< All Reports | Print this Section | Back to Contents > Chapter 3 - Train a maths tutor programIn this chapter, the principles from which the Train a Maths Tutor Program was developed are described in Section 3.1. Each week of the five weeks that comprised the Program is described in Sections 3.2–3.6 in terms of the mathematics focus for each week, the IEWs’ reactions to the instruction in the training sessions, their ability to transfer their knowledge to the trialling sessions held with a small group of students, and the implications emerging from each week’s sessions. Underlying principlesThe Train a Maths Tutor Program was developed from eleven principles relating to mathematics and pedagogy knowledge, professional development, and affect. Mathematics and pedagogy knowledge principlesAbstract mathematicsPast conversations in the Community with teachers and parents had clearly indicated that the Indigenous people did not want a watered-down mathematics program for their children; rather, they wanted their children to learn the “white” mathematics. Thus, the program was aimed to help the Indigenous aides develop the abstract mathematics required to understand the decimal number system as structural knowledge. A more particular aim was to train the IEWS to be fully conversant with the mathematics and pedagogic knowledge related to teaching and learning 3-digit whole numbers (i.e., hundreds, tens, ones). Earlier experiences in the Community had provided evidence that most students from Years 3 to post Year 10 had not acquired this knowledge. VET students were generally unable to pass Level 1 mathematics because of this lack of knowledge. Kinaesthetic learningResearch into teaching and learning of mathematics in general and by Indigenous students in particular (see Section 1.3) has shown that mathematics learning is enhanced if activities involve the learner in kinaesthetic activities, particularly whole-body activities. Thus, instruction in two and 3-digit numbers needs to involve work with concrete (e.g., MAB, bundling sticks) and pictorial (e.g., place value charts, hundreds boards) material as well as virtual materials on computers. LanguageThroughout the training and trialling sessions, the IEWs were exposed to the informal and formal language of the mathematics processes being examined and asked for ways in which the terminology may be adapted to suit Indigenous learners. This was done in relation to appropriate representations (concrete, pictorial, and symbolic) and real world contexts. Professional development principlesTraining-trialling cyclesThe Program recognised the need for both training (professional development) and trialling (professional development in authentic situations) components. Thus, the Program incorporated sessions where the IEWs were given the opportunity to try their new mathematics and pedagogy knowledge in a classroom with each IEW working with a small group of students. Just-in-time supportThisis support that is provided at the moment when it is needed. It has been shown to be an essential element of professional development (Baturo, Warren, & Cooper, 2004) in enhancing student’s mathematics knowledge. Thus, the Program ensured that there was a sufficient number of mathematics educators (“experts”) present at each trialling component of the professional development to provide just-in-time support. ReflectionThe opportunity for participants to reflect on and share their trialling experiences has been shown (Baturo et al., 2004) to enhance teacher knowledge and promote positive dispositions which, in turn, enhance student mathematics understanding. Thus, the program ensured that sufficient time was allocated after each trialling session for the IEWs to reflect on, and share, their trialling experiences with their peers. DurationThe duration of the professional development needed to provide sufficient time for detailed and ongoing cycles of planning, implementing, and reflection so that the goals of the Program could be achieved. Yet the duration should not be too long to be a burden to the schools from which the IEWs were withdrawn for the better part of the school day. To reach this balance, Block A was restricted to four weeks even though another week on 3-digit numeration would have been preferable. The time on Block B was restricted to one week because this was the only time possible, a restriction that put pressure on the goals of Block B being achieved. Social principlesSuccessIt was believed that the Program’s impact would rely on the IEWs’ experiencing immediate success in their training and trialling sessions. Thus, the activities developed for them to teach to the Years 8-10 students needed to be mathematically appropriate for the students’ level of mathematics development as well as being highly motivating. To this end, the Program produced worksheets for the IEWs as aide-memoires for the trials. These notes detailed the activities and materials and specified the teaching questions required to lead the students to the desired learning outcome. Group cohesionIt was believed that the success of the Program would not only depend on the quality of content and delivery but also on the quality of the interactions where trying and making mistakes and celebrating success were valued as part of learning, that is, creating a “shame” free environment. Thus, the Program set out to build a cohesive group not only though through the learning experiences but also though sharing social occasions such as morning tea and lunch. These latter occasions took place at the Community Hall where other members of the Community were welcome to join the group. IdentityThe development of the IEWs’ sense of ownership and pride with respect to their role as mathematics tutors was believed to be important to the success of the Program n terms of continued attendance, Community recognition and desire for further training. Therefore, the Program was designed to operate within the Community (see Learning Space) so that the Community would be aware of the IEWs’ commitment and accomplishment. To this end, the Program incorporated a Graduation Ceremony in which the Community was invited to celebrate the IEWs’ success. Learning spaceThe learning space has been shown to impact on the importance attributed by the learners to what is taught/learnt, the style of delivery, the form of interaction, and learning outcomes (Skill & Young, 2002). Thus, the Program was planned to be undertaken outside the IEWs’ work site (the schools) and within their normal hours of employment to show that the Program was so valued by the schools that they would release important personnel for 16 days from their busy routine. The Community Hall was offered as the training site, again indicating that important others (e.g., Council members, Elders) also valued the Program. The external site meant that the IEWs could act as learners instead of teachers, to make as much noise as they liked, to have fun, to ask questions, and to be free of interruptions. Week 1This week comprised training sessions only. (Trialling sessions began in Week 2.) The purpose of this week was to:
This first week was mainly to establish an overview of the types of activities that are needed to help students construct an understanding of place value to 3-digit whole numbers and the materials and language that are used to do this. The following weeks would focus in more detail on the activities undertaken in Week 1. Week 1 Day 1Setting upThis was the most crucial day in the four weeks of the Program. A path needed to be found between depth of content and enjoyment in learning so that the IEWs were challenged (not patronised) but comfortable and desired outcomes could be met. However, despite extensive personal communication with the Community and IEWs during the weeks leading to the commencement of the Program, Day 1 found that the Community Hall venue was unavailable, and the IEWs (apart from two from the primary school) were reluctant to come. Some primary school IEWs were part-time workers and thought they could come only on those days when they were rostered to work. This problem was overcome by paying them to come on their-off-rostered days. With respect to the venue (learning space), Mr Lawrence Weazel, Woorabinda Council, allowed the trainers to use the new Skill Share building that was almost finished. He believed that the Program was so important for the Community that he supplied whiteboards and other materials that were needed and offered to speak to the principals to urge them to release the IEWs for training (if needed). With respect to the IEWs, the trainer went to the high school to cajole the aides into coming for at least the first session. They confided, on the last day of the program, that they were “too scared to come” because they thought they would have do “the big maths”. Ultimately, Day 1 had seven IEWs with three more attending by the end of the week, and another IEW joining the group midway through the second week when he returned to the Community. Overall, 11 IEWs attended all or part of the 4-week Block A component of the Train a Maths Tutor Program. Each IEW was given a hard bright