Statistical Survey : the Proportion of Students Willing to Become Teachers of Mathematics

The article discusses the problems of mathematical literacy of graduates from secondary schools as well as the problems related to the increasing gap between higher and secondary education institutions in Europe and in Lithuania, whereof education system has been undergoing the reform. The results of a statistical survey, which was carried out in Lithuanian University of Educational Sciences, are also presented. The survey focused on the analysis of the answers of the selected population of students to the question ‘What is the proportion of students willing to become teachers?’ The answer to this question has received highly controversial evaluations from specialists of the Ministry of Education and Science of the Republic of Lithuania, who administer financing of universities, on the one hand, and specialists in educational science, who have been conducting the aforesaid surveys and research, on the other hand. The simple statistical analysis and some methods of survey sampling were applied to solve this problem. Differently from the situation of earlier years, the survey results show that nowadays the above-mentioned proportion of students is growing.


Introduction
Since September 2009 the most capable graduates with highest academic achievements from secondary education institutions of Lithuania have been granted the so-called 'study vouchers' and higher education students with a good academic record have received state funding for the whole period of their studies.The authors of the reform expected the system of 'study vouchers' to increase the competition among higher education schools.Moreover, it may be considered as a more transparent way of allocating state funds, when financing is ascribed to a particular individual rather than to an institution.However, there is a big number of study programmes, for which there are no competing ones and in such cases the system 'study vouchers' does not change anything because there is no competition.
The student is able to bring his / her 'study voucher' to any institutions, which may result in a bigger disproportion among higher education institutions and radically distort the formation of demand of the society for the most needed specialities.According to Apynis and Mazėtis (2011), there are only fewer than 20 students in the first year of studies in all the study programmes that train teachers of mathematics and informatics in Vilnius University and Lithuanian University of Educational Sciences.In 2012 the number of such students totals only 17.This is due the decrease in teacher training programmes and forced competition of future teachers of mathematics and other exact sciences with future teachers of choreography, physical training or educators for the study vouchers.
Since the number of 'study vouchers' for places in the study programmes ascribed to the group of education and training studies in 2012 totalled 568 (in 2011 there were 740 of such study vouchers, i.e. this year the state order has been decreased by 23 percent), the negative consequences of the conducted education reforms are obvious: specialities of teachers of mathematics and informatics received only 3 percent of the 'study vouchers'.What is more, in the school year of 2012-2013 as many as 4009 (i.e., more than 14 %) out of the total number (28 158 teachers) of teachers working in Lithuania were those of mathematics and informatics (Statistics Lithuania, 2013).Thus, a lack of young teachers of mathematics and informatics is likely to emerge in 4-5 years.
After the first phases of the reform, the results of student admission to higher education institutions in Lithuania revealed that the implemented model of the study voucher will not facilitate attainment of the set goals, i.e., higher quality studies, optimisation of higher education network and proportional redistribution of the entrants among university and non-university sectors of education (Želvys, 2009).
The goal of the conducted statistical survey is to analyse the students' attitude towards the profession of teacher of mathematics, the study reform, which is currently under implementation (and the so-called 'study vouchers') as well as towards the place of teacher training institutions in this reform.The topicality of this research was conditioned by the 'reasoned' discussion that the bigger proportion of graduates from teacher training universities do not even consider becoming teachers and that their main purpose is only to obtain a university diploma.

Methodology of Research
The aforesaid reform has had a severe effect on the training of the young generation of teachers of mathematics: only in the last three years the number of students, which have chosen the study programs of Mathematics and Informatics at Lithuanian University of Educational Sciences, has drastically gone down (namely by 40 percent), consequently, resulting in a reduction in the number of teachers.Thus, the target population in the first stage of the research included the students of the aforesaid study programs, which totalled N = 294 elements in April 2013.It is obvious, that the success of a first year student's start at higher education institutions is firstly conditioned by a secondary school, whereas fulfilment of his / her further expectations depends on the process of studies; therefore, to ensure comprehensibility of the survey, the questionnaire was supplemented by the following questions: -In your opinion, is level of preparation of pupils for studies and life in secondary schools decreasing?-Do you regret choosing studies in our university?Seeking to estimate the parameters of the population (for example the proportion of the students, who plan to work at school after graduation of the university), the simple random stratified sample was selected.The procedure of forming the sample of this kind may be divided into several stages (Krapavickaitė, Plikusas, 2005;Särndal et al., 1992): division of the population into a number of distinct non-overlapping subpopulations called strata, sample size determination, sample size allocation and selection of simple random samples from every stratum.Each of these stages will be discussed in a more detailed way.

Stratification of the population
Since the estimates of the population parameters in the case of the stratified sample are more accurate when the strata consist of the elements that are as similar as possible from one or another perspective (for example, in terms of attitude, similar values of quantitative and qualitative variables), the population was divided into the three strata on the following basis: 3) First and second year students of the Master study programme of Mathematics and Informatics (this stratum consists of N 3 = 33 elements).We think that under the above indicated stratifi cation of the population, elements are more similar to each other within strata than between them.

