Practical Application Activities in Mathematics

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Practical Application Activities in Mathematics 1

Can students understanding of mathematics concepts improve if the curriculum contains

practical application activities ?

By: Juan Bottia, Aaron OBrien, Tiffany Rampey

National Louis University

ESR 505 - Graduate Research: Mixed Methods

Instructor: Dr. Erika Burton

October 12th, 2014

Practical Application Activities in Mathematics

Introduction


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science. Some of the reasons were because this subject required students to use higher cognitive

skills, and they were never fully developed due to AYP (Adequate Yearly Progress) indicators

that had more emphasis on language arts and mathematics. Based on this results, the study

encouraged the integration of English, math, social studies, and science curriculums to enable

students to transfer learned skills and information across subjects. According to the research by

Drake and Burns (2004), an integrated curriculum helps students make connections to real life or

even across disciplines. It also offers repetition and a fulfilling learning environment.

Purpose of Study

The goal of this study was to find trends that indicated that practical application activities

in mathematics positively impacts the learning of thirty third grade students. The findings were

analyzed through interviews, quantitative assessments, and one rubric and a survey. This study

used theNumber and Operations in Base Tendomain of the CCSS (Common Core State

Standards) to assess the samples knowledge on this strand. Standards3.NBT.A.1 (Use place

value understanding to round whole numbers to the nearest 10 or 100) and 3.NBTA.3 (Multiply

one-digit whole numbers by multiples of 10 in the range from ten to ninety {e.g., 9 80, 5 60}

using strategies based on place value and properties of operations) were used in the assessments.

Problem Statement


practical application activities?

Literature Review

Introduction
http://www.corestandards.org/Math/Content/3/NBT/#CCSS.Math.Content.3.NBT.A.1http://www.corestandards.org/Math/Content/3/NBT/#CCSS.Math.Content.3.NBT.A.1http://www.corestandards.org/Math/Content/3/NBT/#CCSS.Math.Content.3.NBT.A.1


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Mathematics is a subject that is linear and clinical by nature. However, we are convinced

that this discipline cannot be taught in isolation in the elementary grades. Before students reach

middle school, we think that they should learn the basics of mathematics through practical

application activities that involve circular knowledge of other subjects. Similar to a narrative,

mathematics should be taught to elementary school children with elements that surround their

environment. Subjects like science and social studies, which teaches students about their society

and its intricate interactions, lend themselves to use a student-centered approach to teach

mathematical concepts in elementary school classrooms. We hypothesized that this curriculum

integration practice will provide a student-centered approach that will have a positive impact on

our samples foundation of number sense, their written articulation of the process through which

they solve mathematical problems, and their academic vocabulary.

Student-Centered Mathematics

The literature around creating a child centered math curriculum is difficult to find. Most

of the research that has been done seems to be in regards to professional development for

teachers. In an article entitled Connection Levers: Supports for Building Teachers' Confidence

and Commitment to Teach Mathematics and Statistics Through Inquiry by Katie Makar at the

University of Queensland in Australia (2007), teachers who were open to the idea were given the

support and resources necessary to attempt teaching their math curriculum through inquiry as

opposed to more traditional methods. The research indicates that teachers who commit to an

inquiry based approach, who focus their methods on being responsive to students, and who are

reflective in their learning found success in teaching in this new way. In another article entitled

Going Beyond the Math Warsin the journal Teaching Exceptional Children (2010), research was

done to address the discrepancy of instruction between regular education and special education


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teachers. The article clearly states that students respond well to a student-centered, self-directed

type of learning, based on many different educational studies, but that most special education

teachers are instructed to teach with a more direct instruction approach. The article concludes

that due to the integration of special education students into regular ed classrooms, most students

benefit from a combination of teaching approaches. Overall, the research around student-

centered learning is vague, but we hope our current study can add depth to the conversation.

Articulation of Mathematical Processes

Much research has been conducted to address the most effective and impactful ways for

students to articulate their learning when working mathematically. One study, conducted by

Krawec, Huang, Montague, Kressler, and Melia de Alba (2012) studied effective ways for

middle school students to learn how to articulate their mathematical thinking when problem

solving. The study intended to evaluate the effectiveness of a program entitled Solve It! which

teaches systematic strategies for approaching word problems. Students are taught a four step

process to be followed when approaching any word problem. The study found that there were

significant gains in the ways students communicated their mathematical thinking after using the

Solve It! program, especially students with learning disabilities. Krawec et al (2012) stated

that there needs to be further research to understand how the Solve It! program affects students

of different cognitive abilities. It clearly has a positive impact on students with learning

disabilities, but the study did not go deep enough to understand how different students are

affected differently. This study serves as a positive sign that we will be able to increase student

understanding of basic base tens skills through the use of systems that would be consistent across

the board.


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Dependency to Use Technology

Something that we investigated was the idea that students are too dependent on

technology in their number sense development. According to a study by Heidi Pomerantz (1997)

there are many myths surrounding the use of calculators in the mathematics classroom. One of

the myths was that Calculators are a crutch: They are used because students are too lazy..

(Pomerantz). Pomerantz argues that there is barely any mathematical thinking in rote

computation. Rather a real understanding of math is displayed by being able to comprehend what

a math question is asking. Used correctly, calculators enhance learning and thinking. In another

study by Campbell and Stewart (1993) their research showed that students who use calculators

developed their basic understanding of mathematical operations instead of hindering it. Another

study by Suydam and Brosnan (1993) showed that using calculators promoted achievement,

improved problem-solving skills, and increased understanding of mathematical ideas. Through

the research that we went through it would appear that the use of technology in the development

of number sense was not a hindrance but rather a powerful tool that enabled higher order math

thinking.

Academic Vocabulary in Mathematics

In regards to academic vocabulary in mathematics, a well-known molecular biologist

named John Medina explained in his book,Brain Rules, that the capacity of memory is only

about 30 seconds (2008). Therefore, if the information is not being repeated more than once then

it will most likely be forgotten. It is to no avail that the more often information is repeated and

the more interesting it is, the more likely it is to stay in the long term memory. Consequently, if

disciplines and their corresponding vocabulary are taught in isolation, content knowledge

becomes an archipelago of ideas without a real life connection. A study by Files and Adams


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(2010) compared the integrated and isolated instruction for vocabulary development of English

Language Learners from a university. The statistical analysis demonstrated that the retention of

vocabulary was more effective when it was targeted in isolation and also taught in context. Both,

led to higher learning rates when they were taught together.

