The Effects of Tutoring with Concrete Manipulatives and Real Life Concepts

Abstract: 

Place value is a concept many elementary students struggle to master. The fourth grade class I observed was no different. I gave a pre-diagnostic test to determine the exact concepts with which students were struggling. I assessed six students who were performing below grade level and determined that they had not mastered the concept of place value. Students who have not mastered place value cannot fully master other basic math skills. I began a seven week tutoring period with basic place value. The students were able to make progress in math by building a foundation for later learning. The use of concrete manipulatives, such as blocks or counting cubes, and the connection to real life concepts helped the students relate the information to their lives and made it possible for them to visualize the mathematical processes that were occurring with certain mathematical functions.

Table of Contents: 

    Introduction

    My first real experience as teacher began with the mathematics-tutoring project. I was given the responsibility of determining each student’s needs by reviewing the pre-diagnostic assessment and designing lesson plans that would build on the students’ last successful encounter with math. Creating lesson plans appropriate for the students’ needs that emphasized student understanding rather than telling the students about math proved to be a challenging and rewarding task. My goal was to motivate and engage the students and promote active problem solving by students who would evaluate and connect their learning practice to a “meaningful environment” of mathematical understanding (Tournaki, Young, & Kerekes, 2008, p. 41). I wanted to place importance on manipulatives and authentic learning situations that imitate situations of dealing with mathematics. I believe students must understand the relevance of learning in order for the learning to have importance. To promote understanding of math, students need to be taught through real life examples and situations. Throughout the nine week tutoring process, the six students were presented with real life concepts and concrete manipulatives that allowed for a deeper understanding and mathematic perspective.

    Pretest

    The pretest consisted of material from the Texas Essential Knowledge and Skills for the third grade. The students were in the fourth grade, but were not performing at grade level. I chose to test the students at a third grade level to determine where the students last had success in math. I tested the students on numbers, operation and quantitative reasoning, patterns, relationships and algebraic thinking, geometry and spatial reasoning, and probability and statistics. The students took the pretest on a computer and were allowed to have scratch paper to work out any problems. Each pretest contained the same 20 questions in a different order. The students had 45 minutes to complete the test. After reviewing the test, I was able to determine that each student had an incomplete understanding of place value.

    The students were tested on their knowledge of numbers and operations. Numbers and operations begin with the understanding of place value, which was not expressed during the pretest. Understanding place value is necessary for grasping addition, subtraction, multiplication, and division. If students do not fully understand place value, they cannot be successful in mathematics. “A number of studies have noted that lack of number sense and possibly weak phonological processing ability are the factors most highly correlated with math learning difficulties” in elementary students” (Tournaki, et al., 2008, p. 56). I decided to concentrate on the students’ abilities to process numbers and build on their foundations by creating activities that consisted of real life concepts and concrete manipulatives. One of the challenges I faced was making what I was teaching relevant to the students. I needed to explain and give examples of when and where they would use place value in the future. I also wanted to make this learning experience interesting and enjoyable.

    Instructional Activities and Reflections

    Sample Activity 1

    The first week of tutoring, I began my lesson by allowing the students to play with the manipulatives. I learned from my math methods course professor that when students are given something new, such as blocks or counting cubes, they will play with them regardless of the teachers’ instructions. It is better to give the students a minute to play, and when the minute is up, tell the students that now they get to learn. I believe in telling the students they “get” to learn rather than they “have to” learn something. By making the learning experience positive, the students will be more apt to participate. After students played with the manipulatives, I gave the students a furniture advertisement and asked them to create the numbers they found in the ad with counting cubes. The students were able to use the manipulatives to create the numbers.

