I started this post yesterday, and then read a blog post by Kyle Pearce that filled in some of the gaps in my own thinking. Once again, I feel indebted to the #MTBoS: every day, there is someone in the community that pushes my thinking, or shares something that illuminates a new path.


Anna’s Arrival

Six-year-old Anna arrived from Denmark last week. She was placed in a first grade classroom. The classroom teacher, Ms. N, received little advanced notice about the placement, and even less about Anna’s background. All she knew is that she spoke “limited English,” and would be receiving ELL services.

Ms. N loves teaching math, and she loves exploring student thinking. Ms. N is also amazingly industrious: she immediately set about administering some assessments to determine what Anna already knows. By the time I walked in on Anna’s second day, Ms. N had already probed Anna’s thinking around the problems from our district’s Grade 1 BOYA (Beginning of Year Assessment). She also set Anna up with her own individualized center on the classroom’s desktop computer.

screen-shot-2017-01-22-at-11-02-54-am“Anna doesn’t have one-to-one correspondence,” Ms. N told me as I walked over. “I’m really worried.”

I watched Anna struggle with the first task on Illuminations’ Ten Frames Interactive, which asked her to determine how many spaces in the ten frame were empty. She looked stuck, and she seemed bewildered by my questions and gestures.

Et, To, Tre…
Learning from a Student

I googled how to count to 10 in Danish. The first search result featured a large Danish flag at the top. Anna’s eyes lit up. “Denmark!”

Screen Shot 2017-01-22 at 8.56.35 PM.pngAnna led me through counting to 10. I took somewhat phonetic notes on a piece of scrap paper. En, to, tre… the first few were easy. Anna coached me through my presumably terrible pronunciation of numbers like 7 (spelled syv, pronounced soo/seuw). She has remarkable patience for a 6 year old dropped into a new country.

From there, we tried the task again. Anna carefully counted each red circle or blank space, using her pointer finger to keep track. She whispered each number in Danish: en, to, tre… I was surprised that she did not seem to have any trouble with one-to-one correspondence. I almost always agree with Ms. N’s student “diagnosis.”

Mapping the Critical Learning Phases

This reminded me of Kathy Richardson’s Critical Learning Phases. (I use the NZ Continuum a lot, too, but KR’s Critical Learning Phases fits my work in K-1 better.) I could not yet determine if Anna had a truly robust understanding of counting, but I wanted to get a sense for the next phase: comparing, and changing numbers.

I quickly drew a ten frame on a piece of construction paper, and collected some counters. I modeled a basic “changing numbers” activity for her: there are 5 counters on the ten frame. Now change it to 3. Now 7.

Anna seemed to have one-to-one correspondence when she counted, but struggled with changing the numbers. To decrease the quantity, she would often wipe the ten frame of counters entirely and then add back in the desired quantity.

What’s My Role, Exactly…?
Learning from a Colleague

I told Ms. N that this exercise seemed to be a good fit for Anna, and that she needed to build up an understanding of how to increase or decrease quantities to “change numbers.” I worry about making recommendations like this to teachers, especially a strong teacher like Ms. N, even when they ask for it. Ms. N already has so many great ideas percolating; it would probably be much better to share observations and then ask questions. Isn’t it my job to empower these teachers? If they think there’s a “right” way, and that I am the authority, won’t that hinder the development of their practice (as well as mine)?

Ms. N had, in fact, already thought a bit about this. (I should have known!) She showed me a task Anna had worked on yesterday. “Build 4 on the ten frame. Now add 3. How many counters are there now?”

“Hmm. This looks like addition. We might need to go one step back,” I told her. Too directive, again.

The interaction bothered me for the rest of the day. Ms. N is a fabulous teacher. She is also in her second year in the classroom. I feel like there’s more for me to learn about content – and pedagogical content knowledge – and it’s also my job to help teachers further their learning. How do I help Ms. N see the difference between the two tasks — gently? 