Conclusion

In reviewing the associated literature, curriculum integration with mathematics seems to

be most successful when students have a strong number sense, are able to articulate their

mathematical thinking, and work on continually building their math vocabulary. Overall, the

literature on our topic is missing any substantial research on math integration with other subjects,

particularly when looking at the way integration affects student learning. Our research aims to

fill this gap in information.

Research Questions



How can we teach students to articulate the process through which they solve

mathematical problems?

To what extent is the development of intellectual vocabulary in mathematics important?

What types of teaching practices, in mathematics, most closely align with the primary

learning objectives for elementary school age children?

Is the teaching of math in isolation more difficult for students to comprehend?

How can we develop a program in which mathematics become student-centered?

How can we develop a number sense to overcome the dependency to use technology?


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Methodology

Participants

The sample population for this study was thirty 3rd grade students from Juan Bottias

classroom at Orchard Place Elementary. Orchard Place Elementary is part of the Community

Consolidated District 62 located in the city of Des Plaines. The sample was contacted by the

instructor through a letter of consent, which explained the purpose of the study in addition to

other key factors of the study.

Data Collection

In regards to data collection, we used the scores of NWEA and TenMarks to benchmark

the samples comprehension of mathematical concepts. Both of these assessments provide a

variety of questions chosen randomly to give students the opportunity to demonstrate

comprehension of concepts.

We also used a standardized rubric to assess our sample's mathematical reasoning skills.

This rubric cannot be used through NWEA or TenMarks, however, it can be applied in the

assessments available in Pearsons enVision math program. Aside from a standardized rubric, we

also used a survey and interviews to gather data for this study. Our goal was for the data in each

to delineate the types of teaching practices, in mathematics, that the sample prefers.

Hypothesis

We hypothesized that practical application activities in mathematics are not only

preferred by the sample, but are also beneficial for their understanding and mastery of number

sense. We also wanted to investigate that effective real-world application of mathematics should

be derived from the samples social studies and science curriculum. We hoped to see positive


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results in the assessments in TenMarks which should have had a laser-like focus on the Number

and Operations of Base Ten strand. Furthermore, we hypothesized that the standardized rubric to

assess the articulation of mathematical reasoning would indicate the importance of using

academic vocabulary in mathematics. Additionally, we wanted to see that the interviews and

survey indicate that students like to learn best through practical application activities in

mathematics.

Data Analysis

Qualitative Analysis

In order to analyze our qualitative data for analysis, we used a color and letter coding

system to find commonalities among our samples responses. First we created a code of words

that tied into our research questions. These coded words were organized into categories

according to the different research questions from our study. Each category was then color

coded. Within these color-coded categories, we created sub categories that demonstrated more

detail for each category.

Once our categories and color codes were set, we used the code to go through all the

transcripts from the student interviews. In order to maintain discretion, we gave each of our

students a number and kept their responses anonymous. Our study included the interviewing of

four students, so they were given the labels S1-S4. We changed the text color of their responses

according to our coding system. We did the same thing with the survey questions that asked

students to respond by explaining their thinking.

Quantitative Data

In order to analyze our quantitative data we used the scores of NWEA (Northwest

Evaluation Association) and TenMarks to assess the samples ability level in mathematics.


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Twenty-five students from the sample completed this assessment at the beginning of the school

year. This study used these assessments to analyze the samples performance in Numbers and

Operations.

All the students in the sample were given a survey, which was later coded and graphed

for further analysis. We also used a standardized rubric to assess the mathematical reasoning

skills of five students in the sample. To choose these five students randomly from the sample, we

gave all participants a number from one to thirty. From here, we randomly chose a starting

number, and counted up 10 participants. Every 10 participant was chosen.

Benchmark

Three times a year the Northwest Evaluation Association (NWEA) uses the Measure of

Academic Progress (MAP) to benchmark the sample in mathematics and English language arts.

Figure 1 shows the results of the fall 2014 MAP math assessment. Twenty-five students from the

sample took this assessment at the beginning of the school year (Fall 2014). 72% scored low or

low average in the Number and Operations strand. The deficiency in number sense affected the

performance in the Operations in Algebraic Thinking strand and the Measurement and Data

strand. The overall score indicate that 76% of the twenty-five students in the sample are low or

low average in 3rd grade mathematics.

(Figure 1)


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Figure 1:Math Assessment results (2014 NWEA MAP Report)

The sample was also benchmarked in TenMarks. This online program allows instructors

to choose specific skills to assess. The questions in TenMarks are drawn from a bank that

contains over one hundred questions pers skill. Figure 2 shows that the sample scored an average

score of 22.7% in the Number and Operations in Base Ten domain.

(Figure 2)

Figure 2: TenMarks Assessment/Number and Operations in Base Ten (Fall 2014)


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Both quantitative assessments demonstrate that the samples math deficiency is in

Number and Operations. From this point forward, our data analysis is aimed to search for trends

that indicate better structures to teach number sense.

Research Findings

1. Can students understanding of mathematics improve if the curriculum contains


Figure 3 shows that 100% of the interviewees thought that the math equation was easier

to solve than the word problem. Even though both asked students to use the same mathematical

process to get the answer, all participants still said that solving the equation was easier.

Figure 3:Preferences (4 students interviewed)

One of theparticipants explained that to solve the equation you just have to add and not

really think. This evidence demonstrates that equations in isolation do not have much contextual


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meaning to students. Another participant said that the first one was easier but it really doesnt

tell you what a fraction is. All the participants knew how to solve the equation, but when the

same problem was placed in a complex scenario, they did not have the metacognitive strategies

and stamina to solve the problem. The obstacle these students face is that all standardized math

assessments that they will take throughout their schooling assesses knowledge through complex

scenarios. So for example, instead of simply asking students to round a number to the nearest

ten, the student is actually given a grocery list with the prices of each item. The problem will ask

to round all the numbers to the greatest place value and then make an estimate of the total. If the

student is not exposed to such format, most likely the answer will be incorrect.