    “Mathematical manipulatives offer students a way of understanding abstract mathematical concepts by enabling them to connect the concepts to more informal concrete ideas” (Uribe-Florez & Wilkins, 2010, p. 364). The manipulatives provided support for understanding place value because the students built the numbers using hundreds, tens, and unit cubes. As Uribe-Florez and Wilkins note, “Manipulatives by themselves cannot bring about understanding (364).” The students must relate their knowledge and use of manipulatives to real life concepts. The furniture ads provided real life examples and elements to which the students could relate. I believe the real life concepts also made the activity enjoyable. The students were not drilled with definitions or functions, but instead given the freedom to control and monitor their own learning. This act is supported by theorists, Jean Piaget, Lev Vygotsky, and Jerome Bruner. They “advocated that children must construct their own knowledge through interaction with the physical and social environment” (Insook, 2009, p. 250).

    I observed the students as they worked and checked their cubes when they created the price of an item in the furniture ads. The students were also asked to say the number aloud to me. All of the students were able to make the numbers with the manipulatives, but some students were not saying the number aloud correctly. The students’ oral mistakes are also common and made by many adults. I did not want to correct the students, but listen to their reasons and discuss the misconceptions that could follow as a result.

    Student learning and reflection. In one example, students were pronouncing the number “877”as eight-seven-seven instead of eight hundred and seventy-seven. This sparked a reflecting and teaching moment. I asked the students, “If you worked in a furniture store and I were buying a couch that cost $877, and I paid $8.00 plus $7.00 plus $7.00, would you accept my payment?” The students all said no because that was not the price of the couch. I responded by saying that they told me the cost of that couch from the ad was eight-seven-seven. The students then explained that they meant eight hundred seventy-seven. I asked them if the way we say numbers makes a difference. All six students decided that it does make a difference. The students were able to reflect and make a connection to real life and learned that place value and the way we read and say numbers matter.

    Teacher learning and reflection. From this tutoring session, I learned that modeling is a key factor in student learning. The students have overheard teachers and adult say numbers incorrectly, such as eight-seven-seven or eight-seventy-seven. Taking a shortcut does not help the students, but rather hinders their learning. I learned that students will mimic their teachers and repeat mistakes or incorrect behaviors. This newfound understanding of students is cross-curricular and should be applied in every classroom.

    Sample Activity 2

    The following week, I elaborated on the furniture ad activity and had the students choose two magazine clippings from my course material bag. The students read the price of the first piece of furniture and created that number with the counting cubes, then used the counting cubes to create the price of the second piece of furniture. When the students had created both numbers, the students added while using the manipulatives. I wanted to start addition with two two-digit numbers with regrouping so I could see what the students were capable of completing with 100 percent accuracy.

    Student learning and reflection. The students enjoyed this activity and were able to add the two two-digit numbers with regrouping. The students learned how to use their manipulatives to create the addition problems and they completed the addition with the furniture ads accurately. Some students tried to count out 67 units and add them to 87 units instead of six rods and seven units and adding those to eight rods and seven units. The students learned how to correctly stack the numbers and regroup using the manipulatives and were able to correctly answer the problems. The students expressed understanding by stating they could see what really happens when they add numbers together.

    Teacher learning and reflection. While observing the students, I realized the importance of re-teaching. The previous week, the students practiced building numbers up to three digits. However, when it came time to create two digit numbers to add, the students started counting out units instead of building with rods and units. I now understand that I must give specific instructions and reteach when necessary. I also learned that one assessment cannot determine exactly what a student knows or does not know. The pretest gave me an idea of where the students were academically in math, but when actually working with the students while they solved problems, I gained a better understanding of what they were capable of doing. The students were able to correctly answer the addition problems even though the pretest indicated that the students had not fully mastered addition. This is a learning experience I will look back on when I am a teacher. Students have bad days or make careless mistakes that can determine the outcome of an assessment. I know I cannot determine a student’s ability using one test. I believe the tutoring experience helped me learn this lesson. Tutoring also helped me see how flexible teachers should be. The lessons must be for the students to help them learn, not because the lessons are convenient for the teachers.

    Sample Activity 3

    During the next tutoring session, I asked the students to pretend they were at the zoo. It was a hot day and it was the students’ job to hand out free water to the thirsty people. The students were given two dice and asked them to roll them. I told the students that the first number would represent the number of people who wanted water. The second number would be the number of cups of water on hand. I told the students that the goal was to make sure that every person got a cup of water.

