An Investigation of Prospective Mathematics Teachers’ Knowledge of Basic Algorithms with Whole Numbers: A Case of Turkey

The aim of this qualitative case study is to investigate prospective mathematics teachers’ subject matter knowledge of the underlying concepts of standard and nonstandard algorithms used to solve the problems with whole numbers. Twenty three prospective mathematics teachers enrolled in the Elementary Mathematics Education Program of one of the most successful universities in Turkey were the participants of the study. The data was collected through four tasks containing basic algorit hms. More specifically, the Ones Task assessed participants’ understanding of the underlying place value concepts of standard algorithms. The Andrew Task and the Doubling Task required participants to conceptualize and interpret nonstandard strategies. In the Div ision Task, participants were expected to provide in-depth explanation for the difference between multiplication and division and between partitive division and measurement division. The content analysis method was used to analyze the data. The results of the study revealed that more than half of the prospective mathematics teachers had knowledge about the place value of 1 in addition and subtraction, and also multiplication. However, most of the prospective teachers could not explain the underlying principl e and the meaning of the nonstandard algorithm in subtraction. Similar to their knowledge on subtraction, prospective teachers’ knowledge on division was limited.


Introduction
In the last few years, there has been an increasing interest in knowledge needed for effective teachi ng.Many researchers hav e focus ed on inv estigating the answer to one of the constantly evolving ques tions related to knowledge needed for effective teaching.The central focus of these ques tions is related to the knowl edge needed about a subject which is to be taught and the way of teaching it (Grossman, 1990;Ma, 1999).The questions have been tried to be answered by considering them from different perspectives (Ball, Thames & Phelps, 2008;Gess-Newsome, 1999;Grossman, 1990;Rowland, Huckstep & Thwaites, 2005;Shulman, 1986;1987).Shulman (1986), one of the pioneers of teachers' knowledge, i dentified pedagogical content knowl edge (PCK), curricul ar knowledg e (CK) and subject matter knowledge (SM K) as knowledge needed for effective teaching.Shulman described PCK as the combination of content knowledge and pedagogical knowledge.According to him, CK is the knowl edge of the topics, learni ng outcomes, activities, books, and instructional materials in a s pecific curriculum.It is also related to knowledge of the organization of the topics and the relationship of the topics in a specific grade level with other disciplines.Lastly, Shulman defined SMK as knowi ng the facts, principles, and issues, knowing why thes e facts, pri nciples, and issues are taught, and pres enting jus tifications for mathematical explanations and procedures.Shulman emphasized the importance of SMK by stati ng that "The teacher need not only und ers tand that something is so; the teacher must further understand why it is so" (p.9) Building on Shulman's framework, Ball, Thames and Phelps (2008) propos ed an extended and a more detailed framework called Mathematical Knowledge for Teaching (MKT).In th eir framework, they divided each category (SMK and PCK) into three sub-categories.In order to refer to general mathematical knowledge that is not s pecific to mathematics teachers, they proposed Common Content Knowledge.The second category, Specialized Co ntent Knowledge (SCK), is mathematical knowledge unique to teachers.As the last category, Horizon Content Knowledge is defined as "an awareness of how mathematical topics are related over the span of mathematics included in the curriculum" (Ball et al., 2008, p.403).On the other hand, Shulman's PCK is speci alized into knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of content and curriculum (KCC).KCS is described as the integration of knowledge about mathematics and knowledge about students, whereas KCT is the combination of knowledge about teachi ng and knowledge about mathematics.Las tly, KCC refers to the i ntegration of knowledge about content and knowledge about curriculum.Furthermore, Rowl and, Huckstep and Thwaites (2005) generated the "Knowledge Quartet" framework for mathematical knowledge for teaching.The first category, foundation, is related to pedagogy and us age of mathematical terminology, and the second category, transformation, is the knowledge needed to plan teaching.The thi rd category, connection, on the other hand, concerns the coherence of the planning and teaching and the sequencing of topics of instruction within and between lessons.Lastly, contingency category refers to unexpected clas sroom events.
In relation to the knowledge types that the teachers should have to teach the subject in an effective way, Hiebert and Lefevre (1986) defi ned conceptual and procedural knowledge.These knowl edge types have served as a widely used framework to examine teachers' knowledge.Conceptual knowledge is described as the "knowledge that is rich in relationships" (p.3), while procedural knowl edge is defined as the "rules or procedures for solving mathematical problems" (p.7).Put differently, conceptual knowledge involves understanding of basic arithmetic facts, mathematical ideas, and procedures by being able to interpret and perform them correctly in different situations.However, procedural knowledge includes solving a problem via procedures, rules, formul as, algorithms and symbols (Engelbrecht, Harding & Potgieter, 2005;Rittle-Johnson, Siegler & Alibali, 2001).
Among the frameworks for teacher knowledge, the present study was grounded on Shulman's SMK.Because of i ts importance for high quality teaching, many research studies were conducted to investigate teachers' SMK related to various mathematics topics such as functions, division, fractions, and probability (Ball, 1990;Ev en, 1993;Isiksal, 2006;Lucus, 2006;Nilsson & Lindstrom, 2012;Philipp, Schappelle, Siegfri ed, Jacobs & Lamb, 2008).Among these studies, Philipp et al. (2008) investigated the effect of a professional development program on K-3 teachers' mathematical content knowledge of basic algorithms with whole numbers.They concluded that the prospective teachers had procedural knowledge related to basic algorithms with whole numbers.In other words, the prospective teachers had a tendency to calculate basic algorithms rather than conceptualize them.As basic algori thms with whole numbers is an important subject to teach and learn mathematics conceptually (Van de Walle, 2003), teachers need to be able to interpret and explai n the underlying meaning of the algorithms to empower students mathematically (Thanheiser, 2009).In order to determine whether teachers' mathematical content knowledge of basic algorithms with whole numbers is sufficient to teach the subject effectively, it is significant to explore teachers' knowledge of basic algori thms with whole numbers.As prospective teachers are the future teachers, studies conducted with them have a vital role in mathematics education research.For these reasons, teachers' knowledge of recognizing and expl aining the underlying concepts, knowledge of applying nonstandard algorithms, knowledge of reasoning students' solution strategy, and knowledge of recognizing the probl em structure are the focus of the current study.Within the scope of this focus, the aim of this study is to explore subject matter knowledge of prospective mathematics teachers.The study addresses the following research questions: 1) What is the nature of the prospective mathematics teachers' subject matter knowledge of the underlying concepts of standard algorithms used to solve the problems with whole numbers?
2) What is the nature of the prospective mathematics teachers' subject matter knowledge of the underlying concepts of nonstandard algorithms used to solve the problems with whole numbers?
3) What is the nature of the prospective mathematics teachers' s ubject matter knowl ed ge of the measurement-division problem?

