New research on how learners in special education intuitively and accurately use indirect addition for subtraction

For far too long, the assumption has been that learners with disabilities cannot benefit from constructivist mathematics instruction.  This assumption, in my opinion, is based on the highly erroneous idea that kids with disabilities cannot think for themselves, but must be spoon-fed methods.  In my experience, this is not true.  First, kids with disabilities are a highly diverse group of learners, with different strengths and needs. Second, all learners construct knowledge based on their own experiences and knowledge. Unfortunately, many researchers in special education mathematics seem to misunderstand constructivist mathematics as discovery learning rather than the carefully designed and scaffolded instructional sequence that it is. Here is some new evidence to support that idea, particularly about subtraction.

This work is great news for everyone who teaches subtraction! Put most simply, it means that kids can successfully add on to solve a subtraction problem, and they do so in two situations.  First, when the two numbers are close together, so 62 – 59 would be a great situation for adding on.  Second, kids add on when the problem is put in a context that encourages that kind of thinking.  So, for example,

If my mom is 78 and my dad is 69, what is the difference between their ages?

I started at page 45 in my book and now I am at 52.  How many pages did I read?

Either the context or the problem type suggesting adding on, which is also called indirect addition.  In the terms of Cognitively Guided Instruction, the problem type is Join, Change Unknown.
Here is the nitty gritty of the research:
Peltenburg, M., Heuvel-Panhuizen, M. van den, & Robitzsch, A. (2012). Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure. Educational Studies in Mathematics, 79(3), 351–369.

Here is the abstract:

In this study, we examined special education students’ use of indirect addition (subtraction by adding on) for solving two-digit subtraction problems. Fifty-six students (8- to 12-year-olds), with a mathematical level of end grade 2, participated in the study. They were given a computer-based test on subtraction with different types of problems. Although most students had not been taught indirect addition for solving subtraction problems, they frequently applied this procedure spontaneously. The item characteristics were the main prompt for using indirect addition. Context problems that reflect an adding-on situation and problems that have a small difference between the minuend and subtrahend most strongly elicited the use of the indirect addition procedure. Moreover, indirect addition was identified as a highly successful procedure for special education students, and the best predictor of a correct answer was found in combination with a stringing strategy.

Other work by a different group of researchers has supported this finding.

Peters, G., De Smedt, B., Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2014). Subtraction by addition in children with mathematical learning disabilities. Learning and Instruction, 30, 1–8.

Here is the abstract:

In the last decades, strategy variability and flexibility have become major aims in mathematics education. For children with mathematical learning disabilities (MLD) it is unclear whether the same goals can and should be set. Some researchers and policy makers advise to teach MLD children only one solution strategy, others advocate stimulating the flexible use of various strategies, as for typically developing children. To contribute to this debate, we compared the use of the subtraction by addition strategy to mentally solve two-digit subtractions in children with and without MLD. We used non-verbal research methods to infer strategy use patterns, and found that both groups of children switch between the traditionally taught direct subtraction strategy and subtraction by addition, based on the relative size of the subtrahend. These findings challenge typical special education classroom practices, which only focus on the routine mastery of the direct subtraction strategy.

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