Skills in Third Grade Mathematics
Devon L. Radant
ETL Cluster 12
A Research Project
Graduate Teacher Education Online Program
in Partial Fulfillment
of the Requirements
for the Degree
of Master of Science in Education
Chapter I: Introduction
Description of the Community
The research project was conducted in a suburban
community outside a metropolitan area in a southeastern state. The county within
which this community is located has a population of 673,345 according to the U.S. Census Bureau (2004) for the year 2000. There are 1,359.9 persons per square mile, and the average family income is $60,537. One out of every five residents in the county attends public school as reported in
the county’s website (2004).
Writer’s Work Setting
The writer’s work setting is a public elementary
school, one of 63 in the county. The county school district is the largest in
the state. Enrollment at the local school for the 2002-03 was 668, as reported in the school’s accountability report. Of these 668 students, 3% received services for English Speakers of Other Languages
(ESOL) and 20% were serviced in Special Education programs. Students receiving
reduced-priced lunch equaled 10% of the population. In second through fifth
grades, 27% of students qualified for the gifted program. The racial composition of the population is: 71% white, 14% black,
6% Asian, 5% Hispanic, 4% multiracial, and .1% American Indian. The school
has a good reputation in the community and within the county.
scores on standardized tests frequently meet or exceed state standards. On the
2002-03 Grade 4 Criterion-Referenced Competency Tests (CRCTs), 95% of fourth grade students met or exceeded state standards
in the area of Reading/English Language Arts. Students also performed well in
Math with 91% of those same students meeting or exceeding state standards. Along
with standardized scores, students also perform well at the local level.
Each school within the county also participates
in Local School Plans for Improvement (LSPIs). The objective of this program
is to set annual goals for the school as designed by administrators, teachers, and parent advisory groups. In 2002-03, 89% of students in kindergarten through fifth grade (K-5) performed on or above grade level
on Reading Standards. On Writing Standards, 88% of students K-5 demonstrated
at or above grade level ability. And, in the area of Math Exemplars, 86% of students
in K-5 showed improvement of one or more levels by the end of the year.
In addition, the staff and parents also add to
the school’s success. There are 53 certified teachers; 70% of those teachers
have received a Master’s degree or higher. School faculty also participates
in staff development courses resulting in more than 6,000 hours. Parents support the school by logging more than 5,000 volunteer
hours and taking part in more than 1,000 parent-teacher conferences. The school
also boasts 100% Parent Teacher Association (PTA) membership.
The writer is a third-grade teacher certified
in Early Childhood Education. She works with a team of five on her grade level. As a classroom teacher, she is responsible for teaching math, reading, spelling, writing,
grammar, and science. Students receive social studies and health education from
other third-grade teachers. In addition to the regular math curriculum, the writer also teaches an Academic Knowledge and
Skills (AKS) Implementation Model class in the area of mathematics. Students receive instruction on specified math skills
throughout the year on one of four levels: challenge, enrichment, tutorial, or remedial.
The writer is responsible for planning and implementing instruction within her classroom and assessing student achievement.
also has other roles and responsibilities at the school. She serves as a mentor
teacher to a new teacher on third grade level by advising and participating in mentor-mentee meetings and discussions. Additionally, the writer works with a Local Leadership Team on a Literacy Committee. She is a pal to a student with whom she meets periodically throughout the year. The writer also takes part in various professional leadership opportunities to foster
and improve learning in her classroom.
II: Study of the Problem
The problem-solving skills of third-grade
mathematics students are inadequate.
Third-grade students have problem-solving difficulty
in mathematics. Students at this school traditionally score lower on problem-solving questions on standardized tests than
in other areas of mathematics. On the Criterion-Referenced Competency Tests administered
in Spring 2004, students in first and second grade received lower scores on the problem-solving section than in the other
sections of mathematics. Seeing this trend in the lower grade levels, teachers
in third-grade focus on this skill each year with the goal of improvement.
