Improving 5th Grade Multiplication Performance Through Various Interventions

Colorado Christian University

Developing the Christian MindLearning in Community with Integrity

Cody J. Perry

Improving Fifth Grade Multiplication Performancethrough Constructivist Learning Methods and

Self-Guided Interventions

MACI Program

I certify that I have evaluated this master Practicum and that inmy opinion it conforms to the prevailing standards of scholarly

inquiry and is fully acceptable in scope and quality as aPracticum for the degree Master of Arts.

_________________________________________________________Capstone Professor

IMPROVING FIFTH GRADE MULTIPLICATION PERFORMANCE 2

_________________________________________________________

Wendy Wendover, Ph. D., MACI Director

_________________________________________________________Date

This master’s capstone project has been submitted to Colorado Christian University School of Graduate Studies and is accepted as partial fulfillment of the requirements for the degree of Masters of Arts in Curriculum and Instruction.

Improving Fifth Grade Multiplication Performance through

Constructivist Learning Methods and Self-Guided Interventions

Cody J. Perry


August 11, 2013

Colorado Christian University

Chapter One

Math performance has become a major topic for debate and

discourse among educators in the United States. While

significant emphasis has been placed on preventing illiteracy, it

seems that too many are willing to accept the numeracy issues

present in the U.S. today. Many people attribute their math

problems to inherited abilities or cite their apprehension of

math to justify poor math skills. Yet, mathematics does not have

to be an object of fear and can be grasped and understood by the


majority of students. Math education has been successful in the

past for a percentage of the population, but it fails to reach as

many people as educators would like. Therefore, additional

research and the implementation of improved teaching methods is

necessary in order to improve students’ math abilities. The

curriculum and research presented in this capstone project are

needed to improve skills and can benefit all math students who

are learning multiplication. According to a New York Times

article (2012), students in the United States are performing well

below their foreign counterparts. The article cites data from

the most recent Trends in International Mathematics and Science

Study (TIMSS) that shows 7 percent of American students scored at

an advanced level. In contrast, 47 and 48 percent of students in

South Korea and Singapore scored at an advanced level on the

assessment (Rich, 2012). Clearly, there appears to be a need for

improvement in math education in the United States.

While these statistics seem discouraging, they do not need

to be an indicator of future performance. If other nations have

found success and improved performance, it can also be done in

American schools. This paper explores basic multiplication facts


and concepts that all fifth grade students encounter and that

many find to be difficult or overly challenging. If these

multiplication facts are not properly grasped, students are

denied any significant mathematics growth past the 8th grade

(Caron, 2007). The failure of schools to properly teach

multiplication has presented a major problem that this project

aims to minimize significantly.

There are two main sides of the fierce mathematics debate

about how math should be taught to students. One side of the

argument claims that math instruction should be predominantly

teacher led, where students are taught a specific algorithm and

the resultant steps needed to solve problems. Conversely,

advocates for the other view argue for a student centered

approach where the learner becomes the driving force behind their

education, constructing meaning from the material being studied.

The argument continues in papers and discussions, yet it has not

necessarily contributed to sweeping changes in classrooms across

the U.S. (Jones & Southern, 2003). However, one does not need to

take such a polarizing stance upon the issue. Mathematics

instruction can be a compromise of instructional strategies since


students do not learn exactly alike and their brains process

things in a variety of ways. By incorporating multiple methods

and strategies into this curriculum design, students will be able

to reach a greater level of mathematics success.

Problem Statement

The problem that this curriculum aims to solve centers on a

student’s ability to multiply various numbers. At St. Mary’s

Catholic School, approximately 60% of incoming 5th grade students

are less than proficient with basic multiplication facts, which

inhibits further mathematics progress with multiplication,

division, and higher order tasks. Without a confident and

concrete grasp of multiplication, students are frustrated and

unable to become successful in future math classes. This poses a

problem for students, but is a greater issue for our educational

system at large.

Purpose Statement

Since students struggle to adequately solve multiplication

facts, the purpose of this academic research is to develop a unit

that will increase 5th grade students’ multiplication proficiency


to 90% of basic facts. At present, only 40% of students in the

fifth grade class are considered proficient on standardized test

scores in the area of multiplication. This means that 6 in 10

students are failing to solve facts at a level that would be

considered proficient. Furthermore, math skills must improve

across the board to prevent some of the current trends in math

education and careers. According to 2009 numbers from The

National Science Foundation, 42.5 percent of mathematics graduate

students were non-resident aliens or temporary residents of the

United States (National Science Foundation, 2012). American math

students are in need of improved mathematics skills in order to

change this in the future. Since multiplication is such a

foundational concept in mathematics, this paper aims to solidify

this important cornerstone before it is too late.

Educational Goal

The goal for the following project is to develop a

curriculum that will increase the percentage of students from 40

to 90 percent who can correctly solve multiplication facts with

at least 90% accuracy. Since multiplication and the ability to

quickly solve problems mentally is so foundational to overall


math aptitude, it is necessary to improve this skill to solidify

future math success. Essentially, the curriculum aims to

increase basic multiplication proficiency, freeing up working

memory to tackle other, more complex mathematical tasks, while

simultaneously developing overall multiplication skills and

conceptual knowledge. Students who automatically multiply facts

without the aid of manipulative tools or visual aids can focus

their mental energy on more complex problems and functions.

Furthermore, by improving multiplication skills, students will be

better prepared for additional math operations such as division.

Curriculum Variables and Teaching Factors

Teaching involves much more than standing before a room full

of students and espousing information as quickly as possible.

One must look at many variables and factors before planning any

unit of instruction, such as the type of learning styles

represented, the methods used, physical surroundings, and others.

When designing this unit, the two main variables for

consideration are the methodologies or strategies used to deliver

information and the attitudes that students maintain toward the

information.


Methodology. The methodology that an educator chooses to

use can influence more than just the amount of time that a lesson

will take. The methods and strategies used affect retention

rates, interest levels, and many other factors. In the past,

many teachers have used direct instruction coupled with rote

memorization drills to develop multiplication skills in their

students. However, this method has only been effective for 70

percent of teachers polled and is an ineffective way of

memorizing that takes its toll upon students (Caron, 2007). In

order to improve multiplication success, the project will employ

a pair of self-guided interventions for student practice. In

addition, this project will look at student focused methods of

instruction and learning such as constructivism and

collaboration. This will allow students to work out math skills

and concepts for themselves rather than the teacher attempting to

transmit the information directly. Students will be given the

opportunity to explore math independently, alleviating some of

the perceived pressure and anxiety present during direct

instruction.


Math attitude. Another factor that will influence the

strength and effectiveness of instruction is the attitude toward

math that students possess. If one allows students to be

negative and their math anxiety is persistent, then students will

spend more time using cognitive functions to seek out avoidance

techniques, making them more likely to perform poorly (Ashcraft &

Krause, 2007). Based upon this information it will be necessary

to look at the attitudes and emotional beliefs of students as a

limiting factor. In designing the course, anxiety and ways to

alleviate it will be looked at in order to free up cognitive

function for multiplication and the math processes involved. If

a student can enter class with a positive attitude, believing in

their ability to perform math tasks, they will be more likely to

find success in learning multiplication. The project will look

at performance levels in order to improve math attitudes while

also assessing student anxiety levels to determine proper

strategies to use for improved multiplication skills.

Path analysis. The aforementioned teaching and influencing

factors have been analyzed in a visual model through the use of

path analysis, which can be found in Appendix A. The analysis


looks to determine which factors will have the greatest influence

upon student performance in order to address these items before

implementing the project. By analyzing these factors, one may be

able to determine the elements that need to be changed in order

to garner the greatest performance gains.

Needs Assessment

In developing a new project or curriculum unit it is

imperative to identify and collect information on the prospective

student group (Branch, 2009). In addition, it is of great

importance to look at the resources that are available to the

instructor or teacher before implementing any project.

Learning environment. Branch identifies four main resource

types when designing a course: content, technology, instructional

facilities, and human resources (2009). One must look at the

available resources in each of these main areas as well as any

resources that may be limited or absent. In this manner, one can

prevent many instructional issues that may arise before they have

the opportunity to impede learning. In regards to content

resources, Progress in Mathematics is the district provided textbook

for 5th grade use that will be one resource of many (LeTourneau,


Posamentier, & Ford, 2006). In addition to this text, the course

will also utilize web based resources such as Khan Academy,

teacher generated worksheets, and the SmartBoard with its

accompanying tools and applications. In order to utilize some of

these content items, students will have access to technology

based resources such as district provided laptop computers.

Furthermore, students will be given the option to bring in

Android or Apple iOS based devices for use in the classroom.

Students will also be using worksheets and small, individual

whiteboards to complete the instructional unit.

Facilities that will be used in the project include the main

classroom, which consists of 19 individual desks spread evenly

throughout the room. Each desk faces the large whiteboard and

SmartBoard at the front of the room and are close enough to one

another to allow for open class discussions. There is a large

work table at the rear of the room that students will use for

collaborative work sessions when necessary. Adjoining the main

classroom is a collaborative learning space (CLS) which houses a

whiteboard, three large tables, and a variety of comfortable,

ergonomic furniture for student use. The CLS is an ideal space


for small group discussion and collaborative sessions that allows

the teacher to monitor progress in both rooms as students work.

Human resources are limited at St. Mary’s with the main classroom

teacher typically being the only person available for

instruction. The school does not currently have a

paraprofessional available for 5th grade, but with no students

currently on individual education plans (IEPs) or 504 plans, this

is not a concern at this time. The librarian will be available

for assistance and the media / computer teacher has agreed to use

class time for students to practice upon teacher request.

Instructional environment. When designing new instruction

one must also look at the group of students involved in the

instructional unit. The fifth grade class is made up of 19

students, 10 of which are female and 9 are male. These students

are primarily of Caucasian descent with 2 students of Hispanic

origin, but possessing English as a first language and one

student is African-American. The students come from

predominately upper middle class backgrounds as they pay

considerable tuition to attend St. Mary’s Catholic School. Many

in the class are from a military background and have one or both


parents employed by the United States Air Force at F.E. Warren

Air Force Base. Overall, this group of students is motivated to

perform and when motivation becomes an issue it is quickly

resolved with minor extrinsic motivators.

Developmentally, students are closely matched with minor

needs for adjustment and differentiation. There are currently no

individual education plans (IEPs) or 504 plans in place and all

students seem to be willing to attempt new subjects and learning

units. Differentiation will be accomplished through the careful

grouping of students in sets of 3-4 to accommodate open

discussion and collaboration. Before implementing small groups,

students will need to be coached on expectations and procedures

to prevent wasted time and resources. Students will have these

expectations communicated to them from the beginning. Fifth

grade is often a difficult and transitional time for students and

can be socially awkward for many children. This class had many

social issues last year including a daunting violent student

issue and behavioral expectations will need to be made clear to

prevent further problems from arising.


Academically, students are performing well in all subject

areas but mathematics. Furthermore, standardized test scores

have been acceptable except in math. On recent Measures of

Academic Progress (MAP) tests, 84 percent of students scored at

or below the 54th percentile in relation to the Northwest

Evaluation Association (NWEA) norming study (Northwest Evaluation

Association, 2013). Student growth also failed to meet

expectations in relation to median scores which decreased by four

points from fall to spring 2013. This data indicates a negative

trend in mathematics ability and a change is necessary before the

decline continues to take place. Reading and language skills are

strong overall, with many students excelling in extracurricular

activities such as the spelling bee, essay contests, etc. It

seems that mathematics may be the subject that gives this group

the most trouble. In fact, St. Mary’s experiences an

overwhelming K-8 trend where math is the lowest performing

subject on standardized testing throughout the school.

Application Context

During the design and development process of an

instructional unit it is essential to look at skill areas that


students will need and how they perform in these areas. For the

purposes of this project the four most important skills to

consider are resource management, collaboration, information

processing, and communication skills.

Resource management and collaboration. The students in this

group are very inquisitive and ask numerous questions throughout

class. The challenge will lie in teaching and coaching students

to ask relevant questions that add to the learning experience.

The classroom culture will be designed around discussion and

collaboration with ample time at the beginning given for

conversation about expectations and appropriate questions.

Expectations will be set early and reminders will be made from

time to time to prevent errant discussions. Since this class is

very good at asking questions, strategies that employ inquiry

will be used to improve the unit. If students can continue to

improve their questioning techniques, inquiry methods will help

them draw conclusions about specific subject matter.

Expectations will also include proper behavior and the use of

physical and technological resources to prevent student misuse of

computers and other devices.


Collaboration will need to be taught and improved in this

class before moving forward in the unit. In the past, students

have occasionally struggled with unsupervised work in small

groups or pairs. However, with proper coaching and modeling the

desired results can be obtained. As groups and collaborative

sessions are executed one will need to be cognizant of one’s

ability to closely monitor students as they work. When using the

CLS, the teacher will need to maintain a line of sight or sound

to ensure that students can be redirected when necessary. If the

teacher can adequately model the importance of collaboration and

sharing of academic information it will alleviate many issues

before they have a chance to arise.

Finally, this group of children may struggle to meet

deadlines at the beginning of the instructional unit. In the

past, teachers have allowed assignments to be turned in as late

as two to three weeks after the stated due date. However, this

will not be permitted during the instructional unit. In order to

assess the effectiveness of the unit and instruction, it will be

necessary to evaluate student work on a regular and timely basis.

Therefore, at the beginning of the unit deadlines will be set


comfortably in the future with a small percentage deducted for

late submissions and as time progresses, deadlines will become

more stringent. However, the deduction will remain in effect

regardless of the deadline. By implementing this sliding

deadline, students can gradually become accustomed to turning

things in on time without being overwhelmed.

Information and communication. According to information

from teachers who have worked with this class previously, these

students struggle to process math information quickly. They

generally need more time to process a concept or task in relation

to their peers. Based on personal observations of other students

in the school, a different method of instruction may alleviate

many of these issues. Furthermore, students currently rely too

much on others for solutions and strategies. In order to be

successful in this unit, students will need to be independent

thinkers and will need to make better use of their resources.

They seem to be eager to work independently, but immediately seek

assistance when they meet resistance or difficulty. Students

will be coached and modeling will be provided to show students

how to work independently. Additionally, students who are able


to solve items without the aid of teacher involvement will be

rewarded with verbal recognition to promote self-learning and

independence.

This class communicates as well as most 5th grade students,

with no communication issues or disorders. They possess normal

and effective verbal and written communication skills as

evidenced by recent standardized tests and previous teacher

evaluations. However, their mathematics reasoning and problem

solving skills need to be improved. The school is currently

working to implement a standard mathematics vocabulary using

accepted math industry terminology and this class seems to be

doing well. However, the class tends to discuss things that are

not academically relevant if given the opportunity and must be

redirected to prevent lost time.

Project Terms

The following terms have been identified and defined for the

purpose of this project:

Multiplication facts are the products of numbers from 1 to 12

(i.e. 1 x 1 – 12 x 12).


Working memory is the part of a person’s memory that is

tasked with immediate, short-term tasks.

Rote learning (instruction) is a mechanical habit of memorization.

Constructivist methods are any number of instructional strategies

where a person, with proper guidance, constructs or builds their

own meaning from the material that is to be learned.

Collaboration is the act of learners working together to arrive

at a solution or to explain a mathematical concept.

Math anxiety is the fear or apprehension of mathematics that

leads a student to perform poorly or avoid math.

Automaticity is the ability to quickly solve math facts

without the aid of tricks or manipulative strategies (answers are

nearly automatic).

Taped problems refers to the strategy where students listen to

a tape and attempt to write the answer to a multiplication fact

before the answer is recited on the tape.

Cover, copy, and compare refers to the strategy in which a

student learns by looking at answers to facts, covering them and

answering problems and then comparing their answers to the

original answer sheet that was studied.


