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Chapter 4 – Case studies

This chapter describes the IEWs’ attendance (Section 4.1) and changes in the IEWs’ characteristics (affects and beliefs with respect to mathematics and mathematics education, knowledge of mathematics and pedagogy, and tutoring skill) over the time of Block A of the Train a Maths Tutor Program (Sections 4.2 and 4.3). Tables are used to summarise attendance and pre-post characteristics data collected via interviews and observers’ comments over the time of the Program. Pseudonyms are used for the IEWs in the tables and discussions (Section 4.4). Similar descriptions are given for Block B (Section 4.5). Block A provided much more worthwhile data in terms of the Program’s impact on mathematics teaching and learning than Block B because of its longer duration (four weeks as opposed to one week for Block B) and its major focus on translating training to student classroom trialling.

Block A – Attendance

The researchers’ prior experiences at Woorabinda Aboriginal Community had shown that attendance at any organised function could not be relied on even if personal invitations were issued (where positive responses were given), and community advertising undertaken. This was also the case in this project with respect to parent attendance (see Section 1.3). Therefore, the regular attendance of IEWs in the Train a Maths Tutor Program, particularly in Block A, was an important indication of the success of the Program.

Attendance data

Eleven IEWs attended the Program. These consisted of four of the five possible IEWs from the High School and all (seven) of the IEWs from the Primary School whose teacher allowed them to attend.

Even though they had to be cajoled initially to attend (see Section 3.2.1), once started, the IEWs were willing and active participants in the training and trialling sessions. Observers, particularly the Wadja Wadja High School teachers, continuously commented on the enthusiasm of the IEWs and their willingness to try everything. The feedback from the IEWs also referred to their appreciation of the practicality of the activities and their enjoyment of the training sessions.

Table 8 shows the attendance of the IEWs across the four weeks of Block A of the Train a Maths Tutor Program. Attendance is indicated by a tick, absences by a cross and absences due to personal or contextual circumstances by shading. With respect to personal circumstances, in Week 4, Pam’s son was hospitalised so she was absent from Woorabinda for the last week. Ron started late because he was unaware of the Program having been away from Woorabinda when it started. Mel and Delia missed the first three days of Week 1 because they thought they could attend only on their rostered days at the primary school. With respect to contextual circumstances, Nicole and Ron each missed three days across Weeks 3 and 4 because of a problem in the primary school. In Week 3, Wadja Wadja High School sent a contingent of dancers to a NAIDOC celebration at Biloela on the Thursday. As key organisers and/or dancers, Kim and Victor were absent for two days in Week 3 whilst Pam (dancer) was away for one day.

Table 8: IEWs’ attendance across the four weeks of Block A training and trialling

	Week 1 11-14 August				Week 2 18-21 August				Week 3 25-28 August				Week 4 1-4 September
TUTORS	M	T	W	Th	M	T	W	Th	M	T	W	Th	M	T	W	Th
Katie
Kim					x	x	x	x
Victor									x	x
Pam
Shelley
Nicole
Dixie									x
Rosie											x	x			x
Mel								x				x
Delia								x				x
Ron

Note. Shading indicates absence due to personal or contextual circumstances. Other absences are indicated by x.
When these special personal and contextual circumstances were accounted for, attendance was 90% for Block A of the Train a Maths Tutor Program. This was particularly gratifying considering that: (a) attendance was not compulsory (from the trainers’ perspective); and (b) the Program took most of the school day (Monday to Thursday for four weeks), a factor which made attendance difficult for the primary school IEWs who felt it was a long time to be away from their class. This anxiety was compounded by some teachers putting pressure on their IEWs not to attend the sessions (or to attend some days only) because they were concerned about their ability to handle the class without their IEWs.
Furthermore, the 4-week Program would have been daunting for the IEWs with their own low literacy and numeracy knowledge. In the Program, they were required to learn new knowledge about mathematics and its structure, new pedagogies for effectively teaching this mathematics, and new teaching skills with novel materials, and to trial these teaching ideas with students the day after they were introduced to them.

Observers’ comments

To the observers and the researcher-trainers, the high attendance rate across the four weeks of Block A indicated the IEWs’ thirst for learning to improve their professional capacities so that they could enhance their students’ understanding of basic mathematics concepts and processes. The teacher-observers were surprised but delighted at the IEWs’ persistent attendance, stating that it was a strong indication that they were enjoying the experiences within the Program. Thus high attendance was one of the major successes of the Program.
With respect to Block B, the observers also commented on the eagerness of the IEWs to learn how to use PowerPoint and Excel Programs to develop curriculum materials to support 2- and 3-digit numeration.

Block C – Changes in affects and beliefs

This section reports on the observed effect of the Train a Maths Tutor Program on the IEWs’ affects and beliefs with respect to mathematics (see Section 4.2.1) and to mathematics tutoring (see Section 4.2.2). The section finishes with a summary that synthesises the findings across the three mathematics components of knowledge, pedagogy knowledge and tutoring skill (see Section 4.2.3).

Mathematics affects and beliefs

According to Thompson (1992), positive affects with respect to mathematics are evidenced through motivation (interest and enthusiasm for doing and learning mathematics) and confidence (self concept of capacity to do and learn mathematics). Skemp (1978) believes that positive beliefs about mathematics are evidenced through activities and questioning that stem from viewing mathematics as a relational structured system to be reconstructed by each child rather than as a collection of disparate facts to be remembered. Furthermore, Thompson believes that mathematics is best viewed as organic (a human construction that is continually growing and dynamic). As a subjective social construction which can be reviewed and changed, mathematics can be appropriated by Indigenous peoples for empowerment and emancipation.