    Student learning and reflection. Some of the students did not understand how to make sure everyone got a cup of water while other students knew exactly what I meant and rolled the dice and created a small number first and a larger number second. I walked through the steps with the students who did not understand. I used manipulatives so the students could visualize the numbers. I rolled the dice twice, rolling a seven and a four. I asked the students what number I could create with these two numbers. The students said seventy-four and forty-seven. I asked them what number we should use to represent the people and they decided on seventy-four. I then rolled the dice again and rolled a four and eight. I asked what numbers I could create and the students responded with forty-eight and eighty-four. They decided to use eighty-four. I explained to the students that they chose seventy-four to represent the people and eighty-four to represent the cups of water. I asked the students if these numbers would work for our game. “If we have seventy-four people and eighty-four cups of water, will everyone get a cup of water?” As I was asking the students, I was creating the number with manipulatives and comparing the numbers. The students said yes. “What if we chose forty-eight for the cups of water? Would that have worked?” The students agreed that it would not have worked because some people would not have received any water. The students were then able to create two digit numbers and compare them with other two digit numbers. The students eventually moved up to creating three digit numbers. I changed the items being represented to the number of players and jerseys on a football team, the number of dancers and tap dancing shoes, and the number of people at a baseball game and the number of seats.

    Teaching learning and reflection. This was a learning opportunity for me as well as for the students. I learned students need to be able to see what they are learning and that this will help them in their future whether it is in the next grade level, high school, college, or their everyday adult lives. As Spungin (19960, p. 178) indicates, manipulatives have been “used for many years to help students understand abstract ideas such as number and operation.” The students must be able to make connections that relate to them personally, not because their teacher says they will need to know place value the rest of their lives. Students need visuals and real life situations in order for the connection to be made and to tie the experiences together. I believe I related number and operation and the understanding of place value to the students’ lives by incorporating the furniture ad and having the students provide cups of water to a certain number of people.

    Sample Activity 4

    During a guided practice section, I had the students build different numbers with manipulatives. The guided practice section worked out well. The students did not like building the smaller numbers with the manipulatives, but seemed to enjoy creating the three digit numbers. All the students mastered this technique by creating the appropriate numbers with the manipulatives. I also did the activity where I said “I’m thinking of a number with a 3 in the hundreds place, a 9 in the tens place, and an 8 in the ones place. What’s my number?” The students enjoyed this activity and especially liked it when I said the numbers out of order.

    Student learning and reflection. The activity really focused on place value and the students’ abilities to recognize the order of ones, tens, and hundreds place. The students had to listen carefully to determine what the place number was and the number than went into that place. I did not tell the students how to set up their paper for this activity because I wanted them to use problem solving skills and whatever strategy worked best for them. Most students drew three lines and placed the number in the appropriate place. One student labeled her lines with an H for hundreds, T for tens and O for ones. Another student really surprised me. When I said that I was thinking of a number with a four in the hundred’s place, she would write 400. I saw the student write this, but did not comment on it. I watched to see what she would write when I said that there was a seven in the one’s place. She wrote a seven. I said there is a nine in the ten’s place. The student wrote down 90 then combined the numbers and wrote 497. I was impressed by this and asked the student to share her strategy. She said it made sense to her to write down 400 if a four was in the hundreds place because that is how many hundreds she had. The students discussed this strategy and agreed her strategy made sense and did work. Some students decided to continue the activity with this strategy while others kept using their own method. The students learned from each other during this activity and listened to reasoning of their peers.

    Teacher learning and reflection. From this experience, I learned that students must be given the freedom to teach themselves and each other. I feel the students benefit when they are given opportunities to express their ideas. I understand students must be taught new material and certain steps, but when the students are able to decide how they choose to use the material, they will have a better understanding of the concepts being presented.

