An investigation on the thinking structures and proof-writing levels of future mathematics teachers

An investigation on the thinking structures and proof-writing levels of future


INTRODUCTION
Thinking is defined as activating many mental processes to solve problems (McGillicuddy-DeLisi & Sigel, 1991).These mental processes are creative thinking, reasoning, and evaluation (Tekin et al., 2009).Thinking is a skill that can be improved.Mathematics education is a tool that improves thinking (Umay, 2003).Mathematical thinking can be described as the individual's access to new information by going through various mental processes with the aid of previously acquired mathematical knowledge (Alkan & Bukova-Guzel, 2005).Krutetskii (1976) divided people's thinking structures into three by how they handle mathematical knowledge.These are analytical, harmonic, and geometrical thinking.While people who adopt analytical thinking are prone to verbal items, they have low affinity for visual items.They prefer verbal and logical methods in solving problems.People who have harmonic thinking structures work equally with verbal and visual elements.They want to use both methods together in problem-solving.The last group, people who have geometrical thinking model, is good with visual elements.They need to use visual elements to solve problems (Kozhevnikov et al., 2002;Krutetskii, 1976).
Geometry is a branch of mathematics that deals with phenomena such as point, line, plane, planar shapes, space, spatial shapes and the relations between them and measurements of geometric shapes such as length, angle, area, and volume (Dursun & Coban, 2006).Learning geometry starts with making sense of the physical world and continues with higher-level geometrical thinking that develops in an inductive or deductive system (Ubuz, 1999).Among other, one important purpose of geometry education is to develop reasoning skills (Morali et al., 2006).One of the elements that serve the development of reasoning skill is proof.Proof is the process of judging as true or false a proposition or theorem or a statement in mathematics (Guven et al., 2005).Proof bears several mathematical meanings.However, in the broadest sense, proof is the systematic use of basic axioms, definitions, and theorems to make mathematically clear why a proposition is true (Bell, 1976).From pedagogical perspective, proofs allow students to develop their reasoning skills and increase their creativity by using various methods.The development of proof depends on individuals' acquiring different ways of logical thinking (Guven et al., 2005).

Background
The fact that human mind attends to varying degrees of achievement depending on the relationship between the visualpictorial and verbal-logical components signals the existence of different mathematical thinking styles in mathematical activities (Krutetskii, 1976).It is seen that students with strong analytical thinking skills perform better at problem-solving compared to those with the dominant type of visual thinking style (Lean & Clements, 1981).
It is known that students resort to visual solution strategies to decode the data in problems that they have not experienced before and find complex while they tend to use analytical strategies in dealing with easier problems (Lowrie & Kay, 2001).It is added that students make more effort and have difficulty in managing visual processes than analytical processes (Presmeg, 2006).In short, the literature reveals that different thinking structures decide on the approaches to problem-solving (Delice & Sevimli, 2011;Krutetskii, 1976;Lean & Clements, 1981;Lowrie & Kay, 2001;Presmeg, 2006;Tasova, 2011;Turan, 2011).
Problem-solving refers to elimination of the problem by utilizing the processes of reasoning and proof with the aid of certain operations.There is a relationship between problem-solving skills and the concept of providing proof (Ozdemir & Ozdemir, 2018).According to Waring (2000), the development of students' proof conception is handled at six levels.These levels are proof level 0 (students at this level are unaware of proof), proof level 1 (students with this level are aware of proof, but they regard proof as verification by checking a few cases), proof level 2 (students at this level regard proof as checking with different or randomly selected examples.Also, they take a general example as verification).Proof level 3 (students at this level cannot construct a proof alone, but they can understand the solution of a proof with a certain difficulty level), proof level 4 (students at this level are aware of the need for proofs and they can understand the formation of proofs and use a limited number of familiar proofs).Proof level 5 (students at this level can construct some formal proofs and the proofs they have just seen).In this model, students usually fall into lower-level proof categories (McCrone & Martin, 2004).On the contrary, teachers and pre-service teachers have a limited understanding of what writing proof is, and they perceive experimental assumptions as valid proofs while having difficulties in making proof (Varghese, 2008).

