WEBVTT 1 00:00:00.010 --> 00:00:02.669 So welcome to this video on the research 2 00:00:02.669 --> 00:00:05.819 basis for variation theory. I assume 3 00:00:05.819 --> 00:00:08.220 you've watched the previous video by 4 00:00:08.220 --> 00:00:10.200 Professor fan, and you've got some basic 5 00:00:10.200 --> 00:00:13.019 idea what variation theory is and what 6 00:00:13.019 --> 00:00:16.230 the role of variation can be, especially 7 00:00:16.230 --> 00:00:20.100 in practicing materials for example. Of 8 00:00:20.100 --> 00:00:23.430 course this again sounds perhaps quite 9 00:00:23.430 --> 00:00:26.099 like common sense, and quite logical. But 10 00:00:26.099 --> 00:00:28.859 just like all the other topics, they 11 00:00:28.859 --> 00:00:30.869 actually have quite a sound research 12 00:00:30.869 --> 00:00:33.809 basis as well. I will discuss some of 13 00:00:33.809 --> 00:00:36.630 these articles or these studies that 14 00:00:36.630 --> 00:00:39.750 actually are about variation. And first 15 00:00:39.750 --> 00:00:42.780 example comes from the book by Professor 16 00:00:42.780 --> 00:00:43.260 Fan. 17 00:00:43.260 --> 00:00:47.809 In this book he mentions the origins of 18 00:00:47.809 --> 00:00:50.730 variation theory. An important name in 19 00:00:50.730 --> 00:00:53.850 variation theory is Marton. With his 20 00:00:53.850 --> 00:00:56.840 colleagues, he promoted this theory and 21 00:00:56.840 --> 00:00:59.640 explained what it is. It was later 22 00:00:59.640 --> 00:01:02.640 extended by Watson and Mason, who are 23 00:01:02.640 --> 00:01:06.000 based in the United Kingdom. And in this 24 00:01:06.000 --> 00:01:09.060 theory he proposes that learners must 25 00:01:09.060 --> 00:01:11.729 experience variation in the critical 26 00:01:11.729 --> 00:01:14.520 aspects of a concept within a limited 27 00:01:14.520 --> 00:01:16.979 space and time in order for the concept 28 00:01:16.979 --> 00:01:19.530 to be learnable. The idea behind this 29 00:01:19.530 --> 00:01:22.470 is that invariance structures. So let's 30 00:01:22.470 --> 00:01:24.810 say you have a sequence of tasks, and 31 00:01:24.810 --> 00:01:28.140 some elements vary. They change in the 32 00:01:28.140 --> 00:01:31.850 sequence of tasks that they can actually 33 00:01:31.850 --> 00:01:34.890 aid the learner, and they can actually 34 00:01:34.890 --> 00:01:38.100 help make a concept well known. The idea 35 00:01:38.100 --> 00:01:39.869 that invariance structures during 36 00:01:39.869 --> 00:01:42.210 changing phenomena often denote the 37 00:01:42.210 --> 00:01:44.909 present knowledge acquisition is an 38 00:01:44.909 --> 00:01:46.890 essential part of something called 39 00:01:46.890 --> 00:01:50.130 phenomenology. There are two types of 40 00:01:50.130 --> 00:01:52.590 variation, conceptual variation and 41 00:01:52.590 --> 00:01:55.500 procedural variation. Conceptual 42 00:01:55.500 --> 00:01:57.750 variation as a starting point, that 43 00:01:57.750 --> 00:02:00.149 concepts can be understood from multiple 44 00:02:00.149 --> 00:02:03.659 perspectives. Variation is created in 45 00:02:03.659 --> 00:02:06.960 several ways the first way is called 46 00:02:06.960 --> 00:02:09.629 standard concept variation. And this 47 00:02:09.629 --> 00:02:11.790 happens by varying the concept in a 48 00:02:11.790 --> 00:02:12.210 standard 49 00:02:12.210 --> 00:02:15.390 way by inducing concept by varying 50 00:02:15.390 --> 00:02:18.330 visual and concrete instances. We will 51 00:02:18.330 --> 00:02:21.990 see some examples later on as well. The 52 00:02:21.990 --> 00:02:24.330 main purpose of this type of variation 53 00:02:24.330 --> 00:02:26.670 is to help students establish the 54 00:02:26.670 --> 00:02:28.950 connection between concrete experiences 55 00:02:28.950 --> 00:02:32.070 and abstract concepts. So hopefully you 56 00:02:32.070 --> 00:02:34.620 can again recognize some of the wording 57 00:02:34.620 --> 00:02:38.940 from for example the CPA and model 58 00:02:38.940 --> 00:02:41.280 method. The second way is something 59 00:02:41.280 --> 00:02:44.900 called