Sample size
Sample size determination depends on the accuracy requirements for the estimates of the population parameters.If an estimator (α -probability of error) would be lower than the selected positive value d (absolute tolerable error).In the theory of the fi nite popula- tions statistics it is proved that the sample size n , which meets the aforesaid accuracy conditions, is defi ned by the following equation: .
When no advanced information on the estimated proportion is available, it is suggested writing 1/2 instead of p in the formula (2).
Despite the use of the stratifi ed sample instead of the simple random sample, the sample size is calculated in this paper according to the formula (2).Such choice may be motivated by the following: appropriate stratifi cation of the population results in more accurate estimates of the parameters.Th us, if, in the case of the simple random sample the targeted accuracy is achieved using a certain value of sample size, let us say * n , then in the case of the stratifi ed sample (with the same value of the sample size n * ), the requirements for accuracy are even better met.Setting α = 0.1 and d = 0.06 (which means that the required absolute value of diff erence p p − ˆ with the probability 1 -α = 0.9 is lower than d = 0.06), yields: . 115 = n

Sample size allocation
When population is stratifi ed and the sample size n of the stratifi ed sample is determined, it is necessary to decide how big samples will be drawn from each stratum under the condition that the total of these sample sizes (let us defi ne them by 1 n , 2 n , ... and

H n
, where H -number of the strata) would be equal to n .It should be noted that the union of these samples defi nes the stratifi ed sample used in the survey.
When the strata are of diff erent size and the distributions of survey variables are unknown, strata sample sizes can be calculated using the proportional allocation, i.e. according to the formula: Applying this formula for our data, we obtain: n 1 = 23, n 2 = 79, n 3 = 13.

Selection of the samples from the strata
Further, the simple random samples of the previously determined size were selected from each stratum, the union of which makes up the stratifi ed sample used in the research.Forming the sample of this kind, the samples from the strata are selected separately and independently regardless of the samples selected in other strata.

Survey
Striving for acquisition of the sample data, the method of questionnaire survey was applied.Th e questionnaire form specially designed by the authors contained statements on the changing attitude of students towards teacher's profession, the undergoing study reform and evaluation of the study quality.Th e data obtained during this survey were processed employing Microsoft Offi ce Excel 2010.

Estimator used
A proportion of the elements of the sample, having an attribute of interest, is the standard estimator of the population proportion, which is applied in the case of the simple random sample.In case of the stratifi ed sample (thus, also in this statistical survey), the population proportion is estimated by the following simple only design-based estimator: where h p ˆ -the proportion of the elements in the sample of the stratum h that have a particular attribute.

Growth in the gap between the higher and secondary education institutions
The reformed teaching of school mathematics introduced in the second part of the 20th century resulted in a too formal teaching of mathematics in secondary schools.However, correction of mistakes led to the other extremity -important themes and concepts of mathematics were withdrawn from school textbooks of mathematics, educators rather than specialists of the study subject started writing textbooks of mathematics.Such situation led to emergence of such curiosities as attempts to humanise the Pythagorean theorem.
A big number of professional mathematicians in Europe criticize the quality of teaching contemporary school mathematics.For example, Gunter M. Ziegler, who is a professor at Freie University in Berlin, is dissatisfied with the current quality of teaching mathematics in German schools due to a big number of reasons that are pointed out in his article (Ziegler, 2012).
When experiments with mathematics, the oldest and the most conservative study subject, which is also likely to be considered the most respectable, started, the gap between the level of school and university knowledge significantly widened.This is confirmed not only by the opinions of the representatives of other national universities (Kaminskienė et al., 2013;Stankus, 2008;Stankus, Dagienė, 2010) but also by the conducted research.The graphic presentation of the questionnaire results received to the question 'In your opinion, is level of preparation of pupils for studies and life in secondary schools decreasing?' from the respondents of the described sample is shown in Fig. 1.
The data indicate that approximately 66 percent of students, who are recent graduates from secondary schools, think that the level of professionality of a secondary school is getting lower (Note: hereinafter pointing out that x percent of the students expressed their opinion y, we will have in mind the estimate of the proportion of all the students in the population, who express the opinion y, calculated according to the formula (3) and expressed in percentage form).
On the other hand, the attitude of school students towards fundamental sciences has also been changing.If 25-30 years ago young people thought of the scientist's career as of very prestigious, the results of the recent research (Lamanauskas, Augienė, 2011) revealed that upper secondary school students do not relate their future with scientifi c research activities, researcher's career is considered to be not perspective and of low prestige.