In addition, third-grade teachers continually
voice concerns with this area of mathematics. Teachers state that the problem-solving
questions on chapter tests are the most frequently missed. In observations made
by teachers, they have reported that students often show signs of frustration in class when confronted with these types of
questions. Teachers have also noted that students with reading deficiencies often
have trouble with problem-solving because most of these problems are presented in written form.
Evidence of this problem can be seen based
on students’ performance on exemplars. On exemplars administered at the
beginning of the school year, 26.1% of third-graders performed at the Novice level.
The largest number fall into the Apprentice level with 48.9%. There
were 19.6% of students demonstrating a Practitioner level of knowledge, and only 5.4% scored in the Expert range. Of a population of 19 students:
47% feel confident doing problem-solving
37% have a plan for problem-solving questions
58% feel like they have a lot of strategies to choose from when problem-solving
21% are frustrated when they work on problem-solving
79% believe that computation is much easier than problem-solving
42% think they are able to express their math reasoning clearly
Relationship of the
Problem to Literature
Problem-solving in mathematics is not a new concern
for educators. This particular area has frustrated teachers and students alike
in the past. Ironically, problem-solving should come naturally to you young children
because the world is new to them, and they exhibit curiosity, intelligence, and flexibility as they face new situations. However,
this is not always the case. The challenge lies in building on children's innate
problem-solving inclinations with the goal of preserving and encouraging a disposition that values problem-solving. In recent years, much has been learned about the problem-solving capabilities of elementary school children
and their use of strategies in solving problems. Unfortunately though, problem solving has typically existed apart from core
curriculum objectives. In many situations, it is addressed only after specific concepts and skills are taught and only then
to provide practice for those concepts and skills (Trafton & Midgett, 2001). Several
leading educational institutes, however, recognize the importance of problem-solving.
In fact, the National Council of Teachers of Mathematics has recommended that problem-solving be placed at the center
of mathematics education. Problem-solving has even been called “heart of
mathematics.” (Peixotto, 2000, 2). It should continue to be a core
strand of math learning for all grade levels.
The mathematics community acknowledges the effect
that problem-solving can have for students by developing higher-order thinking, promoting confidence in facing life’s
challenges, and building understanding that is transferable. Students who are
good problem-solvers also display characteristics of being resourceful, willing to explore, persistent, and tolerant to frustration
(Jarrett, 2000). There is sufficient evidence that problem-solving is beneficial
for students; however, most students struggle with this area of mathematics.
This struggle results from several factors. One such factor is the relationship between reading and problem-solving. Many students do not possess a working mathematical vocabulary; they do not know the definitions and therefore
have trouble comprehending what they are being asked to do. In a study
done by Blessman and Myszczak in 2001, they discovered that students did not know the necessary vocabulary to express their
ideas in mathematics. They also made a correlation between the lack of reading
comprehension and the understanding of mathematical concepts. In a similar study, Schoenberger and Liming (2001) helped students
solve multi-step problems and word problems by improving their mathematical vocabulary. Children have to combine reading,
thinking, and computational skills to solve math word problems. For many students, the major problem with word problems
is the words. Solving these problems involves many steps and requires learners
to be good readers as well as proficient at thinking critically, computing, and using a process to solve problems all at the
same time (Forsten, 2004). If students know the vocabulary and can communicate
mathematics effectively, then their problem solving skills will improve (Fletcher & Santoli, 2003). Reading ability can affect a student’s ability to problem solve.
In addition, there is evidence that rote memorization
does not necessarily lead to understanding. James Hiebert has done research over
the past 20 years on problem-solving. He believes that students learn best when
they have to “[grapple] with difficult and absorbing problems” (Jarrett, 2000, 4).
