The Effects of Digital Tools on Third Graders’

Understanding of Concepts and Development of Skills in Multiplication

Esther Jiyoung Yoon

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy under the Executive Committee

of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2015

   

© 2015 Esther J. Yoon

All Rights Reserved

 

ABSTRACT

The Effects of Digital Tools on Third Graders’

Understanding of Concepts and Development of Skills in Multiplication

Esther Jiyoung Yoon

The purpose of this research study was to examine the effectiveness of two digital tools: a

virtual number line (Jumper Tool); and a dynamic hundreds chart (Morphing Chart) in

improving children’s understanding of multiplication and number sense. One hundred twenty-

two third grade students (69 girls), ages ranging from 8 years-0 months to 10 years-3 months (M

= 8.88 years, SD = 0.44) from three New York City public elementary schools, were recruited to

participate in the study. Participants were randomly assigned to one of two math treatment

groups or a reading control group. Students in the Jumper group used a number line tool, while

those in the Morphing group used a morphing hundreds chart. Children’s number sense ability

and understanding of multiplication were tested at pre- and posttest to examine group

differences. Researchers recorded children’s strategy use and a back-end logging system

collected data on accuracy during treatment sessions. No group differences across the Jumper,

Morphing, or Control groups were found at posttests when controlling for pretest performance.

However, the presence of a tool (Jumper or Morphing) during treatment sessions resulted in

better performance than the absence of a tool (No Tool). Strategy use had a significant effect on

session performance as well. Fast and Tool Use responses performed better than Delayed

responses. Additionally, Fast responses were more likely to be correct than those who used an

Advanced strategy. Finally, the results indicated that Fast responses were predictive of

children’s performance on multiplication facts and number sense tests and Tool use was

   

predictive of performance on multiplication facts. These findings suggest that having a tool,

Jumper or Morphing, helped children solve multiplication problems and that tool use is related to

superior mastery of multiplication facts.

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TABLE OF CONTENTS

LIST OF FIGURES ....................................................................................................................... iii  

LIST OF TABLES .......................................................................................................................... v  

CHAPTER 1: THEORETICAL BACKGROUND ......................................................................... 1  

Introduction ................................................................................................................................. 1  

Digital Tools and Learning ...................................................................................................... 2  

Multiplication Models, Problem Types, and Strategies .......................................................... 4  

Study Goals ............................................................................................................................... 26  

Research Questions ................................................................................................................... 26  

CHAPTER 2: METHOD ............................................................................................................... 28  

Design ........................................................................................................................................ 28  

Participants ................................................................................................................................ 29  

Tasks .......................................................................................................................................... 30  

Pretest and Posttest Tasks ...................................................................................................... 30  

Treatment Tasks .................................................................................................................... 35  

Procedure ................................................................................................................................... 35  

Pretest .................................................................................................................................... 36  

Treatment Sessions ................................................................................................................ 36  

Posttest ................................................................................................................................... 40  

Strategies and Coding ............................................................................................................ 40  

CHAPTER 3: RESULTS .............................................................................................................. 44  

Pretest Analysis ......................................................................................................................... 44  

Pretest to Posttest Analysis ........................................................................................................ 45  

Session Performance Analyses .................................................................................................. 48  

CHAPTER 4: DISCUSSION ........................................................................................................ 60  

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Overview of Findings ................................................................................................................ 60  

Limitations ................................................................................................................................. 63  

Future Work ............................................................................................................................... 65  

Implications for Instruction ....................................................................................................... 66  

Implication for Designers .......................................................................................................... 67  

Conclusions ............................................................................................................................... 67  

REFERENCES .............................................................................................................................. 69  

APPENDICES ............................................................................................................................... 73  

APPENDIX A: Missing Number .............................................................................................. 73  

APPENDIX B: Quantity Discrimination ................................................................................... 74  

APPENDIX C: Multiplication Timed Test ............................................................................... 76  

APPENDIX D: Multiplication Conceptual Tasks ..................................................................... 77  

APPENDIX E: Strategy Coding Sheet ...................................................................................... 83  

APPENDIX F: Coding Glossary ............................................................................................... 84  

APPENDIX G: Data Entry Coding Scheme ............................................................................. 85  

APPENDIX H: Nines Finger Trick ........................................................................................... 87  

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LIST OF FIGURES

Figure 1. Main screen in MathemAntics Multiplication software program. ................................. 17  

Figure 2. Screen shot of blank number line jumper tool. .............................................................. 18  

Figure 3. Screen shot of single multiple jump. ............................................................................. 19  

Figure 4. Screen shot of eight jumps of nine on the number line. ................................................ 19  

Figure 5. Screen shot of start of morphing hundreds chart. .......................................................... 20  

Figure 6. Screen shot of morphed hundreds chart into rows on nine. ........................................... 21  

Figure 7. Screen shot of morphed chart into a nine by eight array, showing product of 72. ........ 21  

Figure 8. Screen shot of start of repeated addition tool ................................................................ 22  

Figure 9. Screen shot of repeated addition tool in progress after five multiples of nine have been

added together. ....................................................................................................................... 22  

Figure 10. Screen shot of final product of the repeated addition tool, shows that eight multiples

of nine (seven multiple of nine, plus one more) is 72. .......................................................... 23  

Figure 11. Screen shot of start of 0-99 chart. ................................................................................ 24  

Figure 12. Screen shot of multiples of nine highlighted on the hundreds chart when user clicks

on the number nine. ............................................................................................................... 24  

Figure 13. Mixed Model Design. .................................................................................................. 28  

Figure 14. Examples of Missing Number task. ............................................................................. 31  

Figure 15. Examples of the Quantity Discrimination task. ........................................................... 31  

Figure 16. Examples of researcher-developed multiplication timed test items. ........................... 32  

Figure 17. Number line modeling problems from the researcher-developed MCM tasks. ........... 32  

Figure 18. Multiple choice items for number line problem from the researcher-developed MCM

tasks. ...................................................................................................................................... 33  

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Figure 19. Array problems from the researcher-developed MCM tasks. ..................................... 33  

Figure 20. Area problems from the researcher-developed MCM tasks. ....................................... 34  

Figure 21. Bar model problems from the researcher-developed MCM tasks. .............................. 34  

Figure 22. Symbolic multiplication problem from the researcher-developed MCM tasks. .......... 34  

Figure 23. Multiplication word problem from the researcher-developed MCM tasks. ................ 34  

Figure 24. Ordering convention of treatment sessions by group name. ....................................... 38  

Figure 25. Multiplication software with no tools. ......................................................................... 40  

Figure 26. Multiplication software with jumper tool. .................................................................. 40  

Figure 27. Multiplication software with morphing chart. ............................................................ 40  

Figure 28. BRAINtastic reading software. .................................................................................... 40  

Figure 29. Mean proportion correct on multiplication conceptual tasks at pretest by condition. . 45  

Figure 30. Mean proportion correct on multiplication facts at pretest by condition. .................... 45  

Figure 31. Mean proportion correct on missing number at pretest by condition. ......................... 45  

Figure 32. Mean proportion correct on quantity discrimination at pretest by condition. ............. 45  

Figure 33. Mean proportion correct for Multiplication Conceptual Models tasks ....................... 47  

Figure 34. Mean proportion correct for Multiplication Timed Tests ............................................ 47  

Figure 35. Mean proportion correct for Missing Number task ..................................................... 47  

Figure 36. Mean proportion correct for Quantity Discrimination ................................................ 47  

Figure 37. Mean performance during treatment sessions by tool condition. ................................ 49  

Figure 38. Observed performance for tool condition by order from Session 1 to 4. .................... 50  

Figure 39. Count of coded strategies. ........................................................................................... 55  

Figure 40. Count of new strategies. .............................................................................................. 56  

Figure 41. Accuracy (likelihood of correctness) by strategies. ..................................................... 57  

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LIST OF TABLES

Table 1. Multiplication Problem Types ........................................................................................... 7  

Table 2. Multiplication Models ....................................................................................................... 8  

Table 3. Calculation Strategies for Whole-Number Multiplication Problems .............................. 15  

Table 4. Demographics of Participants in Study ........................................................................... 29  

Table 5. Demographics of Schools ................................................................................................ 29  

Table 6. Summary of Pretest and Posttests Scores (Proportion Correct) by Condition ................ 46  

Table 7. Summary of MANCOVA Analysis on Posttest Scores .................................................. 48  

Table 8. Summary of Session Accuracy by Order and Tool Condition ........................................ 51  

Table 9. Summary of Linear Binary Logistic Regression Mixed Model ...................................... 52  

Table 10. Summary of Linear Binary Logistic Regression Mixed Model with Strategies ........... 52  

Table 11. Post-hoc Least Significant Difference Pairwise Comparisons of Tool Condition ........ 53  

Table 12. Summary of Mean Likelihood of Correctness of Tool Condition by Order ................. 54  

Table 13. Definitions of Strategy Categories for Data Analysis ................................................... 56  

Table 14. Summary of Post-hoc Least Significant Difference Pairwise Comparisons ................. 58  

Table 15. Frequencies of Strategies Used by Tool Condition Cross Tabulation of Strategy Use

and Tool Condition ................................................................................................................ 59  

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ACKNOWLEDGMENTS

I owe many thanks to my advisor, Professor Herbert Ginsburg, for his invaluable

guidance and support. His work and accomplishments have been an inspiration and his advice

was critical to my academic growth and my professional success thus far. I owe him

tremendously for his vision in conducting research on children’s math learning and the use of

technology to support that learning.

I am thankful to the members of my committee. Thank you to Professor James Corter and

Professor Caryn Block for helping me design my study and for welcoming my questions since

early on in this endeavor. They helped me think critically about my measures and how to analyze

my data. Also, I am grateful to Professor Matthew Johnson for patiently listening to my

questions and helping me solve what seemed to be impossible statistical questions. And thank

you to Professor Lisa Son for agreeing to be on my committee at the last minute and for

providing valuable feedback for future research.

I would like to thank the Institute of Education Sciences (IES), National Science

Foundation, Cleveland Dodge Foundation, and the Zankel Fellowship program for providing

generous financial support throughout my research and doctoral studies. I could not have

accomplished this study without the support and wisdom of my fellow MathemAntics team

members: Azadeh Jamalian, Samantha Creighan, Kara Carpenter, Rachael Labrecque, Dana

Pagar, Ben Friedman, and Ama Awotwi. I learned so much from our discussions each week.

Thank you also to my team of dedicated research assistants: Julia (Chen Mu) Xing, Kristin

Slamar, Saba Abebe, Nicole Fletcher, Jessica Lopez, Shimin Kai, Shani Greenman, Krystal

Astwood, Stephanie Realegano, and Jackie Suginaga. I am grateful for your were eagerness to

vii  

learn about conducting research in a school setting and your willingness to collect data for this

study.

I am also indebted to my mentors and professors from University of California, Berkeley.

Dr. Della Peretti has been my biggest support and mentor for more than a decade and without

her, I don’t know where I would be. I thank her for academic advice, but more importantly life

advice. I am also deeply grateful to Professor Paul Ammon, Professor Meryl Gearhart, Professor

Alan Schoenfeld, and Professor Dor Abrahamson for their advice and encouraging me to pursue

my doctorate.

My many friends and colleagues at Columbia University – our shared experiences bond

us together and resulted in lasting friendships. Thank you to my office mate and dear friend, Dr.

