Jones Software Corporation iLikeMath www.j-softech.com
KJones/KbJones Revised 8/10/2015
DDodson/RBrock 5/31/2011
Jones Software and “iLikeMath” E-Flash Cards Yield Significant Results
Dr. Dana Dodson Mathematics Education Indiana University, Northwest Gary, Indiana
Dr. Rochelle Brock Executive Director, Urban Teacher Education Program Indiana University, Northwest Gary, Indiana
1
Introduction
By the time a sixth grader starts his new year in school, he has been assigned over 720 times: “: “Practice your math facts” for homework. This dreaded malady prompted Kevin Jones, PRESIDENT AND CO-FOUNDER OF JONES SOFTWARE CORPORATION, to initiate a call to for action to identify an innovative solution for students K-12 who have struggled with learning their mathematics facts of addition, subtraction, multiplication, and division. This paper is a presentation of the current research on why our students struggle and the importance of success in learning these mathematics facts. This paper also presents the results of an iLikeMath … (formally E Flash Cards) Pilot Program that was conducted in two sixth grade classes in Gary Community School Corporation, which resulted in significant student gains in math.
In October, 2007 while employed as a Senior Financial Advisor with Merrill Lynch, located in Merrillville, IN. Bruce Quint, Merrill Lynch-Resident Director, instructed Kevin to build relationships in underserved communities primarily located in North West IN, in order that Merrill Lynch could support these communities with grants, monetary awards or needed resources. Kevin placed in motion a game plan where Key Community Leaders were identified and recruited to develop opportunities for NW Indiana’s underserved communities. Working in tandem with these community leaders, Kevin became moved and motivated by the challenges that educators were facing due to a lack of resources, technology, manpower, etc. He envisioned how best to leverage Merrill. Lynch’s 2
unlimited resources in addressing the concerns that educators were facing on a daily basis. Afterschool meetings were held with educators to understand and document specific issues and concerns. A majority of the educator’s voiced concern with low student efficiency in math, reading and other core subjects.
In the coming months, Kevin in unison with the educators would develop a
strategy and obtain commitments from all parties involved in creating a model to maximize each students’ long term potential, while focusing on improving their retention memory and learning capability.
Dr. Rochelle Brock, Executive Director, Urban Teacher Education Professor, Indiana University, Northwest and Kevin are professional associates, who Kevin expressed concern with the state of education in the community. Dr. Brock introduced Kevin to Dr. Dana Dodson, Mathematics Professor, Indiana University Northwest, who committed to help and proceeded to conduct multiple interviews with key math coordinators within Gary School Corporation. The objective of the interviews focused on students whose current math placement were below state standards and were testing below thirty percent efficiency. Data suggested that many students struggled with multiplication facts, fluency, along with basic addition and subtraction. In addition, procedural division was below twenty percent efficiency. To validate the antidotal evidence from the interviews, Dr. Dodson and Kevin Jones obtained approval from the Superintendent of Gary School Corporation to conduct math assessment examination of students in grade level third, fourth, fifth and sixth. Dr. Dodson’s results from the assessment examination is listed in this paper. Dr. Dodson conclusion /recommendation to the superintendent centered on students spending twenty to thirty minutes each day in the school computer lab practicing and building upon their individual’s math facts and fluency skills.
3
Working closely with IUN and Gary School Corporation presented Kevin with a unique challenge, where to identify and obtain additional
resources without any city or state funding? Kevin began to recruit professional relationships from his past to bring experience and ideals to this equation. Kenya Brooks, represented one of the first recruits to volunteer and serve in this capacity as an after school mentor for these students. Kenya, theorized, “Who is responsible for a student’s education, when resources, technology, funding and educators are limited? Without a definitive answer to Kenya’s abstract question, Kevin focused on the recruitment of others professional associates to serve as mentors and tutors in the school district. The tipping point was that the school district had no student workbooks or worksheets for mentors to tutor student’s afterschool. This dilemma birthed the creation of Global Flash Learning, Inc. GLF is an educational software company, who vision is to build math skill efficiency for students, in disadvantaged communities.
In 2008, Kevin incorporated Global Flash Learning, Inc. Based on learned data from educators, the software would be
engineered to assist students with drilling and memory exercises in math. Kenya Brooks, increasing her role from mentor to company employee, would be instrumental in deploying the software to school districts in IL and IN. After a successful deployment and several years of experience, GFL discounted business due to management and philosophical difference.