Too much time in the classroom is focused on drill and practice of math facts and computation. Teachers spend more time on memorization and recall instead of critical thinking skills and reasoning necessary
for problem solving (Millard, Oaks, & Sanders, 2002). One problem with the
rote memorization that so many mathematics teachers employ is that it doesn’t allow for students to see past the surface
features of problems and understand what they are being asked to solve. Students
learn by mimicking. This mimicking may lead to success on chapter tests, but
it does not lead to developing real understanding (Fletcher & Santoli, 2003). O’Brien and Moss (2004) write,
fact is that an exclusive emphasis on rote memory and rote performance of computational procedures at a time when every desktop
computer can do billions of computations in a second is downright foolish. Of course, children should learn to add, subtract, multiply, and divide, and they should do so sensibly and efficiently…[but]
today are facing a world that is shaped by increasingly complex, dynamic, and powerful systems of information in a knowledge-based
economy.... Being able to interpret and work with complex systems
involves important mathematical processes that are underemphasized in numerous mathematics curricula” (292).
Research investigations have shown that teaching for meaning and understanding has positive effects on student
learning (Grouws & Cebulla, 2000). However, many students have a lack of training and strategies necessary to successfully
Because students lack training, they have no
background or framework when problem solving. Children simply do not do enough
problem solving. Although they can perform mathematical operations, many do
not really understand what they are doing. (Miller, 1996). When students are confronted with a problem for which they have
no background knowledge to draw on or strategies to employ, they also have no confidence.
They become frustrated with the problem and give up. Knowing that if they
wait long enough, the teacher will provide the answer for them. The goal of many
teachers is for students to simply to get the right answer. They view their role as showing students the proper way to go
about solving the problem. Too often, teachers rescue students in the midst of problem-solving by giving them specific hints
to guide them to the answer or eventually working the problem for them (Mikusa, 1998).
This may help students get the right answer, but it won’t strengthen their problem-solving skills.
Furthermore, teachers have to deal with a lack
of resources when it comes to problem-solving. They have difficulty finding time
to include problem-solving strategies in the math curriculum. A cause for concern
is the critical need for daily problem-solving activities in the classroom. There
is a limited amount of time in each school day, yet teachers have seemingly endless demands placed on it. A lack of teacher training also contributes to the problem (Millard, Oaks & Sander, 2002). Although many teachers agree that math concepts
should be taught through hands-on activities and problem-solving, it’s simply not happening. Even though most elementary
teachers are certified to teach all subjects, most are strongest in language arts and very few favor mathematics. Many do not like to teach math because they feel
inadequate in the subject area (Shafer, 1998). Teachers are also asked
to use textbooks that might not adequately address the area of problem-solving and when it does, it is often in isolation
(Millard, Oaks & Sander, 2002). The problems are not meaningful to the students
because they have no real-life connection; thus, they have no relevance. Teachers
face many obstacles in trying to teach problem-solving skills to students.
Based on the research findings, there appear
to be certain causes that contribute to the problem. Each child is different. What works for one child will not necessarily work for another. With this in mind though, one can focus on the deficiencies that impact a student’s ability to problem-solve. While several causes were stated, the insufficient problem-solving instruction in
the classroom seems to be the biggest factor.
All of the causes given so far are included in
instruction. Students need to have a better working vocabulary of math. In the
classroom, teachers need to focus on problem-solving more frequently and spend less time on “kill and drill” practices. By providing students with instruction in this area, they will build a background
and thus increase their confidence allowing them to work the problems with less frustration.
When teachers are better equipped, the quality of the instruction in their classroom will also be enhanced.
Based on the research findings and the opinions
of educational professionals, the main cause for the inadequate problem-solving skills in third-grade mathematics students
is the ineffective instruction in the classroom.
III: Anticipated Outcomes and Evaluation
Goals and Expectations
The goal of this research project is to increase
the problem-solving skills of third grade mathematics students.
