Students with mathematics difficulty (MD) often struggle with developing fluency with number combinations. Problems with number combinations can lead to difficulties with computation, geometry, algebra, problem solving, and most other mathematics topics. In this article, we discuss fluency with number combinations and its importance for mathematics development. We also discuss why students struggle with learning number combinations. Then, we describe four studies that we conducted to remediate number combinations difficulties. Finally, we offer suggestions for helping students with MD learn number combinations.
Before proceeding, we comment on the use of the term MD. We, similar to other researchers, have defined low performance in mathematics as mathematics difficulty. In our studies, students with mathematics difficulty performed below the 26th percentile on a standardized test of mathematics. Some of these students with MD had an official school diagnosis of mathematics learning disability, but many of the students with MD struggled with low mathematics performance without an official diagnosis.
What Are Number Combinations?
Number combinations are sometimes referred to as basic facts or math facts. We use the term number combinations to show that students can work and solve these problems; that is, these problems do not have to be recalled as a fact from memory. Number combinations comprise 390 addition, subtraction, multiplication, and division number combinations that are some of the basic building blocks of mathematics (Hudson & Miller, 2006). See Table 1 for examples of each of the following types of number combinations. Addition number combinations have addends of 0–9 and a sum of 0–18. Subtraction number combinations have minuends of 0–18 and subtrahends and differences of 0–9. Multiplication number combinations have factors of 0–9 and products ranging from 0–81, whereas division number combinations have dividends of 0–81, divisors of 1–9, and quotients of 0–9. Generally, addition number combinations are more likely to retrieved from memory (rather than calculated) compared with the three other types of number combinations. Because our work on number combinations has focused exclusively on addition and subtraction, we limit our discussion to addition and subtraction combinations in this article.
TABLE 1.
Types of Number Combinations
| Type | Terminology | Examples |
|---|---|---|
|
| ||
| Addition | addend + addend = sum | 3 + 4 = 7 |
| 9 + 6 = 15 | ||
|
| ||
| Subtraction | minued − subtrahend = difference | 4 − 1 = 3 |
| 13 − 8 = 5 | ||
|
| ||
| Multiplication | factor x factor = product | 3 × 2 = 6 |
| 8 × 7 = 56 | ||
|
| ||
| Division | dividend ÷ divisor = quotient | 9 ÷ 3 = 3 |
| 72 ÷ 8 = 9 | ||
Why Are Number Combinations Important?
It is important that students know (i.e., retrieve from memory) or be able to solve (i.e., quickly calculate an answer) number combinations because number combinations are necessary for solving most other types of math problems: computation, money, measurement, geometry, algebra, and problem solving. If students do not know or cannot calculate the answer to a number combination with relative ease, then solving other math problems is more difficult or impossible (Kroesbergen & Van Luit, 2003). Think of how difficult it might be to subtract coins (e.g., 17 pennies minus 9 pennies) or calculate the perimeter of a rectangle (e.g., 8 inches plus 6 inches) if a student does not demonstrate fluency with number combinations. Additionally, students with weak number combinations skill may develop anxiety about mathematics (Hudson & Miller, 2006), which may promote negative attitudes about mathematics and avoidance of situations where mathematics is necessary (as discussed by Laurie Hanich, this issue).
Students Who Struggle with Number Combinations
Many students who struggle with mathematics demonstrate a lack of skill with number combinations (Andersson, 2008; Geary, Hamson, & Hoard, 2000; Hanich, Jordan, Kaplan, & Dick, 2001). For example, second-grade students with MD answered fewer number combinations correctly than students without MD (Hanich et al., 2001). A similar trend emerged with third- and fourth-grade students where students with MD performed significantly lower and made more errors on a test of number combinations than students without MD (Andersson, 2008). These deficits in number combinations skill may stem from difficulties in storing and retrieving number combinations from long-term memory or from deficits in keeping number combinations in working memory (e.g., Geary, Hoard, Byrd-Craven, & DeSoto, 2004; Jordan & Montani, 1997), as elaborated by Daniel Berch in this issue. Because past attempts at remediating number combinations have been equivocal (Kroesbergen & Van Luit, 2003), we conducted four studies to investigate how to help students develop fluency with number combinations skill.
