WEBVTT Kind: captions Language: en 00:00:23.590 --> 00:00:27.340 I'm humbled and thrilled to be here with you today sharing my love for mathematics. 00:00:27.590 --> 00:00:32.169 I'm the proud father of three hockey, baseball and basketball crazed boys and 00:00:32.270 --> 00:00:35.979 have spent my career teaching in Ontario with the York Region District School Board. 00:00:35.980 --> 00:00:41.230 I currently teach in Thornhill as a grade six seven homeroom teacher and math coach. 00:00:42.830 --> 00:00:50.709 My continued evolution as a math educator began with the discovery of the three-act math task specifically donut delight from Kyle Pearce. 00:00:51.260 --> 00:00:58.779 This task completely changed my outlook on what math education can be and quite literally changed the trajectory in my career. 00:00:59.480 --> 00:01:06.580 Over time this is nurtured my belief around the vital role playful mathematics has in supporting the development of mathematical reasoning and 00:01:06.800 --> 00:01:09.249 helping students uncover the beauty of mathematics. 00:01:11.780 --> 00:01:13.780 So what do Bill Gates, 00:01:15.050 --> 00:01:17.050 The Beatles and 00:01:18.080 --> 00:01:20.080 Wayne Gretzky have in common? 00:01:20.570 --> 00:01:25.300 Well without question they're world-famous. Some might even call them masters of what they do. 00:01:25.820 --> 00:01:29.770 They were and are driven by their passion and love for their craft. 00:01:30.319 --> 00:01:32.319 But how did they get there? 00:01:33.440 --> 00:01:40.779 As youngsters pursuing their passion, they played and they played a lot and most importantly they played in a fun environment 00:01:41.030 --> 00:01:43.630 that nurtured their curiosity and passion. 00:01:44.420 --> 00:01:46.899 As we are or well, at least I am Canadian, 00:01:47.660 --> 00:01:51.490 let's look a little bit more closely at how Gretzky developed his skills. 00:01:53.990 --> 00:02:01.240 For Gretzky practicing in a fun and meaningful way for him helped to nurture a lifelong passion and love for the game. 00:02:01.490 --> 00:02:06.429 In a mathematics classroom. This is what we should be striving for in all of our learners. 00:02:07.220 --> 00:02:12.160 As teachers we need to be designing learning opportunities for students to play and have fun 00:02:12.500 --> 00:02:16.419 so they develop a lifelong love and pursuit of mathematics. 00:02:18.500 --> 00:02:22.330 In his article "Mathematics Just Play", author and researcher 00:02:22.520 --> 00:02:28.329 Sunil Singh pens how Gretzky has commented on the current state of the game and how it is over coached. 00:02:28.520 --> 00:02:32.919 With so much emphasis on positional play and drills, all of which are important, 00:02:32.989 --> 00:02:39.949 but only so if they do not impact and jeopardize on the pure fun and joy found in the game. In a math class 00:02:40.170 --> 00:02:43.880 this is akin to the hockey culture and thinking that Gretzky bemoans about. 00:02:44.400 --> 00:02:46.490 Little play and zero fun. 00:02:49.050 --> 00:02:50.220 00:02:50.220 --> 00:02:52.220 In more traditional models of teaching 00:02:52.710 --> 00:02:56.179 students might give what Mike Flynn calls the illusion of understanding. 00:02:56.430 --> 00:03:01.340 It's the idea that kids can seem like they're understanding math by providing correct answers 00:03:01.340 --> 00:03:06.979 but they may not actually understand it and instead have just memorized facts rules and 00:03:07.500 --> 00:03:09.420 procedures. 00:03:09.420 --> 00:03:16.429 So we're going to look a little bit deeper into this by going through a grade five test on fractions decimals and percents. 00:03:17.040 --> 00:03:20.419 For context the pre-assessment classwork and 00:03:20.640 --> 00:03:26.779 practice test questions are all aligned with the types of questions and layout of the test. And as you go through it 00:03:27.480 --> 00:03:30.050 let's have one guiding question at the back of our minds, 00:03:30.750 --> 00:03:34.639 what information is this giving me about this particular student? 00:04:11.590 --> 00:04:13.360 So, what do we now know? 00:04:13.360 --> 00:04:20.760 Well, the student can determine the least common multiple, can order different quantities, and can convert fractions among other things. 00:04:21.280 --> 00:04:28.470 But is this because he understands what he's doing or because he's familiar with the procedures we can make assumptions either way 00:04:28.720 --> 00:04:32.070 But do we really have a sense of this students thinking and understanding? 00:04:32.860 --> 00:04:38.490 I'm not sure we really do because unfortunately often times we equate familiarity with understanding. 