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Using Incremental Rehearsal to Increase Fluency of Single-Digit Multiplication Facts With Children Identified As Learning Disabled in Mathematics Computation

Posted on: Sunday, 30 October 2005, 03:01 CST

By Burns, Matthew K

Abstract

Previous research suggested that Incremental Rehearsal (IR; Tucker, 1989) led to better retention than other drill practices models. However, little research exists in the literature regarding drill models for mathematics and no studies were found that used IR to practice multiplication facts. Therefore, the current study used IR as an intervention to teach single-digit multiplication facts to three elementary students identified as learning disabled in mathematics computation using a multiple-baseline single-subject design. All of the students demonstrated 100% non-overlapping data with a median effect size of 4.79 standard deviation units. Potential implications, suggestions for future research, and limitations are included.

As technology becomes increasingly important in daily lives, the need for mathematical skills and thinking increases as well. Mathematics proficiency has been linked to successful employment and higher income upon employment (Rivera-Baltiz, 1992). However, less than one-third (31%) of 4th grade students in the United States scored at or above the proficiency standard on the 2003 National Assessment of Educational Progress in mathematics (Manzo & Galley, 2003). Moreover, over 50% of children identified with learning disabilities (LD) in United States' schools have Individualized Educational Program goals that address mathematics (Lerner, 2003). This is especially important given that without direct intervention learning disabilities in mathematics tend to persist into adulthood (Miller & Mercer, 1997). Still, interventions for children with mathematics disabilities are far less frequently studied as compared to research on reading disabilities (Badian, 1999; Daly & McCurdy, 2002).

The National Council of Teachers of Mathematics (NCTM, 2000) listed fluent computation as a goal for mathematics instruction, and failure to rapidly recall basic facts a characteristic often associated with mathematics disabilities (Miller & Mercer, 1997). In order to be fluent, a child should be able to automatically compute mathematical facts (Lerner, 2003). Automaticity was described as the goal of most instructional efforts (Slavin, 1997) and is obtained when it is faster to solve the problem through recall than it is to perform a mental algorithm for completing the current task (Logan, Taylor, & Etherton, 1996).

Increasing speed of performance occurs through practice of an individual item (Cohen, Servan-Schreiber, & McCelland, 1992) and increasing the amount of drill and practice is often considered to be the most effective approach to improve learning (Chase & Symonds, 1992). Moreover, Greenwood, Delquadri, and Hall (1984) suggested that opportunities to respond (OTR) are a crucial aspect of academic remediation. Successful retention of new information through rehearsal is directly linked to the number of practice trials (Daly, Hintze, & Hamler, 2000; Logan & Klapp, 1991), which has been shown to be true across academic areas, including mathematics fluency (Skinner, Belfiore, Mace, Williams & Johns, 1997). Thus, automaticity of mathematical facts seems closely linked to OTR for each fact.

An effective method of increasing the number of practice trials for new mathematics facts is the use of drill rehearsal models (Burns, 2004). Research has consistently demonstrated that teaching basic skills through drill tasks led to increased retention (Burns, 2004; Cooke, Guzaukas, Pressly, & Kerr, 1993; Cooke & Reichard, 1996, Koegel & Koegel, 1986; Roberts & Shapiro, 1996; Roberts, Turco, & Shapiro, 1991) and subsequently increased performance of more advanced skills (Dehaene & Akhavein, 1995; Jones & Christensen, 1999; Tzelgove, Porat, & Henik, 1997). Some children may lack prerequisite skills for higher-order tasks and must first master the basic information in order to move to higher levels. This seems especially true for mathematics, given its hierarchical nature. Moreover, Skinner and his colleagues (Cates & Skinner, 2000; Skinner, Fletcher, Wildmon, & Belfiore, 1996; McCurdy, Skinner, Grantham, Watson & Hindman, 2001) demonstrated the benefit of interspersing known, or easier, items within mathematics assignments to increase student preference and on-task behavior.

