Teaching K–3 Multi-Digit Arithmetic Computation to Students with Slow Language Processing

Abstract

The purpose of this article is to present three related studies that build on each other to demonstrate first the need and then the efficacy of the Blended Arithmetic Curriculum (BAC) to help students overcome both slow language processing and the environmental effects of being a student in an urban school district. The author’s underlying theory is that K–3 students with slow language processing may be good at complex reasoning, but still struggle with retrieving basic computational facts. Nonetheless, if they did not learn their facts, these students would struggle with K–3 multi-digit arithmetic computation, and ultimately struggle with their hypothesized strength: seeing numeric patterns as would be needed in university level computation. To teach arithmetic facts conceptually, the author developed a paper and pencil curriculum that first teaches complex multi-digit addition with regrouping using a limited number of facts such as 5 + 5, 9 + 1, 1 + 9; 7 + 7, 7 + 8, 8 + 7, and 8 + 8 in problems such as 197 + 108 = 305 so that fact retrieval and computation are fast and accurate. At the end of 2nd grade, urban students with learning disabilities solved 42 two-digit by two-digit problems with 92 percent accuracy in an average of 7 min. The results matched those of suburban students and were significantly faster and more accurate than general education students in the same urban school.

1. Overall Introduction

The purpose of this article is to present three related studies that build on each other to demonstrate first the need and then the efficacy of the Blended Arithmetic Curriculum (BAC) to help students overcome both slow language processing and the environmental effects of being a student in an urban school district. Study A was a Teacher Action Research study conducted by Oldrieve. Study B was a replication study of Study A. Study C was designed to further demonstrate the efficacy of the Blended Arithmetic Curriculum both on improving mathematical computational fluency and accuracy, as well as showing that these gains would also improve results on more expansive school district proficiency assessments. Both Study B and Study C were overseen by Oldrieve’s dissertation co-chair, Dr. Barbara R. Schirmer and Dr. Donald Scipione, a former physicist who had become president of a firm that programmed online applications for truck and doctor shift scheduling as it had begun branching into elementary school academic content.

The Blended Arithmetic Curriculum was developed by Oldrieve to help students who have IEPs indicating they have learning disabilities in K–3 math. Additionally, the participating students were enrolled in urban, muti-ethnic, low socio-economic school districts, and the study was designed to determine if the Blended Arithmetic Curriculum could benefit students enrolled in Urban Learning Disabilities (ULD), Urban General Education (UGE) and Suburban General Education (SGE) classrooms. The goal was to help slow-language-processing individuals avoid falling behind in K–3 math in the first place.

The results of the Teacher Action Research study were first presented to the superintendent of the urban district and then used to help Oldrieve convince Donald Scipione, the president of a local internet software firm with a background in writing successful local and national grants, to team with him to develop the Blended Arithmetic Curriculum into a complete program with student workbooks, teacher plan books, computer-based conceptual activities, and professional development sessions. The ultimate goal of Scipione was to take the paper and pencil Blended Arithmetic Curriculum and develop computer programming so that an online version could individualize instruction for every student. Clements, Sarama, Baroody, and Joswick [1] have pointed out that students do better with “Learning Trajectories” if what they are taught is only one learning trajectory above their current knowledge-which is exactly what an assessment expert and a well-designed computer program can determine. Furthermore, Clements, Vinh, Lim, and Sarama [2] argue that learning trajectories targeting STEM education are particularly effective for students who live in poverty, are members of linguistic and ethnic minority groups, or are children with disabilities in a STEM field.

Scipione and Oldrieve, with the oversight of Barbara R. Schirmer, a member of Oldrieve’s dissertation committee for a study of Oldrieve’s phonemic awareness, spelling, and writing program, then conducted a replication study of the original Teacher Action Research project. When the results of the larger replication study found that a different set of second-grade Urban General Education students had similar results to the UGE second graders, Scipione and Oldrieve were able to convince principals and teachers to sign on to three local grants that would allow teachers and students to test-pilot the Blended Arithmetic Curriculum. In turn, the results of the test-pilot program helped Scipione’s internet software firm win a four-year Small Business Innovation Research (SBIR) grant from the Special Education division of the U.S. Department of Education.

Once the local and federal grant projects began, assessments measuring progress were conducted by Oldrieve and overseen by Scipione and Schirmer. Thus, the follow-up study’s assessments and student data collection procedures were approved by Oldrieve and Schirmer’s university’s Human Subjects Review Board. The results of these studies were presented at several math and science conferences [3,4,5,6], but were never published.

The reason for submitting this article to Computation is that studies A, B, and C demonstrated the effectiveness of the curriculum modification that can help prepare slow-language-processing K–12 students for the complexities of college and graduate levels of “Computation”. The second reason is that shortly after being hired as an assistant professor of K–3 Literacy Assessment and K–3 Phonics and Word Study, Richard M. Oldrieve and Cynthia Bertelsen [7] helped conduct a large n study that supports the underlying theory of the Blended Arithmetic Curriculum.

Theoretical Foundation of Blended Arithmetic Curriculum

While taking an introductory class in Animal Behavior as an undergraduate, Oldrieve began speculating that differences in cognitive processing speed are a key factor in different special education categorizations such as specific learning disabilities (SLD) and autism. Furthermore, the author speculated that being slow at language processing would be a disadvantage when trying to memorize arithmetic facts but would be counterbalanced with the benefit of being able to solve complex problems.

Wiig, Semel, and Nystrom [8] as well as many other researchers such as Denckla and Rudel [9,10,11]; Fawcett and Nicolson [12]; Korhonen [13,14]; Spring and Perry [15]; Torgesen [16]; as well as Wolff, Michel, and Ovrut [17] had long ago noticed that slow times on RAN of Objects assessments predicted failure in early language development, early reading, and many other learning differences, and it was no surprise when the National Early Literacy Panel [18], combined the findings of many researchers and confirmed this relationship.

