The Efficacy of the Concept-Rich Instruction with University Pre-Service Teachers in a Tanzanian Context Using Vygotskian Perspective

This paper presents the findings of a pilot study on concept-rich instruction with university pre-service teachers in a Tanzanian context. The concept-rich instruction is an instructional approach which is used to develop students’ understanding of a mathematical concept (Ben-Hur, 2006). I conducted a pilot study to determine the efficacy of the concept-rich instruction to university pre-service teachers in Tanzania using Vygotskian perspective. I used a reflective journal and pre-test questionnaire to collect data while implementing the CRI in a daylong research meeting. After the pilot study, it was found that the concept-rich instruction helped preservice teachers to develop their understanding of a concept taught at schools in different ways, including defining a concept in multiple ways and relating a concept with local materials available in their daily environment. The findings have implications in the teaching and learning of the mathematical concepts that are taught at schools to the university pre-service teachers.


Introduction
I conducted a pilot case study with nine Pre-Service Teachers (PSTs) at one of the public universities in Tanzania in October 2016 to determine the efficacy of the Concept-Rich Instruction (CRI) in a Tanzanian context. CRI is a model of teaching which actively engages learners with a mathematical www.scholink.org/ojs/index.php/wjer World Journal of Educational Research Vol. 6, No. 3, 2019 374 Published by SCHOLINK INC. concept in the class (Ben-Hur, 2006). The model was used to encourage PSTs' participation in learning of a mathematical concept-pi through small group and class discussions to develop their understanding of a concept, because participatory approach was rarely practiced in their university mathematics class.
The reason of using pi as an example to engage PSTs with a concept through CRI is that pi is a concept which is taught in Tanzanian primary and secondary schools. But, a concept is not taught to the PSTs in their university mathematics class, because the university lecturers make assumption that PSTs studied a concept while they were in schools. As the result, they leave no chance for PSTs to explore a concept in their university mathematics class. Vygotsky's (1978) Sociocultural Theory was used in a research meeting to encourage participation in learning by allowing social interactions to take place among PSTs in small groups while learning a concept. The reason is that I believe social interaction plays a major role in learners' acquisition and understanding of knowledge while collaborating with knowledgeable others. The theory was also used in a research meeting by using local resources to engage PSTs with a concept, which were not considered as thinkable resources in teaching and learning of a concept(s) in their university mathematics class. The theory was also used to allow PSTs to reflect on their ongoing learning of a concept in a research meeting.

Description of Concept-Rich Instruction
CRI is a model of teaching which intends to develop learners' understanding of mathematical concepts.
The model engages learners with the concepts by encouraging them to "learn and think mathematically" in small and large groups (Ben-Hur, 2006, p. vii). Despite the fact that the model is among of the instructional approaches, it is a concept-oriented because learners are encouraged to participate in learning while focusing on a concept itself (Ben-Hur, 2006).
CRI involves five components, which all contribute to the students' learning of a mathematical concept.
The five components are meaning, decontextualization, recontextualization, practice and realization.
 Encapsulation conceptual understanding in words and symbols focuses to develop learners' understanding of a concept by defining it and representing the definition(s) of a concept in words and symbols.
 Decontextualization focuses on developing learners' understanding of a concept through experiencing a variety of its applications.
The CRI model emerged when schools and mathematics teachers in the United States of America were looking for improved models of teaching that could be used to actively engage learners in learning mathematics. The major reason for looking at the alternative form of instruction was because mathematics teachers in the United States wanted to achieve first-class status in mathematics education in its schools. Ben-Hur filled the gap by developing CRI model and practicing it in mathematics classes for learners from Grades 6 to 8 in the United States. Later on, the model became popular in mathematics classes with learners in the larger North American context due to its potential to develop learners' understanding of mathematical concepts and facilitate learning achievements (Ben-Hur, 2006).

Creation and Implementation of the Concept-Rich Instructional Activities
I created the concept-rich instructional activities before engaging PSTs with a mathematical concept in a daylong research meeting (see Table 1). The activities focused on meaning, decontextualization, recontextualization, realization and practice of a concept. After that, I implemented the instructional activities in a research meeting. PSTs worked on the activities in small groups followed by group presentations and class discussions. Class discussions were important to elaborate individual and group understandings of a concept through interactions with knowledgeable others. Practice related to a concept was done by PSTs in small groups while working on the activities focused on meaning, decontextualization, recontextualization, and realization of a concept. Practice related to a concept was also done by PSTs while presenting their small group activities in a research meeting. objects? (Think, pair, and share).
How can you find the value of pi using Archimedes approach for better approximation of the value? Please explain your answers (Think, pair, and share).
Recontextualization of a concept Three What concepts do we need to know before and after learning pi?
Please explain your answer(s). Think, pair, and share.
What are the applications of pi in mathematics and other fields such as science? Please explain your answers. Think, pair, and share.