yellow plastic folder to collect handouts from the ensuing weeks of training and trialling. Apart from the practicality of doing this, it was also a means of establishing the Social Principles of success, group cohesion, identity and learning space. In the second week, each IEW would be given the concrete materials (e.g., bundling straws, MAB, place value charts, calculators, coloured pens, scissors) required for the training and trialling sessions. Mathematics education trainingThe first session of this day started late and yet managed to cover the essential background of the decimal number system of the Western world. The overarching purpose of this session was to help the IEWs make sense of the structure and language of the number system, and to understand the historical newness of Indigenous cultures to this particular number system. The origin of the term “decimal” was explained and related to the months of the year so that December was originally the 10 month. The IEWs were fascinated by this and it led to a discussion of how the other months were named. The term “decimal” was then linked to the 10 digits (0-9) of the number system and the IEWS were invited to think of a number that wasn’t made up of one or more of these digits. The term “digit” was explained and this led naturally to a discussion of the development of all number systems, namely, the need to have a system of keeping track of how many cows a person might have or how much land they might own. This need mainly arose when people gave up nomadic ways to settle in a community. This aspect was related to their own Indigenous culture and how this is a relatively new situation in terms of the history of other cultures. Once the need for numbers was generated, the next step was to use body parts to indicate “how many”. This was usually to use the 10 fingers (called digits) first before other body parts. Later in the development of number, number names were attached to the digits and finally number symbols were developed. To show how difficult it is for some students to reconstruct the decimal number system, an activity was undertaken to construct a new system of numbers that was foreign to both Indigenous and non-Indigenous people. We (trainer and IEWs) decided that the giant in Jack and the Beanstalk was counting when he said, “Fee, fie, foe, fum.” The IEWS were asked to say how many golden eggs were in each of the nests shown in Figure 7. Figure 7. Constructing numbers names for a new number system
Unlike many other groups of non-Indigenous educators with whom this activity has been done, the IEWs offered “feety” (inherently multiplicative) as the next counting number rather than “fee-fum” (inherently additive). They enjoyed this activity and had no trouble continuing the number names to “fumty-fum”. When asked how they would show “feety” in symbol form, they realised they would need a name and symbol to show zero. This they found difficult to do and felt justified when told that the name and symbol for “nothing” took another 400 years to come into use after the other symbols, 1-9, were known. The connection to the difficulties students have with numbers with zero/s was also made. This they recognised from their experiences with their own students. Thus the point was made that when teaching new places in the number system, avoid using zeros initially. Other number systems such as the Roman, Egyptian and Babylonian systems were shown and contrasted with the decimal (Hindu-Arabic) system. From this the IEWs developed a sense of the simplicity of the decimal number system in that, with just 10 different symbols, very large and very small numbers could be written because of its place value feature which earlier number systems had lacked and is the reason why the decimal system is still in use today in the Western world. The second session of the day was devoted to understanding the principles involved in counting (see Appendix A, Interim Report) and then to undertake activities that would promote rote and rational counting. This was aligned with a discussion on the need to have students construct knowledge through doing as it helped to develop a kinaesthetic memory of the physical actions undertaken in the learning process. A sequence of actions put forward was: whole body actions, hand manipulations of physical materials and virtual (computer) activities. The IEWs played whole-body counting games such as starting in a line with one player out the front calling out letters of the alphabet. Players take as many steps forward as the number of times that letter appears in their name. The associated mathematics was that students must take a step to count, thus associating counting with “jumps” (a difficulty many students have when working with number lines). We also played counting games such as Snakes and Ladders and Ludo. The culminating activity of the day was to reproduce Snakes and Ladders so it didn’t start from 1 or involved counting in 2s, 5s, or 10s. Discussion also centred on the starting rules, for example, not always having to throw a “6” to start because of the erroneous probabilistic thinking that can develop from this. Week 1, Day 2Setting upAll seven IEWS who had attended Day 1 turned up voluntarily for the second day of training. One more IEW joined the group and missed only one other day across Block A training. Mathematics education trainingGroupingThe first session focused on the grouping process which is required for understanding place value. An activity was undertaken with counters and then Unifix cubes where the IEWs started with a number of ones and then made into groups (e.g., 8 counters, making groups of three; 11 Unifix cubes, making groups of 4). The trainer modelled the language to be used for establishing this process with their students whilst the IEWs made the groups. Players took turns in being “the teacher”. Grouping activity example: Do you have enough ones to make a group of 3? [Yes] Show me. How many groups of 3 do you have? [One] How many loose ones? [Five] Do you have enough ones to make another group of 3? [Yes] Show me. How many groups of 3 do you have now? [Two] How many loose ones? [Two] Do you have enough ones to make another group of 3? [No] How many groups of 3 do you have? How many ones? The IEWs then compared their arrangement of the groups with other arrangements. A discussion ensued on the need to consider where the groups should be placed in relation to the ones and how a Place Value Chart (PVC) is useful in establishing the relationships of the positions. For ensuing grouping activities, a PVC was made from an A4 sheet of paper with a line drawn down the middle and headed Groups/Ones. This activity was repeated with the IEWs working in pairs with one acting as tutor and the other acting as student. Roles were reversed for the following activity The above grouping activities were repeated with Unifix cubes and a discussion led to the understanding that the Unifix were better than the discrete counters because they could be joined together and the groups would remain intact. A discussion linking to yesterday’s session about avoiding zeros was used to guide their choice of numbers used for regrouping. For example, avoid activities initially where there were no ones left after grouping. These grouping activities led to a discussion of developing the notion of the Tens and Ones places in the decimal number system and how the PVC should then be drawn up to reflect these positions. Further grouping activities were undertaken focusing on making groups of tens. Pedagogic theoryPayne and Rathmell’s (1977) model for developing concepts (see Appendix A, Interim Report) was drawn on the whiteboard and each component (real-world problem, representation, language, symbol) was discussed. The point was strongly made to delay introducing symbols as long as necessary, and to focus all initial learning on the representation–language interactions. At this stage, all the different types of representations for grouping activities were viewed and the foregoing grouping activities were undertaken with all materials focusing on making groups of 10 on a new PVC labelled Tens/Ones. Thus the grouping activities focused on the following materials, in this order: counters (separate, difficult to join, and has the attribute of colour which can be distracting); Unifix cubes (separate but can be easily joined, but also has the distraction of colour); bundling straws or sticks (separate, but can be easily grouped with a rubber band, no other attribute); MAB (separate ones and tens, ones must be physically traded for tens, has attribute of size only). The IEWs quickly realised the advantages of bundling sticks when teaching young children elementary place value. This session culminated in playing an activity with dice, bundling straws, and the Tens/Ones place value chart where each player rolled the die and put out that number of ones in the ones place on their PVC. Play continued until a player had made 4 tens. Thus the IEWs needed to know not only how to make 10 ones into 1 ten, they also need to know when to do this. A similar activity was undertaken with MABs and the