The Influence of Mathematics Vocabulary Instruction Embedded Within Addition Tutoring for First-Grade Students With Mathematics Difficulty

Abstract

Researchers and practitioners indicate students require explicit instruction on mathematics vocabulary terms, yet no study has examined the effects of an embedded vocabulary component within mathematics tutoring for early elementary students. First-grade students with mathematics difficulty (MD; n = 98) were randomly assigned to addition tutoring with an embedded vocabulary component, addition tutoring without the embedded vocabulary component, or business-as-usual control. At posttest, students who received addition tutoring without vocabulary demonstrated greater gains than control students on addition fluency. On a measure of mathematics vocabulary, students in the active tutoring conditions demonstrated improved performance on mathematics vocabulary over control students. Results indicate that exposure to addition tutoring with or without an embedded vocabulary component positively improves mathematics vocabulary performance.

Keywords

mathematics vocabulary addition intervention

Mathematics language is connected to students’ conceptual understanding of content knowledge and skills (Capraro & Joffrion, 2006), and developing mathematics vocabulary may promote conceptual thinking about numbers and operations (Dunston & Tyminski, 2013). Vocabulary terms such as “more than,” “goes into,” and “variable” are all connected to the symbolic representations (i.e., –, ÷, x) students use to solve problems in elementary and secondary school (Powell, 2015). As emphasized by the communication process standard by the National Council of Teachers of Mathematics (NCTM; 2000) and the standards for mathematical practice of the Common Core (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), explicitly connecting mathematics language to mathematical concepts may help students better comprehend mathematics. Focused instruction on mathematics vocabulary may be especially beneficial for students who experience mathematics difficulty (MD), as these students regularly demonstrate lower mathematics performance compared with typical performing peers (Andersson, 2010). In the present study, we investigated the efficacy of explicit mathematics vocabulary instruction embedded within addition tutoring for students with MD.

In the line of research on vocabulary acquisition and student understanding of vocabulary, few studies have focused on students with or at-risk for MD. This is disappointing as approximately 3% to 6% of school-age students struggle with a diagnosed mathematics disability (Shalev, Auerbach, Manor, & Gross-Tsur, 2000). Of those students identified with a mathematics disability before fifth grade, 95% continue to struggle with mathematics at the high school level (Shalev, Manor, & Gross-Tsur, 2005). An even greater number of students struggle with low mathematics performance throughout elementary and secondary school without ever receiving a formal disability diagnosis. Students with diagnosed mathematics disability and students who demonstrate low mathematics performance are often referred to as students with MD (Vukovic, 2012). In this article, we categorize students falling at or below the 11th percentile on a measure of addition fluency as experiencing MD. Cutoffs for classifying MD typically fall below the 35th percentile (e.g., Bryant et al., 2011; Fuchs, Powell, Seethaler, Cirino, et al., 2010), and our stringent cutoff of students scoring at or below the 11th percentile is in line with current literature (Martin et al., 2013).

Students with MD are more likely than students without MD to have deficits in several mathematical areas. In the early elementary grades, students with MD may experience difficulty with counting, number combinations, computation, and solving word problems (Geary, Hamson, & Hoard, 2000; Reikerås, 2009). Students with MD, however, can improve mathematics performance when exposed to explicit instruction (e.g., Fuchs et al., 2008; Jitendra et al., 2009) and instruction connecting mathematical concepts and procedures (Miller & Hudson, 2006). In addition, early intervention is necessary for alleviating MD in middle school, high school, and later life (Dowker, 2005).

Many students with MD also experience difficulty with reading (Andersson, 2010; Jordan, Hanich, & Kaplan, 2003). Reading ability influences mathematics performance on verbal calculations and word problems (Jordan, Levine, & Huttenlocher, 1995). In Bryant, Bryant, and Hammill’s (2000) study on the characteristics of students with MD, teachers frequently rated students as having difficulty with word problems and the language of math. Although research indicates phonological processing is a predictor for mathematics achievement (Fuchs et al., 2005; Vukovic & Lesaux, 2013), less is known on how other elements of language influences students with MD and how instruction can improve understanding. The role of vocabulary for students with MD, particularly mathematics vocabulary acquisition, is relatively unknown in terms of mathematics understanding and performance.

Vocabulary

Reading vocabulary acquisition is often associated with improved reading comprehension (Joshi, 2005). Improving reading vocabulary can lead to increased comprehension for a wide range of student populations, including students with disabilities (Carlisle, Kenney, & Vereb, 2013; Ebbers & Denton, 2008) and English learners (Lesaux, Kieffer, Faller, & Kelley, 2010). Positive relationships between vocabulary acquisition and comprehension, based on vocabulary matching curriculum-based measures and content area achievement tests, are also found in content areas such as social studies (Espin, Busch, Shin, & Kruschwitz, 2001). This relationship, however, has yet to be explored in mathematics, particularly for students with MD.

