D: Responding/Gearing to
5.4 Analysis and results 1 Introduction
The data collection of the large scale study involves the data of 269 students over 11 Pabos (table 5.1). According to the procedure described earlier, the initial assessment has been scored for the level of theory use and the final assessment for nature and level of theory use. The initial assessment consisted of four situations, each aimed at one of the categories for the nature of theory use. For the purpose of scoring, the final assessments had been divided into a total of 1740 meaningful units, on average seven units per student (table 5.5).
Table 5.5 Statistics units in final assessment
N Valid 246 Missing 23 Mean 7,07 Median 7,00 Mode 6 Std. deviation 1,794 Minimum 4 Maximum 13 Sum 1740
For nature as well as level of theory use, each student has been scored on the number of theoretical concepts used. Here, a division into six categories was made for both the initial and the final assessment.44
The scores of the numeracy tests have been included in the data collection both for the individual problems and in total. This is also the case for the students’ own evaluation on a five-point scale of the difficulty of each problem. A variable ‘personal evaluation index’ (PEI) has also been created (see also section 5.3.4.2), which is the difference between 2M-S, where M is the total of the difficulty scores and S the total of the problem scores. PEI gives a positive result if students overrate themselves and a negative result if they underrate themselves. It might be possible to see PEI as a measure for self-confidence.
In the following overview all variables are mentioned that have data stored in SPSS. The variables:
Pabo (Primary Teacher Training College), class, group size, study year, type of study, prior education, gender, practical experience, number of concepts (pedagogical content knowledge, general pedagogical knowledge, different concepts, for begin and end, comparing number of similar concepts (pck, gpk) start and end, level initial assessment, number of units, nature of theory use (also in percentages), level of theory
use (also in percentages), number of combinations (also in percentages), assessment numeracy, difficulty, Personal Evaluation Index (PEI).
Using various SPSS (version 15.0) tools, equations have been created to make the variables and data accessible for analysis.
Next, an account is given of the analysis and the results for each research question. 5.4.2 Analysis and results of the first research question
The first research question is:
In what way do student teachers use theoretical knowledge when they describe practical situations after spending a period in a learning environment that invites the use of theory?
Considerations for research question 1
The expectation is that, where the nature of theory use is involved, students will relatively often ‘factually represent’ (category A). A number of arguments can be provided for that assumption. To start with, factual description – as the first in the inclusion relationship – has an obvious function as the start of a description. Moreover, it seems likely that a number of students will not get beyond factual description in their reflection. Those will mainly be the students that score highest in category A. It is plausible that this will largely involve students in lower years or students with a lower level of prior education, primarily because the pedagogical (content) jargon of lower year students will generally be less developed than that of higher year students. Without a sufficiently large pedagogical repertoire it is harder to let reflection rise above the level of factual description. Moreover we expect that factual description will occur more in the group of students with a lower level of prior education than in students with a higher level of prior education, because factual description requires less cognitive abilities than explaining and responding to situations.
The two final statements – regarding a limited jargon and a lower level of prior education – are also valid for interpreting (category B). In addition, experience teaches that students at the start of their study have a tendency to be faster to reach a judgement about the teacher or the students they are observing than students in later years. This is an added reason to expect interpreting to occur more with first year than with later year students. To explain teaching situations (category C) students must possess a sufficient pedagogical repertoire and cognitive ability. It is therefore likely that we will find students who tend to apply this type of theory use among the later year students or among students with a higher level of prior education.
Theory-enriched practical knowledge in mathematics teacher education
Responding to situations (category D) is likely to be scored relatively little, since that activity requires a high cognitive level and some creative input.
Regarding the use of concepts the study distinguishes between general pedagogical and pedagogical content concepts. Students tend to react spontaneously, and primarily in general terms to teaching situations. This is understandable, since the general pedagogical jargon is aimed at the whole of the action taken by teacher and students and is more often used in both course and teaching practice. Often, intervention by the teacher educator or by student peers is required in the discourse to focus on content- specific aspects of the situation that has been observed.
The expectation is therefore that students will more often use general rather than content-specific concepts in their reflections on teaching situations. It is also plausible that students who explain or respond to situations more, will also use more theoretical concepts, and vice versa; if you have more theoretical concepts at your disposal, there is more of a chance for explaining or ‘responding.’
The above considerations lead to three hypotheses regarding the nature of theory use.