…and what is going on with Anna’s counting that would make Ms. N think that she lacks one-to-one correspondence? Ms. N is very observant, so my assumption became that either (1) Anna had some other conception about counting that Ms. N confused for one-to-one correspondence or, I suppose, (2) that Anna did not consistently demonstrate one-to-one correspondence.

Counting and Cardinality
Learning from the #MTBoS

Enter: Kyle Pearce‘s post about Counting and Cardinality, in which he identifies 9 Principles of Counting that resonate with him and his experiences.

  1. Stable Order
    Anna has stable order through at least 10 in Danish, and I would guess through at least 20. Some of the tasks she missed on the district BOYA involved numbers that went higher. She is still working to establish stable order to 10 in English. Maybe this is what Ms. N saw?
  2. Order Irrelevance
    Anna always started counting from the same place on the number line. At one point, I moved the counters around and encouraged her to start at another place, and she seemed confused. For the first time, she counted from another direction, and then checked by re-counting from her usual place.
  3. Conservation
    Anna demonstrated conservation of a quantity.
  4. Abstraction
    Anna counted cubes that represented fish in the BOYA, so I am not concerned.
  5. One-to-One Correspondence
    The million dollar question–even though I observed Anna using one-to-one correspondence to count each object once, her classroom teacher reported otherwise.
  6. Cardinality
    Anna demonstrated cardinality, using the Danish words for the numbers 0-10.
  7. Subitizing
    Not thoroughly assessed, but I would venture to say that Anna is able to subitize quantities of 2, but not larger. Also: she did not use a benchmark of 5 on the ten frame to support her calculations.
  8. Movement is Magnitude
    Anna demonstrated this understanding when counting up, but not counting back.
  9. Unitizing
    Anna did not demonstrate an understanding of unitizing, and seemed uncomfortable with numbers >10.

So: maybe Anna does have a secure understanding of one-to-one correspondence, and maybe she doesn’t. Either way, I think there are several key areas we need to develop further with her: stable order >10 (especially in English), order irrelevance, subitizing, movement is magnitude, and unitizing.

And that’s just for counting! From there, we can look again at her current understandings about changing numbers and early conceptions about addition, and…?

I think Kyle’s post can help us design some goals for Anna. Counting is the foundation she will need to develop more additive strategies. It sounds like we may need to develop somewhat independent experiences for Anna that will build her stable order, order irrelevance, and conception that movement is magnitude. We want her able to count forwards and backwards, even if it’s Danish right now.

Meaghan Mutchmor responded to Kyle’s post with a screenshot from a document outlining some instructional strategies to develop these key principles. Maybe this will help us, too?

Where do we go from here?

From this experience, I am also thinking about:

  • Collaboration, especially with early career teachers
    I am unsure exactly how to proceed with Ms. N. I have been impressed with her work with Anna thus far. Moving forward, I want her to feel empowered to make the instructional choices, and I also want to help her refine her choices. She has asked me to conduct clinical interviews and develop plans for other students in her class, but for Anna she merely said “she’s new and behind and doesn’t speak English. Ahh!” Our collaboration feels blurred. I guess I’ll just ask her how Anna is doing, and if she wants any help brainstorming? I won’t be offended if she says no. Anna is in great hands.
  • The importance of making connections with students
    Anna ran up to me in the hallway the other day and gave me a hug. I hadn’t seen her since our session together last week. She may just be an affectionate and effusive child, but I think she may feel an extra bond to me because I took the time to learn from her. It also meant that I could gather more accurate information.
  • Learning from the MTBoS
    There is so much going on in the MTBoS! How can I harness the collective curiosity and wisdom of the MTBoS to help me and my students?
  • Making Plans for new Students who are “not at grade level”
    Anna is not our only new student who seems “behind,” and many of them are mid-year ELL students. We just received a new sixth grader who does not seem to have done much work with fractions — and just missed the fraction operations unit. We have a new fifth grader whose only strategy for multi-digit addition is the traditional algorithm, which he frequently mis-applies or forgets. There are others, too. What do we do for them? Our school does not currently have strong structures to support them mathematically. (Thank goodness for our rock star ELL teacher.)
    Still… what do we do with Anna next?