Conclusion

Mathematics instruction has gone through many changes in the

past, but many may be superficial at best. Many people today

face the inability to quickly and easily multiply numbers, making

more complex mathematical problems nearly impossible. It is due

to this lack of multiplication skill coupled with the resultant

anxiety that may have led to the widespread avoidance of math

related activities. Only 70% of teachers will admit that the old

techniques of rote learning are effective for most students

(Caron, 2007). This admission highlights the fact that teachers

possess the knowledge that math education is not as effective as

it should be. Up to this point this knowledge has not resulted

in positive, extensive change in mathematics instruction. It is

the aim of this curriculum project to close the gap of

multiplication performance while alleviating fears that students

may possess. If students can confidently multiply numbers

without overloading working memory, then they may be less likely

to avoid math in the future while learning more complex problems

more easily. The goal of the implementation of this curriculum

is to help students become better at multiplication so that


future math classes are greeted with eagerness and met with

success.

It has become permissible to perform poorly in mathematics

with many people blaming genetics or a lack of innate ability.

Many people even seem to take pride in the fact that they perform

poorly in math. While the United States continues to see

stagnant results, nations around the world are educating nearly 5

in 10 students to an advanced level of mathematics performance

(Rich, 2012). If the current trend continues, more jobs and

degrees may be lost to people that choose not to make the US

their home. With a greater understanding of math and lower

levels of anxiety we can stem this concerning trend and begin to

place more students in math related degrees and careers.

Outsourcing our math related jobs cannot be an option in the

future and blame cannot be placed upon students. Teachers must

recognize their responsibility to teach more than merely most

students. Teachers cannot be tolerant of educating less than all

of their students and must strive to improve mathematics

instruction for everyone. The following project aims to change


this complacency by developing a curriculum that will improve

multiplication skills, thereby improving future math success.

Chapter Two

From kindergarten classrooms to collegiate lecture halls,

math education continues to be debated as it remains at the

forefront of academic discourse. However, this long and decades

old discussion has not yet led to dramatic improvement in math

performance for many American students. The debate that rages on

between rote memorization and constructivist methods of learning

needs to arrive at a compromise rather than leaving students in

the divide. The changes will take time in the classroom, but

they are necessary and will require courage from teachers as

students need instructional changes to be successful in the

future. Teachers must take time to teach the material properly

and not merely speed through items just to cover them

superficially (Ward Hoffer, 2012). If something is not changed


in mathematics instruction our nation faces many issues,

including the loss of jobs and increased degrees being conferred

to the citizens of other nations.

Math has become a subject that is difficult for many and

conquered by few. Nearly one third of students who enter college

must take refresher courses before taking basic college

mathematics courses as a part of their normal course load. In

addition, almost as many students who start as freshmen fail to

graduate with their class at the end of high school (Steen,

2007). This alarming trend does not need to continue and can be

changed with a properly implemented action plan. A math

education can be viewed as a simple pyramid that has been built

using blocks. As the pyramid is built, a foundation is laid for

other blocks to be built atop the foundational layer. As each

subsequent layer is laid upon the first, a strong and beautiful

pyramid is formed that can stand up to myriad pressures.

However, if the foundation is missing blocks then the foundation

cannot support the weight of other layers and the entire pyramid

will eventually fail and collapse. This is comparable to many

students’ math abilities today. If a student does not grasp


multiplication from an early age, every block that they attempt

to add to their mathematical pyramid will add pressure to the

structure until the inevitable happens. As the foundation

becomes more unstable, the level of anxiety increases, making a

student more prone to a widespread avoidance of math (Ashcraft &

Krause, 2007). By creating a curriculum that brings together

improved memorization techniques and concept attainment one can

aid students in building a strong mathematical foundation where

new concepts and algorithms can be built. Once students

understand that their base is strong and steady they will be less

likely to avoid math and will have a better working memory to

focus on other complex cognitive mathematical tasks.

Problem

The National Council of Teachers of Mathematics (NCTM)

states that there are certain items at every grade level that are

integral to mathematics success before moving on to the next

grade level. These focal points give teachers an idea of the terms

or skills that students should have in their repertoire before

they have completed a specific year of instruction. At present,

incoming fifth graders at St. Mary’s are struggling with


mathematics, especially multiplication facts and concepts.

However, the NCTM states that fifth grade students should be

focused on learning division skills that concern whole numbers

(National Council of Teachers of Mathematics, 2006). One cannot

begin to address this skill until the issue of multiplication has

first been resolved. Since math is cumulative in nature,

students cannot become masters of division until they have a

solid understanding of multiplication, the inverse operation of

division.

On the most recent Measures of Academic Progress (MAP)

assessment, 16 of 19 incoming fifth grade students scored at or

below the 54th percentile in mathematics (Northwest Evaluation

Association, 2013). Furthermore, over the span of the academic

year, student growth did not meet school expectations. The

existing hypothesis for this lack of growth and performance is

that students lack a solid mathematical foundation to perform

well on such an assessment. Since the foundation is fragile,

student anxiety levels may be elevated, resulting in a lack of

confidence which limits a student’s ability to build upon their

skills. In order to be successful in fifth grade, in accordance


with the NCTM’s focal points, a new approach is necessary (National

Council of Teachers of Mathematics, 2006). Since most of the

fifth grade students in this group are unable to reach

multiplication goals, more needs to be done to reach higher

performance levels.

Purpose

In order to accomplish goals that are expected of students

in fifth grade it will be necessary to significantly improve this

group’s multiplication skills before embarking upon further

instruction. Therefore, it is the goal of this curriculum to

increase students’ multiplication accuracy to at least 90% of

multiplication facts, while also developing a conceptual

understanding of multiplication. The project aims to combine

self-guided interventions to develop automaticity with

constructivist methods and collaboration to help students explain

their reasoning on more complex problems. It is imperative that

improvements be made before current issues continue to be

repeated. For instance, less than 32% of eighth graders in the

United States are at or above a proficient level on standardized

mathematics tests (Cole & Wasburn-Moses, 2010). These are not


acceptable numbers and this project will look at changes in

teaching methodology in order to elicit an improvement in

performance.

Education Goal

At the current success rate of 40 percent, it is unlikely

that the fifth grade class at St. Mary’s will be mathematically

successful without the immediate improvement and strengthening of

skills. Therefore, the curriculum aims to increase overall

student success rates from 40 percent to 90. By improving

automaticity of multiplication facts, the curriculum will free up

working memory. This can negate or minimize the devastating

effects of math anxiety as well as allow working memory to be

used for more complex mathematical functions (Ashcraft & Krause,

2007). Consequently, by minimizing anxiety, educators can

prevent the avoidance technique of rushing through problems that

inhibits learning and concept attainment. Hurrying through

problems to prevent anxious feelings can be detrimental to the

learning process and may not be as likely to occur when anxiety

levels are decreased (Ashcraft, 2002).


Since students will have developed the ability to quickly

recall answers to multiplication facts and their anxiety levels

will be low, teachers may see the effects of this positive

situation compounded. By having quick recall, students will have

more cognitive energy and ability to perform more complex

functions (Poncy, Skinner, & Jaspers, 2007). Since students will

have lower levels of anxiety and will be less likely to avoid

mathematics, they may seek to perform math problems and tasks as

they proceed in their education. If their cognitive capacity is

not inhibited by unnecessary facts and figures, they may be more

successful, which will promote greater satisfaction and

confidence. As this continues, anxiety may continue to decrease

and the negative perspective that many students had may change.

By avoiding mathematics less and taking their time, students can

enjoy the rewards of this curriculum design.

Curriculum Variables and Teaching Variables

The way mathematics has been taught has not necessarily

changed significantly in the last few decades as students use

many techniques similar to those of their parents and previous

generations. In addition, many students struggle so much that


they may develop a negative attitude or view of mathematics that

affects every future math class they take. It is these facts

that the project attempts to overcome by using different methods

and changing student attitudes.

Methodology. Mathematics instruction works for many

children though not enough to be considered truly successful.

The National Council of Teacher of Mathematics states that

teachers need to teach math to develop deep understanding of

concepts and focus on learning that is student centered

(Shirvani, 2009). When teachers stand in front of a classroom

and ask students to repeatedly recite multiplication facts they

are not necessarily targeting the strengths of their students.

For a long time students have been subjected to the type of

teaching where teachers dictate information to students to be

absorbed like a sponge. However, this may not be the best method

for long term learning and students may be at a disadvantage if

subjected to this method of teaching for much longer. Steven

Reinhart states, “If my students were to ever really learn

mathematics, they would have to do the explaining and I, the

listening” (2000, p. 478). He further elaborates that students


who are able to teach concepts are those who will learn and

remember the concepts far in the future (Reinhart, 2000). Nearly

anyone can remember certain items for a limited time for a test

or quiz, however this is not the true purpose of education.

Students must take the knowledge they have gained and apply it

for the remainder of their academic careers and lives.

Therefore, it is of vital importance to find the methods of

teaching and modalities of learning that will allow students to

gain the greatest long-term success when designing the

instructional unit.

Constructivism. Previously, many teachers have employed the

use of direct instruction to deliver the information contained in

most math classes. A teacher would stand in front of the

classroom and explain the logic and reasoning behind the process

being used and then demonstrate the process. Students would be

given time to ask questions as the teacher would attempt to

transmit the knowledge to their students. However, many experts

agree that this may not be the best way for students to come to a

deep understanding of math concepts and facts. Students must

actively participate in the learning that is taking place and


must construct understanding from the inside rather than from an

external transmittal of data (Shirvani, 2009). Therefore, the

instructional unit will employ a number of activities that will

allow students to explore the mathematics information for

themselves. This will allow students to work through the

problems and concepts on their own, finding their own plan of

action as they choose what works best for their style of

learning. Since teachers cannot give their understanding to

students, the unit gives students the opportunity to explore

their own thought process to determine the meaning as they aim to

find understanding and comprehension of the math concepts

(Steele, 2001).

Collaboration. In addition to the use of constructivist

learning principles, the curriculum unit aims to utilize

extensive collaboration in order to improve student success and

understanding. Just as scientists and mathematicians in the

working world collaborate with colleagues before publishing their

findings, students will work with one another to develop greater

meaning and understanding. When students work together and have

a chance to explain their thinking, it gives greater weight and


clarity to the words that are being used (Steele, 2001).

Furthermore, if a teacher listens to students as they express

their thinking, he or she can help to guide, clarify, and hone

the processes that students are going through. By listening to

students’ discourse and collaboration, teachers are able to clear

up misunderstandings before they become deeply ingrained in a

student’s mind (Burns, 2005). Throughout the unit, students will

work together to give everyone a voice in the process of

developing an understanding of math concepts and knowledge.

Through this collaborative process, other students and the

teacher can identify erroneous thought patterns and help to guide

students to a more fruitful path.

Self-guided interventions. Many people can recall the use of rote

memorization drills, which elicit negative thoughts for many

people today. Rather than using this method that has proven

effective for less than the nation’s goals, this curriculum unit

will implement self-guided interventions such as cover, copy, and

compare or the taped problems method. These methods give

children access to the answers to multiplication facts as they

are learning them. This allows students to relax and internalize


answers as they also work on solutions for harder problems that

may require more complex cognitive functions. Rather than being

worried about their math facts as well as their assignment for

the day, students are free to focus on learning at a pace that

works for them. According to Thomas Caron, rigid memorization

drills are not necessary for learning multiplication; practicing

the facts numerous times is all that is needed for students to

develop automaticity with their facts (2007).

These self-guided interventions are analogous to the manner

in which children learn their first language. Parents do not

attempt to teach children specific words through the use of rote

memorization drills, nor would one even think to do so. Children

learn words in their proper context as they are used repeatedly

over time. The word or its definition are not withheld from a

child on a regular basis in a high-stakes manner to check for

memory ability. Eventually, the child begins using the word as

it is intended and one knows that he or she has learned the word

and its assigned meaning. Yet, this is not the manner in which

we have asked students to learn multiplication facts in the past.

By using self-guided interventions, the project hopes to achieve


similar results to those of previous research studies. One study

of a student with low cognitive functioning found that both

interventions increased accuracy by 56 to 73 percent (Poncy,

Skinner, & Jaspers, 2007). A second study, which consisted of

six, fifth grade math students (3 male, 3 female) found that the

taped problems intervention increased facts per minute that were

answered correctly by an average of 11.25 per student. This

number was increased further when multiple assessments were given

on the same day (Bliss, et al., 2010). Both interventions show

great promise in improving math performance, and students will be

given a choice between the two with teacher guidance. The unit

will implement daily use of both interventions to ensure that

students receive ample practice to improve automaticity of

multiplication facts. Just as children learn to use words of

everyday items and concepts without realizing it, the students

participating in these interventions will become adept at

multiplication without the strenuous effort put forth in the

past.

Math attitude. The importance of a student’s attitude

cannot be underestimated when designing a course or instructional


unit. Preconceived notions and perceptions can have a monumental

effect upon the way one acts and reacts in a specific

environment. If a student enters the math classroom expecting

failure and looking to avoid all work assigned then the work of

the teacher is going to be that much more difficult. If a

student does not have the desire to make an attempt at learning

math, then it may not matter what methods or strategies a teacher

chooses to employ. However, if one can help a student to forge a

positive and proactive attitude, then the cognitive hurdles that

present themselves can often be overcome with the methods and

strategies mentioned previously. Motivation resides within

students, but teachers can play a pivotal role in designing an

environment and curriculum that is conducive to positive

motivational forces. By viewing and monitoring performance

levels and taking steps to minimize math anxiety, one can aim to

improve math attitudes to the point that students will find the

motivation necessary to succeed in math.

Performance levels. One must be cognizant of students’ previous

performance levels as well as the levels of performance that they

achieve while going through the unit. Some students believe that


if they have struggled in the past that they will also struggle

in future math classes. This stems from a belief that math is

based solely upon ability and not upon effort (Ashcraft, 2002).

If this is the case a student may believe they are born without

an inherent ability to perform well in math and this will always

be true for them. It is important to communicate this error of

thinking to students, but equally important to know how students

have done in the past. If a teacher knows which students have

struggled in the past, then he or she can plan accordingly to

increase confidence levels and prevent similar thought patterns

in the future. Additionally, as the unit progresses one must

always monitor performance levels to ensure that students do not

fall far behind, which can cause further issues with attitude and

confidence levels. It is imperative that the teacher recognizes

when students may be on the precipice of inadequate performance

and implement re-teaching or another method to help students get

back on track.

Anxiety. The fear of mathematics seems to infiltrate every

school and level of academia in America. Math anxiety has become

an epidemic for our schools, teachers, and students. This issue


prevents many of our children from reaching their potential

abilities in mathematics. Anxiety and the attitude that our

students possess are very closely linked. In addition, cognitive

processes, including working memory are affected negatively when

math anxiety levels are high (Ashcraft & Krause, 2007). As math

anxiety increases, students tend to develop negative views of

math and attempt to avoid math in the future. This may include

speeding through problems or avoiding taking classes altogether

(Ashcraft, 2002). Ashcraft has pointed out that it must be

qualified that the exact causes of math anxiety have yet to be

identified (2002). This is an important point to make as the

attitude of the teacher plays an important role at this point in

time. If the instructor can alleviate the effects of anxiety by

assessing students and implementing strategies to combat fear,

math instruction can be more successful. This may also give

students the proper tools to avoid anxiety in future math classes

as they grow and mature in their academic careers.

Assessment. When students enter a new classroom, teachers do

not usually have a clear picture of what students possess in

terms of knowledge and skills. However, teachers generally solve


this problem by administering an assessment that gives a picture

of what a student may or may not know. This can also be done in

regards to a student’s attitude and views toward math. At the

beginning of the instructional unit, each student will take an

assessment that aims to determine their math attitudes. This

assessment will take place when they are also being assessed on

their conceptual knowledge and math skills. This assessment will

allow the teacher to identify which students may need additional

intervention and help in the area of anxiety to prevent issues

that may arise from a fear of math. One cannot simply look at a

student and determine whether he or she will have a fear of or

aversion to numbers. As a matter of fact, anxiety is so

difficult to determine that it has even been found that anxiety

is only weakly related to overall intelligence levels (Ashcraft,

2002). Since anxiety is linked to lower grades, negative self-

image, and lower levels of intrinsic motivation, it is important

to assess students’ attitude and feelings at the beginning of the

unit to begin to implement strategies that can lower the

detrimental impact of math anxiety (Ashcraft, 2002).


