Ways to Visualize Numbers
I sorted through some papers this last week and found something from a few years back, when we were teaching my daughter math at home. We were using some learning materials out of the UK specifically for dyscalculia, including Ronit Bird's ebooks, “Exploring Numbers through Dot Patterns” and “Exploring Numbers through Cuisenaire Rods.”
These ebooks were huge for my daughter. At this point she was twelve years old, but really lacked a foundational sense of number.
After we had worked through these two ebooks there was one day where we decided to do a fun activity. She was really into secret codes and messages at that point, so I made a chart one through ten. I said, "Let's draw a couple different ways that you understand numbers now, two different visuals for how you see numbers. This will be like a secret code for these numbers."
Shown here is the first page, numbers one through ten. First my daughter drew each of the dot patterns, one through ten. These are the doubles and near-doubles dot patterns from Dorian Yeo, and the ones that Ronit Bird uses in her ebooks, as well as Jane Emerson and Patricia Babtie in their books. Anyway, my daughter drew the dot patterns that went with the numbers, and then she also drew the Cuisenaire Rods length and color.
image from YouTube video below
This was pretty easy for her, she had no problem. But then I gave her the next page, numbers eleven through twenty. And I remember she looked at me and she was like, "How are we going to do those?" I could see her mind was thinking and I didn't answer the question for her.
Then it was like a light bulb went on! "Oh! There is a TEN in each of these numbers. A ten plus another number!"
Here she was twelve years old. Talk about the assumptions we make about what dyscalculics understand! Even though she was in sixth grade at this point, she had never understood that there was a ten in every teen number.
So one of the things that dyscalculics lack is that sense of numbers, and numbers as groups or sets. They can count on in ones, and it's just like a big pile of ones to them. But when you start thinking about numbers and that there are numbers inside of other numbers, or that the numbers have relationships to each other, it gets much more exciting and makes a lot more sense for them.
As you can see in this video, or image below, this is how she drew two ways to visualize eleven through twenty. You can see that on the dot pattern side she drew a line around the number ten, which is a dot pattern five, twice, and then she did a one. So you can see all the way down she had a ten and then another number. Same with the Cuisenaire Rods, you'll see the colors. The orange line is a representation of the number ten in Cuisenaire Rod language, and so she has a ten and a one, a ten and a two, a ten and a three.
image from YouTube video below
It was very exciting as she went down this worksheet filling them out, coloring them in. It was just such a new moment. And there were so many of these moments when we were going through this material. Things that I thought she just understood, and a lot of things that many of us just innately understand about numbers. Usually, students don't even have to be taught in school because they just get it. But for dyscalculics, they have to be explicitly taught this. They also have to experience it.
So the first thing we did was not drawing these numbers and different ways to visualize them. The first thing we did was to PLAY with dot patterns. We had glass beads and rocks; we were manipulating and moving around physical items. Same with the Cuisenaire Rods, we were using them to work with numbers in new ways, in a very visual way.
I share all this just to show you a little snippet of what it was like as I was learning more about my own daughter's lack of numerical sense, and also as she started to gain that sense of numeracy. It was very exciting!
I want to encourage you, in your own journey, to challenge the assumptions you make about what your child understands. Schools will often assume that a student has these foundational knowledge steps down, but most dyscalculics don't. That is why they struggle with their current grade level expectations. There are foundational steps and concepts missing - maybe ones even before their grade level, maybe even ones not necessarily taught in school because most students innately understand them.
These are the things dyscalculic learners really need to experience, in order to really grasp and understand these mathematical concepts.