Vocabulary can be acquired directly, as teachers highlight a word and facilitate students’ understanding of the word’s meaning. Vocabulary can also be acquired indirectly, as is the case when students develop an understanding of the terminology over time through its use within contexts. Most mathematics vocabulary learning is indirect (Capraro & Joffrion, 2006). Many students, however, benefit from direct, or explicit, vocabulary instruction. Jitendra, Edwards, Sacks, and Jacobson’s (2004) review of reading vocabulary instruction for students with learning disabilities found that keyword or mnemonic approaches, cognitive strategy instruction (e.g., semantic features analysis), direct instruction, and activity-based methods were generally effective in improving vocabulary understanding. Jitendra et al.’s findings support instructional approaches that explicitly teach content vocabulary for students with learning difficulties. Similarly, Bryant, Goodwin, Bryant, and Higgins’s (2003) review of reading vocabulary instruction for students with learning disabilities found that engaging students interactively with memory devices (e.g., mnemonics) and graphic depictions (e.g., semantic maps or grids) paired with explicit instruction was most promising in promoting word-meaning knowledge and reading comprehension of passages. Of the vocabulary interventions reviewed, Bryant et al. suggested effective vocabulary should include multiple exposures to words across time, be incorporated into classroom teaching on a frequent basis, be delivered in small groups, and be limited in terms of the number of words introduced.

With explicit vocabulary instruction, students learn new vocabulary terms that can, in turn, increase conceptual understanding (Carlisle et al., 2013; Goerss, Beck, & McKeown, 1999; Terrill, Scruggs, & Mastropieri, 2004). Explicit vocabulary instruction includes activating students’ prior knowledge and relating new word meanings to known words and concepts. Teachers should explicitly introduce unfamiliar vocabulary, discuss confusing terms, and encourage students to use mathematics vocabulary in their questions and conversations to ensure understanding (Dunston & Tyminski, 2013). Across all academic content areas, little is known on the effects of explicit vocabulary instruction for students in early elementary grades, despite the importance of early intervention (e.g., Coyne, Simmons, Kame’enui, & Stoolmiller, 2004; Pullen, Tuckwiller, Konold, Maynard, & Coyne, 2010). To our knowledge, there is limited research on mathematics vocabulary instruction for students below third grade.

Academic Language

Typically, academic language is defined as the vocabulary, grammatical structures, and linguistic functions that students engage with and must master to be successful in content areas (Cummins, 2000). Academic language differs by academic discipline (Nagy & Townsend, 2012), and can also be defined by the grade-level language expectations in the content areas and is examined in four language domains: oral exposure, written exposure, oral production, and written production (Ernst-Slavit & Mason, 2011). An understanding of academic language is directly related to academic achievement (Townsend, Filippini, Collins, & Biancarosa, 2012). Traditional views of academic language focus on separating content-specific terminology, which may be considered decontextualized and cognitively demanding, from less demanding conversational terminology, however, the role of context is highly debated (Bunch, 2006). Other scholars believe that academic language encompasses the context it is used in, both formally and informally, to develop meaning (Schleppegrell, 2007). For this purpose of this study, academic language will be considered within the context of academic vocabulary in mathematics instruction.

Academic vocabulary is defined in many different contexts in the literature. Two of the most common forms are domain-specific (i.e., content-specific), with terms that are specific to one type of mathematical concept or process (e.g., “hypotenuse”), and general academic vocabulary, with terms whose meaning can change across content areas (e.g., “variable”; Baumann & Graves, 2010). Both general and domain-specific academic vocabulary should be considered beyond the basic meanings, and instruction should include the context and relationship to other terms associated with the target concept (Nagy & Townsend, 2012).

Mathematics Vocabulary

Monroe and Panchyshyn (1995) broke academic language related to mathematics into four categories: (a) technical words which have one meaning (e.g., “trapezoid”), (b) subtechnical words which have multiple meanings (e.g., “degrees”), (c) general words common in everyday language that have some type of meaning in mathematics (e.g., “simplify”), and (d) symbolic words in which amounts are represented by abstract numerals or symbols (e.g., “plus” as “+”). Existing research on mathematics vocabulary has focused primarily on technical words (e.g., Harmon, Hedrick, & Wood, 2005; Pierce & Fontaine, 2009) and symbolic words (e.g., Gilmore, McCarthy, & Spelke, 2007; Lipton & Spelke, 2005), which are often found in upper elementary and secondary grades.

Several researchers provide suggestions for teaching mathematics vocabulary. Selection of vocabulary is important, and Baumann and Graves (2010) suggested teachers focus on general and domain-specific vocabulary as well as vocabulary related to interpretation of mathematics symbols. Providing explicit vocabulary instruction on mathematics symbols, especially symbols like the equal sign (=) which is frequently misinterpreted by students with MD, is necessary for improved mathematics performance (Powell & Fuchs, 2010). After vocabulary is selected, Pierce and Fontaine (2009) suggested providing student-friendly definitions for vocabulary. Students should have extended opportunities to encounter words and engage in meaningful processing of such words. Bay-Williams and Livers (2009), as well as Monroe and Orme (2002), suggested providing explicit instruction of mathematics vocabulary and opportunities for students to encounter mathematics vocabulary in everyday and context-related situations.