Figure 4 shows the results of one of the word problems in the survey. This word problem

involved subtracting whole numbers. 33% of the sample incorrectly answered this word problem.

Even though the sample knows the procedures on how to subtract whole numbers, they again

struggled to do this when a complex scenario was added instead of an isolated equation.

Figure 4: Word Problem Results

In our opinion, in order to get students exposed to complex scenarios more frequently, it

is necessary to integrate the curriculum of other subjects. For example, this school year the


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sample studies matter in science. This module, Measurement Matter, could be integrated with

the Measurement and Data domain of the CCSS (Common Core State Standards). This will help

the instructor to create practical application activities that can have real world meaning. In Figure

11 it demonstrates that only 3.6% of the sample would not like to learn about social studies and

science during math. Both, social studies and science, are subjects that innately get students to

think at higher-levels. Also, hands-on learning can be easily applied to its content. This could be

one of the reasons why students have a positive outlook toward those subjects.

While this study was being conducted, the sample was reading biographies in social

studies. Students read about famous individuals while also searching for evidence that indicated

how they made the community stronger. In the survey we decided to incorporate a question that

encouraged students to applied mathematical concepts to one of the biographies they read during

social studies. The question asked students to think about the ways Babe Ruth used math

throughout his career. Most of the sample directed their answers toward counting things like

money, home runs, and hit. However, there were certain students that were able to apply deeper

mathematical concepts. One of the students explained that Babe Ruth could have calculated the

ball speed. In our opinion, this is the type of example that makes mathematics come alive in a

classroom. It is within this rational where instructors can apply practical application in

mathematics in order to make it more relevant for students. Figure 5 shows that 43% of the

sample was unable to answer this question:


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Figure 5: Math in Social Studies

2. How can we teach students to articulate the process through which they solve

mathematical problems?

One factor that was evident in the data of the interview, survey, and the standardized

rubric was that the lack of math vocabulary affected the sample from articulating mathematical

processes. Question 3 explores more in depth the impact of academic vocabulary in math.

However, after the data was analyzed it was evident that one of the pillars that support the

articulation of mathematical processes is vocabulary. Other pillars were found through the

standardized rubric.

In the interview 50% of the participants were able to compare the Hindu-arabic and

Babylonian numeral system because they used words like differentbases, and also numerals.

The other 50% could not answer this question mainly because they did not recall the key

vocabulary to articulate this answer This was also evident in Figure 6. 50% of the sample could

not satisfactorily express what they did to solve the word problem because they didnt use words

like subtracted, took away, difference, added, and addition.


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Figure 6:Articulation of Mathematical Reasoning

The information from the standardized rubric indicated that most of the students scores

were brought down because of their inability to interpret data in the word problem Due to this

significant obstacle, 60% were unable to apply the math concepts into formulas. 80% of them

could draw conclusions. However, the heavier load of a word problem lies within the data

interpretation and how it can be manipulated into a formula. If this is not achieved, then the

students cannot achieve a score over 70%.

Figure 7: Standardized Rubric


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To teach students to better articulate their mathematical reasoning, it is critical to create

an environment that lends itself for this type of metacognitive practice. Everyday students need

to be exposed to specific academic vocabulary, which can help them articulate their reasoning to

solve word problems. For example, if students are learning to solve multiplication word

problems. Then it is important to expose them to words like array, multiplier, multiplicand, and

groups.Additionally, students also need to use mathematical models to interpret the data in a

word problem. These models act as a scaffold to help students create formulas. The combination

of targeted vocabulary, models, and formulas should facilitate the articulation of mathematical

processes.

3. To what extent is the development of intellectual vocabulary in mathematics

important?

Something that we found in the interviews was that intellectual vocabulary was integral

for the students understanding of mathematics. For example, in a student interview, one students

was asked about the difficulty between an equation and a word problem, the student responded

saying, The (word problem) was more difficult because I didnt understand the words Just

looking at this, it would seem that word problems make math more difficult; however the student

had more to say about the word problem. When asked which question helped them understand

fractions better the student went on to say, The (word problem) because it has words, and

because the words gives you more details about the fraction.


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Figure 8

Figure 8 shows that the 3rd grade students did well when asked to find the difference

between 8 and 3 as well as the product of 5 groups of 2; however when asked to use the words

multiplicand or multiplier, none of them could explain the equations using those words. A lot of

times math is a subject that focuses on rote memorization rather than comprehension. When

students understood some of the vocabulary, as they did in figure 9, then their answers became

more consistent.


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Figure 9

It will be vital to investigate how much math vocabulary crosses over into science and

social studies vocabulary. In the samples elementary school the classroom teachers teach

science, social studies, and math everyday. We think it might be beneficial for students if they

could be exposed to the same vocabulary across the board.

As already described in question 2, the importance of vocabulary doesnt only help

students answer basic math questions, but it also helps them articulate mathematical processes.

This is a vital skill to have in order to have success in a curriculum molded by the CCSS

(Common Core State Standards). As described on the standards, students must use assumptions,

definitions, and previously established results to construct arguments. In our opinion, these

arguments cannot be satisfactory if the student lacks the academic vocabulary to explain the

procedures and findings.


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4. What types of teaching practices, in mathematics, most closely align with the

primary learning objectives for elementary school age children?

Figure 10 shows that the samples favorite tools to learn in math are flashcards, iPads,

cubes, tests, and base ten blocks. Four out of five of these tools are manipulatives.

Figure 10:Favorite Tools

This is evidence that the sample prefers to learn about math through a hands-onlearning

approach. This correlates with the positive effects of the integration of subjects like science into

the math curriculum. Experiments in science gives teachers the opportunity to connect higher-

order thinking skills, hand-on learning, and mathematical concepts in one unit. In subjects like

science and math, students can have the opportunity to observe and record data. Additionally,

they can measure, develop, discuss, evaluate, and even justify the merits of their explanations.

These are all hand-on learning practices that students prefer to use in order to learn about their

surroundings.