In doing an extensive literature search as to the brain-based structural reasons for this predictability, the author found that MacLeod [19] speculated that differences in Rapid Automatic Naming abilities might be accounted for by the fact that some individuals might devote more “parallel processors” to a given problem than others, and those who devote more parallel processors might take longer to solve said problem.

The parallel processing hypothesis suggests that there are subtle advantages to slow processing that can be missed by educators because of the obvious disadvantages. For example, in early literacy and numeracy, an individual who tends to use lots of parallel processors to solve a problem might be quite creative at storytelling and be very good at comprehension and mathematical problem solving, but these advantages will get lost if the student fails to learn the arithmetic facts and computation necessary for solving algebra problems. Or, as in Oldrieve’s dissertation study [20] on the effectiveness of a pattern-based method for helping Urban General Education kindergarteners learn phonemic awareness, spelling, and writing [21], slow language processors will not reach their potential in learning how to read and write literature. In theory, if a teacher and/or curriculum modification help to teach these slower-processing students phonics and computation, then the students’ comprehension, creativity, and problem solving might become more obvious.

Bodner and Guay [22] speculated that a slow-language-processing individual who was exposed to a “nouveau” type of problem that they had never solved before would take longer to solve the problem than someone who was typically a fast processor. Nonetheless, Bodner and Guay reasoned that, like their fast-processing peers, slower-processing individuals would tend to be fast and efficient at solving routine tasks they’ve seen and solved before. In MacLeod’s model, slower-processing individuals who had learned how to solve a certain type of problem would need fewer processors to solve problems they had already developed a routine to solve. In contrast, Oldrieve speculates that there is a tipping point where, if the problem is complex enough, the fast-processing individual may not even be able to solve it no matter how much time they are given, while the slow-processing individual could solve it given unlimited time.

When first hired as an assistant professor of K–3 Literacy Assessment as well as K–3 Phonics and Word study, Oldrieve joined the COSMOS Teaching and Community for math and science professors and graduate students in order to better understand the brain science behind learning differences. A year later, a subset of the group began to look at the role of 3-D visualization ability in learning college-level math and science, as had been done by previous researchers [23,24]. The science and math professors, as well as the science and math education professors, ultimately wanted to test-pilot interventions that would increase the 3-D visualization ability of some individuals so that they could become more proficient at solving complex mathematical and scientific reasoning [25,26]. To serve as a pre- and post-test, the research team settled upon Bodner and Guay’s [22] Purdue Spatial Visualization Test: Rotations (PSVT:R).

Oldrieve, from his previous dissertation work, then suggested that the team should add Wiig, Semel, and Nystrom’s [8] assessment of Rapid Automatic Naming (RAN) to the test battery. Fortunately, others in the sub-group thought administering the combination of PSVT:R and RAN of Objects using paper and pencil answer sheets would be worthwhile. Subsequently, the governing board of the Northwest Ohio Consortium of Math and Science Educators awarded a USD 3000 grant to the COSMOS research team.

Members of the COSMOS Teaching and Learning Community, along with 8 to 10 of the reading department’s graduate students, tested 391 students enrolled at a Midwest Community College and a Midwest University. The students were enrolled in 24 sections of 9 different classes. Each class was presented with a copy of the same MS PowerPoint that included instructions for the participating students, an embedded count-up timer for the RAN of Objects, and a count-down timer for the PVST-R. Professors were not supposed to administer the assessments to their own classes; instead, they were supposed to go into a fellow researcher’s class and administer the assessments, while the Reading Department’s graduate students helped collect and eventually score the assessments. (For a deeper description of the assessment procedures and protocols, see Richard M. Oldrieve and Cynthia Bertelsen’s article [7]).

As was predicted and expected by all the members of the team, students enrolled in junior and senior level math and science classes tended to have higher scores in the 3-D visualization ability measured by the PSVT:R than most other participants—though the arts classes of Music Theory IV, print-making, and 3-D art also had higher than the grand mean scores on the PSVT:R. In fact, the class averages for all of the STEAM classes were all above the 60 that Sorby [24] and her colleagues Medina, Gerson, and Sorby [25] have as well as Boersma, Hamlin, and Sorby [26] found to be a minimum score necessary to do well in engineering courses.

Furthermore, as was hypothesized by Oldrieve, the Cosmos team’s study found a statistically significant multiplier that indicated that for each second slower a university or community college student performed on the RAN of Objects, the greater the odds that the student earned a higher final score in the class in which they were tested.

Additionally, when PSVT:R vs. RAN of Objects scatterplots were created for each of the four high school content areas of high school science, math, language arts, and social studies teachers, the results were separated into three different quadrants.

• Future high school Science teachers tended to be highly visual and fast.
• Future high school Math teachers tended to be highly visual and slow.
• Future high school English Language Arts (ELA) teachers tended to be low visual and slow language processors.

Furthermore, especially considering the small sample size and an expected alpha of p = 0.1, these differences were statistically significant from moderate to very strong on the planes that were relevant (see

Table 1. Comparing statistically significant differences and similarities between future teachers of math, science, and English language arts (ELA).

Comparison of Teachers	F-Value	p-Value
PVST: Math and Science	F = 0.548	p = 0.474
RAN Object: Math and Science	F = 5.183	p = 0.044 1
PVST: Math and ELA	F = 14.913	p = 0.001 1
RAN Object: Math and ELA	F = 1.198	p = 0.289
PVST: Science and ELA	F = 10.068	p = 0.007 2
RAN Object: Science and ELA	F = 3.146	p = 0.099 3

¹ Statistically significant at an alpha of 0.05; ² Statistically significant at an alpha of 0.01; ³ Statistically significant at an alpha of 0.10.