discussion after the game revealed that they thought the MABs much more difficult than the bundling straws as they couldn’t make the ones into tens – rather, the ones had to be physically traded for another block. The second session focused on games such as “lose 4 tens 3 ones” which were played with bundling straws and then MABs and again the difficulties of trading MAB tens for ones were made apparent. Week 1 Day 3This day focused on reading and recording 2-digit numbers. (Comparing numbers was a secondary focus.) It started with a discussion of Rathmell’s model and recalled yesterday’s session where we focused on the sequence of materials used to represent numbers. Reading and comparing numbersThe IEWs were asked to use bundling straws to show a number larger than 20 (to avoid the language of “teen” numbers) with tens and ones so that there was something in each place on the PVC (avoiding zero). The person beside them was then asked to tell the group what number the IEW had made and how s/he knew. This led to the discussion that we build numbers from right to left but we read them left to right. It also raised the need for children to count in tens and then count on the ones (e.g., for 32: ten, twenty, thirty, thirty-one, thirty-two). The IEWs were then asked to say who had constructed the number with the largest/smallest value and how they could tell (to develop the notion that when comparing numbers, we compare like places from left to right). The IEWs were then asked to work with a partner with one acting as tutor and one as student to undertake activities such as: Show me thirty-seven. How do you know you have thirty-seven? Recording and comparing numbersThe IEWs were asked to build a number on their PVC with bundling straws, again with no zeros or teens and then to the place. that was “worth more” (the tens) and how they could tell. They were then asked to place a playing card (with 10s and picture cards removed from the deck) on the PVC below the straws to represent the number of tens and the number of ones. Next they were asked to cover the bundling straws so that only the digits were showing and read the number shown by the cards. We then discussed how, in a number such as 35, children could tell which digit was worth more. This led to a discussion of the role of concrete materials in helping students construct a picture in their mind (mental model) of the relative sizes of tens and ones. They thought that their students didn’t do enough work with concrete materials. The IEWs were then asked to build 44 on their PVC and then show the number with the cards. When the straws were covered, there were two cards showing 4 and a discussion centred on how students could tell which 4 was worth more if they didn’t have a mental picture of the materials in their mind. Thus, the activity produced the need to avoid not only zeros and teens in early place value work but also the same digits. At this stage a discussion was held as to which 2-digi tnumbers would be easiest for young students to relate the language to the numbers they already knew (i.e., the single-digit numbers 1-9). On the board, the trainer wrote the following numbers: 17, 96, 25, 82, 12, 37, and discussed that only 96 and 82 used the same sounding digits as what they already knew (nine, eight). The smallest number, 12, was actually the most difficult as it doesn’t even have a “teen-sounding” part and is a completely new number name. The use of cards initially provoked a reaction from the oldest IEW who said that there was “too much gambling” in the Community. However, by the end of this session, she was feeling comfortable with its use. We discussed how ordinary cards could be made with the digits if they felt uncomfortable with using playing cards but this decision needed to be made in conjunction with current theory which promotes using materials that are familiar to students as they will more likely be motivated to learn. The second session comprised a counting game in which representing, reading, and recording 2-digit numbers were linked. Three players were to work together – Player 1 in the group was to build numbers with bundling straws bundling straws on a PVC, Player 2 was to record the numbers in digit form on a small PVC on paper, whilst Player 3 was to enter the starting number on the calculator and to “count” by ones using the calculator keys. They were to see who had reached the number with the larger value when the trainer said, “Stop!” Before playing this game, everyone was made familiar with entering numbers on a calculator and using the operation keys. They were given time to play with the calculator. The IEWS loved this tool and were assured that each person would be given a turn at using the calculator during the game. After each game, each team was asked to say how each counting process was the same but different and what were the difficulties in playing. Most said that the player with the bundling straws “went too fast”. We discussed how it was important to have someone recording the numbers of paper as neither the straws nor the calculator showed the record of counting. The calculator users were asked how they knew they had to use + 1 when this was not showing on either the bundling sticks PVC or the recording PVC. This elicited a discussion on the power of the calculator in helping students abstract the mathematics embedded in the concrete activities. The game also focused on the odometer principle embedded in counting (i.e., the change that occurs when a place is filled by 9) and the patterns that emerged from this were highlighted and the IEWS asked to predict what would come after 69, 89, 379. The IEWs wanted to keep playing although the session had finished so we agreed to play again on the following day. Week 1 Day 4As promised, we started the first session with the counting activity undertaken the day before with the bundling straws and PVC, the recording sheet with PVC, and the calculator. However, this time we were going to start with a number (e.g., 75) and count backwards. No help was given to the people with the calculators but it was interesting that all realised in time that they had to use the operation that was the opposite (inverse) of addition. The game was further adapted to counting forwards and backwards by 10s. This led to a discussion of which place changed first when counting by tens and which place remained unchanged. It was contrasted with which place changed first when counting by ones and led to the IEWs predicting that if we counted by hundreds, the hundreds would change first and the tens and ones would remain unchanged. They wanted to try this prediction to see if they were correct so the following activity was first done in order to establish why we need to switch to MABs when dealing with hundreds. First, the IEWS were asked to represent 99 on the PVC with bundling straws. They were asked to add one more one to the Ones place and asked whether they now had enough ones to trade for a ten. This they did and were asked how many tens they now had and whether they had enough tens to trade for a hundred and where would they put the resulting hundred. This led to two discoveries: (1) the bundling straws were unwieldy when rebundled into 10 groups of 10; and (2) they needed a new place on the PVC to the left of Tens to accommodate the new place, Hundreds. This they did and the game activity was repeated with 99 represented by MAB. Two main points emerged from this training session. First, given ample time just to play and explore any new material should be allowed for. This was reinforced by the IEWs’ behaviours where this aspect of material use had been curtailed because of time constraints. Without exception, playing with the materials was one of the first things the IEWS did, particularly with the Unifix cubes, the MAB and the calculator. The second point reinforced the need to establish rote counting (language) to 99 before undertaking place value activities relating to tens and ones, and to establish rote counting to 999 before undertaking place value work relating to hundreds, tens and ones. The last session of the week was to establish the fact that the decimal number system is based on periods of three digits and that their value depends of their place in a number. The remaining concept to be developed was the multiplicative property inherent in the decimal number system, namely that adjacent places are related by 10. To make a number 10 times larger in value, we need to multiply by 10 and, conversely, to make a number 10 times smaller in value, we need to divide the number by 10. Figure 8. Activity to develop understanding of the period patterns in whole numbers