Azadeh Jamalian. She encouraged me to persevere and always offered a helping hand whenever

I needed her. I offer many thanks to Jenny Kao, with whom I started this journey and am now

graduating with this semester. The writing process and defending our theses at the same time

made this road much smoother. Finally, thank you to my friends from Biology, Physics,

Chemistry, and Anthropology at Columbia: Vitalay Fomin, Daniel Flanigan, Dat Hoang, and

Meghan Chidsey. Thank you for giving me another perspective on research, the many coffees,

and enduring these years of studies with me.

Last, but not least, thank you to my mother, Alice Bog Soo Yoon, my sisters, Grace Yoon

and Suzanne Yoon Molloy, and my brother, David Yoon, for their endless support throughout

my life. I especially thank my mother for making many sacrifices while raising four children on

her own – sometimes working two or three jobs to make ends meet – and for never doubting my

potential to succeed. Also, thank you to my nephews, Max and Oscar Yoon Molloy, for letting

me teach them about math and being my test subjects at all times.

viii  

My father, although not with us today, dedicated his life to supporting our family. He

inspired me to work hard and always remember the importance of family. I was the first woman

in my family to earn a master’s degree and now am the first person to earn a doctoral degree.

This dissertation is dedicated to him.

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To my dad, David Chang Suk Yoon

1  

CHAPTER 1:

THEORETICAL BACKGROUND

Introduction

Mastery of multiplication concepts and procedures is vital to children’s mathematical

achievement in the middle and upper elementary grades. Research shows that children’s early

understanding of math strongly predicts their later success in math (Clements & Sarama, 2007).

However, the most recent report from the National Assessment of Educational Progress (2013)

shows that 58 percent of 4th graders and 64 percent of 8th graders scored at or below basic levels

of math performance. Also, previous research has found that many primary grade children lack

conceptual understandings of multiplication (National Mathematics Advisory Panel, 2008;

O’Brien & Casey, 1983). According to the Common Core State Standards Initiative (National

Governors Association Center for Best Practices & Council of Chief State School Officers [NGA

Center & CCSSO], 2010), by the end of third grade, children need to memorize multiplication

facts up to 100, interpret multiplication equations, and solve word problems involving

multiplicative comparisons. Children need to develop an understanding of arithmetic and we

need a better way of helping children master multiplication – both facts and concepts.

Digital tools, or virtual manipulatives, in math software can help children master their

understanding of mathematical concepts and skills. Several research studies on the usefulness of

virtual manipulatives have shown that children can learn mathematical concepts from

manipulating objects on a computer screen as well as or better than when handling physical

objects (Johnson-Gentile, Clements, & Battista, 1994; Moyer, Niezgoda, & Stanley, 2005;

Reimer & Moyer, 2005). While there are many types of concrete and digital tools that have been

designed to teach math, few tools have been designed to foster the use of specific strategies to

2  

improve overall performance on multiplication tasks. This research study empirically examines

two such tools designed to help children master concepts and skills in multiplication.

Digital Tools and Learning

There is increasing potential for children to use computers in elementary school settings

(Clements, 2002). Computer games, web-based lessons, and applications for handheld devices

are now a regular medium for teaching in many schools. Although there is an abundance of

software available, very little offer a comprehensive or research-based curriculum (Ginsburg,

Carpenter, & Labrecque, 2011). One exception to this is a program called Building Blocks

(Sarama & Clements, 2002), which includes printed materials and physical manipulatives for

early childhood classrooms. An evaluative study showed that this program was effective for pre-

kindergarten students, especially for those with low socioeconomic status (Clements & Sarama,

2007). Therefore, it is possible for children to gain valuable experience and learn necessary

mathematical content from engaging with at least some software programs on the computer.

In an effort to connect everyday knowledge of mathematical ideas with formal

representations, researchers have proposed the use of physical objects (Fuson & Briars, 1990;

Moyer, 2001) or graphical displays (Moyer-Packenham, 2005; Moyer-Packenham & Suh, 2012;

Sarama & Clements, 2002). The use of pictures is a common teaching tactic, as one can witness

in many elementary textbooks and everyday practice in classrooms. Since the creation of digital

media and computer games, many designers have attempted to capture these manipulatives and

pictures in a virtual environment. Ohlsson (1987) describes these as interactive illustrations,

where computer graphics allow for the design of visual qualities of an illustration as well as the

structure of the actions associated with them to help the learner grasp the mathematical objects

3  

and to observe the laws which regulate their behavior. Interactive illustrations and the language

associated with these pictures and symbols are tools for communicating and transforming

mathematical understanding and thinking processes.

The affordances of digital tools offer researchers an opportunity to rethink the ways in

which representations influence learning (Noss & Hoyles, 2009). Actions performed with digital

tools provide ways for children to make conjectures about mathematical ideas in ways that are

not as easy to do with physical models. Also, children can explore the use of visual models and

strategies more efficiently because of the precision and flexibility of digital models. For

example, using a computer, one can more easily cut a shape into a variety of different

combinations than by using paper and pencil (e.g. cut a hexagon into two trapezoids, two

triangles and two rhombuses, six triangles, triangle and pentagon, and many more). With the

click of a mouse or using a touchscreen, one can manipulate a shape or visual model more

quickly and precisely than when doing these manipulations by hand. This is especially useful for

young children who may still be developing fine motor skills.

Several studies have demonstrated that learners can grasp important mathematical

concepts from manipulating objects on the computer screen as well as or better than when

handling physical objects. One study, by Johnson-Gentile et al. (1994), showed that students

using computers worked with more precision and exactness, demonstrated a higher level of

geometric thinking, and retained their learning better than those using paper and pencil. Reimer

and Moyer (2005) conducted a study on fractions with third-graders, where participants

interacted with virtual manipulatives on the computer. Results showed that students working

with the computer manipulatives made significant gains in conceptual knowledge of fractions.

Additionally, student interviews and attitude surveys indicated that the virtual manipulatives

4  

were easier and faster to use than paper-and-pencil methods, enhanced student enjoyment, and

helped by providing immediate and specific feedback. Another study (Moyer et al., 2005)

compared how kindergartners learned with physical versus virtual manipulatives on the

computer. Children who worked on the computer made more patterns and used more elements of

their patterns, and only those who worked with the software created new shapes. These studies

show that using computer-based manipulatives has many advantages and are appropriate for use

in an educational setting.

Multiplication Models, Problem Types, and Strategies

Cognitive scientists and mathematics educators believe that students make sense of

computational procedures by constructing mental models (English, 1997; Grouws, 1992).

Building an understanding of how to use appropriate models in various contexts is essential in

children’s mastery of mathematical operations. In order to mathematize, children must learn to

see, organize, and interpret math problems through the use of mathematical models (Fosnot,

2010). Children may begin modeling math problems using cubes or a drawing, and eventually

these models can be used across different mathematical scenarios. For example, children can use

a number line across various operations, such as addition, subtraction, multiplication, division,

fractions, and more. Although some models have general uses, other models work best for

specific contexts. Multiplication models, such as arrays, number lines, 0-99 charts, and area are

useful in solving problems. Children may learn how to use these models on paper; however,

digitizing these tools can provide a more dynamic learning experience than pencil and paper.

This section discusses important multiplication models, problem types, and strategies to

consider when designing digital tools and software for teaching children multiplication. The

5  

researcher differentiates between models and strategies. Models include external structures that

can be manipulated, while strategies are the actions or mental procedures that one uses to solve

problems. Also, models are discussed as they relate to multiplication problem types.

Models and Problem Types. One of the greatest challenges in teaching multiplication is

helping children understand that multiplication has a variety of meanings and is not just a

sequence of isolated facts. Multiplication involves computation and concrete representations

(Lampert, 1986), where one must assess the mathematical scenario, mentally model the problem,

and apply a procedure or strategy to come to a solution. Multiplication problem types or

situations can be classified according to the structure of the quantities involved and the relation

between them (Nesher, 1988; Vergnaud, 1988). Greer (1994) classified multiplication of whole

numbers into four categories: equivalent groups (see 1a in Table 1), multiplicative comparisons,

(see 1b in Table 1), rectangular arrays (see 1e in Table 1), and Cartesian products (see 1d in

Table 1). Wallace and Gurganus (2005) outlined multiplication problems as five types: repeated

addition or grouping and partitioning, (see 1a in Table 1), scalar or multiplicative comparisons

see 1b in Table 1), rate (see 1c in Table 1), Cartesian products (see 1d in Table 1), and area (see

1e in Table 1). Providing experiences with the different meanings of multiplication, especially in

contextual situations, is extremely useful.

Consider the five types of multiplication problems types described in Table 1. Relating

these problem types to visual models can be useful in teaching children how to compute more

efficiently. A variety of models, such as pictorial representations, the bar model, realistic or

physical representations, the number line, array or area model, hundreds chart, symbolic

representations, and the multiplication table, are presented in Table 2. These models can be

6  

useful for conceptualizing and solving various problem types. The following is a discussion on

how specific models are useful for solving particular multiplication problem types.

7  

Table 1

Multiplication Problem Types

Problem Type Description Example

1a) Repeated Addition or Equal Grouping

A specified number of items is repeatedly arranged (grouped) a given number of times. The groups coexist, with no item in more than one group. One factor describes the number of objects in each group; the other factor describes the number of groups. The product is the number of objects contained in all the groups.

There are five plants in a garden. How many plants are there in nine gardens?

5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 9 x 5 = 45 plants

1b) Scalar or Multiplicative Comparisons

Some quantity (number of items) occurs a given number of times (the scalar multiple). The product is a multiple of the original quantity. The scalar multiple expresses a relationship between the original quantity and the product. This kind of problem is often referred to as a multiplicative comparison.

Mark has 5 marbles. His brother has nine times as many marbles. How many marbles does Mark’s brother have?

5 x 9 = 45 plants

1c) Rate A value or distance is associated with a unit. The product is the total value or distance associated with all the units. Because this problem involves a number of measured units, a number line often provides a good representation.

Josh walks five miles an hour. How many miles does he walk in nine hours?

5 x 9 = 45 miles

1d) Cartesian Product

Two disjoint sets exist and the size of each set is known. Each of the objects in one set is paired with each of the objects in the other set. The pairings do not occur simultaneously. The product indicates the number of possible pairings.

Judy’s Cafe offers three kinds of meat and two cheeses for their sandwiches. How many different sandwiches are possible?

3 x 2 = 6 possible combinations

1e) Area or Rectangular Array

A rectangular region is defined in terms of units along its length and width. The product is the number of square units in the region.

A sheet of wrapping paper is three feet wide and eight feet long. What is the area of the sheet of paper?

3 x 8 = 24 square feet

Note. Adapted from Wallace and Gurganus (2005).