Strategies. When one has a general idea of overriding

attitudes and fears, the process of improvement can begin. There

are many steps one can take to minimize the impact that math

anxiety has upon a student’s math performance and the retention

of information. In order to make the unit more successful, the

teacher will use different strategies to help students lessen

their feelings of anxiety. One such strategy that will be used

often is gradual and repeated mathematics success (Furner &

Berman, 2003). By using self-guided interventions students will

see their own success on a daily basis, which will build

confidence and lower levels of anxiety. Another strategy that

the instructional unit will employ is the removal or easing of

certain anxiety inducing parameters such as time limits. Since

instant recall of math facts is so important to math success, one

does not want to leave roadblocks to success in the way (Tait-

McCutcheon, Drake, & Sherley, 2011). Many students find strict

time limits to be intimidating and this limits a student’s

potential accuracy. The instructional unit will remove many time

limits and ease others in order to foster a relaxed environment

that looks to determine a student’s knowledge without the fear of


a deadline. There are other strategies that can be used that

have been mentioned in specific lesson plans. Furthermore, the

curriculum design will attempt to alleviate anxiety by creating a

variety of learning environments, making math relevant to

student’s lives, asking students to participate in their

assessments, and by emphasizing original thought and finding

one’s own path to an answer (Furner & Berman, 2003). By

employing these strategies one should see a marked decrease in

math fears when a second attitude assessment is given at the end

of the unit during lesson 15. Students can also implement these

strategies later to continue to improve math performance in

future endeavors.

Needs Assessment

When developing a needs analysis an educator needs to

recognize specific requirements to ensure that the instructional

unit is as effective as possible. These requirements will be

viewed in the context of both the learning and instructional

environments. By identifying needs, a teacher can attempt to

meet them before they become an obstacle to instruction.


Learning environment. In order to effectively communicate

and teach the concept of multiplication and its accompanying

facts, it is necessary to utilize the available space as well as

various technological resources. Some studies have shown that

classroom environment has a direct correlation to performance on

mathematics concepts through the use of innovative strategies

(Ogbuehi & Fraser, 2007). While the study is limited in scope,

it indicates that using space and innovative techniques may lead

to improved performance for math students. Since this group of

students will enter the year markedly lower than many of their

peers, it will be worth the effort to look at implementing

different methods, new in the eyes of the students, in order to

improve math knowledge and performance.

While there are no major issues in the class such as IEPs or

504 plans, there remains a lack of math knowledge that needs to

be overcome. Since the project aims to use constructivist

methods to promote learning and foster independence, students

will seek information by using technological resources such as

the laptop computer and tablet device. Students will work to

learn for themselves rather than reactively and idly accepting


the teacher’s oration as truth followed by rote work on an

assignment (Inch, 2002). Students will become active

participants in their education rather than being quiet and

passive as they struggle to stay attentive through lectures and

problems being worked at the front of the classroom. According

to current brain research, information must be worked with and

tested, not merely transmitted and received (Sousa, 2010). In

addition to using technology, the class will also make use of the

CLS and main classroom to engage in small group discussion and

collaboration. As Marilyn Burns indicates, small group

collaboration allows students afraid of speaking in large groups

a chance to build confidence in their abilities (2005). As

students discuss and collaborate, the teacher will monitor for

understanding and ask guiding questions in order to assist

students in arriving at the best destination.

Instructional environment. The group of students that this

curriculum is designed for is matched closely, with no individual

outperforming any others considerably. Since all of the students

come from similar socio-economic backgrounds and speak English as

a primary language, no major instructional changes will need to


be made prior to implementation of the project. Differentiation

will be accomplished through small collaborative groups made up

of 3 to 4 students each, with groups changing periodically.

Burns explains that collaboration serves to allow students to

learn from one another while also revealing a child’s reasoning

behind an answer (2004). Before beginning the task of

collaboration, expectations will be set and discussed in detail

to prevent issues such as lost time due to talking, horse play,

etc.

According to previously cited NWEA testing data, this class

is struggling with math concepts and algorithms. This continues

a negative trend from the previous testing phase. In previous

years, these students were given multiplication tables and asked

to rehearse items over and over until they learned them.

However, based upon testing this does not seem to have worked as

well as intended. In order for practice, or rehearsal, to be

effective, it needs to be elaborative, or connected to something

from a student’s long term memory (Banikowski, 1999). Therefore,

as students are working to memorize multiplication facts through

the previously mentioned methods, they need to be connecting this


information to real-world concepts and tasks. The NCTM advocates

this by stating that problem solving should be more realistic and

life-like (National Council of Teachers of Mathematics, 2000).

By connecting real-world elaborate rehearsal to memorization

techniques students should be able to bridge the gap between

working memory and long term memory to more effectively and

permanently learn multiplication.

Since students’ writing skills are so advanced and they

enjoy the task so passionately, the project will also incorporate

writing into the unit. Writing gives students an opportunity to

reflect on their thoughts and allows them to see discrepancies in

their logic as they write. In addition, it gives the teacher

insight into a student’s thought process that they may not be

comfortable revealing orally (Burns, 2004). By catering to

students’ strengths one can build confidence in math ability and

improve the overall experience for students.

Application Context

Application and academic skills and how they will affect the

curriculum unit are an important facet of design and

implementation. One must look at current research and expert


analysis to determine the best practices to address the four

skill areas that were identified in chapter one.

Resource management and collaboration. Collaboration will

be a fundamental part of the project and will add greatly to the

learning experience. Collaborative work is a major principle of

the standards set forth by the NCTM (National Council of Teachers

of Mathematics, 2000). Collaboration is an integral part of the

scientific community and mathematicians often work together. The

teacher will even refer to the students as mathematicians as they

will be doing the scientific work of discovery in this unit.

Studies have shown that the instructional practices of teachers

in the classroom affect the performance of their students in this

respect. Teachers who used more group and pair collaborative

work saw more success than those who used predominately class

wide discussion (Schoen, Cebulla, Finn, & Fi, 2003). The

expectations for collaborative work will be communicated early

and often to ensure proper use of this strategy. Moreover,

students will be informed of the benefits of the use of

collaboration and its real-world application in order to further

alleviate math anxiety (Furner & Berman, 2003).


As previously stated, students have often struggled in the

past with turning assignments in on time. While there will be a

limited amount of homework in order to avoid confusion that can

be caused by help from parents and other family members, there

will be deadlines for assignments. If students are unable to

turn an assignment in on time, they will incur a 5% deduction

from that assignment’s grade as a consequence for the late

submission. If a second assignment is turned in late, students

will receive a 10% deduction and so forth. As the instructional

unit progresses, deadlines will become more strict, but the point

deduction policy will remain the same. However, the maximum

deduction for any assignment will be no more than 20 percent.

Parents will be notified of this policy, in writing, to prevent

any issues that may arise if a student fails to turn work in on

time.

Information and communication. Due to information obtained

from previous teachers it may be necessary to adjust a few

instructional times and due dates, which will depend upon the

amount of time it takes students to adjust to the new

instructional unit. Students rely heavily on teacher assistance


at this point in time, but will be taught and coached to work and

think independently. Students must be taught to think for

themselves in order to be successful in math and move forward

into more complex classes and concepts (Furner & Berman, 2003).

The teacher will provide modeling to guide students in the proper

use of resources to find and utilize information. In the past,

students viewed it as cheating when anything aside from their

textbook was used to help them arrive at an answer. These

misconceptions will need to be discussed to help students think

and perform independently. The class will continue to use

standard math vocabulary in class and students will be expected

to use correct mathematics vernacular rather than jargon and

commonly accepted terms. For example, students will need to use

terms such as multiplication table instead of times table and subtract in

place of take-away. This is an on-going school wide endeavor and

will continue as students explore the multiplication unit.

Conclusion

Mathematics education and its mastery does not successfully

reach enough students at St. Mary’s. Currently 60% of fifth

graders at the school are struggling to keep up with their peers


and are at risk of falling further behind if changes are not

implemented soon. The goal of this project is to increase

multiplication fact accuracy to 90% for at least 90 percent of

the students in the class. However, the all-encompassing aim is

for students to be able to multiply facts while also

understanding the significance of what they are doing. Mastery

of multiplication and mathematics cannot be viewed as the

memorization of facts alone, but it must also include a solid

understanding and knowledge of the concepts behind these facts

(Shirvani, 2009). It is the intention of this project to bring

together constructivist methods of learning with collaboration

and self-guided interventions to improve overall multiplication

performance. Rather than focusing only on the material at hand

and the scores that students receive on multiplication table

tests, instruction will focus on the learner and his or her

specific needs. This unit will require the teacher to learn,

adapt, and change as much as or more than the students involved.

Through the careful marriage of multiple methods and strategies,

this project posits that it is possible to improve math


performance while enhancing student attitudes and limiting the

effects of anxiety.

The evidence for changing the way math is taught is

overwhelming. While some claim that the current methods have

worked for a long time and small advances are being made, this is

not enough to warrant complacency. Seven of ten students being

successful in multiplication can be improved upon (Caron, 2007).

In fact, in a large synthesis of numerous techniques and

strategies, Slavin and Lake found that some of the most

successful learning and teaching programs were those that

utilized cooperative learning (2008). The debate about direct

instruction coupled with rote memorization continues, but the

costs continue to mount. Mathematics remains the subject failed

most by students and math scores on standardized tests have not

increased significantly using the methods that most people recall

from years past (Steen, 2007). Students at St. Mary’s typically

continue to struggle with mathematics as they progress into the

6th grade and beyond. The issue begins much earlier and must be

solved by building a strong mathematical foundation. Once this

foundation has been built on simple and concrete ideas such as


multiplication, one can begin to focus on concepts like division

of whole numbers and other suggested functions and operations.

However, without a solid base to build upon, students may never

have the confidence and skills necessary to add to their

mathematics toolbox. The following curriculum project aims to

build that solid foundation, while also improving perceptions and

attitudes for all students. Students will increase understanding

of the underlying concepts and become more accurate with

multiplication facts. Additionally, students will have better

recall, freeing up working memory to process more complicated

mathematical processes and tasks.

Chapter Three


The following mathematics instructional unit will be taught

to the fifth grade class at St. Mary’s Catholic School in

Cheyenne, Wyoming for the 2013-2014 academic year. At the

current time, only about 40% of the students in the class are

proficient in multiplication, whereas the school’s goal for the

group is 90%. Therefore, this set of students is in need of

additional multiplication instruction in order to reach the

intended goal before the first major testing session in October

2013. Furthermore, Geary (2000) has stated that the biggest

indicator of success in math as a secondary student and as an

adult is a child’s performance during the elementary school

years. This indicates that this is an incredibly important time

in these students’ academic lives and one must ensure that they

do not miss the opportunity to cultivate the multiplication

skills that are needed for their future. In order to make the

most of this opportunity, this instructional unit will combine

constructivist methods of learning with self-guided interventions

to improve multiplication performance. Additionally, some

lessons will be differentiated to include times of guided


instruction for those students that may need it or choose it as a

course of action.

This original curriculum design will begin and end with a

twofold assessment that will evaluate student knowledge of

multiplication as well as their attitudes about mathematics. The

purpose of the design is to improve performance by implementing

specific teaching methods while also lowering levels of anxiety

in students. The attitude assessment that will be administered

will help identify attitudes that may be prohibitive to learning.

This will aid the teacher in implementing interventions to help

students in need. Since anxiety can harm memory and the

cognitive process, it will be beneficial to limit anxious

feelings as students progress through the unit (Ashcraft &

Krause, 2007).

Once the initial assessment has taken place, each lesson

will follow a similar course of action. Students will practice

their multiplication facts for 15 minutes each day using one of

two self-guided interventions. The cover, copy, and compare

intervention allows students to study a set of multiplication

facts for a set amount of time before covering the answers. The


students then attempt to solve problems that involve the studied

facts. When the problems are completed, students compare their

solutions to the answers that they studied in the beginning. By

contrast, the taped problem intervention allows students to

listen to a prerecorded voice that recites various multiplication

facts. Students attempt to answer the recited problems before

the solution is read aloud. Once students have completed the

multiplication practice for the day, the class will engage in a

lesson that builds upon these facts. Over the course of the

unit, students will take the basic facts and add more complex

concepts until they are ultimately multiplying triple digit

numbers. The unit will also incorporate technology based lessons

to integrate the skills that students will need for future

endeavors. At the end of the curriculum unit, students will be

asked to reverse roles and become teacher for a day, challenging

them to put their knowledge into action.

Once students have had a chance to work through all of the

facets of multiplication and become the teacher for a day, they

will be assessed again. Furthermore, students will have a chance

to review their initial attitude assessment to see the changes


that may have taken place over the course of the unit. The post-

assessment will be evaluated to ensure that students have learned

the basic multiplication facts with at least 90 percent accuracy.

Additionally, it will be evaluated to determine whether students

have developed a conceptual framework of multiplication and can

successfully solve more complex problems. Finally, it will be

evaluated to determine whether anxiety levels were minimized

through the use of this instructional unit.

Lesson One

Title: Pre-Assessment with Math Attitude Assessment

Description / Purpose / Direction: The first day of instruction

for the curriculum unit will focus on exploration and assessment.

The goal of this lesson is to determine a baseline for students

in terms of knowledge as well as confidence levels and attitudes.

The day will start with an introduction and general discussion of

multiplication to break the ice. Students will be encouraged to

ask questions and voice concerns about the unit before taking the

assessment. Once the basic multiplication discussion has taken

place, the teacher should communicate class expectations,

including behavior, due dates, and performance expectancies. The


remainder of the class period will be used for assessment, goal

setting, and for students to explore the tools that will be used

for the remainder of the unit.

Objective:

1. All 5th grade students will fully complete the pre-

assessment and math attitude assessment in the allotted time

(approximately 50 minutes).

2. Given a goal setting worksheet, each student will set three

personal goals for the multiplication unit.

Standard: All lessons in this curriculum design adhere to

Wyoming state standards for fifth grade. Specifically, standard

5.NBT, numbers and operations in base ten: perform operations

with multi-digit whole numbers and with decimals to hundredths

(Wyoming Department of Education, 2012).

Time: One 90 minute period.

Materials / Technology: Pre-assessment (see Appendix B), pre-

assessment answer key (see Appendix Q), student attitude

assessment (see Appendix C), goal setting worksheet (see Appendix

D), pen or pencil, tools that will be used for duration of the

unit (computers, textbook [LeTourneau, Posamentier, & Ford,


2006], self-guided multiplication interventions: cover, copy, and

compare; taped problems; or computer games).

Procedure / Sequence:

Class will open with a discussion of multiplication where

students will be given time to ask pertinent questions.

This time will also be reserved for a discussion of overall

class expectations, including behavior, due dates (late

policy), and performance.

Students will then be given the pre-assessment and math

attitude assessment at their own desks. Students will

complete the assessments without the aid of calculators,

computers, etc.

Once tests have been completed, the class will discuss the

regular daily class schedule for the curriculum unit. When

students have had a chance to ask questions and clarify

misconceptions and misunderstandings, they will be given

time to explore the tools that will be used for the rest of

the unit (self-guided interventions, computers, textbooks,

SmartBoard, internet based games, etc.).