As students’ problem-solving skills improve,
the work setting will be changed. Students will become more comfortable in problem-solving
situations. They will approach these problems with more confidence and with less
frustration. With the strategies that students have learned, they will undertake
problem-solving with a plan. They will be able to articulate their reasoning by using a strengthened math vocabulary. Because of their increased understanding, students will perform better on problem-solving
Of a population of 19 students:
· 75% of students will increase their exemplar performance by one level
· 35% of students will achieve Pracitioner level or higher
· More than 60% will feel confident doing problem-solving
· More than 50% will have a plan for problem-solving questions
· More than 70% will feel like they have a lot of strategies to chose from when problem-solving
· Less than 10% will feel frustrated when they work on problem-solving
· Less than 60% will believe that computation is much easier than problem-solving
· More than 60% will think they are able to express their math reasoning clearly
Measurement of Outcomes
Outcomes will be measured through the use of
the same exemplar that was given at the beginning of the year and from the problem-solving survey from which the baseline
data were obtained.
IV: Solution Strategy
Discussion of Solutions
Elementary students struggle with problem-solving
in mathematics. Much has been written about how to improve this inadequacy. One such way to improve is by stressing reading comprehension in math lessons. Mathematical vocabulary is also essential. Students
need to have a working vocabulary in math if they are expected to be able to explain their mathematical reasoning (Fletcher
& Santoli, 20003). In addition, educators need to teach students how to read
word problems. Students need explicit instruction on how to interpret the mathematical language found in word problems. The
text structure, vocabulary, and purpose are unique and quite different from other types of reading. One suggestion is for students to use graphic organizers, highlighting tape, or self-stick notes to identify
key information in the problem (Forsten, 2004). Once students better understand
what the problem asks, then they can go about solving it.
Math problem-solving, however, should go beyond the routine use of operations. Instead, “problem
solving should engage students in rigorous and complex tasks that require them to think, reason, communicate, and apply their
understanding” (Peixotto, 2000, p.2). Computation may be required, but more important, imagination, flexibility, logic,
and reasoning are needed (Holly, 1999). When using problem-solving as the center of learning, the emphasis moves from "doing"
the activity to "thinking" about relationships among mathematical ideas and how they connect to help students make sense of
these ideas (Trafton & Midgett, 2001).
classrooms should be organized in such a way that students can think and talk about their work, share ideas, and question
other students. Students must also have time to reflect on their work. They should be encouraged to think for themselves. Requiring
students to illustrate every problem and then to use the illustration to help solve
it is one possible solution. Some problems may also necessitate a written explanation of the solution
(Miller, 1996). Writing is often overlooked in the math curriculum, but it is
a reflection for students. All of this leads to problem-solving success (Millard, Oaks, & Sanders, 2002).
possible solution for problem-solving success is the classroom environment. When
students are able to work collaboratively, they are better able to explore a problem and therefore develop a richer understanding. They leave feeling a sense of learning and achievement (Coad, 2003). Students should be given the freedom to use their own approach and explore a variety of solutions. Open-ended problem-solving is effective in promoting deep mathematical understanding
(Jarrett, 2000). Kathy Richardson argues that how a student arrives at an answer
is just as important as the actual answer itself (2004). No solutions should
be given so that students will look to themselves as the mathematical authority, thereby developing the confidence to validate
their work. Students should also be encouraged to use alternative approaches
in reaching their conclusions (Holly, 1999). In classrooms that stress sense-making,
reflection, and communication in problem-solving, children learn to listen to one another, respect the thinking of others,
and become confident in their capabilities as mathematics students. They develop
confidence in themselves as problem solvers and become mathematical risk-takers. (Trafton & Midgett, 2001). Teachers, then, should make the classroom environment one that encourages problem-solving and finding solutions
in a variety of ways.
The teacher’s role is also very important
in problem-solving. Along with providing the right environment, she also needs
to model positive attitudes about problem-solving so that her students can develop the same.
It is also essential for the teacher to demonstrate a lack of knowledge sometimes.
When a teacher doesn’t know the answer, she can then model the techniques of good questioning and learning behaviors
that gives students an understanding of mathematical processing and the value of thinking (Millard, Oaks, & Sanders, 2002).