The aforementioned positive benefits of drill rehearsal models led to the development of specific drill approaches. Two examples are Drill Sandwich (DS, Coulter & Coulter, 1989) which involves 50% known and 50% unknown items and Incremental Rehearsal (IR; Tucker, 1989), which uses a gradually increasing ratio of known to unknown items reaching, at the final stage of implementation, 90% to 10%. A comparison of DS, IR, and traditional drill (100% unknown) found that IR led to significantly better retention than the other two models after 1,2,3, 7, and 30 days (MacQuarrie, Tucker, Burns, & Hartman, 2002). It was hypothesized that the better retention was due to IR providing more OTR than the other models. Further, virtually no relationship between number of items retained and verbal intelligence was found when using IR (MacQuarrie et al., 2002).

It seems clear that increasing OTR and interspersing known and unknown items are effective practices when rehearsing new items. It was theorized that including known items within the drill task would lead to an optimal level of challenge in that the learning task would not be too difficult or too easy (Gickling & Thompson, 1985). However, an optimal level of challenge can also be established by taking into account individual rates of learning and by not presenting more items than the child can successfully learn (Gravois & Gickling, 2002). Burns (2001) further operationalized this aspect of an optimal challenge by describing an assessment method with which the acquisition rate, or the amount of new information that can be rehearsed and later remembered during one instructional session, can be measured. In the acquisition rate model, new items are individually introduced and rehearsed in a drill format until three errors occur while rehearsing one new item. After three errors occur instruction stops and the items rehearsed to that point are again individually presented to the child and a correct response is prompted. The number of items to which the child can correctly respond to (e.g., words read orally or multiplication facts answered) are considered his or her acquisition rate. Delayed alternate-form reliability estimates of this method for assessing acquisition rates exceeded .90 for third grade students (Burns, 2001).

Although research on drill models is growing, most studies addressed acquisition and retention of reading words and the participants for most studies were not children with disabilities. Therefore, the current study was conducted to expand previous research on IR and drill tasks by examining the effect of using IR to teach unknown single-digit multiplication facts to children identified as LD in mathematics. It was hypothesized that using IR to teach single-digit multiplication facts would lead to increased fluency of single-digit facts as compared to baseline measures. This was evaluated using a single-subject multiple-baseline design with fluency of computation with single-digit multiplication facts as the dependent variable.

Method

Participants and Settings

Three children identified with a learning disability in mathematics computations, according to Michigan special education regulations, were randomly selected for participation in the study. Michigan regulations state that a child is diagnosed as LD if he or she demonstrates achievement scores that are below average for his or her age group, exhibits a severe discrepancy between intelligence and achievement, and the discrepancy is not caused by other disabilities or by environmental, cultural, or economic factors (Michigan Department of Education, 2002). Each student received mathematics instruction in a special education resource room within the same district in central Michigan, but within three different elementary schools. All children identified as LD in mathematics computation in these three resource rooms were identified as potential participants. Next, the three special education teachers identified those students labeled as LD in mathematics who lacked adequate fluency with single-digit multiplication facts to the degree that interfered with computation of double-digit problems. Finally, one name was selected from each of the lists generated by the three teachers by assigning a number to the students and selecting a number from a random number table.

Student 1 was a Caucasian male, Student 2 was an African- American male, and Student 3 was a Caucasian female. All three students were 8years old, were in third grade, and received mathematics instruction for 1 hour each day in a special education resource room. Each student also participated in special education instruction for 1 additional hour due to deficits in reading (Students 1 and 2) and written expression (Student 3). The remaining instructional time each day was spent in a third-grade general education classroom. Although socio-economic data were not available for theindividual students, 25.5% of the students in the school district that they attended were eligible for the federal free or reduced lunch program. Student scores from the most recent multi- disciplinary evaluation team assessment found Wechsler Intelligence Scale (3rd ed.; 1991) intelligence quotients that fell within the average range (age-based standard scores between 90 and 110). Moreover, the students' age-based mathematics computation standard scores on the Woodcock-Johnson Test of Achievement-Revised (1989) fell between 75 and 85.