The IEWs were accorded pre-post affects and beliefs ratings with respect to mathematics by the researcher who had been involved in the training and trialling components across the four weeks of Block A. These ratings were confirmed or modified by the other researcher and observers based on interviews discussions, observations, videos, field notes and recollections of the IEWs’ behaviours when they first started the training and in the 4th week of the training.

Table 9 provides the consensus of the ratings given pre and post. The mean for each affective and belief characteristic shows an educationally significant increase of two or more from pre to post.

Table 9: IEWs’ pre and post ratings accorded in Block A for mathematics affects and beliefs

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

With respect to affects, the researcher-trainers rated the IEWs as quite low for motivation and confidence at the beginning of the Program and high-very high at the end of Block A of the Train a Maths Tutor Program. The low motivation rating was accorded to indicate the IEWs’ reluctance to attend on Day 1. However, subsequent high attendance may mean that the IEWs’ reluctance was due more to lack of confidence than lack of motivation. The three IEWs who were most motivated initially (Katie, Nicole, Dixie) were from the primary school. Katie was also the most confident initially and this was attributed to her being in the final year of the RATEP course.

With respect to beliefs, all IEWs began the training believing that mathematics is a collection of facts and processes to be remembered and that it was a static body of knowledge. It was gratifying to find that, over the four weeks, all but one IEW began to change their beliefs to relational and organic perspective of mathematics. This change was evident in their behaviours and questioning in the training sessions in the fourth week. Pam, whose beliefs remained fairly constant was by far the oldest IEW and it was difficult for her to change her beliefs about mathematics.

Table 9 shows that the IEWs’ affects with respect to mathematics improved to a higher level than their beliefs. This was attributed to factors such as group cohesion that the four weeks promoted and to the ethos of the learning environment where successes were acknowledge and mishaps supported. Furthermore, it is well known in the literature (e.g., English & Halford, 1995; Schoenfeld, 1986) that success in mathematics leads to changes in beliefs.

Mathematics tutoring affects and beliefs

As for mathematics, positive affects with respect to mathematics tutoring were evidenced through motivation (interest and enthusiasm for tutoring students) and confidence (self concept of capacity to successfully tutor students). Positive beliefs with respect to mathematics tutoring were evidenced by tutoring that was constructivist (facilitating knowledge construction through materials and questions as opposed to directly transmitting knowledge) and structural (oriented towards meanings and connections, not focusing on procedures alone - Thompson, Phillip, Thompson, & Boyd, 1994).

Pre and post ratings were assigned to each IEW with respect to their mathematics tutoring affects and beliefs in the same way as ratings were assigned for mathematics affects and beliefs. Again it was evident to all that there had been large positive changes from pre to post.

Pre-post ratings

Table 10 summarises the pre and post ratings accorded the IEWs with respect to mathematics tutoring affects and beliefs in Block A. The means for each of the affective and belief characteristics show an educationally significant increase of two or more from pre to post. Most pre ratings averaged between 1 and 2.5 (very low-low) and all post ratings averaged above 4 (high-very high).

Table 10: IEWs’ pre and post ratings accorded in Block A for mathematics tutoring affects and beliefs

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

As for mathematics affects, the IEWs who started Block A with the highest rating (3) with respect to mathematics tutoring affects were from the primary school and again Katie (RATEP trainee) was the only IEW to exhibit reasonable confidence in her ability to tutor in mathematics.
Therefore, it was gratifying to see the increased motivation and confidence in teaching mathematics that were exhibited by the other IEWs. Shelley was a huge success story. She was an IEW at the high school and had almost no confidence in herself as a mathematics tutor initially (and hence was reluctant to teach mathematics) but this changed across the four weeks. The change was attributed to the support of both the trainers and the tutor cohort themselves. During the training and the reflections after the trials, the IEWs learnt to laugh at, and value, their mistakes as well as their successes. This behaviour cannot be under-rated as it was considered by the trainers to be a major contributing factor to the success of the entire Program.

Overall, Table 10 shows that the changes in the IEWs’ affects and beliefs were both significant and similar to each other. These changes were very evident to observers even in the trials, many of the IEWs were at ease in the tutoring sessions and used questioning beyond that provided in the trial aide-memoire to elicit their students’ knowledge. Their motivation and confidence improved and their natural tutoring style revealed an innate belief in the constructive approach to teaching and to the structural nature of mathematics learning.

Observers’ comments

The effectiveness of Block A in terms of affects and beliefs was strongly reinforced by the comments of observers and delayed interviews with teachers and administrators. A common response by observers, whether they were critical friends, visiting scholars, teachers or the researcher/trainers, was delight at the “thirst for knowledge” in the behaviour of the IEWs during training. This reaction was seen as very different to their observed behaviour in the classrooms before the training where they appeared uninterested in learning. One observer remarked that it was “like a dam had burst and there was a pouring out of interest”; another said it was “like rain falling on land after a drought”. The position of all observers was summed up by critical friend, Lorrie Maher, who claimed that the successful outcomes of the program were evidenced in the IEWs’ increasing enthusiasm over the time of the project, their attendance when their work situation allowed and their wanting to do subsequent professional learning workshops.

