WEBVTT Kind: captions Language: en 00:00:41.239 --> 00:00:45.120 Ahoy there! Welcome to The Treasure Chests: A Bounty 00:00:45.120 --> 00:00:49.710 of Fractions, Decimals and Percentages. In this interactive session we will share 00:00:49.710 --> 00:00:52.920 with you and adventure on the seven seas that led us to creating the treasure 00:00:52.920 --> 00:00:58.049 chest problem which was published in the most recent edition of The Gazette. 00:00:58.049 --> 00:01:01.290 We will share our experience with developing tasks to support students 00:01:01.290 --> 00:01:05.460 with applying proportional reasoning and demonstrating effective problem-solving 00:01:05.460 --> 00:01:10.110 skills and we'll share with you the steps we took from brainstorming initial ideas, 00:01:10.110 --> 00:01:15.090 to linking mathematical concepts and curriculum expectations, to engaging 00:01:15.090 --> 00:01:18.690 students in the learning, to monitoring and consolidating the learning, and 00:01:18.690 --> 00:01:23.580 reflecting on our next steps. Along the way we'll invite you to pause and engage 00:01:23.580 --> 00:01:27.300 in the process yourself so that you too can play around with developing 00:01:27.300 --> 00:01:33.929 meaningful and fun problems. First though why don't you take a moment to figure us 00:01:33.929 --> 00:01:37.800 out? Using this engaging math task, we share a little bit about ourselves 00:01:37.800 --> 00:01:42.079 through the beauty and bounty of mathematics. 00:01:44.700 --> 00:01:50.979 So what did you learn about us? How might you use this in your own classroom? 00:01:50.979 --> 00:01:54.429 For Ken and I, the "figure me out activity" is a fantastic way to get started at the 00:01:54.429 --> 00:01:58.119 beginning of the school year. One reason being is because the students get to 00:01:58.119 --> 00:02:02.560 learn about one another and they also get to learn about their teacher. 00:02:02.560 --> 00:02:06.310 In addition to that the teacher also gets to learn a lot about the students who 00:02:06.310 --> 00:02:10.840 are just entering their classrooms. And it is more engaging than creating just 00:02:10.840 --> 00:02:15.280 the name card with interests in the four corners. Not only that there are a 00:02:15.280 --> 00:02:19.269 variety of other reasons why Ken and I love the "figure me out activity". 00:02:19.269 --> 00:02:23.950 It allows us to get to know our learners. In addition to that we get to engage 00:02:23.950 --> 00:02:27.760 students in mathematics. It gets the students talking and interacting with 00:02:27.760 --> 00:02:31.660 one another because once their figure me out is complete, they need to interact 00:02:31.660 --> 00:02:34.930 with one another and work collaboratively to help solve each 00:02:34.930 --> 00:02:38.739 other's questions in order to learn more about their partner. And rather than just 00:02:38.739 --> 00:02:41.920 looking at the figure me out and instantly learning about their partner, 00:02:41.920 --> 00:02:47.560 they actually need to engage in some work in order to find out more about who 00:02:47.560 --> 00:02:51.220 the person is that's standing right in front of them. It also helps the students 00:02:51.220 --> 00:02:56.920 take ownership of their learning and their classroom. In my classroom what 00:02:56.920 --> 00:03:01.120 I've done in the past is taking photos of the students and then pasted their 00:03:01.120 --> 00:03:06.220 top half of their body on top of their poster and then these posters get posted 00:03:06.220 --> 00:03:11.920 in the classroom walls and now they're reflected in your own classroom. 00:03:11.920 --> 00:03:16.180 This also serves as a diagnostic or formative assessment and you can see the type of 00:03:16.180 --> 00:03:20.290 operations that students are comfortable with. You can see some big ideas come out 00:03:20.290 --> 00:03:24.940 in the mathematics that they share and then you can also see the quantity or 00:03:24.940 --> 00:03:30.310 the amount or a number that the students are actually working with. And this will 00:03:30.310 --> 00:03:36.069 help to inform where the mathematics instruction takes goes next. Lastly it 00:03:36.069 --> 00:03:39.519 inspires to students to think about mathematics differently because we know 