The three special education teachers were Caucasian females with 5 to 15 years teaching experience. Mathematics instruction during baseline and treatment (when intervention was not being delivered) involved teaching multi-digit multiplication with base-ten blocks and rehearsing single- and double-digit problems with untimed worksheets. No other rehearsal of single-digit multiplication facts occurred during the baseline or treatment phases. It should be noted that use of the base-ten blocks and untimed worksheets began 3 weeks before baseline data were collected, but because the worksheets were not randomly generated and were untimed, no data were available from this period.

Dependent Variable

Because fluent computation is an important goal for mathematics (NCTM, 2000) and difficulty with fluency is often a characteristic of mathematics disabilities (Miller & Mercer, 1997), student progress was monitored with weekly curriculum-based measurement (CBM; Deno, 1985) fluency probes. Once each week, on a consistent day of the week, each student in the study was presented a single sheet of paper containing 35 randomly selected single-digit multiplication facts written vertically and presented in seven rows with five problems in each. Probes were developed from the website www.mathfactscafe.com in which the number of rows (seven), number of items in each row (five), and range of digits for each item (0 to 9) were assigned, and items were randomly generated from those parameters. Different probes were generated each week and for each student. Students were individually administered the probe by a student researcher and were given 2 minutes to work. Administration procedures used for these probes were taken from Shinn (1989) to assure consistency in administration and to use a data-collection approach with well-documented reliability and validity (Marston, 1989).

The number of digits correct was counted and converted to a digits correct/minute (dc/m) metric by dividing the number by two. The resulting dc/m for each assessment was recorded and graphed. The first four CBM assessments served as baseline data for Student 1 and baseline consisted of the first five points for Students 2 and 3. Thus, these probes were administered and graphed before the treatment condition began. Approximately 25% of the CBM probes were also scored by the author and the dc/ m score for each was correlated with the score obtained by the student researcher. The resulting Pearson Product Moment Correlation coefficient of .99 suggested adequate interscorer reliability for the probes. Interobserver agreement (IOA) was computed by counting the number of items that were scored the same by the two scorers and dividing by the total number of items in those probes. The resulting IOA was 98.8% and suggested adequate agreement.

Experimental Treatment

Participants were taught unknown single-digit multiplication facts while other students participated in mathematics instruction in the special education resource room. Each individual classroom was assigned a student researcher who taught the unknown facts to the students in the experimental condition using a one-on-one format. The experimental treatment was administered twice each week for the duration of the project, with the student researcher and student seated at a table within the special education resource room, but away from the other students. Each session required between 10 and 15 minutes to complete.

Single-digit multiplication facts were identified as known and unknown by writing each of the 100 single-digit facts on a 3 X 5 white index card using black ink. The answer for the fact was not included. The cards were then randomly arranged by shuffling them and presented to the child one at a time. Facts for which the correct answer was given by the student within 2 seconds were identified as known. Those for which the student gave an incorrect answer, no answer, or the correct answer after exceeding 2 seconds were identified as unknown. Finally, unknown facts were confirmed to be unknown before being taught to the child by presenting the new fact to the student and again asking for the correct answer. If the child did not give the correct answer within 2 seconds, the fact continued to be identified as unknown and was taught to the child.

Multiplication facts were taught by student researchers using IR (Tucker, 1989). Each fact was written horizontally on a 3 X 5 index card without including the answer (e.g., 4 X 7 = ?). The first unknown fact was presented to the child as the administrator verbally provided the appropriate answer. Next, the child was asked to orally restate the fact and provide the correct answer. Finally, the unknown fact was rehearsed within the sequence of presentations shown in Table 1. After completing this sequence, the first unknown fact was treated as the first known, the previous ninth known fact (the last one presented) was removed, and a new unknown fact was introduced. Therefore, the number of cards always remained 10 and the ratio of unknown to known in the 9 step for each new item was 10% unknown to 90% known.