8  

Table 2

Multiplication Models

Model Example Problem Types

2a) Pictorial

Repeated Addition

Scalar or Multiplicative Comparison

2b) Realistic Manipulatives

Repeated Addition

Scalar or Multiplicative Comparison

2c) Bar

Repeated Addition

Scalar or Multiplicative Comparison

Rate

2d) Number Line

Repeated Addition

Scalar or Multiplicative Comparison

2e) Area or Array

Area

Array

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2f). Tree Model

Cartesian Product

2g) Hundreds Chart

Repeated Addition

2h) Symbolic 9 x 5 = 45

5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 9 x 5 = 45

All

2i) Multiplication Table

Trivett (1980)

Multiplication Facts and Computation

10  

As recommended by many textbooks, such as Singapore Math (2007), TERC

Investigations (Pearson Scott Foresman TERC, 2008), and Everyday Mathematics (University of

Chicago School Mathematics Project, 2007), many teachers begin instruction of multiplication

with repeated addition or equal grouping. Lampert (1986) states that this is useful when using

multiplication as a counting operation, where 9 x 5 means the total number of members in nine

groups, each containing 5 members (e.g. 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 9 x 5 = 45). Several

multiplication models are useful for representing this type of problem. Textbooks display

pictorial representations, such as crayons or marbles, to represent groups of objects (see 2a and

2b in Table 2). In some cases, particularly in Singapore Math, equal grouping is displayed in the

form of a bar model (see 2c in Table 2). The bar model is useful for repeated addition, rate, and

multiplicative comparisons. Also, the bar model is recommended for representing addition,

subtraction, fraction, and algebraic problems (Fong, Ramakrishnan, Choo, Bisk, Clark, &

Kanter, 2009). In this way, it is seen as a universal model for teaching across different

mathematical content areas. Similarly, Carpenter, Fennema, Franke, and Empson (1999) argue

that rate and multiplicative comparisons can extend to the multiplication and division of

fractions. For this reason, a number line (see 2d in Table 2) or bar model can be useful for

teaching across math content areas. Both of these models display equal parts as they relate to one

another in a linear fashion.

Another useful tool for modeling repeated addition problems is the hundreds chart (2g in

Table 2). TERC Investigations at grade 3 has several lessons dedicated to skip counting and

uncovering patterns in the chart (Pearson Scott Foresman TERC, 2008). This chart can be useful

for showing patterns (e.g. highlighting multiples of five shows that all of them have a 5 or 0 in

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the ones places) and for identifying common multiples (e.g. twelve is a multiple of 2, 3, 4, and

6).

Rate problems involve a value or distance associated with a unit. The product is the total

value or distance associated with all the units (e.g. Josh walks five miles an hour. How many

miles does he walk in nine hours? 5 x 9 = 45 miles). Since these problems involve a number of

measured units, a number line often provides a good representation. Jumps on a number line

model increments of distance, such as 5 miles, 9 times, which results in 45 miles. Carpenter et al.

(1999) stated that although these problems are somewhat more difficult than those involving

countable objects, children can still use counters in sets to model these problems.

Similarly, counters are useful for modeling scalar problems, even though the multiplier is

not visible. Scalar problems involve some quantity (number of items) that occurs a given number

of times (the scalar multiple). The product is a multiple of the original quantity (e.g. Mark has 5

marbles. His brother has nine times as many marbles. How many marbles does Mark’s brother

have? 5 x 9 = 45 marbles). The scalar multiple expresses a relationship between the original

quantity and the product. This kind of problem is also referred to as a multiplicative comparison.

In a study with Israeli children, Peled and Nesher (1988) found that scalar multiplication was

more difficult to conceptualize than equal groupings. The difficulty with these problems is that

the multiplier is not an identifiable quantity. For example, plants are measurable, but the other

quantity (e.g. nine times as many) describes the relationship between the measurable quantities

(Mark’s plants and Mark’s brother’s plants). Children need to understand the meaning of terms

like “nine times as many.” Equal groups and rate problems do not use this type of language.

Carpenter et al. (1999) asserted that array and area problems provide a very different

context for developing multiplication concepts than do equal grouping, rate, and scalar problems.

12  

For example, in their observations of several classrooms, children did not construct arrays when

doing multiplication problems unless the problem called for them. Also, unlike equal grouping,

rate, and scalar problems, area and array problems are symmetrical in nature, where the role of

factors is interchangeable (e.g. in the array in 2e of Table 2, neither 3 nor 8 is the clear

multiplier). Thus, arrays are useful for illustrating commutativity (Greer, 1994), where 3 x 8 = 8

x 3. Finding the area of a rectangle and finding the number of items in a rectangular array

involve the same basic conception of multiplication, but the primary difference is that area is

made up of square units and arrays are made up of discrete objects (e.g. rows of muffins in a

muffin tray or rows of desks in a classroom).

Similar to array and area problems, Cartesian problems are symmetrical in nature. These

problems consist of cross products entailing a number of combinations. Two disjoint sets exist

and the size of each set is known. Each of the objects in one set is paired with each of the objects

in the other set and the pairings do not occur simultaneously. The product indicates the number

of possible pairings (e.g. Judy’s Cafe offers three kinds of meat and two cheeses for their

sandwiches. How many different sandwiches are possible using one meat and one cheese? 3 x 2

= 6 possible combinations). Cartesian problems can be solved by making all the possible

combinations and counting them up or it can be thought of as four groups of two or two groups

of four. It is helpful for students to model these problems through the use of a tree diagram (see

2f in Table 2). Carpenter et al. (1999) stated that children have difficulty with making all the

combinations (e.g. turkey swiss, turkey cheddar, ham swiss, ham cheddar, baloney swiss baloney

cheddar, roast beef swiss, roast beef cheddar) in a systematic way and only make some and not

all of the combinations. Therefore, a systematic way to organize these combinations, such as

with the tree model (2f in Table 2), is helpful in solving such problems correctly.

13  

The multiplication table, commonly known as the times table, is a grid with factors along

the x- and y-axis, and the product of those two factors written in the intersection of the two

factors (see 2i in Table2). Trivett (1980) recommended the use of a multiplication table to help

children uncover patterns in numbers, in addition to practicing and memorizing their

multiplication facts. He also hypothesized that studying the table can be useful for other

mathematical topics, such as fractions. Teaching mastery of multiplication facts does not

necessarily require only rote memorization. Making connections between conceptual

understanding and computational fluency is important for all children.

Teaching children about the various problem types and helping them think strategically

will enable children to more successfully master multiplication. Mulligan and Mitchelmore

(1997) suggested that children acquire a repertoire of intuitive strategies and the strategy they

employ to solve a particular problem reflects the mathematical structure they impose it on. The

next section discusses various strategies that children might use to solve multiplication problems.

Strategies. It is important to identify what strategies children are using when solving

multiplication because this can shed light on their progress in mastering necessary concepts and

skills. As stated before, strategies are the actions or mental procedures that one executes to solve

a problem. Researchers have observed children using counting, skip counting, repeated addition

(Mulligan & Mitchelmore, 1997; Wong & Evans, 2007) and quick response (Baroody, 1993;

Siegler, 1988) to solve multiplication problems in various settings. Similar to addition and

subtraction, children solve multiplication problems by directly modeling the action and

relationships in a problem by using manipulatives, and over time, direct modeling is replaced

with more efficient strategies based on counting, adding, and the use of derived number facts

14  

(Carpenter et al., 1999). Counting of objects, physical or virtual, is considered to be the least

sophisticated of the strategies listed. In a series of interviews of teachers in grades one to three,

teachers reported that students used multiple strategies to solve problems and that they

discouraged children from using less sophisticated ones, such as counting on fingers (Siegler,

1988). As one advances in mastery of multiplication, the use of physical objects to represent

quantities in calculations is replaced by an increased ability to create abstract representations

linguistically, visually, mentally, or by an increased ability to memorize or use algorithms

(Levine, Jordan, & Huttenlocher, 1992).

In discussing strategy, the researcher differentiates between a mental strategy (done in

one’s head) and the manipulation of a tool (pencil and paper or digital). Consider the list of

strategies, adapted from Mulligan and Mitchelmore (1997) in Table 3. Repeated addition can be

done mentally or by writing out the problem symbolically in the form of an equation. It can also

be done using a digital tool or visual diagram. Adding factors one-by-one is a laborious task,

especially when the multiplier is very large. Doubling (e.g. 3 + 3 = 6, 6 + 6 = 12, 12 + 12 = 24,

etc.) comes in very handy for these types of problems. Skip counting is another strategy. One can

skip count mentally, or using a model, such as the number line, bar model, or hundreds chart,

which can be very useful for keeping track of this counting. Using one’s fingers to keep track of

how many times one skip counted is also very helpful. A derived fact, or using known fact to

help solve a problem (e.g. 12 x 9 = 10 x 9 + 2 x 9 = 90 + 18 = 108), is another strategy that

children may use to solve problems. The “nines finger trick” is another strategy that children use

to solve multiplication problems involving nines (see Appendix H).

15  

Table 3

Calculation Strategies for Whole-Number Multiplication Problems

 

Strategy Definition

1. Fast Answer in less than 3 seconds 2. Repeated Addition Adding each factor one-by-one (e.g. 3 + 3 = 6, 6 + 3 = 9)

or doubling (e.g. 3 + 3 = 6, 6 + 6 =12). 3. Skip Counting Count by multiples to find answer (e.g. 3x4, "Three, six,

nine, twelve."). 4. Derived Fact Use a known fact to help solve a problem (e.g. 12 x 9,

child says, “10 times 9 equals 90, 90 plus 18 equals 108." or 12 x 9, child says, “11 times 9 equals 99, 99 plus 9 equals 108.").

5.Nines Finger Trick Child uses fingers to figure out multiples of nine. 6. Tool Manipulation of a digital tool or use of a diagram, such as

hundreds chart, multiplication chart, chart paper. 7. Counting Direct counting of physical objects, pictures, or diagrams. 8. Delayed Answer is stated without any expressed strategy in 3

seconds or more.

Note. Adapted from Mulligan and Mitchelmore (1997).

16  

Finally, children will often solve a problem by simply stating an answer to a

multiplication problem without using any apparent strategy. There has been some debate on what

a quick response means. Does a quick response mean the answer is from a memorized fact?

Research studies measuring children’s strategy use code behavior based on what children

express. Siegler (1988) defined a retrieved response as an expressed answer in four seconds or

less, whereas Baroody (1993) argued that these quick, correct answers may stem either from

accessing a relationship or retrieving facts from memory. Quick responses can be from memory

or a known relationship such as the commutative property (e.g. “I know 3 x 6 = 18, so 6 x 3 =

18.”). Instead of focusing so much on whether these quick responses are indeed from memory, in

this study, quick response is defined as an automatized or fast strategy, where one can produce a

response correctly in three seconds or less. A delayed response is one that displays no other

strategy and is expressed in more than three seconds.

As discussed in previously, children use various strategies to solve multiplication

problems. Using models can help children organize their thoughts and more accurately execute a

strategy. Digitizing multiplication models offer a way for children to learn, practice, and acquire

skills in solving multiplication problems. The next section discusses the design of multiplication

tools within MathemAntics, a comprehensive mathematics software program developed by

Ginsburg and colleagues, which is the focus of the present studies.

Teaching and Learning of Multiplication Procedures and Concepts through the Use of Digital Tools

Focusing on the operation of multiplying two numbers or memorizing facts without

understanding multiplicative situations and the quantities involved, narrows children’s focus and

gives the wrong impression about the need to understand what it means to multiply. Further,

17  

children may not understand how to apply multiplicative computations to real world problems.

Digital tools in Multiplication, such as the Number Line Tool, Morphing Hundreds Chart,

Repeated Addition Tool, and 0-99 Chart can help children understand what multiplication means

and provides models to illustrate the underlying concepts in multiplying.

In the context of school-aged children, there seems to be an overemphasis on facts and

procedures, which can result in an impoverished understanding of concepts of multiplication

(Smith & Smith, 2006). Posing questions and providing adequate opportunities to practice using

procedures, along with conceptual knowledge to support the operations, can lead to mastery of

multiplication.