. 4
Jones Software Corporation (JSC) was incorporated in July, 2015, by Kevin Jones and Kenya Brooks, both former associates of Global Flash Learning. Building upon GFL experience, JSC’s expanded their vision to provide “Adaptive Learning Tools & Resources”, to remedial students to improve the mathematics facts and fluency of students that have struggled to meet state educational objectives. JSC, software incorporates a Multi-Tier Support System, based on procedural steps and is a Cloud Base – Algorithm System developed in multiple languages. Employing, Early Educational College Professors, Educators and Web-Design Developers who are qualified to build software based on behavior driven educational results. JSC is uniquely positioned to bring to the market resources to meet specific educational needs.
Kenya Brooks, President and Co-Founder, has knowledge in the industries of Global Trading, Medical Technology, Pharmaceutical Sales, and Television Media coupled with Global Public Relations. Kenya majored in Mass Communications and Political Science from University of South Florida. She has worked in a leadership and mentoring capacity with several organizations including the Rotary Club Chicago 1, Executive Club of Chicago (Young Leaders), James Jordan Foundation (A-team), Advisory Council Cook County Sheriff Department, Women in Christian Media, Chicago Foundation for Women, Annual Bud Biliken Day Parade, Chicago City Classic, The World Trade Center Chicago, International Chamber of Commerce, Stedman Graham and Associates Leadership training, Cook County Advisory Women's Council and the State of Illinois (of the Late) Judy Barr Topinka Women's Committee. Kenya is a sitting Board Member of Illinois
Fatherhood Initiative, and former Executive Director of the Family Christian Center. Kevin Jones is JSC’s Chief Executive Officer and has an extensive background in financial planning. He is a sitting Board Director for
5
the Grant Park Music Festival, located in Chicago, The ECIER Foundation, located in Crown Point IN, and the Urban Teacher Education Program Policy Board at Indiana University Northwest. Kevin Jones is a 2000 graduate of DePaul University and has worked as a Senior Wealth Manager Advisor with Merrill Lynch, Financial Advisor, Edward Jones Investments, and A Regional Sales Manager with AT&T and an Administrator with Xerox. JSC has enlisted consultants in education and business who bring over 100 years of experience and knowledge. Currently JSC’s math software is operating in multiple states encompassing School District 159 located in Matteson, IL. Banner Alternative School District located in Chicago, Gary School Corporation, Gary IN., Alternative Achievement Academy, Posen IL. School District 152.5, located in Hazel Crest, IL. and a number of other school districts. JSC is committed to educational advancement of all children and making a difference in communities with the greatest need. As such they have developed a relationship with the Gary Crisis Center in Gary, IN. The Gary Crisis Center offers Alternative Housing, as an emergency, short-term shelter for runaway, homeless, pushed out, abused or neglected youth, between the ages 6 to 18. The Electronic Flash Cards software was donated to the Gary Crisis Center and training was provided to the staff so residents would have the opportunity to work on math skills.
6
iLikeMath (formally…Electronic Flash Cards
The Electronic Flash Cards include single and double-digit mathematics facts using addition, subtraction, multiplication and division of positive and negative whole numbers. Students are encouraged to work accurately and fluently using the Electronic Flash Cards, which targets the student’s problem areas, monitors their progress and includes a real-time record of success for each session the student is working. In a class of 25+ students, not all students would be on the same level of mathematics fact proficiency. Hence, the Electronic Flash Cards program has flexibility for the teacher to set the entry level for each student. Students who lack mental fluency to add double digits would be able to start in the program with adding double digits. Students needing single digit multiplication fluency would be able to start in this area of the Electronic Flash Cards program. As the program monitors the students’ progress and sends a report to the teacher, it also moves the student forward in the program to practice and gain fluency in all single and doubledigit mathematics facts involving addition, subtraction, multiplication, and division. Reports of the student’s progress are sent to the teachers via email, assessment reports can be archived for future documentation or be sent to the classroom printer at the end of each session.