The delayed-post interviews were particularly supportive of the changes in affect and beliefs. One High School teacher, commenting on the change in her IEW as a result of the Train a Maths Tutor Program, stated that he appeared much more confident in working with students and, as a result, was now much more active in the class, proactively moving around assisting students with their difficulties. The school administrators argued that the growth in IEWs’ confidence had benefited both students and IEWs, stating:

The IEWs have been more successful in their work with students and far more confident in the way they deal with students generally in the classroom. The kids have also responded positively.

As described in Chapter 3, the Train a Maths Tutor Program focused on reducing the shame of being successful and making mistakes when learning mathematics and teaching students. The reduction of this shame factor was also observed and reported to be a success by observers.

Summary across affects and beliefs

The IEWs’ motivation to learn about mathematics and their confidence in their ability to do so changed dramatically and positively over the 4-week duration of training and trialling within Block A of the Train a Maths Tutor Program as did their beliefs about the relational and organic nature of mathematics. Equally dramatic and positive were the changes in IEWs’ motivation to tutor and their confidence to do so. This is evident in Table 11 which summarises the pre-post means and mean changes for affects and beliefs with respect to mathematics and mathematics tutoring.

Table 11: Pre-post means and mean changes for affects and beliefs with respect to mathematics and mathematics tutoring

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

The positive changes to all affects and beliefs were very similar, but Mathematics beliefs had lower pre-ratings than the other three characteristics (Mathematics Affects and Tutoring Affects and Beliefs). This reflects the instrumental and positivist views of mathematics all IEWs held at the start of the training where they exhibited little bits of unconnected knowledge of mathematics (see Section 4.3.4) and therefore they did not show evidence of knowing mathematics as relational and organic.

The increases in affects and beliefs were important for the cognitive characteristics of the IEWs. As Alexander, Jetton and Kulikowich (1995) claim, increase in interest and motivation has a positive effect on acquiring new knowledge and competence, which, in turn, leads to further increases in affect. This would seem to be the case for the IEWs in Block A of the Train a Maths Tutor Program.

Block A – Changes in knowledge and skills

This section reports on the observed effect of the Train a Maths Tutor Program on the IEWs’ mathematics knowledge (see Section 4.3.1), mathematics pedagogy knowledge (see Section 4.3.2) and to mathematics tutoring skill (see Section 4.3.3). The section finishes with a summary that synthesises the findings across the three mathematics components of knowledge, pedagogy knowledge and tutoring skill (see Section 4.3.4).

The training sessions provided the mathematics and pedagogy knowledge; the trialling sessions provided classroom application of the knowledge gained in training. The mathematics knowledge that was the focus of Block A training and trialling were the numeration processes of reading, writing, counting and seriation (1 more/less, 10 more/less), and place value relationships with respect to 2- and 3-digit whole numbers. The mathematics knowledge was then translated to mathematics pedagogy knowledge and then both knowledges were translated to tutoring skill in practical classroom application.

It was evident to all observers and trainers that there were positive increases, equally as dramatic as for affects and beliefs in the IEWs’ cognition with respect to mathematics and mathematics pedagogy knowledge and the transfer of these cognitions to tutoring skill across Block A and, similar to affects and beliefs, moderate increases in cognitive abilities in Block B.

Mathematics knowledge

As for affects and beliefs, the IEWs were accorded pre-post mathematics knowledge ratings (see Table 12) by the researchers and observers based on interviews and discussions, direct observations, videos, field notes and recollections of the IEWs’ behaviours when they first started the training and in the 4th week of the training. With respect to 2-digit whole numbers, Table 12 shows that the IEWs’ beginning knowledge ranged from Low to Medium and, for this reason, the study of 3-digit numbers was delayed until the final week of Block A. The post ratings indicate, however, that the IEWs’ understanding of 2-digit numbers (as evidenced in the trialling sessions) improved significantly in terms of their ratings.

Table 12: IEWs’ pre and post mathematics knowledge ratings accorded in Block A

Note Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

The pre-post ratings with respect to 2-digit number understanding showed a significant increase overall in terms of the ratings accorded. Most IEWs who were unable to complete the 2-digit numeration activities early in the Program were, by the end of the Program, accurate and fluent with these activities. The pre-post ratings with respect to 3-digit number understanding also showed a significant increase overall in terms of the ratings accorded. However, the ratings indicate that the IEWs would have benefited from another week in which their understanding of 3-digit numbers could have been strengthened. Because of the extended time spent on 2-digit numbers, less time was available to focus on all the processes related to 3-digit numeration. To off-set this situation, the virtual activity development in Block B focused on comparing and ordering 3-digit numbers.

The strengths of most IEWs’ mathematics knowledge were in reading and writing numbers; the weaknesses were in place value, counting, seriation, comparison and multiplicative relationships.

Mathematics pedagogy knowledge

Table 13 provides the IEWs’ pre and post mathematics pedagogy knowledge ratings with respect to 2-and 3-digit whole numbers.
As can be see from the Table 13, growth in knowledge in both categories was educationally significant with respect to the ratings accorded with a change in rating of almost 3 in each instance. Katie, who was undergoing a Regional Aboriginal Teacher Education Program (RATEP) was the only IEW who was accorded a pre-rating above 1 and this was for 2-digit pedagogy knowledge. However, even after two years in RATEP, Katie did not know the appropriate materials to use to help students understand 2-digit numbers and did not know the teaching questions to use to focus the student on the activity with the materials.