). For example, future science and math teachers were similar in visualization but significantly different in language processing speed; future math and language arts teachers were similar in language processing speed but differed significantly in visualization; and future science and language arts teachers were significantly different from each other in both language processing speed and visualization ability. What supports the theory that those who are slow language processors might struggle early on with phonics and memorizing math facts due to their slow processing speed is that future math teachers were those who taught the languages of algebra, geometry, trigonometry, calculus, and/or statistics while future language arts teachers became good at teaching the languages of reading and writing literature.

2. Assessing the Effectiveness of the Blended Arithmetic Curriculum: Studies A, B, and C

2.1. Description of the Blended Arithmetic Program

The Blended Arithmetic Curriculum was developed to help kindergarten to third grade students with Individualized Educational Plans targeting arithmetic computation who also lived in low-performing urban school districts (i.e., students with at least one cognitive exceptionality and one socioeconomic disadvantage).

The Blended Arithmetic Program reverses the standard K–3 method of teaching the entire set of addition facts ad nauseum and not introducing multi-digit addition problems until students master all of their addition facts. To help slow-language-processing students who struggle with memorizing facts but who were hypothesized to be good with higher level conceptional thinking, the Blended Arithmetic Curriculum helps each student learn their arithmetic facts by using a limited quantity of patterned arithmetic facts repeatedly in the conceptually more complex framework of multi-digit addition. Then, the program uses the same process for teaching the concepts and arithmetic facts of subtraction, multiplication, and division.

For example, the first step to teaching addition in the Blended Arithmetic Curriculum is teaching counting up to ten and adding-on by one in simple one-digit format so students can begin to learn how to “count on by one” from a number such as six without having to start counting at one, pass six, and then say “seven” [27].

2 + 1 = 3; 7 + 1 = 8; 9 + 1 = 10

Then after the students master the adding on one skill, students need to take a similar amount of time to understand the reverse is also true:

1 + 2 = 3; 1 + 7 = 8; 1 + 9 = 10.

The second step is to get students to be able to count by ones up to 100—both verbally and through writing. It’s by writing the numbers that students can begin to conceptualize the visual patterns in math. Counting by 10s and 100s helps the students be prepared for multi-digit numbers. Having the students write out their counting also cuts down on a common problem in K–3 education: a whole class may seem to be counting to 100 and beyond, but close observation and listening can easily find that some students are only mouthing nonsense. Furthermore, L. Fuchs et al. [27] found it helped to teach students to be more strategic when counting facts prior to memorizing facts.

The third step to teaching addition in the Blended Arithmetic Curriculum is teaching counting up to ten from zero and counting down from ten to zero. While also teaching the concept of zero and adding zero to any single-digit number.

2 + 0 = 2 7 + 0 = 7 9 + 0 = 9 as well as 0 + 4 = 4 0 + 6 = 7 0 + 10 = 10.

Fourth, students are taught to solve 2-digit addition problems that do not involve carrying.

16	39	21	10	81	19
+11	+10	+17	+43	+15	+60

Fifth, students learn the facts of adding 5 + 5, 9 + 1, and 1 + 9. This prepares students to carry from the ones to the tens column and from the tens to the hundreds columns.

15	19	21	49	95	95
+15	+11	+19	+11	+ 5	+15

Finally, students are taught the limited facts of 7 + 7, 7 + 8, 8 + 7, and 8 + 8, and, as Clements and Sarama [28,29] claim is a crucial step, begin to “subitize” them and not have to count.

17	18	77	88	78	78
+17	+17	+18	+18	+71	+78

Part of the point of choosing 7 + 7, 7 + 8, 8 + 7, and 8 + 8 to teach two-digit by two-digit addition is that “counting” the facts out takes a long time. Thus, students taking longer than their peers makes it easier for a teacher to determine which students have subitized the facts and which ones are still counting. A second reason is that the numbers can be seen by the students as “hard”, and thus being successful makes a student feel proud. The third reason is more subtle. For example, at first glance, in a problem such as:

77

+88

the student may assume that the answer should be 155, but by “carrying” from the ones column to the tens, the 7 in the tens column becomes an 8, and thus the fact to be solved becomes 8 + 8 instead of 7 + 8.

2.2. Introduction to Studies A, B, and C

Oldrieve used Arhar, Holly, and Kasten’s [30] Teacher Action Research process of Action, Observation, and Reflection, to develop and improve what would become the Blended Arithmetic Curriculum. Serendipitously, Oldrieve found that the Blended Arithmetic Curriculum also helped these urban students with learning differences become more proficient in multi-digit arithmetic computation and, in the process, memorize their addition, subtraction, multiplication, and division facts.

Arhar, Holly, and Kasten also noted that, according to Lawrence Stenhouse [30], after a teacher uses Action Research to improve their own classroom curriculum, the next step is to conduct a formal study and disseminate the results. In this paper, studies A and B were used to improve the Blended Arithmetic Curriculum and to demonstrate that it was effective compared to what other teachers were using. Study C was the formal study to prove the effectiveness of the BAC when implemented by teachers other than the author. The research questions for Studies A, B, and C were:

• Is there a difference between the speed and accuracy at which students in suburban and urban schools can complete a worksheet of 42 two-digit by two-digit addition problems?
• Can the Blended Arithmetic Curriculum improve the speed and accuracy of students in urban schools to the degree that their performance matches that of students in suburban schools?
• If the Blended Arithmetic Curriculum proves to be effective at helping students finish their arithmetic computation faster and with more accuracy, would this help students do better on school district and/or state proficiency exams?

The results of the Teacher Action Research Study A and the Replication Study B answered research question one in the affirmative and, to some degree, answered question two in the affirmative. Study C answered questions two and three in the affirmative.

3. Materials and Methods: Studies A, B, and C

3.1. Demographics of the Urban and Suburban School Districts

At the time of Studies A, B, and C, the participating urban schools were located in what were labeled as “empowerment zones”—economically depressed sections of a city for which a Federal Block Grant is awarded to redevelop businesses, schools, and housing.

