To longitudinally explore children's developing beliefs towards mathematics, we asked 207 children to define “math” and “reading” at grades 2 and 3 and coded for spontaneous references to likability or difficulty of math (or reading) in their definitions. We found that children attributed more difficulty to math than to reading despite their relatively neutral comments on the likability of either subject. Children described math and reading with comparable degrees of specificity, but girls' definitions were more specific than boys'. Relative to their peers, children with mathematics learning disability (MLD) provided less specific definitions overall, were more likely to describe math as more difficult than reading, and were more likely to show a decrease in likability ratings of math (but not reading) from grades 2 to 3. Grade 2 ratings predicted math ability at grade 3, more so than predictors from grade 3. These findings, although based on informal analyses not intended to substitute for validated assessments of disposition, support the notions that distinct aspects of dispositions towards math emerge in early childhood, are revealed through casual discourse, and are predictive of later math achievement outcomes. This further supports current interests in developing formal measures of academic disposition in early childhood.
A productive disposition towards mathematics is an essential component of mathematics proficiency [
The recent interest in measuring beliefs about mathematics in early childhood (e.g., [
Until recently, dispositions towards math and related constructs, such as math anxiety, were often not evaluated before middle school [
To explore how productive learning dispositions are cultivated in early childhood, we utilized an expectancy-value model of achievement motivation [
What is the relevance of spontaneous conversations of this type in the context of evaluating children’s beliefs about math? First, early conversations about math provide a mechanism by which adults can deliberately attend to the potential influences they have on their children’s or students’ math-oriented beliefs. Second, in addition to using conversations to nurture the development of a healthy math disposition, conversations can be monitored for indicators of a child’s emerging beliefs. In other words, early conversations are useful as a platform for adults’ messages to children, but also for attending to children’s comments—spontaneous or structured.
Why are children’s early comments about math potentially important? In view of emerging evidence that early beliefs about mathematics are related to later achievement outcomes, efforts to steer children from paths towards negative outcomes should begin in early childhood, when children’s beliefs may be more malleable rather than deeply rooted. Consequences of negative beliefs towards math are frequently described in terms of secondary-school behaviors, such as avoidance of elective mathematics courses after grade 10 (age 15 years) [
Children’s math ability self-perceptions become well established in elementary school (e.g., [
There are a number of ways to evaluate math beliefs. Ratings of children’s math self-perception and their enjoyment of mathematics are objectively measureable in elementary school aged children [
What aspects of children’s early dispositions towards mathematics are revealed during open-ended discussions, and are these beliefs related to formal math outcomes? To address these questions, in the present study we used a simple and straightforward approach to elicit children’s spontaneous comments about math. Importantly, we avoided explicit prompting for information about beliefs about math in order to focus on children’s spontaneous comments. Although findings to emerge from this approach do not reveal causal pathways to successful mathematics, evidence of the mere presence of relevant information from spontaneous speech has implications for the role of early conversations about mathematics among children and their parents, teachers, and care providers in shaping or supporting a child’s disposition towards mathematics.
Participants were drawn from a larger longitudinal study of mathematics achievement and mathematics learning disabilities during the primary school aged years, described elsewhere in more detail [
During each year of the study, children were tested individually by one of five female examiners. Testing occurred during two or three sessions up to 45 minutes each, in a room separate from the classroom or other distracting activities. The testing battery included a range of standardized and experimental assessments. The measures relevant to the present report are two measures of mathematical ability, and a vocabulary probe administered to glean information about each participant’s beliefs about mathematics.
The TEMA-2 is a standardized measure of formal and informal mathematical ability normed for use with children ages 2 to 8 years, 11 months. Items from the TEMA-2 include basic knowledge items such as counting principles, calculation and fact retrieval items, and items testing place value concepts or word problem solving. Total correct scores are converted to age-referenced standard scores for which test-retest reliability is 0.94 [
We administered the TEMA-2 to participants in the longitudinal study, from kindergarten through grade 3. In the present study, we used the standard scores from all four grades to assign children to one of three groups: children with mathematical learning disability (MLD), children with low mathematics achievement (LA), and children with age appropriate math achievement (typically achieving or TA). For the participants who exceeded the upper age level for the TEMA-2 ceiling at grade 3, we calculated prorated standard scores using regression-models to predict age 9 years and age 9.5 years outcomes, based on data from over 200 children who completed the TEMA-2 during all four years of the study. The criteria for these participant groups, described elsewhere in detail [
The Woodcock Johnson Psycho-Educational Battery-Revised [
During select years of the longitudinal study, the test battery included a standardized expressive vocabulary test from the Stanford Binet Fourth Edition [
Children’s responses were recorded verbatim and transcribed into a central database with child identifiers removed. This ensured that coders were blind to the children’s identity, gender, or math achievement status. Each response was coded independently by two trained coders. Coding responses were compared, and discrepancies were resolved in a coding meeting with the study PI and both coders present. Therefore, all definitions were double coded and, if there was disagreement, by three coders. Coding reliability is reported subsequently.