Memorizing mathematics facts are abstract and meaningless without the proper grounding in the operational definition and concrete manipulation of materials to ground a student in the comprehension of mathematics. Before a student can gain fluency and appreciate the abstract fact that 3 + 4 = 7 they must spend time arranging 3 items and 4 items to create the total of 7 items. Some students will gain fluency with this fact by doubling the 3 and adding one (3 + 3 + 1 = 7). The JSC program is not intended to shortcut the learning 7
time necessary for a student to gain the concrete understanding of mathematics. The curriculum in early elementary schools works to develop a student’s understanding of the mathematics operations. However, students are not building the fluency needed to progress into higher levels of mathematics. In addition, teachers are slowed down when students take time to count on their fingers or look up the product from a multiplication table.
The “Fluency” Goal
The National Research Council’s report, Adding It Up, describes procedural fluency to include the skills necessary to carry out mathematical procedures flexibly, accurately, efficiently and appropriately (2001). Fluency in the 21st century has replaced “mastery” in the standards and can make the difference between a student’s ability to succeed or struggle in mathematics class. Fluency and fluently appears 29 times in the Common Core State Standards for mathematics in the K-12 curriculum indicators, which is a gauge of its educational importance.
The Common Core State Standards (CCSS) expect students to fluently know the sums of single digit numbers by the end of grade 2 and the products of single digit numbers by the end of grade 3. By the end of fourth grade, students are expected to fluently add and subtract multi-digit whole numbers and by the end of fifth grade students are expected to multiply multi-digit whole numbers. Yet students continue to struggle through middle school, high school and college courses because they have failed to fluently learn their addition, subtraction, and multiplication and division facts. This is not an easy task. After the facts have been discovered, 8
conceptualized and applied, they must be memorized and retained for fluent and immediate use. Many students fall short of the “fluency” goal (Caron, 2007). Robert Gagne (1983) recognized the need for fluency and emphasized that the processes of computation that underlie all problem solving must be “not just learned, not just mastered, but automatized.” (pg. 18) In this study and in agreement with the work of R. Oldrieve (n.d.), “fluency” is defined as the combination of computational rate and accuracy. Students without multiplication-fact fluency spend more time determining routine answers and less time on more meaningful applications. Students who know their facts build on these fundamental concepts, ultimately benefiting their subsequent mathematical development (Wallace and Gurganus, 2005).
Fluency in the Mathematics Standards
The third grade core standard is the foundation upon which higher-level problem solving skills depend. Everyday life necessitates the mastery of multiplication facts, from purchasing a weekly lunch ticket to determining how many square feet of carpet is needed to cover rooms of varying sizes in a newly constructed home. Subsequent applications of mathematical concepts will be uncertain unless the foundation is solid. Elementary teachers across the nation are responsible for this difficult task. The methods used to teach multiplication facts are determined in large part by a school’s curriculum. Some of the most common practices include: time tests, flash cards, dice or number cubes and mathematics manipulatives. Most educators agree that fluency is best developed through drill and practice (Hasselbring, Goin, and Bransford, 1988). Students who have failed continually for years at remembering multiplication facts soon develop a resistance to even try (Caron, 2007). 9
Without the ability to retrieve facts directly or automatically, students are likely to experience a high cognitive over-load as they perform a range of complex tasks. Additional processing demands result in inefficient methods such as counting (versus fluency) often lead to procedural errors (Cummings and Elkins, 1999). Potential difficulties extend well beyond operations on whole numbers. Finding common multiples when adding fractions with unlike denominators or factoring algebraic equations are two examples from secondary-school mathematics where fluency in mathematics facts can facilitate successful performance (Woodward, 2006).
Students with learning disabilities (LD) struggle developing number sense (Robinson, Menchetti, & Torgesen, 2002), have poor longterm memory, and are often tactile learners (Learning Disabilities Association of America, 2004). To compensate for their weaknesses, teachers provide helpful multi-sensory aids and strategies to compute answers, including finger counting, foot tapping, number lines, or picture drawing. Although these supports are helpful, additional instructional strategies are needed to help students with LD become fluent with basic computation.