In the year Study C was being conducted, the Ohio Department of Education [31] reported that the statewide average for median income was USD 29,411 and the number of students receiving reduced or free lunch was 28.8%. For the urban district as a whole, the median household income was USD 21,015 and 80.9 percent of enrolled students were receiving reduced or free lunch while the median household income in the suburban school district was USD 41,960 and the number of students receiving reduced or free lunch was 2.6%.

The racial demographics of the students enrolled in the urban school district as a whole were: 71.01% African Americans, 19.22% European Americans, 8.05% Hispanic Americans, 0.79% Asian Americans, 0.35% Native Americans, and 0.58% Multi-Ethnic. While the students enrolled in the suburban school district as a whole were: 95.61% were European Americans with the remaining students being a mix of 3.22% Asian Americans, 0.68% African Americans and 0.49% Hispanic Americans [31].

3.2. Population of Students in Study A

At the end of the school year, 5 teachers in an urban school district and 5 teachers in an upper-middle-class suburb were recruited to participate in Study A. Both the urban and suburban school district had two participating schools. In the urban school district, one school had 3 teachers assessing their students and another school had 2 teachers assessing their students. In the suburban school district, one school had 2 teachers assessing their students and another school with 1 teacher assessing their students.

Since the class for students with learning disabilities that were using the Blended Arithmetic Curriculum was small to begin with, and not all of the students in the class for students with learning disabilities had Individualized Education Plans that mandated that they should be working on addition, only 7 students with learning disabilities participated in the assessment in the first year of Study A. To build up a statistical database for the urban students with learning disabilities, the same assessment using the same protocol was administered to the students working on addition at the end of three subsequent school years.

3.3. Population of Students in Study B

The results of the Teacher Action Research convinced Scipione, the president of a local internet software firm, to team up with Oldrieve. Scipione wanted to write a federal grant to develop the Blended Arithmetic Curriculum into a commercialized curriculum, but first Scipione wanted to conduct a replication study to determine whether the results of the Urban General Education students in Study A were generalizable to more of the school district. Study B included eleven teachers and their students nested within four schools nested within the same “empowerment zone” of low socio-economic status and an ethnically diverse population in Study B, though they were not the same schools.

For the Study B replication study, permission slips were collected from 38 students, and a review of the personal records of the 38 students revealed that all the participating students were African Americans and were receiving either reduced or free lunches. Since only 38 students enrolled in 11 classes brought in permission slips, it can be deduced that there were an average of 3.45 students per class who brought in a permission slip.

3.4. Population of the Students in Study C Demographics of the Urban School District Participation in the Pilot Study

The results of Studies A and B did help Donald Scipione and Richard M. Oldrieve win a federal Small Business Innovation and Research (SBIR) grant from the U.S. Department of Education to develop the Blended Arithmetic Curriculum (BAC) into a curriculum. After the completion of the BAC’s teacher plan books, student workbooks, and conceptual online games for a 16 chapter unit on addition, a one-year test pilot program was planned.

As originally organized, the population of Study C was supposed to include second graders enrolled in general education and special education classes in the same suburban school as the original Teacher Action Research project of Study A. But due to some unfortunate events, the teachers in the suburban school district dropped out before the end of the first quarter.

One of the original urban schools assessed in Study B agreed to participate; the others did not. Six general education teachers participated in the study, as did two special education teachers. The students in this school differed from the overall school district because the school was located in a mixed neighborhood at the borders of the Asian Community, Eastern European neighborhoods, and an African American neighborhood. Due to the low percentage of parents who agreed to let their students participate in Study B, the research team decided that one possible reason for this low participation rate was the complexity of the permission slip and the fact that it requested researcher access to detailed personal data such as demographic, academic, assessment, and free lunch status. Consequently, the team thought it would be best to merely ask parents permission to test pilot the Blended Arithmetic Curriculum and take unit assessments.

As was feared, several teachers in the participating urban school agreed to participate but did not fully implement the Blended Arithmetic Curriculum. Students in one of the general education classrooms and the students in both of the special education classrooms were only taught the Blended Arithmetic Curriculum when their teachers were being observed by Oldrieve and/or when they were teaching a demonstration lesson. The results of the students in both special education teachers’ classes will not be presented in this article, though the results of the students in the half-hearted general education teacher’s class were included in the results for the general education students.

The other five teachers utilized the Blended Arithmetic Curriculum in full. They attended once a month interactive teacher-in-services where teachers would give their opinions on what was working and what was not, and they gave suggestions as to how to improve the lessons. Several teachers even volunteered to be observed by their peers when they were teaching a lesson to their students and then allowed their colleagues to critique their teaching during the next inservice.

3.5. Teaching Methods Used in Studies A and B

For the Urban General Education and Suburban General Education students given the end-of-the year post-assessment, Oldrieve did not examine the actual teaching materials used by the participating teachers, nor were any of the lessons taught by the participating teachers observed by Oldrieve. Thus, the teaching materials and methods employed by the teachers are based on a blackbox approach. Nonetheless, the Ohio Department of Education’s 2nd grade math standards still require that second-grade students be taught 2-digit by 2-digit addition.

3.6. Teaching Methods Used in Study C

The Addition Unit of the Blended Arithmetic Curriculum begins by focusing on adding-on by one and zero in simple one-digit format so students can begin to learn how to count-on [26,27,28].

2 + 1 = 3; 7 + 1 = 8; 9 + 1 = 10, as well as 1 + 4 = 5; 1 + 6 = 7; 1 + 9 = 10

Next, students learn to solve 2-digit addition problems that do not involve carrying. Third, they move on to adding 5 + 5, 9 + 1, and 1 + 9 to create the need for carrying from the one’s to the ten’s column and from the ten’s to the hundred columns.

