^{a}

^{*}

^{a}

Understanding numerical magnitudes is a foundational skill that significantly impacts later learning of mathematics concepts. The current study tested the idea that encoding of “10ness” is crucial to improving children’s estimation of two-digit number magnitudes. We used commercially available base-10 blocks for this purpose. The children in the experimental condition were asked to construct two-digit numbers by laying down the precise combinations of 10- and 1-blocks horizontally (e.g., three 10-blocks and seven 1-blocks for 37). Two control conditions were also included. In one control condition, children used 1-blocks only. In another control condition, children used one 10-block and as many 1-blocks as necessary. After working with the experimenter for only 15 minutes twice, the children in the experimental condition were significantly more accurate on the estimation task than those in the control conditions. The findings confirmed the importance of encoding 10ness as a unit in making accurate estimates of two-digit number magnitudes. The importance of encoding other units in the base-10 system is discussed.

Early acquisition of numerical magnitudes has been identified as a significant predictor of later mathematics learning (

Of particular relevance to the present study are interventions in which numerical board games were used to strengthen the association between Arabic numerals and their magnitudes. The board game for numbers 1 to 10 had the numerals written in the squares of equal size from left to right (

In both these games, one critical feature is how children were instructed to count as they moved their token. Unlike conventional games, children playing these board games were asked to

As important as a linear ruler representation may be, encoding numerical magnitudes by an increment of one alone may not be sufficient in encoding magnitudes of larger numbers. We thus speculated that children would need to encode “10ness” in order to develop an accurate linear representation of numerical magnitudes for two-digit numbers. Our idea was inspired by Miura and colleagues’ cross-cultural studies (e.g.,

Although Miura and colleagues did not test to see if these two groups of children would show similar or different performance on the numerical estimation task,

Based on these findings, we inferred that transparent counting systems influenced the way children mentally organized two-digit numbers, which, in turn, helped to develop accurate two-digit number magnitudes. The goal of the current study, however, was not to verify this hypothesis between English- and East Asian-speaking children. Rather, we wanted to test the hypothesis that English-speaking children could benefit from instruction that focused on encoding “10ness.” We reasoned that the irregular counting system of English might make it difficult for young children to map between spoken words and their magnitudes. However, training to represent two-digit numbers like the way Chinese-, Japanese-, Korean-, and Welsh-speaking children naturally do in everyday life might strengthen English-speaking children’s mental representations of two-digit numbers and their understanding of numerical magnitudes. When given a 0-100 number line, English-speaking kindergartners’ estimates of numerical magnitudes were better described as logarithmic (i.e., overestimating smaller numbers and underestimating larger numbers). It was not until second grade that English-speaking showed a linear pattern of estimation (

Thus, it is plausible to predict that teaching English-speaking children to encode 10ness using base-10 blocks horizontally would help them to form a linear representation of numerical magnitudes. In proposing a two-linear model of magnitude representations (one for single-digit numbers and another for two-digit numbers),

Our study differs from

Participants were 31 first-graders (12 boys and 19 girls, _{age} = 7 years 1 month, age range: 6 years - 8 years 6 months) recruited from an after-school program of a public elementary school and a parochial language school, both of which were located in the same Central Coast city of California. These schools served children from predominantly low to lower-middle income families. All participating children received free or reduced meals. The ethnic composition of the public school was 55% Latino/Hispanic, 32% Asian, and 13% Caucasian.

For the pretest and posttest, we used a closed number-line task (e.g.,

Placement of students to conditions did not follow a random assignment procedure due to classroom activities in which students were engaged at a time. At the pretest, teachers brought one student at a time and the students were assigned to conditions as they were brought in.

All participants met with the experimenter individually for the pre- and posttest sessions. The training sessions were conducted in a small group of three students. Teachers brought in three students deemed available at a time. Three students were not always the same over the two training sessions but the teachers made sure to bring in the students who were assigned to the same condition. Each of the four-phases of the study (i.e., pretest, Training 1, Training 2, and posttest) was completed within the same week over the four-week period in the same school year. That is, all participants completed each session in the same week. All sessions were held in either their classroom or an unoccupied room nearby.

The number-line estimation task required children to estimate where a particular number should be located on a 0-100 number line. The experimenter showed each child a sheet of paper with the closed number line drawn on it as well as an index card with a numeral written on it. The experimenter then pointed to the 0 and 100 positions and said, “if this is where 0 goes and this is where 100 goes, where will

Children’s estimates were fitted to a linear function (^{2} was used to measure linearity. If ^{2}_{Lin} is larger than ^{2}_{Log}, it means that estimates are more linear than logarithmic (

The children in this condition were provided with multiple 10-blocks and multiple 1-blocks. The experimenter first demonstrated how to “show” (construct) 37 using base-10 blocks (see

The children in this condition were provided with a single 10-block and multiple 1-blocks. The demonstration and training were identical to the Multiple 10 condition, except that only one 10-block and multiple 1-blocks were used. Thus, for the demonstration of constructing 37, the experimenter first placed a 10-block and then 27 1-blocks horizontally as she counted “ten (pause), one, two, three, …, twenty-seven, (pause), thirty-seven” (

The children in this condition were provided with 1-blocks only. The demonstration and training were identical to the other conditions, except that only 1-blocks were used. For the demonstration of constructing 37, the experimenter placed 37 1-blocks horizontally as she counted “one, two, three, …, thirty-seven” (

A preliminary analysis of linearity found no statistically significant gender differences at the pretest, ^{2}_{Lin} for girls was 47.68%, which was comparable to the boys’ average of 43.67%. Ethnicity was also non-significant at the pretest, ^{2}_{Lin}s were 49.19%, 44.82%, and 44.25%, for Asian-, Latino-, and Caucasian-American children respectively. Therefore, the data from girls and boys, as well as thee ethnic groups, were combined for subsequent analyses.

