Abstract
To understand mathematics, children must interpret mathematics symbols. In this study, we designed a brief assessment of mathematics symbols for children in the elementary grades. For each of 23 symbols, children identified the symbol, provided the meaning of the symbol, and used the symbol. We assessed 297 children in Grades 1, 3, and 5. Internal consistency reliability was established across grade levels. Results indicated the overall symbol knowledge of children increased across grade levels, and children demonstrated higher scores on questions related to the identification and use of the symbol rather than explaining the meaning of the symbol. Across grades, symbol knowledge was a significant predictor of mathematics computation, although the variance accounted for was greatest in first grade and least in fifth grade.
Mathematics can be presented nonsymbolically or symbolically. Early nonsymbolic representations involve quantities, usually displayed as shapes or pictures representing the quantity (e.g., 5 circles represent five; Moomaw & Dorsey, 2013). Nonsymbolic representations can also be hands-on materials utilized to represent mathematical concepts (i.e., enactive representations; Bruner, 1966). Symbolic representations involve numerals (e.g., 3, 19) and symbols (e.g., $, =, /). Early mathematics understanding is nonsymbolic, as children identify pictures of quantities and determine which quantity is greater or less (Barth, La Mont, Lipton, & Spelke, 2005). Until formal schooling, children demonstrate stronger performance with mathematics presented in nonsymbolic forms than symbolic forms (Moomaw & Dorsey, 2013) most likely due to more experiences with mathematics in nonsymbolic forms.
Children begin to map nonsymbolic representations onto symbolic representations in the early elementary grades (Mundy & Gilmore, 2009). The connection between nonsymbolic and symbolic representations is strong: Non-symbolic performance measured before school or at the beginning of school predicts symbolic mathematics performance in subsequent grade levels (Mazzocco, Feigenson, & Halberda, 2011). Even though the mathematics curriculum of elementary school adjusts to be presented more symbolically, children in the elementary grades continue to demonstrate robust performance on nonsymbolic items (Driver & Powell, 2015).
With the start of formal schooling, mathematics textbooks and assessments present mathematics in both nonsymbolic and symbolic forms. During instruction, children are encouraged to practice mathematics using enactive (i.e., hands-on) and iconic (i.e., pictorial) representations to understand mathematics in symbolic form (Bruner, 1966). Beyond early numeracy measures (e.g., Purpura & Napoli, 2015), mathematics assessments expect children to provide answers using symbolic mathematics notation. For example, children are presented with 7 + 6 = __ and expected to interpret the numerals and symbols and respond with a symbolic sum (i.e., 13). Nonsymbolic understandings of mathematics may help children with understanding the conceptual underpinnings of a problem, but interpretation of numerals and symbols is necessary to respond to formal mathematics assessments (Sáenz-Ludlow, 2007).
Mathematics symbols are reliable and have a stable form (Hiebert, 1988; Pimm, 2002). In the elementary grades, symbols either represent quantities (e.g., 4.12) or describe actions or operations (e.g., 19 > 12; Hiebert, 1988). With symbols, the form and meaning is consistent from across grade levels and mathematics content areas (MacGregor & Stacey, 1997). For example, the plus sign has the same form and meaning whether a child is adding 3 + 2 or 1,495 + 2,036 or ½ + ¾. In addition, the plus sign, when used within the plus-minus sign (±) of mathematics, continues to hold the meaning of addition. Mathematics symbols have been used for thousands of years (Pimm, 2002). In formal mathematics (i.e., the mathematics taught in school and used in society), symbol understanding is necessary to interpret mathematics related to both arithmetic and algebra (Arcavi, 1994; Van Amerom, 2003). That is, very few high-stakes mathematics items and no mathematics textbooks are symbol free. Therefore, it is necessary to understand which symbols children can identify, explain, and use (Arcavi, 1994). A knowledge of mathematics symbols is likely connected to overall mathematics competence.