15	19	21	49	95	95
+15	+11	+19	+11	+5	+15

Next to get the students to subitize facts [28,29] and not just count them out, students are taught the limited facts of 7 + 7, 7 + 8, 8 + 7, and 8 + 8, and begin to use them in 2-digit by 2-digit addition problems:

27	48	77	88	78	78
+17	+17	+18	+18	+71	+78

From here, students are first taught a small set of conceptual facts, such as even 2’s addition, and then these facts are mixed into the set of 2-digit by 2-digit addition problems as well as word problems so that they get practice using them in context instead of solely on flash cards. Examples would include:

10	96	78	82	74	72
+12	+12	+12	+16	+72	+78

By the end of the Addition Unit, students will have learned all their addition facts in the context of 2-digit by 2-digit computation and word problems.

3.7. Mechanics of Assessing Students in Study A

For the post-assessment of Study A, all teachers were given a worksheet comprised of 42 two-digit by two-digit addition problems. Divvied up between the 42 problems were a random mix of problems that required carrying and non-carrying and that contained a mix of single-digit numbers in which 0 and 1 were added, as well as all the addition facts greater than 1 (see Appendix A for the worksheet). The teachers were also given a protocol on how to administer and time the assessment (see Appendix B) and a set of prepared instructions they were to read to students (see Appendix C). The teachers then gave the students the worksheets and recorded how long a given student took to complete the paper, rounded down to the minute as opposed to minutes and seconds. Then, when a student handed in their completed paper, the teacher would give the student a wordfind to keep them busy until their classmates finished. There was no time limit on how long a student could take to complete the paper.

3.8. Mechanics of Assessing Students in Study B

For the post-assessment of Study B, all teachers were given the exact same worksheet comprised of 42 two-digit by two-digit addition problems as was used in Study A. One change to the protocol was that both the classroom teacher and researcher A served as proctors. A second change was to impose a 30 min time limit to ensure all participating classes could be easily scheduled in buildings employing 40 min class periods and to leave time for taking attendance, giving instructions, and for distributing sharpened pencils, the 42-problem assessments, and word finds for those who finished early.

3.9. Mechanics of Assessing Students in Study C

In Study C, there were three types of assessments that were used.

First, chapter assessments looked similar to those used in Studies A and B in that they measured 2-digit by 2-digit addition fluency and accuracy, but in contrast to the formal assessment, the chapter assessments focused only on facts taught in the given chapter plus facts that had been introduced in previous chapters. An example of a chapter assessment for Study C would be the Chapter 8 exam, which covers the facts of adding on 2 to even numbers such as 0 + 2; 2 + 2, 4 + 2; 6 + 2; and 8 + 2 and their inverses of 2 + 0, 2 + 4, etc. Thus, on the Chapter 8 exam, about 50% of the problems contained facts from the even 2’s chapter, and the other 50% contained a mix of problems from previous chapters, such as adding on 1, adding on zero, and the limited facts of 5 + 5, 7 + 7, 7 + 8, 8 + 7, and 8 + 8.

Second, the end-of-the year assessment had been intended to be the same as the end of the year assessment used in Studies A and B. Because of the same beginning-of-the year difficulties that caused the teachers in the Suburban School District to drop out of the test pilot study, most of the teachers only completed around 13 or 14 of the 16 chapters in the program. Thus, for the final post-test of the year, students in a class were given a specially designed worksheet that contained a mix of addition facts—only those facts that taught in the chapters that were completed by the students in a given teacher’s class. For example, if Teacher A had taught and students had mastered all the facts taught up through chapter 12 out of the 16 chapters in the addition plan book, the assessment for students in Teacher A’s class included all the facts taught in chapters 1 through 12 but did NOT include any facts from chapters 13 through 16. Similarly, if Teacher B felt their students had mastered the material up through chapter 14, then chapters 1 through 14 were included in the assessment given to Teacher B’s class, while chapters 15 and 16 were left off.

The reasoning was that the students’ speed and accuracy would probably change if they were given assessments that included facts that they had not learned. While there was hope that the students would perform well on the school district’s assessments due to their achieving speed and accuracy on the facts and 2-digit problems from the twelve chapters they had completed—thus breaking the vicious cycle of learned helplessness from consistent failure in math and in turn replacing it with a cycle of growth, resilience, and a mentality of “success breeds success”.

Third: The school district’s second grade proficiency exam was used because the school district (and presumably the U.S. Department of Special Education) was most interested in whether the second-grade students who were being taught by the Blended Arithmetic Curriculum would pass the urban school district’s 2nd grade proficiency exam at a higher rate than previous years, using the same exam and protocols as every other 2nd grade student enrolled in the urban school district.

4. Results of Studies A, B, and C

4.1. Results of Studies A and B

In both Study A and Study B, a visual analysis highlights differences and similarities between the various scores obtained by the UGE students in both Studies A and B. Figure 1 and Figure 2 show that both the students in UGE Study A and UGE Study B displayed an extreme bi-modal distribution—there is a cluster of students who scored above 75 percent accuracy, there is another cluster of students who scored between 10 and 40 percent, and there is an equally distributed valley of students who scored somewhere in between.

Conversely, Figure 3 and Figure 4 show that the students in SGE and ULD are right-skewed, with most students scoring above 80 percent and a small tail of students scoring under 80 percent. The difference in accuracy between the Urban General Education students in both Study A and Study B and those who were in Suburban General Education or the Urban Learning Disabled shows that the urban students are not mastering their addition facts in the grade level in which they are expected to master these facts. This will be very problematic when they need to use these facts to master subtraction, multiplication, and division.

Similarly, a visual analysis highlights differences and similarities between the completion times. Figure 5 and Figure 6 show that both the students in UGE Study A and UGE Study B displayed a relatively flat distribution of times that ranged between 5 and 30 min, with a huge jump when the time limit was reached. Note that in practice, none of the teachers in Study A allowed the assessment to go on until all students completed all of the problems. Uncompleted problems were counted as wrong. Thus, in Study B, a time limit of 30 min imposed a standardized stop time the students knew in advance; the time limit also ensured that the assessment’s instructions, distribution, administration, and collection could take place within a typical 40 min classroom period.