Struggles with Number Sense
10
Individuals with learning disabilities experience difficulties in two common areas of mathematics: reasoning and calculation (Augustyniak, Murphey, & Phillips, 2005). Within these areas, problems include limitations in number sense, counting ability, number production, mathematics fact knowledge, understanding of mathematical procedures, and problem solving. Students with learning disabilities experience difficulty learning and recalling basic mathematics facts. Although the reason for these difficulties is not clear, weak number sense is one possibility (Robinson, Menchetti, & Torgesen, 2002). Individuals with good number sense have an understanding of what numbers mean. In other words, they understand that a number represents a specific amount and can recognize patterns in numbers and solve mathematical problems. For some, number sense is acquired easily. For others, intense instruction and thorough practice is necessary. Without sufficient number sense, simple mathematical tasks, such as solving basic mathematics facts, can become overwhelming. Individuals with LD also have difficulty with long-term memory (Learning Disabilities Association of America, 2004). When deficits in long-term memory occur, retrieving information stored over a long period of time becomes challenging. Students with long-term memory deficits fail to retrieve basic mathematics facts with speed and accuracy.
One explanation for deficits in mathematics achievement for students with LD is the teaching method used to teach mathematics (Learning Disabilities Association of America, 2004). Generally, mathematics instruction favors auditory and visual learners, yet many students with learning disabilities have difficulty perceiving one or both areas. Typically, these students require instruction through a multi-sensory approach.
Predictor of Success 11
Richard Oldrieve (n.d.) studied the mathematical computational fluency on urban and suburban 2nd and 3rd grade students with a 42 two-digit by two-digit addition problems worksheet. The urban students were from an economically depressed section of a city whose school district was comprised of 71% African Americans. The suburban students were 96% European Americans. Highly significant results appeared in the completion rates of these groups. Urban students mean average time of completion on their worksheet was ~24 minutes which was significantly slower than the suburban students whose mean average completion time was ~8 minutes. The results suggest the urban students had a focus on accuracy with ~30 seconds spent to count out 7+8 in the ones and/or tens column for each and every problem. The suburban students spent ~10 seconds on each problem with no time allotted for counting. Oldrieve concluded in his study that it was possible to teach students accuracy without speed, but students who achieved the combination of speed and accuracy were able to continue improvements in accuracy. An overall correlation was shown that indicated speed accounts for 32% of the variance in accuracy. Oldrieve also found the results of this computational fluency study in the second grade to mirror the 10th grade state mathematics proficiency exams and graduation rates in both the urban and suburban school districts. Gloria Ladson-Billings (1997) concurs with this mirror effect and “the telling statistics on the life chances of African Americans [to] suggest that whenever we can improve the schooling experiences for African American students, we have an opportunity to reverse their life chances.� (p. 697) Ladson-Billings encourages teachers of mathematics to stop using their subject area as a curriculum sieve to separate the highly-educated from the poorly educated but as a net to gather more and more students to improve mathematics performance. This paradigm shift must include Haberman’s (1991) pedagogy of poverty and a culturally relevant pedagogy.
12
Principles of a culturally relevant pedagogy include: expect all students to perform at high levels of competence, provide instructional scaffolding to bridge what students do know to their new learning, engage students during classroom time in mathematics instruction and not allow them to stray off the instructional path, provide challenging content and extend students’ thinking and abilities beyond what they already know, and teachers must not only know their subject matter, they must be able to cultivate and maintain a strong interpersonal relationship with their students. Classroom practices must make sense and hold the students interest to engage them in the processes of mathematical thinking. Fluency in basic facts and skills is necessary to engage students in developing strategies for solving problems and understanding. The National Council of Teachers of Mathematics states, “Developing fluency requires a balance and connection between conceptual understanding and computational proficiency” (NCTM 2000, p. 35). According to Campbell and Robles (1997) when skills such as multiplication facts are taught for conceptual understanding and connected to other mathematical concepts and real world meaning, students actually perform better on standardized tests and in more complex mathematics applications. As far back as 1935, Brownell and Chazal’s early work in mathematics fact instruction initiated a debate over the best approach to teach facts that has continued to the present day (Woodward, 2006). Augustyniak et al. (2005) found that using instructional strategies with auditory and visual input have been beneficial to students with mathematics deficits. Therefore, the purpose of this study is to evaluate the effectiveness of a daily mathematics intervention computer program on the student’s mathematics course grade
13
Research Description
This study employed a quasi-experimental, two group comparison design to evaluate the effects of a computer mathematics facts program. Pre and post-test data were collected from two groups of 18 elementary students over a 12-week period.
Setting
This study was conducted in an urban elementary school in Northwest Indiana where 97% of students were Black, 1% Multiracial and the remaining 2% White, Hispanic, and Asian. The socioeconomic status of the elementary school was reported by the Indiana Department of Education (2010) as 46% paid lunch, 4% reduced lunch, and 50% free lunch. All data collection took place in the participants’ elementary classroom.