Students will reconvene to evaluate the pre-assessments that

were taken earlier in class. Students will evaluate their

own paper using a different colored pen supplied by the

teacher. Each student will mark their own incorrect answers

(it will be communicated before the grading process that the

grades will not ultimately be entered into the grade book as

the post-assessment will replace this grade) and record

their own number correct and percentage at the top of their

paper. Once all papers have been graded, the teacher will

collect the assessments and students will be given a goal

setting worksheet to complete.

Using the goal setting worksheet, each student will set

three specific goals for the multiplication unit. They will

also set smaller goals that will help them reach their three

main goals. For example, if a student correctly answered 35

percent of their multiplication facts, they may choose to

set a goal to answer 95 percent at the conclusion of the

unit. The student will then set goals to perform the cover,

copy, and compare intervention for 15 minutes per day and to


improve their Friday practice test by 35 percent each week

to reach their intended goal.

As students finish their goal setting worksheets they will

bring them to the front of the classroom and prepare for

their next class.

Assessment: Students will complete both the overall pre-

assessment and the mathematics attitude assessment for the unit.

Both of these assessments will be evaluated before the next class

session. The pre-assessment will be evaluated by students, while

the attitude assessment will be evaluated by the teacher. Overall

findings of the attitude assessment will be briefly shared and

discussed with students at the beginning of lesson two. Students

will also set three goals for the multiplication unit on the goal

setting worksheet.

Teaching Notes / Stratagem: The attitude assessment will be

evaluated by looking separately at the even and odd numbered

responses. A high score on the odd numbered responses and a low

score on the even numbered responses may indicate a reluctance to

be involved in class and/or a high level of math anxiety. These

students will need to be encouraged frequently throughout the


unit to build confidence and to attempt to reverse the negative

effects of math anxiety (Furner & Berman, 2003). A high score on

even numbered responses and a low score on odd numbered responses

may indicate the opposite circumstance. These students will be

eager to get started and may exhibit a penchant for mathematics.

The results as a whole should be shared with students, but

students should not be told that they have a math phobia or math

anxiety. A teacher’s own anxiety or acceptance of a student’s

anxiety can often create more problems, therefore the teacher

should inform students that even though some of them may have

concerns to begin with, everyone will develop confidence as time

goes on (Furner & Berman, 2003). The math attitude assessment

should be used for teacher knowledge to determine best practices

and will be used at the end of the unit for student comparison.


Lesson Two

Title: What is Multiplication? Web Quest

Description / Purpose / Direction: Before starting the steps of

multiplication, students will have an opportunity to explore the

vast world of multiplication for themselves. This begins the

unit by focusing on the student and his or her needs rather than

focusing on where a teacher may desire for students to begin

(Shirvani, 2009). For lesson two, students will be asked to

complete a web quest in which they will look into the history and


origins of multiplication. In addition, students will look at

how multiplication is used in the real-world, what professions or

trades use multiplication (both in simple and complex manners),

and facts about multiplication that students find compelling.

The web quest will be documented and turned in for points, but

the ultimate purpose is to give students a foundation and

appreciation for the use and necessity of multiplication before

beginning to work with numbers.

Objective: All 5th grade students will complete a web quest

within two days and earn a 3 or greater in each of the rubric

categories (History / Origins, Applications, and Facts).


Materials / Technology: Laptop computers, SmartBoard, textbooks,

optional resources (i.e. Nook, Kindle, iPad, etc.), self-guided

multiplication interventions, web quest rubric (see Appendix E),

paper, pen or pencil.


5-10 minutes – The teacher should discuss attitude

assessment from lesson one and highlight positive elements


from the evaluation that was taken. The teacher should share

ideas for alleviating fear with students that will help them

improve their math performance throughout the unit. These

include but are not limited to: Remember that all

mathematicians make mistakes (even professionals); take

different approaches; if one is struggling, take a break or

change the environment for a moment; celebrate success; and

remember that a person’s value is not tied to their ability

to do math (Furner & Berman, 2003).

15 Minutes – Independent multiplication fact practice time:

Based upon test scores from the pre-assessment, students

will choose (with discussion and direction from teacher) an

intervention. Some students may choose the cover, copy, and

compare method or taped problems method. Based upon

previous testing data most of the students in this class

will most likely fall into one of these two strategy groups,

but there may be a few who will be able to use computer

based games if they are already testing well in the 80-90%

accuracy range.


15 minutes – Students will reconvene to be given

instructions and expectations (including the due date and

late submission policies) for the web quest assignment.

This time should be reserved for explaining the process of

performing a web quest as well as discussing the rubric, its

specific metrics, and their point values. Students should

also be given an opportunity to ask clarifying questions at

this time.

40 minutes – Student work time: Students will utilize their

resources and tools to work independently to complete the

web quest. However, students will be free to ask their

peers questions about search strategies, information they

may have found, etc. Students will work as collaborators

and should see one another as team members and fellow

mathematicians. Students should aim to finish the

assignment in this forty minute period. They may have other

time throughout the day, but this is ample time to complete

the task. The intent is to get a very basic idea of

multiplication and its applications and uses.


10 minutes – Students will reconvene for the final ten

minutes of class to perform an exit pass. The exit pass

will consist of 20 multiplication facts as well as the

following questions:

o What is the most interesting item you found in your

research today?

o What is one thing that you want to know, but were

unable to find?

Students will complete the exit pass before moving on to the

next class.

Assessment: Students’ web quest documents will be evaluated

using the rubric designed for this unit (see Appendix E) per

stated objective.

Teaching Notes / Stratagem: As students are working on their web

quest they should be allowed to move into the collaborative

learning space (CLS) and around the classroom as long as they

remain productive. Students should feel free to move as long as

they are working within classroom expectations. During student

work time, the teacher should visit with students about the


research process and ask questions to determine what avenues are

being pursued. The teacher should offer guidance to help

students get back on track when they may have veered off course.

Lesson Three

Title: Two-Digit Numbers Multiplied by Single-Digit Numbers

Description / Purpose / Direction: Since students now have

rudimentary knowledge of the origins of multiplication and have

seen some of its uses and applications, one can begin to build

upon the basic facts that students have been practicing in the

first few days. Students will continue to practice the basic

facts throughout the duration of the entire unit to reinforce and

strengthen their knowledge, but they will also gradually build

upon them. Lesson three starts the simple construction of a

multiplication pyramid with two-digit numbers being multiplied by


one-digit factors. Students will engage in a short class

discussion to delve into the possible strategies that can be used

to perform this type of multiplication before moving into fact

practice and into smaller collaborative groups for the day. The

purpose of today’s instruction is to allow students to take the

first small, logical step in the process of multiplication.

Objective: All 5th grade students will correctly solve 30 or

more of 35 two-digit by one-digit multiplication problems prior

to the start of lesson six.


Materials / Technology: Self-guided multiplication interventions,

textbook, computers, whiteboard, SmartBoard, pen or pencil,

multiplication worksheet (see Appendix F).


5 minutes – Class discussion that will include expectations

for the day and a reminder about the deadline for the web

quest from lesson two.

15 minutes – Independent multiplication fact practice.

20 minutes – Collaborative group time: students will break

into their collaborative groups to discuss and solve a


variety of example problems that have been written on the

whiteboard. Students will use their resources, including

personal computers, tablets, and books to solve as many of

the problems as possible.

15 minutes – Check in: All students will reconvene for a

class discussion and check in with the teacher. Students

will offer some of their answers as they explain the manner

in which they arrived at their solution. At this point, the

teacher can show Khan Academy on the SmartBoard to offer

additional assistance, if needed, or if students are

struggling to grasp the concept (Khan Academy, 2013). Once

students seem to have a grasp of the material, the teacher

should release students to continue with independent work.

25 minutes – Independent work: Students will work on

multiplication worksheets independently. If students do not

find that this is enough time to finish or they choose to

spend more time to complete the activity, this is

permissible as the assignment will not be due until before

lesson six.


10 minutes – Wrap-up: The teacher will have students put

things away and provide a reminder of approaching due dates.

At this point, students will want to double check to make

sure they have these dates documented in their homework

planner. This time can also be used as a question session

for students if needed. Finally, the teacher should remind

students of Friday’s practice quiz and the goals that

students have set so they can evaluate their progress.

Assessment: Students will complete multiplication worksheet with

at least 30 of 35 questions correct.

Teaching Notes / Stratagem:

As students are working in their collaborative groups, the

teacher should closely monitor progress to ensure that

students are not simply providing answers to one another. If

this is taking place, coaching should be performed to help

students understand the benefit of guiding one another to

understand the process.

If students hit a roadblock when performing the worksheet,

they may seek assistance from a classmate to overcome the


difficulty. However, if the teacher sees the same student

doing this repeatedly, re-teaching should take place.


Lesson Four

Title: Application of Two-Digit Multiplication

Description / Purpose / Direction: Lesson three provided

students a chance to practice the basic skill of multiplying two-

digit numbers by single digit numbers. In lesson four, students

will move into the realistic application of the skills that were

learned and rehearsed in lesson three. The goal is for students

to connect skills that they have gained to something that can be

seen in the context of a real world situation or scenario.

Instead of looking at a word problem from a book that students

may never connect with or understand, this lesson will help

students see that multiplication has a real connection to

people’s lives.

Objective: Given a real-world multiplication problem, all 5th

grade students will develop and explain an accurate solution that

meets at least a 3 on all rubric categories.



computers, pen or pencil, paper, real-world solution rubric (see

Appendix G).



5 minutes – Discussion of plans for class, including

expectations and pending due dates.


20 minutes – The teacher will distribute rubrics to students

and explain the assignment. Students will be given an

opportunity to ask questions regarding the assignment before

proceeding into independent work time. Teacher will then

distribute the various problems to students. Each student

will be given a unique problem that will require additional

research before being able to develop a solution. The

following problem is an example provided for reference:

o Mr. and Mrs. Johnson are interested in selling eggs to

earn extra money. Based upon where they live, it is

only legal to own 5 birds, but they are unsure of what

species to buy. They are thinking about either ducks,

geese, or chickens. Which of these options would give

them the most eggs? How many eggs would each option

give? How much money would each option earn?


40 minutes – Student work time: Teacher will monitor student

work and offer suggestions as students research their

answers and formulate an answer based on that research.

Students will come up with an accurate answer based on their

findings and explain how their answer makes sense based on

their method of research and in relation to multiplication.

10 minutes – Reflection time: On a separate sheet of paper,

students will write a short reflection paragraph that

answers the following two questions:

o If you could change one thing about today’s assignment,

what would it be?

o What has been the best part about this unit so far?

Assessment: Students will be given a unique real-world problem,

while their solution and explanation will be evaluated according

to the provided rubric.


Students may struggle with the research process to find the

information they need. If this is the case, one may spend

time discussing broad internet searches and how to narrow

student searches. In addition, as the teacher is monitoring


student work time, suggestions should start out broad to

elicit ideas and then become more specific as students work

to find their intended information.

The reflection activity will provide valuable information

for the teacher regarding future implementation of this

lesson and similarly taught lessons. Furthermore, the

second question may help students to see that they are

enjoying the unit more than they thought, lowering the

impact of anxiety (Furner & Berman, 2003).


Lesson Five

Title: Two-Digit by Two-Digit Multiplication

Description / Purpose / Direction: Lesson five adds the next

logical step of multiplying two-digit numbers with two-digit

factors. Students will now be taking the information and skills

they have already honed and practiced and will be adding an

additional step. When doing this type of multiplication students

can become frustrated if they do not find an algorithm that works

for them. In the past, many teachers may have forced an

algorithm upon students, but this lesson will allow students to

explore the process on their own to find what works for them.

Once students are comfortable and confident with their chosen


method, they can build upon it indefinitely or attempt to learn

new ways of solving problems.

Objective: All 5th grade students will correctly solve at least

31 of 36 two-digit by two-digit multiplication problems by the

start of lesson seven.


Materials / Technology: Untimed multiplication fact quiz (Math-

Drills.com, 2013), pen or pencil, whiteboard, personal

whiteboards, computers, multiplication worksheet (see Appendix

H).


10 minutes – Discussion of class activities and expectations

for the day.

10 minutes – Class will begin with an untimed multiplication

fact quiz (100 facts), obtained from Math-drills.com that

will be different for each student. Even though the quiz is

untimed, students should realistically be given no more than

about ten minutes. If a student needs more than that, they

can continue working on the quiz as other students begin

work on independent multiplication fact practice.


15 minutes – Independent multiplication fact practice: As a

reward for a week of hard work and effort all students will

be given the option of online games for multiplication

practice.

10 minutes – Introduction of concept: The teacher will

provide multiple examples of two-digit multiplication

problems on the whiteboard. Students will write these

examples on their personal whiteboards or on a sheet of

paper to prepare for collaborative time. Prior to moving

into collaboration, the teacher will explain that there are

multiple ways to solve these types of multiplication

problems quickly and simply. These include the traditional

method or algorithm, matrix or grid method, and the partial

products method. The teacher will write the names of these

methods on the board and offer to demonstrate all three for

the students that want to learn in that capacity. At that

point, students will be given the option to go into the CLS

to work in collaborative groups to explore the different

methods or to stay in the main classroom to receive guided


instruction and large group discussion to learn about the

three methods.

20 minutes – Exploration of methods: Students will explore

methods of solving problems in the classroom or CLS. The

teacher should work with the students in the classroom to

explain all three methods, followed by students working to

solve the practice problems provided by the teacher.

25 minutes – Independent work time: Students will work on

their multiplication worksheet and complete it before the

beginning of lesson seven.

Assessment: Students will correctly answer at least 31 of 35

multiplication problems per objective before lesson seven.


The teacher must have a thorough knowledge of multiple

algorithms for solving multiplication problems and will need

to be able to explain the processes to students. Since all

students learn differently, one must recognize that a

student may prefer one method over another. Teachers should

allow students to choose their own method and not force them


to use a method even if one is more labor intensive than

another.

Time suggestions for exploring algorithms and independent

work can be changed according to student need. If students

grasp the methods quickly and can begin working on the

worksheet, times should be adjusted accordingly. However,

if students need more time to explore and discuss the

methods, time can be taken from independent work time to

ensure that students are successful when they begin their

independent work.


Lesson Six

Title: More Two-Digit Multiplication with Collaboration

Description / Purpose / Direction: Since two-digit

multiplication adds new steps that can bring added confusion and

frustration, it is necessary to ensure that students have a

strong understanding before moving into new material or

additional steps. Therefore, lesson six aims to build greater

confidence through continued practice, while also allowing

students to challenge one another. As they develop questions and

answers and solve others’ questions, students will be taking on

the roles of teacher and student, making the lesson more

memorable.

Objective:

1. Given a sheet of paper all 5th grade students will develop a

20 question worksheet and answer key within 30 minutes.

2. After exchanging their worksheet with another student, all

students will correctly answer 18 of 20 problems within 25

minutes.



Materials / Technology: Self-guided multiplication interventions,paper, pen or pencil.


5 minutes – Discussion of class plans and expectations.


10 minutes – Detailed explanation of project: The teacher

will explain the process for the day’s activity, including

expectations for behavior and problems for the worksheet

(two-digit by two-digit, no repeated problems, etc.).

30 minutes – Worksheet development: Students will be given

a sheet of paper. They will each spend this time developing

a set of 20 multiplication problems that will be given to

another classmate. This 20 problem set must also have a

corresponding answer key which will be completed on a

separate sheet of paper.

25 minutes – Students will exchange their paper with a

student of the teacher’s choice and complete the worksheet

that was designed by their classmate. Each student will

have 25 minutes to complete the worksheet and must answer at

least 18 of 20 problems correctly.


5 minutes – In the last few minutes of class, students will

ensure that they pair the answer key with their worksheet

and that both are turned into the teacher so that the

assignment can be evaluated for a grade. The teacher will

double check to make sure that all papers have been

received.

Assessment: Students will be assessed twice. First, students

will be assessed based on their ability to properly write a

worksheet on the parameters set by the teacher. Secondly,

students will be assessed on the number of questions answered

correctly on their completed worksheet.