Four Studies Designed to Improve Fluency with Number Combinations
To investigate ways to help students with MD with number combinations, we conducted four studies over a four-year span, one each year. Each study was carried out with third graders. The four studies, which focused on acquisition and fluency performance on addition and subtraction number combinations, constituted a program of research in which each study was designed to address questions that arose in the prior studies. In this way, a linear progression exists across the studies as we learned more each year about number combinations instruction and learning. See Figure 1 for a flowchart of the four studies. These four studies are described in greater depth in Fuchs, Powell, Seethaler, Fuchs, et al. (2010). Students in all four studies struggled with mathematics (i.e., performing below the 26th percentile on a standardized mathematics computation test of addition, subtraction, multiplication, and division) although their underlying sources of the mathematics difficulty may have varied. Students in all four studies were identified as having MD in the same manner with the same standardized test.
Figure 1.
Flowchart of Number Combinations (NC) Studies
Study 1: Practice, practice, practice
The first iteration of number combinations tutoring included three daily activities: computer drill-and-practice, flash cards, and a review (Fuchs et al., 2008). With computer drill-and-practice, students were presented with number combinations for 7.5 minutes. A number combination “flashed” on a computer screen (i.e., 5 + 4 = __). It disappeared within a second or two; then the student typed in the entire combination (i.e., 5 + 4 = 9) from memory. As the student typed in the number combination, a number line picture filled in at the top of the screen. For addition problems, the first addend was represented in blue boxes and the second addend was represented in yellow boxes. For subtraction problems, the minuend boxes filled in with a yellow color, and the subtrahend was represented by Xs over the minuend boxes. If a student answered the number combination correctly, they heard applause and earned a point toward treasures that collected in a treasure box on the screen. If the student answered incorrectly, they were required to retype the combination until it was answered correctly. At the end of 7.5 minutes, the student’s score (i.e., number of correctly answered combinations) was presented on the computer screen. After this computer drill-and-practice, students answered number combination flash cards, presented by the tutor, for 2 minutes. At the end of 2 minutes, the tutor and student counted the correctly answered flash cards and placed the score on a graph. After 3 consecutive sessions of answering 35 number combinations flash cards correctly, the tutor then switched to number line flash cards for the flash card activity. The number line flash cards represented combinations similar to those presented during computer drill-and-practice. Students were asked to state the combination represented by the number line. Similar to number combinations flash cards, students graphed the number of correctly-answered cards after 2 minutes. Each session concluded with a paper review of 15 number combinations. Students had 2 minutes to answer as many number combinations as possible.
Conditions
We compared this number combinations tutoring to three other tutoring conditions: (a) double-digit computation and estimation, (b) double-digit computation and estimation with number combinations tutoring, and (c) word identification. In double-digit computation and estimation tutoring, students worked through double-digit addition and subtraction problems via a computer program, answered double-digit flash cards, and answered double-digit problems on a paper review. Double-digit computation and estimation with number combinations tutoring included the components of both double-digit computation and estimation tutoring and number combinations tutoring. Word-identification tutoring students participated in computer drill-and-practice on sight words and read passages aloud for fluency practice; this was a control condition. All students received tutoring for 15 weeks, 3 sessions per week, 15–30 minutes per session. Tutoring sessions were delivered during the school day at times designated by the classroom teacher. Students were tutored individually in locations outside of the student’s classroom (i.e., hallway, empty classroom, library, or conference room).
Results
(See Table 2 for results from the four studies. Effect sizes (ES) are reported for significant results.) Students participating in number combinations tutoring demonstrated significantly stronger improvement in number combinations compared to students in the double-digit computation and estimation condition (ES = 0.69), the double-digit and number combinations condition (ES = 0.81), or the word identification condition (ES = 0.78). From this first study, we learned that students who were tutored on number combinations alone demonstrated stronger improvement than those students who received number combinations tutoring in conjunction with tutoring on double-digit computation and estimation. We also learned that the combination of computer drill-and-practice, flash cards, and paper review appeared to improve fluency with number combinations.