Most suggestions for teaching mathematics vocabulary are based on vocabulary acquisition strategies from reading (Rubenstein & Thompson, 2002), and only a handful of studies evaluate mathematics vocabulary instruction empirically. For example, Monroe and Pendergrass (1997) provided mathematics vocabulary instruction to 58 fourth-grade students. Half of students received mathematics vocabulary instruction related to measurement using definitions, and the other half received instruction using graphic organizers. Students in the graphic organizers conditions performed significantly higher at posttest.

Purpose, Research Questions, and Hypotheses for the Present Study

The empirical evidence related to instruction of mathematics vocabulary for all students is scant. Although several researchers provide suggestions for selection of mathematics vocabulary terms for instruction and other researchers provide frameworks for providing mathematics vocabulary instruction to students, these suggestions are based on strategies for teaching reading vocabulary, have not been validated for mathematics, or have not been researched for students with MD. Therefore, the purpose of the present study was to determine the efficacy of mathematics vocabulary instruction for students with MD.

We embedded mathematics vocabulary instruction within addition tutoring. Prior work on types of mathematics vocabulary and the teaching of vocabulary via explicit instruction provides the basis for several research questions and hypotheses. We asked the following questions.

Research Question 1 (RQ1): What is the efficacy of addition tutoring with embedded mathematics vocabulary on mathematics vocabulary knowledge?

Mathematics vocabulary instruction is important for understanding of mathematics (e.g., Roberts & Truxaw, 2013), but research teams have not explored the efficacy for mathematics vocabulary instruction students with MD. We hypothesized that, when given a measure of mathematics vocabulary terminology, addition + vocabulary, students would outperform students in the addition tutoring group. We also hypothesized that the indirect use of vocabulary in the addition tutoring condition (Capraro & Joffrion, 2006) would provide an advantage over control students.

Research Question 2 (RQ2): Does improved mathematics vocabulary transfer to improved addition knowledge?

Mathematics vocabulary demands on high-stakes assessments are significant (Pierce & Fontaine, 2009) and improved understanding of vocabulary may positively affect mathematics performance (Bay-Williams & Livers, 2009). We hypothesized that students who receive a tutoring package that embeds explicit vocabulary instruction with addition tutoring (i.e., addition + vocabulary tutoring) will outperform students who participate in addition tutoring without vocabulary instruction (i.e., addition tutoring) as understanding vocabulary terms may improve conceptual understanding of numbers (Dunston & Tyminski, 2013). We expected both tutoring conditions to have significantly stronger addition outcomes than students in the control condition, but that the addition + vocabulary condition would outperform the addition condition.

Method

Participants

Across 18 elementary schools in two school districts in the mid-Atlantic region, we administered as assessment of Addition Fluency (Fuchs, Hamlett, & Powell, 2003) to 940 first-grade students in 58 classrooms. Based on district guidelines and Institutional Review Board protocol from the university, students participated in the screening assessment if their parents did not choose to opt out of the study. We recruited students scoring a 0 or 1 (out of 25 items) on Addition Fluency for a total of 84 students. Due to tutoring capacity, we also randomly selected 26 students (out of 93) scoring a 2 on Addition Fluency. We categorized these 110 students (from 48 classrooms in the 18 schools) as experiencing MD. The 110 students performed in the lowest 11% of students from the full sample.

The 110 students were randomly assigned to one of three conditions: (a) addition tutoring with an embedded vocabulary component (n = 39; referred to as “addition + vocabulary” tutoring), (b) addition tutoring without an embedded vocabulary component (n = 38; referred to as “addition” tutoring), and (c) business-as-usual control (n = 33). For random assignment, we stratified by classroom to disperse research assistant (i.e., tutor) resources and to not overburden classroom teachers. We wrote a random number formula for a spreadsheet program, and divided the students into three groups based on random number. Approximately 35% of students (i.e., lowest 35% of random numbers) were assigned to addition + vocabulary, 35% of students (i.e., random numbers, range = 36%–70%) were assigned to addition tutoring, and 30% of students (i.e., highest set of random numbers 71%–100%) were assigned to business-as-usual control. This distribution maximized students in active tutoring while maintaining a large enough control group (see Powell et al., 2015). Between pre- and posttest, four addition + vocabulary, three addition, and five control students moved classrooms or schools and were unavailable for posttesting. Complete data were therefore available for 98 students: 35 addition + vocabulary, 35 addition, and 28 control students.

Students did not differ on demographics (gender, race/ethnicity, special education status, English learner status, or retained status) as a function of condition. Students also did not differ based on age as a function of condition (addition + vocabulary = 7 years 1 month; addition = 7 years 0 months; control = 7 years 2 months). The two school districts did not allow collection of individual student information about reduced and/or free lunch. During the 2012 through 2013 school year, the average percentage of students receiving reduced and/or free lunch across the 18 schools was 43.7% (range = 9.2%–82.3%). To describe students, teachers provided students’ reading and mathematics performance levels as “above grade level,” “at grade level,” or “below grade level.” Analyses indicated no significant differences among conditions on teacher ratings.