The Multiplication software program offers children the opportunity to utilize various

visual models alongside computing the product of two factors. The main screen of the software

program (Figure 1) displays the symbolic multiplication problem, the repeated addition problem,

and the objects, plants and gardens, displayed as sets in the formation of arrays.

 

Figure 1. Main screen in MathemAntics Multiplication software program.

 

18  

In addition to this, there are four tools that the user can employ. These include the

Number Line Jumper Tool (Figures 2 – 4), Morphing Hundreds Chart (Figures 5 – 7), Repeated

Addition Tool (Figures 8 – 10), and the 0-99 Chart (Figure 11- 12). All of these tools are useful

in different ways, and the researcher hypothesizes that these tools can be effective in teaching

children both concepts and skills.

The Number Line Jumper Tool lends itself to skip counting and repeated addition, which

are two ways that multiplication is related to sequencing of numbers and addition. Children can

make a jump across by indicating first which multiple to jump by (Figure 3) and then the

computer automatically makes jumps of X. The computer automatically jumps by X, minimizing

the possibility of making an error, and also indicates the corresponding symbolic representation

above (Figure 4).

Figure 2. Screen shot of blank number line jumper tool.

 

19  

Figure 3. Screen shot of single multiple jump.

   

Figure 4. Screen shot of eight jumps of nine on the number line.

 The Morphing Hundreds Chart visually displays multiples in rows in an area, or array,

model, which is continuous and relates to other areas of mathematics, such as geometry. To

begin, the Morphing Hundreds Chart presents itself like an ordinary hundreds chart (Figure 5), as

20  

a ten by ten chart. However, this tool offers a way to rearrange the chart into rows of multiples

that correspond to the problem posed (e.g. 9 x 8 = nines arranged into eight rows). After a child

clicks on “9” the chart rearranges itself into rows of nine. Finally, the child can count out 8 rows

to find the product (Figure 7). This chart can be manipulated in ways that are much more

efficient than drawing, cutting paper, or using blocks to illustrate such a transformation.

 

Figure 5. Screen shot of start of morphing hundreds chart.

21  

 

Figure 6. Screen shot of morphed hundreds chart into rows on nine.

 

Figure 7. Screen shot of morphed chart into a nine by eight array, showing product of 72.

The Repeated Addition Tool is a calculation tool that helps children keep track of their

addition of multiples (Figure 8). The child enters the sum of multiples in the box below (e.g. 9 +

9 = 18, 18 + 9 = 27, 27 + 9 = 36, etc.). As children add each multiple, the tool keeps track of the

22  

sum of multiples already combined and points to the remaining multiples that need to be added

(Figure 9). This tool also has feedback at each multiple, so children are forced to check their

mistakes if they do not add correctly. The computer will not let the user go on if he/she does not

add each multiple correctly.

 

Figure 8. Screen shot of start of repeated addition tool

 

Figure 9. Screen shot of repeated addition tool in progress after five multiples of nine have been added together.

23  

 

Figure 10. Screen shot of final product of the repeated addition tool, shows that eight multiples of nine (seven multiple of nine, plus one more) is 72.

Finally, the 0-99 Chart is very similar to a hundreds chart, but instead of starting at 1, the

chart starts at zero. The reason for this is to display all the numbers with zero in the ones place in

the first column. This chart is different from a paper chart because instead of a child highlighting

each multiple, the chart automatically highlights multiples for the user. For example, if a child

clicks on “9” all of the multiples of nine are highlighted in one color (Figure 12). It is up to the

user to find which multiple of nine is eighth in order to find the product of nine times eight.

24  

 

Figure 11. Screen shot of start of 0-99 chart.

 

 

Figure 12. Screen shot of multiples of nine highlighted on the hundreds chart when user clicks on the number nine.

Each of these tools is useful for applying strategies and using models to solve

multiplication problems. They were created to help tie the concepts and procedures of

25  

multiplication together. They provide a way for children to see and also manipulate models that

help them conceptualize and compute the product of two numbers in ways that are more efficient

and might not be possible on paper, particularly in the case of the Morphing Hundreds Chart.

26  

Study Goals

The aim of this research study is to examine the effectiveness of two digital tools with the

MathemAntics Multiplication software program: a virtual number line (Jumper Tool); and a

dynamic hundreds chart (Morphing Chart) in improving children’s understanding of

multiplication and number sense. To measure the overall effectiveness of the software and the

individual tools, three groups (Jumper, Morphing, and Control) are compared to one another.

Because the Jumper Tool enables children to explore multiplication as a linear function and the

Morphing Chart enables children to manipulate an area model, the hypothesis is that there will be

differences in performance between the two treatment groups from pre- to posttest and during

treatment sessions. Also, because the Jumper Tool might be easier to understand and relate to

commonly used models, such as a number line, it is expected that those children who use the

Jumper Tool will perform better than when using the Morphing Chart. Additionally, it is

expected that strategies, such as using known multiplication facts and efficient counting

strategies will help children solve problems more successfully than other strategies, such as

adding multiples one-by-one.

Research Questions

1) Is the Multiplication software program effective in improving children’s conceptual and

procedural understanding of multiplication and/or general number sense?

2) Are there differences in performance when given two different tools, Jumper or Morphing,

while using the Multiplication software program?

27  

3) Are specific strategies, such as known multiplication facts and efficient counting strategies,

predictive of children’s success on multiplication and/or number sense tasks? If so, which

strategies?

 

  28  

CHAPTER 2:

METHOD

Design

In order to investigate the effectiveness of the various multiplication models, this study

employed a mixed model design, incorporating a pre-posttest between subjects with condition

(Jumper, Morphing, and Control) as main factor and within subjects with order (Jumper then No

Tool, No Tool then Jumper, Morphing then No Tool, and No Tool then Morphing) as main

factor, and had three parts: (1) pretest, (2) four treatment sessions (3) posttest. Figure 13

summarizes the overall design and procedure.

Figure 13. Mixed Model Design.

 

29

Participants

One hundred twenty-two third grade students (69 girls), ages ranging from 8 years-0

months to 10 years-3 months (M = 8.88 years, SD = 0.44 ) were recruited to participate in the

study. All children were recruited from three New York City public elementary schools (School

A = 85, School B = 12, School C = 26). The three schools represent a wide range of students in

New York City. Each school is located in different areas of Manhattan: Upper East Side, Lower

East Side, and East Harlem.

Table 4

Demographics of Participants in Study

(N = 122) Participants Female Age

Schools n n (%) M SD

A 85 41 (48) 8.85 0.42 B 12 6 (50) 9.16 0.52

C 26 22 (85) 8.87 0.45    Table 5

Demographics of Schools

School n Free Lunch

n (%) White n (%)

Asian n (%)

Hispanic n (%)

Black n (%)

ELL n (%)

A 789 32 (4) 544 (69) 103 (13) 79 (10) 24 (3) 16 (2) B 785 769 (98) 24 (3) 16 (2) 243 (31) 495 (63) 141 (18)

C 373 283 (76) 52 (14) 194 (52) 93 (25) 30 (8) 78 (21)

Note. Data retrieved from the National Center for Education Statistics (2012).

 

30

Participants were recruited through fliers sent home by their teachers and through parent

meetings at schools. Parents consented to having their children video or audio taped and to

sharing the videos in an educational setting. Children also signed assent forms before pretests

were administered. Principals and teachers were compensated with gift cards for school supplies.

Random assignment was performed within classrooms, where each class was randomly divided

into the three treatment groups – Jumper, Morphing, and Control – and within the treatment

groups, children were assigned to two different ordering conventions – Tool then No Tool and

Tool then No Tool. Finally, three participants dropped out of the study due to one child opting

out on her own, and two others who were repeatedly absent due to family vacations during the

study.

Tasks

  This section gives a brief description of the pretest, intervention, and posttest tasks. The

pre- and post-test tasks measured children’s knowledge of number sense, conceptual

understanding of multiplication, and mastery of multiplication facts. The treatment tasks

consisted of Multiplication software items and a reading software program. Additional

information on how these tasks were administered is described in more detail in the Procedure

section.

Pretest and Posttest Tasks

Missing Number task (MN). The MN task is a standardized measure from an assessment

program called mClass:Math (Lee et al., 2010). The tests from mClass:Math are standardized,

paper-based assessments that were designed to assess number skills and understanding for

children from kindergarten to third grade. The MN task evaluates a student’s recognition of basic

 

31

and complex number patterns and familiarity with printed numbers (Figure 14). See Appendix A

for full measure. Those who can quickly and accurately identify the number that is missing

demonstrate that they can recognize and make use of fundamental patterns within a number

sequence.

Figure 14. Examples of Missing Number task.

 Quantity Discrimination task (QD). The QD task, also a standardized measure from mClass

math (Lee et al., 2010) determined an individual’s ability to understand the quantitative value of

a number (Figure 15). See Appendix B for full measure. The student identified the number or

operation in the pair with the highest value.

 

Figure 15. Examples of the Quantity Discrimination task.

Multiplication Timed Test (MT) task. The MT task examined children’s fluency with one

hundred multiplication facts from 2 x 2 to 12 x 12 (see example problems in Figure 16). This

task was designed by the researcher, drawing from similar facts tests used in the textbook

program Everyday Mathematics (University of Chicago School Mathematics Project, 2007) and

from recommendations stated in third grade Common Core State Standards Initiative (NGA

Center & CCSSO, 2010). See Appendix C for full measure.

 

32

 

Figure 16. Examples of researcher-developed multiplication timed test items.

Multiplication Conceptual Models (MCM) tasks. The MCM tasks consisted of 18 items. These

included six number line problems (Figure 17 and 18), six array and area problems (Figure 19

and 20), two bar model problems (Figure 21), two symbolic multiplication problems (Figure 22)

and two word problems (Figure 23). These items were designed to assess children’s

understanding of multiplication problems with different models, such as number line, arrays,

area, bar model, symbolic multiplication, and in the absence of a visual model. See Appendix C

for a full set of items.

 

Figure 17. Number line modeling problems from the researcher-developed MCM tasks.

 

33

 

Figure 18. Multiple choice items for number line problem from the researcher-developed MCM tasks.

 

Figure 19. Array problems from the researcher-developed MCM tasks.

 

34

 

Figure 20. Area problems from the researcher-developed MCM tasks.

 

Figure 21. Bar model problems from the researcher-developed MCM tasks.

 

 

Figure 22. Symbolic multiplication problem from the researcher-developed MCM tasks.

 

 

Figure 23. Multiplication word problem from the researcher-developed MCM tasks.

 

35

Treatment Tasks

Multiplication software. The Multiplication program, a part of the MathemAntics software

program, developed at Teachers College, Columbia University, required students to solve

multiplication problems. A central feature of the Multiplication activities is the availability of

various multiplication tools, which help children organize their thinking and calculate the

product of a pair of numbers. These problems range from 3 x 4 to 12 x 12 and were selected at

random, using a randomization software program built into the Multiplication Software Program

(See Figures 1 through 5 in previous section).

BRAINtastic Reading Success software. BRAINtastic Reading Success is a reading software

program, published by EdAlive®. This program is designed to help children develop higher order

thinking skills and literacy skills. The activities consist of reading passages with multiple-choice

questions about cause/effect, main idea, character analysis, and order of events. These activities

were designed for children in grades 3, 4, and 5.

