# Number Sense

### By: Bethany Sterritt

## Introduction

As each student enters the 4th grade from different backgrounds, the mathematics that they each have been taught is similar. Each student was taught the 3rd grade Common Core Standards the year before. And as a 4th grader, the expectation is to be taught the 4th Grade Common Core Standards. The 4th Grade Common Core standards place a lot of emphasis on number sense. Each child’s number sense is of a varying degree when they enter 4th grade. Research has shown that every student is capable of learning and acquiring number sense when given a supportive environment with a number sense intervention (Yang, 2003). Number sense is not measured by puzzles and games, but by numbers representing quantities, spoken number names, and number inherent relationships to each other (Connell, 2012). In the 4th grade number sense is highly emphasized throughout the year. It is apparent through testing, observation, and small groups which student can grasp a number and which ones cannot. Number sense does not just assess that a student can memorize a number or a fact, but that they can explain how a problem should be solved and why they solved it that way. For example, one reason students can’t solve problems because many students don’t understand that numbers are flexible. However, numbers can be broken apart and put back together and help them with problem solving. If 4th grade students were instructed daily on number sense concepts, such as the idea that numbers can be broken apart and put back together, how would this improve their strategies for problem solving?

## MethodologyThe purpose of this study is to examine student’s problem solving strategies through daily Number Talks with fractions while building their number sense and mental mathematics skills. An important part of number talks is allowing the students to share their answers and help each other through the problem solving process. Hintz and Kazemi (2014) believe that sharing answers through discussions keeps the students engaged and interested. In previous studies the students who have a number sense concept will make greater strides in mathematics in high school and college (Connell, 2012). While trying to provide this foundation for my students, the research question I will answer in my study is the following: If 4th grade students were instructed daily on number sense concepts with fractions such as the idea that a fraction can be broken apart and put back together, how would this improve their strategies for problem solving? Does engaging in number talks improve students’ perceived problem solving ability? My study is an approach that will assess quantitative data. I will be assessing how mental mathematics strategies on fractions affect the student’s progress on using strategies to problem solve fractions mentally. I will be analyzing the students’ data about mental mathematics and number talks. I want to get a true reflection of the results of doing number talks daily. The IKAN summative assessment will be given to the 21 students. The students will be assessed using the IKAN assessment. The IKAN assessment is a set of 12 fraction questions where the students are timed for answering each one. We can then gather data for what they know and don't know. This data gives us standards and areas to focus for each student. I will then do a weekly assessment to see how the mental math strategies for fractions helped them improve in their math skills to problem solve for these solutions. | ## Data AnalysisEach student will be given the IKAN assessment and leveled based on their mental mathematics, computation fluency, and number sense. With these results, I will plan problem solving strategies for students that do not have any K-3rd grade standards mastered for fractions. Through the initial pre-assessments for the IKAN I can determine this. I will cover these strategies through number talks and small group centers. The students will continue their number talks strategies throughout the six weeks. The IKAN assessment will be given four times throughout the six weeks which will be about every 7 days. I will now have new results for their mental math strategies on fractions to further their knowledge. I will look for growth in each student. If a student was on a 2nd grade level for the initial assessment, the goal is for that student to be on grade level by the end of the 6 weeks. All data from the IKAN will be analyzed to see how students are progressing and what can be done to improve their problem solving mathematics skills for fractions and ratios. The focus of my research is on fraction Number Talks. I spent six weeks working with my fourth grade students. Prior to teaching, I gave the first IKAN assessment. I administered this assessment without any pre-teaching to determine the student’s knowledge baseline, what they retained from being taught in 3rd grade and prior to this year in 4th grade. The IKAN assessment has 5 stages within which the student scores can be categorized. The students stages range from stage 4 to stage 8. If a student does not qualify for at least a stage 4, then the student is not at a stage. Every two weeks I gave the students another IKAN assessment. There are 4 IKAN assessments total and the students took all of these. Starting in the first week, I looked for areas of weakness and need to see what should be taught the two following weeks. | ## Strategies Used During Number Talks
3/8 and 3/5 3/5 is greater than 3/8 because cutting something into 5th will make a bigger piece than cutting something into 8th. Since a 5th is larger than an 8th and both of the fractions have a numerator of 3, 3/5 is larger than 3/8.
2/3 and 4/10 2/4 is greater than 4/10 because half of 4 is 2 and therefore 2/4 is equal to a half. Half of 10 is 5 which means 4/10 is less than a half. A half is greater than a fraction that is less than a half so 2/3 is greater than 4/10.
5/6 and 3/5 The first strategy most students use to solve this problem would be greater than a half or less than a half. Since both of the fractions are greater than a half, another strategy needs to be used. 5/6 is greater than 3/5 because 5/6 is 1/6 away from a whole and 3/5 is 2/5 away from a whole. 1/6 is less than 2/5 therefore 5/6 is closer to a whole than 3/5 which makes it a greater fraction.
I taught my students to change fractions from a mixed number to an improper fraction. All year long we have done this by expanding the fraction. This week I reviewed expanding the fraction and showed them the multiplication trick. Example: 2 x ¾ The way we solve this is to look at the numerator and keep it the same. Since it is a 4, we will add 4/4 +4/4 + ¾. The reason I added 4/4ths twice is because there are two wholes. This was the strategy that the students knew. I introduced the students to one that involves the algorithm. Since we have 2 wholes and the numerator will be whatever the denominator is, we can multiply 4x2 and get 8. This represents the 8 from 8/4. Next we can add the 3 because it would be like adding ¾ to 8/4. Our answer is 11/4. |