Another area that needs to be enriched is the
problems themselves; they lack real-world connections. Students retain material
when it is relevant to them. In “Going Beyond the Right Answer,”
Kathy Richardson writes, “If children are going to be successful in the study of mathematics throughout their schooling,
it is vital that the mathematics they learn be meaningful to them” (2004, p. 53).
The problems and tasks that are used in problem-solving are crucial. They need to be mathematically rich and appropriate
for the goals of the lesson. The use of familiar, everyday contexts enables children
to connect their informal, out-of-school knowledge with school mathematics (Trafton & Midgett, 2001). Students are often
asked to complete isolated problems from a specific page of text that have little or no connection to their lives. Because many textbooks lack problem-solving questions that relate to the students, Millard, Oaks &
Sanders discuss using the newspaper and/or the Internet as a source for real world problems focusing on such topics as averages,
ratios, large numbers, and percentages (2002). Students need to be able to draw
on previous experiences when problem-solving.
Developing background knowledge will help students
with problem-solving too. Many students lack problem-solving experiences and
therefore are at a disadvantage when faced with such problems. Students needed
to be provided a framework for problem solving. Learning through problems is
powerful. As students make sense of mathematics, they develop a deep, connected understanding of content and learn essential
skills. This approach also causes them to develop as capable, confident problem solvers (Trafton & Midgett, 2001).
to accomplish this is by teaching students the steps to problem-solving. George
Polya is a famous mathematician who developed four steps to help make problem solving easier.
They are: understand the problem, plan a strategy, do the plan, and check your work (Millard, Oaks & Sanders, 2002). Forsten (2004) also suggests using a step-by-step approach. Word problems can appear
overwhelming and difficult to many students. Problem-solving can be made manageable by breaking it into a series of steps.
In addition, start with easy problems and gradually increase their complexity. Teaching strategies doesn't work if they are
used only now and then. Students must see the need for the strategies and have numerous opportunities to practice using them.
Similarly, students need to be provided with
instruction on strategies that are available to use when problem-solving. Teachers
need to incorporate specific strategies for problem-solving. They need to be
introduced one at a time. Examples of strategies employed most frequently in
elementary math include: guess and check, draw a picture, choose an operation,
make a chart or table, find a pattern, and work backwards. Even though these
strategies are taught individually, students need to know that problem-solving usually requires a combination of these strategies
(Millard, Oaks & Sanders, 2002).
solution requires examining the teaching techniques of Japanese educators. It
is well known that Asian students in general do well in mathematics on knowledge level, conceptual, and problem-solving questions. Some noteworthy techniques of Japanese teachers include: connecting present problems
to previous problems and solutions, teacher functioning as a guide, student contributions determining the content and flow
of the lesson, embedding concepts within the lesson, acting out the problem, using manipulative aids, and introducing formulaic
statements at the end of a lesson in summary. One other intriguing aspect of
the lesson examined in “Mathematics as Problem Solving: A Japanese Way,” is the time spent on one problem. Sawada cites that time must be allowed for investigating the nature of the problem,
generating several solutions, discussing those solutions, and reflecting on what has been learned (1999). According to this model, it is acceptable to spend an entire class period on only one problem.
Evaluation of Solutions
In order to be successful problem-solvers, students
need to have a working mathematical vocabulary that they can use when explaining their reasoning. Students need the opportunity to think, talk, share, question, and even write about the problems they are
solving. Collaboration and reflection are also important practices. Teachers need to provide both a positive environment and attitude toward problem-solving. The problems themselves need to be relevant to the student population.
Students need both a plan and strategies for problem-solving. It is also
important for educators to look to the teaching styles of successful teachers and model them.
