Scenario: Holly teaching estimating
Grade (student age): Grade 4 (age 9-10)
Contributed by: Tracy Weston, University of Alabama, USA
Context –curricular, professional, other
As is the case in the United States, North Carolina has state mandated goals and objectives for mathematics instruction at each grade level. The North Carolina Standard Course of Study (NCSCOS) indicates that in grade 4 students are to “Develop number sense for rational numbers 0.01 through 99,999,” which includes “understanding of place value (hundredths through ten thousands)” and “make estimates of rational numbers.” They also are required to “develop fluency with addition and subtraction of non-negative rational numbers with decimals through hundredths” including “estimating sums and differences.”
This school used the EnVisions curriculum, as mandated by the district (local authority). Students in this school were not grouped by ability level for math instruction, and therefore the class was a heterogeneous mix of abilities. The lesson was taught by Holly, an undergraduate pre-service teacher, and the lesson took place in a school-based placement in February of her final semester (which ended in mid-April) in the teacher education program.
Holly, an undergraduate student-teacher, taught a lesson on rounding and estimating with a Grade 4 (pupil age 9-10) class. The lesson began with students seated at their desks. Holly planned to teach the full lesson with the interactive white board and had prepared slides ahead of time to “work through” with the students. These slides included the problems that she had made ahead of time—both “naked number problems” and word problems. However, the interactive whiteboard “froze” as she was stating the first sentence of the lesson. At this point, Holly said an audible, “Uh oh.” She tried to get the interactive whiteboard to respond for about 15 seconds. It is still not responding through the whiteboard or through the computer that is hooked up to it, and Holly says to the class:
Holly: I’d like you to write in your notebooks, please. Listen because I’m going to dictate it to you: One and 64 hundredths. (she repeats the number two more times) I would like you to round this to the nearest whole number.
As students begin to work, she goes back to the interactive whiteboard and tries for approximately five seconds to get it to work, but it does not respond. She continued to try different things as she asked a student:
Holly: Anna, what did you get?
Holly: Can you explain to us how you got that? (to herself, giving up on the board)) Okay, we don’t have a board. Can you explain to us sweetie?
Anna started to explain her answer. The students were very concerned about the status of the interactive whiteboard, which Holly realized, and said:
Holly: Hold on—guys I know, I’m doing it on my computer, but this (the board) isn’t registering. There’s nothing I can do about that. It’s not working. But that’s okay. We will do without it. So, Anna, again, one and 64 hundredths, tell me how you rounded that number.”
Anna explained what she did. Holly then said, as she walked to the back of the room:
Holly: Very good. I am going to grab an individual whiteboard so I can show you this.
Holly returned to the front of the class with a whiteboard that was 24 inches wide and 18 inches high. She wrote “1.64” on the whiteboard and held it at head-height off to her side, so it faced the class. Holly then explained how to round to 2, walking to and fro across the front of the class, and holding the whiteboard up and facing the class all the while.
Next, Holly wrote “13.47” on the whiteboard and asked students to round this number to the nearest tenth. She asked a student for the answer and wrote “13.5” underneath, so the whiteboard looked like this:
Before asking the next question, she again briefly checked the interactive whiteboard, which was still not working, to which she said, “Nope, okay, back to the board.”
Next Holly asked students to estimate the sum of 3.5 and 2.2 to the nearest whole number. She wrote this on the whiteboard, still held at head height off to her side facing the class, as follows:
As a student shared her answer, Holly made the following notations on the whiteboard, so that it looked as follows:
3.5 = 4
2.2 = 2
Later in the lesson, Holly asked the students word problems. She did this by narrating the word problems as best she could based on her memory of what she had written on the slides that she could not access. As she narrated the word problem, she wrote down the numbers on the whiteboard.
Knowledge Quartet Coding Commentary
Contributed by: Tracy Weston, University of Alabama, USA
Knowledge Quartet Dimension: Connection
Knowledge Quartet Code: Responding to the (un)availability of tools and resources (RAT)
Scenario: Holly teaching estimating
In the post observation interview, Holly explained her intentions for the lesson:
It was going to be interactive on the board, so (the students) would be coming up to the board.