00:03:39.519 --> 00:03:43.150 that a lot of students think that mathematics only exists in the classroom 00:03:43.150 --> 00:03:47.799 and that it's something that comes from a textbook. But what Ken and I want to do 00:03:47.799 --> 00:03:51.849 is to share the beauty of mathematics and how it exists within our lives not 00:03:51.849 --> 00:03:58.650 just within a school context. The figure me out activity built a range 00:03:58.650 --> 00:04:02.610 of skills that we want to instill in our students. Take a moment to think about 00:04:02.610 --> 00:04:07.530 the skills that you were using when trying to figure Justin and I out. 00:04:07.530 --> 00:04:10.740 To name just a few, this task builds critical thinking 00:04:10.740 --> 00:04:17.190 skills, effective communication skills, visual literacy skills necessary for 00:04:17.190 --> 00:04:22.350 interpreting and representing ideas. It builds problem solving skills and 00:04:22.350 --> 00:04:26.610 important math skills. As well it encourages different forms of production. 00:04:26.610 --> 00:04:34.140 For example, writing, illustrating or graphing. We've created a slide here for 00:04:34.140 --> 00:04:38.280 you to try this yourself. Why not share this with your students online and 00:04:38.280 --> 00:04:41.340 encourage them to contribute their ideas for what they've learned about you. 00:04:41.340 --> 00:04:46.560 Perhaps they could use a chat room or any other form that you're using. 00:04:46.560 --> 00:04:51.330 Then they can try to make their own. This is a great activity that Justin and I have used 00:04:51.330 --> 00:04:56.610 many times so we wish you well in giving this a go. Ok, so now you know a 00:04:56.610 --> 00:05:01.860 little about us but who are we really? I mean, what do we know about math and why 00:05:01.860 --> 00:05:07.350 should you listen to us? Good question. My name is Justin Hui and I am an 00:05:07.350 --> 00:05:10.020 elementary school educator with experience teaching in the primary, 00:05:10.020 --> 00:05:14.130 junior and intermediate divisions with additional experience as an English 00:05:14.130 --> 00:05:18.180 language learner teacher and a primary special education resource teacher. 00:05:18.180 --> 00:05:22.080 At the moment I am a curriculum consultant with the York Region District School Board 00:05:22.080 --> 00:05:27.150 supporting teachers in the new teacher induction program. My name is Ken 00:05:27.150 --> 00:05:30.300 Pettigrew and I'm a principal in the curriculum and instructional services 00:05:30.300 --> 00:05:33.300 department in the York Region District School Board. I currently have 00:05:33.300 --> 00:05:40.530 responsibilities for mathematics, outdoor education and healthy schools. We know 00:05:40.530 --> 00:05:44.460 that there can be mixed feelings about mathematics. One of our goals is to pass 00:05:44.460 --> 00:05:48.390 on our love for playing with and thinking about numbers. We engage all 00:05:48.390 --> 00:05:52.530 learners by having a little fun through video making and this is the start of 00:05:52.530 --> 00:05:55.220 our math journey. 00:05:56.930 --> 00:06:02.130 Hey Justin, do you want to go grab a smoothie? Sure as soon as I'm done putting up this chart 00:06:02.130 --> 00:06:07.650 paper that I worked on together with my students. Math. I mean come on who really 00:06:07.650 --> 00:06:13.860 needs math. We all need math. I don't need math and look at me. Math is 00:06:13.860 --> 00:06:22.900 everywhere. Math is everywhere? What is math? Where is math? Who is math? 00:06:22.900 --> 00:06:28.900 Hi sir. What would you like? Can I get this please? 00:06:28.900 --> 00:06:32.800 Sure. It will be $1. 00:06:32.900 --> 00:06:36.680 That's all I got. 00:06:36.680 --> 00:06:40.500 Here you go. 00:06:40.600 --> 00:06:42.600 Thanks for shopping. 00:06:49.219 --> 00:06:58.019 This is all I've got. The bottle costs $1 and your handing him $20. 20 minus 1 00:06:58.019 --> 00:07:06.920 is 19. See its math. Oh. 00:07:06.920 --> 00:07:10.829 Yeah I'd like to call to complain. I ordered a pizza from you guys. 00:07:10.829 --> 00:07:16.769 I asked for half pepperoni, half cheese, half Hawaiian, and half veggie but you 00:07:16.769 --> 00:07:25.409 sent me two pizzas. I'm not paying for two pizzas. I'm phoning to complain. Wait. 00:07:25.409 --> 00:07:29.069 before you complain here's the important information. Here we have half a 00:07:29.069 --> 00:07:33.899 pepperoni pizza and half a vegetarian pizza. Half plus half makes one pizza. 