00:04:39.580 --> 00:04:40.600 00:04:40.600 --> 00:04:47.549 In his article "A Few Simple Beliefs", Mark Chubb states that we have all become educators because we want our students to be successful 00:04:47.860 --> 00:04:50.940 and we want to do our best to help them do well. 00:04:51.550 --> 00:04:58.410 However, we're often so eager to get the results that we want that we don't take enough time to allow our students to think, 00:04:59.200 --> 00:05:05.759 to explore, to make sense of the math, to realize why we're learning what we're learning. 00:05:06.190 --> 00:05:08.850 In our eagerness to have our students get answers 00:05:08.850 --> 00:05:12.479 we often miss the developmental pieces that our students need to be successful. 00:05:12.789 --> 00:05:17.339 And if the goal is to reliably get answers for questions involving 00:05:17.650 --> 00:05:22.769 computation, the best and most reliable means to do this is to pull out a smart phone or calculator. 00:05:23.260 --> 00:05:26.429 But we can't be teaching just for the sake of getting answers. 00:05:27.039 --> 00:05:34.919 To support the development of mathematical reasoning we must be teaching for understanding and this begins with playful mathematics. 00:05:38.500 --> 00:05:44.910 The Ontario Ministry of Education defines mathematical reasoning as a process that supports deeper understanding of 00:05:45.310 --> 00:05:51.419 mathematics by enabling students to make sense of the mathematics their learning. This involves exploring phenomena, 00:05:52.599 --> 00:05:54.340 developing ideas, 00:05:54.340 --> 00:05:56.369 making mathematical conjectures, and 00:05:57.099 --> 00:05:59.099 justifying results. 00:05:59.229 --> 00:06:07.139 The ability to think and reason is an essential factor in the development of an individual's ability to understand and apply mathematics. 00:06:07.900 --> 00:06:11.429 While students may be able to solve problems with a memorized procedure, 00:06:11.800 --> 00:06:15.990 they may not have developed the necessary skills towards building a deep knowledge, 00:06:16.599 --> 00:06:18.130 understanding, and 00:06:18.130 --> 00:06:23.070 appreciation of key math concepts. They might just be good at steps or procedures but might not 00:06:23.440 --> 00:06:29.279 understand the math behind what they've done. And I believe that's what we saw in the example from a few moments ago. 00:06:32.860 --> 00:06:35.820 Developing reasoning skills with young learners is a complex business. 00:06:36.460 --> 00:06:42.930 They need to learn to become systematic thinkers and also acquire the ability to articulate such thinking in a clear, 00:06:43.450 --> 00:06:45.450 succinct and logical manner. 00:06:46.210 --> 00:06:54.120 Nrich Maths identifies a five-step progression in reasoning. A spectrum that shows us whether children are moving on in their reasoning trajectory. 00:06:55.240 --> 00:07:00.900 Children are unlikely to move fluidly from one step to the other but rather they flow up and down the spectrum 00:07:01.480 --> 00:07:05.489 settling on a particular step that best describes their reasoning skills at any one time. 00:07:07.780 --> 00:07:11.309 So let's now take a look at how we can take a simple puzzle and 00:07:11.590 --> 00:07:15.660 generate a discussion to support the development and assessment of student reasoning. 00:07:15.970 --> 00:07:20.279 But first we're gonna pause for a few moments so you can take the time to try to determine 00:07:20.470 --> 00:07:23.100 what fraction of the shape appears to be shaded? 00:07:35.020 --> 00:07:39.210 A critical move for teachers is taking the time to engage in the task before students do. 00:07:39.210 --> 00:07:41.940 By making this a regular part of our practice 00:07:42.190 --> 00:07:48.959 we're giving ourselves an opportunity to really focus in on the four principles of teaching and assessing for understanding. 00:07:48.960 --> 00:07:50.960 Now we'll have a better sense of what to look for. 00:07:51.760 --> 00:07:58.499 To recognize student learning or misconceptions as they happen so we can be responsive to the needs of students in the moment. 00:07:59.710 --> 00:08:05.100 So now we're going to go ahead to the next slide and we're going to take a quick listen to some student thinking and reasoning. 00:08:05.100 --> 00:08:14.100 00:08:14.600 --> 00:08:20.100 The way I got this was technically, I ignored this line. 00:08:20.100 --> 00:08:24.100 And if I added up all the rectangles. 00:08:24.100 --> 00:08:31.100 And then I looked at this one specifically and I saw that a piece of it wasn't missing. A piece of it was missing. 