    Sample Activity 5

    Based on the students’ observed needs, I made the decision to move on to multiplication. The students were learning about multiplication up to seven times seven. The students knew the steps for multiplication, but they occasionally made mistakes when working through problems. When I was going over how to use manipulatives to multiply, the students told me they were supposed to have the facts memorized and could not use the manipulatives to find the correct answer. We talked about this because I had seen their teacher demonstrate how to multiply by putting a number of circles into a number of rows. I told the students that it is convenient to have the facts memorized, but it never hurts to check the answers by using manipulatives to have a deeper understanding of what is exactly happening when multiplication occurs. During the structured practice, I was demonstrating how to group the cubes to represent different multiplication problems. The students understood how to group the cubes, but when it came time to count, they refused. They tried to do mental math and multiply in their heads. I told the students that the manipulatives could be counted to find the answer. I do not think the students fully understood what they were doing with the cubes or why they would draw the circles in their class. I told the students they could multiply the numbers in their head, but they still should check their answer with the total number of cubes. The first group of students realized they were making mistakes when only multiplying in their heads. I asked the students to reflect on which answer they thought was correct, and all three thought the number produced using the cubes was correct. Their reasoning for this was because they could see the cubes and they physically counted each cube. The second group had a better understanding of multiplication up to seven times seven. Both groups answered the independent questions correctly while using manipulatives so I felt they were learning as well as applying their previous knowledge.

    Student learning and reflection. The manipulatives helped the students check and correct their answers. Many students thought they answered some questions correctly by only doing the problems in their heads, but when they checked, they realized they had made a mistake. A few students even said they did not know the times tables as well as they thought. They also said the manipulatives helped them see where they made mistakes and helped them find the correct answer. The students previously expressed that they did not like the manipulatives when they used them for addition or subtraction, but they found the manipulatives useful when multiplying.

    Teacher learning and reflection. During this week of tutoring, I learned the importance of having students show their work and reflect and discuss the learning processes that occurred. When I asked one student to describe how she found an answer, she told me she had her own system. This “system” consisted of the student multiplying a number by 5 then adding or subtracting. For example, if the problem is four times six this student will multiply four times five then add a four. When I heard her say this, I was wondering why she did not add six to find the answer. She then said “I bet you’re wondering why I didn’t add six.” I paused and waited for her explanation. “Because it doesn’t work, I have four groups of five, so I need to add one more to each group to have six in all.” I was thinking, “Yes, this does work, but to me that is much more complicated than four times six.”

    I believe students should have strategies that work for them, but I told her that her method may not always work on every problem. I told her she can use her method, but she needed to use the manipulatives as well and she could check answers. I thought she was really thinking about the process of multiplication because she came up with this strategy. I was glad she told me so I could understand her learning and thought process. By using the manipulatives and by learning the way the student found her answers, I was able to see how different strategies help different students. There are different ways students can work problems. I want my students to be comfortable and not afraid to tell me what works and does not work for them. I want to help them build on their skills.

    I also believe repeated addition really helped the students understand how to use the manipulatives when multiplying. I think some students do not understand that three times three is really three plus three plus three. They can physically see that the two are connected by using manipulatives because there are three groups and three manipulatives in each group. This correlates with the student having her own way of finding the answers. She knew that she could do four times five then add four more because of the groups. I believe the manipulatives are good tools and they helped my lesson plan flow and function this week.

    Sample Activity 6

    The following week, the sixth week of tutoring, the students were learning about multiplication up to twelve times twelve. The students were familiar with multiplying with manipulatives because they did it the week before up to seven times seven. I felt confident that learning occurred based on my observations of the different strategies the students were using to solve the problems. This week I made the manipulatives accessible, but did not insist that the students use them. I wanted to see the different strategies the students would use to solve multiplication problems if they had the option of not using manipulatives.

    Student learning and reflection. The students had different methods of solving problems. One student used her “system” of multiplying by five and adding the rest. One student used her fingers, especially with the nines. One student drew circles and put dots in each circle. One student did as much mental math as he could and then either counted his fingers or drew out the rest. I observed the different strategies and then asked the students to check their answers by using manipulatives. A few of the students answered the questions correctly; if they did not, they were able to check their work and see where their mistakes occurred. The student who did most of the problem mentally answered most of the questions incorrectly. I suggested the student write down everything he was doing in his head. The student was able to reflect on the steps he was taking and see where he was making mistakes.

