WEBVTT 00:00:00.506 --> 00:00:09.506 [Music] 00:00:10.006 --> 00:00:11.796 My name is Brian Bottge. 00:00:12.106 --> 00:00:17.036 I am Endowed Chair and Professor at the University of Kentucky in the Department 00:00:17.036 --> 00:00:19.396 of Special Education and Rehabilitation Counseling. 00:00:20.736 --> 00:00:22.936 When we think of concrete representations, 00:00:22.936 --> 00:00:24.766 a lot of times we think about math 00:00:25.096 --> 00:00:30.526 because math teachers use-in elementary grades-Cuisenaire rods and so forth. 00:00:30.526 --> 00:00:34.686 But concrete representations can be used in other areas as well, 00:00:34.866 --> 00:00:40.906 such as computer simulations now, for example, but also simulations, 00:00:41.646 --> 00:00:43.926 group activities in social studies and history. 00:00:44.766 --> 00:00:48.346 So, concrete representations are very important 00:00:48.666 --> 00:00:52.366 and can be used across a number of content areas. 00:00:53.096 --> 00:00:59.116 In our research just recently, we've used fraction strips where we say 00:00:59.116 --> 00:01:06.886 to the students-and these were 8th grade students-we give each student four long paper strips 00:01:06.886 --> 00:01:12.726 and we say that those strips are representative of candy bars, so take one strip and say, 00:01:12.726 --> 00:01:16.816 "You have to divide your strip into two pieces, and how would you do that?" 00:01:16.816 --> 00:01:23.446 So they fold the strip in half and they label the first crease 1/2 and the second crease 2/2. 00:01:23.446 --> 00:01:27.776 And then they go on and they say, "Okay, take your second strip and divide that into four. 00:01:28.576 --> 00:01:33.366 Pretend that you have to give a piece to each of four friends. 00:01:33.696 --> 00:01:34.486 How would you do that?" 00:01:34.486 --> 00:01:40.756 So they fold the paper strip in half and then they fold it again and label the creases 1/4th, 00:01:40.986 --> 00:01:46.386 2/4ths, 3/4ths, and 4/4ths, and they do the same for 8ths and they do the same for 16ths. 00:01:46.766 --> 00:01:50.736 And then, at the end of this exercise, the teacher asked the students, 00:01:50.736 --> 00:01:56.486 "What did you notice about the size of the denominator as we folded the strips?" 00:01:56.486 --> 00:02:00.426 And what they will say, of course, is that even though the number-16, for example, 00:02:00.426 --> 00:02:04.336 on 16ths-is very large, the actual size, 00:02:04.476 --> 00:02:08.485 the quantity of the candy bar that they are receiving, is very small. 00:02:08.556 --> 00:02:11.156 So, you can use those fraction strips 00:02:11.286 --> 00:02:15.926 to represent equivalent fractions, adding, subtracting fractions. 00:02:16.366 --> 00:02:21.626 One of the primary reasons for using concrete representations is 00:02:21.766 --> 00:02:27.406 to help students build mental models or pictures of what it is they're learning. 00:02:27.406 --> 00:02:32.556 For example, with the fraction strips, they are learning that if you divide a fraction strip 00:02:32.556 --> 00:02:34.946 into four pieces, well, those are fourths. 00:02:35.616 --> 00:02:41.976 Obviously, we can't use concrete representations all the time and that eventually we are going 00:02:41.976 --> 00:02:49.416 to have to move to the abstract so that students don't think that fractions are only used 00:02:49.416 --> 00:02:51.966 with candy bars, are only used with linear measurement. 00:02:52.176 --> 00:02:56.586 So, we will have to eventually move from the concrete to the abstract. 00:02:56.586 --> 00:03:02.606 But I guess the point is, is that many students need to have a firm foundation 00:03:02.676 --> 00:03:06.596 in the concrete before they move to the abstract. 00:03:06.736 --> 00:03:13.566 One of the ways we've tried to connect the concrete to the abstract, both in computer simulation 00:03:13.566 --> 00:03:21.746 and in actual hands-on, is with one of the Adventures of Jasper Woodbury called "Kim's Komet," 00:03:22.656 --> 00:03:32.426 that-these are 10- to 12-minute video scenarios where the students try to figure out how 00:03:32.506 --> 00:03:35.846 to help the people in the video solve their math problem. 00:03:36.426 --> 00:03:42.506 Kim's Komet is one of the problems that I really like because it teaches pre-algebraic concepts 00:03:42.506 --> 00:03:50.796 such as line of best fit, function, variables, formulas, graphing-those types of skills. 00:03:51.656 --> 00:03:59.516 So, the object of this particular problem is for the students to help Kim figure out where 00:03:59.516 --> 00:04:05.406 on a ramp to release her car to be able to be going a certain speed at the end of the ramp 00:04:05.406 --> 00:04:08.966 to negotiate several tricks, five different tricks. 00:04:09.296 --> 00:04:15.736 And students time their cars, figure out rate, graph the rates, and then, based on their graph, 00:04:15.736 --> 00:04:19.526 they are able to predict where they should release the car to navigate each of the tricks. 00:04:20.146 --> 00:04:25.926 The speeds for navigating the tricks aren't given to the students until the end and so, 00:04:26.066 --> 00:04:29.266 the day of the "grand pentathlon," as it's called, 00:04:29.266 --> 00:04:33.566 is very exciting for students because they use their own cars that they've built 00:04:33.766 --> 00:04:36.326 and they release them at certain heights on the ramp 00:04:36.326 --> 00:04:40.046 where they think their car will navigate the trick at the end of the ramp. 00:04:40.396 --> 00:04:44.076 The students learn a lot about pretty sophisticated math concepts. 00:04:44.076 --> 00:04:50.196 I want to emphasize that this project isn't just fun for the kids. 00:04:50.326 --> 00:04:52.306 It isn't just motivating for the kids. 00:04:52.576 --> 00:04:56.706 They also learn a lot about variables. 00:04:57.136 --> 00:05:00.776 They learn about measurement error; they learn about reliability. 00:05:00.876 --> 00:05:08.276 They learn how to figure out the relationship between distance, time. 00:05:08.516 --> 00:05:14.366 They're able to predict-using their graphs-speeds, 00:05:15.116 --> 00:05:18.706 how to label x-axis, y-axis, and what that means. 00:05:19.306 --> 00:05:26.806 So there are a lot of concepts that are packed into this teaching unit 00:05:27.366 --> 00:05:30.556 that are not taught individually. 00:05:30.756 --> 00:05:35.896 They are taught in a meaningful way much like you would encounter them in real life. 00:05:36.446 --> 00:05:40.836 [Music]