Through the tutoring project and additional research, I have determined multiple ways to assess students on their math skills and improve their comprehension. During each tutoring session, I chose eight third-grade students to test and tutor on different math concepts. I believed the students’ test scores would improve after the tutoring sessions. Each session was structured using the Mastery Model which incorporated the use of manipulatives as well as guided and independent practice. The Mastery Model allowed me to create a more authentic and organized RTI that met each student’s needs. Incorporating RTI into the process was imperative because it enabled me to analyze the students’ learning as well as assess my own teaching. The posttest results did not show a significant change from the pretest; however, observing the students in the tutoring sessions showed how hands-on activities allow students to view abstract concepts in a more concrete way.
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As I began the first semester of my student teaching, the thought of teaching math was overwhelming. My lack of confidence prevented me from having a positive outlook. After previous struggles with mathematics, I worried not only that I would not be able to teach, but also that I would not be able to help children who fear math like I do. I was eager to learn how to boost my confidence in teaching math and the students’ confidence in learning math. This question coincides with the purpose of this paper which is to present a case study of a tutoring project in which I participated as a tutor of mathematics. The case study involved the analysis of Response to Intervention (RTI) notes from the project. There was no research question or hypothesis associated with the study, only descriptive responses that examine the tutoring process and learning.
The initial step in the tutoring process was having the students take a pretest which allowed me to determine the basic skill levels of the students as well as which Texas Essential Knowledge and Skills (TEKS) were problematic for them. The diagnostics showed me that the highest score was a 65 and the lowest score was a 25 on a 100-point scale. Although the results included a wide range of scores, I noticed the students struggled with many of the same TEKS which included:
- 2.1(A), use concrete models of hundreds, tens, and ones to represent a given whole number (up to 999) in various ways;
- 2.1(C), use place value to compare and order whole numbers to 999 and record the comparisons using numbers and symbols (<, =, >);
- 2.4(A), model, create, and describe multiplication situations in which equivalent sets of concrete objects are joined;
- 2.3(D), determine the value of a collection of coins up to one dollar; and
- 2.5 (B), use patterns in place value to compare and order whole numbers through 999.
Because of the diagnostic results, I decided to group all of the students together for the tutoring sessions. The TEKS that were not yet mastered included those with a focus on place value which is essential to moving on to addition, subtraction, multiplication, and division. Not understanding place value could also be the reason why the students missed most of the questions involving money. I believed the students struggled with questions that asked them to use patterns in place value to compare and order whole numbers through 999, not only because they had not mastered place value, but also because the concept involved algebraic thinking which might have been too difficult for the students to understand at this stage. After making these determinations from the first sessions, I created my first intervention plan that involved the Mastery Model.
The Mastery Model consists of different steps that incorporate a variety of teaching strategies. The first step is the orientation during which the teacher introduces the concept through a song, piece of art, or a story. This process enables students to activate their prior knowledge either from other subjects or from previous math lessons. It also engages the students and attracts the interest of students who are more visual, auditory, or kinesthetic learners. I spoke to the students for only a short period of time. The instruction part of the model was more of a recap of the orientation in order to make sure all of the students understood what we would be learning. Incorporating practice is another strategy that I used during the tutoring sessions. This strategy included structured, guided, and independent practice. During structured and guided practice, the students practiced the concept through activities that included the whole class or partners. Independent practice enabled the students and me to see what the students knew and how they could apply it. During this practice, the students usually answered math problems that I could score and look over. The last step of the mastery model was the assessment, which usually consisted of a game. This game could be played with partners, the whole class, or even independently. I tried to make the games more interactive so that the students could work on their communication skills and could learn from one another. This not only helps the students with their math skills, but it also assists students in other subject areas because it is very important for the student to communicate. Communication helps the students organize their thoughts and also enables them to feel comfortable asking questions to clarify and increase their knowledge.