Teaching Notes / Stratagem: If not monitored properly, this

lesson could take more or less than 90 minutes depending on the

length of time students use to make their tests. The teacher

should be prepared to compensate if there is extra time at the

end of class. If this is the case, students may evaluate the

worksheet of the student who completed the assignment they

designed. However, each teacher must be aware of the policies of

their district as some schools prohibit students from grading

another student’s work under any circumstances.


Lesson Seven

Title: Multiplying Decimals – Comparing and Contrasting

Description / Purpose / Direction: In the first six lessons,

students have had considerable practice with multiplication and

some application of the skills that have been learned. As

confidence increases, one skill that can often pose a problem for

students is decimal multiplication. In the past, teachers may

have immediately demonstrated the steps to perform decimal

multiplication and asked students to recall everything they had

already learned plus add the new idea of counting decimal places.

This is often intimidating for students and can become an

obstacle to learning. Rather than starting with practice, this

lesson aims to help students see that the two concepts are more

similar than different to alleviate apprehension and fear. By

easing tension and first giving students knowledge of the


concept, students may be able to make the transition to decimal

multiplication more quickly and easily.

Objective: All 5th grade students will compare and contrast the

steps of decimal multiplication with whole number multiplication

by accurately completing a graphic organizer with at least 3

similarities and 2 differences.



SmartBoard, whiteboard, computers, tablets, textbook, pen or

pencil, graphic organizer (see Appendix I).


5 minutes – Discussion of plans and expectations for the

day.


20 minutes – Demonstration of decimal multiplication: The

teacher will begin the main portion of class with a few

problems on the SmartBoard as an example. He or she should

work them out and explain the problems in very general terms

without getting too specific, highlighting similarities with

what students already know. Once students have seen a few


examples, the teacher should explain the assignment and

distribute the graphic organizers for student use.

30 minutes – Student work time: Students will disperse

throughout the work areas and use their resources (computer,

tablet, books, etc.) to find the similarities and

differences between decimal and whole number multiplication.

They will then fill out their graphic organizer using the

information they find and sketch examples of the

information.

20 minutes – Class discussion of findings: Students will

return to their desks and engage in a discussion of their

findings. The teacher will fill in the information on a

large graphic organizer on the SmartBoard. As each item is

placed on the board, the teacher will explain each one in

detail and offer a few examples and work them out for the

class. Students will also be given the opportunity to work

a few problems on their own. Students may also use this

time to ask questions for clarification.

Assessment: Students will complete the graphic organizer with 3

similarities and 2 differences.


Teaching Notes / Stratagem: There may be some students who

immediately understand the concept of decimal multiplication. If

this is the case, one can recruit these students to help others.

These students can become teaching assistants rather than

potential behavioral issues in the classroom. This will require

coaching from the teacher, but will pay huge dividends as

students need help with future lessons.

Lesson Eight

Title: Decimal and Money Multiplication

Description / Purpose / Direction: Now that students have a

basic understanding of the differences and similarities between

whole number and decimal multiplication, practice with these

problems can begin. So far, students have performed every step

necessary to complete decimal problems except the counting of

decimal places and placing a decimal point into the answer.

Following this lesson, students will understand this concept and


will also be able to successfully perform the steps with problems

that are related to money.

Objective: Given a 20 problem worksheet of decimal and money

based multiplication, all 5th grade students will correctly solve

at least 85% of all problems.



pen or pencil, computers, tablets, textbooks, whiteboard,

personal whiteboards, SmartBoard, decimal / money worksheet (see

Appendix J).




10 minutes – Review of decimal multiplication differences:

The teacher should begin this section by asking students to

discuss the differences between decimal and whole number

multiplication.

30 minutes – Collaborative groups vs. direct instruction:

Students will be given one of two options. Students can

choose to go into the CLS to work in their collaborative


group to investigate and practice the specified problems or

they can remain in the classroom for a guided instructional

unit with the teacher. Students who choose the CLS will use

their resources to work through the steps of decimal

multiplication in their collaborative group, while students

in the classroom will work with the teacher to understand

the process. In the classroom, the teacher will use

examples on the board to show students how to work various

problems. As students gain a better understanding they will

gradually solve more of each problem until they are

independently solving them on their personal whiteboards.

30 minutes – Independent practice: Students will be given a

worksheet with 20 problems to complete. The worksheet will

be due at the beginning of class, the day of lesson ten.

During this time, students will work to complete their

worksheet and may collaborate if needed. The teacher should

monitor students as they work to ensure that students are

placing decimals properly and that answers are labeled

correctly.


Assessment: Students will complete the decimal / money

worksheet.

Teaching Notes / Stratagem: Some students may need to have an

additional strategy in place to overcome the tendency to bring

the decimal straight down as in addition and subtraction of

decimals. To alleviate this practice it has been found

advantageous to use a highlighter to mark the number of digits

appearing after the decimals in the factors. Once students see

the number of bright digits that have been highlighted they can

count that number of digits in their answer and place the decimal

accordingly. This has greatly helped students who have struggled

in the past.

Lesson Nine

Title: Three-Digit Multiplication


Description / Purpose / Direction: Since students have covered

single-digit and two-digit multiplication, three-digit

multiplication is the next logical step in the instructional

unit. Students will continue to work the same steps they have

been using to solve problems. Each student will focus on the

algorithm or method that they have used for two-digit

multiplication and the format of class will remain the same to

maintain continuity and familiarity for students. Though three-

digit multiplication is slightly more complex, if students

continue to follow the steps they have performed in the past,

they will continue to find success. This lesson becomes the apex

of the unit in respect to level of difficulty and gives students

a strong foundation to build future math skills upon.

Objective: All 5th grade students will correctly solve at least

25 of 30 three-digit multiplication problems before the beginning

of lesson eleven.



whiteboard, SmartBoard, computers, tablets, textbooks, individual


whiteboards, pen or pencil, multiplication worksheet (see

Appendix K).




30 minutes – Classroom instruction or collaborative groups:

The teacher will give the option to students to stay in the

classroom or break into collaborative groups to discuss the

three-digit multiplication process. Students will then work

practice problems in small groups in the CLS or as a large

group with teacher. The teacher will then provide practice

problems to solve. The teacher will gradually release

students to work on problems more independently until they

become comfortable with the process.

30 minutes – Independent work time: Students will work to

complete multiplication worksheet.

10 minutes – Closure / wrap-up: Students will reconvene for

a reflection activity, where each student will answer the

following three questions on a separate sheet of paper:


o As we near the end of the unit, what is one thing you

are struggling with the most? (If you have mastered

everything, just write that as your answer).

o If you were to take a 5 minute multiplication fact test

today, how many out of 100 would you get correct?

o If you had to choose one job in mathematics to do in

the future, what would it be and why?

Assessment: Students will complete the three-digit multiplication

worksheet before the start of lesson eleven.


Remind students that this is just an extension of two-digit

multiplication and will only require additional practice to

become adept with this skill. It is important that students

do not overcomplicate three-digit multiplication as this may

increase feelings of anxiety.

Since students have performed all of the necessary steps in

the past to complete this exercise, the assignment should go

fairly quickly for most students. However, if it seems that

students need more time to complete the assignment, the due

date can be adjusted one lesson further in the future.


The reflection piece is meant to give the teacher insight

into what additional information may be needed to help

students. It also serves to put math into a language

context, solidifying meaning for students (Steele, 2001).


Lesson Ten

Title: Check Your Progress / Mid-Unit Review

Description / Purpose / Direction: One of the assessments that

St. Mary’s Catholic School prefers is the Sadlier-Oxford in-

chapter review, Check Your Progress (Sadlier-Oxford, 2013). This

periodic check allows the teacher to take a glance at a student’s

performance before proceeding further in the chapter. It is an

opportunity to go back and reteach concepts and items before

pushing ahead and frustrating students before making a final

effort to finish a chapter or assess learning for a unit. In

keeping in line with the school district, the unit will

incorporate a mid-unit review similar to the Check Your Progress

worksheet. However, the review has been specifically written for

this unit. This lesson will offer students a chance to go back

and be reminded of anything that may have given them trouble in

the past. It will also provide an opportunity for the teacher to


shore up any weak or troubled areas that students may have before

moving into the final week of the unit.

Objective:

1. Given the worksheet, Mid-Unit Review, all 5th grade students will

correctly solve at least 87% of all questions.

2. On a separate sheet of paper all 5th grade students will

choose a multiplication algorithm and correctly explain the

steps necessary to solve a problem using that algorithm or

method.


Materials / Technology: Untimed multiplication fact quiz (Math-

Drills.com, 2013), scratch paper, pen or pencil, Mid-Unit Review

(see Appendix L).



10 minutes – Class will begin with an untimed multiplication

fact quiz (100 facts), obtained from Math-drills.com that

will be different for each student. Even though the quiz is

untimed, students should realistically be given no more than


about ten minutes. If a student needs more than that, they

can continue working on the quiz as other students begin

work on independent multiplication fact practice.

15 minutes – Independent multiplication fact practice: As a

reward for a week of hard work and effort all students will

be given the option of online games for multiplication

practice.

40 minutes – Check Your Progress: The teacher will

distribute the assessment and allow students time to

complete it. Students will be allowed no resources (book,

calculator, computer, etc.) in taking the assessment. The

teacher should monitor students as they complete the

assessment. On the whiteboard, the teacher will write, “On

a separate sheet of paper please explain the steps you would

take to solve a three-digit multiplication problem.”

Students will answer this problem in complete sentences.

20 minutes – After students have completed the assessment,

including the additional question, the teacher will review

the assessment and discuss each section in general.

Students will explain each section to the teacher. If


students have any questions or uncertainties these can also

be clarified during this time.

Assessment: Students will complete the Mid-Unit Review assessment

with at least 87% accuracy and be able to articulate the steps to

multiply three-digit numbers accurately.

Teaching Notes / Stratagem: While the assessment is pushed by

the district, it is a good opportunity to stop to perform a check

for understanding before making a final push to finish the unit.

Students may be hesitant to want to slow down at this point and

this is a good opportunity to help them slow down if necessary.


Lesson Eleven

Title: Area and Volume

Description / Purpose / Direction: Some feel multiplication is

most enjoyable when viewed in light of real world contexts and

applications. Furthermore, the National Council of Teachers of

Mathematics suggests focusing on more true to life problem

solving exercises and activities to improve math retention

(Furner & Berman, 2003). In this unit students learned the steps

to work with numbers to multiply digits in a variety of ways. In

the past, students may have had opportunities to see word

problems developed by a textbook company, yet these do not always


adequately meet real-world criteria. Various research has come

to the conclusion that commercially produced elementary curricula

misses the mark and needs to be supplemented to be effective

(Jones & Southern, 2003). Therefore, this lesson bridges the gap

between textbook word problems and the real-world applications of

multiplication such as area and volume.

Objective: Given a worksheet all 5th grade students will

correctly solve at least 8 of 10 area and volume problems.



area and volume manipulatives, a sheet of paper, pen or pencil,

area / volume worksheet (see Appendix M).




40 minutes - Whiteboard presentation / instruction: The

teacher will give a whiteboard presentation that contains

various quadrilaterals with sides of differing lengths to

show area. It will also show different visual renderings of

cubes to present the idea of volume. In addition, the


teacher should also have an ample supply of manipulative

cubes and 3 dimensional rectangular prisms for students to

handle and measure. The teacher will begin the presentation

with a discussion of the area of an object using analogies

and illustrations. Students will then provide other

examples of area and volume. Once students have a basic

understanding of the differences between area and volume,

the teacher will explain how to find their mathematical

values using the measurements of the length of their sides.

Students will then be given real-world examples (swimming

pool, sod, concrete pad, etc.) to solve as practice in

class.

40 minutes – Independent work time: Students will use this

time to complete the worksheet containing area and volume

problems.

Assessment: Students will complete the area and volume worksheet

with 80% or better accuracy.


It will be important to use many visual and kinesthetic

models to show the application of area and volume. Teachers


can use blocks and other models to allow students to

physically measure to show how the lengths coincide with the

area and volume of the object being used.

One can also use a large analogy to show the concept, such

as the classroom itself. Area can be shown as the amount of

carpet or tile used to cover the floor, while the volume can

be the amount of water needed to turn the classroom into a

giant swimming pool for the summer months.

Lesson Twelve

Title: Teacher for a Day – Design a Game

Description / Purpose / Direction: Lesson twelve asks students

to take what they have learned to create a multiplication game

aimed at fourth grade students. This lesson challenges students

to become the designer and facilitator of a learning unit rather

than merely a spectator or participant in the learning process.

The learning goal for lessons twelve and thirteen is to take what


has been learned and teach the information to others.

Essentially, students will be demonstrating that they have come

full circle and are able to teach and explain the concepts that

they may have struggled to grasp at the beginning of the unit.

Objective: By the start of lesson fourteen, all 5th grade

students will design a multiplication game (with all facts up to

and including 8 x 8) aimed toward fourth grade students that

earns at least a 3 on each category of the provided rubric.


Materials / Technology: Self-guided multiplication interventions;

miscellaneous items for making games such as construction paper,

dice, blocks, etc.; pen or pencil; multiplication game rubric

(see Appendix N); and computers (if student chooses computer

based game).




20 minutes – Explanation of project: The teacher should

distribute rubric, explain the assignment and each area of

the rubric in detail, and address any questions that


students may have. Though the projects are to be completed

independently, students should feel free to bounce ideas off

of one another and act as game testers for their peers.

50 minutes – Work time: Students should have ample work

time to design their multiplication games. Students will

need to use their time wisely, but should be able to finish

if they work diligently. The teacher should monitor

progress and may act as a game tester if requested by

students. Feedback should also be given based on the

criteria contained in the rubric to aid students as they are

developing their games. The teacher should remind students

that they are designing a simple game for fourth grade

students and to recall what they would have wanted when they

were in fourth grade.

Assessment: Students will design a multiplication game aimed at

fourth grade students that is evaluated using a rubric. Game

should be turned in before the start of lesson fourteen.

Teaching Notes / Stratagem: Depending on the situation, the

teacher may want to suggest that students look at other games for

inspiration. However, since creativity is a rubric


consideration, students will need to be mindful not to copy too

much from another game. Other games should be used to elicit

ideas, not to serve as carbon copies.

Lesson Thirteen

Title: Teacher for a Day - Lecture

Description / Purpose / Direction: In lesson twelve, student

mathematicians designed a game, whereas lesson thirteen asks

students to actively teach the information they have come to know

intimately. Students will explain their thinking in their own

words to their peers. No longer can a student just do the work


in their head and put an answer down on paper and move to the

next problem. This exercise will ask students to go beyond

showing their work and giving an answer by challenging them to

take the additional step of revealing their thought process. By

connecting their thought processes to verbal and oral

communication, students will come to understand their own

thinking in more profound ways (Steele, 2001).

Objective: Given three to five minutes of teaching time, all 5th

grade students will:

1. Explain multiplication in their own terms.

2. Demonstrate one way to perform multiplication.

3. Discuss one job or task that uses

multiplication in real-life.



pen or pencil, paper, notecards, whiteboard, SmartBoard (if

needed for presentations), teaching presentation checklist (see

Appendix O).




30 minutes – Explanation of assignment and preparation time:

The teacher should explain the assignment to students and

should attempt to alleviate fears. Furthermore, copies of

the grading checklist should be distributed to students so

they understand how they are being assessed. Once students

have been made aware of the grading criteria, the remaining

time should be used to prepare for and practice the teaching

presentations.

55 minutes – Teachings presentations: The teacher will

randomly choose students one at a time to present their mini

teaching units. Students will be given no more than 5

minutes to present their mini-unit. If a student reaches

five minutes, the teacher should stop the presentation to

allow another student to give their presentation. If every

student takes the allotted 5 minutes this would take 95

minutes. From previous experience, this is not the norm,

but if it should occur, the teacher has the option of

extending class by 40 minutes, allowing presentations during

lesson 14, or permitting them to carry over into the class

period following lesson 15.


