# Internship Log

**Internship Log**

**Log 1**

The professor used a wide range of teaching methods to teach the prospective teachers. Each student in the classroom taught a 15 minutes lesson. The professor encouraged the students to use manipulative materials when teaching. The students used a wide range of manipulative materials including base ten blocks, fingers, cuisirine rods etc. Using manipulative materials enabled the prospective teachers to overcome challenges they face when learning mathematics and also teaching (Marshall & Paul, 2008). Learning mathematics concepts is a difficult task for many prospective teachers, and this affects their ability to teach mathematics. Prospective teachers require materials to work with in order to master the mathematical concepts. As a result, the manipulative materials enabled them to overcome the problems.

Each prospective teacher taught various things including addition, subtraction, fraction, multiplication to their tables. The base ten blocks and fingers enabled the prospective teachers to deliver the content to their tables and also for the student being taught to comprehend the concepts. The base ten blocks enabled the prospective teachers and students taught to perform addition, subtraction, and other concepts. The base ten blocks are collections of interlocking blocks used to show ones, tens, hundreds and thousands in a solid way. The student being taught learned through play that ten single block make a ten strip, then 10 ten strip make a mat of one hundred and 10 mats of 100 make a thousand blocks. The Base ten blocks enabled students to visualize the meaning of bigger numbers. This improves their understanding of the concept being taught and enables them develop vital skills (Ball, 1988).

**Log 2**

The prospective teachers were supposed to understand the mathematics they planned to teach to their tables. Understanding the mathematics enabled them to select the teaching materials and also deliver the content to the students well. In addition, understanding the mathematics enabled them to determine how the manipulative materials may be utilized to support the development of the mathematics concepts. Additionally, the prospective teachers understood the learner in order to ensure the teaching was successful. They also managed the learning environment well including collection of materials and distribution. The professor encouraged the prospective teachers to think about student thinking. Students come to school with sizeable knowledge. Some knowledge is correct and some not (Marshall & Paul, 2008). The knowledge is based on their daily experiences and what they have been taught in different settings.

The knowledge is organized according to the subject matter including mathematics. The prospective teachers were supposed to understand their students’ pr conception in order to determine the instructions to provide (Van de Walle, Karp & Bay-Williams, 2012). The instruction provided to students depends on students’ preconception. The instructions can promote conceptual growth or add student knowledge if his concept is in line with the concept being taught. Most of the time, teachers teach concepts that are hard for students to learn as the students’ preconceptions are not in line with the concept taught. Thus, the professor encouraged the prospective teachers to understand the student thinking in order to ensure they understood the challenges students faced when learning. The prospective teachers managed to determine whether the student’s preconceptions were in accordance with the concepts they taught. The professor encouraged prospective teaches to use diagnostic method to determine what the students knew about the concept. The diagnostic method enabled the prospective teachers to identify the learners learning processes, concepts that students found it hard or easy to master and the mistakes the students made. For instance, the prospective teachers learned that students learn better through manipulative materials as they do not make a lot of mistakes and understand the concept (Marshall & Paul, 2008)..

**Log 3**

The professor showed students how to teach addition using traditional methods and invented methods. Addition can be a hard concept for learners to grasp. Teachers have used traditional methods to teach students addition, and this has made it difficult for them to understand the concept. Teachers did not use technology or hands on activities to teach students, and this affected grasping of the concept. Traditional education encouraged direct instructions and lecturers. In this case, the teachers provided direct instructions to students or lectured them. Moreover, traditional learning encouraged students to learn through listening and observing. The students listened to the teacher as he taught and observed him as he performed addition of numbers. In addition to that, the instructions were based on textbooks, lectures and person written assignments. The teachers used textbooks to teach students addition. They also expected the students to complete written assignments. The professor demonstrated how teachers teach mathematics using traditional teaching methods by using instruction based on text books and written assignments. For example, the professor taught the students how to add 20+50 using the traditional teaching methods. Using traditional teaching methods to teach addition has led to rote learning and hindered students from acquiring problem solving skills and critical skills. Students do not get an opportunity to practice how to add numbers and enhance their skills. The traditional teaching methods have affected the student academic performance and made learning boring. The students do not enjoy learning as they do not participate in the learning process actively. Instead, they relied on the teacher to learn addition of numbers, and this affected their motivation. The students have a poor attitude and no interest in learning as they do not actively participate in the learning process (Lee, 2011).