00:07:33.899 --> 00:07:39.989 Then you have half a cheese pizza and half a Hawaiian pizza that makes another whole pizza. 00:07:39.989 --> 00:07:52.100 Ine pizza plus one pizza equals two pizzas. There's the math. Oh. Never mind. 00:07:52.100 --> 00:07:56.000 Math is everywhere. We use it every day. 00:07:56.129 --> 00:08:01.029 Justin you were right! Math is everywhere. It is everywhere. We've got to do 00:08:01.029 --> 00:08:04.629 something about this. We do. We've got to spread the message. We have to get people 00:08:04.629 --> 00:08:20.790 to see the math. Get people to use the math and get people to be the math. 00:08:20.969 --> 00:08:24.449 We appreciate that you have chosen to join us for the next 40 minutes or so. 00:08:24.449 --> 00:08:28.319 Take a moment to review the goals for this session. We have captured the goals 00:08:28.319 --> 00:08:32.759 through the P.O.P process. Setting our purpose, establishing the expected 00:08:32.759 --> 00:08:38.759 outcome and defining the process. Our purpose is to learn about the treasure 00:08:38.759 --> 00:08:43.259 chest problem and the process for creating the task. Our outcome is to 00:08:43.259 --> 00:08:46.350 develop a deeper understanding of fractions and proportional reasoning and 00:08:46.350 --> 00:08:51.209 to leave with a task to challenge your students and be equipped to create your 00:08:51.209 --> 00:08:56.160 own tasks. And the process that we're going to engage in is by having fun and 00:08:56.160 --> 00:09:02.160 engaging in reflective conversation. So let's get started. Had you joined us live 00:09:02.160 --> 00:09:05.399 in Oshawa this year, we would have engaged collectively in the following 00:09:05.399 --> 00:09:09.839 clothesline activity. But let's not let any physical distancing be a barrier to 00:09:09.839 --> 00:09:14.189 participating in a task to ignite our thinking. We're about to show you a 00:09:14.189 --> 00:09:19.050 number of cards. If you have access to the slide deck, you can manipulate these 00:09:19.050 --> 00:09:24.269 cards to move them around. We've also provided a link to Estimation 180 where 00:09:24.269 --> 00:09:30.000 you can find similar clothesline math activities. We are providing you with 16 00:09:30.000 --> 00:09:33.839 cards at the bottom of the slide. There are eight pairs of fractions represented 00:09:33.839 --> 00:09:38.939 there. In a moment pause the video to match up the pairs. In the classroom you 00:09:38.939 --> 00:09:41.490 could hand out a card to each student and then have them mingle around the 00:09:41.490 --> 00:09:45.329 room in search for their partner. Once they have partnered up, students could 00:09:45.329 --> 00:09:48.769 then talk about how the information on their cards represents the same fraction. 00:09:48.769 --> 00:09:52.829 For today, match up the cards replacing the fraction shown in standard 00:09:52.829 --> 00:09:56.850 notation on the top or red line. By standard notation we mean fractions 00:09:56.850 --> 00:10:01.110 written numerically with a numerator and a denominator. Place its pair directly 00:10:01.110 --> 00:10:05.040 under it on the bottom or blue line. If you are adventurous you could always 00:10:05.040 --> 00:10:10.019 download the cards and place them along pieces of string. While you are engaging 00:10:10.019 --> 00:10:13.800 in this activity, we invite you to think about your students and to consider how 00:10:13.800 --> 00:10:17.250 they would approach this task. What assumptions might they make? What 00:10:17.250 --> 00:10:20.759 questions might they have? What misconceptions are might surface? Pause 00:10:20.759 --> 00:10:24.620 the video now to give this activity a go. 00:10:28.850 --> 00:10:34.110 Now we've done this activity many times with many groups of learners. One of the 00:10:34.110 --> 00:10:39.180 fraction pairings that seems to raise a lot of questions is this one. Many 00:10:39.180 --> 00:10:42.930 students will look at the image on the left and assume that it represents the 00:10:42.930 --> 00:10:48.180 fraction of one half. After all half of the rectangles on the card are shaded in 00:10:48.180 --> 00:10:53.130 red. Of course that assumes that we are looking at this through the lens of an 00:10:53.130 --> 00:10:57.630 area model where the whole is represented by twelve rectangles and 00:10:57.630 --> 00:11:04.230 half of the area is shaded in red. But is there another way of looking at it? What 00:11:04.230 --> 00:11:08.910 if we looked at this through the lens of a set model. In a set model the whole is 00:11:08.910 --> 00:11:14.370 represented by a collection of items. In this case we have six individual shapes. 