Repeating periods patternsThe IEWs were each given the following place value chart and a set of 10 small digit cards showing the digits 0-9. They were asked to look at the PVC and discuss what they thought the “H, T, and O” referred to, and to find ways in which the chart was the same and different. They were then able to do the following activity. Use the digit cards to show me two hundred and sixty-four ones on the Place Value Chart. Now show me two hundred and sixty-four thousands on the Chart. [Most slid the three cards to the left; others picked up the cards and replaced them one at a time. This may have indicated that they had not perceived that neither the digits nor the order of the digits should change but their position should. Those who had slid their digits were asked to say why.] Now show me two hundred and sixty-five millions on the PVC. [All slid the digits to the Millions period.] Now slide the digits to the left again. How would you read this number? [All said two hundred and sixty-four billions.] Figure 9. Activity to develop understanding of the multiplicative relationships in the decimal numbers system
Abstracting multiplicative patternsThey were given the following PVC which showed places smaller than one and were given a sticker to put on their chart to separate the whole numbers from the fraction parts. They also used the digit cards from the previous activity. A brief description was given of how the decimal fractions were derived and how the decimal fraction names are derived from the whole-number place names. Show me 4 ones on the PVC with the digit cards[All could.] Show me 4 tens. [All slid the digit one place to the left.] Which way did you move the 4 – to the right or to the left? [All said to the left] How many places to the left? [All said one.] Did you make the 4 larger in value or smaller in value? [All said larger in value.] The activity was then replicated with the calculator. Show me 4 ones on the calculator. [All could.] Now do something without using the digit keys to show 4 tens on your calculator. [Many did not know what to do; a few added 36.] For those who have 40 showing on your calculator, did the 4 move to the left or the right? [Left] How many places to the left? [One] If you add 36 again, will the 4 move one more place to the right? [Some thought “yes”; some were unsure.] Try it. Did it work? [No] So adding is not part of the pattern for making digits move to the left. What else can we try? [Shelley said to multiply by 10.] Let’s try Shelley’s idea. Enter 4 on the calculator, then multiply by 10. Has the 4 moved one place to the left. [Yes] Multiply by 10 again …. Did the 4 move another place to the left? [Yes] Keep going until you run out of room for the numbers. Reversing the procedureEnter 400 on the calculator. How can you make 4 hundreds move to 4 tens? [A couple said to take away 10 but the majority said to divide by 10.] Try it. …..Did taking away help? [No] Did dividing by 10 help? [Yes]. Keep going until you have 4 ones. So the side-by-side places are all related by 10. Show what you just did with the digits card on the PVC. Which way did you move the digit? Did it become smaller or larger in value? Move it one more place to the right? Is it smaller than one or larger than one? Try moving the 4 ones to 4 tenths on your calculator. This led to a very brief discussion of the decimal fractions. Figure 10. Multiplicative relationships in decimal numbers and metric measures
ExtensionThe multiplicative patterns activities were extended to show multiplying/dividing by 100 and 1000. The patterns of × 1000 were linked back to the Periods Patterns place value chart to show that multiplying but 1000 makes the digit/s jump from one period to another (e.g., 5 ones in the ones period to 5 ones in the thousands period and 5 ones in the millions period). These multiplicative relationships were shown in the following diagram which was built up on the board. This session drew attention to the inverse of addition (subtraction) and multiplication (division) as well to helping the IEWs construct an understanding of the multiplicative relationships between the places. The relationship of 10 was linked to the “decimal” in the name of the number system. The relationship of 1000 was linked to the period patterns and to metric measured (e.g., 1000 metres = 1 kilometre) Week 1 summaryAs evidenced by the foregoing activities, this week was a heavy theoretical week in terms of the mathematical structure of the decimal number system. It also raised the IEWs’ awareness of the teaching theory related to the major components of concept development (materials, language and symbol), the sequence of materials used to model whole numbers, the role of the PVC in developing an understanding or place value and the role of the calculator in abstracting the mathematics from the concrete materials and actions on those materials. The teaching theory with respect to materials was reduced to the following four main steps.
Thus the Learning/Teaching Principle of abstract mathematics was met with respect to pedagogic knowledge and mathematics in a way that exceeded our expectations. Throughout the week, the IEWs were inspiring in their thirst for knowledge. Their enthusiasm was evidenced in the way they were ready to start each day on time, in the way they attempted each activity, and in the way they listened attentively to various explanations. There is no doubt that they found the week’s training challenging but enjoyable, and felt proud of their efforts. With respect to group identity, it was obviously formed quite strongly during this first week. They helped each other whenever required, laughed kindly at each other’s attempts (there was no “shame” in either being successful or not). During the morning teas and lunches, the trainer was made to feel part of this group identity. For the trainer, it was probably the most rewarding professional development ever undertaken. Week 2Week 2 was also a pivotal week in the Train a Maths Tutor Program as it represented a change from continuous training to a cycle of training and trialling (the IEWs being trained by the researchers as well as trialling tutoring ideas with students). In this week, the IEWs were given their own set of all concrete materials that would be needed to help the students construct an understanding of the number activities to be undertaken in all trials. This material was theirs to keep after the Train a Maths Tutor Program so that they would always have the appropriate materials on hand when working with students in their school contexts. This section will describe the first week of this train-trial cycle. It will start with a description of the cycle, particularly how the relationships between the components of the cycle affected the development of the Program. Then it will describe the two cycles in the week under headings that represent the components of the cycle: pedagogic focus, mathematics focus, training, trialling, and reflections. The train-trial cycleWeek 2 incorporated two train-trial cycles. Cycle 1 was undertaken on Monday and Tuesday; Cycle 2 was undertaken on Wednesday and Thursday. The first day of each cycle (Monday and Wednesday) focused on training the IEWs; the second day (Tuesday and Thursday) focused on reviewing the tutoring materials, trialling the tutoring ideas with students, and discussion of the trial. In this way, the cycle met the Professional Development Principles from Section 3.1: training in tandem with trialling, just-in-time support (the reviewing of the tutoring materials just before the trial and expert availability throughout the trial), and reflection and sharing (the discussions of the trials). Based on their knowledge of the Years 8-10 students’ impoverished understanding of 3-digit numbers and their knowledge of the prerequisite learning that needed to be established, the trainers developed an initial plan where the major focus of training and trialling in Weeks 2 and 3 would be to establish an understanding of 2-digit numbers whole numbers (tens, ones). To accommodate different learning rates, these activities included extension to 3-digit whole numbers for those students who showed understanding of 2-digit numbers. In Week 4, with the groundwork of 2-digits numbers established, the focus was on extending the place value concept to 3-digit whole numbers (hundreds, tens, ones) as well as the numeration processes (identifying, reading, recording, counting, comparing, ordering). The training day of each cycle thus prepared the IEWs in both the mathematics of the trialling session to follow, the pedagogic techniques (how to use both the materials and questioning effectively), and pedagogic theory relating to the sequence of teaching, the common errors students make, and how to ”peel back” to their established knowledge. Peel-back usually meant to delay recording and spend more time establishing the materials-language link of Payne and Rathmell’s (1977) model of concept development (see Appendix A, Interim Report, Week 1). In Week 2, it could also mean revisiting the students’ ability to rote count to 10 and to 99. This meant that the training day had to cover more than the particular mathematics that was to be the focus of trialling. The mathematics had to go beyond the functional requirements of the trial to the abstractions that lie behind the activities to be undertaken in the trial. This included all abstractions necessary to cover extensions and “peel backs” as well as the core mathematics. The time to be spent on this learning had to be of adequate duration to ensure that the IEWs’ understanding would stand