As I read the conference program for the CGI (Cognitively Guided Instruction) Conference coming up this weekend in LA, I was reminded that while math education typically doesn’t include kids with disabilities, CGI has always been an exception. There are several great presentations this week that focus on kids with disabilities doing mathematics, including ones by Jeannie Behrend and Lio Moscardini whose research is discussed below. Here is a quick round-up of CGI research articles that included kids with disabilities (all these articles seem to focus on learning disabilities, which means that we need more work on CGI with other disability categories).
Behrend, J. L. (2003). Learning-Disabled Students Make Sense of Mathematics. Teaching Children Mathematics, 9(5), 269–73. (sorry, can’t find a free copy)
in this article written for teachers, Behrend documents in detail how two elementary school students, identified as learning-disabled, benefited from CGI instruction in story problems. She discusses how one student began the study by guessing, and how he learned to make sense of the story problem rather than guessing. She also discusses how some students needed extra support to visualize the story problem, and how direct modeling supported students in understanding their own errors.
This research article analyzed student participation over one school year in a relational thinking CGI routine with equations in an inclusive third grade classroom. Analyzing video taken weekly during that year, we found that in the first half of the year, 6.5% of student shares were students with IEPs. In the second half of the year, 23.9% of student shares were students with IEPs, much closer to the percentage of kids with IEPs in the classroom (28.6%). We also found that kids with disabilities were able to understand equations relationally, sometimes by using direct modeling. So the mathematical thinking was quite sophisticated, even as some kids continued to use connecting cubes to communicate that thinking. Kids used manipulatives and notebooks as resources to help them present their thinking to the class. It took time for kids to learn to present their thinking to the class, and their initial attempts took a lot of class time. Yet, we saw lots of development in their relational thinking, as well as their confidence in presenting their mathematical thinking to the class.
This study was conducted in special education elementary schools in Scotland. After 8 hours of CGI professional development for teachers, the study tracked the problem-solving of 2 kids in each classroom (total of 24) using teacher notes, children’s work and researcher’s field notes. They found that the students were “able to develop their understanding of mathematical concepts through actively engaging in word problems without prior explicit instruction and with minimal teacher adjustments.”(p. 130). There is lots of great stuff here, including how it was difficult for some kids at first to solve word problems without being told what to do, but how much they grew as they were expected to make meaning themselves. Similar to the Behrend article, some kids began by guessing and wanting to know the operation, but that decreased over time. This also addresses the issue of kids counting by ones, how teachers trained in CGI problems can call on strategies to move children to more sophisticated groupings, such as base ten problems.
I have to add in a quote I loved from this article:
The ultimate goals is that students develop a disposition towards their mathematical learning which involves a sense of themselves as learners who construct mathematical meaning through engaging in mathematical activity, rather than experiencing mathematical instruction as the acquisition of isolated facts and procedures. This study has shown that for the participating pupils with learning difficulties this is a realistic and reasonable expectation, but the realization of this expectations is fundamentally dependent on the knowledge and beliefs of the teacher (Moscardini, 2010,p.136).
On my copy of the article, next to this quote, I wrote BOOM. And it sets us up well for the final article I am highlighting, which is by the same author but about developing teachers:
I love this one too. This study provided general education teachers in Scotland two days of professional development in CGI, and tracked their developing understanding of children’s thinking and how the teachers used that knowledge to support all learners. Teachers were interviewed before and after, and they provided lesson plans and assessment data from their classes. Teachers shifted their thinking about effective instruction in mathematics, and became better at noticing children’s thinking. And while half the teachers reported initially that they did not feel comfortable teaching children who were struggling in their classrooms, their beliefs shifted in this area as well:
. . . by learning about children’s mathematical understanding the teachers felt more equipped to support particular children in the context of the classroom rather than using this knowledge as a mechanism for their removal (Moscardini, 2014, p. 76)
The study brings up the question of who can be an expert in the mathematical learning of kids with disabilities? ONly special educators? Or mathematics educators as well? What kind of knowledge creates inclusive pedagogy? What kind of knowledge separates children?
Themes that emerge from this set of articles:
- Kids with disabilities can successfully solve CGI story problems and develop complex thinking about mathematical relationships.
- Kids with disabilities may need extra help visualizing story problems. Moscardini (2010) also recommends “chunking” the presentation, or restating the problem slowly so that kids can think through each part, and possibly model it as you talk.
- Kids were able to make sense of their own mistakes through their own direct modeling, or the direct modeling of their friends, in ways that would not be possible without direct modeling.
- Some kids with disabilities have extensive practice with skill and drill mathematics, or being told what operation to use. For those kids, they need time to adapt to new expectations. They might guess quite a bit at first, or have trouble starting. If you continue to expect the kids to make meaning for themselves, they will eventually rise to the occasion.
- Some kids with disabilities have little experience sharing their thinking. They may need the support of the teacher, of a notebook, and of manipulatives to begin sharing their thinking.