## Methodology

The purpose of this study is to examine student’s problem solving strategies through daily Number Talks with fractions while building their number sense and mental mathematics skills. An important part of number talks is allowing the students to share their answers and help each other through the problem solving process. Hintz and Kazemi (2014) believe that sharing answers through discussions keeps the students engaged and interested. In previous studies the students who have a number sense concept will make greater strides in mathematics in high school and college (Connell, 2012). While trying to provide this foundation for my students, the research question I will answer in my study is the following: If 4th grade students were instructed daily on number sense concepts with fractions such as the idea that a fraction can be broken apart and put back together, how would this improve their strategies for problem solving? Does engaging in number talks improve students’ perceived problem solving ability?

My study is an approach that will assess quantitative data. I will be assessing how mental mathematics strategies on fractions affect the student’s progress on using strategies to problem solve fractions mentally. I will be analyzing the students’ data about mental mathematics and number talks. I want to get a true reflection of the results of doing number talks daily.

The IKAN summative assessment will be given to the 21 students. The students will be assessed using the IKAN assessment. The IKAN assessment is a set of 12 fraction questions where the students are timed for answering each one. We can then gather data for what they know and don't know. This data gives us standards and areas to focus for each student. I will then do a weekly assessment to see how the mental math strategies for fractions helped them improve in their math skills to problem solve for these solutions.

## Data Analysis

Each student will be given the IKAN assessment and leveled based on their mental mathematics, computation fluency, and number sense. With these results, I will plan problem solving strategies for students that do not have any K-3rd grade standards mastered for fractions. Through the initial pre-assessments for the IKAN I can determine this. I will cover these strategies through number talks and small group centers.

The students will continue their number talks strategies throughout the six weeks. The IKAN assessment will be given four times throughout the six weeks which will be about every 7 days. I will now have new results for their mental math strategies on fractions to further their knowledge. I will look for growth in each student. If a student was on a 2nd grade level for the initial assessment, the goal is for that student to be on grade level by the end of the 6 weeks. All data from the IKAN will be analyzed to see how students are progressing and what can be done to improve their problem solving mathematics skills for fractions and ratios.

The focus of my research is on fraction Number Talks. I spent six weeks working with my fourth grade students. Prior to teaching, I gave the first IKAN assessment. I administered this assessment without any pre-teaching to determine the student’s knowledge baseline, what they retained from being taught in 3rd grade and prior to this year in 4th grade. The IKAN assessment has 5 stages within which the student scores can be categorized. The students stages range from stage 4 to stage 8. If a student does not qualify for at least a stage 4, then the student is not at a stage. Every two weeks I gave the students another IKAN assessment. There are 4 IKAN assessments total and the students took all of these. Starting in the first week, I looked for areas of weakness and need to see what should be taught the two following weeks.

## Strategies Used During Number Talks

**Example of comparing fractions with a common numerator:**

3/8 and 3/5

3/5 is greater than 3/8 because cutting something into 5th will make a bigger piece than cutting something into 8th. Since a 5th is larger than an 8th and both of the fractions have a numerator of 3, 3/5 is larger than 3/8.

**Example of comparing to a half:**

2/3 and 4/10

2/4 is greater than 4/10 because half of 4 is 2 and therefore 2/4 is equal to a half. Half of 10 is 5 which means 4/10 is less than a half. A half is greater than a fraction that is less than a half so 2/3 is greater than 4/10.

**Example of comparing to a whole:**

5/6 and 3/5

The first strategy most students use to solve this problem would be greater than a half or less than a half. Since both of the fractions are greater than a half, another strategy needs to be used. 5/6 is greater than 3/5 because 5/6 is 1/6 away from a whole and 3/5 is 2/5 away from a whole. 1/6 is less than 2/5 therefore 5/6 is closer to a whole than 3/5 which makes it a greater fraction.