Realization of a concept Three
Where is pi in our daily environment? Please explain your answer(s) Think, pair, and share.
What is the importance of pi? Please explain your answer(s).
Think, pair, and share.
What are the applications of pi in our daily life? Please explain your answer(s). Think, pair, and share.

Concept-Rich Instructional Sessions
The instructional sessions took place on Saturday in the same boardroom in the university. The boardroom provided comfortable space and a large table around which the PSTs were able to sit comfortably and within easy access to each other.
There were three sessions in a daylong meeting. The first session started from 9:30-11:00. The second session started from 11:30-1:30. The third session started from 2:30-4:30. Each session took a duration of two hours. The three sessions were spaced for tea and lunch break. Tea break was 30 minutes while lunch break was 1 hour.

Data Collection Methods
This pilot study was used to gather data before the main investigation in another study, which focused on how CRI model revealed PSTs' expressions of their understanding of a mathematical concept. The data was collected using two methods in a daylong research meeting. The data was collected using an open-ended pre-test questionnaire before conducting a pilot study in a research meeting. Reflective journal was also used to collect data during conducting concept-rich instructional sessions. This means that reflective journal was used in session one, session two, as well as session three in two different ways. A reflection was used to gather information about individual participants' ongoing learning of a concept. A final reflection was used to gather information from the individual participants about what they learned in a research meeting focused on their understood, surprising, unfamiliar and familiar ideas about pi.
There were 7 items in a pre-test questionnaire, which were similar with items in the reflective journal.

Result
The efficacy of the CRI model was seen through analyzing data collected from a pre-test questionnaire and reflective journal using Ben-Hur (2006). I focused on a research question: Can CRI model work to Tanzanian university PSTs for development of their understanding of a mathematical concept through participating in a research meeting? I compared the individual responses in a pre-test questionnaire with that of a reflective journal in each item to create categories. I was able to determine the efficacy of the CRI model to the university PSTs in the Tanzanian context. The efficacy of the CRI was observed in a variety of ways, including the following mathematical ideas in the subsections below.

Meaning of a Concept
CRI helped PSTs to develop multiple ways of defining pi. PSTs defined a concept in different ways, including pi as a ratio of two parameters (circumference and diameter) of a circle, pi as ratio of two measures (circumference and diameter) of a circular object, and pi as the number of diameters required to complete the circumference of the circle or circular object. The responses about the meaning of pi are as described in Table 2 below.

Responses in a Reflection
Pi is the ratio of circumference of circular object to its diameter -2 Pi is the ratio of circumference of a circle to its diameter 1 6 Pi is the number of diameters required to complete the circumference of a circle/circular object -3 From the percent of the PSTs were not aware to give the definition of pi, which is defined as the ratio of circumference of a circle/circular object to its diameter. But after attending the pilot study sessions all PSTs became aware to define pi in multiple ways. Also, for justification, 11 percent of the PSTs in a pre-test questionnaire was able to justify the meaning of pi by writing, "for any circumference of a circle dividing by its diameter, a constant number will be acquired which is approximated equivalent to 3.14".
CRI helped PSTs to develop multiple ways of the meaning of the value of pi. PSTs provided the meaning of the values of pi in the class in a variety of ways, including the value of pi is approximated equivalent to 3.14 for easy mathematical computations, the value of pi is an irrational number which goes to infinity (3.14…), and the value of pi is represented by the symbol π for accurate mathematical computations. The PSTs' responses of the meaning of the value of pi are as described in Table 3 below. It is approximated to 3.14 for easy computation and memorization 1 9 It is irrational number (3.14…) -3 It's symbol, π for accurate mathematical computations -2 From the Table above, the number of responses in both pre-test questionnaire and reflection and the number of PSTs of the pilot study are not the same. Before conducting the session, about 11 percent of the PSTs was able to give the correct value of pi as approximated to 3.14. The remaining about 89 percent of the PSTs responded to this item by writing the value of pi as 3.14 or . Among these 89 percent of the PSTs responded in a pre-test questionnaire by arguing that the value of pi is whereby when taking 22 and divide by 7 the answer is approximated to 3.14. However, after conducting the pilot study, all PSTs responded correctly for the value of pi as approximated equivalent to 3.14. As such, I learned that, there were misconceptions of the value of pi among participants before attending this pilot study. goes around the circumference of a circle or circular object. From these three strategies identified, the first two strategies were identified before conducting the sessions, and the third strategy was identified by the PSTs after attending the pilot study sessions as we can see in Table 4 below. able to respond to this item correctly. But, after conducting the sessions, all PSTs were able to respond correctly to this item in a reflection. In doing so, I learned that the pilot study sessions helped PSTs to identify different strategies for finding the value of pi using real objects/local materials.