Design of the Study
As this research study aims to gain a deeper insight into Turkish prospective mathematics teachers' subject matter knowledge, the most appropriate research method was a qualitative case s tudy desi gn.Creswell (2007) stated that the purposes of case studies are to analyze a case or mul tiple cases within a bounded system and to develop a richer and deeper description about the cases.The cas es were prospective mathematics teachers whos e boundaries were being in the fourth year of the teacher education program in one of the most successful universities in Turkey.

Participants
In order to achi eve the aim of the s tudy, the mos t appropriate method for selecti ng participants was purposive sampling.The important criteria for s electing the participants were to select the most knowledgeable and accessible prospective teachers.Based on thes e criteria, 23 prospective middle school mathematics teachers enrolled in the Middle School Mathematics Education program of a state university located in Ankara, Turkey were selected as the participants of the study.The students attending this program take mathematics cours es (e.g.Calculus, Discrete Mathematics, and Analytic Geometry), mathematics education cours es (e.g.Instructional Principles and Methods, and Nature of Mathematical Knowledge for Teaching), and education courses (e. g.Educational Psychology, and Turkish Educational System and School Management).They mostly take mathematics courses in the firs t 2 years and the mathematics education courses in the subsequent years.At the time of data collection, the participants had already taken Methods of Teaching Mathematics I-II and Nature of Mathematical Knowl edge for Teaching.Students taki ng the Methods of Teaching I-II course can understand the basic concepts rel ated to school mathematics and recognize connections among mathematical ideas in the mi ddle school mathematics curriculum, analyze students' misconceptions related to mathematics topics, and design and perform lesson plans and activities for mathematics instruction using different teaching strategies.Furthermore, students completing the Nature of Mathematical Knowledge for Teaching course can distinguish between the types of mathematical know l edge, describe the misconceptions that students may have and generate strategi es to overcome these misconceptions, and evaluate their conceptual and procedural knowl edge by explaini ng their underlying reasoning.The boundary of the cas es, who are prospective mathematics teachers, is that they have taken thes e courses and they are senior students in the Middle School Mathematics Education program of one of the most successful public universities in Turkey.