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In the interview 75% of the participants prefered to learn about math using hands-on

learning tools. One of them recalled having fun doing math while doing experiments and also

using flashcards. Another participant mentioned that he had fun using blocks and measuring

things. The interview also asked the sample about an idea they have in order to make math more

meaningful. 50% of the participants mentioned that blocks or base ten blocks made math a fun

subject. It was interesting that one participant in the interview mentioned that taking more tests

would be fun if they could use blocks and other tools instead of just using paper. This was an

interesting statement because it took us into the idea of using performance tasks in math. We

think that students should definitely be given the chance to demonstrate what they have learned

about math through flexible learning paths. Again, we think that subjects like science and social

studies provide the right platform to assess learning in such way.

5. Is the teaching of math in isolation more difficult for students to comprehend?

This question is multifaceted and proved difficult to fully answer in such a short study.

The two routes we took to gather data on this subject were to look at student opinion on the

topic, as well as their performance on isolated versus integrated questioning.

The first approach we took to answering this question was to ask our sample how they

felt about learning math while integrated with either science or social studies. This question was

both hypothetical and vague. The students werent presented with examples, thus our results are

subject the their interpretations of the question. As you can see in the chart below (Figure 5), the

majority of the students surveyed, 16 out of 30, said they would like to have math integrated with

science and social studies All the Time. The other large portion of the respondents, 7, said

they would like integration Sometimes. Only one of the students showed no interest in an

integration of these disciplines. This data insinuates that the students surveyed have previous


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experience with an integrated curriculum, and enjoyed learning math in such a way that allowed

for practical application of their skills.

Figure 11: Science, Social Studies, and Math

Our second approach to this research question was to present students with a new

mathematical concept in two ways; one asked them to learn the concept in isolation, whereas the

other took a more practical and integrated approach using a word problem format. The students

interviewed had keen reflections regarding the use of math both in isolation and in practical

application . When asked which problem was more difficult (with the first being the problem in

insolation and the second being integrated), the first student said, The second one was more

difficult because in the second one it is hard to understand. It says one out of 7 and that is

difficult to understand. This student was also asked the follow-up question, Which one do you

think will help you better understand fractions?He or she responded, The second one because

it has words, and because the words gives you more details about the fraction. The student was


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able to recognize, as were two of the other students surveyed, that even though the word problem

may have been more difficult, it was likely more beneficial for their learning, as they were able

to gain a deeper understanding of the fractions and their uses. In the limited interviews we

conducted, it seemed as though students preferred the more straight-forward and isolated way to

learn a new concept, but were aware of the benefits of the integrated approach.

6. How can we develop a program in which mathematics become student-centered?

This question is critical in developing a mathematics curriculum that is practical and

integrated. In order for students to see the practicality of learning math in context, it is important

for them to be invested in the process and also be able to recognize the uses in their daily lives.

We collected quantitative data in this area by asking the sample to indicate ways in which

they use math in their daily lives. (Figure 6) They were able to select as many answers as they

thought applied, and were also able to choose other as long as they indicated what they meant

by entering a response. Most of the respondents to this question stated that counting was one of

the math concepts used in their daily lives. Handling Money was another response that most

students could identify as a math tool used in their daily lives. Most of the students who

responded with other wrote in a response of either multiplication or some variation of

statement saying that they use math to complete problems in math class.


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Figure 12: Math Skills Used Daily

In order to collect qualitative data on this topic, we asked the sample three separate

questions regarding their personal interest in math. The questions in the student interview were,

Think of an example of when youve had fun doing math. Why did you have fun? and What

kind of math is the most fun for you, and why? and finally, What is one idea you have for

making math more fun? Throughout the student responses to these question we found five

themes that were reoccuring: games, facts, collaboration, hands on learning, and technology.

Generally, as students answered these questions, they had several reasons or memories about

math being enjoyable. Student three told us, I had fun using blocks and measuring things. I had

fun because I got to work with my friends. Similarly, student four responded by saying I like

doing centers and computers to learn about math. It is fun also to learn math in science, in

experiments. Like yesterday we measures our desks. Flash cards are fun too. Consistent with

students this age, most of the respondents seemed to discuss things that happened in their recent

memory, indicating that there are a large amount of things they enjoy in math. These students are

likely not including the entire breadth of things they have ever enjoyed. Additionally, in


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analyzing our interview responses, the most common response in regards to student math interest

had a theme of hands on learning. While this is probably appreciated by students in every

discipline, it seems to be most effective for learning in the math realm.

7. How can we develop a number sense to overcome the dependency to use technology?

A good sense of numbers is developed by discussing students understanding of numbers,

encouraging them to think about numbers and use operations in new ways, exposing them to

different representations of numbers, and linking mathematics they use every day to the

mathematics they are learning in the classroom. Calculators should just act as a tool to help the

metacognitive process when solving math problems. As much as technology can assist students

in their learning, it is still just a tool and not an answer machine. Figure 13 shows the responses

of students who were asked if they needed calculators to solve difficult word problems. A little

over half of the students believed that they needed calculators for complex word problems. One

student said, "If it is a math word problem just type the numbers, but first you need to know the

type of math you are doing. So first you have to think." Just like a carpenter who uses power

tools to get his job done quickly and accurately, the same can be said of using technology when

using math.


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Figure 13

Future Research

The fact that this study was short in length and was a first attempt for all three as

researchers, it is clear to us what we would extend this research if we were to do it in the future.

Our primary objective would be to compare two groups of students learning a completely new

math concept. One of the groups would learn the concept in a more traditional way, being

introduced to the idea in isolation with traditional algorithms. The other group would learn the

concept in a more contextual and integrated way. Ideally, one teacher would teach both groups

of students, so as to avoid communication discrepancies in the study. This type of research

would be most effective if the concept were taught early in the year, assessed directly after the

unit, then assessed again at the end of the school year to see how concept retention related to the

method of learning.


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Additionally, this study could be improved upon if we could interview each one of the

students that were surveyed. Due to time constraints, we were only able to interview 4 students,

but extending these interviews to the entire scope of children would provide more reliable

qualitative data.

