tag:blogger.com,1999:blog-1907702537884089718.post8713132047512788585..comments2020-12-16T14:18:17.007-08:00Comments on Exit 10A: 22? 30? 50? 100?Joe Schwartzhttp://www.blogger.com/profile/02304083254248927187noreply@blogger.comBlogger40125tag:blogger.com,1999:blog-1907702537884089718.post-41045392620645552272016-01-18T07:24:50.934-08:002016-01-18T07:24:50.934-08:00Hello! This post was recommended for MTBoS 2015: a...Hello! This post was recommended for MTBoS 2015: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to tina.cardone1@gmail.com whether or not you grant us permission to include your post. Thank you, Tina and Lani.Tina Cardonehttps://www.blogger.com/profile/00549943329133396794noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-81868798472611283382015-12-23T03:37:31.800-08:002015-12-23T03:37:31.800-08:00Thanks so much for your suggestions. I like them ...Thanks so much for your suggestions. I like them all, especially having Alex compare 2 dogs with 2 sets of five dogs. Doing that would bring his misconception into vey sharp relief. I also plan on switching up our counting routines. <br />I'm glad you stopped by and took the time to comment. It's becoming more and more apparent that we (elementary, MS, and HS teachers) can really learn a lot from each other.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-28159420061296429372015-12-22T17:27:22.697-08:002015-12-22T17:27:22.697-08:00Thanks Graham. I've compiled a list of tasks ...Thanks Graham. I've compiled a list of tasks I need to work on with Alex and am adding yours to that list. If Alex only knew how many smart, thoughtful, and dedicated math educators are taking the time to help him out! But I know, and so does his teacher, and we are both extremely grateful.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-23285139644921081772015-12-22T09:56:51.443-08:002015-12-22T09:56:51.443-08:00I love this post and the comments only make it bet...I love this post and the comments only make it better. There's so much to take in here but I'll add my 2 cents to this million dollar post:<br /><br />1. Glad to hear Alex is in a subitizing rich environment. If you showed him 2 dot cards (6 & 8) would he be able to tell you which one has more? Which has less? Just curious.<br /><br />2. Touching on what James Jerrell (and many others) shared in their comments about counting and hierarchal inclusion. I wonder if Alex has this understanding through 5? Through 10? Through 20? He could have this understanding but only to certain quantities. You might want to see where his understanding falls on the number sense trajectory. https://goo.gl/NOoVoo<br /><br />3. I agree with Tracy and rote counting but also having him start and stop at specific number. Start at 8 and stop at 13. Does he need to count up to get to 8? Does he count past 13? If so, he doesn't quite own rote counting.<br /><br />Just some thoughts but I'll return to the shadows to watch this amazing conversation continue. Thanks Joe for providing such a thought provoking post and to everyone else for making me smarter.Anonymoushttps://www.blogger.com/profile/08525114028095675402noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-35293382487956652092015-12-20T16:49:41.491-08:002015-12-20T16:49:41.491-08:00Hi, I'm a bored high school teacher who stumbl...Hi, I'm a bored high school teacher who stumbled across this post. This might be out of my area of expertise, but I have some experience counting with all the young children in my life (relatives mostly). Anyway, take my advice with a rain of salt, but my first impression while reading your scenario was quite along the lines of what Marilyn's post said. Two things Alex needs to understand. First, how much does each "dog" represent. When he counted by fives he did fairly well (except the fact that he skipped some of the higher numbers, a common mistake that suggests he needs more mastery at that particular skill), but he was wrong because 1 dog is not 5 dogs. Of course they could be--we use symbols to represent groups of things all the time. Maybe show him 2 dogs and a group of 10 dogs arranged in two groups of 5. Compare and contrast them to the two groups of 5 with the two groups of 1 dog each that make up the 2 dogs. Second, when we count things we do it 1 to 1. He obviously gets this since he demonstrates his attempting to do so--even when counting by 5s :) --but perhaps didn't realize he was counting several dogs more than once when they were in the pile.<br /><br /><br />Another thought, try giving him a number and asking him to build a set with that many dogs (start low). See what he does. If he's successful, try making sets multiple ways, i.e. Have him make sets of 15 one by counting by 1s and another set counting by 5s (or something along those lines). You can then compare the groups.<br /><br />You could also try a variation of your counting doors routine but instead of alternating, tell him you'll switch every five doors (or whatever) ex: you...1,2,3,4,5, Alex...,6,7,8,9,10Thaslamhttps://www.blogger.com/profile/16890604336141114110noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-25016616596764105522015-12-20T08:21:38.497-08:002015-12-20T08:21:38.497-08:00Hi Joe!