**Log 4**

In lesson 4, the professor taught students how to teach mathematics using invented methods. A wide range of methods have been developed to improve teaching of students addition concepts. The invented methods enable students grasp the addition concept and improve their performance. The addition concept can be hard to understand if teachers do not use solid hands on methods to teach students. Students benefit from the use of invented methods including hands on activities. The activities enable students maintain focus and change their attitudes towards mathematics. Using traditional methods to teach students addition affects their attitude as students do not show interest in learning mathematics. Also, the hands on activities enable students develop crucial problem solving skills. The professor used base ten blocks to teach addition to students. Base ten blocks are important in demonstrating what happens when one carriers numbers when adding. A base is a method used to express numbers using place value. Each whole number has a value that is ten times bigger than the number on the right side as base ten is the standard for counting numbers. The professor ensured the prospective teachers understood the base ten blocks. The teacher introduced a unit that represented the ones place value. Then stack 10 units and demonstrated how the equaled a single rod. Each rod presented the tens place value. The professor placed 10 rods side by side which equaled one flat. The flat represented the hundreds place value. The professor allowed the prospective teachers to practice addition of numbers using base ten blocks to be able to teach 2^{nd} to 4^{th} grade children addition later. The professor gave the students two numbers including 16 and 12 to represent using blocks. The students combined the two groups units into a single group. The professor emphasized that the students were adding when they combined the two groups and the outcome was sum (Lee, 2011).

** Log 5**

The professor showed the prospective teachers how to teach subtraction. The professor used small group instructions to teach subtraction. He encouraged the prospective teachers to work in small group and this improved grasping of the subtraction concept. Small group instructions promote variation of teaching methods. It presents a variation in the presentation of materials. Teachers’ present materials according to the instructional processes as teaching methods differ. Also, different students respond differently to a certain educational method. The professor encouraged prospective teachers to engage actively in learning so as to motivate them. The professor used invented methods to motivate students and enable them learn how to teach subtraction. Teaching subtraction to elementary students is vital as it enables them develop number sense by subtracting whole numbers. Also, the students develop confidence in utilizing subtraction as a problem solving skill. Thus, the professor ensured the prospective teachers were capable of teaching subtraction to elementary student by linking their preconception and subtraction concept. In this lesson, the professor taught prospective teachers how to teach subtraction using conceptual models. The models included take away, missing addend and comparison. The prospective teachers learned how to use the take away model of subtraction to teach students (Lee, 2011). The take away model is as follows:

Suppose a quantity b is taken away from the original quantity a with the same units, the remaining quantity is a-b.

Also, the professor taught the teachers how to utilize the comparison take away model. The comparison model enables students determine how larger one number is than the other. The difference refers to the result of the subtraction. The comparison model states that: a-b is how much a is than b. Lastly, teachers learned how to use the missing addend model. The model shows the relationship between addition and subtraction. The model states that:

Suppose one wants to find the whole number z such that z+y=x, then z is the missing addend. The value of z is =x-y. For example, 4+_=11 (Lee, 2011).