To our knowledge, no assessments have evaluated solely how children interpret the range of symbols presented within the elementary grades. Several studies, however, have focused on specific symbols or sets of symbols. For example, Fagnant (2005) asked first-grade children to match additive problem-solving situations with equations. No children correctly used the minus sign (−), indicating early difficulty with interpretation of this symbol. Heath (2010) explained that children have difficulty with interpretation of the greater than (>) and less than (<) symbols. Similarly, Hattikudur and Alibali (2010) demonstrated that, without instruction and practice, not all third- and fourth-grade children provided correct definitions for the greater than and less than symbols. In addition, children have exhibited difficulty with the equal sign (=), likely the most researched sign in the elementary and middle school grades (e.g., Alibali, 1999; Li, Ding, Capraro, & Capraro, 2008; Powell & Fuchs, 2010).
For symbols used in the later elementary grades, Christou and Vosniadou (2012) indicated that children have difficulty with the negative symbol (-) because no symbol is typically used to represent positive numbers. Similarly, Lamb et al. (2012) described the minus sign as being used within subtraction, to represent negative numbers, or to mean the opposite, which could cause confusion for some children. Symbols, like the negative symbol and minus sign, with multiple interpretations may be especially difficult with children with learning difficulties (Grobecker, 2000). And as described by Saxe, Taylor, McIntosh, and Gearhart (2005), children may require instruction on the formal symbols of mathematics (e.g., fraction bar or fraction slash), if children have developed their own informal notation for specific mathematical concepts. Importantly, children learn formal notation using symbols rather quickly (Hewitt, 2012), but it appears that, without continual instruction and practice on the meaning of the signs, children may have difficulty explaining and using mathematics symbols.
Purpose and Research Questions
We developed a brief measure of mathematics symbols to investigate the symbol understanding of children across the elementary grades. We wanted to establish the internal consistency reliability of a whole-class administered measure that was solely focused on mathematics symbols. We also aimed to learn how children identified symbols, provided the meaning of symbols, and used symbols and whether symbol knowledge was related to computation performance. Our research questions were as follows:
Method
Participants
Participants (N = 289) were sampled from six first-grade classrooms (n = 104), five third-grade classrooms (n = 65), and seven fifth-grade classrooms (n = 120) in a school district in a state in the Southwest of the United States. We selected Grades 1, 3, and 5 for the following reasons: Formal symbols of addition, subtraction, and comparison are introduced in first-grade standards; formal symbols of multiplication, division, and fractions are initially mentioned in third-grade standards; and formal symbols of algebra begin to appear in fifth-grade standards (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Table 1 displays demographic information.
Demographic Information, Means, and Standard Deviations.
Note. WRAT = Wide Range Achievement Test.
Calculated within each grade level.
Measures
Mathematics computation
To measure children’s mathematics computation, we administered the Math Computation subtest of the Wide Range Achievement Test (WRAT4; Wilkinson & Robertson, 2006) to children in first, third, and fifth grade. We selected Math Computation because it is brief and appropriate for elementary children and because mathematics computation is a strong predictor of overall mathematics competence (e.g., Mabbott & Bisanz, 2008). With Math Computation, children had 15 min to answer 40 written computation problems of increasing difficulty. The examiner read directions aloud and children worked independently. Children answering more than four written problems correctly (i.e., all children in this sample) were awarded 15 points without administration of an oral arithmetic section. Maximum score was 55. Cronbach’s α was .66 at first grade, .78 for third grade, and .79 for fifth grade.
Mathematics symbols
To measure the mathematics-symbol knowledge of elementary children, we developed Mathematics Symbols. We selected the mathematics symbols terms for the measure by conducting a thorough search of two common first-, third-, and fifth-grade mathematics textbooks and textbook glossaries (EnVisionMATH and GoMath!). Based on this search, we identified 26 mathematics symbols related to numbers and operations and selected 21 elementary symbols to create the measure. The five elementary symbols not included were brackets ([ ]), degree (°), multiplication dot (•), parentheses [( )], and remainder (R). We did not select brackets, the degree symbol, or the remainder symbol because of the nonoperator nature of the symbols. We believed the multiplication dot would be too confusing compared with the decimal point, and a brief pilot testing of items confirmed this. In the same way, a brief pilot test with several children indicated that all children provided definitions for parentheses related to reading not mathematics. To alleviate concerns about ceiling effects, we included two symbols introduced in middle school: the plus/minus sign and the square root symbol. Overall, the Mathematics Symbol measure included 23 symbols. Table 2 provides a list of the 23 included symbols and describes whether the symbol was explicitly named in a textbook at first, third, or fifth grade.