Objective
Previous research suggests that teacher candidates do quite bad at writing proofs.Proving is a process comprised of many skills.Prospective teachers are supposed to learn and use proofs in most undergraduate courses.When it comes to schools, the ability to prove underpins the teaching curricula.Still, identifying the factors that determine the prospective teachers' levels of writing proofs may justify the activities that can be undertaken to elevate these levels.In this direction, thinking, which is a phase of the proof-writing process, can be associated with proof writing.In this study, it is aimed to reveal what kind of relationship exists between prospective teachers' thinking structures and their level of writing proofs.For this purpose, answer was sought to the following research questions.
1. What are the prospective mathematics teachers' thinking structures like? 2. What are the prospective mathematics teachers' levels of writing proofs?
3. What relationship exists between the prospective mathematics teachers' thinking structures and their levels of writing proofs?

METHOD
This study was carried out as a case study.This research method aims to collect in-depth information about the research topic and to understand the construct from all aspects.Special case studies are particularly suitable for individual studies.In those studies, selected case is reviewed in depth and comprehensively.The data are collected in a systematic way and attempt is made to discover the relationship between the variables (Cepni, 2006).

Study Group
The study was carried out with 82 prospective teachers enrolled in the Department of Secondary School Mathematics Teaching of a state university located in the Eastern Black Sea Region of Turkey.The breakdown of the participants is given by their grade year in Table 1.

Mathematical processing instrument
In this study, two data collection tools were used.The first one is mathematical processing instrument (PI) developed by Presmeg (1986).It is made up of three parts intended to measure the thinking structures determined within the framework of Krutetskii (1976) mentioned earlier.It was designed to find out teachers and students' preferences of visual and non-visual methods in solving non-routine mathematics problems.The tool consists of three parts as A, B, and C. Part-A is solely targeted at students and part-C for teachers, but the other part, that is part-B, is designed for both students and teachers.There are 12 problems in part-B and there are six problems in each of part-A and part-C.Since this study was conducted with prospective mathematics teachers, part-B and part-C were used excluding the first part.MPI provides a list of solutions with three to six probable solutions to each of the 18 problems in parts B and C.An answer key follows on which respondents mark the option, which corresponds to their own solution in the list.One of the items in part-B and part-C are shown in Figure 1 as an example.

Geometry proof-writing test
This tool was developed by the researcher and consists of five proofs that are also covered in the curriculum about the geometric shapes circle and circular region.During the development of the data collection tools, two proficient academicians were consulted for expert opinion within the scope of validity studies.One of the experts is a mathematics teacher and the other is a lecturer specializing in the field of geometry.Thanks to the experts' feedback, the spelling and mathematical errors in the form were corrected.A proof situation in the proof-writing test is shown in Figure 2.

Analysis of mathematical processing instrument
The version of MPI used in this study is comprised of two parts including a total of 18 questions.In addition, the instrument features in a list of solutions with three to six possible solutions to each of the 18 questions.There is also an answer key so that respondents can mark the option showing their own solution.Each of the problems can be solved with visual or non-visual methods.For respondents who apply an authentic solution option, the list provides a space for an open-ended answer.In the analysis of the MPI, each visual solution is scored two points and each non-visual solution is scored as zero, regardless of whether the answer is correct or wrong.Problems solved with a somewhat ambiguous method gets 1 point.The lowest overall score in the MPI is 0 and the highest score is 36.Figure 3 shows the solution and scoring criteria in the answer key to question B-2 as an example.
Figure 4 demonstrates the respondent's answer to B-2 and the score of two points given to the solution as it was resolved by using the visual method.(Presmeg, 1986) As a result of the analysis of the data obtained from MPI, the students' thinking preferences were grouped in three categories within the rationale of Krutetskii.In other words, they were classified as analytical, geometrical, and harmonic thinkers based on their scores.Even though it is difficult to draw sharp borders between the students' analytical, harmonic, or geometrical thinking styles, different methods are available for assessing the scores obtained from the MPI.In the current study, the method of Tasova (2011) was used.Tasova (2011) calculated the difference (distribution range) between the largest and smallest value in the MCI data and divided the distribution range into three to identify the class range in a way representing the three distinct thinking structures.Then, he used the class range to determine the reference values for each thinking structure, that is the lowest and highest scores for each category.
In this study, the students' scores ranged between six and 33 points.The distribution range was 33-6=27 and thus the class range was found as 27/3=9.The upper limit of the analytical thinking structure, which is the bottom group, is calculated by summing the class range value and the smallest value obtained from the instrument.The lower limit is the smallest score obtained from the MSA.By following this sequence of transactions, the lower and upper limits were calculated for the other thinking structures.As a result, those with scores of six to 15 were rated as analytical thinkers, 16 to 25 were rated as harmonic thinkers, and those obtaining 26 to 33 scores were rated as geometrical thinkers.