Table 13: IEWs’ pre and post mathematics pedagogy knowledge ratings accorded in Block A

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

The strength of most IEWs’ mathematics pedagogy knowledge was in resisting telling the students the answer, that is, they tended to provide cognitive scaffolding to help students construct the appropriate knowledge. Their weaknesses were in their experience and knowledge of the materials and language for instruction (even though they had been employed as tutors to work with students) and knowledge of common student misconceptions. However, by the end of the Program, the IEWs were able to use these materials to tutor students in an effective manner.

Mathematics tutoring skill

The ability of the IEWs to effectively tutor the students during the trials was of major importance in terms of the perceived success of the Train a Maths Tutor Program and the IEWs’ self perception of their own success (which impacted on their willingness and motivation to continue in the Program).

Pre-post ratings

Table 14 provides the IEWs’ accorded pre and post mathematics tutoring skill ratings with respect to 2-and 3-digit whole numbers. The table shows that, in terms of the ratings accorded, growth in tutoring skill was educationally significant for all IEWs (averaging a rating increase of above 2). Table 14 shows that all IEWs, with the exceptions of Pam and Ron, were accorded increases of two or more levels for the post ratings. Pam’s mathematics tutoring skill was inadequate across the trials. This was attributed to the trainers’ inability to help her understand the mathematics underlying 2- digit numeration processes, the sequence of materials to use, and the questions that would elicit the students’ understanding. (Pam was unable to undertake training in Week 4 of Block A where 3-digit numbers were studied and trialled.) Her age and poor health were also possible contributors to her performance in the trials. Ron’s Medium rating was due to his limited attendance (below 50%) rather than to his ot yhe trainers’ ability. In the trials, Ron showed good tutoring skill but he was hampered by incomplete knowledge of the tasks being undertaken.

Table 14: IEWs’ pre and post mathematics tutoring skill ratings accorded in Block A

Mathematics Tutoring Skill

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

Although Rosie and Delia improved considerably (an increase of 2), their absence from training (see Table 8) had an obvious impact (to the observers) on their trialling performance. For example, Rosie had missed all of Week 2 (for personal reasons) which was a foundational week with respect to trialling 2-digit numbers, the second training-trialling cycle of Week 3 (2-digit numbers) and the final trial in Week 4. Because of contextual circumstances, Delia missed the first three days of Week 1 which the foundational week for getting to know the decimal number system, and was absent for the Thursday trials in Weeks 2 and 3. Thus, they were unavailable for observation during several trials.

Observers’ comments

All the observers (critical friends, visiting scholars, researcher-trainers, teachers and research assistants) were unanimous in claiming that they saw dramatic increases in IEWs’ knowledge and skills with respect to mathematics knowledge, mathematics pedagogy and tutoring skill.

The delayed-post interviews were particularly strong in their support of the IEWs’ cognitive growth. As two administrators from the schools argued, knowledge of mathematics and pedagogy is the most important attribute of an IEW. For the administrators, an effective IEW is one who knows:

… the curriculum and doesn’t have to be taught anything before it is taught to the students; able to pick up on what you are teaching the kids and see and understand where the teacher is going with it … an effective communicator with both the students and the teachers.

…who is able to see and diagnose the problems that the students are encountering and able to understand and communicate in the language of the students.

For the administrators, the effectiveness of IEWs was limited if they “haven’t got the education or the knowledge and so don’t really understand what it is they’re supposed to be teaching”, and “don’t have the knowledge of what needs to be taught or the strategies with which to effectively teach the students”. As the administrators stated, the “biggest problem, however, is knowledge”.

The administrators felt that the “one-on-one teaching” and “fact that they knew what they were doing and were well-skilled in what they were teaching” made the Program successful with the IEWs. They thought that the “the rapport that they [the IEWs] developed with the kids and the assistance that they were able to give the students” were an important part of the success of the Program.

Summary of knowledge and skills

The IEWs’ knowledge of mathematics and mathematics pedagogy and their skills in using these knowledges improved markedly over the 4-week duration of training and trialling within Block A of the Train a Maths Tutor Program. This is evident in Table 15 which summarises the pre-post means and mean changes for mathematics knowledge, pedagogy knowledge and tutoring skill.

Table 15: Pre-post means and mean changes for mathematics knowledge, pedagogy knowledge and tutoring skill

2-digit Numbers

3-digit numbers

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).

Table 15 shows that changes in knowledge and skills overall were similar for 2-digit and 3-digit numbers (although starting knowledge of 2-digit numbers was higher than 3-digit numbers). The table also shows that change in pedagogy and tutoring skill increased more than change in mathematics knowledge. As seen in Table 15, pre-ratings in mathematics pedagogy were lower than pre-ratings for both mathematics knowledge and mathematics tutoring skill, while the pre and post-ratings for 3-digit numbers were slightly lower than for 2-digit numbers. In summary, most of the IEWs knew more about 2-digit than three digit numbers and less about pedagogy than content.

As described in Chapter 3, the major focus of much of the Train a Maths Tutor Program was in the use of materials and language to remediate numeration understandings. The growth of IEWs’ knowledge and practice in Table 14 is, therefore, the best indicator of the success of the Program.