TABLE 2.
Results from Number Combinations Studies
| Study | Conditions | Results1 | Effect sizes |
|---|---|---|---|
|
| |||
| 1 | Number combinations tutoring (NC) | NC > DD | 0.69 |
| Double-digit computation and estimation tutoring (DD) | NC > COMB | 0.81 | |
| Double-digit computation and estimation w/ number combinations tutoring (COMB) Word identification tutoring (CONTROL) | NC > CONTROL | 0.78 | |
|
| |||
| 2 | Number combinations tutoring (NC) | NC > DD | 0.31 |
| Expanded Number combinations tutoring w/ conceptual instruction (E-NC-CONC) | NC > CONTROL | 0.50 | |
| Double-digit computation and estimation tutoring (DD) | E-NC-CONC > DD | 0.37 | |
| No tutoring (CONTROL) | NC-CONC > CONTROL NC = NC-CONC | 0.53 | |
|
| |||
| 3 | Number combinations tutoring w/ counting strategies (NC-COUNT) | NC-COUNT > CONTROL | 0.52 |
| Word-problem tutoring w/ counting strategies (WP-COUNT) | WP-COUNT > CONTROL | 0.62 | |
| No tutoring (CONTROL) | NC-COUNT = WP-COUNT | ||
|
| |||
| 4 | Word-problem tutoring w/ counting strategies practice (WP-COUNT) | WP-COUNT > CONTROL | 0.67 |
| Word-problem tutoring (WP) | WP-COUNT > WP | 0.22 | |
| No tutoring (CONTROL) | WP > CONTROL | 0.43 | |
Significant results on number combinations.
Study 2: Providing conceptual instruction
To determine the importance of conceptual instruction for number combinations remediation, the second study compared the number combinations tutoring of Study 1 to an expanded number combinations tutoring with conceptual number combinations instruction (Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009). Number combinations tutoring included only three activities, which were similar to the number combinations activities in Study 1: computer drill-and-practice, number combinations flash cards, and paper review. The expanded tutoring included six activities: conceptual instruction, number combinations flash cards, computer drill-and-practice, number line flash cards, combinations family review, and paper review. (The two flash card activities, computer drill-and-practice, and paper review were similar to activities described above.) With the conceptual instruction, tutors and students worked with manipulatives (i.e., blue and yellow blocks) to show various combinations of a fact family (i.e., 2 + 4 = 6; 4 + 2 = 6; 6 − 2 = 4; 6 − 4 = 2). Students then practiced generating families of number combinations on the combinations family review.
Conditions
We compared the performance of the students in the two number combinations conditions described above, referred to here as number combinations, and expanded number combinations, to students in two competing conditions: (a) double-digit computation and estimation tutoring and (b) no tutoring (i.e., control). The double-digit computation and estimation tutoring was similar to the tutoring described in Study 1. Tutoring for students in all three tutoring conditions lasted 15 weeks, 3 sessions per week, 15–25 minutes per session.
Results
Students in both number combinations conditions performed similarly to one another. Students who received number combinations tutoring (without conceptual instruction) significantly outperformed students in the double-digit tutoring (ES = 0.31) and control (ES = 0.50) conditions. Similarly, students receiving the expanded number combinations tutoring demonstrated significantly higher scores than double-digit tutoring (ES = 0.37) and control students (ES = 0.53). It was interesting to note that students in either number combinations conditions performed similarly even though the students who received expanded tutoring on number combinations spent much more time on the conceptual basis of number combinations, having to work through and think about how number combinations relate to one another. This finding suggests that students with MD needed to learn a plan or strategy for solving number combinations when the answer is not immediately recalled. For this reason, we introduced counting strategies in Study 3.