Participants
Eighteen participants were included in this study from a school population of 609. The race/ethnicity of the participants was 100% African-American.
Participants were grouped into two convenience groups based upon placement by the school administrator. Each group received the traditional mathematics instruction with the treatment group receiving an additional 25 minutes of computer technology program
14
intervention three times per week. The control group was composed of general education students and the treatment group was composed of students with learning disabilities. These participants were selected because they had learning disabilities consisting of auditory, visual, or attention deficits and had exhibited difficulty in the acquisition and fluency of basic computational skills. The traditional mathematics instruction consisted of lessons aligned with the 6th grade learning standards and a daily worksheet practice of approximately 100 single and double-digit mathematics facts problems per school day. The curriculum focus included a learning component to build accuracy and fluency in addition sums and multiplication products via this daily worksheet as a timed test. The treatment group received the same daily mathematics lessons as the control group plus the additional practice time on the computer to build fluency in addition, sums, differences, multiplication products and divisions. The first time the students log onto the computer, the program evaluates their skill level. Each successful lab practice builds on previous skills and tracks the student’s progress. Students receive accolades via animation on the computer screen. In both groups, the teachers administered a pretest in early September and a post test in late November. The pretest and the post test were the same tests and were based on Indiana state standards for 6th grade math. The teacher’s workload with respect to the school's curriculum was the same. The teacher of the treatment group was not required to document the student’s progress or grade their computer work. The computer program was "selfsufficient" in tracking the student’s progress and placing each student’s at the correct location in the program each visit to the program.
District approval was provided for this study, however since Indiana University Northwest’s Human Subjects Review Board 15
permission was not sought, further data could not be collected from either parents or school district officials.
Research Findings
All students scored similarly on the pretest, with a range of scores from 0 to 17 out of a possible 50 in the general education class, with an average score of 4.6 (9.1% correct) and a standard deviation of 5.4. In the special education class, the pretest scores ranged from 0 to 6 with an average score of 2.3 (4.6% correct) and a standard deviation of 1.7. Since the two groups were selected for convenience, a t-test was performed on the pretest scores of the two groups. The pretest scores of the two groups were compared using SPSS, which indicated that the variances are not significantly different from each other, supporting the fact that the two groups came from similar populations.
General Education Class Special Education Class
2 6 9 Pre Test Scores
16
17
Box and whisker plots of the pretest scores of the two groups visually displays the closeness of the data (ranges 0 to 9 with an outlier of 17), the obvious outlier of 17 points on the pretest by one student in the general education class, and the deviation of the scores from their means. The larger standard deviation in the general education class can be observed from the wider box plot as opposed to the tightness of the special education class, and the smaller deviation of their scores from the class mean.
Sixteen weeks after treatment, students were administered the same test. Student scores in the general education class post test scores ranged from 5 to 27 and had increased from an average of 4.5 to 14.8 (29.8% correct) with a standard deviation of 7.4. Students in the special education class who received the daily treatment had post test scores range from 13 to 25 and a greater increase in their class average from 2.3 to 18.9 (37.8% correct) with a standard deviation of 3.9. The variances in the two groups remained similar, with the general education students displaying a wider box and whisker plot as compared to the smaller variance and tighter clustered scores in the special education class. By the end of the study, the general education class had improved by an average of 20.7% correct and the special education class had improved by an average of 33.1% correct.
17
General Education Class Special Education Class
5
13 20 Post Test Scores
25 27
The mean scores from the students’ post tests are statistically significant with a difference of 6.22 points, which is statistically significant at the 0.05 level, implying that a comparable difference would be obtained 95 times in every 100 repetitions of this study.
Summary of Findings
The purpose of this pilot study was twofold: to improve the fluency of two sixth grade math classes in their addition, subtraction, multiplication, and division mathematics facts and to evaluate the effect of GFL’s Electronic Flash Card software program. Woodward (2006) advocates that automaticity in math facts promotes the conceptual understanding and problem solving. This study showed an increase between the pretest and posttest means for both the general education class and the special education class. Results of the present study supported those presented by Woodward (2006) who compared an integrated approach of strategy instruction and timed practice drills and found that students in the integrated approach group performed better. This study also supports the results of 18
Hasselbring, Goin, and Bransford (1988) who emphasized drill and practice as the optimum method to improve fluency and the work of Augustyniak et al. (2005) who found using auditory and visual input beneficial to students with mathematics deficits. The GFL’s electronic flash cards program expects all students to perform at high levels, engages students, and provides challenging content; all supported by Haberman’s (1991) pedagogy of poverty and a culturally relevant pedagogy.