00:11:14.370 --> 00:11:21.589 Each shape has half of it colored in red. In this case we actually have six halves. 00:11:21.589 --> 00:11:27.029 Hmm, does that sound confusing? Well imagine for a moment that you are 00:11:27.029 --> 00:11:31.950 running an art class. You have six table groups and you've given each table group 00:11:31.950 --> 00:11:37.649 a full can of red paint to use for art class. At the end of class each of the 00:11:37.649 --> 00:11:43.529 six table groups has only a half a can of paint left over. So there, what do you 00:11:43.529 --> 00:11:48.660 have? You have six half filled cans of paint or six one halfs. 00:11:48.660 --> 00:11:54.540 If we consolidate the six half cans together, we actually have three full 00:11:54.540 --> 00:11:59.959 cans. This emphasizes the need for us to be open to different ways of thinking. 00:11:59.959 --> 00:12:04.890 Through talk we uncover how students are seeing fractions. How they are 00:12:04.890 --> 00:12:08.880 interpreting what they see and how they are making sense of the concepts we are 00:12:08.880 --> 00:12:13.050 introducing them to. Marian Small reminds us about the power of questions 00:12:13.050 --> 00:12:18.180 so instead of asking, "What fraction is covered in red?" We might ask "How do you 00:12:18.180 --> 00:12:23.010 see this fraction?" That subtle change in question could make a whole world of 00:12:23.010 --> 00:12:25.880 difference for students. 00:12:26.550 --> 00:12:30.819 The pairing that we just saw is a good example of why it is so important that 00:12:30.819 --> 00:12:35.050 we teach about fractions in a wide variety of ways. Since there are different models 00:12:35.050 --> 00:12:39.220 of fractions, students need to be exposed to these in order to have a deeper 00:12:39.220 --> 00:12:43.449 understanding of what fractions truly represent. Let's take a quick look at 00:12:43.449 --> 00:12:48.370 four fraction models. Linear models are where the length is divided into 00:12:48.370 --> 00:12:53.639 fractional parts. Number lines and bar graphs are examples of linear models. 00:12:53.639 --> 00:12:58.029 In area models, one shape or region represents the hole which is then 00:12:58.029 --> 00:13:04.600 divided into fractional parts. In volume models, 3D objects or figures represent 00:13:04.600 --> 00:13:10.720 the whole and the fractions occupy space within the whole. For instance, water that 00:13:10.720 --> 00:13:16.779 takes up half of the glass. In a set model, a collection of objects represents 00:13:16.779 --> 00:13:20.620 the whole and subsets of the objects represent the fractions. For instance, in 00:13:20.620 --> 00:13:25.389 the image on the slide, 2/3 of the pattern blocks have four sides even 00:13:25.389 --> 00:13:30.360 though they are different shapes and sizes, they represent a fraction of a set. 00:13:30.360 --> 00:13:35.230 By exposing students to a wide variety of fractions, we hope to surface and 00:13:35.230 --> 00:13:40.000 address misconceptions and thereby build understanding. Take a look at the 00:13:40.000 --> 00:13:44.709 following three examples of students responses to fractions. Pause the video 00:13:44.709 --> 00:13:49.709 to think about, what's the misconception that's being surfaced here? 00:13:52.700 --> 00:13:58.940 Now let's review each response. In the first example, in the green box, the 00:13:58.940 --> 00:14:03.440 student doesn't recognize the fraction when the shapes are different sizes. They 00:14:03.440 --> 00:14:07.760 assume that the only way to observe one-fourth would be to see one smaller 00:14:07.760 --> 00:14:13.130 square shaded in green. In this case the fraction represents