Throughout these sessions, I incorporated the use of manipulatives which allowed the work to be more hands on and comprehensive. The manipulatives that I used most often were base ten blocks. Base ten blocks are made up of individual units that represent the ones place, rods or groups of ten that represent the tens place, and flats or groups of hundreds that represent the hundreds place. These base ten blocks were applied to many different concepts rather than just solely place value. The blocks ultimately allowed the students to continue to build on the concept of place value.
Response to Intervention–Tracking Progress
Response to Intervention was another method that I used during the tutoring sessions. “Response to Intervention (RTI) integrates assessment and intervention within a school-wide, multi-level prevention system to maximize student achievement and reduce behavior problems” (National Center on Response to Intervention, 2011). This systematic process was conducted through the tutoring sessions, assessments, and weekly narrative responses. Although this project was not a formal RTI, it still had the same intent and goals.
The RTI system is separated into three tiers. In Tier 1, class instruction is differentiated for various types of learners. In Tier 2, some students are pulled out to work together under the guidance and instruction of a teacher or specialist. Tier 3 is implemented when the student or students show little success with Tier 1 or 2. This type of instruction is more explicit and in depth, and it can either be provided to individual students or to small groups of students. The tutoring sessions were most similar to Tier 2 because the students were organized into small groups who were assessed and tutored over a period of time. RTI was incorporated into the tutoring process through weekly responses in which I reflected on the students’ learning, my learning and the effectiveness of the plan. These responses allowed me to understand that it is imperative to consistently assess and improve the students’ learning as well as my own teaching.
Weekly Responses (RTI)
Descriptions of learning among the children. I believe that the approach I used in teaching the concept of “greater than” and “less than” was a different perspective for the students because they all discussed how they learned with the alligator representation where the greater than/less than symbols are seen as the opened mouths of alligators (> or <). The students seemed to understand how the “greater than” and “less than” symbol worked. In the beginning of the lesson, I had the students use manipulatives, such as base ten blocks, for comparisons so that the students could actually see how one shape was greater than the other rather than just using numbers. I had the students create rows of marks representing each number. This enabled them to use one-on-one correspondence and also provided the students with a visual of “greater than/less than.” Overall, this activity helped show the students how the “greater than” and “less than” symbol can be created and understood in a different manner.
Specific details of student learning. Students demonstrated their learning through the practice activities. First, I noticed they had a good understanding of how to use the base ten blocks as they counted up the different amounts that I gave them and separated the blocks into ones, tens, and hundreds. Then the students ordered themselves from least to greatest. At first they struggled because they were not looking at everyone’s numbers, but I saw how they were able to readjust themselves by figuring out which numbers were greater and which numbers were lesser in value. Observing this thinking process of the students enabled me to assess their knowledge. Learning also occurred when the students had to apply their knowledge to real life examples because they were thinking about how “greater than” or “less than” situations would occur in their own lives.
How the intervention plan facilitated students’ learning. I believe that the learning was facilitated by using various approaches to teach the concept of “greater than” and “less than.” The students demonstrated they understood “greater than” or “less than” in both a concrete and abstract way. This is important because it shows that the students truly understand the concept. In addition, I believe using manipulatives, visuals, and real life examples enabled the students to use these various techniques to better understand their learning process.
Description of the learning that occurred for me. I learned that it was important to allow the students to converse during the lesson, talking as much, or even more, than I was. This allowed me to understand how the students were thinking, in addition to observing whether they were engaged or not in the lesson. Another thing that I learned was that it is beneficial to continue to incorporate specific learning strategies such as the base ten blocks. As the students continued to practice with the base ten blocks, their understanding of the concept improved and they became more comfortable using them. It was important to incorporate a variety of techniques so the students could learn in a variety of ways.
