WEBVTT 00:00:02.360 --> 00:00:06.080 Hi, Paul Lockhart compares doing mathematics 00:00:06.080 --> 00:00:11.240 to climbing a mountain. Boy, do I love that analogy. 00:00:12.320 --> 00:00:16.300 Yes, climbing a mountain is difficult. The legs 00:00:16.300 --> 00:00:21.339 are tired. The path disappears. We choose the 00:00:21.339 --> 00:00:26.000 wrong direction and we retrace our steps. We 00:00:26.000 --> 00:00:29.079 may even wonder, why did we start climbing in 00:00:29.079 --> 00:00:32.920 the first place? Mathematics can feel the same 00:00:32.920 --> 00:00:36.460 way. We might begin with a question, but we do 00:00:36.460 --> 00:00:39.840 not know exactly where it will lead. We may try 00:00:39.840 --> 00:00:44.939 one path and discover it goes nowhere. We return 00:00:44.939 --> 00:00:47.979 to where we start and notice something we might 00:00:47.979 --> 00:00:54.840 have missed. And we try again. Eventually, perhaps, 00:00:55.079 --> 00:01:00.509 we reach the summit. We find the answer. And 00:01:00.509 --> 00:01:06.469 there is joy, not just in the answer, but because 00:01:06.469 --> 00:01:11.349 the journey was difficult. But reaching the summit 00:01:11.349 --> 00:01:16.150 is not the only benefit of climbing. Along the 00:01:16.150 --> 00:01:20.609 way, we become stronger. We learn how to navigate. 00:01:21.390 --> 00:01:24.730 We begin to recognize which paths are promising 00:01:24.730 --> 00:01:30.349 and which ones may lead us in circles. And when 00:01:30.349 --> 00:01:35.349 we finally reach that higher ground, yes, we 00:01:35.349 --> 00:01:38.549 can see things that were invisible from below. 00:01:39.609 --> 00:01:43.510 Ideas that once seemed unrelated suddenly appear 00:01:43.510 --> 00:01:48.969 connected. We go from memorizing, say, the law 00:01:48.969 --> 00:01:53.689 of logarithms to understanding why the law of 00:01:53.689 --> 00:01:58.890 logarithms works. The landscape has not changed. 00:02:00.780 --> 00:02:05.939 We have. And this is why making mathematics too 00:02:05.939 --> 00:02:09.580 easy can sometimes take away something valuable. 00:02:10.479 --> 00:02:14.900 Of course, we should explain ideas clearly. Of 00:02:14.900 --> 00:02:17.439 course, we should remove unnecessary confusion 00:02:17.439 --> 00:02:22.759 and make the path accessible. We should not assume 00:02:22.759 --> 00:02:28.569 that all difficulty is bad. Think about it. If 00:02:28.569 --> 00:02:30.969 someone carries us to the top of a mountain, 00:02:31.090 --> 00:02:35.009 yes, we could see the view, but we missed the 00:02:35.009 --> 00:02:38.710 climb. We would not develop the strength or the 00:02:38.710 --> 00:02:41.909 judgment or the confidence that comes from finding 00:02:41.909 --> 00:02:45.689 our own way. There is something similar about 00:02:45.689 --> 00:02:49.770 being given a mathematical answer before we had 00:02:49.770 --> 00:02:53.370 the opportunity to really explore the question. 00:02:55.009 --> 00:02:58.080 Reaching the destination. without experiencing 00:02:58.080 --> 00:03:03.620 the journey, is shallow. And perhaps this is 00:03:03.620 --> 00:03:07.060 also true beyond mathematics. What I have found 00:03:07.060 --> 00:03:10.360 is that some of my most important understanding 00:03:10.360 --> 00:03:16.520 comes slowly, through questions, through mistakes, 00:03:16.860 --> 00:03:23.360 uncertainty, and even perseverance. I would rarely 00:03:23.360 --> 00:03:27.219 choose the difficult path simply because it was 00:03:27.219 --> 00:03:30.439 difficult. But when the path leads somewhere 00:03:30.439 --> 00:03:34.919 worth going, I have found that the struggle can 00:03:34.919 --> 00:03:39.780 be part of the gift. So if you're working on 00:03:39.780 --> 00:03:43.240 something difficult today, perhaps the question 00:03:43.240 --> 00:03:46.960 is not only how quickly can I reach the answer, 00:03:47.099 --> 00:03:51.099 perhaps another question matters just as much. 00:03:52.330 --> 00:03:56.750 What is this climb allowing me to see? And more 00:03:56.750 --> 00:04:00.490 important, who am I becoming along the way?