Analysis and results of the third research question The third research question is:

3a. Is there a meaningful relationship between the nature and the level of theory use? If so, how is that relationship expressed in the various components of theory use and in various groups of students?

3b. To what extent is there a relationship between the nature or the level of the student teachers’ use of theory and their level of numeracy?

Considerations for research question 3a.

There are reasons to assume that there is a relationship between the nature and the level of theory use.

First, there are signals from the small scale study that students who function at a relatively high level do more often tend to explain and respond, and on the other hand factual description and interpreting mostly go with a lower level of theory use.

Also, the analyses resulting from the first and second research questions have shown that the differences in the size of students’ theoretical repertoire are related to differences in nature and level of theory use. Factual description, interpreting, the first level, and to some degree the second level, have a negative correlation with the number of theoretical concepts used, while explaining and level 3 both correlate positively with the number of theoretical concepts used. Additionally factual description and interpretation are related to a lower level of prior education, particularly mbo without mathematics, while explaining correlates with a higher level of education, particularly vwo with mathematics. Finally it is known from literature that teachers who have less content knowledge are more oriented on facts and procedures, while teachers who possess a larger repertoire look for conceptual and problem solving aspects more (Putnam & Borko, 1997, p. 1232 and 1233).

Combining the above considerations leads to the formulation of hypothesis 3.1.

Hypothesis 3.1

The characteristics of factual description and interpreting for the nature of theory use mainly occur on the first and second level of theory use, while explaining and – to a lesser degree – responding to situations are related mainly to the third level of theory use.

Data analysis and results hypothesis 3.1

Table 5.19 gives an overview of the regression coefficients and the accompanying significances of the correlation between nature and level of theory use by students. Looking at factual description (category A), we see that the regression coefficient beta = 0,129 (sig. 0,043) for level 1 and beta = -0,230 (sig. 0,000) for level 3. Interpreting (category B) shows a similar picture for beta and the related significance. The reverse is

the case for explaining (category C). Beta is negative for level 1 (sig. 0,001) and positive for level 3 (sig. 0,000). Barely any system can be found for category D (‘responding to’).

Table 5.19: Correlation between nature and level of theory use A Factual description B Interpreting C Explaining D Respond to Level 1 A1 Sig. 0,043 Beta 0,129 B1 Sig. 0,096 Beta 0,106 C1 Sig. 0,001 Beta -0,214 D1 Sig. 0,506 Beta 0,043 Level 2 A2 Sig. 0,020 Beta 0,149 B2 Sig. 0,015 Beta 0,155 C2 Sig. 0,105 Beta -0,104 D2 Sig. 0,007 Beta -0,173 Level 3 A3 Sig. 0,000 Beta -0,230 B3 Sig. 0,001 Beta -0,212 C3 Sig. 0,000 Beta 0,282 D3 Sig. 0,212 Beta 0,080 So the linear regression analysis gives a clear confirmation of hypothesis 3.1, with the exception of category D (responding to situations) which deviates more than expected. In view of the inclusion relationship it would be logical for D to have a correlation that is similar to C.

Some explanations can be provided for the deviation from the expected result for category D.

First, the learning environment may have played a part. However, amply attention has been given in the common activities and in the expert notes (e.g., The Guide) to the aspects of responding to situations. Therefore, that does not seem to be the most logical explanation, although it is unclear to what degree for example individual learning styles have played a part (Vermunt, 1992). The tendency to take creative initiatives or to spontaneously develop metacognitive activities mostly suits an open, meaning-oriented learning style (Oosterheert, 2001) and not many students have developed that learning style. It is also not clear what earlier experiences by students in relation to ‘responding to situations’ have played a part.

Second, it can be questioned whether the definition of category D in the analysis instrument had been phrased sufficiently unequivocally. Although the random sample did not show problems with that definition, the number of explainers (Cat. C), including at high levels, is remarkable, particularly as there are few ‘responders’ at a high level. It might be the case that in category D the inclusion relationship is insufficiently expressed or is not made explicit enough in the definition. For example, the definition refers to a metacognitive component. Under that header, D is scored among other things when a student asks himself a question. In such a case there is however often no

Theory-enriched practical knowledge in mathematics teacher education

obvious evidence of an inclusion relationship between C and D. Furthermore, the use of theoretical concepts in that kind of reflection is less obvious, and therefore the chance of scoring the highest level is small.

In summary we can come to a second explanation that the analysis instrument may not be optimal for category D, and the definition for ‘responding to’ situations may need to be adjusted.

A third explanation for the deviation from the expected result for category D is the character of the student population that was studied. The students with ‘vwo with mathematics’ as their prior education are concentrated in the first study year, while students who have done ‘mbo without mathematics’ are mainly third year students. Taking into account the analyses of the first and second research questions, it is well possible that this unbalanced spread across the study years results in a different description of category D than might have been expected based on the influences of prior education and study year.

Table 5.20 represents the average percentages that the students scored per category. The twelve average percentages confirm hypothesis 3.1 in still another way. We see for instance that A3 + B3 = 8%, while C3 + D3 = 27%.

Table 5.20 Mean percentages categories A1 to D3

Category A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Valid 246 246 246 246 246 246 246 246 246 246 246 246

Missing 23 23 23 23 23 23 23 23 23 23 23 23

Mean perc. 12 8 5 5 4 3 12 12 18 7 5 9

Std. deviation 16 13 10 10 9 7 16 14 20 12 9 13

This means that the third level of theory use mainly occurs in explaining teaching situations and responding to situations, and that factual description and interpretation hardly occur at that level.