Assessment: Students will teach for three to five minutes in

front of the class and be evaluated based on the teaching

presentation checklist (see Appendix O).


Since student presentations will consume so much time today,

there will be no independent multiplication fact practice

time at the beginning of class.

Remind students to practice the delivery of their teaching

unit to ensure a quality presentation.

Notecards or small aids are permitted, but students should

be encouraged to refrain from reading from them verbatim.

In order to prevent very long presentations that can occur,

the lesson has purposely been designed to take place over

the course of one class period. Students have all the

knowledge necessary to prepare and present on the same day

if they have done their work properly throughout the unit.

Students should be reassured that they are graded based upon

the grading checklist, and not upon public speaking ability.

However, this is a good opportunity to begin building these

skills.



Lesson Fourteen

Title: Review for Assessment

Description / Purpose / Direction: Before moving into the final

assessment students will be given the opportunity to review to

build confidence and to revisit older information. This lesson

will also be an exercise for students to challenge one another to

improve their mathematical skills. As they prepare to be

assessed on what they have learned in the unit, students will

work together to devise questions to challenge other students.

Furthermore, they will attempt to solve the challenges created by

their peers. This review session will keep students’ minds

making new connections and constantly working to solve problems

that they may not have thought of before that are not connected

to a text book or worksheet. By reviewing in this way, students

will become the teacher and student in the same dynamic session.

Objective:

1. Within 30 minutes of class time, student groups of 2-3 will

devise 10 hypothetical real-world problems in an attempt to

challenge other student groups.


2. In 30 minutes, identical student groups will solve at least

80% of the problems created by other peer groups chosen by

the teacher.



paper, pen or pencil.




30 minutes – Problem construction: Students will be broken

up into groups of 2-3. Each group will devise 10

hypothetical real-world problems that are designed to

challenge their peers. These problems should be

multiplication problems and should be based on items that

have been covered in the unit (two-digit, three-digit, area,

volume, etc.)

30 minutes – Solutions: Student groups will exchange

problems and spend the next one-half hour attempting to

solve the problems that were given to them. Each group will

have the goal of solving at least 80% of their problems.


10 minutes – Discussion of lesson 15: Students will

reconvene as the teacher discusses plans for the final

assessment. Students will be given the opportunity to ask

any questions they may have before the final day of the

unit.

Assessment: Student groups will be assessed based on a twofold

criteria. First, each group will be evaluated based on the

number of problems or challenges devised. Secondly, each group

will be assessed on their accuracy rate with the problems given

to them.

Teaching Notes / Stratagem: The teacher should provide a few

simple examples on the SmartBoard or whiteboard before giving

students work time. In addition, even though a standard time

frame for devising problems has been provided, times can be

adjusted if students create their problem sets faster, but need

more time to solve the problems that have been created.

Furthermore, students need to be reminded that they need to have

solutions to the problems that they have created before they can

pose them to their student colleagues.


Lesson Fifteen

Title: Post-Assessment with Math Attitude Assessment

Description / Purpose / Direction: Students have now had

extensive opportunities to develop knowledge and understanding of

multiplication and its various expressions. Students have

explored everything from basic facts to real life applications

such as area and volume. Each student has developed a game of

their own to help other students practice their math facts and

each has taught the information to their peers. It is at this

point that one will measure the efficacy of the instructional

unit through an assessment identical to the pre-assessment given

at the beginning of the unit. In keeping with the goal of the

curriculum unit, students should be at least 90% accurate with

basic multiplication facts. Furthermore, since students need to

have a solid working knowledge of multiplication to succeed in


future math classes, this final assessment will look to see if

students have adequately gained appropriate grade level skills

and information. Lastly, the attitude assessment will aim to

determine if the unit was successful at improving student

attitudes and alleviating math anxiety.

Objective:

1. All 5th grade students will complete post-assessment with at

least 90% accuracy in 55 minutes or less.

2. All 5th grade students will complete the math attitude

assessment in 15 minutes or less and compare new answers to

the answers provided at the beginning of the unit.


Materials / Technology: Post-assessment (see Appendix P), post-

assessment answer key (see Appendix Q), math attitude assessment

(see Appendix B), pen or pencil, scratch paper, goal-setting

worksheets from lesson one.


10 minutes – The teacher should explain the overall

assessment procedure to students and ensure that all


students understand the process before distributing the

post-assessment.

Once all students have received their instructions, the

teacher should distribute the post-assessment. No materials

may be used for the assessment (calculators, books, etc.),

however scratch paper will be provided for student use. If

students need additional scratch paper, they should raise

their hand during the assessment to receive additional

sheets. These sheets should be turned in with students’

tests.

55 minutes – Administer post-assessment. When students are

finished with the test, the test should be handed directly

to the teacher.

10 minutes – Attitude assessment: The teacher will give

students the attitude assessment to complete. Once all

assessments are completed, the teacher will give each

student their initial attitude assessment that was completed

at the beginning of the unit to compare to the one taken in

lesson 15.


15 minutes – Attitude / Anxiety discussion: Class will

discuss the changes that the class experienced from the

beginning of the unit to the end.

The teacher should make sure that all tests are accounted

for before allowing students to move into next class /

period.

Assessment: All post-assessments will be evaluated by the

teacher for accuracy (>90% expected). Students will self-evaluate

their attitude assessment before giving both attitude assessments

to the teacher at the end of the class session. When post-

assessments are evaluated and returned to students they will be

given to students with a copy of the goal setting worksheets that

were completed in lesson one. This will allow students to

compare their goals with their true performance on the post-

assessment.

Teaching Notes / Stratagem: If students are feeling especially

nervous about the post-assessment, it may be advantageous to

perform the attitude assessment beforehand to alleviate some

anxiety and build confidence. If this is the case, one can still


perform the self-evaluation and discussion of the attitude

assessment after completing the post-assessment.

Additional Teacher Resources


The following is a list of resources that teachers may find

useful as they prepare and deliver the lessons in chapter three.

General Resources

http://www.multiplicationtool.org/ - Before beginning the

instructional unit, teachers may want to ensure that they are

prepared for the three main algorithms discussed in the unit.

There are many more methods available and one can do a web search

to find those if a student chooses to solve a method other than

those discussed in this unit. Yet, the above website offers a

tool to work through problems in each method as well as a

detailed explanation for solving problems using each algorithm.

If a teacher needs a refresher on multiplication methods or has

not encountered partial products or the matrix (also called

lattice) method, this is an excellent resource to utilize.

http://education.cu-portland.edu/blog/curriculum-

instruction/4-math-books-for-teachers-who-need-to-brush-up-on-

their-skills/ - Concordia University has compiled a short list of

books to help teachers who feel overwhelmed or unprepared to meet

the needs of their math students. These resources can help

educators improve their own math skills while learning more about

http://education.cu-portland.edu/blog/curriculum-instruction/4-math-books-for-teachers-who-need-to-brush-up-on-their-skills/



http://www.multiplicationtool.org/


the important strategies specific to a strong mathematics

education.

Lesson One

http://www.bizjournals.com/boston/blog/mass-high-tech/2012/09/goa

l-setting-why-90-of-us-dont-do.html?page=all - Teachers may want

to visit this website to develop an understanding of good, solid

goal setting strategies as well as the reasons for doing so.

When a teacher believes in the process of goal setting, students

will be more likely to follow the teacher’s lead as a result.

Lesson Two

http://www.reallygoodstuff.com/exit-passes/p/157234/ - This

is an economical resource to purchase simple and effective exit

passes. However, one can also purchase large index cards to be

used as exit passes.

http://webquest.org/ - Webquest.org is an online resource

that offers a wealth of information for educators looking for

everything from turnkey web quests to those wanting to create

their own.

http://webquest.org/

http://www.reallygoodstuff.com/exit-passes/p/157234/

http://www.bizjournals.com/boston/blog/mass-high-tech/2012/09/goal-setting-why-90-of-us-dont-do.html?page=all

http://www.bizjournals.com/boston/blog/mass-high-tech/2012/09/goal-setting-why-90-of-us-dont-do.html?page=all


https://play.google.com/store/apps/details?

id=yuku.mp3recorder.lite&hl=en

https://itunes.apple.com/us/app/mp3-recorder-free/id548031942?

mt=8

The above smart phone / tablet applications (web addresses

are cited) can be downloaded to Android or Apple devices to

record a voice as an mp3 file. These files can then be played

back on student iPods, iPhones, iPad, Android players, tablets,

etc. This is an up-to-date, innovative way to implement the

taped problems intervention for students. If students have a

device at home they can practice and listen to facts at their own

convenience. Both of the above applications are free of charge

in the Google Play and iTunes stores.

http://www.jimwrightonline.com/htmdocs/tools/mathprobe/

multsing.php - Jim Wright online has provided a free service for

classroom use that allows teachers to generate cover, copy, and

compare worksheets. The teacher simply selects from a list of

choices that even allows one to personalize the worksheets before

the website generates them for student use.

http://www.jimwrightonline.com/htmdocs/tools/mathprobe/multsing.php

http://www.jimwrightonline.com/htmdocs/tools/mathprobe/multsing.php

https://itunes.apple.com/us/app/mp3-recorder-free/id548031942?mt=8

https://itunes.apple.com/us/app/mp3-recorder-free/id548031942?mt=8

https://play.google.com/store/apps/details?id=yuku.mp3recorder.lite&hl=en

https://play.google.com/store/apps/details?id=yuku.mp3recorder.lite&hl=en


Lesson Four

https://docs.google.com/presentation/d/1C3Lte1CFgc4WSw8ydeEhF8BhH

dXBFxGnXKpSCGsH6XE/present#slide=id.i0 – Tasha Bergson-Michelson

and Kathy Glass give a tutorial on improving Google searches.

This may be beneficial for students if they struggle with the web

quest, however it may prove very useful for teachers as well.

Lesson Five / Ten

http://www.math-drills.com/ - Teachers can download a free

PDF file featuring different multiplication facts for students to

use as a worksheet or as this unit does, as a multiplication

quiz.

Lesson Eight

https://www.khanacademy.org/math/arithmetic/decimals/multiplying_

decimals/v/multiplying-decimals - A fantastic, colorful video

that will give students a better understanding of multiplying

decimals. In addition, the instructor gives a good conceptual

discussion of what it means to multiply by a decimal, which will

benefit students in the long run by connecting the concept to

https://www.khanacademy.org/math/arithmetic/decimals/multiplying_decimals/v/multiplying-decimals

https://www.khanacademy.org/math/arithmetic/decimals/multiplying_decimals/v/multiplying-decimals

http://www.math-drills.com/

https://docs.google.com/presentation/d/1C3Lte1CFgc4WSw8ydeEhF8BhHdXBFxGnXKpSCGsH6XE/present#slide=id.i0

https://docs.google.com/presentation/d/1C3Lte1CFgc4WSw8ydeEhF8BhHdXBFxGnXKpSCGsH6XE/present#slide=id.i0


fractions. This video could be used for guided instruction or as

extra help for students who are struggling.

Lesson Eleven

http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolu

me/ - Shodor.org gives teachers and students a virtual

manipulative that allows one to explore and calculate different

areas and volumes. The virtual interface allows changes to all 3

sides and shows the changes to area and volume as the

measurements change.

Chapter Four

Analysis of the Design and Writing Process

The capstone project was met with eagerness and joy as it

was viewed as an opportunity to display the skills and knowledge

that had been developed during the course of study at Colorado

http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/

http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/


Christian University. It was the climax to an already rewarding

experience of learning and growth offered by the MACI program.

Yet, the joy felt was combined with hardship and times of

difficulty as it often is in life. Once started, the process of

writing and developing quickly became a formidable task that

consumed more energy, family time, and effort than one would

first expect. There were many days when doubt crept in and one

began to wonder if all the training and education were enough to

overcome the obstacles that lay ahead. Yet, as each day passed

and another word, another sentence, and another page was

completed, the process became easier. Doubt gave way to moments

of clarity and satisfaction as one began to realize that taking

things slowly would eventually pay off. The capstone project and

its trials brought with it many moments of hardship and

difficulty which were married to numerous moments of victory and

delight. It was an experience that was more than worth every key

stroke and every sleepless night as the final product brought

with it a profound sense of accomplishment and gratification that

could not have been achieved any other way.


When the process was well underway, the project seemed

intimidating and nearly insurmountable, but looking back there

are many opportunities to reflect and consider growth. One can

see the things that were done correctly such as writing about

concepts that elicited passion and writing from personal

experience. However, it is also easy to identify things that

could have been improved upon. The author always needs to adhere

to the format or style that he or she knows and is comfortable

with. It is one thing to know one’s audience and to target the

writing toward that audience, but another thing altogether to

change a person’s style at the expense of the writing. One must

also look back at the assumptions made during the writing process

and evaluate them for the future. One would assume that all APA

references are correct when all papers returned over the course

of two degrees and nearly four years of study were returned

without corrections. However, when that assumption is proven

wrong this can be frustrating. The author realized that the APA

errors had been made from the beginning of the degree process and

were due to a lack of proper research and personal oversight.

This realization served to make one more attentive to detail and


a better researcher and writer in the future. When one sets high

expectations and fails to reach those standards consistently, it

can be a trying situation. Nevertheless, as long as the mistake

results in learning a lesson, then it has been a successful

experience. The capstone project was challenging and frustrating

at times, but since it served as a positive learning experience

it will be very beneficial for future educational and career

pursuits.

Evaluation of the Capstone

The developed capstone project had not been implemented at

the time of this writing. However, based upon previous

experience and implementation of similar interventions in the

classroom, the project will meet with marked success in

comparison to previous curriculum in use at St. Mary’s Catholic

School. Other students have responded very well to the use of

collaborative work and constructivist learning when implemented

in a similar environment to that of the capstone project.

Furthermore, every math class taught by the author last year saw

drastic statistical improvement on standardized test scores which

included an average of 3.6 points per student per session on the


Measures of Academic Progress (MAP) test (Northwest Evaluation

Association, 2013). These results were gained through the use of

methods similar to those in the capstone project. However, as

the capstone project is much more refined and developed, it is

believed it will lead to greater improvement than previous

instructional units. In addition, the self-guided interventions

will be used at St. Mary’s for the first time, which will further

improve performance.

In the future, it would be beneficial to modify the self-

guided interventions to work in iOS and Android based

applications for use in smart phones and tablets. An initial

desire of the project was to implement this idea with the

capstone, but technology and financial resources did not allow

for the completion of this part of the project at the time of

this writing. Additionally, it would be beneficial to look into

the development of innovative applications for student use that

would help to improve multiplication skills. The applications

that are currently available on the Apple iStore and Google Play

Store are aimed at more traditional learning styles and

interventions. If resources can be procured to develop the


applications, this would be an item to add to future

implementations of the project.

Until the project is fully implemented it is difficult to

know or anticipate all changes or modifications that may be

needed. The project will be successful, but as with any teaching

unit there may be adjustments or modifications that need to be

made in the future to increase subsequent success. It is

important that each lesson be followed by a reflective period

where the teacher can take notes or write out thoughts about the

lesson. These notes should be reviewed before any future

implementation of the unit.

Although the unit was not implemented at the time of writing

there were multiple opportunities for peer and colleague review

before completion of the capstone project. The feedback provided

by the university professor was very helpful and allowed the

paper to take shape with precisely tailored guidance and

suggestions. Each chapter was returned with specific comments

that clarified and expanded upon items from the capstone rubric.

This submission and feedback process helped to develop the

project and take it to a level that would not have been possible


if completed alone. The professor also provided positive and

encouraging dialogue, which made things more manageable as the

process continued.