Love this post and the comments even more!...Hi Joe!<br />Love this post and the comments even more! What an awesome learning experience for us all!<br />Do you have the Pre-K-2 version of "Teaching Student'Centered Mathematics" By Van de Walle? Page 178 begins ideas on the role of counting in constructing base-ten ideas. As most comments have said, he agrees to begin with one. Van de Walle said, "Because children come to their development of base-ten concepts with a count-by-ones idea of number, you must begin there. You cannot arbitrarily impose grouping by ten on children. Children need to experiment with showing amounts in groups of like size and eventually come to an agreement that ten is a very useful size to use." He also advises, as does our CA math framework, about using base-ten language, so thirty-five would be three tens and five. We should consistently connect base-ten language to standard language. This idea has made a huge impact on my first graders. There is so much going on with language and counting. He also shares that the idea of using a ten frame is helpful because it not only shows groups of ten, but shows how many away from the next ten. The next hurdle after he "gets" unitizing by tens is going to be equivalent representations. This has been tough for my first graders as well. This year I really dove into the framework and Van de Walle to help inform my instruction even more than in years past.<br />I'm wondering where on Fosnot's contexts for learning Alex would fall? (The number sense, addition and subtraction one). Again, thanks for sharing this post and an even bigger thank yo to everyone who commented! Jamie Duncanhttps://www.blogger.com/profile/11449749666386431744noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-45208359627800457572015-12-20T07:57:52.915-08:002015-12-20T07:57:52.915-08:00I am going to try this with him when I get back to...I am going to try this with him when I get back to school. Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-53039777016688955462015-12-20T07:55:04.867-08:002015-12-20T07:55:04.867-08:00Wow! I have never thought about skip counting bein...Wow! I have never thought about skip counting being the first step in shifting from additive to multiplicative structuring!Jamie Duncanhttps://www.blogger.com/profile/11449749666386431744noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-69682529044438753662015-12-20T07:54:50.735-08:002015-12-20T07:54:50.735-08:00Thanks Cathy and Marilyn. It seems as if I need t...Thanks Cathy and Marilyn. It seems as if I need to first make sure that Alex is secure in the concept of hierarchical inclusion. This is what Michael, in his comment above, is encouraging me to find out. And I shouldn't be surprised that Alex is struggling, due to the cognitive reordering going on. Perhaps it is taking him longer to do that than the rest of his classmates. Developmentally, how long might that take? When do I get concerned? Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-90147024917196035492015-12-19T18:11:35.105-08:002015-12-19T18:11:35.105-08:00Another after-sleeping-on-it thought, related in p...Another after-sleeping-on-it thought, related in part to James Jerrell’s comment about hierarchical inclusion and what I’ve learned from Cathy Fosnot. I’m curious how Alex would respond to these questions. <br /><br />Ask him to put out 8 cubes on a paper. [I chose 8 because when I remove one, the child won’t be able to know how many by subitizing.]<br />Ask: How many cubes did you put on the paper? (8) [Here I look for whether the child has to recount.]<br />Say: Watch as I take away one cube. Remove one cube and place it on the table. <br />Ask: How many cubes are there on the paper now? (7) [Does the child have to recount, or does the child just know.]<br />Say: Watch as I take away another cube. Remove one of the 7 cubes and place it on the table. <br />Ask: How many cubes are there on the paper now? (6) [This is the same as the previous question, a way to check if the child still needs to recount.]<br />Say: Watch as I put one cube back on the paper. <br />Ask: How many cubes are on the paper now? (7) [Similar, but adding 1.]<br /><br />Sometimes I repeat again removing a cube and asking: Can you tell me how many there are without counting? Some kids shake their heads to indicate they can’t, others say they’ll give a guess, some are able to.<br />Marilyn Burnshttps://www.blogger.com/profile/16127025673987723020noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-69133088211431243152015-12-19T15:28:56.461-08:002015-12-19T15:28:56.461-08:00Thanks Jana. Your question about how he arrived a...Thanks Jana. Your question about how he arrived at 22 is one that many others have asked. I believe it's less an estimate than a mental count he made while spending a minute or so looking over the pile. <br /> I think your point about taking things for granted in spot on. I certainly did this with Alex.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-32713852109388829022015-12-19T14:51:47.550-08:002015-12-19T14:51:47.550-08:00Joe,