Solutions for this problem will comprise:
- Stressing mathematical vocabulary in lessons
- Use exemplars that have real-world problems and requiring students to explain their problem-solving in a variety
- Providing weekly opportunities for students to work collaboratively
- Modeling a positive attitude during problem-solving activities
- Discussing and posting in the classroom the steps for problem-solving
- Teaching a different problem-solving strategy each week (guess and check, draw a picture, choose an operation,
make a chart or table, find a pattern, and work backwards)
- Problem-solving strategies posted in the classroom as they are taught
Chapter V: Results
skills of third-grade mathematics students are inadequate.
The following outcomes for a population
of 19 students were projected for this research project.
of students will increase their exemplar performance by one level. This outcome
was not met. 57.9% of students increased their exemplar performance by one level.
of students will achieve Practitioner level or higher. This outcome was met. 68.4% of students achieved Practitioner level or higher.
More than 60% will feel confident doing problem-solving. This outcome
was not met. 50% feel confident doing problem-solving.
More than 50% will have a plan for problem-solving questions. This outcome
was not met. 22% have a plan for problem-solving questions.
More than 70% will feel like they have a lot of strategies to chose from when problem-solving. This outcome was not met. 61% of students feel like they have
a lot of strategies to chose from when problem-solving.
Less than 10% will feel frustrated when they work on problem-solving. This
outcome was not met. 22% feel frustrated when they work on problem-solving.
Less than 60% will believe that computation is much easier than problem-solving.
This outcome was met. 44% believe that computation is much easier than
than 60% will think they are able to express their math reasoning clearly. This outcome was not met. 28% think they can express
their math reasoning clearly.
While only two out of the eight outcomes
were actually met, the research could still be called successful. On the first
exemplar, only 19.6% of students demonstrated a Practitioner level of knowledge. When
given the same exemplar again after six weeks of problem-solving instruction and practice, 68.4% were achieving at the Pracitioner
level. The goal had been 35% and it was met exceedingly. The other area where positive results can be seen in regards to outcomes is students’ views on computation
versus problem-solving. At the onset of this project, 79% of students believed
that computation was much easier than problem-solving. At the culmination of
the research project, that number significantly decreased to 44%. The anticipated
outcome was 60% so this goal was surpassed as well. While students may not “feel”
that they are better problem-solvers, the data indicates otherwise. Even though
the other outcomes may not have reached their target numbers, there were small gains in other areas. In the beginning, 47% felt confident doing problem-solving compared to 50% at the end. Before instruction, 58% felt like they had a lot of strategies to choose from when problem-solving compared
to 61% after instruction.
the biggest hindrance to the implementation was the interruptions in the instructional schedule. Due to several holiday inconveniences and scheduled school breaks, the instruction was not as consistent
planned. In addition, the final exemplar and problem-solving survey were not
administered until after winter break when students had been out of school for two and a half weeks. However, as a result of the project, students were exposed to more math vocabulary, they were given weekly
opportunities to problem-solve using real-life problems and able to work on them individually, with partners, and in groups. Additionally, they have been taught the steps/plan for problem-solving and are equipped
with six specific strategies to utilize when faced with word problems. Posters
of the plan and strategies were posted in the room for students to use as a reference.
RecommendationsConsistency in incorporating problem-solving into the mathematics curriculum is key. Give students numerous opportunities to problem-solve each week; students need
lots of practice to build their confidence. Be sure to include problems that
are relevant to students and allow them to work alone, with partners, and/or in groups.
Use math vocabulary during instruction and post it in the room or have students keep a math dictionary. Give students a plan for tackling word problems by teaching them the four steps involved: understand the
problem, plan a strategy, do the plan, and check your work. Also, make sure students
understand and practice using the six strategies: guess and check, draw a picture, choose an operation, make a chart or table,
find a pattern, and work backwards. Encourage students to use pictures, words,
and numbers to explain their reasoning. Accept alternative methods for solving
the problem too. Providing students with hints may help them get the right answer
now, but it won’t help their problem-solving skills; it will only foster dependence.
Finally, the teacher’s attitude is vital. Students imitate the teacher;
therefore, it is up to the teacher to display a positive attitude about problem-solving for students to emulate.