Block A – Summary of characteristics

In this Section, the group means of pre-post changes are compared across all characteristics (affects, beliefs, knowledge and skill) (see Section 4.4.1) and then the changes across all characteristics are compared for each IEW in terms of their relative position within the group (see Section 4.4.2).

Changes in group means

Table 16 summarises the data from Tables 10 and 14. Table 11 summarised the pre-post means and mean changes for affects and beliefs with respect to mathematics and mathematics tutoring while Table 15 summarised the pre-post means and mean changes for mathematics knowledge, pedagogy knowledge and tutoring skill.

Table 16 shows that the IEWs had higher pre-ratings accorded for knowledge of 2- and 2-digit numbers than for mathematics pedagogy, affects and beliefs. However, this knowledge appeared to comprise little bits of unconnected knowledge of mathematics (see Section 4.3.4).

Table 16: Pre-post means and mean changes for affects and beliefs with respect to mathematics and mathematics tutoring

Note. N/A means “Not Applicable”.

However, the main purpose of the Program was to help the IEWs develop effective mathematics tutoring skills. Table 16 shows that, from this perspective, the Program was highly successful for most IEWs. From the trainers’ perspective, the very positive changes noted in the IEWs’ tutoring skill first and foremost related to the substantial increase in their understanding of 2- and 3-digit numbers. This, in turn, gave them the confidence to apply their knowledge and certainly increase their motivation to do so.

The training sessions were based on the organic and relational aspects of mathematics and modelled a constructivist teaching approach which was aimed at producing structural knowledge. The fact that the IEWs learnt the mathematics themselves reinforced a teaching approach that focused more on providing cognitive scaffolding than procedural scaffolding. This, in turn, was generally modelled by the IEWs in their tutoring sessions with the students.

Changes in individual characteristics

Section 4.4.1 provided an indication of how the group changed across time relative to the characteristics of affects, beliefs, knowledge and skill. This section looks at changes for each IEW, initially across the group and then across attendance and the characteristics.

In order to do this, the data from Tables 7, 8, 9, 11, 12 and 13 were integrated so that changes could be compared more easily. Attendance was rated as >80% (High), 65-80% (Medium), and <65% (Low), based on actual days attended irrespective of whether absences were legitimate. (See Section 4.1 for reasons for absences.) The pre and post ratings across the two components of each characteristic (affect, belief, mathematics knowledge, mathematics pedagogy knowledge, tutoring skill) were separately averaged and rated on a 5-point scale from 1 (Very Low) to 5 (Very High).

Table 17 provides a summary of attendance ratings and characteristic means for each IEW across the 4-week Block A component of the Train a Maths Tutor program. Overall, the table shows that the IEWs’ pre-ratings were generally Very Low to Low for all characteristics, indicating that the IEWs, without intervention, would not have been able to effectively tutor students with learning difficulties in mathematics. It also shows that the ratings improved markedly in all characteristics for most of the IEWs and that all IEWs improved significantly across the Program.

Table 17: IEWs’ attendance mean pre and post-ratings across for all and characteristic the four weeks of Block A training and trialling

Note. Attendance ratings were accorded as follows: <65% (Low); 65-80% (Medium); >80%(High).Pre and Post ratings were accorded as follows: 1-11/2 (Very Low); 2-21/2 (Low); 3 (Medium); 31/2-4 (High); 41/2-5 (Very High).

Across the group

Because effective mathematics tutoring was the major focus of the project, analysis of results across the IEWs will be in terms of relating attendance, affects, beliefs and knowledge to tutoring skill.

Tutoring skill and attendance

Across the IEWs, Table 17 shows that tutoring skill improvement was evidenced in all IEWs but was uneven. Seven IEWs (Katie, Kim, Victor, Shelley, Nicole, Dixie and Mel) reached a rating of High-Very High in tutoring skill by the end of Block A while four IEWs (Pam, Rosie, Delia and Ron) did not.

There was some evidence in Table 17 that attendance was related to final tutoring skill. The IEWs with the lowest final tutoring skills (Pam, Rosie, Delia and Ron) were amongst the low attendees, while most of the IEWs with high final tutoring skills (Katie, Victor, Shelley, Nicole and Dixie) had high attendance. This relationship was not uniform as Kim and Mel had low attendance and high final tutoring skill. However, both of them had high pre-ratings of 21/2 in mathematics knowledge (as opposed to the Very Low ratings of 1-11/2 accorded Pam, Rosie, Delia and Ron) which may have helped their progress.

Affects, beliefs and tutoring skill

With respect to mathematics, all IEWs showed growth in confidence and motivation (affects) and in the relational and organic views of mathematics (beliefs) across the four-week block. Generally, these increases were reflected in tutoring skill. This was not so in all cases; Kim exhibited Very High-High tutoring skills but showed the smallest growth in affects. Kim, however, exhibited substantial growth in her beliefs about mathematics.

Although their absences from some trialling sessions towards the end of the Program made it difficult to determine growth, Pam, Rosie and Delia (the three IEWs with the lowest final tutoring skill) did not exhibit in their tutoring sessions the growth in affects and beliefs exhibited in the training sessions. In Pam’s case, this was attributed to her age and health. In Rosie’s case, this was attributed to her having not attended any training and trialling sessions in Week 2 (the first week of trialling) and for two days in Week 3. Therefore, it was difficult for Rosie to develop not only good tutoring skills and therefore confidence but also the steady growth in rapport with the students that the other IEWs were able to do. In Delia’s case, her absences were compounded by having to share some trialling sessions (because of inadequate number of students) with a more dominant personality (Mel) who had very good tutoring skills. In these situations, Delia’s role was more one of onlooker. Although a change in Delia’s beliefs about the relational and organic nature of mathematics and about the constructivist approach to teaching and structural knowledge as the outcome of learning was positive, Delia tended to exhibit procedural scaffolding to the students she was teaching, indicating that, to her, the end product of learning was procedural rather than structural knowledge.