Realization of a Concept
CRI helped PSTs to develop their understanding of the importance of pi. PSTs addressed that pi is important in a variety of ways, including pi is used to find the: circumference of a circular object, area of a circular object, volume of 3D objects, and pressure of fluid flowing in a pipe. Table 5 below provides the PSTs' responses about the importance of pi. CRI helped PSTs to realize that pi is not an abstract concept. Pi is connected to real things that we see.
PSTs responded to both pre-test questionnaire and reflection about pi in their daily environment as shown in Table 6 below.

Responses in a Reflection
Pi is found everywhere in circular objects 3 9 From the Table above, the number of responses in both pre-test questionnaire and reflection and the number of the PSTs of the pilot study are not the same. Before conducting the pilot study session, 34 percent of the PSTs were able to respond correctly to this item in a pre-test questionnaire, 11percent of the PSTs did not respond to this item and the remaining 55 percent of the PSTs responded to this item in terms of the applications of pi. But after conducting the pilot study sessions, all PSTs responded correctly to this item in a reflection. As such, I learned that the pilot study session was effective in helping the PSTs identify where pi is found in their daily environment.

Recontextualization of a Concept
CRI helped PSTs to develop many concepts needed to know before learning the concept of pi. PSTs provided a variety of concepts needed to know including circumference of a circular object, diameter of the circular object, meaning and shapes of circular figure, measurement, ratio, and relationship between the circumference and diameter of the circular object. From a pre-test questionnaire and reflection, the following concepts were identified as shown in Table 7.  sessions, all PSTs responded correctly to this item in a reflection by mentioning many concepts compared to the concepts that were mentioned in a pre-test questionnaire. As such, I learned that, after conducting the pilot study sessions, PSTs were able to identify more concepts that we need to know before learning the concept of pi as we can see in the table above. CRI helped PSTs to develop a variety of ways of the applications of pi. PSTs were aware about the applications of pi, including a concept is used to find the: circumference of a circular object, area of a circular object, volume of a circular object, modulus of a circular object and pressure of a fluid flowing in a pipe. Table 8 below gives the PSTs' responses about the applications of pi. It is used to find the circumference of a circular object 4 5 It is used to find the area of a circular object 6 9 It is used to find the volume of a circular object 2 8 It is used to find the modulus of a circular object -1 It is used to find the pressure of a fluid that flows in a pipe -3 From the Table above, the number of responses in both pre-test questionnaire and reflection and the number of the PSTs of the pilot study are not the same. Before conducting the pilot study session, about 22 percent of the PSTs did not respond to this item, 11 percent of the PSTs responded incorrectly to this item and 64 percent of the PSTs responded to this item in a pre-test questionnaire. But, after conducting the pilot study, all PSTs responded correctly to this item in a reflection by mentioning many applications of pi compared to the number of responses in a pre-test questionnaire. As such, I learned that the pilot study sessions were effective in helping the PSTs to identify many applications of pi.
CRI helped PSTs to realize a lot of ideas about pi in the pilot study. PSTs' surprises existed in the daylong meeting focused on pi using CRI model in a final reflection. These surprises include the concept of pi is wide, important, not abstract, and applicable. Individual PSTs raised issues that surprised them after attending the pilot study sessions as we can see in Table 9 below. Table 9. Surprising Ideas about Pi

Categories Number of Responses in a Final Reflection
The concept of pi is very wide than what I knew before attending this session 1 The importance of pi 1 Pi is not an abstract idea; it is a concept where by its value is obtained by taking the ratio of circumference of a circle to its diameter.