Conversely, Figure 7 and Figure 8 show that the students in SGE Study A and ULD Study B are left-skewed, with most students taking less than 10 min to complete the 42-problem worksheet. Again, this contrast between the Urban General Education students and the Suburban General Education and Urban Learning Disability class shows that not only are the students not mastering their math facts and multi-digit addition, they are taking much longer to complete the problems, pointing to the fact that the students are counting out the facts and not subitizing them as per what is necessary to follow a proper learning trajectory to higher level math and doing well in science [1,28,29].

Four caveats to the statistical analysis must be mentioned: First, the bi-modal distributions for accuracy in UGE Study A and UGE Study B could be problematic when they are used in one-way ANOVAs [32]. Second, because a series of eight one-way ANOVAs were used to compare groups, a Bonferroni adjustment was made so that the expected alpha was lowered from p = 0.05 to p = 0.00625 [32]. Third, in Study A, one urban general education teacher failed to record times, and thus the N for the percentage correct in UGE Study A is larger than the N for the times in UGE Study A. Fourth, there was no time limit in the Study A administration of the assessment while a time limit of 30 min was placed on the Study B administration so that a 40 min schedule of instructions, assessment, and collection per classroom could be maintained.

As noted above, the students in UGE Study B appear to have been significantly faster than those in UGE Study A, and statistical analysis indicates that in general, the visual interpretations hold up (see

Table 2. PERCENT correct for students in Urban General Education (UGE)-Studies A and B, Suburban General Education (SGE)-Study A, and Urban Learning Disabilities (ULD)-Study A.

Class Type	Number Students	Mean	Median	Std Dev	Min	Max
UGE-A	88	56.17	57.14	33.17	7.14	100
UGE-B	38	58.59	66.67	34.38	2.38	100
SGE-A	55	89.70	95.24	17.88	19.05	100
ULD-A	27	88.62	92.86	12.20	42.86	100

,

Table 3. TIME in minutes for students in Urban General Education (UGE)-Studies A and B, Suburban General Education (SGE)-Study A, and Urban Learning Disabilities (ULD)-Study A.

Class Type	Number Students	Mean	Median	Std Dev	Min	Max
UGE-A	76	23.74	22	11.12	6	63
UGE-B	38	17.11	14	8.67	5	30
SGE-A	55	7.76	7	3.88	3	30
ULD-A	27	5.48	5	3.68	1	16

,

Table 4. PERCENT: One-way ANOVA comparisons with better results listed first.

Class Type and Study	df	F	p
UGE-B vs. UGE-A	1.201	0.189	0.664
SGE A vs. UGE-A	1.141	47.475	0.000 1
ULD-A vs. UGE-A	1.113	24.688	0.000 1
SGE vs. ULD-A	1.80	0.079	0.780

¹ Statistically Significant at an alpha of 0.05 after the Bonforonni adjusted alpha level of p = 0.00625.

and

Table 5. TIME: One-way ANOVA comparisons with better results listed first.

Class Type and Study	df	F	p
UGE-B vs. UGE-A	1.189	26.763	0.000 1
SGE A vs. UGE-A	1.129	104.119	0.000 1
ULD-A vs. UGE-A	1.101	69.666	0.000 1
ULD-A vs. SGE-A	1.80	94.763	0.000 1

¹ Statistically significant at an alpha of 0.05 after the Bonforonni adjusted alpha level of p = 0.00625.

). Unfortunately, the students in UGE Study A and UGE Study B had statistically similar accuracy.

In contrast, the students in SGE Study A were significantly faster and more accurate than those in UGE Studies A and B. Similarly, the students in ULD Study A were significantly faster and more accurate than those in UGE Studies A. Most importantly, the students in SGE Study A and those in ULD Study A had times and scores with means and medians that were nearly identical.

4.2. Results of Study C

At the end of the first year of the pilot study, the Urban General Education students, who were taught using the Blended Arithmetic Curriculum, were administered a 42-problem worksheet that was modified to include the chapters the teacher had taught and the students had mastered. As noted above, the purpose was to see how well the students learned how to accurately compute multi-digit addition problems and how well they had memorized the facts to be fluent without counting. Figure 9 shows the percentage of correct problems for the students in Urban General Education (UGE) who test-piloted the Blended Arithmetic Curriculum in Study C. Then, visually compare them to Figure 1 for students in Urban General Education in Study A, Figure 2 for Urban General Education in Study B, Figure 3 for Suburban General Education in Study A, and Figure 4 for Urban Learning Disabilities in Study A.

For the students in Urban General Education classes who used the Blended Arithmetic Curriculum during Study B, 68% of them scored 80 percent or above. This means they were not quite as accurate as the 85% of students in Suburban General Education in Study A who scored above 85% accuracy or the 80% of the students in Urban Learning Disabilities in Study A. Nonetheless, the bilateral nature of the curve in the Urban General Education classes in Studies A and B is gone. Additionally, 68% scoring above 80 percent is far more accurate than the students in Urban General Education in Study A, for whom only a third scored 80% or higher, and UGE students in Study B, for whom only 40% scored 80% or higher.