an area model. The 00:14:13.130 --> 00:14:18.440 hole is the larger square and the green triangle is still one-fourth of the 00:14:18.440 --> 00:14:25.490 shape. In the second example, in the red box, the student doesn't recognize the 00:14:25.490 --> 00:14:31.790 fraction when the objects in the set are different. Pikachu is one of four objects 00:14:31.790 --> 00:14:39.050 in the set so he represents one-fourth of the objects in this set model. In the 00:14:39.050 --> 00:14:43.550 third example, in the blue box, the student doesn't recognize the fraction 00:14:43.550 --> 00:14:48.410 when the objects are separated by space. In this case the green apples represent 00:14:48.410 --> 00:14:56.270 one-third of the apples in a linear model. Before we set about to design our 00:14:56.270 --> 00:15:01.130 task, we began by asking ourselves, "What do we know about our students?" We knew 00:15:01.130 --> 00:15:04.460 that in order to better support them in their learning, we needed to anticipate 00:15:04.460 --> 00:15:09.320 how they would approach a problem about fractions. We needed to examine what it 00:15:09.320 --> 00:15:13.040 was that they did know and what skills are they using along the trajectory 00:15:13.040 --> 00:15:17.660 toward a deeper understanding. A tool that we turned to was the fractions, 00:15:17.660 --> 00:15:22.850 decimals, and percents landscape of learning from Cathy Fosnot. We have 00:15:22.850 --> 00:15:26.120 provided you with a link to this document. Pause the video and take a look 00:15:26.120 --> 00:15:32.180 at the landscape. As you do think about your students. How might students solve a 00:15:32.180 --> 00:15:37.490 problem involving fractions? What skills might they bring as assets? Which big 00:15:37.490 --> 00:15:42.740 ideas models and landmark strategies might your students use? What misconception 00:15:42.740 --> 00:15:47.930 might interfere with their demonstration of learning? Record any questions you 00:15:47.930 --> 00:15:51.430 might have in the Google Doc. 00:15:53.620 --> 00:15:59.330 So once Justin and I reflected on what we knew about our students, we set about 00:15:59.330 --> 00:16:03.740 to design a problem to captivate their attention. We wanted to challenge their 00:16:03.740 --> 00:16:09.680 thinking and to help them to build new skills. So we followed the plan. 00:16:09.680 --> 00:16:14.420 We started with the curriculum and then we determined what the big ideas were that 00:16:14.420 --> 00:16:19.580 we're going to guide the learning. Next we identified the specific skills that 00:16:19.580 --> 00:16:24.080 we wanted to target. From there we planned the strategies that we were 00:16:24.080 --> 00:16:30.110 going to use to build those skills. And then as we engaged in the problems, we 00:16:30.110 --> 00:16:36.050 provided ongoing and timely feedback to help students refine those skills. 00:16:36.050 --> 00:16:40.190 For the treasure chest problem we identified the following curriculum expectations 00:16:40.190 --> 00:16:44.950 from the grade five number sense and numeration strand. 00:16:51.460 --> 00:16:55.730 From there we want to build students understanding about the relationships 00:16:55.730 --> 00:16:59.750 between fractions, decimals, and percents and we knew that that required them to 00:16:59.750 --> 00:17:07.040 apply an understanding of proportional reasoning. So we wondered about how we 00:17:07.040 --> 00:17:09.470 could get our students to demonstrate the skill of connecting the 00:17:09.470 --> 00:17:13.520 relationships between fractions, decimals and percents. We wanted it to be 00:17:13.520 --> 00:17:17.090 interesting and truth be told we were experimenting with problem solving 00:17:17.090 --> 00:17:21.650 ourselves. Over recess break one day we kept talking about ideas to spark 00:17:21.650 --> 00:17:28.610 students curiosity. That is when the treasure chests were born. Next we needed 00:17:28.610 --> 00:17:32.750 to identify the skills that students needed to apply in order to be 00:17:32.750 --> 00:17:37.220 successful with this problem. We brainstormed what we thought were the 00:17:37.220 --> 00:17:41.210 important skills that fell under the different achievement chart categories 00:17:41.210 --> 00:17:46.280 and then collaboratively