Thursday, December 17, 2015

22? 30? 50? 100?

     Meet Alex.  Inspired by a post from Andrew Gael, Alex (not his real name) and his classmates in first grade have spent three periods over the past month exploring different ways to count collections.

It's been an incredible experience...

...and I fully intend to blog about it.  But not today.

      I spend 4 days a week supporting instruction in Alex's regularly scheduled math class, and see him once and sometimes twice a week for an additional, individual intervention period.  Today was one of those days.
     It was my intention to work with Alex on our identified objective, which is counting forward and back on a number line.  But I thought I'd warm up with a quick counting activity.  I took a bag of small plastic dogs and dumped them out in front of him.

Nothing too crazy.  Just 30 little dogs.

    I asked him first to estimate how many dogs were in the pile.  I could see him squint, and almost hear him counting to himself.  He seemed reluctant to commit, but after a bit of prompting he agreed there were more than 10 and less than 100. He settled on 22 as an estimate, which I had him record on the whiteboard.
    Off to what I believed was a good start, I asked him to describe some of his classroom counting experiences.  After some more prompting (Alex has trouble expressing himself) he was able to relate that he had counted wooden blocks.  He was also able to tell me that he and his partner were successful counting the blocks by 10s.  I asked him how he would like to count the dogs, and he said he'd count them by 5s.
    Taking one dog at a time, he counted (miscounted, actually) by 5s and here's how 14 dogs turned into 100 dogs:

"5, 10, 15, 20, 25, 30, 35, 40, 50, 60, 70, 80, 90, 100."

    He stopped when he got to 100, leaving the other 16 dogs in the pile.  I decided to set aside his miscounting and focus on the set of dogs now in front of us:

Me: How many dogs are there?
Alex: 100.
Me: (Pause.  What now?) Can you count them again for me?  This time one at a time?
Alex: (Counting with one to one correspondence as he touches each dog) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,  11, 12, 13, 14.
Me:  So how many dogs are there?  100?  Or 14?
Alex: Both.  100 and 14.

    I decided it was time to step in with some direct instruction, and I turned our focus back to the original set of 30 dogs.  I tried to explain as best I could that he was counting 1 dog as 5 dogs, and that if he wanted to count the dogs by 5s, he was going to have to first put them in sets of 5.  Which he did.

"5, 10, 15, 20, 25, 30."

Me: How many dogs are there?
Alex: 30.
Me: Count them by 1s now.
Alex: OK.  (Touching each one as he counted) 1, 2, 3, 4, 5, 6, ... 30.
Me: So how many dogs are here?
Alex: 30.

     I had him write that on the board, and when he came back to the table I took his neatly arranged sets of dogs, smushed them all back into a pile, and asked him again: How many dogs are there?  He studied the pile intently, and then, with a little crooked finger, began trying to "air count" them all one by one:

He had no way of knowing which dogs he had already counted,  and which were left uncounted.  He stopped at 50.

Me: So how many dogs are in the pile?
Alex: 50.

     I took a breath. I had an idea.

Me: You told me that when you counted the wooden blocks in class, you counted by 10s.  Try counting the dogs by 10s.
Alex: (Taking one dog at a time and setting it aside) 10, 20, 30, 40,...
Me: (Bad idea. What now?) OK, you can stop.  Let's go back to class.

    We got up from the table, me thinking about how 14 dogs became 100 dogs, how 30 dogs became 50 dogs, and how 4 dogs became 10 dogs.  We walked out the door and started down the hallway, me thinking:  What just happened? and How did things ever come to this?  and, What am I going to do now?  And through all the noise in my head I heard his little voice call out: "One".  
   I looked down, momentarily confused.  He was staring straight ahead with a little smile on his face.
   Again, "One."
   On our walks back to his room, we always play a little game.  We alternate counting by ones, sometimes forward and sometimes backward, and stop when we reach his classroom door.  He wanted to play.
   "One," he insisted.
   "Two," I responded.
   "Three," he said.  We were off, until we got to 88, and he was delivered back into the hands of his teacher.
     So now I'm  trying to untangle this mess.  I know that a lot  was revealed, and it needs sorting out before I can map the way forward.   I have some ideas, but I'll take all the help I can get.




  1. Hi Joe,
    Thanks for blogging about this. I'm curious if a little more context would matter here. For example, what are some items that are easily grouped in fives?
    If the two of you laid your hands on the table and counted each finger, you'd have 20 fingers. However, each group of five fingers is one hand. If we counted the number of hands, would he say four? If we counted the number of fingers, would he say 20? If we skip counted, would he say 5, 10, 15, 20?