One possible explanation for the results of this study could be teacher expectation for fluency as demonstrated by consistent use of the GFL program throughout the fall semester. This teacher made a mathematics fact practice commitment and priority by scheduling it into her school day as opposed to leaving mathematics fact fluency as an at home activity.
One limitation of this study is the small sample size. Knowing this limitation, the participants agreed to continue as long as possible during the fall term, which resulted in 16 quality weeks of implementation. The comparison of the post test mean scores from the two groups support the research to repeat this same test with a larger sample size and additional information which can be gained from Human Subject approval.
All students in this study began in the initial low region of the GFL program. Repetition of this test should include grades 3 and above and track these students over time to evaluate retention and math proficiency.
19
Future Opportunities
GFL is pleased to present this study, which equals the passion, and creation of electronic flash cards in the K-8 classroom with a research study that is statistically significant. Replication of this study is the next step for GFL’s program with a larger sample size from a more diverse population and expanded grade levels. GFL is also interested in tracking the success and longevity of the effects of their assistance in helping students learn their mathematics facts. GFL is interested in collaborating with educators on professional development of their product for the success of all students. For inquiries, please contact: Kevin Jones at kjones@j-softech.com or Kenya Brooks- Jones at kbj@j-softech.com.
20
References Augustinian, K., Murphy, J., & Phillips, D. K. (2005). Psychological perspectives in assessing mathematics learning needs [Electronic version]. Journal of Instructional Psychology, 32(4), 277-286. Billings, G.L. (1997). It doesn’t add up: African American students’ mathematics achievement. Journal for Research in Mathematics Education, 28(6), 697. Campbell, P. F., & Robles, J. (1997). “Project IMPACT: Increasing Mathematical Power of All Children and Teachers.” In Reflecting on Our Work: NSF Teacher Enhancement in K-6 Mathematics, edited by Susan N. Friel and George W. Bright, 179-86. Lanham, Md.: University Press of America. Caron, T. (2007). Learning multiplication the easy way. The Clearing House, 80(6), 278-282. Cumming, J., & Elkins, J. (1999). Lack of automaticity in the basic addition facts as a characteristic of arithmetic learning problem and instructional needs. Mathematical Cognition, 5(2), 149-180. Gagne, R. M. (1983). Some issues in the psychology of mathematics instruction. Journal for Research in Mathematics Education, 14(1), 7-18. Haberman, M. (1991). The pedagogy of poverty versus good teaching. Phi Delta Kappan, 73(4), 290-294. Hasselbring, T. S., Goin, L. I., & Bransford, J. D. (1988). Developing math automaticity in learning handicapped children: The role of computerized drill and practice. Focus on Exceptional Children, 20(6), 1-7.
21
Indiana Department of Education. (2010). Retrieved June 1, 2011, from http://compass.doe.in.gov/AdvSearchResults.aspx Kilpatrick, J., Jane Swafford, J., Bradford J. & Findell, J. (Eds.). (2001). Adding It Up: Helping Children Learn Mathematics Mathematics. Learning Study Committee. Washington, D.C.: National Research Council The National Academies Press. Learning Disabilities Association of America. (2004). Types of learning disabilities. Retrieved February 24, 2008, from http://www.ldaamerica.org/aboutld/teachers/understanding/types.asp. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Oldrieve, R. (n.d.). Structured internalization: An educational theory. Retrieved from http://personal.bgsu.edu/~richaro/page4/files/MATH%20Study%20Results.doc Robinson, C. S., Menchetti, B. M., & Torgesen, J. K. (2002). Toward a two-factor theory of one type of mathematics disabilities [Electronic version]. Learning Disabilities Research & Practice, 17(2), 81-89. Wallace, A. H., & Gurganus, S. P. (2005). Teaching for mastery of multiplication. Teaching Children Mathematics, 26-33. Woodward, J. (2006). Developing automaticity in multiplication facts: integrating strategy instruction with timed practice drills. Learning Disability Quarterly, Fall 2006 (29), 269-89.
22