    Teacher learning and reflection. I wanted the students to learn about their learning by reflecting on their work while using the manipulatives to check themselves. I had the students use manipulatives in the steps they took to solve the problem. I did not tell the students what strategies to use, but instead I told them to solve the problem the same way they would if they were in their classroom. So the student that multiplies by five then adds the rest was allowed to do her method, but I encouraged the use of the manipulatives as she worked. She and the other students were able to check as they worked their own way.

    Another step I took that required the students to reflect on their own learning was having the students write their own multiplication sentences. They each wrote a problem for two of their two peers. At first the students tried to make the sentences confusing so the others would not answer the question correctly. I saw this was happening so I had the students answer their own questions. No one could answer their own questions. I asked them why they could not answer their own questions. They said the questions were confusing and did not make sense. I told the students to think about the clues they look for when they begin to solve a word problem. The students thought about it and then rewrote their sentences. The sentences now made sense and the other students were able to solve their peers’ questions. I believe the students now see word problems in a different way because they understand how important it is to underline the important details and eliminate the useless information.

    Sample Activity 7

    The last week of tutoring was a time for the students to reflect over their learning process as well as apply their prior knowledge. The students played a game that I created for an assessment during the previous weeks of tutoring. In the game, the students drew a picture of an item. The pictures were of household appliances such as a toaster or blender, furniture, and grocery items such as bread, milk, and eggs. The game had all the same components as the first time the students played the game; however, this time the students had to maintain a budget. I told the students they were going to be given a budget which they could not exceed. Only one student had heard of a budget. They did not know how a budget really works. I explained and modeled how to play the game with a budget. The students had to add the amounts of each item they chose, and then had to subtract the amount they had spent from the total budget amount.

    Student learning and reflection. The students had different methods to figure out if they had spent too much or how much they had left to spend. One student would pick everything he wanted, then add everything up at the end. When he had spent too much, he would remove one item from his list. With trial and error, he figured out his method was not the best and converted to different operations. Another student only chose items that she felt were really necessary. She wanted to have more money left over so she could spend it on going to the movies or other fun things she would want to do. There was one student who felt he had to spend every penny of his budget so he bought everything he could. The students also multiplied an item’s amount. I told the students they were planning a family event and supplying the food. They would draw an item from the envelope and a card with a number on it. The students would multiply the item’s amount by the number and the product represented how many items they would need to feed their family. The students had to budget their spending with this activity as well. Every time the students correctly added, subtracted, or multiplied, and maintained their budget, they received a sticker.

    Teacher learning and reflection. I believe the games have value because the students can practice their addition, subtraction, and multiplication skills, as well as practice keeping a budget. When all the students had finished buying their items, I had the students discuss the different methods and strategies used. I did not tell the students the importance of managing and budgeting their money. I wanted the students to do the activity first, then discuss the relevance of the activity for their future. The students learned how to manage and budget. I explained to the students that this is what adults do in their lives to make sure they do not overspend. I asked the students if they thought maintaining a budget was a good idea or a bad idea. After some discussion, the students agreed that keeping up with the amount of money they had and deciding how much they wanted to spend would be a good idea. They thought it would be a good idea because they did not want to spend money they did not have and get in trouble with the police or the store where they bought the items. All the students thought this game would help them when they got jobs as they get older and had to determine how their money should be spent. I thought this activity was a good way for the students to practice their math skills and relate what they are learning to real life.

    Summary

    I learned how important it is for teachers to teach students math through real life situations and the use of manipulatives. The “idea that the use of manipulatives in mathematics classrooms helps students to develop mathematical understanding is supported” by many theorists and literature (Uribe-Florez & Wilkins, 2010 p. 364). Manipulatives allow students to visualize the mathematical processes taking place during addition, subtraction, multiplication, and division. Manipulatives also allow students to have a better understanding of place value. When students are able to create the numbers with counting cubes, they see how each place is valued by different uses of the counting cubes.

