Increasing Student Understanding of Proportions

Michelle McLendon-Eslick

University of Texas Rio Grande Valley

EDCI 6348

Dr. Garcia

September 20, 2017

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Table of Contents

Abstract……………………………………………………………………………………………3

Rationale……………………………………………………………………………………..……4

Purpose………………………………………………………………………………………...…..5

Learning Standard…………………………………………………………………………………5

Research Questions……………………………………………………………………………..…5

Literature Review………………………………………………………………………………….5

Grit………………………………………………………………………………………...6

Methods to Support Content Acquisition…………………………………………………8

Conclusion………………………………………………………………………….……10

Procedures…………………………………………………………………………………...…...10

Participants………………………………………………………………………………….……11

Design……………………………………………………………………………………………12

Pre/Post Test Assessment………………………………………………………………..13

Data Analysis: Pre-Test……………………………………………………………………….…16

Data Analysis: Post Test…………………………………………………………………………16

Conclusion……………………………………………………………………………………….18

Annotated Bibliography…………………………………………………………………………19

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Abstract

This project was conducted to test student understanding of proportions and proportional

reasoning, more specifically, will using hands-on, project based learning, along with the use of

cooperative learning groups and increased understanding of student grit, will students perform

better on a pre/posttest involving proportions and proportional reasoning.

In this paper, you will be introduced to research on the benefits of project based learning

as well as how grit can enhance student achievement levels. You will also learn about

cooperative learning groups and the benefits they can have on increasing student achievement

levels. During the course of this research, I implemented all three into my classroom and the

results were as expected. Not only did cooperative learning groups increase the amount of

mathematical discussions going on in the classroom, but peer tutoring began and turned out to be

highly successful. When you have a classroom full of ELL students, sometimes they are more

comfortable talking to someone who speaks their language and they can, of course, understand

information better in their native language.

As an educator, I want to find the best way to present information to my students so that

they can reach the highest level of achievement possible, and this study has pointed me in the

right direction. If I can continue following the model I used for this lesson, I believe I can get

closer to reaching my goal of reaching every student that walks through my classroom door. I

want all of my students to be and feel successful in mathematics.

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Increasing Student Understanding of Proportions

Rationale

I am lucky to teach in a district that gives me free reign on how I teach my class. At the

beginning of the year I am given the standards for my state and as a department we make a

curriculum map/pacing guide that shows me about how long each unit should take. After that, I

get to present and teach information any way I choose. As long as I fit all of my standards in by

the end of the year then I am good to go. My first year teaching I relied very heavily on my

textbooks. Almost everything came out of the textbook, as it did when I was in middle school all

those years ago, and it baffled me that my students were not performing where I thought they

should be. That summer I really reflected over my year and decide that since my students could

not seem to learn in my traditional teaching style, I ditched my old ways and went a completely

different route my second year. I ditched the textbook, for all of my math courses, and started

doing mini-lessons along with project-based learning. One key idea to making this idea work,

was grit. Grit from me, and from my students. I had to teach them that having a positive growth-

mindset while being persistent in their learning, great things could happen.

More specifically, I chose to focus on proportions because that is a huge part of my 7th

grade mathematics curriculum. In years past, students have struggled to understand proportional

relationships, how to set up proportion problems, and how to solve them. The second focus on

this study is to see how a student’s level of grit affects their overall academic achievement. Grit

has been a huge focus in my district over the course of this school year. Over the summer I

researched on the importance of grit in the classroom and how to foster an atmosphere of a

positive growth mindset and encouraging grit in my students. The final focus of this paper is to

see the effects of project based learning in the classroom as it relates to proportional

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relationships.

Purpose

The purpose of this study is to find out a couple of things. First, what does a students’

overall level of grit play on student achievement in mathematics. Second, given the different

learning styles found within my classroom, would project based learning be an effective method

of teaching proportions? Third, by putting students in cooperative learning groups where they

can have open mathematical discussion, will this help to improve understanding of proportions?