Specific details of my learning. I observed for myself the learning occurring among my students as they reacted to my teaching. In the past few tutoring sessions, I realized that each one is a learning experience because you never know what the students are going to understand and how they are going to react. During a particular session, I saw the importance of having the students engage in discussion because, as they would call out answers, I would ask them why they thought their answer was correct rather than simply responding “good job,” “that’s correct,” or “try again.” This forced the students to explain why or how they got their answer, which allowed me to observe their learning and thinking processes. It was important to be consistent when using manipulatives and to use a variety of techniques. Using the base ten blocks made the blocks seem less foreign to the students and allowed them to continue to use them.
How the intervention plan facilitated my learning. I believe my learning was facilitated because the intervention plan was more student-centered. I did less talking and allowed the students to figure out things on their own. I also learned by simply asking the students questions about what they know and what they would like to work on. This allowed me to learn more about the students’ strengths and weaknesses which helped me develop the next intervention plan.
Reflection on the plan. What worked? Why? The things that did work included having the students continue to use manipulatives, creating more of a group discussion, and allowing the students to move around during the activity. Using the manipulatives allowed the students to continue to see the concepts in both a concrete and abstract way and also made them more comfortable using the manipulatives which might have encouraged them to use the manipulatives in class. Discussing the concepts during the lesson allowed the students to take ownership of their learning and also allowed me to facilitate the students’ learning. Allowing the students to move around during one of the activities enabled them to get their energy out as they were creating a visual image of least to greatest.
What did not work? Why? First, I believe it was probably a bad idea to have the students work in the conference room because they got so excited to be in a new and different place that it was hard to keep them focused on the task at hand. I also think that it did not work to have all six children together because it made it more difficult to work with each student individually. However, during this time, I did not have an alternative because the students were not in school on Monday. Another thing that was hard to do was to get the students to create their own word problems. Most of the students just sat there and said that they could not think of anything, or they created a problem that did not make any sense. As a result, I worked with the students and together we created a problem on the board. Having the students work with partners during the game also did not work because the students were very particular about their partner and they complained if they had to be partners with somebody that they did not like. These distractions prevented them from staying on task and they did not put very much effort into the game they were playing.
What would have made it work even better? Why? I believe that the intervention plan would have worked better if the students had been divided into two groups of three. That would have allowed me to work with the students more individually and to better understand their learning. The next time I teach these concepts, I will take one group of students on Monday and another group of students on Tuesday. Another way I could have improved my tutoring session was to have the students continue the activities or the game the following week rather than trying to get everything in during one session. I believe that this session was rushed and I want the students to have enough time during each activity to really grasp the concept. One last thing that would have made it even better would have been to have the game include the whole group instead of just one partner. With a group game, I believe the students would not complain so much about their partner and would instead focus on their learning and the application of their knowledge.
Description of the students’ learning. During Week 6, we finished up the intervention plan from the previous week which focused on double digit subtraction. The students completed a worksheet and played BINGO. The students were very excited to play BINGO which seemed to make them put extra effort into their work. They are continuing to become more proficient in subtraction with regrouping because they are applying their knowledge to the games and math problems. The students are also continuing to use their manipulatives as aids which shows that they really are beneficial for them.
Specific details of students’ learning. I could tell that learning occurred by observing the students work on their problems. They were showing improvement by answering most of the problems correctly and also by showing they knew how to work through problems. They verbalized their understanding and also showed their work on paper. It was important that the students continue to use the base ten blocks because that indicated that they felt comfortable having material in their hands and also that they better understood the concept. Although they did not use the base ten blocks during BINGO, I thought it was important that they were able to complete the problems without the blocks because I knew that they understood the concept and I believed it was okay that they did not use the blocks every time.
How the intervention plan facilitated this learning. I believe the intervention plan enabled the students to apply their knowledge in a fun and engaging way. All of the students loved playing BINGO which was both an encouragement for them to do their work and a way for me to further assess their understanding of concepts. I believe having them complete a worksheet with subtraction problems with regrouping that included word problems proved helpful because it allowed the students to practice questions that they might see on future standardized tests.