Data Collection Tools
In this study, to explore prospective mathematics teachers' subject matter knowledge related to basic algorithms with whole numbers, four tasks given in the Appendix were used.These tasks, prepared by Philipp, Schappelle, Siegfried, Jacobs, and Lamb (2008), were applied after receiving permission from the authors.Three tasks were related to algori thms and one task was designed to evaluate whether participants recognize probl em structure.More specifically, the Ones Task assessed participants' unders tanding of the underlying place value concepts of the standard al gorithms.Particularly, the aim of the task was to define 1 in addition as regrouping 10 ones into one group of ten, and in subtraction as regrouping one hundred into 10 groups of ten.In contrast to the Ones Task, the Andrew Task and the Doubling Task requi red participants to conceptualize and interpret nonstandard strategies.In the Division Task, participants were expected to provide an in -depth explanation for the difference between multiplication and division and between partitive and measurement division.The participants were asked to expl ain their answer to the tasks in detail to gain an in-depth understanding of their knowledge of basic operations with whole numbers.

Data Analysis
In the data analysis phase, content analysis defined by Strauss and Corbin (1990) was applied.In the first stage of the data analysis, participants' written respons es for each task were examined and coded by two researchers.The data gathered from four tasks were coded separately.This first analysis resulted in combini ng some codes together, removing some of them, or adding some others to the list.At the end of this stage, categories were generated for each task.In the second stage of the analysis, the frequency tables were formed to reveal the number of participants in each category.The final lists of categories and the frequencies are presented in Table 1, Table 2, Table 3 and Table 4.To establish inter-rater reliability, each task was coded by two independent researchers who are exper t in mathematics education.The i nter-rater reliability was calculated and a 90 percent correlation was found between ratings.

Findings
The findi ngs were presented in four parts.In each part, prospective teachers' knowledge of basic algori thms with whole numbers was analyzed based on each task.

The Ones Task
The Ones Task involved standard al gori thm in addition (Problem A) and in subtraction (Problem B) and the participants were required to explain the value of 1 in both problems.Also, they clarified the reason for adding 1 to 5 in Probl em A (Appendi x) and the reason for adding 10 to 2 in Problem B (Appendix).This content i tem was evaluated on a 4-point rubric (0-3 scal e).The score of 0 reflects no explanation regarding place value, while the score of 1 repres ents expl anation for place value with addi tion, not subtraction.Moreover, the score of 2 reflects making an explanation for place value for both problems, but not explaining the reasons for addi ng 1 in the addition problem and adding 10 in the subtraction problem.Finally, the score of 3 represents explaining both the place value for both problems and the reasons for adding 1 and 10 in the addi tion and subtraction problems, respectively.The scores, the expl anation of the place value of 1, the expl anation of the reasons for adding 1 and 10 and the number of participants for each score are presented in Table 1.3,4,5,6,8,9,10,13,17,18,19,20,21,22,23 Among the 23 prospective mathematics teachers, 3 pros pective teachers (13%) (P14, P15 and P16) did not mention the place v alue of the 1 i n each problem.For this reason, they could not explain why i n the addition problem, 1 is added to 5, but in the subtraction problem 10 is added to 2. The examples from their explanations are given below: Participant 14: "The value of 1 is different since 297 and 395 are not the same." Participant 16: "The value is the same since both represent 10. " One prospective teacher (P11) explained the pl ace value of 1 in the addition problem; howev er, she did not clarify its value in the subtraction problem.Similar to P14, P 15 and P16, she could not expl ain the reason for adding 1 and 10 in the addition and subtraction problems, respectively.The explanation of P11 is presented below: "In Problem A, we get 10 ones when we add 9 and 8.The 10 ones equal to 1 tens.Fo r this reason, we add 1 to the tens digit." On the other hand, three prospective teachers (13%) explained that the place value of 1 in each problem represents a different amount.Howev er, they could not clarify correctly, why their amount is different in each probl em.As an example, P12's explanation is given below: "1 doesn't represent the same amount.In here, we decomposed the numbers.In Problem A, 1 represents 10 and we add 1 tens to tens place.However, in Problem B, 1 represents 100 and we add 10 tens to tens digit." As it can be seen, his explanation related to the value of 1 is correct for both the addition and subtraction problem.The following excerpt gives his explanation regarding the reason for adding 1 in the addition probl em and 10 in the subtraction problem.
"In addition, the operations start fro m the ones digit and 10 ones equals to 1 tens.This is added to the tens digit.In subtrac tion, if we don't have enough number, we should deco mpose the numbers.Thus, 10 is added to 2." Regarding the addition problem, P12 explai ned the reason for adding 1; however, he could not clarify why 10 is added to 2 in subtraction.Although he knows that he should decompose the numbers, he does not know how he will decompose it.Therefore, his explanation was missing.
Most of the prospective mathematics teachers (69.6%) expl ained the place value of 1 in the addition and subtraction problem, and why 1 is added to 5 in the addi tion problem, but 10 is added to 2 in the subtraction problem.Among 16 prospective teachers, the explanation of participant 23 is given below as an example.