Symbols and Accuracy Rates.
Note. ID = identification of symbol.
Every mathematics symbol was presented alone in a rectangle and was accompanied by three questions. The first question asked, “What is this symbol?” (i.e., identification of the symbol). The second question asked, “What does this symbol mean?” The third question prompted, “Use the symbol.” These three levels of questions aligned with a question design framework outlined by Haladyna and Rodriguez (2013). The final Mathematics Symbols measure included 23 symbols with a total of 69 questions. During administration, the examiner read directions aloud and worked a sample problem with all children. Then, children worked independently for 20 min. Children received 1 point for each correct answer. The maximum score was 69.
Procedures
The first and second authors collected all data. Both authors had degrees in education-related fields and experience in standardized test administration procedures. Each examiner read verbatim from a test administration protocol during administration. Each measure was administered in whole class format, and data collection occurred at the end of the academic year approximately 4 weeks before the end of the school year. Four scorers (two undergraduates and two graduate students in education-related fields) received a 1-hr training about the coding and interpretation of child responses from the first author. Then, the scorers entered 100% of child responses for each measure. This was conducted on an item-by-item basis into two separate electronic databases. For the WRAT Math Computation, scorers entered whether the child’s response was correct (1) or incorrect (0). The same scoring process was used for Mathematics Symbols. Children received a correct or incorrect score for symbol identification, meaning of symbol, and use of symbol for each of the 23 symbols.
On Mathematics Symbols, the scorers used a scoring guide that provided a range of acceptable answers. For example, with the minus sign, children could have written that the minus sign was the “minus” sign or “take away” sign. When scoring about the meaning, children had to provide at least one accurate definition of the symbol’s meaning. For example, with the plus sign, children could write that the plus sign meant to “add” or “put together” or “bring some more.” Scorers accepted misspelled words in which the meaning of the word could be interpreted (e.g., “ad” was acceptable for “add,” “molply” was acceptable for “multiply.” On the use of symbol scoring, children had to use the symbol correctly but computation errors were ignored. For example, 4 + 5 = 8 was scored as correct, but 4 + 5 = 2 was scored as incorrect. The former example demonstrates an understanding of adding numbers together or adding on, whereas the latter example does not.
Scoring of the minus sign, negative symbol, and fraction bar was more complicated than most symbols. On the measure, we intended for Item D to be the negative symbol, Item L was the minus sign, and Item U was the fraction bar. Some (e.g., Ball, 1993; Ganor-Stern, 2012; Lamb et al., 2012) have described and used the minus sign and negative symbol interchangeably, whereas others (e.g., Bishop et al., 2014; Saxe et al., 2010; Van de Walle, Karp, & Bay-Williams, 2013) have used a different negative symbol (i.e., -) instead of the minus sign (i.e., –). This may be especially important for notation such as -4 – -7. On Mathematics Symbols, the majority of children identified Item D as the minus sign. Rather than strictly score the assessment, which would lead to inaccurate information about child symbol knowledge, we allowed children to provide a definition of the minus sign for Items D, L, or U. Once a child identified one of these items as the minus sign, however, we did not accept minus sign explanations for the other two items. For example, a child who described Item D as the minus sign could not get correct points on Item L or Item U for minus sign identification, meaning, or use. Interscorer reliability of the measures was above 96% for both measures. Cronbach’s Kappa was .81. Discrepancies between the two databases were compared and rectified to ensure 100% accurate scoring.
Data Analysis
To answer our research question about internal consistency reliability, we calculated Cronbach’s alpha (α) using SPSS Version 25 (IBM Corp., 2017). We used a one-way analysis of variance (ANOVA) to investigate our second research question about overall performance on the measure of mathematics symbols. The dependent variable was mean correct response on the measure of mathematics symbols, and the independent variable was grade level (i.e., a categorical variable). We determined effect size (ES) by calculating Cohen’s d. For our third research question about subscale performance, we analyzed the data using paired samples t tests within each grade level. The dependent variable was mean correct response for each subscale. We calculated Pearson correlations for the subscales within each grade level. For the fourth research question about differences among mathematical content areas, we ran ANOVAs with the dependent variable being mean correct response within each mathematical content area. We used grade level as the independent variable, and we analyzed post hoc comparisons using Tukey’s honesty significant difference (HSD) tests. For our fifth research question about the relationship between mathematics-symbol knowledge and computation performance, we utilized a regression model because both the outcome and dependent variable were continuous variables.