Table 1

Steps in Incremental Rehearsal

As stated earlier, each child participated in the IR intervention twice each week on two different days with one session each day. The number of unknown multiplication facts presented during the instructional session varied between children based on each child's acquisition rate. Unknown facts were presented and rehearsed one at a time until three errors occurred while rehearsing one fact as suggested by Burns (2001). Once three errors occurred, instruction stopped and the child returned to the special education resource room activity. The number of items rehearsed before reaching the acquisition rate varied between sessions and between students but ranged from three to six items.

Training

One student researcher was assigned to each participating special education resource room for a period of 15 weeks. Student researchers were Caucasian female undergraduate special education teacher candidates participating in an undergraduate assessment course taught by the author. Before beginning the project, each student researcher participated in 2 hours of training regarding IR procedures and research, and 2 hours of instruction in conducting CBM for different academic areas including mathematics. Next, each student researcher was observed by the author within the first week of being placed within the special education classrooms to assess the fidelity of implementation and deliver guided practice. A second observation occurred during the 13th week to again assess the fidelity of implementation. A checklist containing 16 items was used to assess implementation fidelity during the second observation. Items on the checklist addressed fidelity of treatment condition procedures (implementing IR based on MacQuarrie et al., 2002) and measurement (conducting CBM mathematics fluency probes based on Shinn, 1989). On average, the student researchers correctly completed 96%, with a range of 94% to 100%, of the necessary steps for implementation on both the IR and probes.

Results

The CBM graphs for the three students are presented in Figure 1. Median baseline points for the three students were 3, 8, and 11 dc/ m respectively, the median score for the three final CBM data points were 15, 25, and 27 dc/m respectively. A visual inspection of the data suggests improvements in performance immediately after beginning the treatment condition and continuous increases in performance during the treatment. Thus, changes in level and trend of the data were observed between baseline and treatment for all three students, which implies consistent and reliable treatment effects within single-subject research (Kazdin, 1982).

Effect sizes were computed using the procedure outlined by Busk and Serlin (1992) in which the mean of the baseline data was subtracted from the mean of the intervention data and dividing by the standard deviation of the baseline data. The resulting effect sizes were 17.00 for Student 1, 3.42 for Student 2, and 4.79 for Student 3. Data were also analyzed using Percentage of Nonoverlapping Data (PND), by computing the percentage of data points that did not overlap between the baseline and treatment phases, as recommended by Busse, Kratochwill, and Elliott (1995). The PND was 100% for the three students.

Figure 1. CBM Multiplication Facts Graphs for Students

Discussion

IR appeared to be an effective intervention for increasing fluency of single-digit multiplication facts among these third- grade children identified as LD in mathematics computation. In the introduction to a special series on developments in academic assessment and intervention, Daly and McCurdy (2002) described an increased interest in direct assessment and intervention within school psychology and special education research due to external pressures such as the No Child Left Behind act. The current study could contribute to the research and knowledge-base associated with mathematics intervention, which has significantly lagged behind research regarding reading instruction (Badian, 1999; Daly & McCurdy, 2002) only if confidence in the validity of the inferences exists. Kratochwill (1992) listed 12 research characteristics that improves the validity of inferences from single-subject research and the current s\tudy addressed several of them including objective data, repeated measurement, use of a planned intervention, history suggesting long duration of problem, large effect size, immediate effect within the treatment phase, effect was demonstrated on multiple subjects from different gender and ethnic groups, and reliability of implementation was repeatedly assessed. Perhaps the only two that were not addressed were the use of multiple outcome measures and measures of generalization and follow-up. Although fluency of computation with single-digit multiplication facts, as opposed to assessing the number of taught facts that were retained, could arguably be considered a generalized measure, the current study only examined fluency because that was the outcome of interest. However, future researchers may wish to use a different generalized measure and also include follow-up assessment.