P23:
They are not in the same amount.Terry solved Problem A by considering that 10 ones is 1 ten.However, in Problem B, he thought that 1 hundred equals to 10 tens.In other words, in Problem A, the addition of 9 and 8 equals to 17 and 17 ones equals to 1 ten and 7 ones.Now, Terry got 1 ten and added it in tens.In problem B, 400 indicates 4 hundreds.He took 1 hundred from 4 hundreds.This means that he took 10 tens.Then he added 10 tens to the tens.
Her explanation regarding the reasons for adding 1 and 10 i n the addition and subtraction problems respectively is as follows: In Problem A, the student added 1 tens to 5 in the tens digit because 10 ones equals to 1 tens.But, in Problem B, Terry added 1 hundred to the tens digit because 10 tens equals to 1 hundred.Since she has already had 2 tens, she got 12 tens.
The remai ning 15 prospective teachers agreed wi th P23, and their explanations were similar.To summarize, more than half of the prospective teachers adequately explained the place value of 1 and the reasons for adding 1 and 10 in t he addition and subtraction problems, respectively.

The Andrew Task
Contrary to the Ones Task, the Andrew Task presented in Appendix is related to nonstandard algori thm i n subtraction.Prospective teachers were asked to explain the reasoning behind Andrew's strategy and to solve 432 -162= ◻ using Andrew's strategy.This content item was assessed on a 5 -poi nt rubric (0-4 scale) with scores of 0 and 1 reflecting no expl anation rel ated to place value, scores of 2 and 3 reflecti ng missing or incorrect explanation of place value and lastly score 4 reflecti ng correct explanation of the place value.The scores 0 and 1 differ from each other in application.Score 0 represents incorrect application, while score 1 repres ents correct application of the Andrew's strategy to solve 432-162.Similarly, score 2 differs from score 3 in terms of the correctness of the application.The scores, the correctness of both explanation and application, and the number of participants for each score are presented in Table 2.As it can be seen in Table 2, although three prospective teachers could not explain Andrew's strategy, they applied the same strategy to solve different subtraction algorithms.Apart from thes e three prospective teachers, three prospective teachers' explanations were missing or incorrect and thei r application of the strategy was also incorrect.As an exampl e, the explanation and the application of P9 was pres ented below.Other two prospective teachers (P4 and P 14) explained and applied Andrew's strategy in a similar way.

P9:
Andrew stated that minuend and subtrahend have the same uni ts so he subtracted 63 fro m 23.Thus, he got 40.However, he decreased subtrahend as 2, so he should subtract 2 from 40.Then, he got 40 -2=38.
Although P9 did not mention the place value, his explanation of the algorithm was correct.Based on his explanation, it cannot be determi ned whether he had pl ace value knowledge or not.In addition, his explanation related to the application of Andrew's strategy was not clear.He did not clarify his algorithm with res pect to place value knowledge.
On the contrary, he reported that he subtracted 132 from 432 to make the subtraction easier while applying Andrew's strategy.His explanation was given in the following excerpt.
Furthermore, ten prospective teachers' explanations were coded as missing or incorrect explanation; however, they applied the Andrew's strategy correctly.The explanation and application of P13 were given below.