00:08:32.470 --> 00:08:39.430 And then I saw that over here was identical to this one so 00:08:39.430 --> 00:08:43.400 It would be the same as putting this one here. 00:08:43.400 --> 00:08:49.400 Which would complete the one and it would make one. And since there's 12, 00:08:49.400 --> 00:08:53.400 then it would be 1 out of 12. 00:08:58.460 --> 00:09:03.819 So based on what we just heard where might this student be on the progression of reasoning continua. 00:09:05.750 --> 00:09:13.000 Is she simply describing what she did? Is she explaining her process while offering up some reasons? 00:09:15.770 --> 00:09:21.370 Or is she providing a convincing and confident argument showing a progression of mathematical thinking? 00:09:22.910 --> 00:09:28.360 Is she at the latter stages of the continuum by justifying her strategy and solution? 00:09:29.210 --> 00:09:33.639 Or has she provided sound proof with an argument that is reasoned with mathematics? 00:09:35.630 --> 00:09:37.390 Wherever she falls on the continuum 00:09:37.390 --> 00:09:44.800 we now have clear evidence of her thinking and reasoning and can confidently identify for this student where they're going next. 00:09:46.490 --> 00:09:51.370 And this is where we could consider going next to support the development of this students thinking and reasoning. 00:09:59.300 --> 00:10:01.930 Now let's take a look at the classic towers of Hanoi. 00:10:02.510 --> 00:10:10.060 The towers of Hanoi is a puzzle consisting of three rods and a number of disks of different sizes, which can slide onto any rod. 00:10:10.580 --> 00:10:16.930 The puzzle starts with the disks in a neat stack in ascending order of size on one rod the smallest at the top. 00:10:17.390 --> 00:10:23.259 The objective of the puzzle is to move the entire stack to another rod in the fewest moves possible, 00:10:23.510 --> 00:10:28.749 obeying the following simple rules. Number one - only one disk can be moved at a time. 00:10:29.690 --> 00:10:34.510 Number two - each move consists of taking the upper disk from one of the stacks and 00:10:34.790 --> 00:10:38.230 placing it on top of another stack or on an empty rod. 00:10:39.290 --> 00:10:43.749 Number three - no larger disk may be placed on top of a smaller disk. 00:10:45.950 --> 00:10:47.170 Now not to worry 00:10:47.170 --> 00:10:50.829 if you don't happen to have a class set of Hanoi towers lying around. 00:10:50.990 --> 00:10:52.850 Three pieces of tape and really any 00:10:52.850 --> 00:10:57.999 objects of different sizes will work as we see here with what I did with two of my own kids. 00:10:58.160 --> 00:11:00.939 And not that I happen to have sets of relational rods just lying around 00:11:01.130 --> 00:11:06.309 but as we prepared to transition to a distance learning model, I made sure to have someone home to play with. 00:11:07.610 --> 00:11:11.140 So we're going to take a pause and take a few minutes to get three pieces of tape, 00:11:11.780 --> 00:11:15.309 three objects of different sizes and let's try to move the entire 00:11:16.160 --> 00:11:21.579 stack to another rod in the fewest moves possible by following the rules. 00:11:21.580 --> 00:11:27.280 Again the objective of the puzzle is to move the entire stack to another rod and the fewest moves possible obeying these rules: 00:11:27.800 --> 00:11:30.370 One - only one disk can be moved at a time 00:11:30.830 --> 00:11:31.580 Two - 00:11:31.580 --> 00:11:32.540 each move 00:11:32.540 --> 00:11:37.689 consists of taking the upper disc from one of the stacks and placing it on top of another stack or on an empty rod. 00:11:38.570 --> 00:11:42.670 Three - no larger disc may be placed on top of the smaller disk. 00:11:53.690 --> 00:11:57.429 By again taking the time to engage in the task before students as 00:11:58.100 --> 00:12:05.920 educators we are better equipped to incorporate the principles of teaching and assessing for understanding in order to be responsive to student thinking and understanding 00:12:08.810 --> 00:12:10.760 00:12:10.760 --> 00:12:17.499 And after perhaps a few tries we can see that with three discs the puzzles can be solved in seven moves. 00:12:20.270 --> 00:12:23.290 So now we're going to go to the next slide and we're going to take a quick listen 00:12:23.360 --> 00:12:28.960 to some of my oldest son's thinking and reasoning after moving the entire stack and for context he's 11 years old. 00:12:38.400 --> 00:12:43.700 Can you tell us how you knew Micah to move the rods in the way that you were moving them? 00:12:43.700 --> 00:12:51.700 I moved them in that way because the smallest one it can go either on the second or the third piece of tape. 00:12:52.000 --> 00:12:53.500 So I moved it onto the third. 