Procedure

This study was conducted over the course of six consecutive weeks, in the spring of 2013.

Students were pretested in Week 1, given four treatment sessions during Weeks 2 through 5 (one

session per week, spaced approximately one week apart), and posttested during Week 6,

approximately one week after the last treatment session. Each session lasted for about twenty

minutes and never exceeded twenty-five minutes, depending on if a child needed extra time on

the untimed portion of the tests. All sessions were administered in a separate classroom, assigned

 

36

by the school principal. No researchers from the pre and posttest sessions collected data during

intervention sessions. All researchers who collected session data were blind to the pre- and

posttest items.

Pretest

The pretest was administered in small groups (4-6 children at a time) in a separate

classroom setting, where participants worked in a partitioned workspace, separated by a privacy

board. Children could not see what others were doing. One researcher read instructions to the

children and another researcher passed out papers and helped monitor the students during the

assessments. All of these tasks were paper-based and children handwrote their responses.

The researcher was interested in measuring children’s general number sense as well as

multiplication facts fluency and understanding of conceptual models of multiplication. Four

measures were employed: Missing Number (MN), Quantity Discrimination (QD), Multiplication

Timed Test (MT) task, and Multiplication Conceptual Models (MCM) tasks. For MN and QD,

participants were required to answer as many questions as they could in under 2 minutes, by

recommendation of the assessment manual from mClass:Math. For the MT task, participants had

three minutes to answer as many questions as they could. Children were not timed for the MCM

task and could take as much time as they needed to complete the tasks. Children were

encouraged to complete the items to the best of their ability, to show their work, and not leave

any questions blank. These items were scored using an answer key by a group of trained

researchers at a later time.

Treatment Sessions

Four treatment sessions were conducted one-on-one with each participant. Participants

were blocked by school (A, B, and C) and classroom (A-1, A-2, A-3, A-4, B-1, B-2, C-1, C-2, C-

 

37

3), and then assigned to one of three conditions: Jumper, Morphing, and Control. Traditionally in

a learning study, children would carry out the treatment condition for the entirety of the

treatment, which in this case is four sessions. However, it was possible that that if children had

the availability of the tool for all sessions, children were not likely to use any strategies other

than Tool Use. Therefore, a within subjects manipulation was used for the two treatment groups,

Jumper and Morphing. The within subject groups, Jumper 1 (tool then no tool) Jumper 2 (no tool

then tool), Morphing 1 (tool then no tool), and Morphing 2 (no tool then tool), only differed in

the order in which children played sessions with a tool or without tool (see Figure 24).

A researcher sat next to each participant at every session. Altogether, students played 48

trials, two sessions of 12 trials with a tool and two sessions of 12 trials without a tool.

Participants playing Multiplication were given up to two attempts to answer each multiplication

problem. If the child answered the problem incorrectly, he or she received immediate feedback

from the software program that the answer was incorrect and he or she could try to answer the

problem one more time. If incorrect a second time the correct answer was provided by the

software program and the session proceeded to the next problem.

 

38

Figure 24. Ordering convention of treatment sessions by group name.

In the No Tool condition, students saw a word problem, repeated addition sentence, and a

multiplication sentence (see Figure 25). The multiplication problems in this condition were

identical to the ones that children experienced in the conditions with tools.

In the Jumper condition, students saw the same screen as in the No Tool condition, but

with the jumper tool, in which they move a frog along a number line to make jumps based on the

multiples in the problem (see Figure 26). The Jumper Tool allows children to skip count on a

number line. Children move a frog along a number line to make jumps based on the multiples in

the problem. For example, for 6 x 7, a child clicks on the number six and then moves the frog to

the right to make the required number of jumps of six. The child must find where the frog lands

on the seventh jump to find that six times seven is forty-two. Tool use is not limited to a certain

number of attempts before a final answer is entered and no feedback is provided until after a

final answer is entered.

In the Morphing condition, students were presented with the same problems as the no

tools condition, but with the use of a morphing hundreds chart. This can be manipulated into an

Group Name

Pre-Test

Session 1

Session 2

Session 3

Session 4

Post Test

Jumper 1 All Jumper Jumper No Tool No Tool All

Jumper 2 All No Tool No Tool Jumper Jumper All

Morphing 1 All Morphing Morphing No Tool No Tool All

Morphing 2 All No Tool No Tool Morphing Morphing All

Control All Reading Reading Reading Reading All

 

39

array based on the multiples in the problem (see Figure 27). This tool possesses dynamic,

customizable characteristics, which would not be so easily accessible with a static image.

Children can rearrange the rows and columns to fit specific number combinations. For example,

the chart starts as a 10 x 10 (1 to 100) array, showing each row as a group of ten. For the problem,

8 x 6, the child can rearrange the rows into groups of eight by clicking on the number eight. Then,

the child needs to find the sixth row to find out the product of eight times six. For problems that

are larger than 10 x 10, such as the 11s or 12s, the chart starts out as a 14 x 14 chart (14 rows of

14), so that the child is forced to rearrange the chart into the appropriate array. Tool use was also

unlimited for the Morphing tool and the same type of feedback was provided as in the Jumper

condition.

Finally, the control group played twenty-minute sessions of a reading software program,

called BRAINtastic Reading Success (see Figure 28). All children played the intermediate levels,

which were designed for children in the end of third grade. These children received the same

instruction in the classroom as everyone else and were pulled out at the same times as all the

other participants.

 

40

Posttest

Posttest measures were conducted a week after the completion of the four treatment

sessions. The measures were identical to the pretests and the sessions were administered in the

same fashion as the pretest.

Strategies and Coding

During the one-on-one intervention sessions, trained researchers coded covert and

expressed strategies through observation. The codes were developed based on pilot testing and

Figure 25. Multiplication software with no

tools.

Figure 26. Multiplication software with

jumper tool.

Figure 27. Multiplication software with

morphing chart.

Figure 28. BRAINtastic reading software.

 

41

with guidance from the literature (Baroody, 1993; Mulligan & Mitchelmore, 1997; Siegler, 1988;

Wong & Evans, 2007). The strategies included fast response, delayed response, repeated

addition, skip counting, derived fact, nines finger trick, tools, and counting (See Appendix E & F

for coding glossary and sample coding sheet). Although coding mostly occurred during the

treatment sessions, videos on sessions were also recorded via a software program, Silverback by

Clearleft Ltd, which simultaneously captured the screen with mouse clicks and video recorded

the child’s actions and audio. Two video sessions were coded due to lost recording sheets from

the treatment sessions.

Each researcher was trained to conduct one-on-one intervention sessions by the lead

researcher one month before the start of the training for three two-hour sessions. They coded a

set of videos to test for inter-rater reliability. The videos from the pilot research group, consisted

of four sessions with 15 trials each, for a total of 60 trials. Eleven out of twelve trained

researchers scored 90% or greater agreement with the lead researcher when coding the videos.

The one individual who did not achieve 90% or greater agreement did not ever achieve

agreement of more than 80%, and therefore did not continue assisting with the study.

The researcher defined covert strategies as behaviors that did not explicitly indicate a

specific calculation or counting method. The covert strategies included fast response and delayed

response. Fast response, when a child enters or says an answer in less than 3 seconds, and

delayed response, when a child enters or says an answer in 3 seconds or more, were coded when

no visible or verbal expression of a strategy occurred. When a child responds quickly, under 3

seconds, one can assume that this is not enough time to use a developed, deliberate strategy,

while a delayed response, 3 seconds or more, is enough time for a child to conceivably calculate,

count, or use another mental backup strategy. It is important to differentiate these types of

 

42

responses because the researcher was interested in children’s automaticity with multiplication

facts.

Expressed strategies were defined as behaviors that were visible or stated explicitly.

These include repeated addition, skip counting, derived fact, nines finger trick, tools, counting,

and other (not adding). Repeated addition was coded when a child added a number repeatedly.

This category was divided into three categories: Adding one addend at a time (3 + 3 = 6, 6 + 3 =

9, 9 + 3 = 12, etc.), Doubling (3 + 3 = 6, 6 + 6 = 12, 12 + 12 = 24), and Other (any variation of

repeated addition that was not Adding one addend at a time or Doubling). Skip counting was

coded when a child skip counted to find an answer (e.g. 3 x 4 ! “three, six, nine, twelve”).

Derived fact was coded when a child used a known multiplication fact to help him/her solve the

problems. This strategy was split up into two categories: Tens and Known fact. Tens is defined

as a multiplication procedure that uses a multiple of ten to find the product of the given set (e.g.

12 x 9, child says, “10 times 9 equals 90, 90 plus 18 equals 108.”). Known fact is defined as a

multiplication procedure that uses multiples of another number, not in the pair of factors posed in

the given problem, to find the product of the given set. (e.g. 2 x 9, child says, “11 times 9 equals

99, 99 plus 9 equals 108.”). Nines finger trick was coded when a child used a specific trick with

his or her fingers to figure out multiples of nine (see Appendix I). Tool use is defined by whether

a child visibly used the tool in the software program by clicking, pointing, or using the mouse

over the tool. The use of tools was split into two categories: before and after. Before was coded

when a child used a tool before using another strategy or before stating a final solution. After

was coded when a tool was used after the child already used another strategy. Other (NOT

adding) was coded when a child used any strategy other than the ones specified above. This is

not defined the same as “Other Adding” because if a child used an addition strategy, this was

 

43

coded separately. Finally, the researchers coded strategies for each attempt, since each child had

two chances to answer the problem.

 

  44  

CHAPTER 3:

RESULTS

Pretest Analysis

In order to ensure that groups were equivalent at pretest, a Multivariate Analysis of

Variance (MANOVA) with Condition (Control, Jumper, Morphing) as the independent variable

and four pretests (Multiplication Conceptual Models, Multiplication Facts, Missing Number, and

Quantity Discrimination,) as dependent variables was conducted. The average proportion correct

(out of 16) on the Multiplication Conceptual Tasks for Control, Jumper, and Morphing were .80

(SD = .23), .81 (SD = .26), and .82 (SD = .21), respectively. The average proportion correct (out

of 100) for Multiplication Facts for Control, Jumper, and Morphing were .39 (SD = .25), .35 (SD

= .26), and .34 (SD = .19). For Missing Number, the average proportion correct (out of 22) for

Control, Jumper, and Morphing were .49 (SD = .25), .49 (SD = .27), and .46 (SD = .21),

respectively. The average proportion correct (out of 42) for the fourth pretest, Quantity

Discrimination for Control, Jumper, and Morphing were .46 (SD = .24), .45 (SD = .20), and .44

(SD = .23), respectively. The results confirmed that there were no statistical differences by

condition at pretests, Wilks’ Lambda = 0.99, F(8, 208) = 0.19, p = .99. Figures 29 to 32 show

performance on four pretest measures across conditions.

 

45

 Figure 29. Mean proportion correct on

multiplication conceptual tasks at pretest by condition.

 Figure 30. Mean proportion correct on

multiplication facts at pretest by condition.

 Figure 31. Mean proportion correct on missing number at pretest by condition.

 Figure 32. Mean proportion correct on

quantity discrimination at pretest by condition.

Pretest to Posttest Analysis

1) Is the Multiplication software program effective in improving children’s understanding of

multiplication and/or general number sense?