Hypothesis 1.1
The characteristics of the nature of theory use will manifest to various degrees, with ‘factual description’ as a category with a relatively high frequency.
Hypothesis 1.2
The characteristics of factual description and interpreting for the nature of theory use will occur most often with lower year students or with students with a lower level of prior education, while explaining and ‘responding to’ will mostly occur with later year students or students with a higher level of prior education.
Hypothesis 1.3
Students will mainly use theoretical concepts to explain teaching situations and to respond to situations. This will involve general pedagogical concepts more often than pedagogical content concepts.
Data analysis and results hypothesis 1.1
To be able to research the hypothesis, first the variables ‘percentage X’ (X = A, B, C or D) for the four categories of the nature of theory use are entered. The different numbers of units per students necessitate the creation of a comparable measure.
Quantitatively speaking the first hypothesis can be answered simply by giving the four percentages that arise from the descriptive analysis of the percentages X. The output of that indicates a division into respectively 25, 12, 42 and 21 as the average percentages scored by students in the categories A to D (factual description, interpretation, explanation and ‘responding to’; see table 5.6).
Table 5.6 Statistics mean percentages for the nature of theory use percentage A factual description percentage B interpretation percentage C explanation percentage D ‘responding to’ Valid 246 246 246 246 N Missing 23 23 23 23 Mean percentage 25,30 11,62 41,96 21,11 Std. deviation 25,320 18,077 28,423 20,898 Furthermore, 38% of the students starts the reflective note with a factual description of the teaching situation, 19% even scores category A on both unit 1 and 2.
The results partly confirm the first hypothesis. While factual description (A) does score high (25%), explanation (C) has a score of 42%, which is by far the highest percentage, and category D also scores higher than expected.
Because the percentages mentioned for category A to D relate to the average percentages scored by students (with standard deviations of 18 to 28%), and not to percentages of the population or numbers of students per category, we also look at groups of students where the nature of the use of theory is relatively often aimed at one specific category. This gives us an extra opportunity to look for specific student characteristics that belong with certain characteristics for the nature of theory use. For this purpose we define the concept ‘characteristic dominance’ as that characteristic of theory use where the students scores at least 50% of the total number of units in the category that occurs most often45. The seven
students who score 50% in two categories are left out of consideration (table 5.7).
Table 5.7 Students with two 50% scores for the nature of theory use
number Student nr.
Perc A = perc B = 50% 1 164
Perc A = perc C = 50% 3 75, 163 and 238
Perc A = perc D = 50% 1 107
Perc C = perc D = 50% 2 260 and 262
Total 7
Data selection and frequency analysis provide the following view of the percentages and the numbers of students in the four categories:
Table 5.8 Statistics characteristic dominance perc A ≥50 (FILTER) perc B ≥ 50 (FILTER) perc C ≥ 50 (FILTER) perc D ≥ 50 (FILTER) N Valid 239 239 239 239 Missing 23 23 23 23 Percentages of population 17,6 6,7 43,5 11,7 Number of student teachers 42 16 104 28
Theory-enriched practical knowledge in mathematics teacher education
It turns out that as much as 79,5% of the students dominates in one of the four categories (190 out of 239 students; table 5.8). It may be possible to explain that result from the differences in learning or writing style between students (Kolb, 1984; Vermunt, 1992). The ranking of the category percentages X ≥ 50% matches that of the category percentages X. Here too the relatively high frequency of category C stands out. It turns out that 43,5% of students belongs to the category percentage C ≥50.
The higher-than-expected percentage C and ‘C-dominance’ may be related to the fact that the student population consists of a relatively large number of higher-year students (84% second and third year) and students with a relatively high prior education level (havo – senior general secondary education – with mathematics 36%; vwo with mathematics 19%). We will look further into this conjecture for hypothesis 1.2. The influence of the learning environment may be another factor that has reinforced the explanatory character of students’ reflections.
Data analysis and results hypothesis 1.2
Where factual description is involved, linear regression analysis points to a significant negative correlation between both category A and category percentage A > 50% and students’ prior education (respectively sig. 0,041; beta –0,131 and sig. 0,003; beta – 0,195).
Further analysis on prior education indicates a significant positive correlation between both category A and category percentage A ≥ 50% and the students with an mbo education without mathematics (respectively sig. 0,041; beta 0,130 and sig. 0,013; beta 0,160; table 5.9).
Table 5.9
Correlation nature and prior education Beta Sig.