**Mixed Number to an Improper Fraction**

I taught my students to change fractions from a mixed number to an improper fraction. All year long we have done this by expanding the fraction. This week I reviewed expanding the fraction and showed them the multiplication trick.

Example: 2 x ¾

The way we solve this is to look at the numerator and keep it the same. Since it is a 4, we will add 4/4 +4/4 + ¾. The reason I added 4/4ths twice is because there are two wholes. This was the strategy that the students knew. I introduced the students to one that involves the algorithm. Since we have 2 wholes and the numerator will be whatever the denominator is, we can multiply 4x2 and get 8. This represents the 8 from 8/4. Next we can add the 3 because it would be like adding ¾ to 8/4. Our answer is 11/4.

## Findings

__Findings/Conclusions from the first IKAN assessment:__

· Only 9 students were able to write the fractions for one half and one fifth.

· Only 1 student got the two ordering fractions questions correct.

· Word form for fractions was not a skill that students had retained from third grade.

· The students were taught how to order fractions and many of them used the strategy of finding a common denominator. Since these assessments are timed, the students do not have enough time to change the denominators.

· Areas to focus on for the second IKAN assessment: writing fractions and ordering fractions

· The strategies to teach for ordering fractions are using a common numerator, comparing to a half, and comparing to one. Also, when comparing fractions, it is essential to understand that the larger the numerator, the smaller the fraction.

__Findings/Conclusions from the second IKAN assessment:__

· Eleven students increased from no stage or stage 4 to a stage 4, stage 5, or stage 7.

· The first time the students took the assessment, no student was higher than a stage 4. Now 8 students have increased past a stage 4.

· Three students went down a stage and one student stayed the same.

· Thirteen of the twenty three students are at a stage 4 or higher.

· Six of the ten students that are not at a level are special education students.

· The two questions for stage 4 last week asked the students to write a fraction for one quarter and one third. I think the students would have understood if the question had said one-fourth. The academic language used in this problem was confusing to the students.

· Areas to focus on for the third IKAN assessment: writing fractions, ordering fractions, comparing fractions with a common numerator, comparing to a half, and comparing to a whole

- Two students left the study because they are no longer in my classroom.

__Findings/Conclusions from the third IKAN assessment:__

· Nine students increase from no stage, stage 4, and stage 5 to a stage 4, stage 5, stage 6 or stage 7.

· Only one student went down a stage, but four students stayed the same.

· Fifteen of the 21 students are at a stage 4 or higher.

· Four of the six students that are not at a level are special education students.

· Two of the students who started the fraction number talk with my class are no longer in my class.

· Areas to focus on for the fourth IKAN assessment: writing fractions, changing a mixed number to an improper fraction and recognizing equivalent fractions

__Findings/Conclusions from the fourth IKAN assessment:__

· Eleven students increase from no stage, stage 4, and stage 5 to a stage 4, stage 5, stage 6, stage 7, or stage 8.

· Three students went down a stage, and five students stayed the same.

· Ninteen of the 21 students are at a stage 4 or higher.

· One of the two students that are not at a level is special education students with no motivation.

· Two of the students who started the fraction number talk with my class are no longer in my class.

Overall the growth the students showed from the first IKAN assessment to the final IKAN assessment is tremendous. I think many of the students have put forth great effort in learning new strategies and working hard. There is not one particular group that has shown significant improvement over another. Whether the student is ESOL, Special Education, EIP, or not flagged for an academic need, there has been improvement across the class. There are still six students who are not at a level. These students vary in academic need.

In finding a student's knowledge stage using the IKAN assessment and teaching fraction Number Talks daily, I have seen significant growth in the students problem solving ability. Having a structured program and assessment tool to use definitely helped identify student's weaknesses and strengths. I will contonue to use Number Talks in my classroom and incorporate it into all topics. Students need to be encouraged to think critically and have peers to discuss problems with. Number Talks allows these two aspects to take place in the classroom.

## References

Connell, Mike. (2012). Number Sense: What it is, why it’s important, and how it develops.

*Native Brain*, Retrieved from http://www.nativebrain.com/2012/11/number-sense-what-it-is-why-its-important-and-how-it-develops/

Hintz, A., & Kazemi, E. (2014). Talking About Math. Educational Leadership, 72(3), 36-40.

Yang, Der C. (2013). Teaching and Learning Number Sense- An Intervention Study of Fifth

Grade Students in Taiwan. International Journal of Science and Mathematics Education,1.

Retrieved from http://link.springer.com/article/10.1023/A:1026164808929#page-1