Measures

We administered four measures at pretest. Addition Fluency (Fuchs et al., 2003) was administered in one 15-min whole-class screening session. On Addition Fluency, students have 1 min to answer 25 vertically presented addition facts with sums to 12. The examiner reads directions aloud and then allows students to work independently for 1 min until the timer beeps. The maximum score is 25. Coefficient alpha for this sample was .92. On Subtraction Fluency (Fuchs et al., 2003), students have 1 min to answer 25 vertically presented subtraction facts with minuends to 12. After listening to directions, the students work independently until the timer beeps. Maximum score is 25, and coefficient alpha for this sample was .95. Subtraction Fluency was also administered during the 15-min whole-class screening session.

Vocabulary was administered in a 15-min individual pretest session. On Vocabulary (Powell & Driver, 2013), students answer 14 questions related to math vocabulary. The examiner reads aloud each question and the corresponding answer choices, and this gives students an opportunity to select their answer(s). Technical, subtechnical, general, and symbolic words were included as part of the Vocabulary assessment (Monroe & Panchyshyn, 1995). The first two questions ask students to circle vocabulary terms that mean (a) a lot of something, very large, more than (with answer choices of fewer, bigger, balance, more, add, greater) and (b) very little, less than (with answer choices of fewer, blank, amount, less, add, smaller). Students earn 1 point for each correct answer with a maximum of 3 points for each question. The next six questions ask students to circle mathematics vocabulary definitions. Each question contains a correct definition and two incorrect definitions. Students define (c) number, (d) compare, (e) amount, (f) add, (g) subtract, and (h) balance. Students earn 1 point for each correct answer. The final six questions ask students whether statements are always true, sometimes true, or never true. This format is similar to a symbol knowledge assessment from Matthews and Rittle-Johnson (2009). Students answer questions related to (i) plus sign, (j) take away, (k) same as, (l) equation, (m) minus sign, and (n) equal sign. Students earn 1 point for each correct answer. Maximum score is 18. Coefficient alpha for the MD students was .68.

The other measure administered during the 15-min individual pretest session was Wide Range Achievement Test (WRAT) 4 Reading (Wilkinson & Robertson, 2006). Students are provided with 55 words of increasing difficulty. Students read words aloud until a ceiling of 10 consecutive errors. Students reading less than 5 words correctly name 15 letters. Students reading 5 or more words correctly are awarded 15 points without naming letters. The maximum score is 70. As reported by Wilkinson and Robertson (2006), median reliability for second grade is .96. Criterion validity, as measured against the Wechsler Individual Achievement Test (The Psychological Corporation, 2002), is .71.

Two measures were administered in a 15-min individual posttest session: Addition Fluency and Vocabulary. The other two measures (Subtraction Fluency and Reading) were used to describe the sample and determine any differences among conditions at pretest.

Two research assistants independently entered responses on 100% of the test protocols for each measure on an item-by-item basis into an electronic database, resulting in two separate databases. The discrepancies between the two databases were compared and rectified to reflect the student’s original response. After discrepancies were rectified, student responses were converted to correct (1) and incorrect (0) scores using spreadsheet commands. This ensured 100% accuracy of scoring.

Tutoring

Tutoring began the first week of March and ran until the last week of April. Students received 15 tutoring sessions. Due to a strong snow season, 7 days of school were canceled during tutoring. So as much as possible, we conducted sessions 3 times per week for 10 to 15 min a session. Addition tutoring and addition + vocabulary tutoring sessions were the same number of minutes in duration. Tutors were 16 undergraduate or graduate students in education-related fields. Tutors participated in a 2-hr training to become familiar with and practice the two tutoring programs. Tutors met with the project coordinator at the end of the second and fourth weeks of tutoring to discuss tutoring implementation and to resolve any issues related to student behavior. Each session was scripted as a “lesson guide” to ensure tutors covered materials in a similar manner. Tutors did not read lesson guides verbatim. Instead, tutors became familiar with the content of each lesson guide and delivered the lesson in their own words by following the framework, concepts, and vocabulary of the script.

Addition tutoring

During each addition tutoring session, three activities occurred. First, students participated in a flash card activity. Each flash card displayed pictures or numbers. Tutors used the picture flash cards for Sessions 1 through 5 and addition flash cards for Sessions 6 through 15. With the picture flash cards, the student subitizes or counts two sets of objects with a maximum of six objects in each set (for a maximum of 12 objects on the card). The addition flash card set included all 100 addition number combinations with addends 0 through 9. The tutor showed each flash card to the student, and the student answered by saying the total number of pictures or the total (i.e., sum) to the addition number combination. The tutor showed flash cards for 1 min. If the student answered correctly, the flash card was placed on the table. If the student answered incorrectly, the tutor asked the student to count to find the correct answer. Students counted all or counting on, depending on their fluency with addition and their progression through the tutoring program. After the student remediated an incorrect answer, the flash card was placed in the pile on the table. At the end of 1 min, the tutor and student counted the number of flash cards on the table. The student graphed this flash card score on a graph.