Percentage A (factual description) and prior education -0,131 0,041
Percentage A ≥50 and prior education -0,195 0,003
Percentage A and mbo without mathematics 0,130 0,041
Percentage A ≥50 and mbo without mathematics 0,160 0,013
Percentage A and vwo with mathematics -0,099 0,127 Percentage A ≥50 and vwo with mathematics -0,106 0,103
Percentage B (interpretation) and prior education -0,129 0,043
Percentage B ≥50 and prior education -0,042 0,514 Percentage B and mbo without mathematics 0,092 0,151 Percentage B and vwo with mathematics -0,043 0,498
Table 5.9
Correlation nature and prior education Beta Sig.
Percentage C (explanation) and prior education 0,243 0,000
Percentage C ≥50 and prior education 0,275 0,000
Percentage C and mbo without mathematics -0,202 0,001
Percentage C≥50 and mbo without mathematics -0,246 0,000
Percentage C and vwo with mathematics 0,138 0,031
Percentage C≥50 and vwo with mathematics 0,149 0,021
Percentage D (‘responding to’) and prior education -0,051 0,426 Percentage D≥50 and prior education -0,037 0,573
There is a negative trend for the correlation between the percentage interpretation
(category B) and the students’ prior education (sig. 0,043; beta –0,129).
For explaining (percentage C and percentage C ≥ 50), linear regression analysis points towards a significant positive correlation with students’ prior education, likewise for the specific case of vwo with mathematics. Conversely, mbo without mathematics has, as expected, a significant negative correlation with percentage C and percentage C ≥ 50. For ‘responding to’ (percentage D and percentage D ≥ 50) there is no significant correlation with students’ prior education.
As far as the correlation between the nature of theory use and the year in which students are, linear regression analysis only shows a significant relation for explaining (category C). Other than what was expected, that correlation is negative (table 5.10). A more detailed analysis shows that the correlation is positive for the first year – and negative for the third year. It seems likely that a combination of the following factors can explain these correlations. First, a negative correlation has been found for explaining and mbo without mathematics as prior education, and a positive one for explaining and vwo with mathematics (table 5.9). In addition, the first-year students are mainly vwo with mathematics students and many of the third-year students have mbo without mathematics (table 5.1).
Table 5.10
Correlation nature and year of study Beta Sig.
Percentage A (factual description) and year of study 0,096 0,134 Percentage A ≥50 and year of study 0,079 0,222 Percentage B (interpretation) and year of study 0,010 0,879 Percentage B ≥50 and year of study -0,054 0,408 Percentage C (explanation) and year of study -0,169 0,008
Theory-enriched practical knowledge in mathematics teacher education
Table 5.10
Correlation nature and year of study Beta Sig.
Percentage C and study year 1 0,128 0,046
Percentage C ≥50 and study year 1 0,079 0,226
Percentage C and study year 2 0,070 0,273
Percentage C ≥50 and study year 2 0,086 0,186 Percentage C and study year 3 -0,157 0,013
Percentage C ≥50 and study year 3 -0,136 0,036
Percentage D (‘responding to’) and year of study 0,105 0,101 Percentage D ≥50 and year of study 0,068 0,292
In summary we can say that the characteristics for the nature of theory use that were assumed in hypothesis 1.2 mainly occur for factual description (category A) and explaining (category C). This happens particularly for students with a prior education of
mbo without mathematics (more factual description, less explaining) and students with
vwo with mathematics (more explaining).
As far as the variable year of study – and particularly for years 1 and 3 – the results (table 5.10) confirm the characteristics that were mentioned for the category explaining, keeping in mind the specific composition of the student population (first year mainly vwo with mathematics, many mbo students without mathematics in the third year).
Data analysis and results hypothesis 1.3
Linear regression analysis shows a clear confirmation of hypothesis 1.3 for explaining
teaching situations (table 5.11). A significant positive correlation appears between the percentage C – and percentage C ≥ 50 as well – and the number of theoretical concepts. The significant negative correlation between factual description – and to a lesser degree
interpreting – and the number of theoretical concepts, can be seen as additional support for that confirmation.
No significant relationship has been shown for responding to situations. Perhaps the learning environment has been an influence to the extent of responding to situations shown by students, although it seems unlikely that this influence is dominant, since the meetings and the individual study materials did not just give attention to explaining, but also to responding to situations. It is also possible that responding to situations is a habitual action, something that appears to require no theory.