    Then, what if you both laid your hands on top of each other so it was a pile of four hands and twenty fingers, would Alex still need to count all the fingers? Or another person came and added their hands to your pile, what next?

    I'm not sure my ideas will be successful.

    1. I like this idea. I could even manage to rest a dog on each finger. When we do the counting collection activity in class this kids have things like cups, egg crates, ten frames, ice cube trays, and muffin tins to organize the collections, something he did not have when I worked with him back in my room. I would never think to use a hand or hands in that way but it makes sense to me. I'll see if it makes sense to him!

    2. Hi Joe,

      Thanks for this post! I love reading teacher's descriptions of student's counting strategies/challenges! There's so much going on before addition that people often overlook.

      I agree with Andrew that context (like hands), graphic organizers (like ten frames), or structure (like baggies or cups) would assist in making the move from 1:1 correspondence to counting in groups more concrete.

      Also it's an honor to think I inspired your work in some way! Thanks again Joe!
      Andrew (@bkdidact)

    3. The use of referents, 5 fingers versus 1 hand, is interesting. And kids have lots of experiences with hands! I wonder if something like this would create a troubling situation for the student, like Tracy was saying. I wonder if Alex would find the idea of having 50 fingers (counting each as 5) troubling/hysterical creating an opening to talk about counting fingers versus groups of fingers. Or maybe 10 hands (again, counting each as 5).
      SO interesting!

    4. Andrew, you're absolutely right. I should have provided the organizers and structures that are available in class when we do this activity. And know that what you do is a constant inspiration for me.
      Kendra, I agree that Andrew's idea of using fingers and hands with Alex is a way to provoke a troubling situation and might help him understand the difference between counting items one by one or by groups of 5 and 10.
      Thank you both for taking the time to comment!

  2. Wow. Great blog, Joe. A few thoughts.

    One, has he done lots of rote counting without counting stuff? I wonder if that's part of the problem here? It sounds like he's learned the songs of 1, 2, 3, 4, 5... and 5, 10, 15, 20, 25... and 10, 20, 30, 40, but without connecting those names to quantities. If that's part of the issue, counting collections are definitely the way forward. I'd keep the total number of objects small. No more than 30, maybe start with <15. Count those by ones a ton. Ask your excellent conservation question again, where you mess up the rows and ask, "How many are there now?" You might need a whole lot of repetition here before he gets that concept.

    I think I'd hold off on counting by 5s and 10s as a focus until he's able to count by 1s reliably. I mean, occasional is fine, but there are some big counting concepts that haven't stuck yet.

    Also, organizational strategies like bundling, twist ties, or tens frames might help, for sure. I'm thinking about the strategy of moving each piece you count? That air counting attempt stands out. Actually sorting them into counted and not-yet-counted might help him get the hang of this.

    Finally, I was thinking about some estimation challenges where you start by asking which is more rather than how many. For example, two small piles of stuff. Maybe 5 in one, 10 in the other. Which is more? How do you know? Or which has fewer? How do you know? Let's count and see. Starting to build that sequencing of which numbers are greater than others?

    I'm about to tag a bunch of people for input. I need my K-2 experts!

    1. Thanks Tracy. Lots to ponder in your comments. Yes, he has done a lot of rote counting. Maybe for him the counting collections class activity was too much, too soon. We started with bags of collections of objects ranging from about 40-80. We've allowed the kids to work with partners, and clearly his partner has done most of the heavy lifting.
      As for the air-counting, I've asked him to count small collections (5-20) of objects in the past, and he normally does touch and move them as he counts. This makes me think of what I say before I ask him to do the task. Does he feel he needs permission to do that? For example, "Count this set of dogs. It's OK if you want to touch and move them." As opposed to, "Count this set of dogs." Or maybe saying, "It's OK if you want to move them," is not so much giving permission as it is reminding him to do it?
      I think at heart my problem was that I expected him to have no trouble counting the bears.