Similarly, Figure 10 shows that the students in Urban General Education in Study C took far less time to complete the worksheet than students in UGE classrooms in Study A (see Figure 5) and in Study B (see Figure 6). In fact, the bilateral nature of the curve is gone for these students in UGE classrooms in Study C, and 59% of them took 10 min or less to complete the worksheet compared to the students in UGE classrooms in Study A, where only 10% of the students completed the computation assessment in 10 min or less, and in Study B, where only a third of the students finished in 10 min or less. On the other hand, in Study A, 85% of the students in Suburban General Education classrooms (see Figure 7) and 93% of the students in the Urban Learning Disabilities classroom (see Figure 8) completed the assessment in 10 min or less. Thus, most of the students seem to be on their way to subitizing their math facts and are following a proper learning trajectory towards subtraction, multiplication, and division [1,28,29]. Nonetheless, even the improved results support Dr. Donald Scipione’s goal that including a computer/tablet-based program would allow for individualized instruction for those needing more practice repetitions on a given chapter of addition facts. Thus, students could be held back in order to achieve 85% accuracy in under 10 min before moving onto the next chapter. For example, the computer could help a student master 3’s addition before moving the student to 4’s addition.

More importantly, as shown in Figure 11, the second-grade students in Urban General Education in Study C tripled the number of students who passed the school district’s math proficiency exam to just under 50%, compared to 17% or less passing for the four previous years. Thus, demonstrating that mastering addition is beneficial to doing well on standardized assessments.

5. Discussion of Studies A, B, and C

5.1. Discussion of Study A

Visual (See Figure 1, Figure 2, Figure 3 and Figure 4) and statistical (See Table 2, Table 3, Table 4 and Table 5) analysis of the results indicates that there are distinct differences between how quickly and accurately the students in urban general education (UGE) classrooms could complete the 42-problem assessment as compared to the students in suburban general education (SGE) and students in urban with learning disabilities (ULD) classrooms. Nevertheless, in Study A, the practical differences between the students in UGE and those in SGE and ULD seem to be the most important finding of this study.

The distribution pattern in the upper-middle-class suburb shown in Figure 3 was that 85 percent of the students scored over 80 percent correct on the assessment. Similarly, as shown in Figure 7, 85 percent of the upper-middle-class students completed the worksheet in 10 minutes or less. Amongst these general education students, intervention specialists could easily spot students who were outliers who should be targeted for further testing, possible special education placement, and individualized instruction by a teacher.

Conversely, the distribution of scores for the students in the Urban General Education (UGE) classroom on the assessment in Study A was entirely different. As shown in Figure 1 there was a clump of students who scored between 75 and 100 percent; there was another clump who scored between 10 and 40 percent; and there was a huge valley in between. Intervention specialists would be hard-pressed to determine who should be recommended for further testing and placement because, if anything, the outliers could be considered the small number of students in UGE classrooms who completed the test as accurately and as quickly as the students in SGE classrooms. Furthermore, the small number of students in UGE who completed the test in 10 min or less in Study A indicates that a good number of those who did score well did so by counting—a seemingly problematic strategy that in the short term makes it more difficult for these students to learn multi-digit subtraction, multiplication, and division, and in the long term makes it almost impossible for these students to factor equations in algebra, balance equations, find limiting reagents in chemistry, and do complex computation.

Finally, the students in the Urban Learning Disabilities (ULD) classroom who used the Blended Arithmetic Curriculum matched the distribution and statistical patterns of the students in suburban general education (SGE) classrooms. For example, in Study A, the students in SGE were slightly but not significantly more accurate than the students in ULD. In contrast, the students in the ULD classroom were significantly faster (p = 0.000), though only slightly faster than the students in SGE classrooms. These findings would be consistent with students with learning disabilities tending to be more impulsive than students who are not diagnosed as having learning disabilities. Because of the black box design, the exact reasons why the students in ULD in Study A did so well cannot be explained through data analysis. Nevertheless, what seems important is that this study reveals that not only is there a practical difference between students in urban general education (UGE) classrooms and students in suburban general education (SGE), but that this practical difference can be overcome. There is evidence that students on IEPs who were assigned to urban learning disabilities (ULD) classrooms for math did similarly to the students in suburban general education (SGE).

5.2. Discussion of Study B Compared to Study A

Study B focused on determining whether the results of Study A were still relevant after three additional years were needed to teach the Blended Arithmetic Curriculum to enough students with Individualized Education Plans for Addition from the Urban Learning Disabilities class, and then assess them, to make a meaningful statistical comparison to the Urban General Education and Suburban General Education classes in Study A.

Graphs 1, 2, 5, and 6 show that there were some improvements in both accuracy and time for the UGE students in study B compared to the UGE students in study A, but Table 2 and Table 4 show that only on time were there statistically significant improvements from Study A to Study B. A visual comparison between graphs 1, 3, and 4 show that the SGE and ULD students from study A were visually faster than the UGE students from Study B. Nonetheless, it is the combination of accuracy and speed that matters for being ready for computing K-3 subtraction, multiplication, and division problems and then even more importantly using this computation ability in algebra, geometry, statistics, chemistry, physics and higher levels of mathematics and science.

5.3. Discussion of Study C

In Study C, the Urban General Education students who were taught using the Blended Arithmetic Curriculum were given a 42-problem worksheet at the end of the academic year that was modified to include the chapters a given teacher had taught and their students had mastered.

See Figure 9 for the percentage correct the students in Urban General Education (UGE) scored after test-piloting the Blended Arithmetic Curriculum in Study C and visually compare them to Figure 1 for students in Urban General Education in Study A, Figure 2 for Urban General Education in Study B, Figure 3 for Suburban General Education Study A, and Figure 4 for Urban Learning Disabilities in Study A.

For the students in Urban General Education classes who used the Blended Arithmetic Curriculum (BAC) during Study C’s academic year, 68% of them scored 80 percent or above. This means they were not quite as accurate as the 85% of students in Suburban General Education in Study A who scored above 85% accuracy or the 80% of the students in Urban Learning Disabilities in Study A. Nonetheless, for the Study C students who used the Blended Arithmetic Curriculum, the bilateral nature of the curves in the Urban General Education classes who did not use BAC in Study A and B is gone, plus 68% scoring above 80 percent is far more accurate than the students in Urban General Education in Study A, for whom only a third scored 80% or higher, and UGE students in Study B, for whom only 40% scored 80% or higher.