Description of the learning that occurred for me. I learned that it is important to make sure that every student is on track. During the BINGO game I decided to work with one of the students individually because I knew that the two other students were able to do the problems. This student needed extra help so I took the opportunity to support him as he played the BINGO game. I noticed this was beneficial because he got more of the work finished than he usually did during the tutoring sessions. After this experience, I understood that it was important to take every opportunity to help the children take control of their learning. Seeing the students so excited about the BINGO game taught me that I should incorporate fun and enjoyable activities for the children so that they would become more engaged and be more able to apply their knowledge.
Specific details of my learning. I knew that learning occurred when I was working with the students and observing them play the game and completing the worksheet. Not only did the students show their comprehension, but they also felt a sense of accomplishment when they completed their work. This taught me how important it is to make sure your students not only understand the concept but that they are also proud of themselves. If a student has drive and confidence, they will perform to the best of their ability.
How the intervention plan facilitated my learning. I think that I had more time during the intervention plan to work with the students individually which enabled me to learn more about the students’ learning. This also let me assess the students more thoroughly because they were completing individualized activities where they had to apply their own knowledge. This knowledge helped with assessment of the students’ progress in their learning. Another aspect of the intervention plan that was helpful was that we were continuing the previous lesson. This allowed me to see what the students remembered and it also gave the students more time to talk and complete their activities. In other words, the plan was centered more around the students.
Reflection on the plan. What worked? Why? I believe that both of the activities worked because students were allowed to continue to practice single subtraction and regrouping subtraction. The worksheets had the students apply their knowledge and use their higher order thinking skills to solve word problems. Although they struggled more with the word problems, I believe this was important for me to know in order to determine what else the students needed to work on. The students also continued to practice using the base ten blocks as they completed the worksheet which was another reason the BINGO game was successful.
What did not work? Why? I think it was difficult to get the students to do the worksheet at the beginning. They are not used to coming in and having to get right to work. I think I should have started off with an orientation or opening activity to get the students more engaged and also to help recap what we learned the previous week. One thing about the BINGO game that did not work very well was the fact that I did not have a prize for the winner. The students seemed to expect it, so they were disappointed when they did not receive anything.
What would have made it work even better? Why? I believe it would have worked better if I had incorporated a “warm-up” at the beginning to get the students thinking and to activate their prior knowledge from the previous session. Another thing that would have improved the plan was to have the students play a game where they had to interact more with each other. This would have allowed me to see how the students communicated and voiced their understanding and it would also have allowed the students to learn from one another.
Although the pretest and posttest served as formal assessments, I believe that the formative assessments of the tutoring sessions allowed me to get a more accurate perspective of the students’ success because I was able to work with each student individually and also because I reflected on each session. These reflections taught me that RTI is a very important and helpful tool that I believe all teachers should implement in their classroom because it allows teachers to analyze the students’ learning as well as assess their own teaching strategies. Typing up responses and figuring out what worked and what did not work helped me to improve my sessions from week to week.
Learning From the Process of RTI
Implementing RTI throughout the tutoring process was extremely beneficial and had a positive impact on the overall process. It is a common misunderstanding that RTI should be used only for students who have learning disabilities, but Douglass and Horstman (2011) say that “many researchers affirm that there is evidence to support RTI as a means of monitoring the progress of students with or without disabilities” (p. 24). Determining which strategies were the most effective and how the students responded to them helped me with the RTI part of the tutoring project. I constantly made changes to my lesson so the students could better grasp the concepts. I was able to think on my feet and create modifications during the sessions. I believe that every teacher should incorporate RTI into their classroom based on the three tiers. This strategy will help improve the learning of all students.