Descriptive analyses of pretests and posttests indicated that all groups (Control, Jumper,

and Morphing) improved from pretest to posttests. Overall, children made significant gains from

0.80   0.81   0.82  

0.0  

0.2  

0.4  

0.6  

0.8  

1.0  

Control   Jumper   Morphing  

Multiplication    Conceptual  Tasks    

(Proportion  Correct)  

0.39   0.35   0.34  

0.0  

0.2  

0.4  

0.6  

0.8  

1.0  

Control   Jumper   Morphing  

Multiplication  Timed  Test  (Proportion  Correct)  

0.49   0.49   0.46  

0.0  

0.2  

0.4  

0.6  

0.8  

1.0  

Control   Jumper   Morphing  

Missing  Number    (Proportion  Correct)  

0.47   0.45   0.44  

0.0  

0.2  

0.4  

0.6  

0.8  

1.0  

Control   Jumper   Morphing  

Quantity  Discrimination  (Proportion  Correct)  

 

46

pretest to posttest on Multiplication Conceptual Tasks, F(1, 110) = 6.13, p = .00, Multiplication

Facts, F(1, 110) = 146.43, p = .00, Missing Number, F(1, 110) = 100.33, p = .00, and Quantity

Discrimination, F(1, 110) = 67.02, p = .00. See Table 6 and Figures 33 to 36 for summary of

pretest and posttest scores.

Table 6

Summary of Pretest and Posttests Scores (Proportion Correct) by Condition

Control n = 38 M (SD)

Jumper n = 36 M (SD)

Morphing n = 36 M (SD)

Multiplication Conceptual Task

Pretest .80 (.23) .81 (.26) .82 (.21) Posttest .84 (.17) .87 (.18) .87 (.16) Multiplication Timed Test Pretest .39 (.25) .35 (.26) .34 (.19) Posttest .44 (.27) .43 (.28) .43 (.19) Missing Number Pretest .49 (.25) .49 (.27) .46 (.21) Posttest .55 (.27) .54 (.27) .52 (.24) Quantity Discrimination Pretest .47 (.24) .45 (.20) .44 (.23) Posttest .54 (.25) .60 (.25) .55 (.24)

 

47

Figure 33. Mean proportion correct for Multiplication Conceptual Models tasks

Figure 34. Mean proportion correct for

Multiplication Timed Tests

Figure 35. Mean proportion correct for

Missing Number task

Figure 36. Mean proportion correct for

Quantity Discrimination

To test whether condition had a significant effect on posttest performance, a Multivariate

Analysis of Covariance (MANCOVA) was conducted, with Condition (Control, Jumper,

0.80 0.81 0.82 0.84 0.87 0.87

0.0

0.2

0.4

0.6

0.8

1.0

Control Jumper Morphing

Prop

ortio

n C

orre

ct

Multiplication Conceptual Models

Pretest Posttest

0.39 0.35 0.34 0.44 0.43 0.43

0.0

0.2

0.4

0.6

0.8

1.0

Control Jumper Morphing

Prop

ortio

n C

orre

ct

Multiplication Timed Test

Pretest Posttest

0.49 0.49 0.46 0.55 0.54 0.52

0.0

0.2

0.4

0.6

0.8

1.0

Control Jumper Morphing

Prop

ortio

n C

orre

ct

Missing Number

Pretest Posttest

0.47 0.45 0.44 0.54 0.60 0.55

0.0

0.2

0.4

0.6

0.8

1.0

Control Jumper Morphing

Prop

ortio

n C

orre

ct

Quantity Discrimination

Pretest Posttest

 

48

Morphing) as the independent variable, pretests (Missing Number, Quantity Discrimination,

Multiplication Conceptual Models, and Multiplication Facts) as covariates, and posttests

(Missing Number, Quantity Discrimination, Multiplication Conceptual Models, and

Multiplication Facts) as dependent variables. Although it appeared that there may have been

some differences in posttest scores by condition, the results showed that there were no

statistically significant effects of condition at posttests when controlling for pretest scores,

Wilks’ Lambda = 0.93, F(8, 200) = 1.04, p = .41. See Table 7 for a summary of results by

measure.

 Table 7

Summary of MANCOVA Analysis on Posttest Scores

Source Type III Sum of

Squares df Mean Square F p

Multiplication Conceptual Models 0.04 2 0.02 0.77 .47

Multiplication Facts 216.90 2 108.45 0.89 .41

Missing Number 0.89 2 0.44 0.06 .95

Quantity Discrimination 168.89 2 84.44 2.81 .07

Session Performance Analyses

 2) Are there differences in performance when given two different tools, Jumper or Morphing,

while using the Multiplication software program?

Descriptive analyses were conducted to determine mean performance on all four sessions

by Tool condition (see Figure 37). On average, it appeared that participants were less likely to be

correct on session items when they had No Tool (M = .96, SE = .01) and were more likely to be

 

49

correct when they had the Jumper Tool (M = .98, SE = .01) or the Morphing Tool (M = .98, SE =

.01).

Figure 37. Mean performance during treatment sessions by tool condition.

   

Descriptive analysis of performance from sessions 1 to 4 showed that there might be

differences in performance due to the order in which participants had the availability of the tools.

Those in the Tool First Condition for Jumper and Morphing appeared to be less accurate after the

tool was made available and then taken away. Session performance of the participants in the

Tool First* Jumper condition on average decreased from Session 1 (M = .89, SD = .31) and

Session 2 (M = .97, SD = .16) to Session 3 (M = .89, SD = .31) and Session 4 (M = .86, SD =

.35). Performance for participants in the Tool First*Morphing condition also followed the same

trend, decreasing from Session 1 (M = .96, SD =.20) and Session 2 (M = .99, SD = .12) to

Session 3 (M = .90, SD = .30) and Session 4 (M = .88, SD = .32). However, Session performance

improved from Session 1 (M = .95, SD = .22) and Session 2 (M = .87, SD = .34) to Session 3 (M

= .98, SD =.14) and Session 4 (M = .98, SD = .14) for those in the Tool Second* Jumper

condition. Session performance also improved for those in the Tool Second* Morphing condition

0.96 0.98 0.98

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

No Tool Jumper Morphing Lik

elih

ood

of C

orre

ctne

ss

Session Performance by Tool Condition (Mean Accuracy)

 

50

from Session 1 (M = .91, SD = .29) and Session 2 (M = .93, SD = .26) to Session 3 (M = .95, SD

= .21) and Session 4 (M = .97, SD = .16). Figure 38 shows the observed performance for Tool

Condition and Order for Sessions 1 to 4. Table 8 summarizes the mean accuracy from Sessions

1, 2, 3, and 4 by Order and Tool Condition.  

 

Figure 38. Observed performance for tool condition by order from Session 1 to 4.

0.89

0.97

0.89 0.86

0.96

0.99

0.90 0.88

0.95

0.87

0.98 0.98

0.91 0.93 0.95 0.97

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1 2 3 4

Prop

ortio

n C

orre

ct

Sessions

Performance by Session (Tool Condition*Order)

Tool First* Jumper

Tool First* Morphing

Tool Second* Jumper

Tool Second*Morphing

 

51

Table 8

Summary of Session Accuracy by Order and Tool Condition

Session 1

M (SD)

Session 2

M (SD)

Session 3

M (SD)

Session 4

M (SD)

Overall

M (SD)

Tool First* Jumper .89 (.31) .97 (.16) .89 (.31) .86 (.35) .91 (.27)

Tool First* Morphing .96 (.20) .99 (.12) .90 (.30) .88 (.32) .93 (.24)

Tool Second* Jumper .95 (.22) .87 (.34) .98 (.14) .98 (.14) .94 (.29)

Tool Second* Morphing .91 (.29) .93 (.26) .95 (.21) .97 (.16) .94 (.26)

To test whether or not the presence of the Jumper or Morphing Tools or the absence of

tools differentially affected accuracy on treatment tasks, treatment session data were analyzed in

a linear binary logistic regression mixed model analysis with Tool Condition (No Tool, Jumper,

or Morphing) and Order (Tool then No Tool or No Tool then Tool) as independent variables and

accuracy (correct or incorrect on each session item) as the dependent variable. Interactions

between Tool Condition and Order were also included in this analysis. The analysis indicated

that there were significant differences in accuracy during sessions by Tool condition, F(2, 3312)

= 31.15, p = .00. Although Order did not have a significant effect on accuracy, F(1, 3312) =

1.15, p = .29, there was a significant interaction between Order and Tool Condition, F(2, 3312) =

4.28, p = .01. Results of the linear binary logistic regression mixed model analysis are

summarized in Table 9.

In order to test the main effects of Strategy Used, which is discussed further in research

question 3, a linear binary logistic regression mixed model analysis was conducted with Strategy

 

52

Used included in the model (see Table 10). Strategy Used was not included in the original model

because it was possible that Tool Condition and Strategy Used were confounding variables.

However, further analysis showed that including Strategy Used in the model did not change the

outcome significantly. The analysis indicated that there were significant effects of Tool

condition, F(2, 3248) = 5.01, p = .01, no significant effect of Order, F(1, 3248) = 1.77, p = .18,

significant effects of Strategy Used, F(5, 3248)=7.83, p = .00, and a significant interaction

between Order and Tool Condition, F(2, 3248)=3.39, p = .03, on accuracy during treatment

sessions. The effects of Strategy Used will be discussed in the next section.

Table 9

Summary of Linear Binary Logistic Regression Mixed Model

Source F df1 df2 p

Tool Condition 31.15 2 3312 .00

Order 1.15 1 3312 .29

Tool Condition * Order 4.28 2 3312 .01

Table 10

Summary of Linear Binary Logistic Regression Mixed Model with Strategies

Source F df1 df2 p

Tool Condition 5.01 2 3248 .01

Order 1.77 1 3248 .18

Strategy Used 7.83 5 3248 .00

Tool Condition * Order 3.39 2 3248 .03

Post-hoc Least Significant Difference pairwise comparisons of Tool Condition indicated

that when participants had the Jumper Tool (M = .99, SE = .00) and Morphing Tool (M = .99, SE

 

53

= .00), they were more likely to be correct than when they had No Tool (M = .95, SE = .01),

t(3312) = 4.39, p = .001 and t(3312) = 4.38, p = .001, respectively. However, when comparing

the Jumper and Morphing tools, the difference was not statistically significant, t(3312) = .13, p =

.891. See Table 11 for a summary of pairwise comparisons.

Table 11

Post-hoc Least Significant Difference Pairwise Comparisons of Tool Condition

Tool Comparisons Mean

Difference t df p

Jumper vs. No Tool 0.043 4.39 3312 .001

Morphing vs. No Tool 0.042 4.38 3312 .000

Jumper vs. Morphing 0.001 0.13 3312 .894

In order to explain the significant interaction between Tool Condition and Order, post-

hoc Least Significant Difference pairwise comparisons were included in this analysis. The

analysis revealed that the interaction of Jumper*Order was statistically significant, F(1, 3312) =

2.21, p = .03, while interactions between No Tool* Order and Morphing*Order were not

statistically significant, F(1, 3312) = 0.74, p = .46 and F(1, 3312) = 0.84, p = .40, respectively.