Next, students participated in a tutor-led lesson. The first part of the tutor-led lesson asked students to sort a set of shapes by a given attribute. For example, during Session 1, students found all the triangles. During Session 6, students found all the circles. After the shape sort, the tutor-led lesson worked on representing the numbers 0 through 12 (Sessions 1 and 2), comparing 2 amounts (Sessions 3, 4, and 5), and solving addition number combinations (Sessions 6–15). Students worked with hands-on manipulatives such as motors, linking clips, or unifix cubes. Each session, students solved 10 problems. See Figure 1 for an example of the tutor lesson guide for the problem 7 + 4. In session 7, tutors taught students a counting on strategy for solving addition number combinations (Fuson & Secada, 1986). Researchers have determined this counting strategy effective for improving the number combinations fluency of students with MD (e.g., Fuchs, Powell, Seethaler, Fuchs, et al., 2010).

Figure 1.

Comparison of lesson guides for addition tutoring versus addition + vocabulary tutoring. The italicized section emphasizes the vocabulary component of addition + vocabulary tutoring.

Finally, the students participated in a paper-and-pencil review. The student worked for 1 min to solve six problems. At the end of 1 min, the tutor graded the review for correct answers and allowed the student to take the review back to their classroom.

Throughout each session, students had the opportunity to earn puzzle pieces for following directions, doing their best work, and completing each of the three activities (i.e., flash cards, tutor-led lesson, and paper-and-pencil review). At the end of each session, students counted the number of earned puzzle pieces and colored that amount of puzzle pieces on a 12-piece puzzle printed on paper. When a student colored an entire puzzle, typically after two or three tutoring sessions, the student had the opportunity to select a small prize from a bag of prizes.

Addition + vocabulary tutoring

Each tutoring session for addition + vocabulary tutoring students was almost identical to the addition tutoring program except for two key differences. First, the addition + vocabulary tutoring students participated in a mathematics vocabulary introduction or review each day. This vocabulary instruction occurred at the same point in the lesson that addition tutoring students worked on sorting shapes. We selected 13 vocabulary terms related to numbers, comparing, and addition, and these terms were technical, subtechnical, general, or symbolic words (Monroe & Panchyshyn, 1995). The 13 vocabulary terms for practice were number, count (Sessions 1 and 2); greater, more, bigger (Sessions 3 and 5); less, fewer, smaller (Sessions 4 and 5); add, plus sign, equal sign (Sessions 6 and 7); and equation, total (Sessions 8 and 9). During Sessions 10 through 15, students reviewed vocabulary terms. These terms represent a combination of technical (e.g., equation), subtechnical (e.g., total), general (e.g., count), and symbolic (e.g., equal) words (Monroe & Panchyshyn, 1995).

During the vocabulary instruction, the tutor introduced or reviewed key vocabulary by talking about the vocabulary term and the definition. See Figure 2 for an example worksheet from Session 6. As the tutor and student discussed vocabulary, the tutor provided manipulatives or hand motions to help explain the vocabulary term. For example, when focusing on the terms greater, more, and bigger, the student would hold their arms wide apart. For the terms less, fewer, and smaller, the student would clasp their hands together.

Figure 2.

Vocabulary student sheet.

Second, during the tutor-led lesson, the tutor emphasized vocabulary terms. See the italicized words in Figure 1 to show the difference between addition tutoring and addition + vocabulary tutoring. In addition tutoring, the tutor did not ask about the meaning of vocabulary terms. In the addition + vocabulary tutoring, the tutor would ask questions such as, “What does number mean?” or “What does it mean to compare two numbers?” Frequently, the tutor asked students to use hand motions or manipulatives to explain vocabulary terms. For example, to show addition, students would move their arms from a wide stance and clasp their two hands together. During the tutor-led lesson, the tutor asked for vocabulary explanations for any of the 13 vocabulary terms. The mathematics problems for students in addition tutoring and addition + vocabulary tutoring were identical; the only difference between the two active tutoring conditions was the embedded vocabulary component for the vocabulary tutoring students.

Business-as-usual

The business-as-usual tutoring condition was included to control for history and maturation effects. Students in the business-as-usual did not receive any mathematics tutoring as part of the study.

Fidelity

To evaluate fidelity of implementation of the two tutoring conditions, we digitally audio recorded all sessions. Of the 1,050 sessions, 9.8% of sessions were randomly sampled to ensure comparable representation of tutoring condition, tutoring, and session. One research assistant listened independently to recordings while following a checklist identifying essential points addressed during each session. For each tutoring condition and for each session, there were 24 essential points on the checklist. Examples of essential points on the fidelity checklists include the following: reviewing flash card procedure, doing flash cards, graphing flash card score, introducing vocabulary or shape sort, introducing or reviewing addition concepts, completing addition problems, conducting paper-and-pencil review, and grading paper-and-pencil review. Fidelity averaged 97.62 (SD = 1.99) for addition tutoring and 97.34 (SD = 2.68) for addition + vocabulary tutoring.