Few mathematics fluency criteria exist in the literature, but many scholars use the standard suggested by Deno and Mirkin (1977), in which third-grade children completing mathematics problems at a rate of at least 20 dc/m demonstrate a mastery skill level. Children completing problems at a rate of 10 to 19 dc/m would be classified as an instructional level and those with a fluency rate of nine dc/ m or less would exhibit a frustration level of fluency. The median of the baseline data points fell within the frustration level for two of the children and was at the lowest end of the instructional level for the third. However, the median of the final three data points exceeded 20 dc/m for two of the students and was within the instructional level for the third. Therefore, the fluency levels at the end of the intervention were probably within an acceptable range given the suggested criteria.

The instructional format in the current study involved a one-to- one arrangement, which might not be possible in many general or special education classrooms. Therefore, IR may be considered a relatively intensive intervention used only when warranted by student need. More smallgroup applications of IR may be possible after additional research, but previous and current research utilized the one-on-one approach (Burns, Dean, & Foley, 2004; Burns & Kimosh, in press; MacQuarrie et al., 2002). However, given that the student researchers participated in 2 hours of training regarding IR administration and research, people other than the classroom teacher could probably implement IR. For example, an instructional aide, parent, or even peer may be able to implement the procedures.

Previous IR research reported a required range of 15 to 25 minutes to complete the model (Burns & Dean, 2002), but that was based on teaching each child 10 new items instead of stopping after reaching the child's acquisition rate. Therefore, using acquisition rates to limit the number of new facts taught in each session resulted in fewer items being taught in the current study and required less time than previously reported.

Although the current study provides data to examine effectiveness of an academic intervention for mathematics, it does little to advance the theory of drill practice. Using IR to teach single- digit multiplication facts led to increased proficiency of this skill, but it is unclear as to why. Two competing hypotheses exist to explain the results. Gickling and Thompson (1985) proposed that providing an appropriate level of challenge leads to increased academic outcomes and hypothesized that 70% to 85% known items, with 15% to 30% unknown, represents appropriate challenge. Additional research has questioned Gickling and Thompson's proposed ratio, but has consistently demonstrated that including known items within rehearsal tasks increased student learning (Burns, 2004). The data provided here were consistent with the theory of appropriate challenge in that the ratio of known to unknown fell within that range during all but the first two steps. However, IR also provides a high number of repetitions, which is consistent with the OTR explanation defined by Greenwood et al. (1984) and supported by MacQuarrie et al. (2002). Which component (appropriate level of challenge or OTR) is primarily responsible for the successful outcomes remains unknown and is in need of future research.

The current study provides data that could be useful for practice and focusing future research, but limitations should be noted. First, although there was some control in format for special education mathematics instruction during the intervention phase, instruction external to the study and individual practice at home could have differed between students. Second, one of the strengths of the current study is the naturalistic setting of the research, but that also serves as a limitation. Moreover, the naturalistic setting did not allow for control of consistency of participation. In other words, although the number of total intervention and assessment sessions did not vary much between students, variance did exist in consistency of treatment schedule (e.g., on Monday one week but on Tuesday the next). Finally, the number of data points within the baseline was either four or five. Thus, extending baseline across the latter students would have strengthened the multiple- baseline design.

Although it is important to note the limitations described above, the current data suggest cautious implications for practice and future research. IR appears to be an easily-implemented academic intervention that could assist in effective drill and practice of unknown items, and seems to be worthy of additional investigation.

References

Badian, N. (1999). Persistent mathematics, reading or mathematics, or reading disability. Annals of Dyslexia, 49, 45-70.

Burns, M. K. (2001). Measuring acquisition and retention rates with curriculum-based assessment. Journal of Psychoeducational Assessment, 19, 148-157.

Burns, M. K. (2004). Empirical analysis of drill ratio research: Refining the instructional level for drill tasks. Remedial and Special Education, 25, 167-175.