In addition to the professor’s feedback it was also

important to receive information from teachers with similar

instructional interests. However, due to the time of year and

scheduling conflicts, it was difficult to meet with many fellow

teachers. Of the teachers who were willing to review the work,

some were not able to complete their reviews before the finished

project was ready for submission. The elementary teachers who

were able to complete a review were polled through informal

interviews and discussions. Overall, the project was met with

positive reviews and respondents were confident that the results

of instruction would be promising. The teachers who evaluated

the capstone project found it to be an instructional unit that

could be implemented in their classroom given the information

that was provided. One fifth grade mathematics interventionist,

Jessica Carter, felt that she could have accomplished the

instruction with fewer appendices, but was thankful for their

inclusion (personal communication, August 2, 2013).


In the course of review, the project was also given to the

new principal of St. Mary’s Catholic School (P. Lane, personal

communication, August 1, 2013). He was very helpful and

insightful in his assessment of the instructional unit. His

impression was also very positive and professional. He provided

very effective suggestions for improvement. He recommended

sending out a detailed email (rather than a simple email about

the late assignment policy) to ensure that parents understood the

unit. In addition, he also pointed out that the numbering system

used on one worksheet might have been confusing to some students.

The numbers of the problems were so close to the actual problem

that it nearly made them look like decimal problems. This may

have caused issues for students so the numbering system was

changed to prevent any chance of confusion. After making

suggestions to improve the project, St. Mary’s principal also

noted that the integration of problem solving and real-world

application was especially important and beneficial to the

project. Furthermore, he stated that empowering students to

assume the roles of teachers and leaders would help them to feel

positive and would help them to be ready for future instructional


challenges. Finally, he noted his satisfaction with the attitude

assessment and the way it was integrated into the unit to guide

instruction and improve performance. He believes that it will be

a successful starting point for the year and will be a project

that students will welcome and enjoy.

Personal Reflections

Math has become a topic of disdain and revulsion, even among

some people who have a natural ability to perform strongly in

math. When attending teaching conferences it is easy to find

others who specialize in reading, special education, social

studies, and other areas. However, if one looks for fellow math

teachers, they may be difficult to find. It is easy to uncover

teachers and other adults who claim to hate math or who state

that they are inept with their math skills (Change the Equation,

2013). It is a very sad and frightening prospect that some of

our teachers may be entering the classroom with poor math skills

or high anxiety. According to John Marshall, the issue with math

education reform today is the fact that many young teachers are

ready to jump into the classroom, but due to how they were taught

math they lack the confidence necessary to address what children


truly need (2003). Furthermore, the fact that high levels of

anxiety are potentially infectious is a major cause for concern

in classrooms across the country. Teachers may be modeling the

behavior and attitudes that they hope their students avoid

(Furner & Berman, 2003). The future of math in the United States

may not look very appealing at this time, and if more and more

teachers are entering the work force with similar attitudes it

may be getting worse before it gets better.

Math never had to be this way and if teachers can find a way

to get students to embrace math then the future does not have to

look so bleak. There are many students who have entered math

class without strong skills and have exhibited a hatred for math,

but have left with a healthy respect for math and improved

scores. It does not happen overnight and it is not easy, but it

can be done. However, it is not done by forcing students to

learn the same algorithm nor by rushing through a textbook that

shallowly covers everything from multiplication to polynomials.

It is accomplished through teamwork and collaboration. It is

done with compassion and an understanding that there are often

two or more routes to the same destination. One needs to know


all routes to aid students along the road, and when a student

finally makes it on their own, the satisfaction that everyone

feels is liberating and amazing. The work of the capstone

project was exhausting and long, but it was done with

satisfaction and joy. This is the same feeling that students

feel when they solve a math problem through the process used in

this capstone project. It may take more time and planning on the

part of the teacher. It may also take more math knowledge,

patience, and pedagogical knowledge as well as the ability to

explain and justify one’s methods and strategies to parents and

administration. However, in the end, the benefits greatly

outweigh the price paid and students may become advocates and

lovers of math rather than being another person parroting their

contempt for math as they discuss their poor abilities to all who

will lend an ear.

In the preparation and completion of the capstone, the

lessons learned are not limited to the area of mathematics. How

one interacts with the world around them is shaped by the

capstone process as well. Throughout the entire capstone and

MACI experience, one’s patience, abilities, courage, and faith


are continually tested. It is hard to imagine completing the

process without the help of a supportive family, but even more

remarkable to do so without God’s assistance. Knowing that God

has equipped each MACI candidate with the gifts necessary to

complete the degree has made things easier along the way.

Completion of the capstone has served to strengthen belief and

faith in God’s guiding hand while improving the personal

relationship one shares with Jesus Christ. Furthermore, one

truly realizes the blessing of having a caring family to battle

through obstacles and to share in the triumphs and celebrations.

One can truly experience joy through various trials when God is

present in life. Trust and faith in God allows one to traverse

the capstone project to a successful conclusion.

Conclusion

The time spent completing the capstone project and the MACI

program was rewarding as well as challenging. Time management,

prioritization, organization, and writing skills were all

developed and improved upon over the course of the last 12

months. There were many hurdles that were only overcome with

prayer, determination, perseverance, and the help of God and


family. As each obstacle was met and overcome, it became clear

that this was not the last stop of the educational journey.

With every assignment and paper written the desire to proceed

further in education began to grow significantly. As professors

pushed the writer to grow and improve, the passion to excel

increased greatly. Although the capstone marks the end of a

degree and a journey, it also signals the beginning of a new

endeavor. One can now pursue new opportunities to teach at the

collegiate level, which has been one of the author’s goals for a

long time. In addition, this opens another door to look into

potential doctoral work in the future. Teaching mathematics has

become a passion and it is believed that most people can be

successful in math if given proper instruction. Through future

research and doctoral work it is hoped that changes can be made

in the way math instruction is delivered to the majority of

students in the United States. Math education does not need to

remain stagnant and numeracy issues do not need to permeate

society as they do today.

There is considerable research and dialogue that still

needs to take place to improve math education in the United


States for the future. The polarization over direct instruction

and constructivism needs to reach a place where researchers can

work together to benefit students. Results of self-guided

interventions and the use of collaboration are showing great

promise. However, educators must have strong math knowledge and

skills in order to guide and aid students in reaching their full

mathematical potential. Math education needs to be improved at

the elementary level, but to do this, pre-service teaching

programs must ensure that new teachers are equipped to properly

educate their students. Teachers entering the elementary

teaching field cannot be afraid of numbers or teaching math to

their students. This fear or anxiety can be communicated to

students inadvertently, sending the wrong message to students

without intent. Teachers looking to implement this unit need to

ensure that they are adequately prepared to meet students’

multiplication needs, including addressing different algorithms

to solve complex problems. Having too narrow of a focus on one

algorithm has been identified as one possible source of math

anxiety for many students (Furner & Berman, 2003). Therefore,

it is important that a teacher be able to teach and troubleshoot


multiple algorithms to help students find success in this unit.

It is suggested that possible future study involves looking at

the effectiveness of different algorithms for elementary

students as well as research into the best sequence for teaching

math concepts and ideas. In addition, it would be beneficial to

begin looking at policy research to develop pre-service teacher

programs to develop better math teaching courses for generalist

teachers who will be entering K-6 and K-8 education.

As the MACI program and capstone project draw to a close it

is met with mixed feelings. It is a blessing to be able to

relax and spend more time with family, but at the same time the

program has become such a valuable part of life. Seven years

ago a lost, broken man stood alone wondering what to do with

life, and never could have imagined what possibilities might lay

ahead. Now as that same person looks back it is amazing to see

the changes that God has orchestrated. The road ahead looks

much different today than it did seven years ago and much has

been learned through the MACI program and the capstone process.

Though the journey has been a challenge, filled with obstacles

and difficulty, the blessings and knowledge gained are so


valuable and indispensable. God has been there every step of

the way, guiding the author through every valley and over every

hill. Although the process is complete, the journey of

education and growth never ends and one must continually strive

for daily improvement and excellence, striving to be God’s

faithful servant.

References


Ashcraft, M. H. (2002). Math anxiety: Personal, educational and

cognitive consequences. Current Directions in Psychological Science,

11(5), 181-185.

Ashcraft, M. H., & Krause, J. A. (2007). Working memory, math

performance, and math anxiety. Psychonomic Bulletin & Review,

14(2), 243-248.

Banikowski, A. K., Mehring, Teresa A. (1999). Strategies to

enhance memory based on brain-research. Focus on Exceptional

Children, 32(2), 1-16.

Bergson-Michelson, T., & Glass, K. (n.d.). Narrowing a search to get

the best results. Retrieved from Google Docs:

https://docs.google.com/presentation/d/1C3Lte1CFgc4WSw8ydeEh

F8BhHdXBFxGnXKpSCGsH6XE/present#slide=id.i0

Bliss, S. L., Skinner, C. H., McCallum, E., Saecker, L. B.,

Rowland-Bryant, E., & Brown, K. S. (2010). A comparison of

taped problems with and without a brief post-treatment

assessment on multiplication fluency. Behavioral Education,

19(2), 156-168.

Branch, R. M. (2009). Instructional design: The ADDIE approach. New York:

Springer.


Burns, M. (2004). 10 Big math ideas. Scholastic Instructor, 113(7), 16-

20.

Burns, M. (2005). Looking at how students reason. Educational

Leadership, 63(3), 26-31.

Caron, T. A. (2007). Learning multiplication: The easy way. The

Clearing House, 80(6), 278-282.

Change the Equation. (2013). In a new survey, Americans say, “We’re not good

at math”. Retrieved from Change the Equation:

http://changetheequation.org/press/new-survey-americans-say-

%E2%80%9Cwe%E2%80%99re-not-good-math%E2%80%9D

Cheo, K. (2012, September 11). Goal setting: Why 90% of us don’t do it

effectively. Retrieved from Boston Business Journal:

http://www.bizjournals.com/boston/blog/mass-high-tech/2012/0

9/goal-setting-why-90-of-us-dont-do.html?page=all

Cole, J.E., & Wasburn-Moses, L. H. (2010). Going beyond "The math

wars". Council for Exceptional Children, 42(4), 14-20.

Concordia University. (2013). 4 Math books for teachers who need to brush

up on their skills. Retrieved from Concordia University Online:

http://education.cu-portland.edu/blog/curriculum-


instruction/4-math-books-for-teachers-who-need-to-brush-up-

on-their-skills/

Dodge, B. (2007). Welcome. Retrieved from Webquest.org:

http://webquest.org/

Furner, J. M., & Berman, B. T. (2003). Math anxiety: Overcoming a

major obstacle to the improvement of student math

performance. Childhood Education, 79(3), 170-174.

Geary, D. C. (2000). From infancy to adulthood: the development

of numerical abilities. European Child & Adolescent Psychiatry, 9(2),

11-16.

Inch, S. (2002). The accidental constructivist: A mathematician's

discovery. College Teaching, 50(3), 111-113.

Jones, E. D., & Southern, W. T. (2003). Balancing perspectives on

mathematics instruction. Focus on Exceptional Children, 35(9), 1-16.

Khan Academy. (2013). 2-Digit times 1-Digit number. Retrieved from 2-

Digit times 1-Digit number:

https://www.khanacademy.org/math/arithmetic/multiplication-

division/multi_digit_multiplication/v/multiplication-4--2-

digit-times-1-digit-number.


Khan Academy. (2013). Multiplying decimals. Retrieved from Khan

Academy:

https://www.khanacademy.org/math/arithmetic/decimals/multipl

ying_decimals/v/multiplying-decimals

LeTourneau, C. D., Posamentier, A. S., & Ford, E. R. (2006).

Progress in mathematics. New York: Sadlier-Oxford.

Marshall, J. (2003). Math wars: Taking sides. Phi Delta Kappan,

85(3), 193-200,249.

Math-Drills.com. (2013). Multiplication facts worksheets. Retrieved from

Math Drills :

http://www.math-drills.com/multiplication.shtml#Facts144.

National Council of Teachers of Mathematics. (2000). Principals and

standards for school mathematics. Reston, VA: National Council of

Teachers of Mathematics.

National Council of Teachers of Mathematics. (2006). Curriculum

focal points of prekindergarten through grade 8 mathematics. Reston, VA:

National Council of Teachers of Mathematics.

National Science Foundation. (2012, January). Graduate education,

enrollment, and degrees in the United States. Retrieved from Science


and Engineering Indicators 2012:

http://www.nsf.gov/statistics/seind12/c2/c2s3.htm

Northwest Evaluation Association. (2013). Measures of academic

progress. Portland, OR: Northwest Evaluation Association.

Ogbuehi, P. I., & Fraser, B. J. (2007). Learning environment,

attitudes and conceptual development associated with

innovative strategies in middle-school mathematics. Learning

Environments Research, 10(2), 101-114.

Poncy, B. C., Skinner, C. H., & Jaspers, K. E. (2007). Evaluating

and comparing interventions designed to enhance math fact

accuracy and fluency: Cover, copy, and compare versus taped

problems. Behavioral Education, 16(1), 27-37.

Public Learning Media Laboratory. (2010). Multiplication tool.

Retrieved from Multiplication tool:

http://www.multiplicationtool.org/

Really Good Stuff. (2013). Really good exit passes. Retrieved from

Really Good Stuff: http://www.reallygoodstuff.com/exit-

passes/p/157234/

Reinhart, S. C. (2000). Never say anything a kid can say!

Mathematics Teaching in the Middle School, 5(8), 478-483.


Rich, M. (2012, December 11). U.S. students still lag globally in math and

science, tests show. Retrieved from New York Times:

http://www.nytimes.com/2012/12/11/education/us-students-

still-lag-globally-in-math-and-science-tests-show.html?_r=0.

Roberts, K. (2013, July 20). Math Anxiety Survey. Retrieved from

Mathipedia: http://www.mathipedia.com/legal.html

Sadlier-Oxford. (2013). Check your progress. Retrieved from Progress

in Mathematics:

http://www.sadlier-oxford.com/math/mc_progress.cfm?

grade=5&sp=teacher

Schoen, H. L., Cebulla, K. J., Finn, K. F., & Fi, C. (2003).

Teacher variables that relate to student achievement when

using a standards-based curriculum. Journal of Research in

Mathematics Education, 34(3), 228-259.

Shirvani, H. (2009). Does your elementary mathematics methodology

class correspond to constructivist epistomology? Journal of

Instructional Psychology, 36(3), 245-258.

Shodor. (2013). Surface area and volume. Retrieved from Shodor:

http://www.shodor.org/interactivate/activities/SurfaceAreaAn

dVolume/


Slavin, R. E., & Lake, C. (2008). Effective programs in

elementary mathematics: A best-evidence synthesis. Review of

Educational Research, 78(3), 427-515.

Sousa, D. A. (2010). Mind, brain, & education. Bloomington, IN:

Solution Tree Press.

Steele, D. F. (2001). Using sociocultural theory to teach

mathematics: A Vygotskian perspective. School Science and

Mathematics, 101(8), 404-416.

Steen, L. A. (2007). How mathematics counts. Association for

Supervision and Curriculum Development, 65(3), 8-14.

Tait-McCutcheon, S., Drake, M., & Sherley, B. (2011). From direct

instruction to active instruction: Teaching and learning

basic facts. Mathematics Education Research Journal, 23(3), 321-

345.

Truong, V. (2013, May 30). MP3 recorder free. Retrieved from iTunes:

https://itunes.apple.com/us/app/mp3-recorder-free/id54803194

2?mt=8

Ward Hoffer, W. (2012). Minds on mathematics: Using math worskshop to

develop deep understanding in grades 4-8. Portsmouth, NH: Heinemann.


Wright, J. (n.d.). Multiplication. Retrieved from Jim Wright Online:

http://www.jimwrightonline.com/htmdocs/tools/mathprobe/mults

ing.php

Wyoming Department of Education. (2012, July). Wyoming content and

performance standards. Retrieved from Wyoming Department of

Education: http://edu.wyoming.gov/programs/standards.aspx.