Each definition was coded separately for three attributes: the extent to which the child’s definition spontaneously reflected a like or dislike for math or reading (
Likability scores: coding criteria and real examples of math and reading definitions.
Score | Definition | Math examples | Reading examples |
---|---|---|---|
−2 | Extremely disliked, hated, or dreaded. Includes reference to words such as hate, worst, awful, and so forth. | n/aa | “When you sit and read a book bored out of your mind.” |
−1 | Disliked or avoided. Includes reference to words such as do not like, not fun, boring, bad, and so on. | “Math is something I do not like.” | “Something you do only when you need to do it, you have to read to figure out information.” |
0 | Neutral feeling or tolerated. Includes no reference to an emotion. |
“When you learn math problems.” | “Like when you read a book in school or at home.” |
1 | Liked or enjoyed. Includes reference to words such as play, fun, like, good, and so forth. | “Math is fun, you could do math at school—you could do it anywhere!” |
“You can read for information or just for the fun of it.” |
2 | Extremely liked, loved, or favored. Includes reference to words such as love, favorite, best, and so forth. |
n/aa | “Oh I love to read.” |
Note: aNo responses were coded at this value.
Difficulty scores: coding criteria and real examples of math and reading definitions.
Score | Definition | Math examples | Reading examples |
---|---|---|---|
−2 | Really hard. Includes reference to words such as |
“Something you do in school. Something that’s very, very hard.” | “Like you are reading and you are bad at reading and you read something like hard words.” |
−1 | Kind of hard. Includes reference to words such as the noun form of |
“It is like problems that you have to solve.” | “It is when there is like lots of sentences and you read it to get information.” |
0 | Neutral difficulty. Includes no reference to difficulty, refers to verbs such as work, do, use, teach, or “work with” without implying exertion. | “Like math projects. Doing math homework.” | “You have a book and you read the words.” |
1 | Kind of easy. Includes reference to words such as “not hard” or words implying ease. | “It is easy and you have to do it in your homework.” | “To just lay back and read a book.” |
2 | Really easy. Includes reference to words such as simple, effortless, or speed. |
n/aa | n/aa |
Note: aNo responses were coded at this value.
Specificity scores: coding criteria and real examples of math definitions.
Score | Definition | Examples |
---|---|---|
1 | No response, a circular response, a response unrelated to math, or another uninformative response. | “You do your math.” |
2 | Response is unspecific or only indirectly related to math as a primary school subject. The response may include references to activities performed in relation to math or in a math class but with no discernible reference to math concepts or procedures. | “You play games and stuff.” |
3 | Unelaborated basic concepts or mechanics of math. Includes reference to real numbers, operations, math problems, or learning math. | “It has to do with numbers and sizes and fractions.” |
4 | Elaborated concepts of math. | “Like if you have a word problem, like Jim has 18 apples and eats 3, you use math to solve it.” |
5 | Concept of math as a useful tool. | “Math is something that people do and they have to know math to be able to get a job and do other stuff.” |
Specificity scores: coding criteria and real examples of reading definitions.
Score | Definitions | Examples |
---|---|---|
1 | No response, a circular response, or a response unrelated to reading or otherwise uninformative. | “Reading means you read a lot.” |
2 | Response related to unspecific reading activities, or indirectly related to reading activities. Includes reference to activities that may be performed in relation to reading or for a reading class but without a clear reference to the concept of reading. | “It means that you do reading projects.” |
3 | Unelaborated basic principles or mechanics of reading. Includes reference to sounding out and/or to reading materials or to learning to read. | “To like sound out something.” |
4 | Elaborated concepts of reading. Includes reference to the extraction of meaning from written material through the act of reading. | “Like if you have a book, it has words, and you read the words not to just look and say, but to know what the words say and what the story is about.” |
5 | Concept of reading as a useful tool. Must demonstrate usage of reading or usage of meaning derived from reading. | “To like look at books and some books help you make things and repair your house and some people just read for fun.” |
The full range of possible
The full range of possible
All data were double entered into two excel spreadsheets, subtracted, and reviewed for discrepancies that were then corrected until the subtraction comparisons yielded no errors.