Perception of outcomes
Many students and undergraduates seem to think of mathematicians as old, white, middle-class men who are obsessed with their subject, lack social skills and have no personal life outside maths.Th e student's views of mathematics itself included narrow and inaccurate images that are oft en limited to numbers and basic arithmetic.
Th e paragraph we just quoted is the summary of the study 'Maths Images & Identities: Education, Entertainment, Social Justice' carried out by three British sociologists Heather Mendick, Debbie Epstein and Marie-Pierre Moreau at the 'Institute for Policy Studies in Education' at London Metropolitan University in 2008.It was based on the survey of British students.
A similar attitude has been widely spread in Lithuania.Th erefore, the question 'What proportion of students have intention to become teachers of mathematics?' has become of utmost topicality and the answer to this question has received highly controversial evaluations from the specialists of the Ministry of Education and Science of the Republic of Lithuania, who administer fi nancing of universities, on the one hand, and specialists Th us, it is necessary to identify what goals a student pursues studying at university.Th e answers of the respondents to the question 'What was your purpose of entering the university?' are presented in Fig. 2.

Fig. 2. What was your purpose of entering the university?
As it can be seen from the respondents' answers, the majority of them would like to become qualifi ed specialists or qualifi ed teachers; however, the concepts of qualifi ed specialist or qualifi ed teacher are likely not to be perceived the same way by everybody.It should be noted that the number of students, who are planning to search for a job at school has considerably increased (see Fig. 3).Professor mathematician V. Uspensky from the famous Lomonosov Moscow State University expressed the following opinion regarding experience of learning mathematics at school (Mathematics for the Humanitarians, 2004): 'Th e result of such teaching mathematics in a secondary school is hatred towards this study subject.And this happens not only because of bad teachers of mathematics.Th e society does not have any understanding of what goals are attained while teaching mathematics... Th is concerns not only mathematics but also literature and other study subjects'.Referring back to the Figure 2 'What was your purpose of entering the university?', it is necessary to point to the alarming fact that there is a small number of respondents, who prioritised the ability to learn to think.Who if not a teacher of mathematics has to not only encourage but also to teach school students to think?To think consistently, logically, rationally, reasonably...

Enhancement of the prestige of teacher in the society
Th is is a welcome trend.Th e diagram presenting the answers to the question 'Are you going to look for a job of a teacher aft er graduation?' shows that slightly more than 66 percent of the respondents are planning to choose teacher's job in 2013, which is a joyful trend and was similar in 2010 and 2009 but not in 2008 (Januškevičius, Januškevičienė, 2009) (see Fig. 4).However, the number of entrants to the study programmes of Mathematics and Informatics has been annually decreasing.Making attempts to clarify the reasons why so few graduates chose studies of mathematics, it was natural to start with the question regarding the quality of studies.Th e results of the survey showed that as many as 96 per-cent (!) of students that have chosen the study programs of Mathematics and informatics at Lithuanian University of Educational Sciences do not regret choosing studies in this university (see Fig. 5).It is necessary to pay attention to one more quite dangerous trend -having chosen the model of free competition in the current stage of Lithuanian higher education development and failing to fully fi ll groups of specialities of exact sciences with students, the universities that are fi nanced according to numbers of admitted students will be forced to withdraw these specialities and replace them by others.
1) First and second year students of the Bachelor study programme of Mathematics and Informatics and the Bachelor study programme of Informatics (this stratum consists of N 1 = 59 elements); 2) Third and fourth year students of the Bachelor study programme of Mathematics and Informatics and the Bachelor study programme of Informatics (this stratum consists of N 2 = 202 elements); ISSN 1392-0340 E-ISSN 2029-0551 Pedagogika / 2015, t. 118, Nr. 2 normally distributed, then one can require that the absolute value of the diff erence where z α/2 is 1 -α / 2 -quantile of the standard normal distribution, ( ) θ V -variance of the estimator Čia turi būti Čia turi būti Čia turi būti Čia turi būti.Th e estimated parameter in this statistical research is the proportion p of the population (for example, the proportion of the students, who have already been working and are further going to work at school; the proportion of the students, who think that the level of preparation of school students in secondary schools for studies and life is deteriorating, etc.).If, in the case of the simple random sample, the population proportion is estimated by the sample proportion and the estimator 34 psl.Čia turi būti Čia turi būti Čia turi būti Čia turi būti Čia turi būti is normally distributed, the sample size n , which satisfi es equation (1), equals:

Fig. 1 .
Fig. 1.In your opinion, is level of preparation of pupils for studies in secondary schools decreasing?

Fig. 3 .
Fig. 3. Are you going to look for a job of a teacher a er graduation?

Fig. 4 .
Fig. 4. Are you going to look for a job of a teacher a er graduation?

Fig. 5 .
Fig. 5. Do you regret choosing studies in our university?

Fig. 6 .
Fig. 6.What is your opinion about the undergoing study reform and the so-called 'study vouchers'?