Analysis of geometry proof-writing test
The assessment of geometry proof-writing test was done according to Senk's (1983) analysis method.In her study, Senk (1983) scored questions of proving, as follows: one point was given for the correct drawing, two points for the placement of the given data, and one point for writing the proof.In the current study, the respondents obtained one point for drawing only the shape of the proof situation, two points for placing some of the given data appropriately, three points for placing all the given data appropriately, and four points for placing all the given data and writing the proof completely.The top score for each question was four points.It means that a student who can write all the proofs correctly can get a maximum of 20 points from this test.Apart from this, application of Senk's (1983) criteria for determining the level of writing proofs in this study suggests that a respondent who gets a total score of five or below from the test is considered unsuccessful in writing proofs.On the contrary, a respondent who gets a proof score of 20 or above is considered very successful in writing proofs.The rating scale continues as following: Scores between five and 10 points refer to low success in writing proofs and scores from 10 to 15 points refer to intermediate success in writing proofs.Overall, students with a proof score of up to 10 points are considered low achievers while those obtaining higher scores are considered high achievers.
Answers for sample scoring are displayed in Figure 5 and Figure 6.In the sample in Figure 5, one point was given to the respondent as nothing except for the illustration of the given situation was drawn.In the sample in Figure 6, three points were given as the situation was illustrated along with full placement of the given data, but a complete explanation of the proof situation was missing.

RESULTS
In this section, the study findings are presented under separate headings as "identifying prospective teachers' thinking structures", "identifying prospective teachers' proof-writing levels", "identifying relationship between prospective teachers' thinking structures and their proof-writing levels".

Identifying Prospective Teachers' Thinking Structures
In this study, it was found out that 29 of the 82 teacher candidates have dominant analytical thinking skills, 44 have harmonic thinking, and the other nine have dominant geometrical thinking skills.As Table 2 shows, seven students got six points, five got eight points, six students got 10 and 12 points, respectively, three students got 13 points, and two got 14 points.In the middle band, 16 points were obtained by 10 students, 18 points by 14 students, 20 by 11 students, 21 points by six students, and 22 points were obtained by three students.In the last and the highest band, two students got 27 points, three got 28 and 30 points, respectively, and one student got 33 points from the test.(Senk, 1983) 1.In a circle the center is equidistant from the congruent chords.

Identifying Prospective Teachers' Proof-Writing Levels
The participants' overall scores in proof-writing test are displayed in Table 3.
It was found out that three students obtained three points and four obtained four points from the test, both of which remain below the success threshold.In other words, seven prospective teachers were found to be low achievers scoring five or less.In the next band, seven students got six points, nine got seven points, 11 got eight points, and 12 students got nine points.A total of 39 students were found to have low levels of success (scores from five to 10).Next up, 11 points were recorded by seven students, 12 points by 11 students, 13 points by nine students, and 14 points were obtained by two students, constituting the intermediate level of success.There were 29 students regarded moderately successful in writing proofs (scores from 10 to 15).In last group, that is high achievers in proof-writing, there were seven students all of whom got 16 points or above.