Burns, M. K., & Dean, V. J. (2002, February). Rethinking the instructional level: Impact on assessment and intervention. Paper presented at the annual conference of the National Association of School Psychologists, Chicago, IL.

Burns, M. K., Dean, V. J., & Foley, S. (2004). Preteaching unknown key words with incremental rehearsal to improve reading fluency and comprehension with children identified as reading disabled. Journal of School Psychology, 42, 303-314.

Burns, M. K., & Kimosh, A. (in press). Using incremental rehearsal to teach sight-words to adult students with moderate cognitive impairment. Journal of Evidence Based Practice in Schools.

Busk, P. L., & Serlin, R. C. (1992). Meta-analysis for single- case research. In T. R. Kratochwill & J. R. Levin (Eds.) Single- case research design and analysis: New directions for psychology and education (pp. 187-212). Hillsdale, NJ: Lawrence Erlbaum.

Busse, R. T., Kratochwill, T. R., & Elliott, S. N. (1995). Meta- analysis for single-case consultation outcomes: Applications to research and practice. Journal of School Psychology, 33, 269-285.

Cates, G. L., & Skinner, C. H. (2002). Getting remedial mathematics students to prefer homework with 20% and 40% more problems: An investigation of the strengths of the interspersing procedure. Psychology in the Schools, 37, 339-347.

Chase, D. H., & Symonds, P. M. (1992). Practice vs. motivation. Journal of Educational Psychology, 84, 282-289.

Cohen, J. D., Servan-Schreiber, D., & McCelland, J. C. (1992). A parallel distributed processing approach to automaticity. American Journal of Psychology, 2, 239-269.

Cooke, N. L., & Reichard, S. M. (1996). The effects of different interspersal drill ratios on acquisition and generalization of multiplication and division facts. Education & Treatment of Children, 19, 124-142.

Cooke, N. L., Guzaukas, R., Pressley, J. S., & Kerr, K. (1993). Effects of using a ratio of new items to review items during drill and practice: Three experiments. Education and Treatment of Children, 16, 212-234.

Coulter, W. A., & Coulter, E. M. (1990). Curriculum-based assessment for instructional design. Unpublished manuscript.

Daly, E. J., Hintze, J. M., & Hamler, K, R. (2000). Improving practice by taking steps toward technological improvements in academic interventions in the new millennium. Psychology in the Schools, 37, 61-72.

Daly, E. J. & McCurdy, M. (2002). Getting it right so they can get it right: An overview of the special series. School Psychology Review, 31, 453-458.

Dehaene, S., & Akhavein, R. (1995). Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 21, 314- 326.

Deno, S. L. (1985). Curriculum-based measurement: The emerging alternative. Exceptional Children, 52, 219-232.

Deno, S. L., & Mirkin, P. K. (1977). Data-based program modification: A manual. Reston, VA: Council for Exception Children.

Gickling, E., & Thompson, V. (1985). A personal view of curriculum-based assessment. Exceptional Children, 52, 205-218.

Gravois, T. A., & Gickling, E. E. (2002). Best practices in curriculum-based assessment. In A. Thomas & J. Grimes (Eds.) Best practices in school psychology IV (Volume 2; pp. 885-898). Bethesda, MD: National Association of School Psychologists.

Greenwood, C. R., Delquadri, J., & Hall, R. V. (1984). Opportunity to respond and student academic performance. In W. Heward, T. Heron, D. Hill, & J. Trap-Porter (Eds.), Focus on behavior analysis in education (pp. 58-88). Columbus, OH: Charles E. Merrill.

Jones, D., & Christensen, C. A. (1999). Relationship between automaticity in hand writing and student's ability to generate written text. Journal of Educational Psychology, 91, 44-49.

Kratochwill, T. R. (1992). Single-case research design and analysis: An overview. In T. R. Kratochwill & J. R. Levin (Eds.) Single-case researchdesign and analysis: New directions for psychology and education (pp. 1-14). Hillsdale, NJ: Lawrence Erlbaum.

