## Thursday, April 21, 2016

### Dot Crazy

Last month, on the Friday before our spring break, I borrowed Shannon's third grade class.  Inspired by the work both Simon Gregg and Steve Wyborney are doing with dot images...

...I was anxious to see what would happen when our kids were presented with a similar task. I chose this potentially over-ambitious image from Amanda Bean's Amazing Dream, a book I had used as part of a Do the Math multiplication intervention:

 I wanted them to count how many bushes there were.

We decided to start by modeling the activity with a simpler dot image.  I spent a few minutes playing around with Steve Wyborney's cut slides, and brought one into class for the kids to work on, asking them to figure out how many dots there were and record their thinking with some equations. First working independently...

...and then sharing out on the SmartBoard.

After this brief intro, we gave them copies of the image from the book, some blank paper, and told them to figure out how many bushes were in the picture.  They could work alone, or choose a partner.  Once they had done it, they were to take another copy and figure it out again, but in a different way, and see if they got the same answer.  If they had time, they could try it again.
What happened next was pretty amazing.  Right and wrong, neat and disorganized, straightforward and complex, in pencil and in marker, in rows and in columns; it was like we had turned on a fire hose.  Math started gushing out all over the place.  In fact, I don't think I've ever seen so much math happen in one place in such a short period of time:

Shannon and I stood back and watched, offering feedback when asked.  The activity really hit the elusive sweet-spot: our high achievers loved the challenge of finding multiple ways to compose and express the total, while the strugglers enjoyed engaging with the picture (maybe they were just relieved it wasn't another word problem) and didn't seem to mind that the correct total was elusive.  The persistence level was off-the-chart.  The kids were so involved it was hard to get them to stop.
We closed by having some volunteers to share their work, and before leaving I asked the students to take a few moments to reflect on the experience.  Some selected comments:

Sensing we had a winner on our hands, I spent some time over spring break creating more dot images (thanks again to Steve Wyborney), and Shannon has left them out for the kids to work on:

Here's why I like this activity:
• It promotes both additive and multiplicative thinking, and is a great formative assessment.
• It puts writing equations into a meaningful context.
• It's a task with a low barrier to entry and a high ceiling, so every student can engage at his or her own level.
• It's simple to implement, and takes virtually no explanation or prep work.  Big return on a small investment.  Everybody wins!

1. I always use the dot images for the ten-minute start of the lesson, but I really like the idea of finding a more complex image and asking individual children to take time to explain how they see it - one or multiple ways. I might lift that image directly from you, Joe...!

1. Some teachers have begun to implement the dot image routine, not to the extent that you have, but we've made a start. Your students should have no trouble transitioning to the more complex image. Steve made me 12 x 12 cut slides to play with, and that's how I'm generating new images. They're actually kind of fun to make!

2. I also did the dot activity with two struggling learners from a third grade class. It was a great to see how each of them "broke up" the dots in different ways based on their abilities. One students immediately started to say "I can make groups of __ " and the other student wanted to show how she broke it up by using repeated addition. In the end they learned from one another and that is priceless. Thank you for bringing this activity to my attention. I will definitely be using it in the future.

1. You're welcome, and thanks for taking the time to comment. One of the (many) great things about this activity is that it has such an open middle. Many pathways to a solution, and many opportunities for kids to learn from one another.

3. Such a great activity which extends wonderfully to what we do at the high school level with Fawn Nguyen's Visual Patterns!

1. Great connection Sharon, and one that would've never crossed my mind. Thanks for sharing!

4. I did a very similar activity in my Education class in college and it was indeed very enlightening to see how different students approached this problem. Afterward, we were shown footage of how grade school students tried to solve the problem. The first student comment in the pictures accurately reflect the reactions of the students in the footage after learning where they went wrong. Solutions to simple activities like this one never cease to amaze me.

1. Thanks for the comment Justin. I am also amazed by the fact that it's often the simple, low- or no-tech activities that have the greatest benefits. I was also surprised here that even the students that made mistakes had such positive things to say about the task. There's a lesson in there somewhere.

5. Dot images are a great way to develop early multiplication strategies and build connections between addition and multiplication. When used in specific ways, they can also be used to develop big ideas in multiplication (e.g., the distributive, associative and commutative properties). The images shown in this blog, however, could be improved by having dot arrangements that could be more easily subitized. To use the “Amanda Bean’s Amazing Dream" image as an example, the center 5 x 6 array could be visually marked by the designer in a way that highlights the parts (e.g., if the there were a slight separation or a line marking to indicate the first two rows and the first 3 columns). This would mean that visually the image would vertically cut in half and each half would clearly be subdivided into a (2 x 3) + (3 x 3) array. Building in these potentially realized suggestions would support the use of the distributive property, that, 5 x 3 = (2 x 3) + (3 x 3), which is a big idea in multiplication. Other big ideas (equivalence and the associative property of multiplication) might also arise as students explore how a 6 x 5 = (2 x 3) x 5 = 2 x (3 x 5). For a beautiful example of how this might be done, see the Context for Learning unit, “Groceries, Stamps, and Measuring Strips, created by Frans Van Galen and Cathy Fosnot.

6. Thanks for your insight. How important is it for the students to be their own designers, to visually mark the arrays to highlight parts themselves, in ways that make sense to them? Maybe they need to see an example, such as the one you suggest, first? My instinct is to let them make sense of it on their own. Thanks also for pointing me towards the Context for Learning unit.