What a great post about Alex’s experience w...Joe, <br /><br />What a great post about Alex’s experience with counting and estimation! So intriguing to understand a child’s sense-making and thinking about number, counting, cardinality. <br /><br />Many of the others who have commented already mentioned many points so I would reiterate their comments and the readings mentioned. His understanding is fragile and will become more solid with more experience I was interested that his estimation was rather close to the actual and would love to understand that more. It makes me wonder if they have a sense about a quantity that they can’t articulate. <br /><br />A noticing I had was that Alex didn’t seem to what’s referred to as “conservation of number,” which is the idea that 30 is 30 is 30 whether the collection is arranged in a 5 x 6 or 1 x 30 array, a small scrunched pile, a large spread out pile, or the collection is counted by 1s, 2s, 5s, etc. This, too, is a concept built over time. I mention it because I took that for grated when I was teaching 1st grade and it quite honestly freaked me out when I first experienced it. Just is a great reminder how many experiences young mathematicians need counting and how much I can take for granted. <br /><br />Look forward to hearing a follow-up post about Alex. <br /><br />Jana<br />@jsanchezmath<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-82726794554583285862015-12-19T14:27:27.294-08:002015-12-19T14:27:27.294-08:00Thanks Krisitn. I'll try my best to answer bo...Thanks Krisitn. I'll try my best to answer both questions. As for his estimate of 22: I did not ask him how he arrived there. He spent several minutes looking at the pile, and I could see him scanning back and forth, mentally trying to count each one without lifting a finger or touching the dogs in any way. Reflecting back, I'm not sure what his idea of "estimate" really is. Is it possible to argue that "22" wasn't even a true estimate? Just the answer he arrived at by counting them all in his head? What's interesting now that I think of it, is that later, looking at the same pile and counting it in almost the same way, he came up with a count of 50. So I'm not sure what's in there to connect to his counting. Maybe with a ten frame in front of him he could visualize how many in that pile it would take to fill it up? And would there be enough left to fill another? A third?<br /> As for his work in class: Alex's teacher and I have been focused on the management/organization part of the counting collection activity. I realized that I really did not have a good sense of how he was performing during the class time. If I had, what happened back in my room probably wouldn't have happened. He did know that when he and his partner counted the blocks by 10s, they came up with the correct answer, but for all I know his partner did the work and he watched. And anyway, as some others have pointed out, he may not really understand hierarchical inclusion.<br /> Now that we have the routine and organization established, we need to begin focusing on specific kids and their individual development. Perhaps this means that the activity needs to be differentiated, with some collections growing in number and some being reduced. There's so much here to think about! It's truly incredible how one small task, like counting 30 dogs, can open such a tremendous discourse! <br /> Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-30986253682190644402015-12-19T14:05:56.855-08:002015-12-19T14:05:56.855-08:00Thanks Jaimee. Alex's teacher does a good job...Thanks Jaimee. Alex's teacher does a good job incorporating kinesthetics into her counting routines, for example I've seen them do jumping jacks and "wiggles" while counting. I like your idea that this could be explored more with Alex, especially in comparing quantity. Like the more he jumps, the more tired or out of breath he becomes. Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-36800560333411686772015-12-19T09:49:27.784-08:002015-12-19T09:49:27.784-08:00Love this post and loving the comments just as muc...Love this post and loving the comments just as much! Awesome conversation and I third, fourth, or whatever number recommendation it would be, Kathy Richardson's book! <br /><br />I am curious about two more things in re-reading this post this morning....