with students, we developed a set of success criteria 00:17:46.280 --> 00:17:52.640 to help guide students throughout the problem-solving process. Take a moment to 00:17:52.640 --> 00:18:01.070 review the success criteria. You'll notice that we identified key skills in 00:18:01.070 --> 00:18:05.570 the four different categories of the achievement chart. We knew that under 00:18:05.570 --> 00:18:08.990 knowledge and understanding it was important that students demonstrated 00:18:08.990 --> 00:18:13.280 that they understood that fractions, decimals, and percents were all different 00:18:13.280 --> 00:18:17.570 ways of representing values. We also knew that it was important for them to 00:18:17.570 --> 00:18:22.730 understand that each of these could be converted to the other. It was also going 00:18:22.730 --> 00:18:26.150 to be very important that students understood that fractions could be 00:18:26.150 --> 00:18:32.660 simplified. In the thinking category we knew that it was going to be important 00:18:32.660 --> 00:18:36.590 that students be able to show that they could develop a plan of action to focus 00:18:36.590 --> 00:18:41.000 on the different aspects of the problem. We wanted to see that students were able 00:18:41.000 --> 00:18:45.290 to apply critical and creative thinking skills when defending how reasonable 00:18:45.290 --> 00:18:50.840 their answer was. In terms of communication we wanted students to be 00:18:50.840 --> 00:18:55.040 able to represent their thinking about fractions, decimals, and percents in a way 00:18:55.040 --> 00:19:00.710 that was meaningful to them. This particular problem required that 00:19:00.710 --> 00:19:03.890 students demonstrate what they knew about numbers 00:19:03.890 --> 00:19:08.750 and proportional relationships in order to solve the problem. We wanted to know 00:19:08.750 --> 00:19:12.440 that they could synthesize the learning and explain how they could apply the 00:19:12.440 --> 00:19:18.530 learning to new situations. The time spent on co-developing the success 00:19:18.530 --> 00:19:22.640 criteria with students was vital because it really helped set them up for success 00:19:22.640 --> 00:19:28.130 when they engaged in the problems and those that followed. We decided to 00:19:28.130 --> 00:19:31.610 present students with a scenario based on the idea that we were a band of 00:19:31.610 --> 00:19:36.470 pirates who had found two treasure chests. We were willing to split the 00:19:36.470 --> 00:19:40.370 treasure two ways. We would keep a fraction of the treasure and give our 00:19:40.370 --> 00:19:44.510 band of pirates, our students, the remaining fraction as any respectable 00:19:44.510 --> 00:19:49.310 pirate would. However we extended the offer in a form of a puzzle. We gave 00:19:49.310 --> 00:19:54.740 students four options to choose from. Take a look now at our options. Notice 00:19:54.740 --> 00:19:58.640 that the options on the left represent the fraction of the treasure in each of 00:19:58.640 --> 00:20:03.080 the two chests that we would keep. While the options on the right represent the 00:20:03.080 --> 00:20:06.590 fraction of the treasure in each of the two chests that the students would get. 00:20:06.590 --> 00:20:12.110 In a moment pause the video to engage with the problem. Think about how your 00:20:12.110 --> 00:20:15.230 students would approach it. We have provided a link to some digital 00:20:15.230 --> 00:20:20.350 manipulatives if you would like to represent your thinking in that manner. 00:20:23.770 --> 00:20:29.320 Of course every problem comes with an additional challenge. Once students had 00:20:29.320 --> 00:20:32.770 worked their way through the initial problem and began to make some decisions 00:20:32.770 --> 00:20:36.280 about which treasure option they would choose, we presented them with some new 00:20:36.280 --> 00:20:42.370 information. Now we revealed the value of the treasure in each chest. We asked them, 00:20:42.370 --> 00:20:46.440 "Now that you know what is inside the chest, how might your choice change?" 00:20:46.440 --> 00:20:52.980 Defend your thinking. Pause the video for a moment to record your own thinking. 