Then, when the interactive whiteboard stopped working at the very beginning of the lesson, she said,
I was thinking, “Oh, what am I going to do now?!”
I didn’t have the interactive whiteboard, but thank goodness I had the white board. It was helpful to be able to show them. If I didn’t have that, I think I probably would have taken a big piece of construction paper and wasted all this paper, but I would have to show them somehow. So, at least I had the white board.
I will say, you know, once I eventually found my bearings and my heart rate dropped a little bit, and it went down to normal, I got at the same concepts I wanted to get at. I didn’t get them at the same rate or in the same order, because the interactive whiteboard would have guided me a little bit more, because I had set it up the way I wanted them to think about it. So for example, Nathan brought up a really good point about using the word “about” [meaning approximately] in the word problem I dictated [since they were calculating an estimated sum, not an exact sum of the numbers] and it was something that when I was saying it, I wasn’t reading it, so I forgot to say “about.” So I was glad he brought that up, because it was something I wanted to talk about anyway. So, thank goodness he did, because I wouldn’t have remembered it probably until after the fact. So if I had the interactive whiteboard it would have been on the slides and it would have clued me in to talk about it with the kids.
And also, I was trying to dictate the word problems to the class, and also write it on the small board. It was hard. Even the problems I had for them to practice, just with rounding, just the decimals, they increasingly got more difficult, which was something when I was up there, I was trying to think about that, but I didn’t want to wait for too long and have the students sitting here. They’d get really antsy, so I was trying to just think on the fly. But on the slides they started with very simple decimals and increasingly the decimals got more challenging.
It was really hard because I was thinking, “Okay, I remember these decimals”, and I’m dictating something, thinking of something in my head, but in the end I did try to get at the same ideas with using tenths and hundredths. I wanted them to see that even if the decimal goes to the hundredths place, you still only look at the tenths. And so I tried to explain that to them, even though I didn’t have it on the interactive whiteboard. So that was the same. I tried to do a couple word problems with them, and again they didn’t get to see the word problems just now, but I tried to say it to them because I wanted them to see how this is something they could encounter in every-day life. And so, in that sense, there was the same basic concept taught just in a slightly different manner.
At one point Holly said that her alternative of using the small white board was problematic because there “was just not much space.” The interviewer then asked her if she would have formatted her notations differently if she had more space, such as if the interactive whiteboard had worked, or if she just would have written everything the same way, but larger in size.
I would have done it different because I had (on the slides) the decimal and then I had space under the decimal for the rounding answers. So, I would have the decimals on top, underneath the decimals it would have been what they each rounded to, and the addition would have gone horizontally, because these are numbers that they should not be needing to add vertically, so I was going to do it horizontally. Subtraction would have been done probably vertically, because most of the subtraction problems I gave them were double digits. Probably, in the end I would have written it both (horizontally and vertically). Basically it was going to be original decimals horizontally on top, rounded decimals horizontally underneath.
For example, on the slides she had planned on using Holly wrote out the problem and was planning to add the rounded numbers as follows:
3.5 + 2.2
4 + 2 = 6
As a side note, this formatting seems like it would have eliminated some notational errors Holly made in regards to the equal sign, for example when she wrote 3.5 = 4 on the smaller whiteboard. Holly made a brief mention of this in the post-observation interview, when she said:
I didn’t like having to write the decimal and putting equals in it and then it was going off the board.
It is unclear if she realized the notational error, or if she didn’t like it because there wasn’t enough space.
When asked if she learned anything from teaching this lesson, the first thing Holly said was:
How to be flexible, and to prepare when technology doesn’t work.
This is offered as an example of RAT because the unexpected lack of the interactive whiteboard clearly changed the way Holly taught this lesson, although she tried very hard to do her best by using a smaller whiteboard and walking around and holding it up for the duration of the lesson.
This scenario suggests that Holly was working at the limits of her mathematical knowledge for teaching. When planning she had been able to find examples that increased in difficulty and developed pupil learning. However, when this planning became unavailable to her through a failure in technology she was not able to draw on this knowledge in action (Schön). Rather she appeared to be trying to remember what was in her IWB presentation.