Results
Internal Consistency Reliability
With our first research question, we explored the internal consistency reliability of Mathematics Symbols at Grades 1, 3, and 5. The 69-item Mathematics Symbol measure demonstrated strong internal consistency reliability at each grade level. Cronbach’s α was .79 for first grade, .88 for third grade, and .87 for fifth grade. Across grades, Cronbach’s α was .95. In terms of internal consistency reliability of the subscales of the measure, Cronbach’s α was .83 for the identification subscale, .86 for the meaning subscale, and .86 for the use subscale.
Performance by Grade
We investigated the Mathematics Symbol performance of children in Grades 1, 3, and 5 with our second research question. Table 1 presents means and standard deviations. Expectedly, there were significant differences across grade levels in terms of symbol performance, F(2, 286) = 309.83, p < .001. First graders performed lower than third graders (ES = 2.44) and fifth graders (ES = 3.57); third graders performed lower than fifth graders (ES = 0.84).
Performance on Subscales
With our third research question, we focused on how children perform on the subscales about identification, meaning, and use of symbols (see Table 2). Because we determined grade-level differences on Mathematics Symbols, we present these data disaggregated by grade level. The average score on Mathematics Symbols was 8.63 for first-grade children, with a range of 0 to 21 points. On the subscales, first graders scored highest on identification, followed by use, then meaning. Identification scores were significantly higher than the use subscale, t(103) = 5.49, p < .001 (ES = 0.57), and meaning subscale, t(103) = 8.13, p < .001 (ES = 0.84). In turn, use was significantly higher than meaning, t(103) = 2.54, p = .013 (ES = 0.22). The identification subscale was correlated with the meaning subscale (r = .45, p < .001) and use subscale (r = .44, p < .001). The strongest correlation was between meaning and use (r = .60, p < .001).
At third grade, children scored an average of 23.65 points (range = 3–43). Scores for the identification and use subscales were not significantly different, t(64) = 0.44, p = .658 (ES = 0.04). Scores on the identification subscale were significantly higher than the meaning subscale, t(64) = 6.57, p < .001 (ES = 0.64), and the use subscale was significantly higher than the meaning subscale, t(64) = 4.99, p < .001 (ES = 0.55). The identification subscale was correlated with the meaning subscale (r = .70, p < .001) and use subscale (r =.73, p < .001). The correlation between the meaning and use subscales was .61 (p < .001).
Fifth-grade children scored an average of 29.96 points, with a range of 8 to 49. Similar to third grade, scores from the identification and use subscales were not significantly different, t(119) = 0.52, p = .606 (ES = 0.04). Identification subscale scores were significantly higher than meaning subscale scores, t(119) = 3.67, p < .001 (ES = 0.26); the use subscale was significantly higher than the meaning subscale, t(119) = 3.16, p = .002 (ES = 0.22). The correlation of the identification and meaning subscales was .70 (p < .001). Identification was correlated with use at .69 (p < .001), and meaning was correlated with use at .72 (p < .001).
Performance by Mathematical Content
Our fourth research question focused on the differences among mathematical content areas. We categorized each symbol into mathematical categories: addition and subtraction (i.e., plus and minus), multiplication and division (i.e., multiplication, asterisk, obelus, slash, long division bracket), comparison (equal, not equal, greater than, less than, less than or equal to, and greater than or equal to), money (i.e., dollar and cent), rational numbers (i.e., decimal, fraction bar, percentage, ratio), or algebra (i.e., approximate, plus/minus, square root, negative). Table 3 presents mean differences, standard errors, and confidence intervals.
Comparisons Among Grade Levels by Mathematical Content Area.
Note. CI = confidence interval.