<br /><br />I am first really curious about the estimation...I would love to hear how he decided upon 22, since it is not far off. Did he say how he got it Joe? I am wondering if somewhere in that thinking, seeing 10's came up? If it did, could we then assume he has a general sense of the "size" of a bundle of 10 but not able to connect that to counting them individually? <br /><br />I am gonna go with this thought for a minute here....so, let's assume he can "see" an estimate of ten visually, but cannot connect the counting....could you do some more estimating and checking with counting, but not counting all, just the pieces he sees. For example, after he said 22, and let's say he said it looks like a little more than two tens. You could ask him if he could pull out one of those tens without counting (like a game a bit) and then have him check by counting. I love the idea of Tracy's of putting that counting on a 100s chart or ten frame. Again, this is making assumptions I obviously don't know about him, but I just find it so crazy that his estimate is so close! It leads me to believe he is thinking something in there that we could connect to his counting. <br /><br />My second curiosity is not so much about his counting in this post, but the counting experience he had in the classroom with a partner. When he described the experience to you, he said "he and his partner were successful in counting by 5's and 10's." It made me wonder what the roles looked like between he and his partner. Did Alex count all the tens and then followed his partners lead in counting the groups? So although working together, all of the counting on Alex's part was by 1's to 10 and then his partner combined them. That would explain why his estimate could be close and then the count could be off. Group work is so interesting like this and I would be really interested to see him engaging with his partner in a counting activity.<br /><br />Thank you Joe for opening up this convo, it is full of so many wonderful talking points!<br />-Kristin<br /><br />kgrayhttps://www.blogger.com/profile/00385187692877248447noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-39466639072858073672015-12-19T08:14:25.268-08:002015-12-19T08:14:25.268-08:00I'd love to see those recording sheets too!I'd love to see those recording sheets too!Tracy Zagerhttps://www.blogger.com/profile/18078005798782089280noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-61777779174777282792015-12-18T21:25:12.998-08:002015-12-18T21:25:12.998-08:00I am not sure that this suggestion will help your ...I am not sure that this suggestion will help your particular kiddo but it worked last year for one my first grade girls. She wasn't connecting quantity to number, struggled most of the time with 1 to 1 correspondence but could rote count well. We started her intervention with a quick whole body counting routine. She would jump with me a certain number of times. Small amounts but she had to say only one number ask she jumped. At first we did it together and then she had to teach her jumps and her counting. She learned to slow down and compare one set of jumps to another. It was like she had to "live" her numbers. Just an idea. Thanks for sharing your conundrum. We've all been there and it's so valuable to see all the great ideas commented above! Jaimeenoreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-81594279980400699952015-12-18T13:16:25.310-08:002015-12-18T13:16:25.310-08:00Thanks so much. My knowledge of the theory is lim...Thanks so much. My knowledge of the theory is limited, and your suggestions are really helpful. Early this morning I realized that the post was heavily influenced by something Michael Pershan wrote:<br />https://problemproblems.wordpress.com/2015/12/07/three-minus-one/<br />You're right, we've all experienced moments like this. What makes them difficult is that they take us by surprise, so our reactions are more prone to be unguarded. <br />Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-18100597367244634282015-12-18T11:57:18.957-08:002015-12-18T11:57:18.957-08:00Hi Joe,
Your ability to honestly relate the nuanc...Hi Joe, <br />Your ability to honestly relate the nuances of your work is so helpful. I think we've all experienced moments in teaching such as you've described, particulary your observation that kids may think we are displeased with them when we are actually perplexed by them! Here are some resources which might prove helpful: <br />Page 45-66 (chapter 3) of this text: http://bit.ly/1Pe191F (Contemporary Perspectives on Mathematics in Early Childhood Education) This one reminds us of just how complex early math learning actually is. Another go to: "Teaching Number in the Classroom with 4-8 year olds" http://bit.ly/1IZ7TPY Also, you'll find a lovely description of what it means to struggle with counting in, "Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction" The description begins on page 32, and offers not only a sense of what might be going on, but also some ways to help children deepen their understanding of numerosity. http://bit.ly/1Qy9oHn <br />Meanwhile, I'm so glad Alex has you helping him make sense of the world, and finding ways to express his understanding. Turtle Gunn Tomshttps://www.blogger.com/profile/14821223688546846237noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-20000430476896495612015-12-18T10:21:47.755-08:002015-12-18T10:21:47.755-08:00Thanks Marilyn. Your observation that students ca...Thanks Marilyn. Your observation that students can seem to understand something when they are working with a teacher is really important. I sometimes think they're just telling us what they think we want to hear and don't really own the learning for themselves. The implications are really scary because it means that we not only have to worry about kids that give us wrong answers, but those who give right answers as well! So it also means we have to be sure we're asking the right questions.