  3. Do you have Kathy Richardson's book How Children Learn Number Concepts handy? Take a look through chapter 1. Some of the things you describe relate exactly to certain critical learning phases that Alex isn't grasping yet. It sounds to me like Alex doesn't yet trust quantities and needs some work with smaller quantities first and counting a lot by 1s. Even if he can count accurately by 1s, there are other light bulbs that haven't gone off for him yet. For example, he was unsure about estimating. What happens when he counts by 1s? Does he react to his estimate at all? Does he spontaneously revise his estimate? Without a solid foundation in counting by 1s, and learning to trust that the number stays the same every time it is counted, it wouldn't be surprising that counting by 5s would lead to some incorrect answers and even contradictory answers.

    1. Thanks Brian. Clearly I need to look at Kathy Richardson's book. You and Tracy agree that he needs more work counting smaller quantities by 1s and experiences that solidify the idea that the quantity is the same every time it is counted.
      Admittedly I don't know that much about how these basic number concepts are acquired, and I don't have all that much experience working with kids of this age. So it was all very shocking and I appreciate your guidance!

    2. It's a great book. She pulls it all apart so you can think about each element. You'll leave in awe of all the things kids learn at Alex's age!

  4. HI Joe,

    Thank you for this post! I'm in my second year at first grade and I've seen this problem many times with my own students. There seems to be a disconnect between rote counting numbers and a concept of numbers as quanitiy and written numbersin my kiddos. I started trying to count collections in as a small group activity (between 25 and 60 items) and it failed miserably. I put the task away, but it still nagged at me. Then last weekend I attended a class at CMC north by Melissa Maygar. She shared the recording sheets she used and now I have been off and running with counting collections. It has been going fairly well, and my kids really like using ten frames or hundreds charts to organize their thinking. I am going to have my kids continue counting collections with organizers until they are really solid (probably to march or April)

    It seems like Alex may struggle with permanence, which is more a developmental issue than a counting issue. He also may struggle with his visual processing or tracking that has been undetected. Either way, a ten frame will help with all of these developmental problems if such is the case.

    Thanks for writing about first grade. I am always looking for others who are facing the same kinds oacademic challenges I see with my students.

    1. 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.
      And I'd love to see those recording sheets.

    2. I'd love to see those recording sheets too!

  5. Joe, great post! First, I would like to second Tracy’s suggestion to read Kathy Richardson’s book(s). Having moved from teaching high school pre-calculus to leading K-5 math, her book opened my eyes to numerical development. I think Tracy was spot on by connecting his misconception of skip counting to his discomfort with quantity. One of the biggest misconceptions in K-1 math education is that because a child can count a pile of objects using one-to-one correspondence, keep track of all of the objects, and then name the amount counted, he/she has an understanding of quantity. Rather, it is possible that the child has simply begun to mimic counting activities he/she has seen in the classroom. A way I determine if a child has an understanding of quantity is by assessing his/her understanding of hierarchical inclusion (read Constance Kamii for more). I assess this by having the student count a pile and stopping her in the middle after counting, say, 8. She has said aloud “1, 2, 3, 4, 5, 6, 7, 8, 9, 10…” and I stop her and ask, can you show me 8? Where is 8? If she points to 8th counter she touched, then she is not relating quantity to the counting sequence. In this case, I would bet that Alex hasn’t constructed hierarchical inclusion. The goal, of course, would be for Alex to acknowledge that the 1st 8 counters he touched was, in fact, 8.
    I would argue that he doesn’t have a firm grasp on what “5” actually is. In fact, I’m not sure the estimation question can even drill down to his understanding of quantity. When you ask, “how many do you think are there?” before counting, is he relating it to his concrete understanding of “22-ness” or is he guessing that “when I complete all the steps that I’ve been taught, i.e. pointing at each object, assigning a number name to each, and landing on the last object, I will say 22”? I think there is a small, but important difference between that and relating it to an understanding of what 22 objects looks like.
    My recommendation would be to build his understanding of the number system in two ways. First, subitizing (both perceptual and conceptual) is a great way to build a sense of quantity. He can begin to construct his understanding of what “3” “4” and “5” look like (perceptual), and then use that knowledge to determine larger quantities (conceptual). Second, hierarchical inclusion is the beginning of the conceptual understanding of “counting on”. One great (Kathy Richardson) activity is called “build a train race”. You set a benchmark number (say, 30) and you roll 1 number cube and you build a train with that number of unifix cubes. After each gets a turn, you roll again. Whatever you roll, you add on to your existing train. The first person to get to your benchmark number wins. This builds an understanding of hierarchical inclusion because each person will want to know how big their train is and how close they are to 30. Instead of counting and recounting every time you add to the train, eventually the child will recognize that the blocks they’ve counted each time have kept the same value and there is no need to count them again. Eventually, they will begin to count on.
    I would hesitate to expand upon the concrete understanding of skip counting until he as a firmer grasp on quantity. In fact, skip counting using concrete objects is the beginning stages of multiplicative thinking, which I’m not sure he (or many 1st graders) are prepared for yet. Hope this helps and I welcome any push-back because I’m learning right along with everyone else.