the test of tutoring. This meant looking at incorrect conceptions as well as correct. This was particularly important if the IEWs had weaknesses with the content. The training had also to cover the pedagogy of tutoring mathematics, the kinaesthetic use of physical materials and the associated language/questions for instruction. The materials could be provided to the tutor and integrated directly into the tutoring activities, but the questioning had to be contingent on the responses of the student. Learning such contingent questioning was a complex matter. The training had to be completed in a manner that built group cohesion, and in a learning place that maximised positive feelings within the IEW and built confidence towards the tutoring. The relationship between the two days meant that the Program had to be emergent and dependent on previous outcomes. That is, the tutoring material had to be dependent on the extent to which the prior training day covered content and pedagogy, while the focus of the training had to be dependent on the success of the previous trial. Thus, although a sequence of activities to be followed could be planned, the speed at which this plan was implemented depended on the knowledge, learning and tutoring skill of the IEWs and the knowledge and learning of the students. To achieve the cycle, the first day was divided into two sessions of training, 9-11am and 11.30 to 1pm and the second day into three sessions, 9-10am reviewing, 10-11am trialling, and 11.30-1pm reflecting and cleaning up. The first day of each cycle was held in the Community’s Skill Share building. The second day was held in a spare classroom in Wadja Wadja High School into which students could be brought. The afternoon was spent talking with observers, and planning and preparing materials for the next day. Week 2, Days 1 and 2Pedagogic focusThe overarching aim of thetrials (and particularly the first trial) was to ensure that the IEWs experienced the Social Principles of success and group cohesion (see Section 3.1). To this end, the activities for the students were designed to be fun so that they would be motivated to participate and cooperate, thus making the IEWs’ task easier. This was particularly so for the primary school IEWs who had not experienced teaching older students (Years 8-10). Mathematics focusWhen planning the trialling activities, the trainers balanced the needs of the students with the needs of the IEW tutors (so as to meet the Professional Development principle of training in tandem with trialling. (See Section 3.1). Thus, the activities for the students needed to be motivating, worthwhile (capable of enhancing construction of appropriate knowledge) and not too complex for this stage of the IEWs’ professional development. Prior knowledge of the Year 8-10 students’ poor mathematics performance with 2-digit whole numbers was the basis for focusing on these numbers in the first and second weeks in order to lay the foundation for 3-digit whole numbers in Week 4. The first and second activities to be undertaken with the students (and therefore with the IEWs) in the first cycle of Week 2 were designed to determine whether they had an understanding of the 100 Board. To this end, the students were first of all asked to fit jigsaw pieces (parts of the 100 Board) together to re-form the 100 Board (see Appendix A, Interim Report, Week 2, Activity 1). Next they were asked to cut out a jigsaw piece on a blank 100 Board, and write the missing numbers (see Activity 2). These activities required an ability to identify 2-digit numbers. The third activity was designed to elicit the extent to which the students understood the place value of 2-digit whole numbers by taking the top two cards from a deck of cards (with 10s and picture cards removed) and deciding which number they would nominate as the tens digit and which number they would nominate as the ones digit. The student would then identify the resulting 2-digit number on the 100 Board and cover it with a counter. The first person to cover three numbers in a row, column or diagonal was the winner. (See Activity 3.) This game also involved strategic thinking TrainingThe IEWs appeared to benefit from these activities. The IEWs from the secondary school and the upper-grade IEWs from the primary school had never seen a 100 Board, so the jigsaws were an excellent tool for developing their counting skills and to help them construct a mental model of the 100 Board. Most had to have the patterns of counting by 1s (across the rows) and by 10s (down the columns) drawn to their attention. Discussion also focused on the “teen” numbers and how students found these difficult because the language pattern in the number names is different from all other 2-digit number-name patterns. (A brief discussion on this had been undertaken the previous week but it needed to be reinforced.) TriallingOf the eight IEWs who undertook the first trial of Week 2, one had not been trained in the use of the materials, language, and questioning from the day before. However, this IEW was a RATEP teacher from the primary school and she was able to gain enough information from the review session immediately before the trial to enable her to undertake the trial efficiently and successfully. Each IEW had at least two students during Trial 1 in Week 2. One pair of IEWs combined their students and took turns in teaching. ReflectionsThe students enjoyed the activities immensely but it was obvious that they needed more teaching with respect to counting to 99, with place value (tens and ones), and with identifying the patterns inherent in the 100 Board. During the reflection over morning tea, the IEWs, particularly those from the primary school, indicated their surprise that the Years 8-10 students found these tasks difficult initially. The primary school IEWs were somewhat shocked to learn that these students whom they had taught in the primary years had not advanced. They realised that they needed to work more efficiently with their students who were at-risk and this realisation provided another impetus for their wanting to learn more about mathematics teaching and learning. Week 2, Days 3 and 4Pedagogic focusAs for Cycle 1 (Week 1, Days 1 and 2). However, in this session, students would be using specific mathematics aids (100 Board and calculators) so the IEWs were reminded of the need to ensure that students were familiar with the materials before undertaking activities. This would mean including time for the students to explore and ask questions about the materials. An important teaching principle of reversing was embedded in the set of activities in this second cycle. For example, Activity 2 required the students to focus on one number and to identify the numbers adjacent to the left, tight, above, and below. In other words, they focused on a “middle” number. In Activity 3, this was reversed as student were given the adjacent left, right, above and below numbers and had to identify the “middle” number. Mathematics focusThe major purpose of this train-trial cycle was to establish an understanding of the patterns embedded in the 100 Board, namely, that adjacent numbers in rows are related by 1 (either 1 more or 1 less) and that adjacent numbers in columns are related by 10 (either 10 more or 10 less). The calculator was needed to abstract the mathematics of these relationships, namely, +1, – 1, + 10, – 10. They also needed to ensure that students understood the language, rows and columns in relation to the 100 board. In establishing the patterns within the rows and columns through the number name language (see Appendix A, Interim Report, Week 2), the IEWs were reminded that the teen numbers do not meet the language patterns. Therefore, it was suggested that they ask the students either to read the numbers across the rows but to skip the second row or to begin from the “twenties” (3rd row). They could then come back to the teen numbers if time permitted to point out that the 2nd row of numbers do not fit the patterns of names but do fit the pattern of digits. The second activity of the trial was to have the student discover the mathematical relationships of adjacent numbers in the rows and columns and to verify this with their calculator. The verification was a means of assessing whether the students had abstracted the more/less language as +, –. The third activity was designed to evaluate whether the students could reverse the understanding developed in Activity 2 (see Pedagogic focus). The IEWs had to construct the “window” in Figure 11. Figure 11. The window used in Week 2, Trial 2