Learning Standard

Oklahoma Academic Standards (OAS) Mathematics 7th Grade: 7.A.2.1-Represent

proportional relationships with tables, verbal descriptions, symbols, and graphs; translate from

one representation to another. Determine and compare the unit rate (constant of proportionality,

slope, or rate of change) given any of these representations.

Research Questions

The questions I will be asking are: Will students be able to accurately represent a

proportional relationship in a variety of ways? What role does grit play in the academic success

of my students? Does project based learning increase student’s abilities to represent proportions

in a variety of ways?

Literature Review

In many classroom across the country over the last year or two, there have been a few key

terms or ideas that have appeared. We have been hearing ideas like growth-mindset, positive

attitude, and grit. These ideas have caused a whole movement in the way teachers,

administrators, and students perceive their educational experience. “Mathematics education in

America’s public schools has been heavily influenced by both the traditionalist and

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constructivists theories. It is important to understand, though, that whatever the viewpoint, the

aim has always been the same; to enhance students achievement in mathematics,” (Carter, 15).

In my personal classroom, I take more of the constructivist approach, and working on those three

components has been an ongoing process. I have started my second year using the Positive

Pledge by John Gordon to help my students realize that our outlook and attitude can play a huge

part in our success or failures. Using Joan Boaler and Carols Dweck’s research on the effects of a

fixed-mindset versus a growth-mindset, I have implemented more open ended, thought

provoking lessons and activities that will help solidify student understanding.

As a 7th grade mathematics teacher in Oklahoma, one objective my students really

struggle with is representing and understanding proportional relationships. “It is well known that

teaching and learning fractions, ratio, and proportionality in the middle grades are very complex

processes,” (Adjiage and Pluvinage, 151). While complex, they are major concepts in

mathematics that needs to be studied and understood on a conceptual level. As we know, each

concept in mathematics builds upon previous objectives that have been taught and learned. While

we know that a fraction is a ratio between two numbers, do our students understand that? My

students see a fraction and automatically shut down because, as a student put it last week,

“fractions are the devil.” If teaching fractions, ratios, and proportionality are done in a scaffolded

way, where we can show the correlation between all of these components, then students will

develop a stronger and deeper conceptual understanding of the objective and then be apply to

apply their learning in a meaningful way.

Grit

One powerful predictor of student success is grit. Grit is a person’s “ability to work hard,

endure struggle, fail, and try again,” (Williams) which can be essential to long-term success.

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While grit sounds like an amazing thing for a person to have, this raises even more questions.

How does one get grit? What are the effects of grit on an individual basis? As an educator, how

can I build grit in my classroom? Since grit is described as perseverance and passion towards

one’s goals, we need to make sure our students have a goal. If we want them to work for

something, they need something worth working for. “Some research observations indicate the

relative importance of these character strengths, as perseverance, persistence, and grit appear to

be strong indicators for success and achievement,” (Hokanson, 110). If we can help teach

students the importance of setting goals and how to stick with it, then grit will begin to grow in

those students. According to Erik Laursen, we can begin teaching grit to our students by reading

about it. “Another way to help students learn about grit is to ask them to research and report on

people who have demonstrated perseverance in the face of hardship,” (Laursen, 22). So, now our

students know what grit is, but how can we develop this perseverance in our students? “To

develop grit in our students, Duckworth says we first need to develop interest,” (McGlynn and

Kelly, 24). Interest, which can be difficult sometimes, especially when you teach math to middle

schoolers who have already decided they do not like mathematics. “Having a purpose in life

entails a commitment to an ultimate life goal that serves to organize and plan the individual’s

daily and long-term activities, and individuals oriented towards a set of life goals tend to

demonstrate consistency to their choices over several years,” (Hill, Burrow, Bronk, 258). This is

just further proof that setting goals and having a purpose is such a powerful indicator or grit. This

is where using hands-on learning can help. Also, by using real-life math examples, students can

see that mathematics has real applications. Grit is what can help students push through the

struggles of fractions, ratios, and proportions.