Procedure

Whole-class screening was conducted in one 15-min session, the second or third week of February. During this session, we administered Addition Fluency and Subtraction Fluency. Individual pretesting took place during one 15-min session during the last week of February or first week of March approximately 1 to 5 days before tutoring began. We administered Vocabulary and Reading during the individual pretest session. Individual posttesting occurred in one 15-min session 1 to 5 days after the last tutoring session was conducted. All control students were pre- and posttested in the same time frame as tutoring students. At posttest, we administered Addition Fluency and Vocabulary.

Data Analysis

To assess pretesting comparability of tutoring conditions, we applied ANOVAs to the screening data, pretest data, and teacher rating scale data using tutoring condition (addition + vocabulary tutoring vs. addition tutoring vs. control). We used chi-square analyses to determine demographic differences among the three conditions. To assess learning as a function of tutoring conditions, we ran ANOVAs on improvement. Post hoc tests of least significant differences were run to determine which condition, if any, demonstrated significantly greater scores on the outcome measures. We calculated effect sizes (ESs) by subtracting means and dividing by the pooled standard deviation (i.e., Hedges’s g), as outlined by the What Works Clearinghouse (2011).

Results

See Table 1 for screening and teacher rating scale data. See Table 2 for pre- and posttest scores on the two outcome measures by tutoring condition.

Table 1.

Student Demographics and Teacher Rating Data by Tutoring Condition.

	Addition + vocabulary tutoring		Addition tutoring		Control
	(n = 35)		(n = 35)		(n = 28)
Variable	n	(%)	n	(%)	n	(%)
Males	16	(45.7)	17	(48.6)	17	(60.7)
Females	19	(54.3)	18	(51.4)	11	(39.3)
Race/ethnicity
African American	7	(20.0)	7	(20.0)	6	(21.4)
White	18	(51.4)	17	(48.6)	16	(57.1)
Hispanic	2	(5.7)	5	(14.3)	5	(17.9)
Other	1	(2.9)	2	(5.7)	0	(0.0)
School-identified disability	8	(22.9)	6	(17.1)	1	(3.6)
English learners	3	(8.6)	6	(17.1)	5	(17.9)
Retained	4	(11.4)	3	(8.6)	3	(10.7)
	M	(SD)	M	(SD)	M	(SD)
Subtraction fluency	3.77	(4.00)	3.23	(4.18)	3.75	(3.34)
WRAT reading	19.89	(4.99)	21.89	(5.64)	23.32	(8.45)
Teacher rating scales
Reading level^a	2.64	(.62)	2.52	(.63)	2.44	(.64)
Math level^a	2.57	(.50)	2.42	(.56)	2.33	(.55)

As determined by teacher: 1 = above grade level, 2 = at grade level, 3 = below grade level. WRAT = Wide Range Achievement Test.

Table 2.

Pre- and Posttest Means and Standard Deviations.

	Addition + vocabulary tutoring		Addition tutoring		Control
	(n = 35)		(n = 35)		(n = 28)
Measure	M	(SD)	M	(SD)	M	(SD)
Addition fluency
Pretest	1.06	(0.97)	1.03	(0.92)	1.07	(0.90)
Posttest	4.80	(3.18)	5.74	(3.23)	4.29	(2.68)
Vocabulary
Pretest	7.94	(2.76)	7.54	(3.11)	8.71	(2.83)
Posttest	9.74	(3.25)	9.77	(2.83)	9.07	(3.31)

Pretest Comparability

At pretest, there were no significant differences among tutoring conditions on any screening measure: Addition Fluency, Subtraction Fluency, or Vocabulary. There was a significant difference between addition + vocabulary tutoring and business-as-usual control students on WRAT Reading (p = .036). Students in the control condition demonstrated significantly higher reading scores than students in the addition + vocabulary condition. Interestingly, there were no significant differences among conditions based on teacher ratings of reading and mathematics performance.

Improvement on Outcome Measures

To investigate RQ1, we compared the improvement of students in the three conditions on mathematics vocabulary. On Vocabulary, improvement as a function of tutoring condition was significant, F(2, 95) = 3.60, p = .031. Follow-up tests indicated addition + vocabulary tutoring students outperformed control students (p = .048; ES = 0.49), and addition tutoring students outperformed control students (p = .011; ES = 0.64). Addition + vocabulary and addition tutoring students performed comparably (p = .529; ES = 0.16).

To investigate RQ2, we compared the performance of students on an assessment of addition. On Addition Fluency, improvement as a function of tutoring condition was not significant. Follow-up tests indicated that addition tutoring students significantly outperformed control students (p = .049) with an ES of 0.48. Addition + vocabulary and addition tutoring students performed comparably (ES = 0.31), as did addition + vocabulary and control students (ES = 0.19).