Limitations

There were several limitation on this study, starting with the fact that all three of the

researchers had no experience doing a study of this kind. The research project was laid out

clearly, but as first-timers it was difficult to write research questions that lead us to the kind of

results we were aiming to obtain. Our overall research topic was one that lends itself to a longer

study, perhaps even over the course of several years. With the limited data we collected, it is

difficult to say that the results of our study are at all conclusive. In addition to the fact that our

study was conducted over a short period of time, we were also limited by the fact that we only

had one classroom of student from which to draw our information. The demographic of this

class was representative of the school in which our study was conducted, but may not be an

accurate portrayal of all students in the area. Finally, it was difficult for the researcher who

conducted the surveys and interview to find the time to conduct all the necessary research while

also teaching his prescribed curriculum. Despite these limitations, we were still able to make

conclusions on the research that we had the time to conduct.

Conclusion

Our hypothesis stated that practical application activities in mathematics are not only

preferred by the sample, but are also beneficial for their understanding and mastery of number

sense. According to our quantitative and qualitative data there is evidence that the sample


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prefers to learn math through practical application practices. The assessments from NWEA and

TenMarks also provided evidence that the sample struggles the most with the Number and

Operation strand, which is closely aligned with the development of number sense. We can also

conclude through this study that in order to develop number sense, it is critical to get students to

use manipulatives, along with complex scenarios that can help students develop the articulation

of mathematical processes.

The standardized rubric validated our next hypothesis, which indicated that the

articulation of mathematical reasoning would indicate the importance of using academic

vocabulary in mathematics. Based on our quantitative and qualitative data, vocabulary is an

important element in order for students to successfully articulate their mathematical reasoning.

This study uncovered the importance of learning the skill of interpreting data to formulate

equations. This is a vital process for students to successfully articulate the mathematical

processes used to solve a word problem.

Since our sample prefers to learn about math through practical application activities that

involve hands-on learning, we think that integrating the social studies and science curriculum in

math can provide the learning environment utopia that can help students master critical math

skills in the elementary grades. This curriculum integration provides the full package for

academic vocabulary acquisition, hands-on learning, real-world connections, practical

application activities, modeling, communication, and collecting, analyzing, and interpreting data.

This type of pedagogy ensures that students rely more on their metacognitive abilities rather than

thinking that answers to complex problems could be found solely in calculators.

Appendix

Appendix A: References


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Campbell, P. F., & Stewart, E. L. (1993). Calculators and computers. In R. Jensen (Ed.),

Early childhood mathematics: NCTM research interpretation project (pp. 251-268).

New York: Macmillan Publishing Co.

Cheng, Q., & Wang, J. (2012). Curriculum Opportunities for Number Sense

Development: A Comparison of First-Grade Textbooks in China and the United

States.International Journal For Mathematics Teaching And Learning,

Drake, S., Burns, R., & Association for Supervision and Curriculum Development, A. A.

(2004).Meeting Standards through Integrated Curriculum. Association for

Supervision and Curriculum Development.

File, K., & Adams, R. (2010). Should Vocabulary Instruction Be Integrated or Isolated?.

TESOL Quarterly: A Journal For Teachers Of English To Speakers Of Other

Languages And Of Standard English As A Second Dialect, 44(2), 222-249.

Gopalsingh, B. (2010, August 1). Teacher Perceptions of High School Students

Underachievement in Science. Online Submission,

Krawec J., Huang J., Montague M., Kressler B., Melia de Alba A. (2012). The effects of

cognitive strategy instruction on knowledge of math problem-solving processes of

middle school students with learning disabilities.Learning Disability Quarterly,

36(2), 80-92.

(2012). Grade 3 Number & Operations in Base Ten | Common. Retrieved October 16,

2014, fromhttp://www.corestandards.org/Math/Content/3/NBT/.

Makar. (2007). Connection levers: Supports for building teachers' confidence and

commitment to teach mathematics and statistics through inquiry.Mathematics

Teacher Education and Development,8, 48-73.
http://www.corestandards.org/Math/Content/3/NBT/http://www.corestandards.org/Math/Content/3/NBT/http://www.corestandards.org/Math/Content/3/NBT/http://www.corestandards.org/Math/Content/3/NBT/http://www.corestandards.org/Math/Content/3/NBT/


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Medina, J., Medina, J., & Medina, J. (2008).Brain rules: Brain rules for baby. Seattle,

Wash: Pear Press

Pomerantz, H. (1997). The role of calculators in math education research. Retrieved

August 31, 2005, fromwww.educalc.net/135569.page.

Suydam, M. and Brosnan, P. (1993). Research on mathematics education reported in

1992.Journal for Research in Mathematics Education, 24, 329-377.

(2010). Going beyond the "math wars". Teaching Exceptional Children, 42(4), 14-20.

Appendix B: Letters of Consent

INFORMED CONSENTINTERVIEW/SURVEY PARTICIPANT

Dear parent or guardian,

Your child has been selected to participate in a study conducted by myself and two other graduate

students. The study is entitled:Practical Application Activities in Mathematics: Can studentsunderstanding of mathematics concepts improve if the cur ri culum is integrated with the content area

subjects?

The purposes of the study is to gain insight about the following areas in math: (1) The types of teaching

practices, in mathematics, that most closely align with the primary learning objectives for elementary

school age children; (2) the importance of vocabulary development in mathematics; (3) the articulation of
http://www.educalc.net/135569.pagehttp://www.educalc.net/135569.pagehttp://www.educalc.net/135569.page


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mathematical processes;(4) curriculum integration in mathematics; (5) dependency of technology to solve

mathematical equations; (6) developing a curriculum that is student-centered.

I consent for my daughter or son to participate in a research project conducted by myself, Juan Bottia, and

two colleagues Aaron OBrien and Tiffany Rampey, both educators enrolled in the NLU (National Louis

University) graduate course entitled: Educational Inquiry and Assessment

I understand that my childs participation will consist of one survey conducted online with the use of

Google Forms. The answers to this survey are confidential and will only be visible to the three educators

conducting this research. However, the results of the research will be shared with the class instructor and

other graduate students enrolled in this course. I also understand that my child may be additionally

chosen for a short math interview with his or her teacher to gather further data.

I understand that my participation is voluntary and can be discontinued at any time without prejudice until

the completion of the research report.

I understand that only the researcher(s), Juan Bottia, Aaron OBrian, and Tiffany Rampey will have

access to a secured file cabinet in which will be kept all transcripts, taped recordings, and field notes from

the interview(s) in which I participated.