Knowledge and tutoring skill

With respect to both mathematics and pedagogy, Table 17 shows that, in general, knowledge and tutoring skill were closely allied. In the training sessions, Table 17 also shows that pedagogy knowledge was the area of largest growth for all IEWs. All began the four-week block with a pedagogy that was based on telling or questioning but, by the end of the block, three IEWs (Victor, Shelley and Mel) had acquired Very High mathematics pedagogy with respect to the role of language, appropriate materials and sequencing and four IEWs (Katie, Kim, Nicole and Dixie) had acquired High pedagogy knowledge.

Across attendance, characteristics and IEWs

In summary, the main findings from the Table 17 data was that although the IEWs were different in the ways the ratings distributed across the characteristics, high rating growth in one characteristic tended to result in high ratings elsewhere and that the Program was successful in improving the ratings of most IEWs. The results do not highlight any one characteristic (including attendance) as being more important than others in terms of tutoring skill. Rather, as shown by Table 17, there is a need for all characteristics, indicating an interplay of affects, beliefs and knowledge with respect to tutoring skill.

Relationships

With regard to attendance and characteristics of the IEWs, Table 17 provides some tantalising evidence of relationships and non-relationships between attendance and the characteristics (affects, beliefs, knowledge and skill). First, there is some indication that attendance and affect impact on knowledge and skill. In general, high attendance resulted in relatively higher increase in ratings in affects, beliefs, knowledge and skill and vice versa for lower attendance, while high affect ratings and high knowledge and skill tended to go together. However, there were exceptions. Kim Mel and Ron had low attendance but reasonable knowledge and skill, but they started with better mathematics knowledge than the other three IEWs with low attendance (Pam, Rosie and Delia). Kim and Delia were particularly interesting as they had similar reasonable motivation and confidence re mathematics and mathematics tutoring but different positions re knowledge and tutoring skill.

Second, Table 17 provides some indication that ratings in mathematics relate to ratings in mathematics pedagogy and tutoring skill, particularly in terms of post-ratings. This relationship was particularly strong for the two extremes in the IEWs: Pam, Rose and Delia (relatively low pre-ratings and changes in ratings) and Victor, Shelley, and Mel (relatively high pre-ratings and changes in ratings).

Improved ratings

When Pam (who was quite elderly and in ill-health) is omitted from the analyses of the characteristics, all but two IEWs (Rosie and Delia) were rated above 3.5 in final tutoring skills. The main difference between those who scored excellently and those who made less progress was in the use of materials. Although all the IEWs were given materials in the planned activities, some did not “peel back” to the materials when needed in more abstract activities. For example, when comparing 2-digit numbers with playing cards, some IEWs used the place value chart and bundling straws or MAB to help those students whose understanding of the comparing process was tenuous. Others tended to ask which digit was larger without providing a visual representation of the tens and ones. (thus reverting to normal practice). Knowing how to use materials does not necessarily translate to knowing when to use the materials.

Three interesting cases were Kim, Shelley and Ron. Kim had ratings of High to Very High for all characteristics including tutoring skill but was not motivated to teach mathematics. (She was no longer tutoring students in the following year.) Apart from reasonably good knowledge of 2- and 3-digit numbers and a perfect attendance record, Shelley rated Very Low initially on all characteristics and of all the IEWs, Block A had the greatest positive impact on Shelley. (Shelley had left Woorabinda by the time of Block B and has not returned in the 12 months since.) Ron was an interesting case as he had the lowest attendance but was rated as 3.5 on final tutoring skills. Ron, however, had Very High affects and beliefs with respect to mathematics tutoring allied with High mathematics and pedagogy knowledge.

Block B – Attendance and characteristics

Block B of the Train a Maths Tutor Program ran for one week (four days) only so is not seen to have the significance of Block A. However, changes in characteristics were identifiable across the week. This section provides attendance details (Section 4.5.1), IEWs’ ratings of the Program and self ratings with respect to characteristics (Section 4.5.2), changes in characteristics (Section 4.5.3), and a summary of findings (Section 4.5.4).

4.5.1 Attendance

Eight IEWs attended all or part of the training and development sessions in Block B. Table 18 shows their attendance across Block B, indicating attendance by a tick, absences by a cross, and absences due to personal or contextual circumstances by shading. The table shows that, excluding the shading, attendance was almost 80%, and 88% (if Shane who came for the first day only is not included).

Table 18: IEWs’ attendance across the one week of Block B training

Note. * indicates IEWs who had participated in Block A Training. Shading indicates absence due to contextual circumstances. O indicates unexplained absence.

In Block B, seven of the eight IEWs in Block B were from Woorabinda primary school and one from the Wadja Wadja High School. Yet, in Block A, this distribution was reversed; four IEWs were from the primary school and seven were from the high school. The one IEW from the high school, Linda, was relatively new to the school and had quite good computer knowledge.