00:20:57.940 --> 00:21:02.019 As students engaged in the problem we circulated the room to monitor their 00:21:02.019 --> 00:21:06.879 progress. An important element of problem solving is that time is given for 00:21:06.879 --> 00:21:10.929 students to engage in productive struggle. That's the time where students 00:21:10.929 --> 00:21:15.429 explore and experiment with ideas. Engage in discussion and debate and use their 00:21:15.429 --> 00:21:21.009 mistakes to motivate and pass. Our rule throw productive struggle is to provide 00:21:21.009 --> 00:21:26.080 the right feedback at the right time to support students along in meeting the 00:21:26.080 --> 00:21:31.929 learning goal. Depending on how far they are from that goal, the type of feedback 00:21:31.929 --> 00:21:37.000 we provide will differ. Students who are far away from the goal may require an 00:21:37.000 --> 00:21:41.559 example to provide a more concrete understanding of the concept. This might 00:21:41.559 --> 00:21:46.600 require some modeling followed by some guided practice. Students who are a 00:21:46.600 --> 00:21:50.740 little closer to the goal may need a scaffold such as a connection to a 00:21:50.740 --> 00:21:55.360 previous task or encouragement to further develop an idea or to take a 00:21:55.360 --> 00:22:00.009 look around the room to see where they can find that information. Students who 00:22:00.009 --> 00:22:04.750 are very close to the goal may need a reminder such as double-checking the 00:22:04.750 --> 00:22:12.759 success criteria or reviewing their representation to ensure clarity. 00:22:12.759 --> 00:22:16.330 Pause the video for a moment to look over some of the suggested prompts at the three 00:22:16.330 --> 00:22:20.529 levels of feedback. You may want to consider what additional feedback would 00:22:20.529 --> 00:22:22.919 you include. 00:22:27.080 --> 00:22:32.130 Feedback is integral to supporting students in reaching the learning goal. 00:22:32.130 --> 00:22:37.080 So when they're farther away we want to provide an example for them and this 00:22:37.080 --> 00:22:43.200 might take the form of really providing them with an example or perhaps showing 00:22:43.200 --> 00:22:48.660 them and then having them try along with you or identifying a next step for the 00:22:48.660 --> 00:22:53.250 student and let's try it together so that you can see that they understand 00:22:53.250 --> 00:22:57.270 what to do. This may be putting manipulatives into their hand and 00:22:57.270 --> 00:23:03.000 actually showing them. As students get closer to the learning goal the feedback 00:23:03.000 --> 00:23:07.679 we provide may be more in the form of a scaffold where you might direct them to 00:23:07.679 --> 00:23:12.750 information that is available around the room that will support them or we may 00:23:12.750 --> 00:23:17.280 ask them to go back and add some new ideas something that maybe had been 00:23:17.280 --> 00:23:22.320 practiced before or been part of the lesson earlier or it may be a prompt to 00:23:22.320 --> 00:23:27.990 ask them to further develop an idea or tell me more about. If students are 00:23:27.990 --> 00:23:32.070 pretty close to attaining that learning goal, then perhaps all they require is a 00:23:32.070 --> 00:23:37.230 reminder. So you might just be asking them to remember to double-check a few 00:23:37.230 --> 00:23:41.640 things or to double-check the success criteria because you know that they're 00:23:41.640 --> 00:23:49.110 very close. Learning takes place during the consolidation. It is here that the 00:23:49.110 --> 00:23:53.010 connections between ideas are made explicit. Take a look at this work sample 00:23:53.010 --> 00:23:57.960 from a pair of students. What do you notice? What rationale have these 00:23:57.960 --> 00:24:02.630 students provided? What does that reveal about what they know about fractions? 00:24:02.630 --> 00:24:06.270 What is clear is that these students have demonstrated that they see the 00:24:06.270 --> 00:24:10.140 relationships between fractions, decimals, and percents and that they can justify 00:24:10.140 --> 00:24:13.140 the reasonableness of their answer through connecting what they know about 00:24:13.140 --> 00:24:15.679 fractions. 