Yuku. (2013, June 21). Hi-Q MP3 Voice Recorder (Free). Retrieved from

Google Play: https://play.google.com/store/apps/details?

id=yuku.mp3recorder.lite&hl=en

Appendix A

Path Analysis


Appendix B

Multiplication Unit – Pre-Assessment


Name:___________________________Date:______________________________

Directions: Please complete this entire test to the best of yourability. If you get stuck you can skip a problem, but make sure you come back to the problem to complete it.

Section One: Multiplication Facts

7x12= 3x10= 2x11= 11x9= 2x2=12x11= 4x8=

12x7= 5x9= 3x12= 7x10= 6x9=10x12= 2x7=

3x7= 9x7= 7x7= 11x9= 5x7= 10x2=4x12=

5x8= 7x9= 11x11= 2x12= 4x4= 12x3=2x2=

2x10= 10x10= 7x12= 2x3= 5x11= 4x7=2x5=

3x9= 8x8= 8x2= 8x6= 7x7= 3x3= 5x7=

4x3= 6x2= 11x8= 6x4= 2x9= 3x7=4x9=

8x3= 5x4= 2x4= 12x12= 1x6= 5x9= 2x8=


3x9= 9x9= 11x4= 6x6= 9x7= 9x5=7x4=

3x12= 8x9= 7x8= 1x12= 4x8= 6x5=3x3=

Section Two: More Multiplication

1. 4 x 40 = 2. 9 x 60 = 3. 3 x 700 = 4. 8 x 500 =

5. 20 x 4000 = 6. 20 x 40,000 =

7. 164 8. 279 9. 312 10. 673 11. 51212. 482

x56 x34 x284 x406 x700 x47

13. $35.24 14. $15.26 15. 17.6 16. 19.31 x 16 x 10 x 4 x

5


17. Alana has 17 boys and girls in her class this year. Her birthday is coming next month and she would like to give everyone a treat at the end of the day. She has decided that she will give Twinkies since they are making a comeback. If they come in packages of six and she wants to give each student two (2) and the teacher one (1), how many total Twinkies will she hand out and how many boxes will she need toask her mother or father to purchase?

18. If a rectangle has one side with the length of 3 inches and another side with the length of 5 inches, what is the areaof the rectangle? Hint: Draw and label a rectangle to help solve the problem.

Section Three: Reasoning and Short Answer

19. Please describe what multiplication is and how it relates toaddition.


20. Using a problem from earlier in the test or your own exampleplease explain the steps to solve a multiplication problem. Whatis the answer to a multiplication problem called?

Appendix C


Math Attitude Assessment **(NOTE: Adapted from Math Anxiety Survey [Roberts,2013])**

Name:___________________ Date:____________________________

Please circle the number that best describes your feelings on a scale from 1 to 5. A 1 means that you really agree with the sentence and a 5 means that you really disagree with the sentence. For instance if the sentence said thatyou liked to eat dirt, you would probably circle a 5, because people do not like to eat dirt and they would not agree with that statement.

1. I feel badly when I get ready for math class 1 2 3

4 5

2. I am confident about my math skills 1 2 3

4 5

3. I feel uneasy about going to the board in class 1 2

3 4 5

4. I enjoy math as a whole and look forward to it 1 2

3 4 5

5. I am afraid to ask questions about math 1 2 3 4

5

6. I am eager to learn new math ideas 1 2 3 4

5

7. I always worry about being called upon 1 2 3 4

5

8. I never think about what might go wrong 1 2 3 4

5

9. I may get math now, but worry I won’t later 1 2 3

4 5


10. I tend to take notes and keep them for a long time 1 2

3 4 5

11. I tend to zone out (daydream) during class 1 2

3 4 5

12. I look forward to tests and study for them 1 2

3 4 5

13. I fear math tests more than any other kind 1 2

3 4 5

14. I feel good about my ability to do math 1 2 3

4 5

15. I am unsure of how to study for math tests 1 2

3 4 5

16. I feel that my math ability is based on what I do 1

2 3 4 5

17. I get it in class, but it is hard at home 1

2 3 4 5

18. I am looking forward to multiplication this year 1

2 3 4 5

19. I fear that I may not be able to do 5th grade math 1 2

3 4 5

20. I am willing to work hard and do my best in 5th grade 1

2 3 4 5


Add the total of odd numbered questions and write on this line___________________

Add the total of even numbered questions and write on this line___________________

Appendix D

STUDENT GOAL SETTING WORKSHEETName:___________________________

Date:_______________________________

Example: I can currently solve 40% of multiplication problems in 10 minutes.

I would like to be able to correctly solve 90% of multiplication problems in 5 minutes or less. In order to do this, I must do cover, copy, and compare for 15 minutes per day and play multiplication games on the computer at home for 30 minutes per day. I will also try to take a practice test at least once on weekends.

Goal 1:

Current Condition:_____________________________________________________________

Where I would like to be:________________________________________________________

What I need to do to get there: ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Goal 2:




What I need to do to get there :_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Goal 1:



What I need to do to get there: ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Appendix E

Multiplication Web Quest Rubric

Name:________________________________ Score:________Grade:________


Category 1 2 3 4

History / Origins

Facts

address the origins or history of multiplication in any way

somewhat discusses the history or origins of multiplication

adequately discusses the history or origins of multiplication

y discusses the history or origins of multiplication

Applications / Jobs / Careers

Student names one or less job or application of multiplication

Student names 2 jobs or applications of multiplication in the real world

Student names 3-4 jobs or applications of multiplication in the real world

names more than 4 jobs or applications of multiplication in the real world

Student fails to discuss or address any pertinent facts

Student discusses one or two facts concerning multiplication

Student discusses 3-4 multiplication facts

Student discusses myriad facts pertaining to multiplication

Appendix F

Two-digit by One-digit Multiplication

Name:__________________________Date:___________________________


1)63 2) 53

3)47

4)14

5)39

x8

x7

x1

x2

x2

6)83

7)55

8)34

9)72

10)82

x6

x3

x6

x1

x9

11)51

12)25

13)65

14)72

15)51

x4

x4

x4

x5

x7

16)27

17)96

18)62

19)18

20)53

x8

x2

x8

x9

x8

21)83

22)16

23)51

24)34

25)39

x1

x9

x1

x7

x1

26)79

27)77

28)23

29)92

30)96

x x x x x


6 7 7 3 3

31)11

32)84

33)92

34)25

35)89

x4

x9

x3

x2

x3

Appendix G

Real World Problem RubricStudent Name:_________________________ Score:__________

Grade:_________

Category 1 2 3 4

Solution

Solution isinaccurate or missing

Solutionis partially inaccurate

Solution is accurate

Solution goes aboveand beyondthe requirements of the assignment

Explanation

Explanationis incorrect or cannot be understood

Explanation is partially correct or hard to understa

Student'sexplanation is correct and understandable

Student's explanation exceeds expectations


nd

Comments: -__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Appendix H


(Worksheet Works, 2013)1

Appendix I1 No page number is available. A different worksheet is generated each time. This is an example only. Worksheets are intended for classroom use.


Decimal vs. Whole Number Multiplication

$36.56 639Name:________________________________

Date:_____________________________

Please name at least 3 similarities and 2 differences between decimal and whole number multiplication using the graphic organizer below. Note: Sketch an example to help you remember each one.

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

Decimal vs. Whole

Similiarities

Differences


_________________________________________________________________

_________________________________________________________________

__________________________

Appendix J

Multiplying with Decimals and Money

Name:__________________________________Date:________________________

All problems are multiplication. Please multiply and remember the differences that are present when multiplying with decimals. Do not forget about the sign when dealing with money ($).

21 5.9 0.31 11 1.21. 6.1 2. 1.5 3. 0.86 4. 2.7 5. 5

6.$1.5

6 7. 23 8. 96 9. 5610.

$3.65

6 2.1 1.2 0.5 5

11 1.8 12 12 13 $0.4 14 71 15 57


. . . 9 . .88 1.2 29 7.3 0.3

16. 89

17. 0.69

18. 1.2

19.

$6.89

20. 3.6

0.9 0.43 4.2 7 5.9

Appendix K

Three digit number multiplication

(You may use a separate sheet of paper if necessary)

Name:_______________________Date:______________________

1. 481

2. 631

3. 174

4. 949

5. 944

6. 221

3 2 5 6 8 5

7. 505

8. 793

9. 541

10. 213

11. 477

12. 228

41 85 82 25 89 45


13. 839

14. 189

15. 849

16. 704

17. 904

18. 732

15 75 11 15 75 34

19. 683

20. 977

21. 355

22. 986

23. 954

24. 581

554 337 349 124 352 428

25. 699

26. 628

27. 111

28. 635

29. 526

30. 231

141 468 235 234 225 128

Appendix L

Mid-Unit ReviewName:____________________________ Score:__________

Grade:______

1. 2077

2. 398

3. 357

4. 459

5. 56


x8

x42

x203

x512

x52

6.

125

7.

1.2

8.

0.2

9.

6.3

10. 15

x2.3

x9.8

x0.6

x6

x6

11. 12

12. 112

13. 365

14. 111

15. 56

x12

x10

x12

x11

x1.1

16. John and Mary have 14 children with another one on the way. Next year they are planning to build a new house for their large family. The builder has suggested at least 800 square feet for mom and dad and 200 for each child in the new house. How many square feet will their new home be? If their home will cost $100 per square foot, how much will their new home cost them to build?


Appendix M

Area and Volume

Name:_________________________

Please solve all problems. Remember to pay attention to what thequestion is asking you to do. Draw a picture or diagram if needed.

1. John and Stella have a square back yard with a length of 48 feet. Stella would like to have the yard covered with sod, which is $2.00 per square foot. How many square feet do they need and how much will it cost them to buy the sod?

2. Miguel has a dance floor that needs a new surface. It measures 50 feet by 36 feet. What is the area of the dance floor? A friend has said he has 1500 square feet of flooring left over from another job. He said that he will it install for free if Miguel will provide him with free dance lessons. Will this be enough to recover Miguel’s floor? If not, how much is still needed?

3. If a football field is 100 yards by 42 yards, how many square yards does the groundskeeper mow every Saturday


before the big game?

4. A football field is being covered in 6 inches of ice to hostan NHL ice hockey game this winter at Soldier Field in Chicago, IL. What is the volume of ice that it will requireto cover the football field?

5. Harry just found out that a cubic centimeter (cc) or cm3 is the same thing as one ml. He has a vase that is ten centimeters cubed (each side measures 10 cm) and he would like to know what its volume is. He is also curious if thiswould hold the amount of liquid that his Gatorade bottle contains (890 ml). Can you help Harry?

6. The basketball court at Samantha’s house has many cracks that cause the ball to take some strange bounces. The courtmeasures 50 feet by 40 feet and she wants to try to fix the court. Her dad said that putting down new concrete is too expensive. Sam found a liquid sealer that will cover and help to smooth out the cracks, but it costs $75.00 per bucket. Each bucket covers 300 square feet. Her dad said they only have $1000.00 to spend. Can they use the sealer


to fix the court? If so, how many buckets will they need tobuy and how much will it cost?

7. If the area of the surface of a pool is 2500 square feet andit is 10 feet deep, what is the volume of the pool?

8. If the volume of a cube is 1000 ft3 what is the measurementof one of its sides?

9. Elizabeth found out that one can of paint will cover 250 square feet. The room that she wants to paint is a rectangular room with each wall measuring 20 feet by 12 feet. She is not painting the ceiling. What is the area ofeach wall? What is the total area of the room? How many cans of paint will she need to finish the room? What if shechooses to paint two coats?


10. Please explain how multiplication is used to find the area of a square or rectangle. Secondly, explain how multiplication is used to find the volume of a 3 dimensionalobject such as a cube. Use an example in each of your explanations.

Appendix N


Teacher for a day – Multiplication Game RubricStudent Name:______________________________ Score:_________

Grade:______

Category 1 2 3 4

Completenessof Facts

Student was missing numerous multiplication facts

Student was missing afew multiplication facts

Student had all required multiplication facts

Student included more thanrequired multiplication facts

Age Appropriateness

Game was far beyond fourth grade level (more than 2-3 years)

Game was slightly beyond fourth grade level (1-2 years)

Game was at fourthgrade level

Originality / Creativity

Game was not original and did not show signs of creative thought

Game showed some signs of creative thought and originality

Game was original and showed good signs of creativity

Game was novel andvery creative


Comments: -__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Appendix O

Teaching Checklist

Student Name:________________________ Score:________Grade:_______

Below is a breakdown of your specific teaching presentation for the multiplication mini-unit.

Item Yes NoPossible Points

Points Earned

Explains multiplication in own terms 20 Demonstrates multiplication 20


Discusses real-life application 20 Time-limit:1.Under 5 min- no points 2.Between 3-5 min - all points 3.Over 5 min / cut-off - 1/2 points 10 Total 70

Comments: ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Appendix P

Multiplication Unit – Post-Assessment

Name:___________________________Date:______________________________


Directions: Please complete this entire test to the best of yourability. If you get stuck you can skip a problem, but make sure you come back to the problem to complete it.


7x12= 3x10= 2x11= 11x9= 2x2=12x11= 4x8=

12x7= 5x9= 3x12= 7x10= 6x9=10x12= 2x7=

3x7= 9x7= 7x7= 11x9= 5x7= 10x2=4x12=

5x8= 7x9= 11x11= 2x12= 4x4= 12x3=2x2=

2x10= 10x10= 7x12= 2x3= 5x11= 4x7=2x5=

3x9= 8x8= 8x2= 8x6= 7x7= 3x3= 5x7=

4x3= 6x2= 11x8= 6x4= 2x9= 3x7=4x9=

8x3= 5x4= 2x4= 12x12= 1x6= 5x9= 2x8=


3x9= 9x9= 11x4= 6x6= 9x7= 9x5=7x4=

3x12= 8x9= 7x8= 1x12= 4x8= 6x5=3x3=


1. 4 x 40 = 2. 9 x 60 = 3. 3 x 700 = 4. 8 x 500 =

5. 20 x 4000 = 6. 20 x 40,000 =

7. 164 8. 279 9. 312 10. 673 11. 51212. 482

x56 x34 x284 x406 x700 x47

13. $35.24 14. $15.26 15. 17.6 16. 19.31 x 16 x 10 x 4 x

5


17. Alana has 17 boys and girls in her class this year. Her birthday is coming next month and she would like to give everyone a treat at the end of the day. She has decided that she will give Twinkies since they are making a comeback. If they come in packages of six and she wants to give each student two (2) and the teacher one (1), how many total Twinkies will she hand out and how many boxes will she need toask her mother or father to purchase?

18. If a rectangle has one side with the length of 3 inches and another side with the length of 5 inches, what is the areaof the rectangle? Hint: Draw and label a rectangle to help solve the problem.


19. Please describe what multiplication is and how it relates toaddition.

20. Using a problem from earlier in the test or your own exampleplease explain the steps to solve a multiplication problem. Whatis the answer to a multiplication problem called?


Appendix Q

Pre / Post Assessment Answer Key



84 30 22 99 4 132 32

84 45 36 70 54 120 14

21 63 49 99 45 20 48

40 63 121 24 16 36 4

20 100 84 6 55 28 10

27 64 16 48 49 9 35

12 12 88 24 18 21 36

24 20 8 144 6 45 16

27 81 44 36 63 45 28

36 72 56 12 32 30 9


1. 160 2. 540 3. 2100 4. 4000

5. 80,000 6. 800,000

7. 9,184 8. 9,486 9. 88,608 10. 273,238 11. 358,400 12. 22,654

13. $563.84 14. $152.60 15. 70.4 16. 96.55

17. 35 Total Twinkies. Alana will need at least 6 boxes of Twinkies.

18. 15 in2



19. Answers will vary. Make sure students include all pertinentinformation.

20. Answers will vary based on the algorithm students choose to use to explain their problem. A product is the answer to a multiplication problem.

Improving 5th Grade Multiplication Performance Through Various Interventions

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