First we asked whether qualitative features of children’s definitions of math and reading differ as a function of gender or MLD status. We carried out three analyses of variance (ANOVAs), each based on a 2 (subject area: math versus reading) × 2 (grades: 2 and 3) × 2 (gender) × 3 (MLD status: TA, LA, and MLD) design, with repeated measures of the first two factors. The outcome variables were likability codes, difficulty codes, or specificity codes from children’s spontaneous definitions of math and reading. Note that the means reported in the text are marginal estimated means whereas means reported in the figures are observed means.
We found no main effects of grade, MLD status, or gender on likability codes,
Two-way interactions emerged for grade × MLD status,
Observed means for likability scores, from grades 2 and 3, depict the three following interactions: grade × MLD status, grade × subject area, and grade × MLD status × subject area.
Even the strongest interaction to emerge from this analysis was associated with a small effect size, for a three-way interaction between subject area, grade, and MLD status,
In summary, at grades 2 and 3, boys and girls do not spontaneously make reference to extremely positive or negative sentiments about math or reading when defining either term, but there is a slight tendency for more negative comments about math (versus reading) among children with MLD, and, to a lesser degree, among children with low achievement in math.
We also evaluated spontaneous comments about the difficulty versus easiness of math or reading in our second set of repeated measures ANOVAs. Here, the main effect of subject area was significant,
There was no main effect of grade or gender on difficulty ratings,
Observed means for difficulty scores in grades 2 and 3 depict the significant main effects of subject area and MLD status, and the significant three way interaction between grade, gender, and subject area.
In sum, at both grades 2 and 3, both boys and girls make references to math or reading as difficult (or as requiring “work” or exertion), but there is a tendency for math to be described as slightly more difficult than reading, and a tendency for children with MLD to rate both math and reading as more difficult compared to ratings assigned by their non-MLD peers.
It is possible that likability and difficulty ratings simply represent a general reporting tendency, rather than specific constructs. For example, each measure may reflect a positive or negative disposition towards academic subjects in general, in which case likability codes for math and reading should be positively correlated, and difficulty codes for math and reading should also be correlated. Alternatively, the measures may reflect an even broader tendency for positive or negative reporting in general, in which case likability and difficulty codes should be correlated with each other. Finally, if the codes represent stable, subject-domain sentiments, then math (or reading) likability scores should be correlated across grades.
To explore which of these alternatives is supported, we ran three sets of four correlations, using 12 Spearman rank tests, with alpha adjusted to 0.004 based on multiple correlations (.05/12). With respect to the three alternatives posed above, we found only weak, partial support for the notion of a general academic valence bias, based on the weak positive correlation between likability codes for math and reading observed at grade 2 (
There was less support for a broad reporting disposition bias, because likability and difficulty codes were not correlated with each other,
There was far more support for subject-specific likability (or difficulty) over time, suggesting that our codes are indicative of stable, and subject-domain beliefs, at least over the short term (from grade 2 to 3): reading likability at grade 2 was correlated with reading likability at grade 3 (
Observed means or definition codes among total study sample (
Math | Reading | |||
---|---|---|---|---|
Mean (SD) | Range | Mean (SD) | Range | |
Grade 2 | ||||
Likability | 0.01 (0.14) | −1 to 1 | 0.03 (0.30) | −2 to 2 |
Difficulty | –0.29 (0.52) | −2 to 0 | –0.11 (0.33) | −2 to 0 |
Specificity | 2.85 (0.473) | 1 to 4 | 2.96 (0.63) | 1 to 5 |
Grade 3 | ||||
Likability | 0.00 (0.14) | −1 to 1 | 0.06 (0.39) | −2 to 2 |
Difficulty | –0.32 (0.50) | −2 to 0 | −0.21 (0.43) | −2 to 1 |
Specificity | 3.03 (0.66) | 1 to 5 | 2.97 (0.61) | 1 to 5 |
Note: Standard deviations shown in parentheses.
The third set of repeated measures ANOVAs concerned the specificity of children’s descriptions of math or reading, with codes reflecting noninformative to elaborate descriptions (exemplified in Tables
Gender and MLD status each contributed significantly to variance in the specificity of definitions. Girls gave slightly more specific definitions of mathematics or reading (mean = 3.00) than did boys (mean = 2.75),
The only interaction to reach statistical significance was a two way interaction between MLD status and gender,
Observed means for specificity scores depict significant main effects of MLD status and gender, and the significant MLD status × gender interaction. Subject area is not significant.