TrainingAs for Cycle 1, the IEWs appeared to benefit from these activities in Cycle 2. No IEW had undertaken these activities before and it was obvious that they too enhanced their understanding of the mathematics involved in counting patterns. The reversing teaching tool was novel to them and they appreciated the difficulty in switching from one perspective to another but, at the same time, realised its power in establishing a mathematical concept. Their enjoyment of these activities was obvious in the tenacity they showed in “getting it right”. All agreed that the window activity was “a good one, eh”. TriallingOf the seven IEWs who undertook the second trial of Week 2, all had been trained in the use of the materials, language, and questioning from the day before. More students had turned up for this session so each IEW had at least three students. However, not all students were the same as those who had attended the first trial. ReflectionsThe students found these activities challenging but were highly motivated by the materials (particularly the calculator and the window) to succeed. The IEWS themselves recognised that they needed to have spent more time in making sure the students had enough time to become familiar with materials and in establishing the language patterns (particularly for those students who had not attended the first trial). Over morning tea, the IEWs spoke about their individual experiences and were relieved to find out that most of the others had experienced similar difficulties. They expressed again their surprise at how much difficulty the Years 8-10 students had with “little numbers” and recognised the need not to rush students through early mathematics learning. They also recognised that older students didn’t feel “shamed” because they enjoyed the activities and came to be successful. Week 3Week 3 was organised in the same way as Week 2, that is, there were two train-trial cycles, each of which ran over two days. Once again, the two cycles are described under headings: pedagogic focus, mathematics focus, training, trialling, and reflections. Week 3, Days 1 and 2Pedagogic focusAs for Week 2 Mathematics focusThe major purpose of this train-trial cycle was to establish an understanding of the place value of 2-digit numbers. This required the IEWS to be familiar with Place Value Charts and bundling straws, with the process of grouping ones to make tens, and with the process of recording 2-digit numbers on a mini PVC. All of this work had been done in Week 1 when the IEWs were learners but was reviewed in this cycle in terms of the IEWS as teachers. The first activity focused on ensuring the students knew the materials, namely, the PVC and the tens and ones represented by bundling straws. (See Appendix A, Interim Report, Week 3 for the activities undertaken in this trial.) The second activity of the trial was to have the student discover how the ones were formed to make tens. The third activity required the students to represent with bundling straws on the PVC the numbers that the IEW said using informal language (e.g., 3 tens 7 ones). The students were then required to count by tens and ones to say how many altogether. For example, after showing 3 tens 7 ones, the students were expected to point to the tens and say ”ten, twenty, thirty” and then continue to count on the ones (thirty-one, thirty-two, etc). The materials-language link was strengthened through reversing the activity (e.g., the IEW would ask the student to show a number such as “forty-five” on the PVC with bundling straws). The fourth activity focused on having the students understand the link between representation of a number (materials), the language (number name) and the symbol (number recorded in digit form). The final activity in the trial was to represent the counting process with MAB and a PVC, with numbers recorded on paper, and with a calculator. The tutor represented a number with bundling straws on a PVC, the student wrote the number in digit form, and said the number. The tutor put out 1 more one on the PVC; the student wrote the number and read it. Play continued past 9 ones so the student could see how the materials changed and how the PVC “looked”. When 10 ones were grouped to make another ten and there were no ones in the Ones place on the PVC. As a first variation, the tutor and student swapped roles. As a second variation, the tutor worked with the straws and PVC whilst the student counted with the calculator. TrainingThe work done in representing 2-digit numbers with bundling straws on a PVC in Week 1 had obviously mad an impact with the aids as they were very comfortable with the regrouping process. Nevertheless, the accompanying questioning to direct the students needed to be revised. For Activity 4 (recording numbers), a discussion was undertaken on the need to supply the student with a mini PVC to offset the common error students make, that is, recording a number such as “twenty-five” as 205 because they write the number as they hear it, for eaxmple, “twenty” (20) and “five” (5) to get 205, instead of 25. This generated a lot of discussion as it was an error they had seen made by several students and had not known what caused it or how to remediate it. Activity 5, calculator counting, had been done in Week 1 so the IEWs were familiar with it. However, as tutors, they needed to understand the role of each material so that they could guide the students. Thus, they had to realise that, in MAB and calculator counting, the numbers constantly changed so there was no visual record of the counting process. Therefore, the recording on paper was essential to look back at the counting and to abstract the understanding that, when counting by 1s, the ones digit changed first. The role of the MAB and PVC was to enable the student to “see” the counting process but the connection had to be made explicit (e.g., connecting putting out 1 one on the PVC to +1 on the calculator). TriallingOf the ten IEWs who undertook the first trial of Week 3, three had not been trained in the activities to be presented in the first training session of the cycle. All three IEWs were from the primary school and the principal could not release them on a Monday because of a problem within the school. However, two of these had worked with similar activities in Week 1 and the pre-trial review of the activities helped the third IEW. Nevertheless, a trainer sat beside him to help when needed. More students again turned up for this trial. (The class teachers thought it was because the students who had been there the week before had said it was fun.) ReflectionsThe students undertook the activities wholeheartedly so did not appear to think they were too “babyish” for them. They again loved the calculator counting and it was difficult to end the session (and to retrieve all the calculators!). The teachers who had been observing commented on the power of the calculator in these activities and also commented on how professional the IEWs were. During morning tea reflections, the IEWS thought that the trials had gone well. They were obviously feeling good about their teaching efforts. This newly-found confidence was heartening and this trial seemed to be when the IEWs made the transition from learner to teacher. Week 3, Days 3 and 4On these days, IEW attendance was down because of the NAIDOC celebrations in nearby Biloela and the IEWs were either helping the schools organise for this or actually participating in the dance activities. Pedagogic focusAs for Week 2 Mathematics focusThe major purpose of this train-trial cycle was to consolidate 2-digit numbers and to introduce 3-digit numbers. Activity 1 required the students to understand the ungrouping process that underlies the formal subtraction pencil-and-paper algorithm. Activity 2 involved playing a trading game to consolidate understanding of the regrouping process and when to use the process. Activity 3 introduced the students to a need for a new place beyond tens (i.e., hundreds) and the need for a new type of material (MAB) as 100 bundling straws are too unwieldy to manipulate. Activity 4 played a game to establish the grouping process which is more abstract with MAB than with bundling straws. Activity 5 was a calculator counting game which was isomorphic in structure to that played with 2-digit numbers in Week 3, Day 4, Activity 5. The final activity (Activity 6) was just for fun and for the students to apply their new knowledge of the counting process abstracted from the calculator counting activities, namely, when counting by ones, the ones place changes first; when counting by tens, the tens place changes first; and so on. TrainingThe IEWs were well-prepared for the activities taken in the two trials. They appeared to recognise the isomorphic structures in the processes of representing, reading, recording, and counting 2-digit numbers and 3-digit numbers. They applied the questioning associated with grouping and ungrouping 2-digit numbers to 3-digit numbers with ease. TriallingAlthough eight IEWs undertook the second training session in Week 3, three only attended the trialling session. This drop in attendance was due to the NAIDOC celebrations. Fortunately, there was a concomitant drop in student attendance, again because of the NAIDOC celebrations. The students enjoyed the activities but it was apparent that they needed to be re-taught how to read 3-digit numbers. ReflectionsThe again enjoyed the calculator counting. The IEWs who were present for the trialling all commented that their students had difficulty with reading 3-digit numbers. (This was It was then we decided as a group that trialling should focus on this process in the first trialling session of Week 4. The teachers who had been