Another item of concern that greatly influences student achievement is motivation.

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“Educators readily confess that they are confused and ill prepared to address what they regard as

student disinterest and lack of effort,” (Turner et al, 719). When students are coming to the

classroom unmotivated, lacking grit, with a negative and fixed mindset, it is easy to feel like

teachers are fighting a losing battle, but it all comes back to setting goals and making the

learning authentic and real for the students. To increase student motivation we must foster

student competence, autonomy, and belongingness as well as making sure students are aware of

the meaningfulness of mathematics (Turner et al, 720).

Methods to Support Content Acquisition

Let us explore the idea of real-life application and interest in mathematics. One way to

make mathematics more interesting is my exploring the idea of project-based learning. “PBL is

defined as students working collaboratively to design solutions for authentic and meaningful

questions and problems in the real world,” (Holes and Hwang, 449). While project based

learning may be harder to implement for younger children, but for secondary aged students who

are working on strengthening their problem solving skills, project based learning will help

students understand the real life applications of mathematics. It makes math feel real instead of it

just being something they have to do inside of a classroom an nowhere else. In order to spark

interest we have to make mathematics real to our students. We have to show them that

mathematics is all around us and that with grit, we can all be good at mathematics.

One of the main struggles I see is a child walking into my classroom whom has had bad

mathematical experiences in the past, and they feel defeated before we even begin. In Dweck’s

book, Mindset, she identified two types of mindsets; the fixed mindset and growth mindset. A

fixed mindset is when a person looks at a problem and thinks they cannot do it so why even try.

They have a predetermined idea about their abilities and tend to not take any risks because they

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are too afraid of failing. A person with a growth mindset sees a challenge or mistakes as a

learning experiences. These are the students with grit, perseverance, and determination. Dweck,

along with Joan Boaler, worked on Mathematical Mindset which applied the idea of fixed and

growth mindset to the mathematics classroom. The book describes how there is not a math gene.

We are not born good or bad at math. In fact, anyone can be good at mathematics if they have the

right mindset and put the work into it. It also outlines several learning activities to help inspire

growth mindset and help students’ gain conceptual understanding of mathematics. Fostering a

growth mindset is not something that will happen overnight, just like building grit in a student,

but if we can start planting those seeds early, and sew them over the years, then the result will be

stronger, more confident, and capable mathematics students.

One strategy I have started using in my classroom are what I call a learning co-op, or

learning pairs. It is no real shock to any teacher this day and age when someone says they have a

class filled with 30 plus students, outdated textbooks, if any at all, IEPs, behavior issues, and

ELL students. This is happening every single day in classrooms across the country. In any given

classroom we have a variety of learning styles and levels. You have the students who are high

achievers and catch on to lessons quickly, and we also have those who require a lot of

remediation and extra help. With all of this going on in a single classroom, how do we engage all

learners and ensure they are all getting what they need? “Many mathematics teachers have

adopted cooperative group work as a daily classroom practice, along with standards-based

mathematics curricula and pedagogies based on constructivist view of learning,” (Esmonde,

247). When students work in a cooperative learning group, they have the opportunity to talk

about mathematics and learn from each other. There have been many times when I have tried to

teach a lesson and have it fall upon a few deaf ears. Then I would ask another student who had

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mastered the objective to help their peer and then like magic, the student understood it. “The

equitable implementation of cooperative learning in mathematics classrooms depends not just on

what teachers do; students’ learning depends on how they interact with one another,” (Esmonde,

248). In order for cooperative learning groups to be successful, proper communication must be

taught and there must be mutual respect amongst those in the classroom. I always tell my

students that I do not expect everyone to be best friends, but in side this classroom we are a

family. We will respect each other and their contributions to the classroom environment and we

will speak to each other in a way that we would want to be spoken to. Everyone’s opinion is

valued. Everyone’s learning is valued. Equity must be found within the classroom in order for

cooperative learning to be successful.

