WEBVTT 00:00:00.046 --> 00:00:07.955 [Music] Welcome to Monitoring Progress While Solving Problems. 00:00:09.046 --> 00:00:12.416 I am Sybilla Beckmann, Josiah Meigs' Distinguished Teaching Professor 00:00:12.416 --> 00:00:13.946 at the University of Georgia. 00:00:14.226 --> 00:00:18.926 I was a member of the panel for the Problem Solving guide as well 00:00:18.926 --> 00:00:22.526 as a member of the panel for the Response to Intervention guide. 00:00:24.076 --> 00:00:27.476 So what does monitoring and reflecting 00:00:27.476 --> 00:00:29.856 on problem solving look like in the classroom? 00:00:29.856 --> 00:00:35.066 Well, overall, the key thing is that we want students to be thinking 00:00:35.146 --> 00:00:39.076 about their thinking and thinking about problem solving. 00:00:39.516 --> 00:00:46.006 What we want is something that is more focused on making sense of problems 00:00:46.006 --> 00:00:52.106 and of solution methods and something that is more reflective and goes deeper 00:00:52.106 --> 00:00:55.616 into the problem-solving process. 00:00:55.616 --> 00:01:00.006 Here is a problem about some clay in a bowl: There was some clay in a bowl. 00:01:00.006 --> 00:01:05.896 After I took out 2/3 of it, there was 1/4 of a cup of clay left. 00:01:05.896 --> 00:01:08.036 How much clay was in the bowl at first? 00:01:08.106 --> 00:01:13.356 So let's think about understanding this problem. 00:01:13.706 --> 00:01:17.216 When we read a problem like this, it's very tempting for students 00:01:17.216 --> 00:01:20.656 to simply do something with those numbers -- 2/3 and 1/4. 00:01:20.996 --> 00:01:26.336 But it's very important, a very key first step is to really understand what 00:01:26.336 --> 00:01:28.546 that problem is asking and what is it about.' 00:01:28.546 --> 00:01:34.566 Often it's good to just restate that problem in your own words. 00:01:35.176 --> 00:01:38.056 I can restate this as, "I had some clay, 00:01:38.566 --> 00:01:41.806 I took some clay out, and what do I know? 00:01:41.806 --> 00:01:43.906 I know how much is left." 00:01:43.906 --> 00:01:48.356 So that's the overall big picture of what's happening in the problem. 00:01:49.716 --> 00:01:52.836 When students solve a problem like this, it's good for them 00:01:52.836 --> 00:01:58.206 to be asking themselves and each other questions about what they want to know 00:01:58.206 --> 00:02:01.696 and what they already know from the problem statement. 00:02:02.196 --> 00:02:04.626 We know that's what we're looking for in this problem. 00:02:04.626 --> 00:02:08.265 We want to know how much clay did we start with in the bowl. 00:02:08.985 --> 00:02:13.446 I know I took some out, I know that amount that I took out was 2/3 of it, 00:02:13.906 --> 00:02:17.426 and I know that when I am done taking that clay out, 00:02:17.426 --> 00:02:20.136 I have 1/4 of a cup of clay left. 00:02:21.336 --> 00:02:24.526 So at this point, students could start thinking 00:02:24.526 --> 00:02:28.296 about how might they start to solve this problem. 00:02:28.796 --> 00:02:32.466 And a student might right away think of subtraction 00:02:32.466 --> 00:02:35.416 because something is being removed from the bowl. 00:02:35.836 --> 00:02:39.186 "Maybe I can subtract 1/4 from 2/3." 00:02:39.776 --> 00:02:43.866 And at this point, students should be asking themselves and each other 00:02:43.946 --> 00:02:49.026 if that makes sense: "Well, that doesn't quite fit because, look, 00:02:49.026 --> 00:02:51.776 we took out 2/3 of the clay. 00:02:52.086 --> 00:02:54.276 That would be the amount that's taken out. 00:02:54.396 --> 00:02:56.676 It shouldn't be 2/3 minus 1/4. 