    1. 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.
      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".
      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.

  6. I am obviously entirely off-course in my response Alex's counting, as I was applauding that he could so intuitively scale the dogs so that each one equals fives. This kind of scaling an important skill, but I do get that being able to count by ones is a foundational piece that he needs. I watched children who just do not trust numbers, like what Brian mentioned in his comment, watching them recount 2+3=5 every time because they just don't trust that each time it will come out the same. I disagree that you have gotten yourself into a tangled mess. I see that what you are doing is engaging him in conversations about numbers and letting him feel safe in these conversations. For now, just being able to have a place to be comfortable with numbers is an excellent starting place. If I were in your shoes I would be thinking about working with the dogs by having him divide them into two equal groups. Start with only four dogs and work your way up. He has a sense of relationship thinking, so work on that and maybe one day you can ask him to make uneven piles. Then playing games with dice might be another enjoyable way into counting. Start with just one die. (I think I learned most of my number sense by playing Yathzie.) good luck. Have fun.

    1. 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.
      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.

  7. hi, you are being so thoughtful about what you are seeing this student do, and how best to respond. I am wondering if it is too early to ask him to count by 5s or 10s as he has not consolidated one to one correspondence with counting (as you noted when he tried to count the unorganized pile) Have you looked at Learning Trajectories (Google Number Worlds Learning trajectories)? If you want him to count by 5s or 10s I think I would try the 10 frame- he could place the dogs in each box. Would he see then that if he counted rows he could count by 5s? That would be interesting. Good luck and thanks for posting this!

    1. 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.

  8. I’ve had this same thing happen with students. They seem to know that the last number in the sequence they say when counting objects tells how many there are (cardinality), but they use this understanding when counting by 1s, 2s, or 5s, tagging one object at a time. I’ve struggled with helping. And I’ve tried different approaches. (I like the ideas suggested, especially with the hands.) And while a student may seem to understand when we’re working together, I worry that they still cling to their misconception. I’ve sent your blog to friends who have much more experience with younger children and hope that I’ll receive some responses. And I hope that I’ll interest them in your blog, since these are friends who aren’t (yet) participating in the blogosphere. Stay tuned.

    1. 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.

  9. 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.

    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: 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.


    1. 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?

    2. Wow! I have never thought about skip counting being the first step in shifting from additive to multiplicative structuring!

  10. Hi Joe,
    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:
    Page 45-66 (chapter 3) of this text: (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" 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.
    Meanwhile, I'm so glad Alex has you helping him make sense of the world, and finding ways to express his understanding.

  11. 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:
    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.

  12. 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!

    1. 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.

  13. 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!

    I am curious about two more things in re-reading this post this morning....

    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?

    I am gonna go with this thought for a minute, 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.

    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.

    Thank you Joe for opening up this convo, it is full of so many wonderful talking points!

    1. 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?
      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.
      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!

  14. Joe,

    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.

    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.

    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.

    Look forward to hearing a follow-up post about Alex.


    1. 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.
      I think your point about taking things for granted in spot on. I certainly did this with Alex.

  15. 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.

    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.]
    Ask: How many cubes did you put on the paper? (8) [Here I look for whether the child has to recount.]
    Say: Watch as I take away one cube. Remove one cube and place it on the table.
    Ask: How many cubes are there on the paper now? (7) [Does the child have to recount, or does the child just know.]
    Say: Watch as I take away another cube. Remove one of the 7 cubes and place it on the table.
    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.]
    Say: Watch as I put one cube back on the paper.
    Ask: How many cubes are on the paper now? (7) [Similar, but adding 1.]

    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.

    1. I am going to try this with him when I get back to school.

  16. Hi Joe!
    Love this post and the comments even more! What an awesome learning experience for us all!
    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.
    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!

  17. 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.

    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.

    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...,6,7,8,9,10

    1. 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.
      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.

  18. 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:

    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.

    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.

    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.

    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.

  19. 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.

  20. 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 whether or not you grant us permission to include your post. Thank you, Tina and Lani.