Conclusion of Literature Review

In short, in order for students to truly understand proportions, fractions, and ratios, we

need to encourage growth mindset, grit, student motivation, and cooperative learning groups. If

we can grow our students in these areas, then they will truly be unstoppable. It starts with

positive mathematical experiences from a young age, providing meaningful and authentic

learning experiences where students can learn the importance of mathematics, and encouraging

mathematics discussions in the classroom. When students can do that, then we have truly made a

difference in their lives and in the world.

Procedures

I came up with this project design because I know that for the last two years my students

have struggled with understanding proportionality. Students have had a very difficult time

understanding fractions and ratios, which directly link to proportionality and slope of a line. In

the past, I did mostly lecture style teaching while students took notes, and then students would

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practice solving proportion problems, first as a class and then on their own. So, you could say, I

had a traditionalist view of teaching at that point in time, but I quickly noticed that style was not

working for my students. This year, I have decided to try and take a more constructivist

approach, where students are working for their learning. I wanted to make the learning more

meaningful for them, more authentic and engaging.

To implement the project, I started by going in depth about what a fraction is, and it

surprised me that students really could not explain a fraction beyond “there is a numerator and a

denominator.” Students did not seem to realize that a fraction was a ratio or comparison between

two numbers. They also did not know that a fraction is merely a division problem. Needless to

say, I had to spend more time on my proportions lesson than I had originally planned because we

went back to the beginning, the basics, because students lacked the conceptual understanding

needed to fully comprehend proportionality.

Participants

The study is being conducted in a 7th grade mathematics classroom. I am one of nine

math teachers for the 7th grade in a school with over 1,000 7th grade students. I have fifty-two 7th

grade, on-level mathematics students and each student was given the pre-test before we got into

the proportions unit.

I administered a pre-test over representing proportions to my 7th grade, on-level

mathematics student, as I suspected, their scores were low. I have a total of 53 on-level students,

meaning they are taking traditional 7th grade mathematics, non-advanced course. Of those 53

students, 39.6% are Limited English Proficient (LEP), 1.9% are gifted, 3.8% have a 504 or an

IEP, and 20.8% are in a school program for at-risk youth to keep students off of the streets and

out of trouble and provides extra tutoring and homework help for those students as well as life

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skill courses, anger management, etc. which we call the Carrera program. The ratio of male to

female students is 1:1, and various ethnic backgrounds and over 60% of our students are on the

free and reduced lunch plan.

I am sure some of you are wondering why my 504/IEP percentage is so low. I teach in a

school district of 16,000 students where we average about 1,200 students per grade level. WE

have thirteen elementary schools that all feed into my building, the 6/7th Grade Center. In my

building, teachers and students are put on teams. Each team contains a core teacher for math,

science, geography, and language arts. Students stay on their own team all day with the

exception of elective courses, which students have two of those per day. The way teams are set

up, we have one team that takes care of our emotionally disturbed students, and one that takes

care of the English Language Learner Sheltered students. I happen to be on the ELL Sheltered

Team, and as Newcomers test out, they transition into Sheltered where they spend a year or two

before moving into a mainstream classroom setting. I do not usually have IEP students unless

they are also ELL. Let me also add, just because I am on the ELL team, it does not mean I have

all of our ELL students. Those who have been moved into the mainstream classes can be found

on any of the nine teams in 7th grade and they are pulled in for ALA resources from outside of

their team.

Design

The instrument used to collect the needed data is shown in diagram 1.1 and 1.2. It was

designed to dig deeper into student understanding of proportionality. The tool consists of six

questions which show proportions in different ways, as the 7th grade math standard requests. This

tool was designed by myself and some fellow colleagues, other teachers in my math department,

and we have made all of our test follow this same pattern for the year. Students are given a

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problem and they must decide if it represents a proportion or not. Then we take their knowledge

further and students must explain their answer choices. Having students explain their reasoning

is what truly lets me know if they comprehend the objective or if they just had a “lucky guess.”