non-standard concept variation, 60 00:02:44.900 --> 00:02:47.310 highlights the essence of a concept by 61 00:02:47.310 --> 00:02:49.230 contrasting the concept with a 62 00:02:49.230 --> 00:02:54.150 non-standard example. So for example if I 63 00:02:54.150 --> 00:02:56.910 have some triangles and I put a square 64 00:02:56.910 --> 00:03:00.290 in the middle of it. It is quite clear 65 00:03:00.290 --> 00:03:03.090 hopefully that the square is not a 66 00:03:03.090 --> 00:03:06.780 triangle. So this variation will aid the 67 00:03:06.780 --> 00:03:09.330 learning. This stresses the teaching 68 00:03:09.330 --> 00:03:11.400 strategy that examples should not only 69 00:03:11.400 --> 00:03:14.100 be the normal ones, but also the 70 00:03:14.100 --> 00:03:18.390 non-standard ones. Then finally the third 71 00:03:18.390 --> 00:03:21.510 or non-concept variation uses non 72 00:03:21.510 --> 00:03:24.060 concepts for example counter examples to 73 00:03:24.060 --> 00:03:27.170 reinforce a concept. Then there's also 74 00:03:27.170 --> 00:03:31.070 procedural variation which progressively 75 00:03:31.070 --> 00:03:33.990 involves unfolding mathematical 76 00:03:33.990 --> 00:03:36.750 activities in procedural variations. 77 00:03:36.750 --> 00:03:39.600 Students can arrive at a solution to a 78 00:03:39.600 --> 00:03:42.090 problem and form connections among 79 00:03:42.090 --> 00:03:44.730 different concepts step by step from 80 00:03:44.730 --> 00:03:47.250 multiple approaches. The word procedural 81 00:03:47.250 --> 00:03:50.310 here is quite important. This type of 82 00:03:50.310 --> 00:03:52.770 variation is also created in several 83 00:03:52.770 --> 00:03:55.740 ways. The first way addresses the 84 00:03:55.740 --> 00:03:58.800 formation of concepts, and the process of 85 00:03:58.800 --> 00:04:02.220 unfolding concepts. The second way uses 86 00:04:02.220 --> 00:04:05.670 scaffolding for problem-solving. Multiple 87 00:04:05.670 --> 00:04:07.560 variations analyses of the 88 00:04:07.560 --> 00:04:09.900 configurations of a problem do not only 89 00:04:09.900 --> 00:04:13.020 help students clarify the process of 90 00:04:13.020 --> 00:04:15.180 solving the problem and the structure of 91 00:04:15.180 --> 00:04:17.820 the problem, but they also are an 92 00:04:17.820 --> 00:04:19.770 effective way of experiencing 93 00:04:19.770 --> 00:04:22.230 problem-solving and enhancing the 94 00:04:22.230 --> 00:04:25.260 competency of problem solving. The third 95 00:04:25.260 --> 00:04:28.590 way establishes a system of mathematical 96 00:04:28.590 --> 00:04:31.860 experiences. You can see overlap again 97 00:04:31.860 --> 00:04:34.170 with the themes that we covered in 98 00:04:34.170 --> 00:04:37.050 previous weeks. You have heard the word 99 00:04:37.050 --> 00:04:40.320 scaffolding which is actually quite well 100 00:04:40.320 --> 00:04:42.330 known from Bruner's work, and we 101 00:04:42.330 --> 00:04:45.000 mentioned Bruner when we were discussing 102 00:04:45.000 --> 00:04:48.120 the week about concrete pictorial and 103 00:04:48.120 --> 00:04:50.760 abstract. You can see some examples of 104 00:04:50.760 --> 00:04:53.910 each type here as well. There is an 105 00:04:53.910 --> 00:04:57.720 example where there's one key point but 106 00:04:57.720 --> 00:04:59.880 there are different methods in which it 107 00:04:59.880 --> 00:05:00.630 is solved. 108 00:05:00.630 --> 00:05:03.960 Or there is one key point but there are 109 00:05:03.960 --> 00:05:06.300 different numbers. You can see the 110 00:05:06.300 --> 00:05:08.850 variation directly on the screen here. 111 00:05:08.850 --> 00:05:12.810 because the numbers are changed, the 112 00:05:12.810 --> 00:05:16.170 numbers are varied. And this is part of 113 00:05:16.170 --> 00:05:18.960 this procedural variation. And then you 114 00:05:18.960 --> 00:05:22.380 can also have one key point with 115 00:05:22.380 --> 00:05:24.840 different applications. I think the 116 00:05:24.840 --> 00:05:28.170 underlying theme here is that you vary 117 00:05:28.170 --> 00:05:31.530 the ways in which you present the tasks. 