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-36879102771811008962015-12-18T10:14:14.598-08:002015-12-18T10:14:14.598-08:00Thanks for your kind words and suggestions. Some ...Thanks for your kind words and suggestions. Some others have suggested that it might be too early to ask him to count by 5s and 10s. He is familiar with ten frames from his work in class, and I agree that using one (or in this case multiple ones) would have been a much better choice. Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-49097287939528768232015-12-18T10:10:53.393-08:002015-12-18T10:10:53.393-08:00Thanks for being so encouraging! I work with anot...Thanks for being so encouraging! I work with another student who, when setting aside objects in sets of five, will count and recount the set numerous times, as if in the intervening seconds something about the set has changed. <br /> I was a little nervous that he might have sensed my panic and thought I was displeased with him, and that he thought that by initiating the little counting game we play on the way back to class he might get back into my good graces. Of course I want him to feel that when I take him from class back to my room to work he feels safe and comfortable. In this instance it was a case of me not knowing exactly what he could and couldn't do that led to what happened. Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-15307310305243245522015-12-18T10:07:18.182-08:002015-12-18T10:07:18.182-08:00As I said in my previou post, I've reached out...As I said in my previou post, I've reached out to friends. Here's a response I received from Cathy Fosnot, which I'm posting with her permission. More to come.<br /><br />Yes, it is about unitizing a group and cardinality, which I wrote about in Young Mathematicians at Work: Constructing Early Number Sense, addition and subtraction (Heinemann, 2001). I also filmed it for our new online platform: www.newperspectivesonline.net. I really can’t take credit however as Piaget wrote about it first. Cardinality is a huge big idea for young kids--the idea of understanding that the number you end on is a set, not the name of the last cube, and that inside n is n-1, and inside n-1 is n-1-1, etc: the idea of nesting, or hierarchical inclusion (Kamii). Once cardinality and hierarchical inclusion are understood children can function quite well with a system of ones, but then they try skip counting and grouping (the reach beyond the grasp as they seek efficiency) and things often fall apart at first. With a system of ones they know that one word is needed for each object (1-1 tagging and synchrony), but when they are skip counting they are now counting groups of objects. Here they have to unitize a group (make a group a new unit) and count groups with the same words they use to count the objects in the groups. Disequilibrium!! So we often see children grappling with how to skipcount and count groups with meaning, until the disequilibrium is resolved. Unitizing is actually multiplicative structuring: 3 groups of five; 3 x 5; or 5, 10, 15 (not 1, 2, 3) and so for children the shift in structuring is from additive to multiplicative structuring, and this requires time and a great deal of cognitive reordering on their part. <br /><br />Cathy <br />Marilyn Burnshttps://www.blogger.com/profile/16127025673987723020noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-89523340092673304282015-12-18T10:03:36.732-08:002015-12-18T10:03:36.732-08:00Thanks for your thoughtful comments and suggestion...Thanks for your thoughtful comments and suggestions. Alex's teacher has done a good job integrating subitizing routines into her math class. But I need to spend more time on those types of tasks when I meet with him during our intervention periods. <br /> I think his answer of "22" had everything to do with the fact that he was attempting to count them all in his mind's eye and arriving at 22 (which after all is not so far away from 30) and nothing to do with any innate sense of "22-ness".<br /> Thanks also for your suggestions on how to assess and then remediate weaknesses of understanding about hierarchical inclusion. I am adding those to our menu of activities.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-38314204744557884272015-12-18T09:55:37.046-08:002015-12-18T09:55:37.046-08:00Thanks for sharing your thoughts. Alex's clas...Thanks for sharing your thoughts. Alex's class has spent several periods counting collections and it has gone well for many of his peers. We have provided the students different organizers (ten frames among them), but I am not sure he has availed himself of these tools. So he may not have been solid even with the counting supports. You are also certainly on to something with processing. He struggles in other areas and with other tasks, and this is a weakness and area of concern for his teacher.<br /> And I'd love to see those recording sheets.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.com