The lack of IEW participation from the high school was mainly due to the turmoil in the school due to disagreements between the School Council and staff, which saw the principal and all teaching staff but two not re-employed in the following year. As well, the changes in attendance were also due to: (a) the turnover of IEWs in the weeks between Blocks A and B (three of the original 11 were no longer in Woorabinda); (b) the lack of advertisement for Block B (not advertised to the extent as Block A); (c) the perception that Block B did not have the same prestige as Block A (Block B was one week only and lacked a graduation ceremony); (d) the training times of Block B (1-4 p.m. each day was not seen as convenient by the IEWs); and (e) the technophobia (i.e., fear of computers) of some IEWs.

Similar to Block A attendance (see Section 4.1), the observers (researchers, teachers, research assistants, critical friends, visiting scholar) commented that the high attendance rate indicated the success of Block B. It was obvious to the observers that the IEWs attending were pro-computers and eager to know how to use PowerPoint and Excel. In particular, the teachers believed that the high attendance was a strong indication that the IEWs were enjoying the Program.

Ratings and self-ratings

At the end of Block B of the Program, six of the eight IEWs (Dixie, Mel, Linda, Ron, Dick & Trevor) were interviewed with respect to their experience with computers, their perceptions of Block B of the Program, and their beliefs about the future. The IEWs gave ratings on a 5-point scale (1 – Very Low to 5 – Very High) to their beliefs and self ratings with the same 5-point scale on their abilities.

Table 19 summarises the IEWs’ responses to the interview and ratings and self-ratings with respect to the one week of Block B computer training.

Table 19: IEWs’ responses and self-ratings with respect to the one week of Block B computer training

Note: The IEWs ratings and self ratings were by five-point scale (1-very low to 5-very high).

* Mel was not working at the time of the training and therefore these questions were not applicable to her situation.

** Dixie’s classroom had no computers hence her response of N/A.

Although not as strong as the growth in rating in Block A, the self-ratings show a reasonable increase in the IEWs’ perceptions of their own computer knowledge and skills for the one week of training of Block B. Most of the IEWs rated themselves at High to Very High with respect to the mathematics-educational uses of computing covered in the training.

From their responses to the interview questions, the IEWs indicated that they were happy with the computer training, wanted further training, thought that computers were useful for teaching mathematics, and felt able to use the virtual materials with students. Although not as dramatic as for Block A, this feedback indicated improved affects and beliefs. The IEWs also generally supported computers as learning tools. Some comments made by the IEWs were: Computers motivate the students because they are in colour and they can choose what to do (Dixie); It is easier and provides hands-on and written opportunities simultaneously - the graphics are motivating (Linda).

Changes in characteristics

In this section, the pre-post ratings accorded the IEWs in Block B are provided, discussed and analysed in terms of computer learning and teaching. Ratings were accorded to those IEWs whose attendance was >50%. These were for: (a) affects and beliefs with respect to computers and computer learning and teaching, and (b) knowledge of PowerPoint and Excel and ability to develop worthwhile mathematics activities. As for Section 4.2, the IEWs were accorded pre-post ratings on a 5-point scale with 1 Very Low and 5 Very High. This was done firstly by the researcher who led the training in Block B. These ratings were confirmed or modified by the other researcher and observers based on interviews discussions, observations, videos, and field notes.

Computer as teaching tool

The affects with respect to computers were motivation and confidence - motivation to understand how to use computers and confidence in ability to do this (similar Section 4.2). The beliefs with respect to computers were learning-teaching value and student control - value that IEWs placed on the computer as a learning and teaching tool and control envisaged by IEWs in terms of whether students controlled the computer (as opposed to being controlled by the computer). Table 20 provides the pre-post ratings accorded the IEWs for these affects and beliefs. Rosie, Dick and Shane are not included because they absent for two or more of the four training days.

Table 20: IEWs’ pre-post ratings for affects and beliefs with respect to computers

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).
N/A= not applicable (≥50% absence).

With respect to affects, Table 20 shows that the IEWs were very motivated towards computer usage at the beginning of the week and even more so by the end of the week. It also shows that their confidence level rose dramatically across the four days of Block B. However, the interview revealed that no IEW had computer access at home and only three IEWs (Linda, Dick, Trevor) were able to access a computer at work although Linda (high school) could only access a computer if classes weren’t timetabled. With such limited access, it is surprising that the IEWs’ initial confidence level was as high as it was. However, the interview also revealed that all but one IEW (Dixie, primary school IEW) had had some form of computer training although this had taken place at high school or in a TAFE course (Linda). None had had training in PowerPoint or Excel.

With respect to their beliefs about the computer as a teaching tool, Table 20 shows that, initially, most IEWs did not think the computer had much educational value in terms of mathematics. They nearly all said that students at both the high school and the primary school used the computer mainly for playing “thrill and kill” drill and practice games (CD-ROMs) or listening to music. The IEWs’ mean rating of 2.4 (Low) for the computer’s educational value in terms of teaching and learning mathematics had changed to 4.4 (High-Very High) by the end of the week. With respect to their beliefs about student and teacher control of the computer (i.e., using the computer as just another mathematics tool), Table 20 shows that, initially, no IEW had thought of the computer in terms of control. In their experiences, the teachers showed students how to use the computer for word processing or allowed them to use the computer as an entertainment tool (playing games, listening to music) if they finished set work early, as a “baby-sitting” tool (particularly after school) or the aforementioned drill and practice games. Thus, in the IEWs’ experience, teachers bought the software and computers did the “teaching”.