118 00:05:31.530 --> 00:05:36.330 You sequence them in a cunning way. You 119 00:05:36.330 --> 00:05:40.530 scaffold them and that helps the learner 120 00:05:40.530 --> 00:05:43.710 to learn the concepts. So I think this 121 00:05:43.710 --> 00:05:48.120 also shows how procedures go hand in 122 00:05:48.120 --> 00:05:50.580 hand with understanding, something we 123 00:05:50.580 --> 00:05:54.480 covered in the last lecture. We can go 124 00:05:54.480 --> 00:05:57.300 into a little bit more detail. And Marton, 125 00:05:57.300 --> 00:06:01.170 Runesson and Tsui actually gave four 126 00:06:01.170 --> 00:06:05.660 concepts that underpin variation theory. 127 00:06:05.660 --> 00:06:08.760 One of them is contrast. In order to 128 00:06:08.760 --> 00:06:10.560 experience something a person must 129 00:06:10.560 --> 00:06:12.990 experience something else to compare it 130 00:06:12.990 --> 00:06:14.880 with. Think of the example with the 131 00:06:14.880 --> 00:06:17.460 squares and the triangles. The second one 132 00:06:17.460 --> 00:06:19.890 is generalization. In order to fully 133 00:06:19.890 --> 00:06:23.370 understand what "three" is, we must also 134 00:06:23.370 --> 00:06:26.640 experience varying appearances of three. A third 135 00:06:26.640 --> 00:06:29.160 point is separation. In order to 136 00:06:29.160 --> 00:06:31.230 experience a certain aspect of something 137 00:06:31.230 --> 00:06:34.230 and in order to separate this aspect 138 00:06:34.230 --> 00:06:37.680 from other aspects, it must vary while 139 00:06:37.680 --> 00:06:38.849 other aspects 140 00:06:38.849 --> 00:06:41.509 in variant. And then the fourth one, 141 00:06:41.509 --> 00:06:44.580 fusion. If there are several critical 142 00:06:44.580 --> 00:06:46.709 aspects that the learner has to take 143 00:06:46.709 --> 00:06:49.709 into consideration at the same time, they 144 00:06:49.709 --> 00:06:53.119 must all be experienced simultaneously. 145 00:06:53.119 --> 00:06:57.209 So for example, perhaps you have a graph 146 00:06:57.209 --> 00:06:59.939 of something and you have an image of 147 00:06:59.939 --> 00:07:01.649 something and you present them both at 148 00:07:01.649 --> 00:07:03.809 the same time. If you have a collection 149 00:07:03.809 --> 00:07:06.209 of triangles, you can hear I love 150 00:07:06.209 --> 00:07:08.219 triangles, I've given several examples 151 00:07:08.219 --> 00:07:10.979 with them, you can apply these four 152 00:07:10.979 --> 00:07:12.059 aspects as well. 153 00:07:12.059 --> 00:07:14.429 You have triangles that actually have 154 00:07:14.429 --> 00:07:18.259 the same structure, for example 155 00:07:18.259 --> 00:07:20.909 equilateral triangles at the bottom 156 00:07:20.909 --> 00:07:23.059 right and the top left. 157 00:07:23.059 --> 00:07:25.919 You've got right-angled triangles. You 158 00:07:25.919 --> 00:07:27.599 see one on the left and at the bottom 159 00:07:27.599 --> 00:07:31.499 left. But there also are differences. You 160 00:07:31.499 --> 00:07:32.819 could argue there are differences in 161 00:07:32.819 --> 00:07:34.619 color of course. There are differences in 162 00:07:34.619 --> 00:07:37.499 angles. There are differences in acute 163 00:07:37.499 --> 00:07:41.069 and obtuse angles. So taking into account 164 00:07:41.069 --> 00:07:43.499 this variation between the different 165 00:07:43.499 --> 00:07:47.039 objects will actually help you implement 166 00:07:47.039 --> 00:07:50.999 and clarify the concepts. I hope these 167 00:07:50.999 --> 00:07:52.800 examples have further emphasized what 168 00:07:52.800 --> 00:07:55.469 variation theory is and what research 169 00:07:55.469 --> 00:07:58.409 underpins it. It is important to realize 170 00:07:58.409 --> 00:08:01.289 that variation is not only integrated in 171 00:08:01.289 --> 00:08:04.019 asian lessons, but also in their 172 00:08:04.019 --> 00:08:06.569 textbooks. And you can seem for more 173 00:08:06.569 --> 00:08:08.909 examples in one of the interactive tasks 174 00:08:08.909 --> 00:08:12.000 that we have included in the course.