** Log 6**

The professor taught the prospective teachers how to use physical models when teaching subtraction to students. The physical models included base ten blocks and colored chips. First, the teachers learned how to use base ten blocks to help students understand the subtraction concept. The professor grouped teachers into several groups in order to demonstrate the concept and enhance their understanding. He assigned the teachers a number to represent using the blocks. For example, the professor assigned the teachers number 23. The teachers represented the number using units only. After that, the students took away 7 units from 23. The professor ensured the prospective teachers understood what they were doing. The teachers knew the results of the subtraction were the difference between the units. The teachers also used rods and units to subtract. The professor showed them how to borrow ten units in order to replace a single rod. In addition, the professor taught the prospective teachers subtraction using colored chips. The colored chips helped teach conceptualize the subtraction of integers. The black chips represented positive integers and red chips negative integers. Dragging a black chip on top of a red chip made the pair disappear showing the sum is 0. This lesson equipped the prospective teachers with knowledge on how to use manipulative materials to teach mathematics especially subtraction. Using the physical models improve the teachers understanding and motivated them to learn and teach mathematics. It also equipped them with skills needed to use manipulative materials to teach students (Lee, 2011).

**Log 7**

Moreover, the professor taught the prospective teachers how to teach multiplication concept to students. The professor taught the teachers how to teach multiplication to their students using a wide range of methods. The teachers understood some of the mistakes that students make when multiplying numbers. Some students fail to multiply numbers correctly because they do not line the numbers up correctly. Some students multiply out the first number and then multiply the second line and add a zero there. The students do this to avoid mixing the numbers. Additionally, the students do this as it helps them keep the numbers inline. This ensures the multiplication is correct. Also, the professor highlighted the challenges that prospective teachers cause when teaching if they do not understand the importance of place values in multiplication. Some prospective teachers do not understand the role of place value in multiplication of numbers.

The teachers use different concepts to refer to place values. The conceptual language they use affects students understanding of the multiplication concepts. Some prospective teachers use “the tens place” to refer to the procedure. They also use add a zero to refer to concepts. This causes confusion among the students as they do not understand the concepts. Some teachers encourage students to use zeros as the place holders. The teachers encourage students to put zeros when multiplying numbers so as to align the numbers and remember. Helping the students understand the place values and mistakes students enabled the teachers to teach elementary students effectively. Also, it equipped the teachers with sufficient knowledge to overcome challenges they faced when teaching elementary students multiplication (Chinnappan, 2012).

** Log 8**

The prospective teachers learned how to use different methods and models to teach multiplication of whole numbers and fractions. Multiplying whole numbers and fractions is hard for many students and hence teachers should use the appropriate method to enable students learn how to multiply. Teachers can use addition and rectangular array method to teach students. In this case, the teacher teaches students how to multiply numbers by adding them. The professor used numbers like 7*6 to illustrate the concept of multiplication through addition. The professor encouraged the teachers to add 7 six times in order to multiply the numbers. Therefore, 7*6=7+7+7+7+7+7. Thus, the teachers learn how to help students show the connection between multiplication and addition.

Moreover, the teachers learned how to use rectangular arrays to assist students visualize multiplication. However, the rectangular array should be based n symbolic interpretation and real life situation. Also, the professor used base ten blocks to show the teachers how to teach multiplication. Teachers can use the commutative property of multiplication when using base ten blocks to multiply numbers. The base ten blocks can be used to multiply numbers by adding them. In this case, one needs to arrange the blocks and then add them to multiply the numbers. Also, one can use the base ten blocks to multiply numbers through rectangular method or area method. The base ten blocks enable students understand the concept and multiply numbers correctly without using zeros as place holders. The professor used base ten blocks to motivate teachers and improve their understanding. The method used to teach students depends on the teacher’s preference (Chinnappan, 2012).