Identifying Relationship Between Prospective Teachers' Thinking Structures and Their Proof-Writing Levels
The frequencies representing each proof-writing level by thinking structures of the prospective teachers are given in Table 4.
Table 4 shows that seven of the prospective teachers with analytical thinking structure were unsuccessful and 22 had low success in writing proofs.Of the prospective teachers with harmonic thinking structure, 17 had low success and 27 had intermediate success in the same task.Lastly, two prospective teachers with geometrical thinking structure could write proofs at intermediate level and seven could write at a high level.
In order to explore the relationship between the prospective teachers' thinking styles and their proof-writing levels, Pearson's correlation coefficient between the two sets of data was analyzed by using SPSS.Table 5 displays the results of this analysis.
According to Table 5, the coefficient of significance between the thinking structures and proof-writing levels of the participants was found to be 0.000.These values were significant at p<0.01.It is also noticeable that Pearson's correlation coefficient was 0.912.Since this value is statistically positive, a positive relationship appeared between the thinking structures and proof-writing levels of the students.Also, this value implies a statistically higher level of positive correlation as it is between 0.71 and 0.99 (Büyüköztürk, 2006).

DISCUSSION AND CONCLUSIONS
This section is devoted to interpreting the findings regarding the prospective teachers' thinking structures and their proofwriting levels and the relationship between these two variables.When the thinking structures of the participants are examined, it can be understood that harmonic thinking is the most common thinking type among the study participants.This type is followed by analytical and geometrical thinking types, respectively.As known, harmonic thinkers are predisposed to both verbal and visual elements equally.They are ready to apply both methods together in problem-solving.Geometrical thinkers are prone to visual elements.Analytical thinkers apt to choose verbal items, while they are unlikely to use visual items.It can be inferred that the prospective teachers in this study tend to couse visual and verbal elements in problem-solving.Other studies revealed that the most common thinking structure among teacher candidates is analytical thinking and the least common one is geometrical thinking (Delice & Sevimli, 2012;Galindo-Moralez, 1994).The ranking of the geometrical thinking structure for this target group can be explained with the heavier emphasis on algebra teaching in the undergraduate curricula.Besides, while separate curricula were taught in mathematics and geometry courses until the year 2015, the two courses were mingled in one single curriculum then.This amendment in 2015 might have affected the distance to geometry.Another explanation for the recessive geometrical thinking structure could be the teaching method used for preparing the students for the university entrance examination.In this teaching method, students have to reach the result in the shortest time possible if they want to succeed in the exam; thus, visual elements are ignored as a part of the teaching method.
When the proof-writing levels in this study are looked at, it becomes obvious that most of the prospective teachers are weak in terms of this skill.The second largest group is comprised of those with an intermediate success level.Only a small number of participants exhibit a high level of proof-writing.Similarly, Varghese (2008) remarkably found that teachers and prospective teachers faced difficulty in constructing proofs, as well as having a limited understanding of what writing a proof is and they perceived experimental assumptions as valid proofs.The related literature also reveals that students' proving levels are generally low (Dickerson, 2008;McCrone & Martin, 2004;Sevgi & Orman, 2020).This seems to be an expected result as the classroom practices of the teachers directly affect the students.
In this study, the participants' proof-writing levels and thinking structures were found to be correlated.According to the results, lower proof-writing levels relate to analytical thinking structure, intermediate level of proof-writing indicates harmonic thinking structure.Lastly, prospective teachers with the highest level of proof writing embody geometrical thinking skills.However, Lean and Clements (1981) concluded that the students with analytical thinking type performed better at problemsolving compared to their peers with the dominant visual thinking type.In contrast, Lowrie and Kay (2001) pointed out that the students implemented visual problem-solving strategies to conceive the data in problems, which were new and complex to them, whereas they tended to use analytical strategies in addressing easier problems.Presmeg (2006) stated that the students exerted more efforts and had difficulty in managing visual processes than analytical processes.These results emphasize the need to attach more importance to works aimed at improving visuality and visual skills.As an example, algebraic/symbol-using skills and their integration with visual skills can be practiced.

Figure 7 .
Figure 7. Participants' thinking skills (Source: Author's own elaboration) As seen in Figure 7, 54% of the participants are characterized with harmonic thinking, 35% with analytical thinking, and 11% geometrical thinking structure.Table 2 details the frequency and corresponding percentage of each MPI score obtained by the prospective teachers symbolizing their thinking styles.

Table 1 .
Number of participants

Table 2 .
Details frequency & corresponding percentage of each MPI

Table 3 .
Participants' overall scores in proof-writing test

Table 4 .
Frequencies representing each proof-writing level by thinking structures

Table 5 .
Pearson's correlation coefficient between two sets of data