Because random assignment was not used to place children into the three conditions, we next compared the pretest scores of the children in the three conditions. The results showed no statistically significant differences among the conditions,

A repeated-measures ANOVA was carried out to examine if training had differential effects on the children in the three conditions. The results showed statistically significant differences among the conditions, ^{2}_{p} = .81. The effect size was substantially large. Further analyses showed that the gains made by children in the Multiple 10 condition were significantly greater than those in the Single 10 block or Multiple 1 condition, both at ^{2}_{Lin} showed that the Multiple 10 condition outperformed the Single 10 condition

When each condition was examined separately, both the Multiple 10 and Single 10 conditions made significant gains from pre- to posttests whereas the Multiple 1 condition did not (see ^{2}_{Lin} for the Multiple 10 condition improved from 52% at pretest to 98% at posttest,

Variable | Cohen’s |
||||
---|---|---|---|---|---|

Pretest | Posttest | ||||

Multiple 10 | .52 (.11) | .98 (.02) | 13.81 | < .001 | 4.17 |

Single 10 | .48 (.06) | .63 (.11) | 5.83 | < .001 | 1.85 |

Multiple 1 | .43 (.02) | .45 (.06) | 1.53 | .162 | 0.48 |

Median estimates of numerical magnitudes at pre- and posttests for the Multiple 10 (a), Single 10 (b), and Multiple 1 (c) conditions.

In fact, children’s estimates at the posttest were best fitted by the linear function, ^{2}_{Lin} = .98, as opposed to the logarithmic function, ^{2}_{Log} = .68. Taking into account that at the pretest these children’s estimates were more logarithmic (^{2}_{Log} = .79) than the linear function (^{2}_{Lin} = .52), it showed a clear trend from log to linear for this condition. This was not the case for the other two conditions. The median estimates made by children in the Single 10 and Multiple 1 conditions were best described as logarithmic both at the pre- and posttests.

Although descriptive, we were interested in visually inspecting if children’s estimates differed at the posttest depending on the numbers they estimated. We reasoned that training provided for children in the Multiple 10 condition should show linearity regardless of number. Training provided in the Single 10 condition should help make more accurate estimates for numbers between 10 and 19. We did not expect any particular pattern for the Multiple 1 condition. It is possible that children in the Multiple 1 condition, only practicing to count by ones, would show estimates in line with (1) overestimation of smaller numbers and underestimation of larger numbers; (2) a two-linear model (

As shown in

Range of estimation for each number at posttest by conditions.

Thus far, our analyses focused on median estimates at the group level. We next considered individual estimates at the posttest to determine if the results obtained earlier would hold up at the individual level as well.

The results for the Multiple 10 condition showed that all of the 11 children’s individual estimates were better described as linear. As for the Single 10 condition, only one child’s estimates were linear and all of the children’s estimates for the Multiple 1 condition were logarithmic.

The primary goal of the present study was to examine if teaching children “10ness” by asking them to place 10-blocks horizontally would improve their estimation of numerical magnitudes. Our rationale was that children using base-10 counting systems show earlier mastery of place value (e.g.,

Children who were taught to use only single 10-block and multiple 1-blocks in the present study made significant gains. However, their median estimates at the posttest were better described as logarithmic than linear. Inspecting individual performance, we found only one child’s estimation was linear in the Single 10 condition. Having just one 10-block appeared to have helped children to form relatively accurate magnitudes of numbers smaller than 20. But their understanding of one 10 did not transfer to larger two-digit numbers.

Although replications to confirm the current findings are in order, we emphasize the fact that the significant improvements resulted from working with children over two brief sessions, constructing just five numbers each. In replicating the current study, it is essential to recruit many more participants. Having about 10 children in each condition limited our ability carry out more detailed analyses.

We also note that we only used a closed number line as our outcome measure. As

As mentioned earlier, our study was designed based on the prior findings showing that children who use regular counting systems develop base-10 like representations of number and master place value earlier than those who use irregular counting systems. The literature on the relation between numerical language and mathematics performance, however, provides counter evidence as well. For example, Towse and colleagues (e.g.,

As should be apparent from above, there are more questions than answers in regards to the relation among numerical language, numerical estimation, and mathematics abilities. The current study at the very least reconfirmed the importance of children making connections between Arabic numerals and their magnitudes. As foundational as encoding the unit of one is to single-digit numbers, the unit of ten is crucial to understanding two-digit numerical magnitudes. Likewise, we speculate encoding the unit of 100 is as essential to comprehending three-digit magnitudes as the unit of .1 is to the decimal numbers in the tenths place. These units might play an important role in developing a complete sense of numerical magnitudes. If our speculation proves to be correct, it has significant implications for how we help children expand their understanding of the number system. Use of number lines, focusing on different units of measure, is likely to help children encode numerical magnitudes of not just whole numbers but also rational numbers.

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no support to report.