Do these exploratory measures predict future or concurrent math performance? We carried out two regression models, each comprised of three predictors of grade 3 WJ-R math Calculation score as the outcome variable of interest. These models were based on predictors obtained at grade 2 or grade 3 (grade level predictors examined separately). When combined, grade 2 likability, difficulty, and specificity codes accounted for approximately 5% of the variance in WJ-R Calculation scores obtained at grade 3,
Model strength improved significantly when grade WJ-R Calculation score was included as a predictor,
To examine whether
To evaluate whether the effect to emerge from grade 2 predictors was specific to math disposition, we carried out parallel regression models using grade 2 or grade 3 likability, difficulty, and specificity codes from
Our research questions examined primary school aged students’ spontaneous comments about mathematics and specifically tested whether these comments reveal emerging dispositions linked to students’ later mathematics achievement. Additionally, we tested the notion that beliefs about math may differ among children with versus without MLD. For comparative purposes, children’s spontaneous comments about reading were also obtained. Our approach involved a straightforward design to elicit spontaneous comments of the kind that may emerge during casual verbal dialogue between young children and their parents or teachers. Finally, by repeating this procedure during two consecutive grades, we were able to test short term stability or reliability of the codes we collected.
Despite the subtlety of our approach, our findings indicate that children’s spontaneous comments about math are informative even if they are no substitute for a structured and validated assessment of disposition. Specifically, we found that subject-specific comments about likability (or difficulty) appear to be stable beliefs, at least from grade 2 to 3. Lower likability ratings (i.e., more negative comments) were evident in definitions of math (versus reading) among children with MLD, and, to a lesser degree, among children with LA in math; but overall, reference to liking or disliking mathematics was absent from most of the participants’ spontaneous comments of math and variability was greater among definitions of reading.
With regard to
Analyses related to
Of the four main predictor variables examined across these analyses—MLD status, subject area, gender, and grade—the only variable to account for variability in all three outcome variables was MLD status. Main effects or interactions involving MLD status emerged for likability, difficulty, and specificity. Although it is erroneous to infer that brief, open-ended questions like the ones used in our study are appropriate for diagnosis of MLD, future and ongoing work is needed to address the dynamic role(s) of a productive disposition and long term math achievement outcomes.
There were no gender differences in likability ratings for math or reading, but boys and girls reported slightly greater difficulty for math than reading at both grades. From grades 2 to 3, boys shifted towards reporting slightly greater difficulty for both math and reading over time, whereas girls shifted towards reporting slightly less difficulty for math over time. This may reflect differences in shift towards behavioral compliance rather than a specific shift in beliefs about mathematics, but neither explanation is supported by our data. The effect was small and warrants more in depth evaluation, especially in light of research that has found primary school aged boys identify more strongly with mathematics and have higher self-concepts than girls despite similar levels of mathematics achievement [
It is quite clear that the effects that emerged in this exploratory study are small. Yet the fact that
Although our findings do not indicate causal pathways, they are consistent with evidence that primary school aged students’ interests or beliefs about mathematics affect their achievement level or reflect risk status for future math outcome [
These findings in early childhood have implications for teachers, parents, and care providers of young children. In general, children in our study expressed relative flat affect with regard to their enjoyment of mathematics, did not hold very elaborate conceptions regarding the usefulness of mathematics, and attributed more difficulty to mathematic than reading. Despite the small effect size that emerged from these findings, these suggest potential negative outcomes when considered in the context of expectancy-value theory. According to the model, these dispositions run counter to facilitating engagement in mathematics activities given the proportional relationship among components of the model (i.e., engagement = success expectancies
There are several limitations to our exploratory study. Although our sample size was large and data collection over time was longitudinal, the time period over which we examined definitions of math and reading was limited to two consecutive years, and the number of students with MLD and LA was limited by the defining criteria of these constructs (e.g., MLD occurring in only ~6–10% of the population, as it was in our study). The deliberately simplistic nature of our data collection was consistent with our goal to evaluate more naturalistic and spontaneous versus prompted comments about mathematics likability, difficulty, and specificity, but children’s comments could be elaborated upon through structured probes and conversations. For instance, most children’s definitions received “neutral” codes for likability, but this does not mean that children did not have beliefs about liking or disliking mathematics.
Our research question did not concern the strength of children’s beliefs about math so much as the likelihood of their emergence during conversation. In addition to expressing (or not expressing) beliefs about math and reading, the children in this study remind us of the importance of listening to what they say, even during casual discourse. That is, adult-child discourse provides a means for parents and teachers to both nurture a child’s positive disposition towards math, and monitor the child’s emerging disposition.
This work was supported by a Grant from the Spencer Foundation awarded to M. M. M. Mazzocco and L. B. Hanich, based on data collected from a study supported by NIH Grant HD R01 34061 awarded to M. M. M. Mazzocco. The views expressed are solely those of the authors. They would like to thank the children who participated in the study, their parents and teachers, the staff at participating Baltimore County Public School elementary schools; and an anonymous reviewer of an earlier version of this paper. They also acknowledge the outstanding contributions of Gwen F. Myers, Project Manager for the longitudinal study, to this work and to the overall longitudinal research program.