## Saturday, November 10, 2012

### The Abstraction of a Histogram

Before the students entered the class, I asked them to tell me how long it took them (to the nearest minute) to get to school this morning. I took a marker and sticky notes and each student told me their number, I wrote it on the sticky and told them to stick it on the board.

Naturally, the students really wanted to round to the nearest 5 minutes. But I emphasized the "nearest minute" part, in an attempt to get more accuracy and less repetition.

So the class is in the room and we're looking at a board filled with sticky notes. For the most part, as a class, we tried to organize, sort, structure, and then label our way into a graphical representation. The end product was a histogram of travel time intervals versus frequency.

HOWEVER, during one of my classes, I tried something a little different. I took two students aside and worked with them as the rest of the class caught up on some work from the previous day.

My goal?
Document the abstraction of a histogram.

It is noteworthy that the two students I chose had shown specific characteristics that I wanted.
• They have shown in the past that they don't give up
• They have shown in the past to struggle more than average with certain acts of abstraction

So, staring at the white board with random sticky notes I wanted to guide the process and see where they took it.

Instruction #1 from the teacher:
"Can you group these stickys in whatever way you want so they are more organized?"

Here's the result:

Instruction #2:
"I can't really see some of the numbers, can you fix that?"

Result:

Instruction #3:
"Oh, I'm wondering why you chose those groups?" and "I'm getting a bit distracted by the circle, the line and the star shape, can we fix that?"

Result:

Instruction #4:
"I noticed you changed some of the groups, why?" and "As I'm looking at these, the order is kind of from left to right, but then I have to go up to the next level and go left to right again. I wish it was just left to right without having to back track." Now, this instruction is MY personal preference because I need that left to right for the histogram, but both students were accepting, although I felt manipulative here.

Result:

So now I'm staring at this new structure and I realize the impact of my previous instruction. Encouraging them to go left to right broke down their group stacks. But I rolled with it and gave them,

Instruction #5:
"Ok, can we add something here so I know what the groups are?"

Result:

Instruction #6:
"Ok, now I see all your groups. It looks like you have 3 groups. Is that true?" They looked at it and changed some things because of my question. Note, they didn't answer me, they just went to work.

Result:

Instruction #7:
"I'm noticing you are having trouble reaching the top right of the board, both to reach the sticky and to even label your groups. Can we fix that?"

Result:

Ok, here I didn't have a one liner. So during

Instruction #8:
I discussed that the numbers look like they're just in a straight line, AND, the labels don't seem to match where the group of numbers are, AND, is there a fix, AND, is this even a problem.

They started stacking stickys on top of (covering) each other.
"Wait, I still want to see them all."

Now they paused, thought about the 'problem' and eventually came up with the following result. It is noteworthy that this seemed to be the most challenging step. Perhaps the most abstracting (verb) they had to do?

Result:

At this point I felt like I could bring in the X and Y axis. I had chart paper ready and asked them,

Instruction #9:
"Can you move your stacks onto the chart paper?"

Result:

Instruction #10:
"Do you mind if I label these for you? What would I put here?" (pointing to the left side of the x-axis) They told me to put 0-10...and then 10-20. The result created the next level of organization, including a category (interval) that was empty.

Result:

To wrap things up I added some vocabulary at the end.
Words like intervals, frequency, axis...also some details regarding spacing and the notable differences with respect to bar graphs.

Some Initial Thoughts:
1. It was interesting to note which instruction created the most difficulty #8
2. Although I tried to get out of the way, I felt I was imposing myself too much in certain spots #4
3. I don't know if getting out of the way is an intrinsic part of my goal (documenting the abstraction of a histogram)
4. The difficulties students experience while abstracting are very student specific. In other words, this would have taken less instruction for others, and perhaps more for others.

Any thoughts are welcome.

Bonus footage.
Here is a partially completed histogram superimposed onto a stem and leaf plot. These were built by a different class and resulted in a "Oh, that's cool" moment.

1. Yeah this is great, Nico. I'm glad you documented the different stages of the abstraction. It's useful to compare the final abstraction (precise, orderly) with the original (clumps, haphazard). It drives home why we use the abstractions we do.

2. Nico, I LOVE the steps you gave! (I try to document my lesson posts like this too, helps me reflect on what happened and hopefully improve.) The use of sticky notes is perfect for this, talk about really being able to "manipulate" the data. Thank you!

3. Very low-tech but stickys go a long way with students as you likely know. Documenting each step definitely works in my favour. I tend to make too many assumptions.
For example, I assumed they would use groups of 10 to start, 0-10, 10-20, 20-30. Nope, I was wrong.
Great to look back and see how my instruction affected their result either intended or not.
Thanks for the comments.

4. Nice progression...I am wondering what your students would do with a post it note of 10...

5. Hey I totally stole your idea -- but used it with my whole class at a time. The kids found it to be pretty interesting and it was a perfect way to transition from mean-median-mode to more complex data representations. Thanks for posting cool stuff!! -Meghan

1. Thanks Meghan.
I thoroughly enjoyed the exercise with my two students and I would think it would take on a new path if done with the a whole class of students. Every fork in the road is noteworthy, so I'm sure you found some valid points to highlight.
I'm glad you and your students found it useful.

2. It would almost adds to the creative activity of accounting task and students will indeed get everything to their mind more efficiently. finance homework help