I understand that the results of this study may be published or otherwise reported to scientific bodies, but

my childs identity will in no way be revealed.

I understand that in the event I have questions or require additional information I may contact the lead

researchers: Juan [email protected],Aaron [email protected], or Tiffany Rampey

[email protected]

If I have any concerns or questions before or during participation that I feel have not been addressed bythe lead researcher, I may contact the NLU course instructor at [email protected]

I understand my rights as outlined above as an interviewee.

Students Name ____________________________________________

Please print clearly

Guardians Name ____________________________________________


Guardians Signature _________________________________________ Date _____________

Researchers Name____________________________________________

mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Researchers Signature_________________________________ Date_________

Return signed to Mr. Bottia

----------------------------------------------------------------------------------------------------------------------------

English to Spanish Translated Version

CONSENTIMIENTO INFORMADO - ENTREVISTA / participante de la encuesta

Estimado padre o tutor,

Su hijo ha sido seleccionado para participar en un estudio llevado a cabo por m mismo y otros dos

estudiantes de posgrado. El estudio se titula:Aplicacin de actividades prctica en Matemticas: Los

estudiantes pueden mejorar la comprensin de matemticas si el plan de estudios est integrado

con el contenido de otras materias?

Los propsitos del estudio son para obtener una perspectiva sobre las siguientes reas de matemticas: (1)

Los tipos de prcticas de enseanza, en las matemticas, que se alinean ms con la mejor instruccin para

ninos; (2) la importancia del desarrollo del vocabulario en las matemticas; (3) la articulacin de los

procesos matemticos; (4) la integracin de otras materias en matemticas; (5) la dependencia en la

tecnologa para resolver ecuaciones matemticas; (6) el desarrollo de un plan de estudios que est

centrado en el estudiante.

Doy mi consentimiento para que mi hija o hijo participe en un proyecto de investigacin llevado a cabo

por m mismo, Juan Bottia, y dos colegas Aaron O'Brien y Tiffany Rampey, ambos educadores estn

inscritos en el curso NLU (National Louis University) posgrado titulado: Investigacin y Evaluacin

Educativa

Entiendo que la participacin de mi hijo consistir en una encuesta realizada con el uso de Google Forms.

Las respuestas a esta encuesta son confidenciales y slo sern visibles para los tres educadores que llevan

a cabo esta investigacin. Sin embargo, los resultados de la investigacin sern compartidos con el

instructor de la clase y otros estudiantes de posgrado matriculados en este curso. Tambin entiendo que

mi hijo puede ser elegido para una entrevista de matemticas con su maestro.

Entiendo que mi participacin es voluntaria y se puede interrumpir en cualquier momento sin perjuicio

hasta la finalizacin del informe de investigacin.

Entiendo que slo el investigadores, Juan Bottia, Aaron O'Brian, y Tiffany Rampey tendrn acceso a un

archivo asegurado en el que se mantendr todas las transcripciones, grabaciones grabadas, y las notas de

campo de la entrevistas en que particip.

Entiendo que los resultados de este estudio podrn ser publicados o informados de otro modo, pero la

identidad de mi hijo ser de ninguna manera revelada.


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Entiendo que, en caso tengo preguntas o necesita informacin adicional puedo contactar con los

investigadores principales: [email protected] Juan Bottia, Aaron O'Brien [email protected], o Tiffany

Rampey [email protected]

Si tengo alguna inquietud o pregunta antes o durante la participacin, usted puede contactar al

investigador principal: Erika Burton, [email protected]

Entiendo mis derechos como estaa indicado en este documento:

Nombre del alumno ____________________________________________

Por favor escriba claramente

Nombre del Guardin ____________________________________________

Por favor escriba claramente

Firma del Guardin _________________________________________ Dia_____________

Nombre del Investigador: ___________________________

Firma del investigador:_________________________________ Dia _________

Por favor regresar al Sr. Bottia

Appendix C: Key for Interview and Survey Coding

UC: Understanding/Comprehension E: Easier

H: Harder

ET: Explaining Thinking EV: Using Explicit Vocabulary

BK: Background Knowledge

F: Formulas

MV: Math Vocabulary N: Necessary for Comprehension

UN: Unnecessary

CI: Concepts in Isolation E: Easier

H: Harder

SC: Student Centered G: Games

F: Facts


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C: Collaborative

HO: Hands On

T: Technology

T: Technology H: Is helpful

N: Is necessary

UN: Is unnecessary

A: Answer C:Correct

I: Incorrect

Appendix D: Coded Student Interviews

If this is a new idea, try your best! Here is how you add fractions:

(These top numbers are added together)1 2 3 2 3 5

___ + ___ = ____ AND ____ + ____ = ____

4 4 4 7 7 7(If these bottom numbers are the same, they stay the same)

Now you try:

1.) 2.)

1 2 3 4

___ + ___ = ____ ____ + ____ = ____

5 5 5 9 9 9

Was this a new idea to you?

S1: Yes.I think it was easy because the first number is the only one that you need to add, and the second

one stays the same.(E)The numbers at the top are the only ones that change

S2: Yes it was, I have heard of fraction but have never learned about fractions.

S3: Yes. I have never learned about fractions.

S4: Yes


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Question: Seven ships sailed across the Atlantic Ocean towards America. If one out of seven

ships arrived in September, and two out of seven ships arrived in October, how many ships out

of the five arrived in September and October?_________ ships

Use that information to complete the number sentence below:

1 2

___ + ___ = ____

5 5 5

Which problem was more difficult? Why? Which one do you think will help you better understand

fractions?

S1: The second one was more difficult because in the second one it is hard to understand .(H)It saysoneout of 7(EV)and that is difficult to understand.(H)(Which one do you think will help you better

understand fractions?) The second one because it has words (EV), and because the words gives you

moredetails (EV)about the fraction.

S2: The first one is easier (E)because you just have to add and not really think (E)about the answer. It

also doesn't have words.(EV)(Which one do you think will help you better understand fractions?) I

think showing a word problem it is easier because you can use an example. Like 1 out of seven ships,

and you can put that as a fraction. (E)

S3:The second one was more difficult because it has words and a story. Word problems are hard

because of the words, and there are words I don't understand .(H) (Which one do you think will help

you better understand fractions?) The first one was easier but it doesn't really tell you what a fractions

is (E). But the other one has more words and details about the problem. Maybe the second one because

it has words, and you can see how fractions can be words.

