WEBVTT 00:00:02.160 --> 00:00:05.320 Consider this problem. How many squares are on 00:00:05.320 --> 00:00:09.800 a standard 8x8 chessboard? The immediate answer 00:00:09.800 --> 00:00:13.880 is 64. There are 8 rows, each containing 8 squares. 00:00:14.740 --> 00:00:18.679 But look again. The board also contains squares 00:00:18.679 --> 00:00:22.539 that are made from 4 smaller squares. It contains 00:00:22.539 --> 00:00:27.559 3x3 squares, 4x4 squares, and so on, all the 00:00:27.559 --> 00:00:31.089 way up to the single 8x8 square. that is formed 00:00:31.089 --> 00:00:33.869 by the entire board. The answer is therefore 00:00:33.869 --> 00:00:37.869 1 squared plus 2 squared plus 3 squared plus 00:00:37.869 --> 00:00:42.409 dot dot dot up to plus 8 squared, which is 204. 00:00:43.109 --> 00:00:46.770 Now, our first answer was not foolish, it just 00:00:46.770 --> 00:00:50.509 simply was not complete. Finding the larger answer 00:00:50.509 --> 00:00:53.590 required us to remain with the problem after 00:00:53.590 --> 00:00:57.350 the obvious answer appeared. We had to notice 00:00:57.350 --> 00:01:00.659 what we overlooked. Then we had to revise our 00:01:00.659 --> 00:01:05.420 interpretation and begin again. This is productive 00:01:05.420 --> 00:01:08.439 struggle. It is the uncertainty that we experience 00:01:08.439 --> 00:01:11.359 while learning something that is within reach 00:01:11.359 --> 00:01:15.879 but not quite yet within sight. Productive struggle 00:01:15.879 --> 00:01:19.719 may involve wrong answers, it may involve frustration, 00:01:19.799 --> 00:01:23.319 and the uncomfortable feeling of not knowing 00:01:23.319 --> 00:01:26.819 what to do next. But it is not evidence that 00:01:26.819 --> 00:01:30.319 we are incapable of mathematics. Very often it 00:01:30.319 --> 00:01:34.000 is the evidence that learning is happening. But 00:01:34.000 --> 00:01:37.340 we should distinguish struggle from suffering. 00:01:37.680 --> 00:01:40.599 A student wrestling with a difficult idea is 00:01:40.599 --> 00:01:44.260 not in the same position as a student who is 00:01:44.260 --> 00:01:49.480 humiliated or excluded or overwhelmed or left 00:01:49.480 --> 00:01:53.480 without needed support. Difficulty can strengthen 00:01:53.480 --> 00:01:56.969 learning. but shame and abandonment usually does 00:01:56.969 --> 00:02:00.689 not. So we should be careful not to call every 00:02:00.689 --> 00:02:04.409 painful experience, quote, good for us, simply 00:02:04.409 --> 00:02:06.810 because something might eventually be learned 00:02:06.810 --> 00:02:10.370 from it. Probably the important question is not 00:02:10.370 --> 00:02:13.849 how can I eliminate every difficult feeling, 00:02:13.930 --> 00:02:17.629 and it's probably not how much pain can I endure. 00:02:18.129 --> 00:02:21.629 Maybe a better question is what kind of difficulty 00:02:21.629 --> 00:02:26.870 is this? And what response does it require? Now, 00:02:26.889 --> 00:02:29.389 sometimes the answer is persistence. Let's try 00:02:29.389 --> 00:02:32.550 another example, or maybe draw a picture, or 00:02:32.550 --> 00:02:35.310 maybe return to a definition. But sometimes the 00:02:35.310 --> 00:02:38.330 answer is assistance. Let's ask someone to explain 00:02:38.330 --> 00:02:42.050 it, or maybe offer a hint, or maybe someone can 00:02:42.050 --> 00:02:45.330 work besides us. And sometimes the answer is 00:02:45.330 --> 00:02:50.090 simply rest. Wisdom probably lies partly in learning 00:02:50.090 --> 00:02:53.569 the difference. Now the chessboard did not become 00:02:53.569 --> 00:02:57.830 clearer because we stared at it miserably. It 00:02:57.830 --> 00:03:00.030 became clearer because we changed the way we 00:03:00.030 --> 00:03:04.090 looked. So think of something that you are currently 00:03:04.090 --> 00:03:07.689 struggling to understand. What have you unsuccessful 00:03:07.689 --> 00:03:11.569 attempts already taught you? And maybe what small 00:03:11.569 --> 00:03:15.490 hint would help you continue without taking the 00:03:15.490 --> 00:03:19.949 discovery away? Difficulty is not always a barrier 00:03:19.949 --> 00:03:23.229 to learning. but sometimes it is the place where 00:03:23.229 --> 00:03:24.169 learning begins.