observing also stated that not being able to read 3-digit numbers was a major stumbling block in their students’ ability to process 3-digit numbers with understanding. During morning tea reflections, the IEWS thought that the trials had gone well. They were obviously feeling good about their teaching efforts. This newly-found confidence was heartening and this trial seemed to be when the IEWs made the transition from learner to teacher. Week 4This week took place in the second last week before the school holidays and already student numbers at the high school were diminishing so that the IEWs had fewer and fewer students to work with. As well, a graduation ceremony, to which the Community had been invited, had been planned for the IEWs on the Thursday (which would have normally been the day for the second trialling session). To counteract this, the second trialling session was undertaken on Wednesday. This meant that there was one training day and two consecutive days of trialling. Therefore, the training had to cover activities for both trials. This section describes the three training and trialling days under the previous weeks’ headings, pedagogic focus, mathematics focus, training, trialling, and reflections, and the day of the graduation ceremony under the headings, collective interview and ceremony. Week 4, Days 1, 2 and 3Pedagogic focusAs for Weeks 2 and 3. Mathematics focusThere were two major purposes for training in this week, namely, to consolidate the Years 8-10 students’ ability to represent, read, record and compare 3-digit numbers and to help students understand the multiplicative relationships embedded in the decimal number system. Because of their inherent difficulties, numbers with zeros or teen numbers were delayed until the final trialling session on Wednesday (but were undertaken in the Monday training session as well as just prior to the Wednesday trial). The focus of Activity 1 for the Tuesday trial was to determine whether students would record a number such as “three hundred and fifty-seven” as 30057 or as the correct 357. The IEWs had seen several students with this misconception but said that they didn’t know how to “fix it”. This activity, called “Cross the River” (see Appendix A, Interim Report, Week 5, Trial 1) involved a set of cards with numbers written in word form and a worksheet with a river and stepping stones with the card numbers recorded on them. Some of the numbers were distracters (such as 30057 for 357 or numbers that weren’t on the stepping stones). Students had to cover the correct numbers to cross the river safely. Activities 2 and 3 were aimed at remediating misconceptions such as those revealed in Activity 1. These activities focused on consolidating how to read and record 3-digit numbers using MAB, PVC, and digit cards. Activity 4 focused on comparing 2-digit and 3-digit numbers through games. The focus of Activities 1 and 2 for the Wednesday trial was to replicate Activities 2 and 3 from Tuesday with teen numbers or numbers with zero/s. Activity 3 was helping students understand the multiplicative relationships in the decimal number system. Training for both trialsBy this stage, the IEWs were very familiar with using the materials to represent numbers and understood the need for students to record numbers on a small PVC. They thought the Cross the River game (with its in-built teaching) and the Wednesday activities with zeros might help students to understand that only one digit is needed to show a place. They remembered the multiplicative relationships activities from Week 1 and still enjoyed working with the calculator, PVC, and digit cards. It was suggested that they might be able to do the activities related to representing, recording, and reading 3-digit numbers (i.e., Activities 2 and 3 for the first trial and Activities 1 and 2 for the second trial) in the Tuesday trial as this would leave them free to focus on the multiplicative relationships in the Wednesday trial. TrialsOf the ten IEWs who undertook the first trial of Week 4, two had not been trained in the activities. Once again, these IEWs were from the primary school and the principal could not release them on a Monday because of an ongoing problem within the school. However, they were familiar with the use of MAB and PVC in representing, recording, and reading 3-digit numbers. However, a trainer sat between them at the trial to help if needed. With respect to the Wednesday trial, only one of the 10 tutors had not been trained so she was happy to work with another IEW. At the Tuesday trial, there were enough students for the IEWs to have at least two students each. However, on Wednesday, there were only three students and the high-achieving Year 10 students were asked to come. The multiplicative activity was appropriate for them as it challenged them and the resulting learning would have enhanced their understanding of place value. ReflectionsThe IEWs discovered that it was difficult for some students to change their misconceptions with respect to recording a number such as 375 as 30075. However, they felt that they had accomplished something with these students as the materials helped them to understand that only one digit per place was needed when recording numbers. They continued to realise that the calculator was a powerful mathematics teaching tool and realised that the “good” students also had gaps in their knowledge. There was a sense of pride in themselves for being able to teach the higher-performing students something which they didn’t know. Week 4, Day 4This was the day of the Graduation Ceremony which was to begin at
Collective interviewMany IEWs indicated that they had been very nervous at the beginning and didn’t want to come because they thought they wouldn’t be able to do the “big maths”. All agreed, however, that they were surprised that they could learn how to teach the mathematics and how it wasn’t too hard when you understood what materials to use and questions to ask. They indicated quite strongly that they would like to do another training Program. They were told that there would be a follow-up week later in the year where we would show them how to develop some of the activities we did over the four weeks as computer activities for the students to do. They thought that this would be a very good thing to know. CeremonyVictor, one of the IEWs, opened and closed the ceremony with a traditional dance. The opening dance was followed by an address by the Chair, Wadja Wadja School Board, to officially open the proceedings, welcome Community members, and acknowledge the IEWs’ “tremendous” achievement in undertaking the four-week Program. Other dignitaries also spoke to acknowledge the IEWs’ achievement. After the speeches, each IEW was called to the front to receive their certificate for the Program from Professor Tom Cooper, QUT and their Train a Maths Tutor bag containing their own set of mathematics concrete materials. (See Appendix B.) On conclusion of handing out the certificates, Mel (another IEW) responded on behalf of all the graduates. Following Mel’s speech, the remaining IEWs were asked to come to the front of the Hall where the Community were invited to acknowledge their achievements through clapping. The IEWs were embarrassed but proud by this public demonstration of their achievement. The Community was then invited to attend the graduation lunch and cutting of the graduation cake (made by a high school teacher and iced by Kylie to represent the place value materials used in the training sessions). The ceremony was very successful; many more community members attended than was expected. The IEWs’ pride in receiving their certificates was evident to all observers, as was the pride of all the community members who watched the ceremony. Victor’s dancing was well received as was the speech by Mel. In a particularly poignant moment, two elders drew all the people at the ceremony together before lunch for a minute’s silence for the Indigenous people who had died in Woorabinda. Week 5Week 5 (Block B) was designed to train the IEWs (and parents) in how to develop computer activities that would enhance the mathematics activities undertaken in Block A and to trial these activities with the students. For the reasons given in Section 1.1.2, the trialling sessions were not undertaken. However, unlike Block B, the computer training lent itself to allowing the IEWs to develop their own computer activities. This week describes the training and the activity development as two two-day cycles (similar to Block A) - one for PowerPoint and one for Excel. It describes the two cycles under headings that represent the components of the cycle: pedagogic focus, mathematics focus, training, activity development, and reflections. The train-trial cycleThe week focused on two training-activity development cycles, with each cycle taking two days. PowerPoint training and activity development was the focus of the first cycle and Excel training and activity development was the focus of the second cycle. (See Table 7.) Although 3 hours was dedicated to each session, the reality was 2-21/2 hours, depending on how quickly the primary school IEWs could get