By the end of the week, the IEWs’ mean rating for control was 4.0 (High). This was attributed to their changed beliefs about the educational value of computers when used as a tool, particularly for activities that cannot be done without computers (e.g., the interaction between data and representation using Excel). Discussion on the final day of Block B also revealed that the IEWs thought it was a good idea to be able to make their own virtual mathematics activities because they could make sure that the activities matched the particular student’s stage of learning.

Computer learning and teaching

The IEWs were also accorded pre-post ratings with respect to affects and beliefs (5-point scale with 1 Very Low and 5 Very High) in relation to computers and learning and teaching. Similar to Section 4.2.2, affects were motivation and confidence to develop virtual materials, and beliefs were with respect to the level of constructivism in the ways the virtual activities were developed and the structural focus of the virtual activities. Table 21 summarises the pre-post ratings for these affects and beliefs for those IEWs who had attended for 3 or more days.

Table 21: IEWs’ pre-post ratings for affects and beliefs with respect to learning and teaching with computers

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).
N/A = not applicable (≥50% absence).

As Table 21 shows, the IEWs initially were highly motivated to apply their PowerPoint and Excel knowledge to developing mathematics activities. This motivation was maintained and increased in some instances across the four days of training in Block B.

With the exception of Linda, who had a good working knowledge of PowerPoint and Excel, Table 21 shows that all but Linda initially exhibited Very Low to Low confidence in their ability to develop pedagogically sound mathematics activities using PowerPoint and Excel. As indicated in Section 3.6, the IEWs were initially unable to transfer their mathematics pedagogy knowledge to the computer environment. However, when shown that the activities were merely replicating (or amplifying in the case of Excel), their confidence increased dramatically to Very High (for those who attended all or most of the sessions).

Table 21 also indicates that the IEWs’ beliefs about the constructivist nature of the computer as a teaching tool underwent a substantial positive change (Very Low-High) as did their beliefs about the computer’s ability to help students construct structural knowledge (Very Low-Medium/High). However, whilst the replication activities undertaken in PowerPoint gave the IEWs confidence in their computer skills and their ability to translate their mathematics pedagogy knowledge to this medium, it was the power of the amplifying activity undertaken in Excel that excited the IEWs the most. The comparison activity undertaken in Excel (see Section 3.6) enabled them to experience firsthand the dynamic and instantaneous interaction between data and representation (structural knowledge), an interaction that cannot be shown on paper.

Changes in knowledge and curriculum development

The IEWs were accorded pre-post ratings with respect to knowledge (5-point scale with 1 Very Low and 5 Very High) of computers in terms of PowerPoint and Excel and their knowledge of how to construct virtual mathematics activities that produce structural knowledge. (Rosie was included in the ratings for PowerPoint only as she had attended both of these days.) Table 22 summarises these ratings. It shows dramatic increases in ratings (+2.7-+3.1) for all four knowledges (of the PowerPoint and Excel applications and using PowerPoint and Excel to develop virtual activities). All the IEWs who attended showed obvious increases in knowledge of programs and how to use them to teach mathematics.

Table 22: IEWs’ pre-post ratings for knowledge of computer programs (PowerPoint and Excel) and how to construct virtual mathematics activities

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High).
NA = Not Applicable (³50% absence).

With respect to knowledge of the computer programs, PowerPoint and Excel, Linda alone had some knowledge but very limited. However, after just 11/2 hours of training in PowerPoint and 11/2 hours of application, the IEWs were able to use textboxes and fonts, draw, copy, paste, move, enlarge, group and ungroup objects, change the colour of textboxes, shapes and lines, and change line thickness and font size to focus students. During the 2-21/2 hours of Excel training, the IEWs learnt to transfer the skills they acquired in PowerPoint as well as learning about the rows and columns layout of Excel, how to change the format of the cells, how to enter data, how to create a graph using the Chart Wizard and how to embellish the chart so that it was more attractive to students. They explored the different graphs in terms of data presentation, were shown how to add labels to the graph, and how to change the scale. They were also given the opportunity to activate the addition formula. Whilst this training did not provide full understanding of either program, it did provide the IEWs with the knowledge they would need to construct suitable replicating and amplifying activities.

With respect to applying this knowledge to creating virtual activities, the IEWs needed to be provided with example of mathematics activities before they felt comfortable with developing their own activities. Once shown, they produced worthwhile activities where they used colour, line thickness, font size to focus students on the salient learning embedded in the activities. They understood the need to keep text at a minimum for their students. This was a non-trivial task as it required knowledge of the pertinent concept to be developed.

Summary of changes

Table 23 summarises the pre-post means and mean changes for affects and beliefs (computers and computer learning and teaching) from Tables 19 and 20 and for knowledge (computers and virtual materials) from Table 22.

Table 23: Pre-post means and mean changes for affects and beliefs with respect to computers and computer learning

Note. Ratings were accorded as follows: 1(Very Low); 2 (Low); 3 (Medium); 4 (High); 5 (Very High). N/A = Not Applicable (≥50% absence in terms of PowerPoint and Excel training).

Overall, Table 23 shows significant improvements (+1.8-+3.1)) in the mean ratings across all characteristics with the exception of motivation which was initially very high. Thus, Block B training, was very successful considering its short duration.

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