00:02:56.676 --> 00:02:59.516 That really just doesn't fit." 00:02:59.516 --> 00:03:03.216 One student might think, "Well, wait a second. 00:03:03.216 --> 00:03:11.296 If I took out 2/3, how can I have 1/4 left because if you take out 2/3, 00:03:11.296 --> 00:03:13.686 shouldn't you have 1/3 left? 00:03:13.686 --> 00:03:18.586 Hmm..." So this might be the kind of point in problem solving 00:03:18.586 --> 00:03:23.766 where students debate with each other and try to come to some agreement 00:03:23.796 --> 00:03:28.056 or some consensus about what is stated in the problem 00:03:28.056 --> 00:03:30.256 and what is it that we are looking for. 00:03:30.316 --> 00:03:36.136 So let's go back and really try to understand the problem one more time. 00:03:36.136 --> 00:03:42.646 And if students go back and read it they see, ""After I took out 2/3 of it, 00:03:43.186 --> 00:03:47.266 there was 1/4 of a cup of clay left." 00:03:47.366 --> 00:03:51.816 The 2/3 and the 1/4 refer to different things. 00:03:52.266 --> 00:03:54.786 They are not both referring to cups of clay. 00:03:55.156 --> 00:03:59.436 The 2/3 is referring to the amount of clay that was in the bowl, 00:04:00.096 --> 00:04:03.366 whereas the 1/4 is referring to a cup of clay. 00:04:05.076 --> 00:04:10.936 So after 2/3 of the clay is removed, it is the case that 1/3 of the clay 00:04:11.336 --> 00:04:14.656 that was originally in the bowl is left. 00:04:14.656 --> 00:04:18.375 We're given in the problem that this 1/3 of the clay that's left 00:04:18.375 --> 00:04:22.756 in the bowl is 1/4 of a cup of clay. 00:04:23.586 --> 00:04:28.876 And this means that what was originally in the bowl is three times as much 00:04:29.176 --> 00:04:33.496 as this 1/4 of a cup, which means that what was originally 00:04:33.496 --> 00:04:40.786 in the bowl must be three sets of 1/4 of a cup, or three pieces, 00:04:40.836 --> 00:04:45.706 each of which is a fourth of a cup, which is what we mean by 3/4 of a cup. 00:04:46.876 --> 00:04:49.736 We have reasoned through that the initial amount 00:04:49.736 --> 00:04:51.716 in the bowl was 3/4 of a cup. 00:04:52.376 --> 00:04:54.766 Is it true that if we take out 2/3 00:04:54.766 --> 00:04:59.446 of that then we'll have a fourth of a cup left? 00:04:59.446 --> 00:05:04.766 1/3 of the initial amount of clay is equal to 1/4 of a cup, 00:05:05.366 --> 00:05:09.356 and that gives us a multiplication sentence. 00:05:09.356 --> 00:05:17.326 1/3 times the initial amount of clay is equal to 1/4, which could be solved 00:05:17.326 --> 00:05:24.466 by dividing 1/4 by 1/3, and that again gives us the 3/4 of a cup. 00:05:24.636 --> 00:05:28.896 So this would be a way to reflect on the problem and then to see it 00:05:29.076 --> 00:05:32.716 in a new light in terms of a multiplication equation. 00:05:34.146 --> 00:05:38.846 We can see that if students simply do the first thing that pops 00:05:38.846 --> 00:05:41.916 into their mind, it may not be correct. 00:05:41.916 --> 00:05:47.006 And unless students have some mechanism by which they are prompted 00:05:47.006 --> 00:05:50.106 to rethink what they have done or to reflect 00:05:50.106 --> 00:05:55.316 on whether it makes sense what they've just done, they may simply rush headlong 00:05:55.316 --> 00:05:59.596 to a solution and not ever reconsider what it was that they did. 00:06:01.596 --> 00:06:04.646 To learn more about Monitoring Progress While Solving Problems, 00:06:04.646 --> 00:06:09.076 please explore the additional resources on the Doing What Works website.