** Log 9**

Further, the professor taught the prospective teachers how to elementary students division. Prospective teachers do not have adequate knowledge in the division, and this has affected student learning. The kind of knowledge the teacher has affects students learning in different easy. Most of the prospective teachers have no content and pedagogical content knowledge needed to teach the topic effectively. Though prospective teachers are capable of representing division of whole numbers, they are unable to represent the division of fractions. They do not make sense of division of fractions. Thus, teachers should be able to teach the concept of division beyond division of whole numbers. They should have sufficient knowledge in division of fractions. The professor identified the causes of division problems. The models used by teachers to teach division affect student learning. The models do not encourage student learning beyond division of whole numbers. Examples of the models used include fair sharing and repeated subtraction. The models differ greatly and are used differently. In fair sharing model, the teachers consider the division of fair sharing. Thus, students assume that the division represents fair sharing of items. The fair sharing model is a traditional teaching model used to teach division of whole numbers. The model prevents division of fractions. The fractions cannot be represented using the fair sharing model as the assumptions and results are invalid. For example, fair sharing cannot be used to find the value of 48/1/6 as one cannot share an item among a quarter people. However, it can be used in whole numbers like 48/6. In this case, the student assumes that 48 sweets will be shared between 6 students. So, how many sweets does each student get? (Rivzi & Lawson, 2007).

**Log10**

The prospective teachers learned how to use the repeated subtraction model and base ten blocks to teach division. The professor used another example to illustrate how to use the model. The prospective teachers were supposed to divide 42/6. In order to find the answer, the teachers asked themselves how many times can 4 be subtracted from 48? The professor discussed with the prospective teachers the problem. The model can be used to divide some fractions, but it is difficult to use the model to divide fractions where the divisor is greater than the dividend. For instance, dividing 1/3 and ½. The fractions cannot be subtracted, and this makes it hard to divide the fractions. In addition to that, the prospective teachers learned how to divide expressions using the rate and ratio model. The teachers represented the expression as a rectangular ray as evidenced in the Vergnard schematic diagram. The teacher and professor together found out a number that had similar multiplicative relationship with as the relationship between the divisor and dividend. For instance, the teachers and professor calculated 4 divided by ½ using the rate and ratio model giving 12. Thus, the prospective teachers acquired sufficient knowledge to help them teach division of fractions. Finally, the teachers learned how to use base ten blocks to teach division. The professor grouped the students into various groups which wired together to solve the problem. The teachers solved 278/4 using the blocks. The base ten blocks gave the teachers hands on activity to learn division (Rivzi & Lawson, 2007).

**Log 11**

The professor taught the prospective teacher how to teach fractions. Students find it hard to comprehend fractions, and this affects their performance. Teachers should have enough knowledge on fractions in order to teach students how to calculate fractions. The professor used hands on activities, and different models to equip the teachers with adequate knowledge to teach fractions. Teachers use different strategies when teaching fractions. Students should under sand fractions well to use them to solve problems. However, teachers and students struggle to understand fractions. Students struggle to understand fractions and teachers feel frustrated as they look for ways to teach fractions to students effectively. The professor examined rules that teachers use to teach fractions. Teachers encourage students to add the numerator and utilize the same denominator. Also, they encourage students to determine the smallest common multiple of the denominators in order to get the common denominator. The students change the addends to have a common denominator. Then they add the numerators. Moreover, students multiply the denominator and numerator with a similar number. They also find the largest common divisor of the denominator and numerator and divide the two using the common divisor. The students multiply the denominator and the numerator. Lastly, they can find the reciprocal of the denominator and using it to multiply. The methods have not been effective as students do not understand the fraction concept due to lack of hands on activities. The students do not understand the denominator and numerator. As a result, the professor proposed other methods to teach fractions (Wu, 2011).

**Log 12**

The professor taught the teachers how to use manipulates and visual models or pictures to teach fractions. Teachers should teach students to visualize fractions and do simple operations using the visual images instead of rules. The fractions become concrete when students visualize them in their minds. Students estimate the answers before calculating the fractions. They also assess reasonable the final result and do most of the simple operations in their mind. Teachers can use different visual models to teach fractions. They can use objects like rectangles and circles to help students solve fractions. The professor used group discussion to teach the teachers fractions. The teachers used different shapes to express different fractions. Each group was assigned either a rectangle of circle. The groups drew the objects and represented their results. Also, the professor drew the rectangles and circles on the whiteboard ands asked students to express the fractions. The rectangles and squares represented a single unit. The students divided the rectangles and squares into several parts. Then used them to solve the fractions. The divisions represented a certain section of the whole unit. Using rectangles and circles enabled the teachers to understand the concept of fractions and also to obtain knowledge. The lesson prepared the prospective teachers to teach fractions effectively by creating visual images of the fractions. It also motivated the prospective teachers to learn fractions and teach fractions. The teachers understood that a fraction 1/x is the quantity formed by 1 part when a whole is divided into x parts, and y/x is formed by y parts of 1/x (Wu, 2011).