The results indicate that those who had the Jumper Tool first were less likely to be correct (M =

.98, SE = .01) than those who had the Jumper Tool second (M = 1.00, SE = .00). This is not

surprising because the participants who had the Jumper Tool in sessions 3 and 4 had more

                                                                                                               1 A GLM Repeated Measures ANOVA was also conducted to test whether Tool Conditions had a differential effect on session performance. Results showed that Tool Condition did have a significant effect on accuracy, F(2,214)=12.13, p = .00. Post-hoc Least Significant Difference pairwise comparison confirmed that the Jumper (M = .93, SD = .15) and Morphing (M = .94, SD = .11) conditions were more likely to be correct than the No Tool Condition (M = .86, SD = .23), t(299)= 3.44, p = .001 and t(299) = 4.04, p < .001, respectively.

 

54

practice than those who had the Jumper Tool in sessions 1 and 2. A summary of estimated

marginal means for Tool Condition by Order are in Table 12.

 Table 12

Summary of Mean Likelihood of Correctness of Tool Condition by Order

Order Mean SE

1 (Tool First)

No Tool .94 .02

Jumper .98 .01

Morphing .99 .00

2 (Tool Second)

No Tool .96 .01

Jumper 1.00 .00

Morphing .99 .01

3) Are specific strategies, such as known multiplication facts and efficient counting strategies,

predictive of children’s success in solving multiplication problems and/or number sense

activities? If so, which strategies?

In order to investigate the effects of strategy on children’s success in solving

multiplication problems accurately, four analyses were conducted to answer the following

questions: a) How often was each strategy used? b) Did strategy affect accuracy during session?

c) Were strategies predictive of posttest performance on multiplication tasks and number sense

tasks? d) Was there an association between the presence of a tool and strategies used, and if so,

was there a difference between the frequency of strategies used in the presence of a tool (Jumper

or Morphing)? Results are reported in this order, however, before discussing these analyses, is a

discussion about descriptive data about how often children used each strategy.

 

55

How often was each strategy used?

Descriptive analysis on the frequencies of how often each strategy was used indicated

that Fast, Tool Use, and Delayed accounted for 91.73% of all items (see Figure 37). Since the

counts for Nines finger trick, Skip Counting, Known Facts, Tens, and Doubling were very low

and considered to be better strategies than adding one-by-one, the researcher combined these to

make a category called Advanced. Adding one-by-one had a very low count, but remained as its

own category because it was considered to be inefficient strategies and most prone to error.

Therefore, causing children to be less likely to be correct compared to Fast, Tool, and Advanced.

Other Adding only occurred 9 times out of 3,629 cases and it wasn’t clear what children were

doing in these cases, so it was combined in the Other category. This resulted in six categories:

Fast, Delayed, Add 1x1, Tool Use, Other, and Advanced. See Figure 38 for the frequencies of six

strategy types used to solve multiplication problems while using the software program and see

Table 8 for a description of the strategy types.

Figure 39. Count of coded strategies.

973 1271

48 75

9 11 38

84 19

1085 16

0 500 1000 1500

Fast Delayed Add 1x1

Doubling Other Adding

Tens Known Fact

Skip Counting 9s Finger Trick

Tool Other

Count of Coded Strategies

 

56

Figure 40. Count of new strategies.

 Table 13

Definitions of Strategy Categories for Data Analysis

Strategy Use Description

Fast Answered in under 3 seconds Delayed Answered in 3 seconds or more & no expressed strategy Add 1x1 Add multiples one-by-one Tool Use Used tool in the software to solve the problem Other Anything other than the defined strategies Advanced Included skip counting, doubling, derived facts, and nines finger trick

Did strategy affect accuracy during session?

In order to test the effects of strategies on accuracy, a linear binary logistic regression mixed

model with strategies as the independent variable, student at random factor, and accuracy as the

dependent variable was conducted. Strategies had a main effect on accuracy during session

973

1271

48

1085

25

227

0 500 1000 1500

Fast

Delayed

Add 1x1

Tool

Other

Advanced

Count of New Strategy Categories

 

57

performance F(5, 3253) = 19.370, p < .01. Figure 39 summarizes accuracy (likelihood of

correctness) by strategies.

 

Figure 41. Accuracy (likelihood of correctness) by strategies.

Post-hoc least significant difference pairwise analysis showed that on average, children were

more likely to be correct when using the Fast strategy (M = .99, SE = .00) compared to the

Delayed strategy (M = .96, SE = .01), t(3248) = 3.54, p < .001. Additionally, children who used

the fast strategy were more likely to be correct when compared to those using Other strategies (M

=0.93, SE = .03), t(3248) = 2.04, p = .05, and Advanced strategies (M = .96, SE = .02), t(3248) =

2.33, p = .02). Also, students who were Delayed in answering were less likely to be correct than

when they used the Tool (M = .99, SE = .00), t(3248) = -2.62, p = .01. Table 12 summarizes the

post-hoc pairwise comparisons for strategy.

0.99 0.96 0.96 0.99 0.93 0.95

0.5

0.6

0.7

0.8

0.9

1

Fast Delayed Add1x1 Tool Other Advanced

Lik

lihoo

d of

Cor

rect

ness

Accuracy by Strategy

 

58

Table 14

Summary of Post-hoc Least Significant Difference Pairwise Comparisons

Strategies Mean

Difference t df p

Fast vs. Delayed .03 3.54 3248 .00

Fast vs. Add1x1 .03 1.53 3248 .13

Fast vs. Tool .01 1.48 3248 .14

Fast vs. Other .06 2.04 3248 .04

Fast vs. Advanced .04 2.33 3248 .02

Delayed vs. Add1x1 .00 0.04 3248 .97

Delayed vs. Tool -.03 -2.62 3248 .01

Delayed vs. Other .03 1.11 3248 .27

Delayed vs. Advanced -.03 -1.53 3248 .13

Add1x1 vs. Tool -.03 -1.20 3248 .27

Add1x1 vs. Other .03 0.92 3248 .36

Add1x1 vs. Advanced .01 0.26 3248 .80

Tool Use vs. Other .06 1.80 3248 .07

Tool Use vs. Advanced .03 1.88 3248 .06

Other vs. Advanced -.03 -0.83 3248 .41

Were strategies predictive of posttest performance on multiplication tasks and number sense

tasks?

In order to test whether or not strategies were predictive of posttest performance, a

multivariate multiple regression, with strategies as independent variables and posttests as

dependent variables, was conducted. The results indicated that strategies were predictive of

performance on three of the four posttest measures: Multiplication Facts, R2 = .444, F(5, 78) =

9.45, p < .001, Missing Number, R2 = .27, F(5, 78) = 9.45, p = .001, and Quantity

Discrimination, R2 = .26, F(5, 78) = 4.17, p = .001. Specifically, the analysis showed that Fast

strategy was predictive of performance on Multiplication Facts, F(1, 78) = 11.68, p = .001,

Missing Number, F(1, 78) = 4.81, p = .03, and Quantity Discrimination, F(1, 78) = 6.89, p = .01.

 

59

Also, results indicated that Tool Use was predictive of performance on Multiplication Facts, F(1,

78) = 4.17, p = .045.

Was there an association between the presence of a tool (Jumper or Morphing) and strategies

used? If so, was there a difference between the frequencies of strategies used in the presence of a

tool (Jumper or Morphing)?

A cross-tabulation of Strategies and Tool Condition was conducted to calculate the

frequencies of strategies used in two Tool conditions: Jumper Tool and Morphing Tool. See

Table 13 for a summary of strategies in each Tool Condition. To test whether there was an

association between Tool Type (Jumper and Morphing) and Strategy Use, a multinomial logistic

regression mixed model with Tool Type as the independent variable, Student ID as the random

factor, and Strategy Use as the dependent variable was conducted. The results indicated that Tool

Type did not have a significant effect on Strategy Use, F(5, 1507) = 1.00, p = .42.

Table 15

Frequencies of Strategies Used by Tool Condition Cross Tabulation of Strategy Use and Tool

Condition

 

Tool

Strategy Used Jumper Morphing

Fast 164 144

Delayed 146 135

Add1x1 2 2

Tool Use 430 471

Other 0 1

Advanced 18 4

 

  60  

CHAPTER 4:

DISCUSSION

This study examined the effectiveness of two digital tools: a virtual number line (Jumper

Tool); and a dynamic hundreds chart (Morphing Chart) in improving children’s conceptual and

procedural understanding of multiplication. It also explored strategy use while solving

multiplication problems and how these strategies affected children’s performance on

multiplication tasks and how this improved children’s number sense in general. This section

interprets the results of each research question, discusses limitations, presents areas for future

research, and offers implications for designers and educators.

Overview of Findings

Research Question 1. The first research question asked whether the Multiplication

Software improved children’s conceptual and procedural understanding of multiplication and/or

number sense. Specifically, the study examined whether those children using the software

outperformed those who did not use the software. The findings indicate that all groups – Control,

Jumper, Morphing – made significant gains from pre- to posttests, however, there were no

significant differences in performance between those in the treatment group and those in the

control group. Also, the two treatment groups did not differ from one another from pre- to

posttest. There were limitations in the design of the study that contributed to a lack of significant

findings from pre- to posttests, which will be discussed further under Limitations.

Research Question 2. The second research question asked whether or not there were

differences in performance in the presence or absence of the Jumper or Morphing tools while

 

  61  

using the software program. The findings suggest that there were no differential effects of

Jumper or Morphing in overall performance while using the software, however, the presence of

either of the tools proved to be more advantageous to not having a tool at all. This was an

expected outcome, as the tool provided children a form of help, similar to a calculator, and as

long as the user knew how to manipulate the tool, they were very likely to the get the correct

answer. In both of the Tool Conditions, participants saw a tutorial on how to use the tool and

then the tool automatically appeared in the screen for each trial. Therefore, it was likely for a

child to use the tool and more likely to be correct when the tool was present compared to when

there was no tool present.

As a follow up to this question, the study also explored whether or not the order in which

children used the tools throughout the sessions made a difference on performance. As discussed

earlier, children either had the availability of the tool in Sessions 1 and 2 (Tool First) or they had

the availability of a tool in Sessions 3 and 4 (Tool Second). The results indicate that there were

no significant differences by order on overall performance, but when analyzing the performance

of the Tool Condition by Order, there were significant differences in the group who used the

Jumper Tool. Participants who used the Jumper Tool in sessions 1 and 2 were 95% likely to be

correct in sessions 1 and 2 compared to those who had the Jumper Tool in sessions 3 and 4, who

were 99% likely to be correct when they had the tool. This result indicates that those who had

Tool First performed worse than those who had the Tool Second, which is not surprising because

those who had Tool Second already completed Sessions 1 and 2, so by the time they completed

sessions with the tool, they had twice as much practice as the group there were being compared

against. These groups are not equivalent, so the results of this test do not indicate whether order

impacted their overall learning. However, it is important to point out that there were no

 

  62  

differences between those in the Tool First and Tool Second groups for No Tool and Morphing.

There were no differences in performance between those with No Tool in Sessions 1 and 2

compared to those with No Tool in Session 3 and 4. This was also the case for those with

Morphing in Sessions 1 and 2 compared to those with Morphing in Session 3 and 4. If more

practice should help children perform better, these results do not support such an argument.

Rather, these results seem to indicate that more practice with items that were already mastered,

do not make much of a difference. It also indicates that the treatment tasks were too easy for this

sample.