away from their school duties. (See Section 3.1.2 for the rationale for using computer activities in mathematics teaching and learning.) The training component of the cycle where the IEWs got to know how to use the particular Microsoft program was undertaken in the high school’s computer laboratory. The first 11/2 hours of the activity-development component of the cycle took place in the classroom where the Block A trials had been undertaken. In this component, various mathematics activities were demonstrated and suggestions provided on how to use colour, line thickness, font size to focus students on the salient learning embedded in the activities. The remaining 11/2 hours were undertaken in the computer laboratory where the IEWs developed their virtual mathematics activity based on the ideas learnt from the first 11/2 hours. The decision to use on-and off-computer sessions was based on prior knowledge of other in-service activities. The attraction of the computer is so strong that it is difficult to get the attention required to make points about the curriculum aspects of the activities to be developed. Before teachers/aides can construct appropriate (in terms of facilitating learning) virtual mathematics, they need to be familiar with what PowerPoint and Excel applications can do and how they can be used to produce virtual activities. To this end, Baturo (2002) had developed a set of slides for each program to develop learners’ proficiency in specific program features that will be needed in constructing mathematics activities. These activities were developed to be fun and non-threatening (see Figure 12 for a sample of PowerPoint activities). Figure 12. Sample pre-construction skill-building activities for getting to know PowerPoint
Figure 13. Samples of aide-memoire for using PowerPoint features
The practice activities were accompanied by a short aide-memoire to remind the IEWs on which menus they would find particular tools or processes (see Figure 13). Thus, no one needed to have any prior knowledge of computers before undertaking the practice session. Before the Block B training and activity development sessions, the aide-memoires and introductory activities for PowerPoint and Excel practice activities were loaded on the server in the computer laboratory. Week 5, Days 1and 2Pedagogic focusAs for Block A, the overarching aim of the training and activity development cycle was to ensure that the IEWs experienced the Social Principles of success and group cohesion (see Section 3.1). To this end, the training material for the IEWs was designed to help them get to know the features of PowerPoint through motivating and fun activities. This was particularly important for the primary school IEWs who had had limited or no experience with working in PowerPoint. Mathematics focusBlock B was also designed to develop activities that were motivating for students as well as structural knowledge eliciting. After the training session in PowerPoint, the IEWs were required to construct (using PowerPoint) a non-trivial mathematics activity based on the activities undertaken in Block A. This part of the training was where theory and virtual practice merged and it is in this stage that we were able to strengthen the IEWs’ subject matter knowledge in the domain of mathematics that was the focus of the activity. Thus, developing virtual activities is essentially a non-threatening form of professional development. TrainingThe IEWs had varying degrees of competence with computers – mainly in word processing or accessing a CD-Rom and playing games. The IEW from the high school (Linda) was the only one who had worked in PowerPoint. No IEW had had any experience in developing virtual mathematics activities. Therefore, before undertaking the formal training in PowerPoint, the IEWs were shown how to access their folders on the server and how to save their finished work to the server. It was fortunate that there were almost as many assistants (critical friend, visiting scholar, research assistant, and tow trainers) as IEWs to provide the just-in-time support that all learners in a computer environment need. The IEWs enjoyed the PowerPoint introductory activities but the session was not long enough to work through the entire introductory activities. The computers in the Wadja Wadja High School laboratory were used mainly by the class undertaking the VET Business course and their use of the computers was limited to improving typing skills and speed using a commercial CD-ROM. The general modus operandi of other classes designated to use the computer laboratory was to let the students play games. Many students also came into the laboratory after school to play games. (This aspect of computer usage was detrimental to enticing students to trial the IEW-developed activities.) Hence, the monitors were not set up to facilitate the use of the Drawing tool or did not display other useful icons such as the Undo tool. These features needed to be activated during the course of the first training session. Activity developmentFor the off-computer component, some of the activities undertaken in Block A were revised with materials and then shown as computer activities. Each IEW was given a CD-ROM with virtual copies of these concrete materials and some sample activities. (See Figure 14.) Figure 14. Sample PowerPoint mathematics activities provided to the IEWs
Figure 15 provides samples of the activities that the IEWs constructed after 2 hours of skill-building in PowerPoint and 11/2 hours of construction-cum-inservice. Figure 11 provides samples of the activities that these teacher aides constructed after 11/2 hours of skill-building in Excel and 11/2 hours of construction-cum-inservice. Figure 15. IEWs’ PowerPoint activities for 3-digit numbers
ReflectionsUnlike some non-Indigenous teachers with whom the same training had been undertaken in a variety of schools, the IEWs were not nervous beginners. Rather, they were all eager to “get to know the computer” even if they had not used one before (as was the case for two IEWs). They were totally engrossed in the development of the activities and seemed to have a natural flair for layout, use of colour and font size to make activities appealing to students. They exhibited pride in their achievement and were happy to show and explain their activities. Week 5, Days 3 and 4Pedagogic focusAs for Days 1 and 2. However, the features of Excel enabled the IEWs to see the power of dynamic interactive mathematics activities. Mathematics focusIn this training-activity development cycle, the IEWs were shown the power of Excel in enabling students to see two almost simultaneous different representations, for example, numerical data and pictorial graphic representations. TrainingAs for Days 1 and 2, the IEWs were shown how to work through the introductory Excel activities. They came to understand that similar tools were used for both PowerPoint and Excel. They learnt how to enter simple formulae for counting and how these could be copied. We decided to develop an activity focusing on the comparison of 2- or 3-digit whole numbers (see Figure 16). To this end, five numbers in random order were entered in an Excel column or row. The IEWs were shown how to select these and how to activate the Chart Wizard to produce a bar graph that would give a graphical representation of the five numbers they had entered as well as for the row of numbers the students would enter. Figure 16. Sample Excel number comparison constructed by the IEWs
During this on-computer component of the Excel activity development, some Year 9 students had come into the computer laboratory to play games after school. They were easily enticed into trialling the IEWs’ activities and were enthralled by the speed at which their ordering of the given numbers was represented graphically. They immediately knew whether their response was correct or incorrect and, because of the graph, were able to correct the number/s out of order. Block B summaryThe virtual mathematics activities the IEWs constructed in both PowerPoint and Excel were attractive, based on non-trivial aspects of mathematics and were easily stored for use by a number of students. Most IEWs became totally engrossed in the development of the activities and seemed to have a natural flair for layout, use of colour and font size to make activities appealing to students. They exhibited pride in their achievement and were happy to show and explain their activities. This was significant, in that the normal use of computers in the community was to play games. The students were highly motivated by the Excel tasks they trialled. Whilst these were important outcomes, the most important was the IEWs’ enhanced subject-matter knowledge gained through discussion of the teaching sequences in which their activities should be embedded. Whilst the Block B training was very successful, it is doubtful whether the primary IEWs would have further opportunities to practise their new-found computer skills as they did not have access to computers at home and the school’s computer laboratory was unusable because of a lack of technical expertise to maintain the computers. |