Lerner, J. (2003). Learning disabilities: Theories, diagnosis, and teaching strategies (9th ed.). Boston: Houghton Mifflin.

Kazdin, A. E. (1982). Single-case research designs: Methods for clinical and applied settings. New York: Oxford University Press.

Koegel, L. K., & Koegel, R. L. (1986). The effects of interspersed maintenance tasks on academic performance in a severe childhood stroke victim. Journal of Applied Behavior Analysis, 19, 425-430.

Logan, G. D., & Klapp, S. T. (1991). Alphabet mathematics: Is extended practice necessary to produce automaticity? Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 179- 195.

Logan, G. D., Taylor, S. E., & Etherton, J. L. (1996). Attention in the acquisition and expression Automaticity. Journal of Experimental Psychology: Learning Memory and Cognition, 17, 179- 195.

MacQuarrie, L. L., Tucker, J. A., Burns, M. K., & Hartman, B. (2002). Comparison of retention rates using traditional, Drill Sandwich, and Incremental Rehearsal flashcard methods. School Psychology Review, 31, 584-595.

Manzo, K. K., & Galley, M. (2003). Math climbs, reading flat on '03 NAEP. Education Week, 23 (12), 1, 18.

Marston, D. B. (1989). A curriculum-based measurement approach to assessing academic performance: What it is and why do it. In M. R. Shinn (Ed.), Curriculum-Based Measurement: Assessing special children. New York: The Guilford Press.

McCurdy, M., Skinner, G. H., Grantham, K., Watson, T. S., & Hindman, P. M. (2001). Increasing on-task behavior in an elementary student during mathematics seatwork by interspersing additional brief problems. School Psychology Review, 30, 25-32.

Michigan Department of Education (2002). Revised administrative rules for special education. Lansing, MI: Michigan State Board of Education.

Miller, S., & Mercer, C. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities, 30, 47- 56.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults in the United States. The Journal of Human Resources, 27, 313-328.

Roberts, M. L., & Shapiro, E. S. (1996). Effects of instructional ratios on students' reading performance in a regular education program. Journal of School Psychology, 34, 73-91.

Roberts, M. L., Turco, T. L., & Shapiro, E. S. (1991). Differential effects of fixed instructional ratios on students' progress in reading. Journal of Psychoeducational Assessment, 9, 308- 318.

Shinn, M. R. (Ed.). (1989). Curriculum-based measurement: Assessing special children. New York: Guilford.

Skinner, C. H., Belfiore, P. J., Mace, H. W., Williams, S., & Johns, G. A. (1997). Altering response topography to increase response efficiency and learning rates. School Psychology Quarterly, 12, 54-64.

Skinner, C. H., Fletcher, P. A., Wildmon, M., & Belfiore, P. J. (1996). Improving assignment preference through interspersing additional problems: Brief versus easy problems. Journal of Behavioral Education, 6, 42.7-437.

Slavin, R. E. (1997). Educational psychology theory and practice (5th ed.). Boston: Allyn and Bacon.

Tucker, J. A. (1989). Basic flashcard technique when vocabulary is the goal. Unpublished teaching materials, University of Tennessee at Chattanooga. Chattanooga, TN: Author.

Tzelgove, J., Porat, Z., & Henik, A. (1997). Automaticity and consciousness: Is perceiving the word necessary for reading it? American Journal of Psychology, 110, 429-448.

Wechsler, D. (1991). Wechsler Intelligence Scale for Children: Third Edition. San Antonio, TX: Psychological Corporation.

Woodcock, R. W., & Johnson, M. B. (1989). Woodcock-Johnson Psychoeducational Battery-Revised. Alien, TX: DLM.

Matthew K. Burns

University of Minnesota

Address Correspondence: Matthew K. Burns, Ph.D., Department of Educational Psychology, University of Minnesota, 178 Pillsbury Drive, Minneapolis, MN 55455.

Copyright Pressley Ridge Schools Aug 2005

Source: Education & Treatment of Children

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