Out of a total of 6 points, first-grade children scored 4.58 (SD = 1.47) on addition and subtraction items. Third-grade children scored 5.68 (SD = 0.75), whereas fifth-grade children scored 5.74 (SD = 0.64). We determined a significant difference among grade levels, F(2, 286) = 40.06, p < .001. Post hoc comparisons indicated significant differences between first grade and third grade (ES = 0.94) as well as first grade and fifth grade (ES = 1.02). There was not a significant difference between the addition and subtraction symbol performance of children in Grades 3 and 5 (ES = 0.09). On multiplication and division items (maximum = 15), first-grade children had scores at 0.54 (SD = 0.99). The average score for third graders was 6.82 (SD = 3.06). Fifth-grade children demonstrated an average score of 9.06 (SD = 2.71). We noted a significant difference among grade levels, F(2, 286) = 380.79, p < .001. All between-grade comparisons were significant with an ES of 2.76 favoring Grade 3 over Grade 1, an ES of 4.18 favoring Grade 5 over Grade 1, and an ES of 0.78 favoring Grade 5 over Grade 3.
In terms of the 18 comparison items, the average score at first grade was 2.02 (SD = 1.53). Third-grade children (M = 4.74, SD = 2.33) scored lower than fifth-grade children (M = 6.43, SD = 2.54). There was a significant difference across grade levels, F(2, 286) = 114.58, p < .001, and all between grade-level comparisons were significant. That is, third grade performed better than first grade (ES = 1.38) and fifth grade outperformed first grade (ES = 2.10) as well as third grade (ES = 0.69). On money items (maximum = 6), first-grade children scored an average of 1.32 (SD = 1.47). Third (M = 3.89, SD = 1.86) and fifth (M = 4.12, SD = 1.55) grade scores were comparable. We determined a significant difference among grade levels, F(2, 286) = 96.79, p < .001, but post hoc comparisons indicated that only the comparisons between Grades 1 and 3 (ES = 1.53) and Grades 1 and 5 (ES = 1.85) were significant. That is, there was not a significant difference between the money symbols performance of children in Grades 3 and 5 (ES = 0.13).
Rational number scores for first-grade children averaged 0.16 (SD = 0.66), whereas third-grade children scored 2.25 (SD = 2.51). Fifth graders scored an average of 4.20 (SD = 2.87) on the 12 fraction items. We identified a significant difference among grade levels, F(2, 286) = 90.91, p < .001. Post hoc comparisons between grade levels indicated significant differences between Grades 1 and 3 (ES = 1.14), Grades 1 and 5 (ES = 1.94), and Grades 3 and 5 (ES = 0.72). Finally, on algebra items (n = 12), all children demonstrated low scores at first grade (M = 0.02, SD = 0.14), third grade (M = 0.28, SD = 0.74), and fifth grade (M = 0.42, SD = 1.12). We noted a significant difference among grade levels, F(2, 286) = 6.84, p = .001. Only one post hoc comparison (i.e., Grades 1–5) was significant (ES = 0.50). The comparisons between Grades 1 and 3 (ES = 0.49) and Grades 3 and 5 (ES = 0.15) were not significant.
Prediction of Computation
Our final research question asked about the relationship between performance on Mathematics Symbols and computation performance on the WRAT4 Math Computation. We conducted a regression analysis at each grade level with WRAT 4 Math Computation as the outcome variable and identification, meaning, and use scores as the predictor variables. Note that these measures were administered within the same testing session, so we use the term predictor as the language used within regression. The term predictor is not intended to indicate a measure administered at one time point with the purpose of predicting mathematics performance at a later time point. Table 4 provides regression results by grade level.
Linear Regression Results.
In first grade, mathematics-symbol knowledge was significantly related to mathematics computation, F(3, 100) = 19.822, p < .001. With this model, 37% of the variance in computation scores was predicted by identification, meaning, and use scores on Mathematics Symbols. Only meaning and use scores were significant predictors in the first-grade model. At third grade, 22% of the variance of mathematics computation was accounted for by identification, meaning, and use in the model. Overall, mathematics symbol performance was significantly related to mathematics computation, F(3, 61) = 5.886, p = .001. An analysis of identification, meaning, and use subscales indicated that only use was a significant predictor of computation. In fifth grade, mathematics-symbol scores significantly predicted mathematics computation, F(3, 116) = 4.719, p = .004, but only 11% of the variance in was accounted for with this model.