P13:
He generalized the "subtract smaller fro m larger" expression.It makes sense since they learn to subtrac t smaller from larger.
His application of Andrew's strategy is as follows: As it can be seen from the explanation of P13, he bas ed subtraction on the "s ubtract smaller from larger" expression.
While explaini ng the reasoning behind the Andrew's strategy, he did not mention place value.His application was in contrast to his explanation because while he was applying the Andrew's strategy to solve 432 -162, he subtracted 30-60 and found -30.Thus, he did not subtract the smaller from the larger.Other nine prospective teachers explained the reasoning behind the Andrew's strategy like P13.
Among 23 prospective teachers, seven of them explained the Andrew's strategy correctly and applied the strategy to solve a subtraction algorithm.An example regarding this type of explanation and application is presented below.

P5:
Firstly, Andrew works on ones, 3-5=-2.Then, Andrew works on tens, 60-20=40.Finally, -2+40= 38 P5's application of the reasoning behind Andrew's strategy is as follows: As understood from the excerpts of P5, she clarified the Andrew's strategy based on place value.Other six participants expl ained the strategy in a similar vein.In the light of thei r explanation, it can be concluded that thes e prospective teachers have knowledge about place value.
As a resul t, 13% of the participants hav e no idea related to the Andrew's s trategy, but they could apply it to solve another subtraction probl em.Moreover, 56.5% of the prospective teachers tried to explain the reasoning behind the strategy; however, their explanations were missing or incorrect.Among these prospective teachers, 13% of them could not apply the Andrew's strategy to subtract 132 from 432, 43.5% of them applied the strategy successfully.On the other hand, 30.5% of the prospective teachers' explanation and application of the Andrew's strategy were correct.

The Doubling Task
Similar to the Andrew Task, the Doubling Task requires participants to conceptualize and interpret nonstandard algori thm.In this task, Todd, a third grade, solved the multiplication problem using a different strategy.Prospective teachers were asked to solve 33x19= using Todd's strategy.This content item was assessed on a 4 -poi nt rubric (0-3 scale).The scores, the explanation of each score, and the number of participants under each score are presented in Table 3.  2,3,4,5,6,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23 Table 3 shows that of the 23 prospective teachers, one of them (P16) used Todd's strategy to solve 33x19; however, his calculation was missing.He doubled and labeled correctly for 2, 4, 8, but he stopped doubling after 8. His solution is presented below.
Moreover, one participant (P7) solved the problem not using the doubling strategy, but her strategy was correct both calculationally and conceptually.Instead of using the doubling strategy, she used the triple strategy.
Apart from P7 and P16, the remaini ng prospective teachers used Todd's strategy to find the result of the 33x19.Some of them doubled based on 33, and the remaining used 19 while doubling.Doubling strategies are presented below.

P3:
P19: As it can be seen from the exampl es, both calculations were correct calculationally and conceptually.It can be concluded that for the Doubling Task, nearly all the participants conceptualized the nonstandard algori thm and applied it to another operation.

The Division Task
In the Division Task, participants were expected to provide an in-depth explanation for the difference between multiplication and division and between partitive and measurement division.In order to determine prospective teachers' knowledge of division, a 3-point rubric was prepared.The rubric, pres ented in Table 4, repres ents participants' understanding of the distinction between multiplication, division, partitive and measurement division.2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 16, 17, 19, 20, 23 1 Understanding of distinctions between multiplication and division, but no distinction between partitive and measurement division 4 (17.4 %) 6, 8, 13, 18 2 Understanding of distinctions between multiplication, and partitive and measurement division 3 (13 %) 10, 21, 22 According to Table 4, most of the prospective teachers (69.6%) did not understand the distinction between multiplication, and partitive and measurement division.All of these prospective teachers solved the problems correctly, but they did not explain their reasoning behind thei r solutions.Howev er, four prospective teachers (17.4%) provided evidence for their understanding of the distinctions between multiplication and division.They stated that the problems in Part 1 and in Part 3 were division problems, whereas the problem i n Part 2 was a multiplication problem.Among the problems in Part 1 and in Part 3, they did not address any distinction between partitive and measurement division.The number of prospective teachers who understood the distinctions between multiplication, parti tive and measurement division is three (13%).As an example, the explanation of P10 is provided below.