Research Question 3. The third research question asked whether there are specific

strategies are predictive of children’s success on multiplication and number sense tasks. Main

findings on session performance indicated that strategy had a significant effect on accuracy when

using the software program. When comparing each strategy to one another, Fast responses,

which are responses answered in less than three seconds, were more likely to be correct than

Delayed, which are responses answered in three or more seconds. Fast Responses were also more

likely to be correct than those who employed an Advanced strategy. Finally, Tool Use responses

were more likely to be correct than Delayed responses. These results were expected because Fast

responders are likely to know the answers from memory and Tool Use responders could use the

tool to solve the problem for them. Therefore, children who know their multiplication facts or

used a tool are most likely to be correct.

Further analyses on whether or not strategies were predictive of posttest performance

were conducted. Results indicated that Fast responses were predictive of performance on

Multiplication Facts, Missing Number, and Quantity Discrimination. As mentioned before, Fast

responders most likely solved the multiplication problems from memory, so it is not surprising

 

  63  

that this strategy was predictive of performance on a timed multiplication test. Missing Number

and Quantity Discrimination measure general number sense performance and the results

indicated that Fast responses were predictive of better number sense abilities at posttest. These

results indicate that there is a positive relationship between Fast responses and performance on

number sense items. This is not surprising as children who are fast at answering multiplication

problems would also be good at answering timed number sense items, where some items include

multiplication items (Quantity Discrimination) and skip counting (e.g. Missing Number).

Results also indicated that Tool Use was predictive of performance on Multiplication

Facts tests. There was a positive relationship between Tool Use responses and mastery of math

facts. This may indicate that using the tool to solve multiplication problems can help children

improve in mastering their facts. This finding was also expected, as practicing math operations

can help children memorize their facts or solve those facts more quickly.

Limitations

This study has several limitations. This study was conducted as a learning study in a

series of grade-level studies from pre-K to third grade, which required a learning study to be

conducted with third graders. In the United States, multiplication is taught at end of second grade

and throughout third grade. By the end of the year, third graders have been taught a great deal

about multiplication strategies and models. Due to scheduling constraints, the study was

conducted at the end of the school year. It was not possible to conduct this study until the end of

the year because teachers and principals would not allow such an intervention to take place until

after the completion of state-wide achievement tests in early April. Therefore, the study did not

take place until the middle of April and continued into the beginning on June. By this time of the

 

  64  

school year, children were quite competent in their ability to solve multiplication tasks and

number sense items. These items proved to be relatively easy for the participants, which was

particularly the case for the Multiplication Conceptual Tasks (MCM), which were designed to be

near transfer tasks for the software treatment. Almost half of the students scored above 90%

correct on those items at Pretest and almost one third of the participants scored 100% correct at

pretest. Therefore, there was a ceiling effect for this measure and there was very little to no room

for growth for most participants. It should also be noted that the MCM models did include items

that directly modeled the Number Jumper, but there were no items that directly modeled the

Morphing Chart. This measure should have included items that were more closely related to the

Morphing Chart.

The treatment tasks also proved to be too easy for end-of-year third graders. It would

have been better to make the session treatment items more challenging or to conduct this study

with second graders. However, due to the commitment that the researchers had made to the

participating schools to work with third classrooms since the prior year, it was not possible to

conduct a multiplication study with another grade level. Also, it was not possible to conduct this

study at the beginning of the year because the school administrators would not allow this. The

best solution under these circumstances would have been to make the software items more

difficult and subsequently make the MCM items more challenging. This would have resulted in a

wider distribution of scores.

There are also limitations to generalizability. The sample size was relatively small (N =

122) and most of these children were from a high income, high achieving school (School A =

85), while only a small proportion of participants were from low income, low achieving schools

(School B = 12, School C = 26). Although just as many permission forms were sent out to both

 

  65  

types of schools, participation at School A was 82%, while participation at School B was 37%

and School C was 24%. Therefore, the sample consisted of a greater proportion of students from

a high-income, high-achieving school compared to those in low-income, low-achieving schools.

This may have contributed to the ceiling effect of the MCM items as well.

In an effort to offer children a similar experience of working one-on-one with a

researcher while using a computer program, Braintastic Reading software was chosen as a grade-

level appropriate program. However, choosing Braintastic Reading did not address mathematical

topics and therefore, was not the best choice for use with the Control group. Instead, paper-based

multiplication activities would have been a better activity for comparison purposes.

Future Work

Future studies should consider modifying the near transfer tasks, such as those in the

MCM tasks, to be appropriately designed for students in a given age group and time of year that

they are participating in such a study. The measures should also include items that are more

closely related to the Morphing Chart. Studies with younger children, or early in the school year,

should be conducted because instruction of multiplication typically begins in second grade and

not in third grade.

Studies using software programs as intervention are valuable because novel tools in a

digital setting can be tested out and backend data can be collected easily while students use

software. Although data mining techniques were not employed in these analyses, log data from

this study could be used to analyze more about children’s behavior while using such software.

As a follow-up study, researchers could explore which students were fast and correct in

the sessions 1 and 2 to see who already knew their facts. Eliminating such cases would allow

 

  66  

researchers to learn more about those who had not mastered their facts yet. Also, order effects

could be explored further. The timing of interventions seems to matter and having time to

explore and invent before receiving explicit instruction about how to solve math problems

appears to have an advantage over a “tell and practice” approach to teaching (Schwartz &

Martin, 2004). Investigating how the timing of when interventions are implemented affects

learning could have important implications for instruction and the design of software programs.

More studies should include information on how children solve problems, rather than just

looking at performance on activities. Exploring conditions that affect the choices that children

make on the type of strategy employed would also be valuable. As a follow up to this study,

more analyses could be conducted on which strategies children choose in the absence of a tool,

after they have already used a tool.

Implications for Instruction

There are varying philosophies on how to teach children math concepts and skills.

Memorizing facts seems to be a useful way to solve multiplication facts, but what do you teach

when children have not memorized their facts yet and they are just learning about the idea of

multiplying? This study indicated that overall performance during the treatment sessions was

better when children had the tool present compared to having no tool at all. It appears that the

tool had a positive effect on performance. It is not clear what aspect of the tool was providing the

most aid to children, but it does appear that having some help was better than having none at all.

Also, tool use was associated with better outcomes in the Multiplication Timed Test, which

indicates that using the tool was associated with better mastery of facts. Since the tool can be

 

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seen as a type of calculator, it appears that practice with the tool can help children master their

multiplication facts and therefore, using calculators should be considered as useful.

The use of math software can also prove to be useful for teaching. Software can certainly

replace worksheets and teachers can leave the scoring up to the software instead of grading a

whole set of papers. Software also has the ability to give students immediate feedback. Typically

a child might complete their homework or test and have to wait a few days before the teacher has

a chance to look over the work and provide feedback, but with software, the child can find out if

he s/he is correct and many programs can even pinpoint the type of errors that children make.

Implication for Designers

Digital tools can help children solve problems in ways that they may not be able to on

paper or in their minds. Software designers can improve the design of educational software by

conducting research on the effectiveness of such tools.

This study utilized backend data for studying children’s interaction with the software.

When designing educational software, it is important to build in backend log systems and it

would be useful to also include backend data analytics that can report performance for users,

teachers, and parents. Although there are many educational apps on the market, very few of these

offer reports on performance. Much of this is due to a lack of backend data collection and

analytical systems.

Conclusions

Although there has been much research conducted on how children learn about basic

number concepts, addition, and subtraction, there are not many studies about children’s learning

 

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of multiplication, nor has there been much research with young children using software. This

study sought out to show that certain models and digital tools were effective in helping children

learn about multiplication. Although the study could not show that these tools had a differential

effect on children’s learning or performance, the study did demonstrate that having a tool or

visual model is better than having none at all. The results of this study also provide evidence that

children who know their facts from memory have better number sense abilities. Therefore,

educators should not discount the value of learning and mastering multiplication facts. Efforts to

help children master their facts, as well as understand concepts, should be a focus of elementary

math education and further research should be conducted on how to best help children reach

mastery. Additionally, this study demonstrates that research on digital learning has to be very

different from traditional learning studies. In traditional learning studies, reinforcement of

learning is compared to the absence of reinforcement, but in the case of software, there are many

more factors that impact learning (e.g. graphics, timing of intervention, game mechanics, types

of feedback, frequency of feedback and tutorials, quality of tutorials and feedback, etc.).

Experimental studies may not capture all these factors, so case studies and qualitative studies

may make more sense in studying learning with software.

 

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APPENDICES

APPENDIX A: Missing Number

 

 

 

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APPENDIX B: Quantity Discrimination

 

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APPENDIX C: Multiplication Timed Test

 

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APPENDIX D: Multiplication Conceptual Tasks

 

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APPENDIX E: Strategy Coding Sheet

 

 

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APPENDIX F: Coding Glossary

Generate graphs and stats about percent of children who employ the following strategies as observed by coders:

• Fast Response: NO explicit strategy use, but student answered in 3 seconds or less • Delayed: NO explicit strategy use, but student answered in more than 3 seconds • Repeated Addition: added a number repeatedly

o Adding on one addend at a time (3 + 3 + 3 + 3…) o Doubling (3 + 3 = 6; 6 + 6 = 12; 12 +12 = 24) o Other: any other type of repeated addition

• Derived Fact: Child uses a known fact to help them solve a problem. (CHOOSE ONLY ONE) o Tens (multiplies a number by 10 to find product)

" ex: For 12 x 9, child says, “10 times 9 equals 90, 90 plus 18 equals 108) o Known Fact (other than 10)

" ex: For 12 x 9, child says, “11 times 9 equals 99, 99 plus 9 equals 108

o Skip Counting: Child skip counts to find an answer. o For 3x4, “Three, six, nine, twelve.” o For 5x9, “five, ten, fifteen, twenty, twenty-five...forty-five.”

o Nines Fingers Trick: Child uses fingers to figure out multiples of Nine. o TOOL: Child uses the tool to solve the problem.

o child visibly uses the tool by clicking, pointing, or using mouse over the tool. o Before: used tool before using another strategy or answering o After: used tool after already using another strategy

o Other (NOT adding): Check this if a child used any other strategy from the ones specified above. Please make notes.

 

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APPENDIX G: Data Entry Coding Scheme

CODE Strategy 1 Strategy 2 1 Fast 2 Delayed 3 Add 1x1 4 Doubling 5 Other Adding 6 Tens 7 Known Fact (not tens) 8 Skip Counting 9 Nines finger trick 10 Before 11 After 12 Other 13 After Fast 14 After Delayed 15 After Add 1x1 16 After Doubling 17 After Other Adding 18 After Tens 19 After Known Fact (not tens) 20 After Skip Counting 21 After Nines finger trick 22 Before Add 1x1 23 Before Doubling 24 Before Other Adding 25 Before Tens 26 Before Known Fact (not tens) 27 Before Skip Counting 28 Before Nines finger trick 29 Delayed Tens 30 Delayed Known Fact 31 Skip Counting Doubling 32 Add 1x1 Skip Counting 33 Doubling Known Fact 34 Delayed Add 1x1 35 Known Fact Other

 

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36 Add 1x1 Other 37 Doubling Other 38 Tens Skip counting 39 Add 1x1 Known Fact

 

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APPENDIX H: Nines Finger Trick

This trick entails children holding up their ten fingers and then folding one finger down to figure out how much nine times “x” is. For example, if you want to multiply nine times eight, you fold down the eighth finger from the left and you are not left with seven fingers to the left and two fingers to the right for an answer of seventy-two. See examples below.