Also remarkable are the strong correlation between explaining (both percentage C and percentage C ≥ 50) and the amount of general pedagogic concepts used, and the absence of any correlation between the nature of theory use and the amount of
pedagogic content concepts. There was an expectation of a difference in use between general pedagogic and pedagogic content concepts, but not this large. This point
requires closer analysis. Perhaps the level of theory use plays a part; we will look into this in research question 2.
We see another ‘opposite’ in the significant negative correlation between factual description and the number of general pedagogic concepts used, as well as a negative trend in relation to the correlation between the percentage interpreting and the amount of theoretical concepts.
In connection with the outcomes of hypothesis 1.2 (table 5.9), it is to be expected that mbo students without mathematics will use fewer concepts and vwo students with mathematics will use more. Linear regression analysis does in fact show a significant negative correlation between students who have mbo without mathematics as their prior education and the number of general pedagogic concepts used (table 5.12). No correlation exists between vwo with mathematics and the number of general pedagogic concepts.
Table 5.11
Correlation nature of theory use and number of used concepts Beta Sig.
Percentage A (factual description) and number of theoretical concepts -0,197 0,002
Percentage A ≥ 50 and number of theoretical concepts -0,197 0,002
Percentage A and number of general pedagogical concepts -0,256 0,000
Percentage A ≥ 50 and number of general pedagogical concepts -0,217 0,001
Percentage A and number of pedagogical content concepts -0,050 0,435 Percentage A ≥ 50 and number of pedagogical content concepts -0,101 0,118 Percentage B (interpretation) and number of theoretical concepts -0,130 0,041
Percentage B ≥ 50 and number of theoretical concepts -0,101 0,120 Percentage B and number of general pedagogical concepts -0,107 0,094 Percentage B ≥ 50 and number of general pedagogical concepts -0,169 0,287 Percentage B and number of pedagogical content concepts -0,116 0,070 Percentage B ≥ 50 and number of pedagogical content concepts -0,106 0,101 Percentage C (explanation) and number of theoretical concepts 0,218 0,001
Percentage C ≥ 50 and number of theoretical concepts 0,159 0,014
Percentage C and number of general pedagogical concepts 0,262 0,000
Percentage C ≥ 50 and number of general pedagogical concepts 0,209 0,001
Percentage C and number of pedagogical content concepts 0,086 0,181 Percentage C ≥ 50 and number of pedagogical content concepts 0,039 0,552 Percentage D (‘responding to’) and number of theoretical concepts 0,054 0,400 Percentage D ≥ 50 and number of theoretical concepts 0,003 0,961 The final result can be explained as follows. The group of students with vwo-with mathematics as their prior education mainly consists of first-year students who have an
Theory-enriched practical knowledge in mathematics teacher education
as yet undeveloped pedagogical (content) jargon. Furthermore, they have not yet gained much experience in reasoning about teaching situations, which is expressed in the fact that these students explain less (see table 5.9) than might be expected on the basis of their prior education.
Another remarkable point is the significant negative correlation between students with prior education mbo without mathematics and the used number of general pedagogical concept in opposition to the significant positive correlation between students with mbo with
mathematics as their prior education and the number of general pedagogical concepts used (table 5.12). It puts the group of students with mbo without mathematics in a special light.
Table 5.12
Correlation pre-education and number of concepts used Beta Sig.
MBO without mathematics and number of general pedagogical concepts -0,142 0,026
MBO with mathematics and number of general pedagogical concepts 0,154 0,016
VWO with mathematics and number of theoretical concepts 0,051 0,424 5.4.3 Analysis and results of the second research question
The second research question is:
What is the theoretical quality of statements made by the student teachers when they describe practical situations?
Considerations for research question 2
As far as the level of the average percentages scored by the students for the levels of use of theory, it is impossible to give a considered opinion. At most it seems reasonable to predict that the average percentage for level 3 will be the lowest, simply because level 3 is the hardest to reach, highest level.
Concerning the use of theoretical concepts, it is obvious to assume that there is a relationship, both in the initial and in the final assessment, between the level of theory use and the number of concepts that is applied. For both the initial and the final assessment, for the teaching situations, respectively the units, the levels of theory use have been determined on the basis of a definition where theoretical concepts are the determining factor for the level (section 5.3.6.4). The more concepts are used, the higher the chance to score level 3 and vice versa. It might also be the case that students who