    The post diagnostic test was given to the students the last week of tutoring. The students were given the same assessment as the pretest under the same conditions. Table 1 contains a row for each individual student. Each student has a pretest and posttest score listed and the overall gain. The students were tested in numbers, place value, addition, subtraction, and multiplication. If students gained place value knowledge, according to the post diagnostic test, this would be indicated by a Pvx in the Concept tested gained column. This was applied to all concepts tested. The xindicates if the student gained or lost in each concept tested. After reviewing the post diagnostic test, I determined five out of the six students I tutored improved their understanding of place value. One student stayed the same with his understanding of place value. I believe the success of the students who improved was a result of the manipulatives and the focus on real life concepts. The manipulatives presented the students with concrete materials to better understand abstract questions. Throughout the tutoring process, I have learned ways to incorporate real life concepts to math activities and lessons that provide the students with hands on experiences that are relatable and meaningful. I feel this experience has prepared me to be an engaging and interactive teacher. I will use manipulatives to engage my students and teach place value and numbers and operation.

    Works Cited

    • Insook, C. (2009). Korean teachers’ perceptions of student success in mathematics: Concept versus procedure. Montana Mathematics Enthusiast, 6(1/2), 239-255.
    • Spungin, R. (1996). Teaching teachers to teach mathematics. Journal of Education, 178(1), 73.
    • Tournaki, N., Young She, B., & Kerekes, J. (2008). Rekenrek: A manipulative used to teach addition and subtraction to students with learning disabilities. Learning Disabilities – A Contemporary Journal, 6(2), 41-59.
    • UriFlorez, L. J., & Wilkins, J. M. (2010). Elementary school teachers’ manipulative use. School Science & Mathematics, 110(7), 363-371. doi:10.1111/j.1949-8594.2010.00046.

    Table 1: Analysis of Gains and Losses on Diagnostic Scores and Concepts Taught

    The data in the table show the diagnostic score, the concepts tutored, and which areas showed gains, losses, or remained the same.

    Student Pre-diag
    core
    Post-diag
    score
    Overall
    gain
     1
    Mcn_Kah_Da
     90 85  -5
     2
    Mcn_Kah_Db
     70 80 10
     3
    Mcn_Kah_Dc
     65 75 10
     4
    Mcn_Kah_Dd
     75 90 15
     5
    Mcn_Kah_De
     80 85 5
     6
    Mcn_Kah_Df
     75 70 -5
    Student Concepts
    tutored
    Concepts
    tested
    gain
    Concepts
    tested
    loss
    Concepts
    tested
    same
     #
    sessions
    attended
     1
    Mcn_Kah_Da
    Num
    Pv
    Add
    Sub
    Mul

    PvX


    AddX
    SubX
    NumX



    MulX
     6
     2
    Mcn_Kah_Db
    Num
    Pv
    Add
    Sub
    Mul


    AddX
    SubX
    MulX
    NumX
    PvX
     7
     3
    Mcn_Kah_Dc
    Num
    Pv
    Add
    Sub
    Mul

    PvX
    NumX



    MultX

    AddX
    SubX
     7
     4
    Mcn_Kah_Dd
    Num
    Pv
    Add
    Sub
    Mul

    PvX
    AddX
    SubX
    MulX
      NumX  7
     5
    Mcn_Kah_De
    Num
    Pv
    Add
    Sub
    Mul

    PvX
    AddX
    SubX




    MulX
    NumX  6
     6
    Mcn_Kah_Df
    Num
    Pv
    Add
    Sub
    Mul

    PvX


    AddX
    SubX
    NumX



    MulX
     7

    Num=Numbers
    PV=Place value
    Add=Addition
    Sub=Subtraction
    Mul=Multiplication
    X indicates if the student test scores showed a gain, loss or no change