P10:
Because in the question [in Part 1],the total number of set is given, but the size of the set is not given.This is partitive division.[The question in Part 2 is] multiplication problem since the total number is asked.
[The question in Part 3], the size of the set is given and the total number of set is asked.Thus, it is a measurement division.
As it can be seen from Table 4, prospective teachers had difficulty in discriminating between multiplication, and partitive and measurement division.Although all of them solved the problems successfully, they did not know the reasoning behind using multiplication and division.

Discussion and Conclusion
The purpose of this study was to inves tigate pre-s ervice mathematics teachers' subject matter knowledge of the underlying concepts of standard and non-s tandard al gorithms used to solve the problems with whole numbers.In line with this purpose, the tasks related to each algorithm were presented to the pros pective teachers and they were asked to give detail about the answers students gave i n the tasks.The resul ts of the study showed that the majority of the prospective students could explain the value of 1 in the addition and subtraction problems and the reasons for adding 1 and 10 in the addi tion and subtraction problems, respectively.However, a few prospective teachers did not know the place value of 1 and the reasons for addi ng 1 i n the addition and 10 i n the subtraction problem.As previously stated, SMK is defined as knowing the facts, pri nciples, and issues, knowing why these facts, principles, and issues are taught, and presenting justifications for mathematical explanations and procedures (Shulman, 1986).As most of the prospective teachers could expl ain the underlying concepts and meanings of standard algorithms, it could be concluded that the prospective mathematics teachers' subject matter knowl edge was adequate to help the students learn and apply the standard algori thms to solve the problems with whole numbers.Furthermore, Hiebert and Lefev re (1986) stated that a teacher who understands the basic arithmetic facts has conceptual knowledge.In this sens e, it could be concluded that in the present study the prospective teachers' conceptual knowledge was adequate to interpret the procedures and to perform them correctly in different situations.This result is in contradiction with the resul ts of the earlier studies (Ma, 1999;Philipp et al., 2008) which found that prospective teachers struggle to explain the al gorithms.
For instance, Philipp et al. (2008) found that prospective teachers had difficulty in understandi ng standard multi -digit addi tion and subtraction.Furthermore, they emphasized that teachers' performance on s tandard multi-digit addition and subtraction was worse than non-s tandard subtraction and multiplication.The prospective teachers in the present study, on the other hand, had not participated in any professional development program; however, they had taken a number of mathematics courses and mathematics education courses such as Methods of Teaching Mathematics and Nature of M athematical Knowl edge for Teachi ng.In those mathematics cours es, the prospective teachers discussed the underlying meaning of the al gori thms in detail and prepared activities related to the al gori thms.Their knowledge of the place value of 1 in addition and in subtraction might be higher than the knowledge of prospective teachers who participated in the previous studies due to these discussions and the activities.
The analysis based on the nonstandard algorithm in subtraction indicated that almost half of the prospective teachers' expl anation related to the underlying concepts of the nonstandard subtraction algorithm was missing or incorrect.Only one thi rd of the prospective teachers explained why the nonstandard subtraction al gorithm made mathematical sense.These teachers based thei r reasoning of the nonstandard algorithm on the ones and the tens.This finding was in agreement with the study of Siegfri ed, Gordon and Garcia ( 2007).In the s tudy of Siegfried et al. (2007), most of the prospective teachers had difficulty in explaining the nons tandard al gorithm because of the lack of specific content knowledge for teaching mathematics.This situation could be valid for the pres ent study in which prospective teachers might not have enough knowl edge to jus tify the reasoning behind the nonstandard al gorithm, whereas they had knowledge to reason the underlying concepts of standard algori thm.Based on Shulman's defini tion of SMK, it could be concluded that most of the prospective teachers lacked SMK.In other words, since explaini ng the underlying concepts of nons tandard subtraction algorithm is a kind of knowledge specific t o mathematics teachers (Ball et al., 2008), the prospective teachers' specialized content knowl edge was not enough to explain this algorithm conceptually.Although thei r knowledge is not enough to explai n the algorithm, these prospective teachers could apply the nonstandard algori thm to the similar subtraction operation.While applying the nonstandard subtraction algorithm to another algori thm, they most probably tri ed the same operations by changi ng the numbers.Interes tingly, but perhaps not surprisingly, almost all prospective teachers applied the nonstandard algorithm in multiplication to another set of numbers.Put differently, these teachers could apply the systematic action sequences to perform the nonstandard algori thms.This could be interpreted as the prospective teachers' having procedural knowledge about nonstandard algori thm in subtraction and multiplication.O n the other hand, Rittl e-Johnson et al. ( 2001) stated that a task that included a nonstandard al gorithm or unfamiliar procedures could be us ed to assess someone's conceptual knowledge.