Discussion
As Arcavi (1994) explained, all children must develop a symbol sense, and this symbol sense is essential for successful performance in algebra. The primary purpose of this study was to design and test a brief measure of mathematics symbols presented in the elementary grades to gauge the symbol sense of children. We designed this measure after classroom educators inquired about assessment resources related to mathematics symbols. Because children must interpret mathematics symbols to demonstrate mathematics competence, it is necessary for educators to know which symbols children can identify, how children describe the meaning of symbols, and whether children can use symbols with mathematical accuracy.
The Assessment
Currently, no assessments exist that survey a collection of the most important mathematics symbols presented in the elementary grades. One assessment (i.e., Test of Mathematical Abilities) does include a subtest named Mathematical Symbols and Concepts (Brown, Cronin, & Bryant, 2013). Some questions ask for specific identification of symbols, but the majority of questions ask about mathematics vocabulary terms. To fill the void related to an assessment only focused on symbols and their meaning and to provide meaningful information to educators, we designed a measure of mathematics symbols that asked children to identify, provide the meaning, and use 23 mathematics symbols. This measure demonstrated strong internal consistency reliability at first, third, and fifth grade. Internal consistency reliability is important as the coefficients demonstrate the degree to which all the different questions on the measure represent a construct (i.e., understanding of mathematics symbols; Henson, 2001).
Mathematics Symbols was administered in a whole-class setting and did not take long to administer and score. These are important considerations for educators who might use the assessment to collect data about child knowledge related to symbols. The whole-class administration ensures that educators can collect data from classrooms with large numbers of children. The timed nature of the task (i.e., 20 min) requires little interruption from the classroom routine for the one-time assessment, and the ease of scoring (i.e., correct or incorrect) means that educators can easily determine performance on the measure. In addition, educators can use information from the written responses to learn of child misconceptions that need to be remediated at the class or individual level. Our intention was that educators could use this measure to gather screening data about which symbols are easier or more difficult for children. Educators could also use this measure to assess mathematics symbol knowledge across the school year (e.g., fall, winter, and spring administration). Future research, however, should determine how well the measure acts as a screener or expected parameters for growth on the measure across a school year.
Child Performance on the Mathematics Symbol Measure
As expected, fifth-grade children outperformed third-grade children, who, in turn, outperformed first-grade children. We interpreted each of the effects for comparisons between grade levels as large (i.e., above 0.80; Cohen, 1992). These results indicate that symbol knowledge develops across the elementary grades. First-grade children demonstrated enhanced performance on identification subscale scores with a medium ES over the use subscale and a large ES over the meaning subscale. This indicates that first graders have the greatest difficulty with symbol questions related to the meaning of a symbol. Following a similar pattern in third grade, we identified medium ESs for identification over meaning and identification over use. By fifth grade, however, patterns for the subscales changed. That is, we detected only small ESs between identification and meaning and use and meaning. As children progress in their mathematics learning across the elementary grades, gaps in symbol knowledge are measured differently among subscales in first grade but this gap closes by fifth grade. In the next sections, we describe the mathematics symbols that were easier or more difficult at each grade level.
First grade
There were seven symbols regularly used in first-grade textbooks. Of these seven symbols, children demonstrated the highest level of accuracy with identification of the plus sign. With the plus sign, 93% of first-grade children identified the symbol, but only 81% provided an explanation of the plus sign meaning to add, combine, or put together. When considering what is most important about symbols, it is likely being able to use the symbol correctly. Correct use would demonstrate an inherent understanding of the symbol in the way that identification and meaning could not. Even so, only 80% of the first-grade children used the plus sign correctly (e.g., 4 + 4 = 8; 2 + 5). First-grade children demonstrated similar levels of performance related to the minus and equal signs, with over two thirds of children identifying the symbols correctly. About 61% of children provided a correct meaning of the minus sign, whereas only 31% of children provided a correct meaning of the equal sign.