Similarly, Figure 10 shows that the students in Urban General Education using BAC in Study C took far less time to complete the worksheet than the students who did not use BAC in UGE classrooms in Study A (see Figure 5) and Study B (see Figure 6). In fact, the bilateral nature of the curve is gone for these students in UGE classrooms in Study C, and 59% of them took 10 min or less to complete the worksheet compared to the students in UGE classrooms in Study A, where only 10% of the students completed the computation assessment in 10 min or less, and in Study B, where only a third of the students finished in 10 min or less. On the other hand, in Study A, 85% of the students in Suburban General Education classrooms (see Figure 7) and 93% of the students in the Urban Learning Disabilities classroom (see Figure 8) completed the assessment in 10 min or less.

More importantly, as shown in Figure 11, the second-grade students in Urban General Education in Study C tripled the number of students who passed the school district’s math proficiency exam to just under 50%, compared to 17% or less passing for the four previous years. Further study would be needed to determine if bringing 85% of the Urban General up to 85% accuracy through individualized human and/or computer instruction would result in higher percentages of students passing mandated proficiency exams.

5.4. Discussion of Limitations to Studies A, B, and C

As was mentioned earlier, the research design conceptualized in Study A reflects a “black box” approach. There was no systematic comparison of teaching materials or observation of teachers teaching lessons. Beyond the overall limitations of the black box design, this particular study had specific design flaws:

• The assessment privileges students who have been drilled using worksheets with a similar style.
• For ease of teacher use, teachers were asked to round down to the nearest minute (for example, 7 min and 59 s would be rounded down to 7 min). Standard statistical methodologies would have required rounding down to the nearest whole minute when the recorded seconds were from 0 to 29 s and rounding up to the nearest whole minute when the recorded seconds were from 30 to 59 s. Since some teachers only recorded to the minute during the 1991 sampling period, it would be impossible to recalculate the student times using the standard rounding procedure. Consequently, all subsequent administrations of this assessment have maintained the same rounding criteria as the original study.

Nonetheless, it is important to point out that this determination to always round down to the nearest minute makes retrospective sense when results revealed that the mean time for the students in Urban General Education (UGE) classrooms was nearly double that of the students in Suburban General SGE classrooms. Furthermore, the task of timing and writing down the times to the second for three students in an SGE class who finished between 5:05, 5:39, and 5:58 may be doable, but if seven students rushed up and turned in their papers in rapid fire succession, writing down “6” at the top of seven papers would be far easier and more accurate than expecting a teacher who may be inexperienced with stop watches to glance down, look at the minutes on a stop-watch, see 6:05, memorize it, and write down 6:05, then click on the stop-watch for the next student, look down to see and memorize 6:09, and then write down 6:09, and so on and so forth for the next five students who turned in papers before the time switched over to 7:00. And in a class of 30 students in suburban general education (SGE) classrooms, where the vast majority of students had become fast and accurate, seven students turning in their papers within a minute could easily happen in the minute before and after the top of the bell curve for time of completion was approached and passed.

Importantly, the original research design had several other advantages:

• The assessment of a worksheet of 2-digit addition problems was simple to administer, grade, and analyze.
• In the United States, most second-grade textbooks, curricula, and proficiency exams expect second-grade students to learn their addition facts and master 2-digit by 2-digit addition. Furthermore, teachers, school administrators, politicians, and the general public would probably agree that second graders ought to master 2-digit by 2-digit addition with carrying—though admittedly, not every school district is teaching the same addition algorithm. Consequently, the 2-digit computational worksheet assesses what most second-grade students are expected to know, regardless of how it was taught or how the student chooses to solve the problem.
• The protocol for administering the test was relatively simple for a classroom teacher to use and for a future independent researcher to replicate. Because the point of the initial assessment was to determine if there were huge differences in how fast and accurate the students in Urban General Education (UGE) were compared to the students in Suburban General Education (SGE). If the differences in means had been 30 s for time of completion and only a few percentage points on accuracy, then why bother changing how and what the teachers were teaching?

5.5. Overall Discussion

One major point of this article is that teaching arithmetic computation to high levels of accuracy and speed in elementary school is vital to ensuring that students from low socio-economic backgrounds—especially those who have an aptitude for reading and writing complex literature and/or solving complex algebra, geometry, trigonometry, and calculus problems—will grow into their gifts and potentially become high school math teachers or language arts teachers, as per the results of Richard M. Oldrieve and Cynthia Bertelsen’s study [7]. This is because the college students studying to become future high school math and language arts programs were found to have had RAN of Objects times slower than the grand mean of their college classmates, which is often a marker for being a slow language processor. Additionally, a statistically significant multiplier was found that indicated that for each second slower that a university or junior college student took to complete the RAN of Object’s assessment, the higher the odds the student would receive a good grade in the class where they were assessed.

Furthermore, it is also important to point out that in Oldrieve and Bertelsen’s study, future early childhood teachers were evenly split between the four quadrants defined by the PVST-R and the RAN of Objects, indicating early childhood teachers as a whole have a broad range of visualization and language processing gifts.

In contrast, in a follow-up study of junior college and university students [33], Richard M. Oldrieve and Cynthia Bertelsen were disappointed to find that future intervention specialists, who are expected to teach all subject areas well, were found to be UNEQUALLY distributed amongst the four quadrants. Instead, seven out of ten present or future intervention specialists were found in the fast processing and low PVST-R quadrant—where only three future high school math, science, and/or language arts teachers were located. Which might explain why in Study C, one urban general education teacher and two intervention specialists did not want to fully implement the Blended Arithmetic Curriculum.

The overall point of this article is that there are ways to improve arithmetic computation in elementary school to ensure that individuals with the intellectual gifts needed to excel in the field of computation are prepared to learn high school-level algebra, calculus, and statistics so that they are ready to master high-level computation when they reach graduate school.