | 0.32 |

^{a}Maximum scores (total possible points) in parentheses. ^{b}Number of correct items per second. ^{c}Percent absolute error.

*The Bonferroni correction method was used;

Correlations among the various mathematical measures are shown in

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

1. Number comparisons | - | .63*** | .31** | .29* | .23* | -.38*** |

2. Ordinal judgments | .68*** | - | .60*** | .45*** | .53*** | -.53*** |

3. Arithmetic fluency | .44*** | .55*** | - | .46*** | .57*** | -.37** |

4. Word problem solving | .32** | .35** | .40*** | - | .55*** | -.51*** |

5. Fraction/algebra arithmetic | .05 | .19 | .46*** | .42*** | - | -.42*** |

6. Number line | -.08 | -.18 | -.30* | -.33** | -.41*** | - |

*

Notably, the two groups barely overlapped in arithmetic fluency: Few Canadian-educated participants were as skilled as the average Chinese-educated participants (see ^{2}(1,

Distribution of arithmetic fluency for the Canadian- and Chinese-educated participants.

Multiple-group path analysis using MPlus Version 7 (^{2}(8)= 7.68, ^{2}(16) = 23.25, ^{2}(8) = 15.57,

*

Our first hypothesis was that ordinal judgments would mediate the relations between number comparisons and arithmetic fluency for both groups of participants, replicating

Direct and indirect effects | Canadian-educated |
Chinese-educated |
||||
---|---|---|---|---|---|---|

β | 95% CI ^{a} |
β | 95% CI ^{a} |
|||

Number comparisons to arithmetic^{b} |
||||||

(a) Indirect through ordinal | .38*** | .24 | .51 | .38*** | .22 | .48 |

Ordinal judgments to number line | ||||||

(a) Total effect | -.53*** | -.66 | -.38 | -.18 | -.37 | .07 |

(b) Indirect through arithmetic | -.05 | -.14 | .07 | -.16* | -.28 | -.02 |

(c) Direct effect | -.48*** | -.67 | -.30 | .03 | -.24 | .22 |

Ordinal judgments to word problem solving | ||||||

(a) Total effect | .45*** | .22 | .62 | .35*** | .10 | .53 |

(b) Indirect through arithmetic | .18** | .05 | .29 | .16* | -.00 | .30 |

(c) Direct effect | .27* | -.01 | .53 | .20 | -.07 | .40 |

Ordinal judgments to fraction/algebra arithmetic | ||||||

(a) Total effect | .52*** | .34 | .69 | .19 | -.05 | .37 |

(b) Indirect through arithmetic | .23** | .06 | .41 | .28*** | .14 | .43 |

(c) Direct effect | .30** | .06 | .57 | -.10 | -.34 | .13 |

^{a}Confidence intervals were calculated with bias-corrected bootstrapping in Mplus (1000 samples). CI = confidence interval;

^{b}There is no direct effect from number comparisons to arithmetic fluency (i.e., total effect = indirect effect).

*

Second, we hypothesized that

Third, we hypothesized that arithmetic fluency would uniquely predict the three mathematical outcomes for the Chinese-educated participants. As shown in

As predicted, we found that ordinal judgments mediated the relations between number comparisons and arithmetic fluency (

Written and spoken number words are symbols that activate multiple associations. The notion that symbol learning involves creation of complex networks of mental associations has been discussed extensively (e.g.,

The proposed Hierarchical Symbol Integration (HSI) model captures patterns of individual differences among measures of symbolic digit knowledge and mathematical performance for adults. The HSI model is based on the notion that performance on fundamental measures of symbolic processing (number comparisons, ordinal judgments, and arithmetic fluency) reflects stages in the ongoing integration of the numerical associations into a symbolic network. On this view, individuals who are more fluent in accessing symbolic number associations would have more integrated symbolic networks compared to individuals who are less fluent in accessing symbolic associations. To address this issue, we evaluated the HSI by contrasting performance of Canadian- and Chinese-educated adults who differed in their mathematical performance (e.g.,

The first prediction of the symbol integration model was supported in that ordinal judgments mediated the relations between number comparisons and arithmetic fluency for both skill groups (see

For Canadian-educated students, the one exception to the pattern of hierarchical mediation was on the number line task. Only ordinal judgments (not arithmetic fluency) predicted number line performance for this group. The 0-7000 number line is presumably a novel task for all of the participants and there are different possible solution strategies that could be applied. Adults are generally assumed to use proportional reasoning to locate numbers on the line (

Although the proposed HSI model provides a framework for understanding patterns of individual differences among adults for basic and advanced mathematical skills, it does not directly address the question of how various associations (number comparisons, ordinal judgments, and arithmetic fluency) unfold over time. Nevertheless, a developmental progression is implicit in the increasingly integrated relations among the core skills and this pattern is consistent with the limited existing work. For example, previous research suggests that cardinal associations (number comparisons) develop first, followed by the ordinal associations (

Another limitation of the framework is that it does not include a complete hierarchy of symbolic skills. Many additional symbolic relations are formed during the course of mathematical learning that go beyond number symbol connections (

In the present paper, we proposed a Hierarchical Symbol Integration (HSI) model for numerical associations and tested the model by contrasting mathematical performance of more- and less-skilled adults. In the present research, we used culture as a proxy for arithmetic expertise. As predicted by the HSI model, we found that arithmetic fluency mediated the relations between cardinal (number comparisons) and ordinal knowledge (ordinal judgments), with partial mediation for the less-skilled Canadian-educated adults and full mediation for the more-skilled Chinese-educated adults. For the Chinese-educated adults, activation of cardinal and ordinal associations may be automatic and thus no longer a source of individual differences in complex mathematical tasks; instead, variations in the accessibility of arithmetic representations supersede variability in cardinal and ordinal associations. These results are consistent with the view that symbolic number knowledge becomes increasingly integrated as individuals experience growth in mathematical expertise. They also suggest that the accessibility of arithmetic associations form a crucial foundation for subsequent complex numerical processing among highly skilled individuals. Presumably, skills should be also integrated during learning to allow the hierarchical associations among different associative connections to evolve in a coordinated way. Accordingly, the findings have implications for designing math education curricula: Children may need to practice and gain fluency with a range of symbolic number associations to develop a strong symbolic network that supports the acquisition of more advanced mathematical skills (

The authors thank Richard Ding for valuable inputs on this manuscript.

The authors have declared that no competing interests exist.