Children accurately identified the dollar sign and cent sign with the same level of accuracy (31%). Meaning and use scores for both symbols were lower than identification scores. The Common Core states that first-grade children record “the results of comparisons with the symbols >, =, and <” (National Governors Association Center for Best Practices & Council of Chief State School Officers, p. 16). Even with this explicitly stated direction, only 13% of children correctly identified the less than symbol; 8% of children correctly identified the greater than symbol. The meaning and use scores for both symbols were lower than identification scores. Our results indicate that many children do not meet expectations as expressed by standards. We did not expect strong performance from first-grade children on any of the other symbols because the other symbols were not detected in first-grade textbook materials. We did, however, note that 25% of children identified the multiplication symbol correctly and 6% identified the division symbol (i.e., obelus).
Third grade
Of the seven symbols regularly used within first-grade textbooks, third-grade children demonstrated variable performance. For the plus sign, minus sign, and equal sign, identification percentages were all above 90%. Similar to first grade, we continued to note that children had difficulty defining the equal sign correctly. Of the other four first-grade symbols, approximately two thirds of children identified and used the dollar and cent symbols correctly. The lowest scores for third graders on symbols introduced in first grade were related to the greater than and less than symbols. Interestingly, for both symbols, identification scores were lower than use scores. Some children can use the symbols correctly but struggle with the language of identifying and providing the meaning of the symbol.
For the seven symbols that first appeared within third-grade mathematics textbooks, 78% of children identified the multiplication symbol correctly and about the same percentage of children used the symbol correctly. Children also did well with two division symbols. Approximately 83% of children identified the obelus correctly, and 77% of children provided correct identification of the long division bracket. Even though identification and meaning scores were similar for these two division symbols, children in our sample had greater difficulty using the long division bracket correctly than the obelus. On the other four symbols introduced within third-grade textbooks (i.e., slash, decimal point, fraction bar, and not equal sign), children identified, defined, and used with less than 25% accuracy. On the remaining nine symbols (i.e., those symbols introduced after third grade), we noted low performance (i.e., less than 10% accuracy) on all symbols except for the percentage symbol. With this symbol, 45% provided correct identification, 22% of children knew the meaning, and 40% used the symbol correctly.
Fifth grade
For the mathematics symbols first detected within first-grade mathematics textbooks, we might expect fifth-grade children to demonstrate 100%, but this was only the case with one symbol—the plus sign. Minus sign identification, meaning, and use was extremely high. Children did very well with identification and use of the equal sign, but less well on providing correct definitions of the equal sign. Children identified the cent sign correctly more often than the dollar sign. The lower accuracy related to the dollar sign is likely attributed to our scoring and children’s use of informal mathematics vocabulary. We did not accept “money sign” as a correct response to the dollar because both the dollar sign and cent sign are signs related to money. Similar to third grade, children identified the greater than sign correctly more often than the less than sign. Use scores for both of these symbols were greater than identification and meaning scores. For the mathematics symbols that first appeared in third-grade textbooks, the obelus and multiplication symbols had the highest accuracy rates at over 95%. Over three fourths of children correctly identified the long division bracket. Similar to the third-grade results, children had greater difficulty with the decimal point, slash, and fraction bar. Very few children identified the not equal sign correctly, provided a definition of this sign, or used it correctly.
For the symbols that we noted as making a first appearance in fifth-grade textbooks, fifth-grade children only performed well with the percentage symbol. The asterisk (for multiplication) and ratio symbol were identified correctly by less than one fifth of children. All other fifth-grade symbols (i.e., approximately equal, greater than or equal to, less than or equal to, negative, plus/minus, and square root) demonstrated very low percentages of correct identification, meaning, and use. As these symbols appear in fifth-grade textbooks, the hope is that children have seen and worked with these symbols by the end of the fifth-grade year.
Symbol Knowledge as Predictor
At first grade, we determined the 37% of the variance on a measure of mathematics computation was predicted by performance on Mathematics Symbols. When analyzing whether identification, meaning, and use subscale scores were significant predictors, only the meaning and use subscales proved significant. This result indicates that identification of the symbol is somewhat important (i.e., “that is the equal sign”) but knowing the meaning of the symbol (e.g., “the equal sign means the same”) and being able to use the symbol (e.g., 5 = 3 + 2) are more important for computation. Given that computation involves interpretation of symbols (i.e., meaning) and using the symbol correctly, this result is not surprising, but does provide direction for educators. Educators should ensure that children be able to define mathematics symbols and use the symbols correctly. Educators could use the Mathematics Symbols measure as one method for data collection about children’s understanding of mathematics symbols.