S4: "The word problem was more difficult because I didn't understand the words (H)"(Which one do you

think will help you better understand fractions?)"I think first students need to learn like the first

questions and then the second one."

Think of an example of when youve had fun doing math. Why did you have fun?

S1: In second grade. We played a game that we had to put cards down and guess a number with our

eyes closed. It was fun because it was a game.(G)

S2: In first grade we had a test on Fridays, and when we passed the test we went to the next one. I like

taking test because it helps me learn more about math facts.(F)

S3:I had fun using blocks and measuring things.(HO) I had fun because I got to work with my friends. (C)


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S4: I like doing centers and computers(T)to learn about math. It is fun also to learn math in science, in

experiments.Like yesterday we measures our desks. Flash cards are fun too. (HO)

What kind of math is the most fun for you, and why?

S1: Multiplication, because it helps me to know more about numbers.

S2: Multiplication because I think it gets easier (E) and you get better as you practice more and more .

S3: Math is fun when you can work with other friends.(C)I like to measure things(HO)and go outside and

learn math there.

S4: Multiplication because you have to know addition and subtraction. Knowing multiplication also helps

you solve word problems with multiplication. (E)

What is one idea you have for making math more fun?

S1: Doing math that helped me learn new numbers with blocks (HO)and stories.

S2: I would teach multiplication first, and explain how multiplication really is I will also use word problems

because then you have to read and use numbers. Because in the real world you have to use both, and it

will be more easier for you.

S3: Taking more test. But it will be fun if in the tests(F) we could useblocks (HO)and other things for

math instead of just doing it on paper.

S4:Play games (G)about math. Play with other friends.(C)Count money games.(G)

Do you need a calculator to help you solve equations? Why?

S1: Sometimes because they are too hard. Like when you have to solve problems in the table of 6

multiplication (N)

S2: No, because my sister did not teach me how to use a calculator and the calculator I use is in my

brain.(UN)

S3: Not really. Well, sometimes, it depends on the problem. Like if the problem has big numbers (N)then

yes.

S4: Sometimes because the answer might be like 1,000 divided by 24 and you need a calculator forthat.(N)

How can a calculator help you solve a word problem?

S1: Because we don't have enough fingers to count big numbers (H)


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S2: If it is a math word problem, just type the numbers but first you need to know the type of math you are

doing.So first you have to think (UN)

S3: It can help you but you need to know what to do first.(H)

S4: If you don't know the answer, the calculator can help you.(H) But you have to decide what to do, so it

is up to you.

Explain how you solved the following equation: 6 x 5

S1: I solved the problem by countingby 5's.

S2: I know it was thirty because if you can skip count by 5.

S3: I counted6 five times.

S4: I addedsix five time.

Explain how you solved the equation in question 13. This time use the words Multiplier and

Multiplicand.

S1: Student cannot explain because she doesn't know the vocabulary.(N)

S2: I do not know(N)those words.

S3:Observation: Student could not explain using the vocab(N)because she doesn't know the words.

S4: Observation: Student cannot explain the answer because he doesn't know the vocabulary (N)

Compare the Hindu-arabic and Babylonian numeral systems?

S1: NO RESPONSE

S2: The Babylonian numbers have different shapes,and also a different base.(EV)

S3: We have abase 10(EV) and they (babylonian) have a base 60(EV)

S4: Observation: Student could not answer the question.

How does knowing about the Roman and Babylonian numeral systems help you understand the

Hindu-arabic numeral system?

S1: Yes, because we get to compare other numbers with our numbers.


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S2: In some clocks there are Romannumerals(EV) andin the beginning of books (BK), so now when I

see them I am going to understand the answers.

S3: Because like Romannumerals (EV)are everywhere and now you know what they mean. The

Babylonian do not exist anymore,but they help us track time with their base. (BK)

S4: The Roman are coolbecause I see them in clocks.(BK) Now I know what they are. I don't know how it

helped me understand, but it has different rules than ours.

Appendix E: Coded Student Survey Story Problems

The following was the portion of our data coding that was used for the story problems included in the

student survey.ET: Explaining Thinking

EV: Using Explicit Vocabulary

BK: Background Knowledge

F: Formulas

MV: Math Vocabulary N: Necessary for Comprehension

UN: Unnecessary

A: Answer C:Correct

I: Incorrect

Survey Question: Carlos went to the store with $20. He bought a pencil for $1.50 and a stapler for $8. How

much money did Carlos have left after leaving the store?

S1:S2:S3: $1.50. The way I did it was I used subtraction!(BK)My answer is $1.50(I)S4: You can use Subtraction like you have 20 and he spends nine dollars so you do 20-9=11 but you still

have 50cents(F)so its 11.50(I)because you add the 50!

S5:I got it for doing the equation and twenty dollars minus eight dollars minus one dollar fifty cents (F) is$1.50(I)S5:Carlos took back$11.50 because I - 9.50 to $11.50.(I)S7:It is 10.50(C). I found it doing subtraction.(BK)S8: $11.50 is my answer.(I)I subtracted $20 from 8 and I got 12 and subtracted 1 dollar and and added

50 to the end and 12-1=1.So thats how I got $11.50(F)


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S9:first carlos gave 20.00$ to the lady he used 9.50 so he got 11.50 in all.because you start to 9 and

count to 20.how many fingers you have is how much money you get plus the 50 cents.(BK) so you have

11.50.(I)S10:Carlos has $11 dollars left. I used a paper to solve the problem. I subtract and draw a picture.(BK)S11: Carlos had $10 because 20-9=11 and 11-1=10(F)so that means that he had 10.00 (I)S12:Carlos brought back $1.78 after leaving the store . It is true because i did $20 plus $1.50 plus $8

(BK) equals $1.78(I).