** Log 13**

The professor examined the use of base ten blocks to teach fractions. Using base ten to teach fractions improves students understanding. Students get an opportunity to use hands on activities to understand fractions. Hand on activities motivates students to learn fraction by overcoming the problems they face when learning fractions. The traditional methods used to teach fractions have affected grasping of the concept and student understanding. The professor used different teaching strategies to teach fractions instead of lecturing. He used group discussions and activities to teach the concept. He explained to the respective teachers how they can base ten blocks to calculate fractions. The professor divided the class into two groups and then counted the number of flat base ten blocks in the class. The professor showed the teachers how the numbers became the denominator of the fraction. The professor divided the flat base ten blocks between the groups. Each group counted the blocks and represented the fraction. The groups indicated the number of blocks they had in total. For instance, if one group had 8 and the total flats 20 the fraction would be 8/20. Thus, the prospective teachers benefited a lot from this lesson as they understood the concept of fractions and how they could apply the concept in future (Wu, 2011).

** Log 14**

The professor did not use group discussion or hands on activities to teach the prospective teachers. He employed a different method that enabled the teachers to acquire knowledge and skills. The prospective teachers attended an online class. The prospective teachers read the fraction body article and write their reflection. The article examined various things about fractions. The article discussed traditional methods and modern methods used to teach fractions. The teachers understood fractions in details. They learned how to solve fractions using base ten blocks and visual models. Allowing teachers to read the article and write a reflection equipped them with various skills. The teachers acquired problem solving skills, critical thinking and analytical skills. The teachers developed problem solving skill by learning how to solve fractions or use fraction to solve problems in real life. In addition, the teachers developed analytical and critical thinking skills. The teachers learned how to examined content critically and analyze it in order to understand the concept. Reading the article helped teachers learn how to be independent and read mathematical concepts independently. Also, the teachers learned how to reason and solve problems independently. Reflection improved the teacher’s memory as they were able to remember the content they read. Reflection improves mastering of the content as students read the material and then reflect on it. Reflection enables teachers to identify their weaknesses and strengths (Van de Walle, Karp & Bay-Williams, 2012).

** Log 16**

The prospective teachers were supposed to teach grade 2 students the mathematical concepts. The teachers taught multiplication, addition, subtraction, division and fractions. The professor required the teachers to practice what they had been taught in class before going to the field. The professor observed the prospective teachers as they taught second grade students and corrected them. The professor identified the mistakes the teachers made while teaching and corrected them. The observations made by the professor are important as they help me teach prospective teachers a methodology class. The observations made provide sufficient information about teaching second grade students. They provide information about the challenges prospective teachers’ face when teaching second grade students. Prospective teachers face various problems when teaching second grade students including lack of sufficient knowledge on the mathematical concepts. Also, the teachers do not know how to use manipulates to teach students, but instead use traditional methods of education.

The observations enable me plan the methodology classes. The information facilitates the development of instructions and identification of teaching strategies. The instructions and strategies are developed based on the needs of the prospective teachers. The strategies and instructions ensure the teachers acquire knowledge and skills needed to teach the teachers. For instance, I am able to determine the effect traditional teaching methods have on teaching mathematical concepts. This in turn, enables me to select the best teaching methods to teach the methodology classes. Further, the information improves provision of instructions to prospective teachers and ensures they are well prepared. Therefore, the information will transform teaching and preparation of prospective teachers to teach second grade students (Van de Walle, Karp & Bay-Williams, 2012).