The amount of variance within the model with mathematics computation as the outcome and mathematics symbol knowledge as the predictor decreased at third grade to 22%. At this grade level, the only significant predictor of computation scores was being able to use the symbol correctly. At fifth grade, the amount of variance was only 11%, and providing the correct meaning of the symbol was the only significant predictor of mathematics computation. This decrease from first to fifth grade, in terms of variance accounted for by mathematics symbol knowledge, is likely due to children becoming more fluent with mathematics facts and computation and relying less on knowledge about mathematics symbols.
Limitations and Future Research
Our measure of Mathematics Symbols was administered to children under timed whole-class conditions. That is, children had 20 min to respond to questions about 23 symbols. During administration, we noted that almost all children finished before the 20 min ended. Many children skipped symbols that were unfamiliar; therefore, most first-grade children finished the assessment in less than 10 min. With that being said, there could have been some children who would have benefited from an extended time or an untimed assessment, so future research should investigate whether accuracy with mathematics-symbols questions is similar to fluency with answering questions about mathematics symbols. It may also be important to conduct analyses using item response theory to determine which symbol items are necessary to include on an assessment. Because this study showed that performance differed considerably across grade levels, it may be important to develop a Mathematics Symbol measure with alternate forms that could be used across the school year to monitor growth of symbol knowledge as it develops. Such a measure could be used by educators for progress monitoring within mathematics.
On our assessment, children responded in a written format. That is, children wrote to identify the symbol, provide the meaning, and use the symbol. Children with writing or reading difficulties may have been at a disadvantage on the assessment because of the written open-ended nature of the responses even though incorrect spelling was not penalized during scoring. Future research may want to accept oral explanations from children, explanations about the symbols using hands-on materials or pictorial representations, and written explanations provided by a scribe. Researchers could also investigate whether an assessment with oral responses differs in a meaningful way from an assessment with written responses. As an oral-response assessment would have to be individually administered whereas a written-response assessment can be administered to a classroom of children at one time, it would be helpful for educators to understand which type of response is necessary for gaining adequate information about symbol knowledge. Similarly, as noted by Driver and Powell (2015) and Sherman and Bisanz (2009), most children are better at nonsymbolic mathematics tasks than tasks presented with symbols. Another limitation of our study is that we only administered an assessment of symbols in symbolic form. Future iterations of this work may include data collection on how children interpret mathematics symbols using both symbolic and nonsymbolic forms.
A future iteration of this assessment may want to focus more on the responses that children provide pertaining to meaning of the symbol and use of the symbol. As meaning and use demonstrated more importance for computation, it may be that an assessment of mathematics symbols does not need to ask children to identify the symbol. Future research may also want to administer other types of mathematics measures (e.g., problem solving, algebra, fractions) to conduct further analyses of the connection between knowledge of mathematics symbols and performance in areas of mathematics beyond computation.
Conclusion
In this study, we learned that it is possible to create a brief measure focused exclusively on the mathematics symbols prevalent in the elementary grades. This measure provided rich information about which symbols are easier or more difficult for children at first, third, and fifth grade. Mathematics symbol knowledge increases across grade levels, yet children are not accurately identifying, defining, and using all mathematical symbols. Because symbols are an important part of mathematics and understanding symbols is necessary for mathematics competence (Arcavi, 1994; Sáenz-Ludlow, 2007), educators should use the measure of Mathematics Symbols to understand how much children know about symbols and what children need to learn.
Footnotes
Acknowledgements
We thank Gena Nelson, Jina Park, Olivia Powell, and Dakota Schiel for assistance with data gathering and scoring. We also extend thanks to the teachers and children who participated in this project.
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by a National Academy of Education/Spencer Postdoctoral Fellowship. Statements do not reflect the position or policy of the funding agencies, university, schools, or persons, and no official endorsement should be inferred.