00:12:53.810 --> 00:12:58.000 Then I knew I have to move the second one onto the second piece of tape. 00:12:58.430 --> 00:13:06.400 And then if I wanted to move the biggest rod I would have to move the smallest rod onto the green rod. 00:13:06.400 --> 00:13:12.400 And I moved the maroon rod over to the third piece of tape. 00:13:12.400 --> 00:13:18.400 And I moved the smallest one here because then I have access to move this one over here. 00:13:18.400 --> 00:13:20.400 Then I just have to move this one. 00:13:25.879 --> 00:13:32.739 So coming back to the progression of reasoning continua how might we best describe Micah's thinking and reasoning? 00:13:33.439 --> 00:13:37.779 Now in the absence of any follow-up questions, he's likely at step 2 or step 3. 00:13:38.029 --> 00:13:42.489 But we would need to follow up as we likely don't have enough evidence yet to make that assessment. 00:13:43.069 --> 00:13:45.069 So where might we go next? 00:13:51.709 --> 00:13:56.469 We're going with four rods, so that's what we did and in the spirit of productive struggle 00:13:56.809 --> 00:14:00.789 he went at it trying to move the entire stack in the fewest moves possible. 00:14:01.490 --> 00:14:08.500 After several tries, he just wasn't seeing it yet. So what moves could we incorporate to support his thinking and understanding? 00:14:15.740 --> 00:14:20.979 We can lower the floor just a little bit to give him an entry point that's more accessible for him. 00:14:21.439 --> 00:14:27.189 So first we went to two rods and decided to create a table just to chart some of the things that we were doing/ 00:14:29.600 --> 00:14:31.490 00:14:31.490 --> 00:14:36.609 And then down to one rod. So now things are beginning to open up for him. 00:14:36.610 --> 00:14:40.479 Now he's able to visualize that perhaps there's a pattern he might be looking for. 00:14:41.089 --> 00:14:48.549 So based on what we see his thinking is developing and he's making a conjecture around the number of moves for different numbers of rods. 00:14:49.550 --> 00:14:56.800 So now based on what we've heard from Micah the teacher moves that we could consider incorporating and what we're seeing on the table, 00:14:57.529 --> 00:14:59.529 where might he fall on the continua? 00:15:01.759 --> 00:15:04.279 How might we respond to what he's doing? 00:15:05.430 --> 00:15:07.380 And what questions or prompts 00:15:07.380 --> 00:15:11.840 could we support or could we incorporate to support the development of Micah's thinking? 00:15:12.360 --> 00:15:18.500 However, if we look a little closely at the chart, we can see that there's a bit of a misconception in there. 00:15:18.750 --> 00:15:22.190 So in the spirit of productive struggle, we can let him play around 00:15:22.980 --> 00:15:30.440 until he realizes his conjecture can't be supported with evidence or we can provide them with a clue or some hints just to manage flow. 00:15:35.640 --> 00:15:40.759 So through the careful selection of playful mathematics we can encourage and support strategic 00:15:41.490 --> 00:15:48.229 mathematical thinking as students investigate strategies for solving problems and deepen their understanding of concepts and ideas. 00:15:49.260 --> 00:15:55.640 When students have opportunities to engage in playful tasks repeatedly and in different and more complex ways, 00:15:55.980 --> 00:16:01.880 we are bridging the gap between conceptual understanding and the development of computational fluency. 00:16:03.240 --> 00:16:11.900 As noted by a hockey legend Wayne Gretzky kids are motivated to practice when opportunities to practice are fun and spark curiosity. 00:16:12.600 --> 00:16:16.520 In this way playful mathematics can present opportunities for practice 00:16:17.070 --> 00:16:24.379 often without the need for teachers to provide the problems. This then provides us as educators with opportunities to observe or 00:16:24.930 --> 00:16:28.700 assess students in order to support the development of mathematical thinking. 00:16:29.820 --> 00:16:32.929 So whether it's a youngster exploring with Magna Tiles, 00:16:33.690 --> 00:16:39.260 or students actively engaging with different quantities or trying to figure out how many donuts are in a massive box, 00:16:40.080 --> 00:16:44.629 engaging students in meaningful and playful mathematics supports the development of 00:16:44.820 --> 00:16:48.859 mathematical reasoning while uncovering for students the beauty of mathematics. 00:16:49.770 --> 00:16:51.770 Thank you.