** Log 17**

The professor observed prospective teachers as they taught third grade students. The professor identified the mistakes the teachers committed while teaching. The information from the evaluation process is important to me as it will help in teaching methodology classes to prospective teachers. Teaching third grade students is hard for many prospective teachers. Most of the prospective teachers struggle to teach third grade students, and this affects student’s performance. The teachers struggle because of lack of enough knowledge on the mathematical concepts. The teachers do not have enough knowledge on some of the concepts taught to third grade teachers. For example, the teachers do not have knowledge in fractions, division, multiplication etc.

Teachers should understand the concepts well to deliver instructions effectively to the students. Also, the prospective teachers do not know the student thinking and this affects learning and teaching. It affects students’ grades as they do not get appropriate instructions. Prospective teachers should understand their third grade students. They should know the preconception of the students about mathematical concepts. Understanding students’ preconception helps develop instructions that improve student understanding. The information will help teach prospective teachers how to understand their student thinking and its importance. Understanding the information will enable me teach the prospective teachers methodology classes and ensure they understand the concepts. This will transform teaching and learning among third grade students. The students will understand the mathematical concepts well and use them to solve real life issues (Van de Walle, Karp & Bay-Williams, 2012).

**Log 18**

The professor also observed the prospective teaches as they taught fourth grade students. Observing the prospective teachers enabled the professor identify the weaknesses and strengths of the teachers. Fourth grade students experience problems when learning mathematical concepts. The students do not understand some of the mathematical concepts including fraction, multiplication and division. Also, the students do not remember the mathematical concepts. Prospective teachers should understand the students to provide the right instructions. They should understand the student weaknesses and strengths. Observing the prospective teachers as they teach reveals the problems the teachers face when using invented methods to teach students. Prospective teachers rely on textbooks and other traditional methods to teach students. Thus, the teachers find it hard to use the invented methods to teach students.

Teachers should have knowledge of the invented methods to teach mathematical concepts using the methods. They should be able to use different types of manipulative. For example, they should be able to use the base ten blocks when teaching. Moreover, the teachers should understand place values and their importance in multiplication, division etc. Understanding the place values will eliminate confusion among the teachers and students. However, most teachers do not understand place values. The information provided will be useful to me when teaching methodology classes to prospective teachers. I will ensure the teachers clearly understand the place value by using the base ten blocks to teach them. The teachers will also apply the place value when multiplying numbers to avoid confusion and understand the concept. In addition, I will ensure the teachers learn how to use invented methods to teach mathematical concepts. Lastly, I will ensure the teachers think about their student thinking while teaching to ensure the instructions are effective and enhance their performance (Van de Walle, Karp & Bay-Williams, 2012).

**Reference**

Ball, D.L. (1988). Knowledge and reasoning in mathematical pedagogy.http://www-personal.umich.edu/~dball/books/DBall_dissertation.pdfon 22/10/2012

Chinnappan, M. (2012). Prospective teacher’s representation of multiplication. Retrieved from http://www.merga.net.au/documents/RP242005.pdf on22/10/2012

Lee, D. (2011). The implication of number sense on the mastery of addition and subtraction concept. Retrieved from on http://scimath.unl.edu/MIM/files/research/Lee_AR_FINALDraft_LA.pdf on 22/10/2012

Marshall, L., & Paul, S. (2008). Exploring the use of mathematics manipulative materials. EDU. COM international conference

Rivzi, N.F., & Lawson, M.J. (2007). Prospective teachers knowledge. Concept of division. International education journal, 8(2), p377-392

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2012). Elementary and Middle School Mathematics: Teaching Developmentally. (8^{th}ed). Pearson

Wu, H. (2011). Teaching fractions according to the common core standards. Retrieved from http://math.berkeley.edu/~wu/CCSS-Fractions.pdf on 22/10/2012

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