5
Applications of pi and its value 1 The more the decimal places in the approximations of the value of pi, the more the accurate is the answer 1 The value of pi is approximated to 3.14 and it is not equal to 3.14 2 The value of pi is an irrational number (3.14…) 3 The value of pi equals to is not valid since there is no apparatus/ tools which can be used to verify this value 1 About 3.14 times the diameter equals to the circumference of a circular object 2 From the from the ratio of two measures (circumference and diameter) of a circular object, can be obtained using Archimedes approach for better approximation of the value, and is not equal . Table 10 below provides the ideas about pi that the PSTs were able to understand after attending the pilot study sessions. The value of pi which is irrational number 4 The value of pi goes to infinity (3.14…) 2 The value of pi is obtained by taking the ratio of the circumference of a circular object to its diameter 7 The concept of Archimedes method for approximation of the value of pi whereby when counting the number of diameters that goes around the circumference of a circle gives the value of pi 2 The value of pi is approximated to 3.14 and it is not equal to 3 For the accurate computations, we better use the symbol for pi which is to represent its value in (mathematical) computations 1 From the Table above, the number of responses in a final reflection and the number of the PSTs of the pilot study are not the same. From this item, I have learned that before attending the pilot study sessions, PSTs thought that pi is 3.14 or only, But after attending this pilot study, they realized a lot of ideas that they were not aware with pi as a concept due to a lot of divergent responses that emerged at the end of the pilot study in a final reflection as we can see in the Table above.
Finally, CRI helped PSTs to realize unfamiliar ideas before attending pilot study sessions. These unfamiliar ideas include using Archimedes approach to better approximate the value of pi, the value of pi is not a fraction ( ), and the value of pi can be obtained by taking the ratio of two measures (circumference and diameter) of a circular object. About 89 percent of the PSTs raised ideas of pi that they were not familiar with them before attending the pilot study sessions as we can see in Table 11 below. However, 11 percent of the PSTs did not respond to this item in a final reflection. object.
ii. The value of pi is not a fraction 2 iii. The more you use the value of pi with many decimal places, the more you get the accurate answer in the computations. Therefore, it is better to use the symbol of pi (π) in the computations involving the use of pi instead of using the approximated value of pi to two decimal places ( 1 iv. Irrationality of pi 1 v. The value of pi is approximated equivalent to 3.14 and it is not equal to 1 vi. The value of pi is obtained by taking the ratio of circumference of a circular object to its diameter.
1 From the Table above, the number of responses in a final reflection and the number of the PSTs of the pilot study are not the same. From these responses among the PSTs, it seems that the PSTs were not aware with a lot of ideas about pi before attending this pilot study. Therefore, from the Table above, I learned a lot of unfamiliar issues about pi from the PSTs, including the following ways. PSTs were not aware that: pi is a concept which is defined as the ratio of circumference of a circle to its diameters; Archimedes method for approximation of the value of pi; the number of diameters required to complete the circumference of a circle is about 3.14 which represents the value of pi; and the value of pi is an irrational number (3.14…).

Validity and Reliability of the Study
The validity of this study was ensured (Creswell, 2014) in the following ways. The first way involved verification and confirmations of the participants' responses. The individual PSTs reviewed their responses for trustworthiness and accuracy and to confirm the dependability of data obtained through a pre-test questionnaire and reflective journal after a research meeting. PSTs were asked to clarify responses, when necessary.
The reliability of this study was ensured based (Creswell, 2014) while focusing on the conducted pilot study and data collection techniques. The findings from the pilot study confirmed that the concept-rich instructional processes were effective, because each participant was asked at the end of the instructional meeting to reflect on the whole day lesson. The recommendations from my pilot study participants ensured the soundness of the created instructional activities.

Discussion
This paper has described the efficacy of the CRI in the Tanzanian context. PSTs were engaged with the concept of pi through CRI model in the pilot study. In doing so, CRI helped PSTs to develop their understanding of pi, such as realizing different ways of defining a concept and multiple ways of finding the value of pi using circular objects. As such, this model is potential to develop PSTs' understanding of the mathematical concepts related to their work of teaching.
Vygotskian perspective also played a conducive learning environment to the university PSTs in a research meeting. First, using local resources in a research meeting was important to develop PSTs understanding of pi. As a result, I propose to use local resources as teaching and learning resources in university mathematics class to help PSTs relate local materials in their academic settings. Also, PSTs might use local resources in their own mathematics classrooms after completion of their university degree program. Second, social interactions among PSTs through collaboration with knowledgeable others in small and class discussions helped to encourage PSTs' participations in learning. Finally, allowing PSTs to reflect on their ongoing learning of a mathematical concept helped individual PSTs to realize what they learned and surprising ideas about pi.

Concluding Thoughts
From the findings of a pilot study, it shows that CRI model was potential for PSTs' conceptual