I noticed that most of the students that I tutored had the greatest difficulty with word problems. They might have been able to do the mathematical procedures just fine, but when they had to start applying them to different scenarios, it seemed to get really confusing for them. I believe that there are two reasons for this. First, they do not know where to begin when they are answering a word problem, and second, they have not had enough practice. In order to help the students effectively practice answering word problems, I believe that the teacher has to model the process and break it up into steps rather than long explanations. The article, The Problem with Word Problems, says that “[t]urning these lists into simple graphic organizers allows students to approach problems in a step-by-step fashion” (Forsten, 2004, p. 23). Throughout the tutoring sessions, the information was organized through the Mastery Model. The Mastery Model incorporates different types of instruction and practice in order to make the learning successful. It is essential for the students to understand why they are taking these steps and how each step affects the solution so they can apply it to everyday situations and incorporate their mathematical thinking. I also believe that this will continue to help students and teachers to assess and understand their ongoing learning and thinking.
The main focus that the article, Using Manipulatives to Teach Elementary Mathematics, is that “educational research indicated that the most valuable learning occurs when students actively construct their own mathematical understanding, which is often accomplished through the use of manipulatives” (Boggan, Harper, & Whitmire, 2010, p. 2). Other benefits of using manipulatives include improvement of students’ short-term and long-term memory of math as well and opportunity to reflect on one’s accomplishments and the prevention of math anxiety.
Theorist James Zull conducted many studies on the brain, discussing the central nervous system and the complicated process of neurons reacting to certain experiences and behaviors. “The more brain areas we use, the more neurons fire and the more neural networks change–and thus the more learning occurs” (Zull, 2004, pg. 72). This reaction enables students to better understand and retain information, showing the importance of limiting anxiety by creating new dendrites in order to build a bridge between fear and pleasure. “Research also indicates that using manipulatives is especially useful for teaching low achievers, students with learning disabilities, and English language learners” (Boggan, et al., 2010, p. 5). It is crucial to implement manipulatives in the classroom especially because students’ learning styles and backgrounds are so diverse.
One potential problem with manipulatives is that they can be used without being internalized. Students might enjoy using the manipulative and might seem more engaged but that does not necessarily mean that they understand the concept they are learning. In the article, Examining the Role of Manipulatives and Metacognition on Engagement, Learning, and Transfer, we find that, “We hypothesize that it is not only the content of the learning materials (concrete versus abstract) but also how those materials are used that is critical to learning complex cognitive skills such as those taught in mathematics and science” (Belenky & Nokes, 2009, p. 103). In this article I learned the importance of not only incorporating hands on activities with students, but also of making sure students are thinking metacognitively as they learn. After working with a group of students, the article concluded that their “research suggests that pairing concrete materials with metacognitive prompts should facilitate procedural fluency as well as conceptual understanding and transfer” (p. 108). I believe that students have to make the connection to what they are learning and how it relates to their prior knowledge. In order to facilitate this connection the students must be provided with these opportunities.
The basic goal of the tutoring sessions was to determine clearly what the students knew or had not yet learned and then to help them with their understanding. I used manipulatives, RTI, the Mastery Model, and different assessments to determine the students’ comprehension, but what helped me the most was feedback from the students. I know that the techniques I used to help teach the students do have an effect on how they learn the concepts, but the most crucial part of the learning process is to make sure that the students have used their mathematical thinking to help them realizehow they figured out a problem. According to Kostos and Shine (2010, p. 223), “It is important for students to be able to demonstrate their mathematical thinking as well as their method of solving a problem.” This article, entitled, “Using Math Journals to Enhance Second Graders’ Communication of Mathematical Thinking” discusses an action research study involving math journals. The journalist’s process of working with the students and using manipulatives seems very similar to my tutoring project, but her approach also included documentation of the students’ mathematical thinking. This is extremely important because it helps the students show their mathematical thinking as they solve a problem rather than just memorizing the method they use to solve a problem. If students just memorize ways to figure out the answer, then they will not truly understand why they used that method and how it relates to the overall math concept.