Considerations for research question 3b.

Question 3b is formulated as follows: To what extent is there a relationship between the nature or the level of the student teachers’ use of theory and their level of numeracy? It has been proposed earlier (section 5.2) that the development of numeracy in the education of primary school teachers in training is intertwined with the development of pedagogical insights and skills (Goffree & Dolk, 1995). The growth in the development of numeracy is seen by Oonk, Van Zanten & Keijzer (2007) as an amalgam of four components, namely the acquisition of elementary arithmetical skills, recognizing mathematics in one’s own environment, being focused on solution processes in solving mathematical problems, and responding to pupils’ solution processes. Along the way mathematizing is entwined with didacticizing. The pedagogical content aspect of

numeracy will develop fully in the later years of teacher training. This matches the fact that students in later years have access to a larger pedagogical (content) repertoire than students in earlier years.

Possessing solid content knowledge is seen in educational circles as an undisputed quality of teachers. Popular wisdom also subscribes to that necessary quality of a teacher: “If you cannot do math well yourself, how can you teach someone else to do it?” Research among secondary school teachers shows that lack of good understanding of the core concepts of one’s own subject can lead to misconceptions about those core concepts in students (Putnam & Borko, 1997; Van Driel & Verloop, 1999). We have already mentioned before (section 5.4.4) the conclusions of Putnam & Borko (1997) in relation to the connection between content knowledge and the orientation – ‘the level’ – of a teacher’s actions. Mandeville (1997) also found, in a large scale study of 9000 students, a positive correlation between the content knowledge of mathematics teachers and the performance of students, with here too the differences occurring mainly in relation to students’ higher order skills.

As far as we know, no studies have been done into this kind of phenomenon in teachers (in training) in primary education. Research has been done into the level of the content skills of Pabo students. In the 1980s the low mathematical skill level of Pabo students was being linked to the mechanistic and insufficiently insight-based mathematics teaching these students themselves had received in primary school (Jacobs, 1986). Recent research into mathematics as a subject shows that it is mainly students with mbo as their prior education who score lower. Their content knowledge is deficient, while students with havo – senior general secondary education – as prior education who do less well, have a ‘maintenance problem’ (Straetmans & Eggen, 2005; Meijer, Vermeulen-Kerstens, Schellings & Van der Meijden, 2006).

Someone with a large amount of proficiency – or even numeracy – for mathematics, is likely to function at a relatively high level of reasoning. In view of the nature of theory use, for that reason a positive correlation between explaining and numeracy is to be expected. In terms of the inclusion relationship that correlation should be present also for ‘responding to,’ although that conclusion is not obvious after the results of the previous analyses of category D.

Taking into account the positive connection that was found earlier between explaining, level 3 of theory use and the number of theoretical concepts used, it is plausible that there will be a positive correlation between level 3 or the number of theoretical concepts and the variable numeracy as well.

Theory-enriched practical knowledge in mathematics teacher education

Hypothesis 3.2

There is a positive correlation between the level of numeracy and these variables: - nature of theory use ‘explaining,’

- the highest, third level of theory use, - the number of theoretical concepts used, and - students’ prior education.

Data analysis and results hypothesis 3.2

The very first thing we note is that linear regression analysis confirms the results of recent studies into the relationship between Pabo students’ own proficiency and their prior education (table 5.21). Particularly strong relationships are the significant negative correlation between numeracy and mbo without mathematics and the significant positive correlation between numeracy and vwo with mathematics. Although not unexpected, these results are remarkable when it is taken into account that the mbo students in this population are mainly third year students and the vwo students are mainly first year students. A negative correlation has also been found between the personal evaluation index (PEI) and students’ prior education (Beta –0,155; Sig. 0,034). This may indicate increasing reticence about assessing one’s own level of numeracy as the level of prior education increases.

Table 5.21

Correlation numeracy and prior education Beta Sig.

Numeracy and mbo without mathematics -0,342 0,000

Numeracy and mbo with mathematics -0,014 0,826

Numeracy and havo without mathematics -0,117 0,069

Numeracy and havo with mathematics 0,180 0,005

Numeracy and vwo without mathematics -0,035 0,587

Numeracy and vwo with mathematics 0,323 0,000

The hypothesis is also confirmed for the nature of theory use ‘explaining’ (table 5.22). The other categories relating to the nature of theory use do not show a significant correlation, including, as expected, for ‘responding to situations.’

There is a positive trend between level 3 and numeracy. That the correlation with level 3 is less strong than that with explaining can be understood from the relationship between explaining and ‘problem solving,’ while the relationship between numeracy and level 3 is less obvious.

The negative trend between level 1 and numeracy (table 5.22) is virtually mirrored with level 3, and also supports the hypothesis.

This is the case particularly for the number of general pedagogical concepts, and not for the number of pedagogical content concepts.

Table 5.22

Correlation numeracy, nature and level Beta Sig.

Numeracy and percentage C (explanation) 0,202 0,003

Numeracy and percentage C ≥ 50 0,201 0,003

Numeracy and level 3 0,135 0,046

Numeracy and level 3 ≥ 50 0,133 0,050

Numeracy and percentage C3 0,235 0,000

Numeracy and level 1 -0,136 0,044

Numeracy and level 1 ≥ 50 -0,129 0,056

Numeracy and number of theoretical concepts final assessment 0,166 0,014

Numeracy and general pedagogical concepts final assessment 0,175 0,010

Within the framework of hypothesis 1.3 and 2.1, the differences between the number of general pedagogical and pedagogical content concepts used have already been pointed out.