When a student can explain their reasoning, put their thinking into words, that is how we can

know a child truly understands what has been put in front of them. A student can get lucky

guessing a multiple choice question, but they cannot guess at an accurate explanation and

representation of the work it takes to find that answer.

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Diagram 1.1- This is page one of the pre/post assessment for proportionality. Each question is followed by an

answer. Students must decide if the answer is correct or incorrect and then back up their answer choice using their

work and mathematical knowledge.

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Diagram 1.2- This is page two of the pre/post assessment for proportionality. Each question is followed by an

answer. Students must decide if the answer is correct or incorrect and then back up their answer choice using their

work and mathematical knowledge.

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Data Analysis: Pre-Test

The data found in my spreadsheet and table indicates that my students are starting with

little to no concept of proportional reasoning or how to display proportions. I expected this to be

the case since, in Oklahoma, 7th grade is usually the first time students are formally exposed to

proportions. My one gifted student performed quite well, which also reaffirmed my theory that

she could be in my advanced mathematics course, but that is a whole other story. Overall, my

students scored where I expected them to on this pre-test.

The results are that 2% of students passed the pre-test with a score of 70% or higher with

no proportion lessons before this course. Another 15% scored a “D” (60-69%) on the pre-test.

83% of the students tested had a 50% or less on the pre-test. Of the students that failed, 13 of

them scored a 0%. All of my LEP students failed the pre-test and 57% of my Carrera children

scored a D or higher on the pre-test.

Diagram 1.3- This is a graphical representation of the pre-test scores for the 53 students tested on representing

proportions. Scores are mostly on the failing side.

Data Analysis: Post-test

When comparing the pre-test and posttest scores, there is a drastic change in scores.

Everyone showed improvement from pre to post testing. If you look at the graph below, you can

see the comparison of scores for each student tested along with the Paired t test results. Of the 53

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students tested, 92% of them scored an 80% or higher, and 100% of students showed growth in

their understanding of proportions and proportional relationships

Diagram 1.4- This is a graphical representation of the comparison between pre and posttest scores. As you can see,

there were great improvements across the board.

Paired t test resultsP value and statistical significance:   The two-tailed P value is less than 0.0001   By conventional criteria, this difference is considered to be extremely statistically significant. 

Confidence interval:   The mean of Group One minus Group Two equals -61.28   95% confidence interval of this difference: From -66.99 to -55.57 

Intermediate values used in calculations:   t = 21.5250   df = 53   standard error of difference = 2.847 

Review your data:  Group   Group One   Group Two

Mean 28.74 90.02SD 23.67 11.93

SEM 3.22 1.62N 54 54

The paired t test results shows there is a statistically significant relationship between

student achievement and the use of project-based learning, grit, and cooperative learning groups.

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I am confident that student scores will meet or exceed my expectations. Just by monitoring my

students, I am hearing more mathematical discussion about proportions and proportional

relationships and they are showing to have a deeper understanding of the content than I have

seen in years past I have not only scaffolded instruction, but I have put students into cooperative

learning pairs as well as provide many opportunities for hands-on learning and lots of discussion

to take place. We have also talked about grit in the classroom and have worked on working

through the problems and not giving up, and I have seen my students persevere through some of

the challenges put before them where in the past they would have quit.

Conclusion

Through the course of this study, it has been proven that by providing meaningful, hands-

on, project based learning, along with the use of cooperative learning groups and increased levels

of student grit, student understanding of proportional reasoning can be obtained. This level of

student understanding is rarely seen in my building, so these results lead me to believe that this

type of instruction and support can drastically increase student achievement in mathematics.

Student responses to learning about grit, and building grit have been fantastic and I have seen

that continue with them even outside the four walls of my classroom.

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Boaler, J. (2016). Mathematical Mindsets: Unleashing Students’ Potential Through Creative

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