Learning subtraction with regrouping is a difficult task for students to grasp conceptually. If each student is given base 10 blocks, we can use the units, rods, and flats to demonstrate how 10 units in the ones place can be regrouped into a rod that moves to the tens place. Manipulatives are much easier for students to understand than the abstract concept of crossing out numbers and changing them. Some might be able to memorize this method and determine the correct answers, but it will not help them think mathematically. If children are not thinking mathematically then they will continue to have difficulty with math concepts that are even more abstract.
During each tutoring session, I would come across a challenge and try to overcome it or I would be working on a challenge that I noticed from the previous sessions. Some of the situations I encountered seemed more challenging because it was something that I could not control, such as only having a limited amount of time to tutor the students. In this situation, I thought about what was best for the students rather than focusing on finishing the intervention plan. If the students did not complete everything I had planned for them, I would make sure to continue the lesson during the next session. I believe that it is more important for the students to really understand the concept and be able to show the teacher their understanding than to just show the teacher that they can rush through an activity. I observed some of the students taking their time to really apply their knowledge and understanding when they used their base ten blocks to determine the answer and even to check their answer. This showed me that the students have really taken ownership of their learning.
Although some of the tutoring sessions did not go exactly as planned, it gave me the opportunity to figure out how to make the next tutoring session better. I view these types of experiences as changes that I made in order to improve the students’ learning rather than challenges. During the first tutoring session, I had all six students attend at the same time. I decided to change it to two sessions of three students for the next time because of the results from the first tutoring session. There were too many students in that first session so they were getting distracted and I was not given the opportunity to work with each student individually. I also noticed that the students were on different levels both academically and behaviorally. One of the students I worked with had a very difficult time listening and making the right choices. Although this was frustrating and disrupting to the other students, I did not give up on him. During the next tutoring sessions I worked with him rather than for him and was able to observe where he was getting confused. In order to help break down the concept and look at it in a more comprehensible way, I had him work with the base ten blocks and use them to complete each problem. I believe that this really did work because he asked for the base ten blocks during the rest of the sessions. Although this might not have solved all of his problems, it was beneficial because it enabled him to stay on the right track rather than falling even further behind.
After experiencing the tutoring sessions and the challenges that went along with them, I learned that it is important to help the students take ownership of the learning. I witnessed the students taking charge by following directions, staying on task, asking questions, and even helping their peers. The students should not come into the tutoring sessions thinking that they are there to just get lectured or to play pointless games; they should realize that they are receiving help that will allow them to be able to do the math problems or figure out the concepts on their own. Tutoring also is not just there for the teacher to tell students the answers or make the information any easier. As a tutor, I made sure to have the same expectations I would with any other student. I just approached the content in a different way and provided the students with more one-on-one time so I could understand their thinking process. This enabled me to determine both their strengths and weaknesses so I could work with both.
I have also learned how essential it is to include different strategies and techniques during the tutoring sessions. Using the Mastery Model enabled me to make sure that all of the students were provided with hands-on and engaging activities that helped the students better understand the math concepts. During these activities, the students also interacted with each other and worked with base ten blocks which allowed them to understand the abstract material in a more concrete way. Overall, this model taught me that it is important to structure learning environments in this way whether it is a small group tutoring session or a large group classroom setting. The Mastery Model also helped me to create a more authentic and organized RTI that met each student’s needs. Another positive experience that I gained from the tutoring project was just learning how to communicate with the students on a more personal level. When teaching an entire classroom, it is difficult to make sure that everyone understands the concept and is on track. The tutoring sessions helped me practice this skill and also helped prepare me for working with individual students in a classroom setting.