00:24:19.350 --> 00:24:23.670 To ensure that all students had access to the learning, we made sure to make 00:24:23.670 --> 00:24:27.420 options available that took into account different learning styles and learning 00:24:27.420 --> 00:24:32.850 needs. Manipulatives were made available both physical and digital. We also had an 00:24:32.850 --> 00:24:36.630 alternative problem-solving page that was more visual in nature in order to 00:24:36.630 --> 00:24:43.890 support students for whom the text might have been a barrier. Just as we engage 00:24:43.890 --> 00:24:48.720 our students in developing success criteria for their learning, we also 00:24:48.720 --> 00:24:53.400 engage in the practice with adult learners. Over time we have met with many 00:24:53.400 --> 00:24:57.330 teachers from across Ontario and we've worked with them to develop a success 00:24:57.330 --> 00:25:02.370 criteria for creating a meaningful problem. Take a look at the success 00:25:02.370 --> 00:25:09.570 criteria here. What do you notice? What do you wonder? Are there things that you might add? 00:25:09.570 --> 00:25:16.710 Think for a moment about what were some of the characteristics of the 00:25:16.710 --> 00:25:23.640 problem-solving process that that we went through. We've identified four areas 00:25:23.640 --> 00:25:30.480 as indicators of a meaningful problem. First we know that it's important that a 00:25:30.480 --> 00:25:35.970 meaningful problem includes multiple entry points, that it's open-ended, it's 00:25:35.970 --> 00:25:39.510 important that all students are able to access the learning and can enter in 00:25:39.510 --> 00:25:45.090 where they are and that they can extend their thinking. We want to make sure that 00:25:45.090 --> 00:25:51.300 our problems have many possible answers and many possible approaches to allow 00:25:51.300 --> 00:25:55.460 different ways of understanding and different ways of knowing to be surfaced. 00:25:55.460 --> 00:26:01.680 We also know that it is important to include relevance. That we want students 00:26:01.680 --> 00:26:06.780 to be able to see the importance, the joy, the beauty, the relevance of the math 00:26:06.780 --> 00:26:12.810 problem. It's also important that a meaningful problem has a focus. The 00:26:12.810 --> 00:26:16.770 rationale. We need to know why are we choosing the numbers that we are or the 00:26:16.770 --> 00:26:22.860 concept that we are? That needs to be at the forefront of the problem. That helps 00:26:22.860 --> 00:26:27.180 us to identify the anticipated responses from students and it helps us to know 00:26:27.180 --> 00:26:32.500 which tools and strategies to guide our teaching and instruction. 00:26:32.500 --> 00:26:37.900 As well it's really important that a meaningful problem is open to students 00:26:37.900 --> 00:26:40.809 with special needs or for English language learners. 00:26:40.809 --> 00:26:46.150 A very important consideration. How might this support you in developing your own 00:26:46.150 --> 00:26:52.030 meaningful problem? Thank you for spending time with us today. We know that 00:26:52.030 --> 00:26:55.360 you have many choices and so we appreciate that you chose to join us. 00:26:55.360 --> 00:27:00.070 You know when all this started we were just beginning on our whole problem-solving 00:27:00.070 --> 00:27:05.320 adventure. We experimented with ideas, make mistakes but leverage those to try 00:27:05.320 --> 00:27:11.559 new ideas and we kept navigating these mathematical waters. This particular task 00:27:11.559 --> 00:27:16.990 really resonated with students. So we took a risk. We decided to contact the 00:27:16.990 --> 00:27:24.880 OAME Gazette to share our problem. After some thoughtful feedback and support 00:27:24.880 --> 00:27:29.950 from the good people at OAME, our problem was published in the current edition of 00:27:29.950 --> 00:27:34.390 The Gazette. We encourage you to check it out and we hope that you will consider 00:27:34.390 --> 00:27:38.950 setting sail on the high seas yourself to share your problem ideas and 00:27:38.950 --> 00:27:45.610 adventures. Please feel free to contribute ideas or questions to the 00:27:45.610 --> 00:27:52.179 Google Doc or to contact us at the address as you see on the screen. We hope 00:27:52.179 --> 00:27:56.590 you've been enjoying the OAME virtual conference this year and we wish you all 00:27:56.590 --> 00:27:58.830 the best.