Discussion

The purpose of this study was to assess the efficacy of vocabulary instruction embedded within addition tutoring for first-grade students with MD. As this study was the first to develop and test a mathematics vocabulary intervention for first-grade students with MD, the study should be considered a pilot study and results should be interpreted accordingly. We compared students who received the embedded vocabulary instruction (i.e., addition + vocabulary tutoring) against two competing conditions: addition tutoring without explicit vocabulary instruction (i.e., addition tutoring) and business-as-usual control. Including the addition tutoring condition allowed us to understand whether embedded vocabulary instruction enhanced vocabulary performance or whether, in fact, improvement on vocabulary was more simply due to addition tutoring. The business-as-usual tutoring condition was included to control for history and maturation effects.

With Vocabulary, we assessed the vocabulary understanding of 19 mathematics terms related to counting, addition, and subtraction. Thirteen of these terms were embedded within the vocabulary instruction embedded within addition tutoring for students in the addition + vocabulary condition, and these terms covered the four areas of mathematics language as outlined by Monroe and Panchyshyn (1995). Addition + vocabulary students demonstrated significant growth over control students with an ES of 0.49. Our hypothesis for RQ1, that vocabulary instruction would improve vocabulary understanding, was corroborated. This finding is similar to vocabulary research outside of mathematics in that explicit instruction on vocabulary improves student understanding of vocabulary (Bryant et al., 2003; Jitendra et al., 2004). Not corroborating our other hypothesis for RQ1, however, was the result of addition tutoring students outperforming control students with an ES of 0.64. We expected addition tutoring students to demonstrate some vocabulary gains because of their participation in addition tutoring, where many mathematics vocabulary terms were used during each lesson. We did not, however, expect addition tutoring students to demonstrate as much growth as addition + vocabulary tutoring students. (Addition + vocabulary and addition tutoring students performed comparably, but addition tutoring students demonstrated slightly higher growth than addition + vocabulary students.)

Although the results on Vocabulary (i.e., addition tutoring students performed comparably to addition + vocabulary students) are not quite as we hypothesized, the significant growth in vocabulary knowledge for students in both conditions over control students is interesting. Because students in addition tutoring (i.e., who did not receive explicit vocabulary instruction) performed comparably with addition + vocabulary students, it is difficult to ascertain whether vocabulary performance gains were due to vocabulary instruction or overall participation in specialized tutoring. It is possible that mathematics vocabulary improves through intensive work in one mathematics area (e.g., addition) provided by a trained tutor using a structured format such as individual tutoring. If this is true, the continuous exposure and use of specific mathematics vocabulary, presented explicitly or implicitly, may be an important component to include in any mathematics intervention for students with MD.

We designed our vocabulary instruction following the frameworks set forth by several researchers (e.g., Carlisle et al., 2013; Dunston & Tyminski, 2013; Monroe & Orme, 2002). We also included aspects of reading vocabulary instruction deemed important in the vocabulary reviews of Bryant et al. (2003) and Jitendra et al. (2004) where multiple exposures, hands-on activities, and explicit instruction were noted as promising practices for teaching reading vocabulary. As the research on teaching mathematics vocabulary is extremely limited, we felt these frameworks were provided us with a reliable starting point. It could be, however, that mathematics vocabulary instruction requires a different instructional framework, and future research needs to investigate the most effective ways to introduce, teach, and review mathematics vocabulary for students with and without MD. It is also possible that the lack of a significant performance difference between the addition + vocabulary and addition students may be attributed to the content of the tutoring. Perhaps this sample of students with MD already learned many of the mathematics vocabulary terms prior of tutoring or perhaps the vocabulary used within the addition tutoring was not appropriate (i.e., too simple or too difficult). It may be that vocabulary from all four categories of academic language, as outlined by Monroe and Panchyshyn (1995), is not appropriate for students with MD when delivered within a brief tutoring intervention.

In the future, we need to determine (a) which mathematics vocabulary require instruction, (b) how to sequence mathematics vocabulary terms, (c) which vocabulary definitions are student-friendly and easy to understand, (d) how to design and deliver effective vocabulary instruction, (e) whether mathematics vocabulary instruction should be embedded within mathematics instruction, and (f) which hands-on manipulatives, pictorial representations, or mnemonics are most effective for mathematics vocabulary instruction (Miller & Hudson, 2006). In addition, mathematics vocabulary instruction for students at the elementary school level may need to be different from mathematics vocabulary instruction at the middle or high school levels (e.g., Dunston & Tyminski, 2013), so a number of studies with students across a range of grade levels are necessary to determine what nuances, if any, exist for students across grade levels.

On Addition Fluency students solved single-digit addition number combinations. We provided 15 sessions of tutoring on addition number combination to students in both the addition + vocabulary and addition conditions. At posttest, students in the addition condition demonstrated significant gains over control students with as ES of 0.48. Contrary to our hypothesis that both addition + vocabulary and addition tutoring students would demonstrate significant gains over control students, addition + vocabulary students did not demonstrate significant growth from pre- to posttest compared with the control students. The difference between addition + vocabulary and addition tutoring students, however, was comparable. Students in both the addition + vocabulary and addition tutoring conditions worked on the same number of addition problems each lesson and the same addition number combinations, and students practiced these problems in the same way (i.e., with the same manipulatives). Although the statistics indicate addition + vocabulary and addition tutoring students performed comparably with one another, the ES favors the performance of students in addition tutoring.