I have learned that it is crucial to provide the students with opportunities to activate prior knowledge and connect their learning to real life situations because it will help the students see how math can be applied every day. If I asked a student what 22 minus 12 was, then most likely they would be able to find the correct answer, but if I gave them a word problem where they had to figure out that how to use subtraction to get the right answer, most students seemed to get confused. I noticed this difficulty with word problems throughout the tutoring process, and it reminded me of what I should not do. Some teachers have the students remember certain words that could help them figure out if they should add or subtract such as “sum,” “difference,” “how many more,” and so on. However, having the students memorize these words will not help their mathematical thinking where they really understand why they are adding or subtracting. This is why I always made sure that the students used manipulatives when they were trying to determine an answer to a word problem. It was also helpful to have them act out the scenario so they could make a better connection. Not only would I have the students act out the scenario, use the manipulatives, and find the answer, but I would have them tell me why they added or subtracted. Overall, this method seemed to help the students reflect on their own thinking and to practice using their mathematical thinking rather than memorization.
The tutoring project was very rewarding because it allowed me to gain experience working with and assessing diverse learners. Having the opportunity to work with the students also showed me the different perspectives the students had of the concepts. As a tutor and pre-service teacher I believe that it was my duty to help the students believe in themselves by showing them that they have the capability to learn and understand math.
During each tutoring session I discovered new things about myself and the students. I enjoyed the fact that it was a learning experience in both situations and that I was able to reflect on myself as a tutor and a pre-service teacher. Although none of the session was perfect and none went exactly the way I planned, they gave me the opportunity to think on my feet and implement modifications that would best meet the needs of the students. I believe that this is an important skill to develop because I will experience situations such as the tutoring sessions when I have a classroom of my own.
In order to help a student solve a problem, teachers cannot simply demonstrate their mathematical thinking; they have to make sure the students provide evidence of their mathematical thinking as well. Using math journals could be beneficial for future tutoring projects or even for math lessons in the classroom. Not only will math journals allow the students to reflect on their understanding, but it will also allow teachers to better understand the learning process and to reflect on how their students were able to understand the concepts. I would consider the weekly responses as a sort of journal I kept throughout the process. These reflections helped me immensely; therefore, I would assume that the students would benefit from writing responses each week as well.
My final recommendation is to provide students with a survey that asks them questions about their overall experience. If I had used the survey in this project, I could have used the student responses to help me to better reflect on my experiences. I could have asked students to fill out surveys after each session because I might have gotten a more honest response on paper than asking them in person. Also I think it would have been beneficial to have had tutoring sessions with each student individually both at the beginning and at the end of the tutoring project. This would have given me a better understanding of each student’s prior knowledge and of their personal growth. As for future tutoring projects, I would recommend that more time be given to work with the students and that teachers be given fewer students to tutor. Extra time and fewer students would enable tutors to get to know the students better so that the intervention plans could be individualized to fit the needs of each student and, therefore, be more effective.
- Belenky, D. M., & Nokes, T. J. (2009). Examining the role of manipulatives and metacognition on engagement, learning, and transfer. Journal of Problem Solving, 2 (2), 102-129. Retrieved from http://docs.lib.purdue.edu/jps/
- Boggan, M., Harper, S., & Whitmire, A. (2010, June). Using manipulatives to teach elementary mathematics. Journal of Instructional Pedagogies, 3, 31-6. Retrieved from http://www.aabri.com/jip.html
- Douglass, L., & Horstman, A. (2011, Fall). Integrating response to intervention in an inquiry based math classroom. Ohio Journal of School Mathematics, (64), 23-30. Retrieved from http://www.ohioctm.org/ojsm.htm
- Forsten, C. (2004). The problem with word problems. Principal, 84(2), 20-23. Retrieved from http://naesp.org/
- Kostos, K., & Shin, E. (2010). Using math journals to enhance second graders’ communication of mathematical thinking. Early Childhood Education Journal, 38(3), 223-231. doi:10.1007/s10643-010-0390-4
- National Center on Response to Intervention. RTI4SUCCESS.org. Retrieved from http://www.rti4success.org
- Zull, J. E. (2004). The art of changing the brain. Educational Leadership, 62(1), 68-72. Retrieved from http://www.ascd.org/Default.aspx