S13:The answer is $11.50(I).I found out because I subtracted $20 from $8. Then I subtracted $1.50(BK) S14:He left with $11.50 (I)because $20 - $8 = $12 and $12 - $1.50 = $11.50. At the last part, I subtracted

$1 - $12 and it gave me $11. Then I subtracted $11 - $0.50 and it gave me $11.50.(F) S15:The answer is $11.50.(I)I found out because I subtracted $20 from $8. Then I subtracted $1.50(BK) S16:Carlos brought back 10 dollars.(I)What I did to answer this question is that you have to subtract(BK)S17:After leaving the store he had eleven fifty left,(I)of buying a stapler and a pencil.S18:22 how i got the to 22 I knew that he had $20 there was a pencil that was $1.50 and he didn't have

enough for that but he did have enough for the stapler and that was for $8 he had 20 and than take away

8 (BK) and now he has $8 dollars(I)thats how I got the awnser.S19:My answer is 11.22(I)

S20:my answer is 11,22(I)S21:eleven dollars and fifty cents(I) because i added eight dollars plus one dollar and fifty cents and i got

nine dollars and fifty cents then i subtracted twenty dollars minus nine dollars and fifty cents and i got

eleven dollars and fifty cents(BK)S22:I don't now.S23:Carlos has $11.50 now(I)because he gave $1.50 then $8 1+8=9 so he has $9.50 so $20-$9=11 so

he has $11.50(F)S24:Carlos had 20$ when he went to the store. Then, he left with 10.50$(C) after shopping at the store. I

got this because I wrote it on a piece of paper but first, I counted it in my head then I ended up with

10.50$. After that, I checked if the equation was incorrect but I guess I didn't. That's how I got the answer:

10.50$.(BK)S25:Carlos has left 7.30(I)andi got the answer by taking away the money.(BK)

S26:Carlos went home with 11 dollars.(I)Because 20 minus 8 minus 1 = 11.(F)S27:he has left 11 dollars and fifty cens.(I)I kwon that is truth because Carlos has 20 dollars and he

wasted nine dollars and fifty cens. (BK)

S28: Carlos got $11.50(I)left because I subtracted $20-8=$12-1.50=$11.50 (F)S29:Carlos has $11.50 now(I)because he gave $1.50 then $8 1+8=9 so he has $9.50 so $20-$9=11 so

he has $11.50 (F)S30:My answer is 12(I)because I sobtrac. (BK)

Questions:How did Babe Ruth use math throughout his career?

S1:

S2:S3:I think he used it by how many homers he made plus his team and world Series!

S4:i don't know?S5:So he could get good grades and play baseballS6:he calculated the balls speed and hit the ball as hard as he can and made 60 home run world record. S7:I DON'T KNOWS8: I think he used math when he spent money for his motorcycle and when he rebuild St.Marys and took

the St.Marys band to the game to play.


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S9:he used math to know and count his baseball homers S10: i think he counted the home runs he did.

S11:he used math when he had to see how much power he needed for his hits S12:He used math because to know how much home runs and points . S13:He used it to count all his money. S14:To count how many homers and home runs he made in his careerS15:He used it to count all his money. S16:To know how many home runs he hit.S17:do not noS18:i don't knowS19:He used math to figure out his angel of hitting the ball(N)S20:he used math to figure out his angel of hitting the ball(N)S21:he used to count his money because each time he played they paid him more money and more S22:A lot of mony!

S23:S24:S25:he is going to school of math and he is learning more math.

S26:He used math by counting how many home runs he did in the season. S27: I DONOT NOWS28: ?S29:i dont knowS30: He used math for adding his home runs

Questions: There were 152 men who tried out to be on the New York Yankees baseball team. Only 38 men

made the team. How many guys did not make the team?

S1:

S2:S3:114.(C)I also used subtraction(BK)S4:114 People didn't make the team(C)because there are 152 men and 38 made the team so 114-

38=114 that should be your answer (F)S5:Because one hundred fifty two minus thirty eight (BK) is 114 (C) thats how much people that didn't

stay to make the teamS6:114 (C)men did not go because I - 152-38=114.(F)S7:IT IS 114(C).doing it agenS8:My answer is 114(C).I got my answer by subtracting 152 from 38(BK)and got 114.It was easy.S9:one-hundred-fourteen people didn't make it(C)S10:in the team theres 114 poeple that could not particapate on the new york yankees(C) S11: 126 people (I) because 5-3=2 and 2-8=6 and there are 100 people left so then you have to add

100+26=126 so that means that 126 people are left.(F)

S12:16 ,because you subtract 152 and 38 and then you get(BK)the answer 16 .(I)S13:The answer is 114.(C)I found out because I counted backwards 38 from 152.(BK)S14:114 becauseI subtracted 152 to 38 and it gave me 114.(BK)S15:The answer is 114(C). I found out because I counted backwards 38 from 152.(BK) S16:114 men failed to make the team.(C)You subtract 152-38.(F)S17:114 guys did not make it on the team.(C)S18:114 i counted back 38(BK)S19:There are 114 left(C)


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S20:there are 1,14 leaf(C)S21:one hundred fourteen(C) because i subtracted one hundred fifty two minus thirty eight(BK)S22:114 men did not(C)S23:the answer is 114(C)because 152-38=114 (F)S24:I was trying to do it in my head but I got confused so I wrote it on a sheet of paper laying next to me. I

did the equation 152-38=?. When I was doing the equation, I used regrouping to help. I used regrouping

when I came across 2-8=?. The number was 3-digits so I put a 1 above 5-3 because I was regrouping the

numbers. I got 3 on 6-3 (5-3). 1-0 was simple for me because its a beginner equation for like Pre-Kinder.

(BK)Finally, I ended up with the answer: 136(I). That is what I got from the equation. I think I maybe

incorrect because regrouping for subtraction(N) is pretty confusing because tens is confusing me. S25:there was 126 that made the team on Yankees baseball team.(I)

S26:126 people did not join the team(I)because 152 minus 38 = 126.(F)S27:one hundred fifty men had not did it(I).Because 152 maines 38.(BK)S28:152-38=126pleople (I)S29:the answer is 114(C)because 152-38=114(F)S30:My answer is 114(C)because i sobtrac.(BK)

Appendix F: Survey


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Appendix G: Rubric to Assess Mathematical Reasoning


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Practical Application Activities in Mathematics

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Transcript of Practical Application Activities in Mathematics