We expected improved vocabulary performance to transfer to improved addition performance as mathematics vocabulary may improve conceptual understanding of numbers and operations (Dunston & Tyminski, 2013). Because the improvement from pre- to posttest for addition + vocabulary students was not significant compared with control students, we cannot confirm a connection between mathematics vocabulary and improved mathematics performance. To control for tutoring time between addition + vocabulary tutoring and addition tutoring, we included a shape sorting activity for addition tutoring students during the time slot that addition + vocabulary tutoring students learned and practice vocabulary terms. We chose a shape sorting activity as it was related to geometry and was not based in numbers or addition. With the shape sorting activity, students were given a set of oral directions (e.g., “Find all the triangles”), and students were asked to complete the activity as quickly as possible. It is possible that the shape sorting activity led to improved Addition Fluency and Vocabulary outcomes in a roundabout manner. Perhaps addition tutoring students demonstrated improved Addition Fluency because Addition Fluency and the shape sorting activity function similarly in terms of administration: A set of oral directions is provided by the tutor, and the student works as quickly as possible. It is also possible that the duration of tutoring was not enough to improve mathematics vocabulary, which, in turn, may influence addition knowledge. Another possible reason for the lack of transfer may be attributed to the researcher-designed vocabulary measure used at pre- and posttest. It is possible this measure was not appropriately gauging student mathematics vocabulary.

Before concluding, we note the limitations of this study. First, we were unable to conduct a full tutoring program due to weather-related school closures. With the gaps in the tutoring schedule (i.e., some students did not see their tutor for 10 consecutive days), we wonder if more consistent implementation of tutoring lessons could change the outcomes of the study. In future research, we would implement tutoring for at least 8 weeks, if not more, to provide students with MD with the intensive duration necessary to create an impact on mathematics understanding and performance (Marston, 2005). Second, we did not collect any maintenance or follow-up data on students. We are unsure if treatment effects were temporary of longer lasting. In the future, we would plan to administer maintenance assessments several weeks and months after the conclusion of tutoring. Third, we only collected one assessment related to addition fluency and one assessment related to vocabulary. Both assessments were researcher-created. In future studies, it would be important to administer standardized mathematics assessments related to mathematics fluency and vocabulary to determine transfer of learning gained during tutoring. Finally, we only assessed students using written responses. Students in both active tutoring conditions participated in a variety of hands-on activities related to learning addition. In the future, we should assess students using written assessments as well as using student interviews and demonstrations to track how students understand mathematics operations.

Implications for Practice

Our results indicate there may be merit in embedding vocabulary instruction within addition tutoring for students with MD, but further work in this area is warranted. Based on the current body of literature related to mathematics vocabulary, we provide the following implications for practice: (a) Students should participate in high-quality mathematics instruction where vocabulary is consistently used and emphasized by the teacher (Schleppegrell, 2007). (b) Teachers should understand the mathematics vocabulary introduced within the mathematics curriculum and how to define vocabulary terms to students (Harmon et al., 2005). All vocabulary definitions should be mathematically correct and student-friendly (Pierce & Fontaine, 2009). (c) Teachers should provide students with activities related to understanding important mathematics vocabulary (Monroe & Orme, 2002). For example, allowing students to work with manipulatives to demonstrate the concepts of addition as “putting together” or “adding on.” (d) Teachers should be cognizant of the ways they use vocabulary (Ernst-Slavit & Mason, 2011). Teachers should refer to mathematics vocabulary in a consistent manner (e.g., using “greater” instead of switching among “bigger,” “more,” and “greater”). For students with MD, consistency of vocabulary may be especially important (Powell & Fuchs, 2010). (e) Teachers should review vocabulary definitions and engage students in discussions about vocabulary definitions often (Pierce & Fontaine, 2009). Students with MD may require more review on subtechnical words that have multiple meanings or general words that have mathematical and everyday meaning. (f) Teachers should provide mathematics vocabulary instruction presented graphically (Dunston & Tyminski, 2013; Monroe & Pendergrass, 1997). As evidenced in other mathematics areas, students with MD benefit from visual representations for organizing information (Ives, 2007; Jitendra, 2002).

Footnotes

Acknowledgements

Special thanks to Elizabeth Abramson, Gabby Allen, Jessica Foster, Larissa Jakubow, Kristen Kipps, Kristina Ormsby, Hannah Smith, Laura Steensma, Bethany Straub, Katie Twilley, Christy Vaught, Kelcey Wall, Olivia White, Kelly Widdows, and Casey Young for testing and tutoring our students. We also extend thanks to the teachers and students who participated in this project.

Authors’ Note

Statements do not reflect the position or policy of the university, schools, or persons, and no official endorsement should be inferred.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the University of Virginia.

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