Study 3: Counting strategies
Number combinations tutoring in Study 3 included instruction on how to use counting strategies to solve addition and subtraction combinations (Fuchs et al., 2009). (See Figure 2 for an explanation of the counting strategies.) Number combinations tutoring included five activities: 1) number combinations flash cards, 2) explicit instruction, 3) lesson-specific flash cards, 4) computer drill-and-practice, and 5) paper review. The number combination flash cards, computer drill-and-practice, and paper review used in this condition were the same as described in Studies 1 and 2. Explicit instruction focused on teaching and practicing the counting strategies along with instruction focused on groups (not families) of number combinations (e.g., the +0 and −0 combinations; the +1 and −1 combinations). The lesson-specific flash cards also focused on the groups; students answered these cards for 1 minute. As students moved on to the next group, they were permitted to take the lesson-specific flash cards home for practice.
Figure 2.
Counting Strategies
Conditions
In addition to the number combination tutoring condition used in this study, the comparison conditions in Study 3 were (a) word-problem tutoring and (b) no tutoring (i.e., control). Word-problem tutoring consisted of teaching students to read, set up, and solve word problems by problem type. Students also learned the counting strategies taught in number combinations tutoring. Students in each of the two tutoring conditions received instruction over 16 weeks, 3 sessions per week, 20–30 minutes per session.
Results
On number combinations, students participating in number combinations tutoring outperformed control students (ES = 0.52). There were no differences between students receiving number combinations or word-problem tutoring, given that word-problem tutoring also included instruction on solving number combinations with counting strategies. Interestingly, word-problem tutoring students outperformed control students on number combinations (ES = 0.62). From this study, we learned that explicit instruction and practice on using counting strategies to solve number combinations appeared to be an important component of number combinations instruction.
Study 4: Daily practice with counting strategies
Because all tutored students, regardless of whether counting strategies and number combinations were the focus of tutoring, improved reliably more than control students on number combinations in Study 3, the goal of Study 4 was to determine how much practice on counting strategies was necessary (Fuchs, Powell, Seethaler, Cirino et al., 2010). In Study 4, students in each of two active tutoring conditions received instruction on word problems. The nature of word-problem instruction in the two conditions was identical. The only difference between the two conditions was that only one condition included daily practice on number combinations, and the other did not. For instance, in both conditions, students received an initial explanation lesson on counting strategies, but in only one condition did students also receive daily practice on solving number combinations with counting strategies. Specifically, in both conditions, students participated in word-problem warm up, explicit word-problem instruction, word-problem sorting cards, and paper review. These activities focused on solving word problems belonging to three word-problem types: total, difference, and change. Students in both conditions also participated in flash cards, but the nature of the flash card activity differed across the two conditions. Children in the word problem tutoring without the practice condition simply read numbers on flash cards. Children in the number combinations instruction with daily practice condition answered number combinations flash cards for 1 minute. If a student answered incorrectly, the tutor asked him or her to use a counting strategy until the student answered correctly. At the end of 1 minute the number of correctly answered flash cards was recorded on a graph. During counting up practice, the tutor asked the student to solve four number combinations using counting strategies. In summary, although students in the word-problem tutoring without daily practice received explicit instruction on using counting strategies to solve number combinations, they did not participate in the number combinations flash cards or counting practice each session.
Conditions and results
Tutoring in both active conditions lasted 16 weeks, 3 sessions per week, 20–30 minutes per session. We also incorporated a no tutoring condition (i.e., control). On number combinations, students who received word-problem tutoring with daily counting strategies practice outperformed students in the control condition (ES = 0.67) and students who received the word-problem tutoring without daily counting strategies practice condition (ES = 0.22). Nevertheless, students who received word-problem tutoring without daily practice outperformed control students (ES = 0.43). From Study 4, we learned that students require daily practice on number combinations even if the time spent on number combinations is only a few minutes per session.
Lessons Learned Across the Four Studies
In Studies 1 and 2, we provided third graders with tutoring focused wholly on number combinations. The instruction included a variety of activities ranging from flash cards to computer drill-and-practice to explicit conceptual instruction to paper reviews. We hoped that a package of diverse number combinations activities would be the best route for remediation of number combinations skill. In Studies 3 and 4, however, we provided brief but explicit instruction on solving number combinations, and we learned that students benefit from learning a strategy to solve number combinations. Students benefitted from counting strategies instruction embedded within word-problem tutoring just as much as tutoring programs focused exclusively on number combinations. So, we learned that number combinations instruction can be provided alongside instruction on word problems, and students still derive benefit on number combinations.