From this point of view, the Andrew task and the Doubling task could also be used to evaluate teachers' conceptual knowledge beside their SMK.As most of the prospective teachers in our study could not explai n the underlying meaning of the nons tandard algori thm in these tasks, it can be concluded that their conceptual knowledge is not adequate to teach these concepts effectively.In other words, the prospective teachers could apply mathematical algori thms and perform them accurately and efficiently (National Research Council [NRC], 2001).Similarly, Philipp et al. (2008) reported that the participants in thei r study had high performance on applying nonstandard algori thm to multiplication even though the algorithms were not familiar to them.
Another poi nt worth discussing is the pros pective teachers' knowledge of mathematics on division.The important point regarding division is that the prospective teachers recognize the difference between multiplication and div ision and between partitive and measurement division.Based on the analysis of the data, only a few pros pective teachers provided justification for the distinctions between multiplication, partitive division and measurement division.It is surprising that more than half of the prospective teachers did not explai n the difference between multiplication, partitive division and measurement division.Furthermore, a few prospective teachers had the understanding of the distinction between multiplication and division, whereas they do not know the difference between partitive division and measurement division.In other words, they know that multiplication requi res finding the total number of objects and division requires finding the number of objects in each group or findi ng the number of groups.However, they did not realize which quantity is the unknown.In more detail, the prospective teachers did not know the distinction between whether the number of groups is the unknown or the number of objects in each group is unknown.This result lets me conclude that most of the prospective teachers' understanding of division in whole numbers is not adequate to teach these algorithms to students deeply.Similarly, Ball (1990) stated that prospective teachers could not explain the underlying principles and meanings of the division with fraction.Accordi ng to Ball (1990), prospective teachers had a narrow unders tanding of division.On the contrary, Philipp et al. (2008) concluded that prospective teachers' mean score in Division Task was high.Thus, they stated that teachers had enough knowledge to explain the difference between multiplication, partitive division and measurement division.
The results of the study rev ealed that more than half of the prospective mathematics teachers had knowledge about the place value of 1 in addition and subtraction, and also in multiplication.However, mos t of the prospective teachers could not explai n the underlying principle and the meaning of the Andrew's strategy, which contains subtraction.Similar to thei r knowledge on subtraction, prospective teachers' knowledge on division was narrow.As a conclusion, the prospective teachers' SMK on the algorithms in standard addition, subtraction, and mul tiplication was found to be high; howev er, thei r SM K on nons tandard subtraction and division was found to be low.In other words, it could be stated that most of the prospective teachers have conceptual knowledge related to addition and subtraction, but they have procedural knowledge on nonstandard subtraction and division.Based on these fi ndings, some implications and recommendations could be suggested.As basic algori thms with whole numbers is a central theme in the mathematics curriculum, it is very important to have this knowledge for effective teachi ng.Therefore, prospective teachers should extend their knowledge on this topic, and instructors should give them opportuni ties in this regard.In order to do this, the class hours of the lessons contai ning content knowledge mi ght be increas ed and the unde rlying principles of the basic algorithms with whol e numbers might be discussed.Moreover, the opportuniti es might be provided to prospective teachers to gain experi ence in basic algorithms duri ng school experience and practice teaching in elementary education courses.
Although the present study provided interes ting findings that contribute to the literature and teaching practices, there is still more to explore.In the present study, the prospective mathematics teachers' knowledge on basic algorithms with whole numbers was analyzed.When the importance of teachers' knowledge on students' mathematical understanding is considered, it is essential to conduct similar studi es with Turkish in-service primary school and mathematics teachers to reveal their knowledge and underst anding of the basic algorithms.Apart from the whole numbers, it would be valuable to investi gate teachers' knowledge on basic algorithms with different number sets.As a final point, data might be collected via follow-up i nterviews to gain an in-depth understanding of pros pective teachers' knowledge of algorithms with whole numbers.

Table 1 .
The 4-point rubric for the Ones Task

Table 2 .
The 5-point rubric for Andrew Task

Table 3 .
The 4-point rubric for Doubling Task

Table 4 .
The 3-point rubric for Division Task