Starting in Study 3, we began to provide students with counting strategies for solving number combinations when they did not automatically recall the answer (i.e., counting strategies). We felt students needed a reliable and efficient backup plan. In Study 4, we isolated the effect of practice within counting strategies instruction and our findings suggest the importance of daily (albeit brief) practice for this population of learners. In this way, we conclude that providing students with explicit instruction (i.e., counting strategies) on solving number combinations and practicing these counting strategies is an important and efficient component of mathematics instruction for students with MD.
Additionally, we note that, across studies, students participated in daily flash card practice and a paper review to improve number combinations fluency. Although we never isolated the contribution of flash card practice or paper reviews, we believe that reviewing number combinations and requiring students to use counting strategies to solve flash cards answered incorrectly and graphing the flash card score, may be another important component of number combinations instruction. Along the same lines, reviewing number combinations on paper may be important for ensuring transfer to paper-and-pencil tasks.
Will All Students with MD Benefit from Number Combinations Instruction?
As already mentioned, in each of the four studies in this program of research, all participants were third grade students who struggled with mathematics. Some of these students also struggled with reading difficulties (RD) (i.e., scored below the 26th percentile on a standardized reading test) and were therefore categorized as having MD+RD. Other students performed above the 40th percentile in reading and were therefore categorized as having MD-only. In each of the four studies, we looked for performance differences between MD+RD and MD-only students. Some prior research has shown performance differences on number combinations between students with MD+RD and MD-only students (Hanich et al., 2001) with MD-only students outperforming MD+RD students, whereas other research has not demonstrated differences between students with MD+RD and MD-only on number combinations (Geary, Hoard, & Hamson, 1999).
Unfortunately, in all but one of our four studies reviewed here, the number of students in each subgroup was too small to determine whether response to intervention differed as a function of whether students experienced MD alone or in combination with RD. Yet, some patterns in the data provide the basis for hypothesizing that relative to students with MD-only, students with MD+RD may require more intensive intervention, and different kinds of intervention, that have more systematic practice with a greater emphasis on language. Large-scale intervention research is, however, needed—with adequately large samples of students with MD with and without RD—to assess the tenability of these hypotheses.
Advice for Teachers and Parents
Based on this program of research on number combinations remediation, many students with MD benefit from explicit instruction on how to solve number combinations. Students should be provided with strategies for solving combinations when the answer is not immediately recalled and should receive many opportunities to practice solving number combinations through a variety of formats (i.e., flash cards, oral quizzes, paper reviews). From what we have learned from our program of research, we feel that the more opportunities students have to see and work with number combinations, the more likely they will improve their number combinations skill. More opportunities does not necessarily mean more time practicing number combinations, because, as we have learned in our studies, it appears that students benefit from well-designed and implemented number combinations tutoring that lasts a few minutes each session just as much as number combinations instruction that lasts 30 minutes each session. We believe students should be provided with explicit instruction on solving number combinations and provided with different outlets to practice number combinations throughout the school year.
Acknowledgments
This research was supported in part by Grant P01046261 from the Eunice Shriver National Institute of Child Health and Human Development (NICHD) to the University of Houston, with a subcontract to Vanderbilt University, and by Core Grant #HD15052 from NICHD to Vanderbilt University. Statements do not reflect the position or policy of these agencies, and no official endorsement by them should be inferred.
Biographies
Sarah R. Powell, Ph.D., is a Research Associate in the Department of Special Education at Vanderbilt University. Her research interests include math interventions for students with mathematics difficulty and student understanding of equality symbols
Lynn S. Fuchs, Ph.D., is the Nicholas Hobbs Professor of Special Education and Human Development in the Department of Special Education at Vanderbilt University. Her research interests center on classroom assessment, reading and mathematics intervention, and the cognitive predictors associated with responsiveness to intervention
Douglas Fuchs, Ph.D., is the Nicholas Hobbs Professor of Special Education and Human Development in the Department of Special Education at Vanderbilt University. His research interests include learning disabilities classification issues, reading intervention, and the cognitive predictors of reading development
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