School maths versus real-world maths

The constructivist or reform mathematics movement tends to set-up a distinction between the kind of mathematics that is often taught in school and the mathematics that is used in real-life. Three kinds of research are used to support this.

- Studies that show that children invent their own mathematical strategies if required to do so in order to earn a living. These include studies of Brazilian street vendors.
- Studies that show that adults who can answer an arithmetic question in a real-life context are less successful when answering similar questions in a formal test.
- Studies that show that children who are taught a standard mathematical procedure will apply it in a nonsensical way and/or generate bizarre answers.

A possibility of learning though discovery is not the same thing as discovery being the most effective way of learning.

... what of the students who, when taught standard mathematical procedures, use them to give bizarre answers or answers that don’t fit the context.
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I think I know what is going on here. For most students, carrying out a long division problem is sufficient to fill-up the working memory, so much so that there is little capacity left over to attend to other aspects of the question. This is why students make supposedly ‘silly’ mistakes in class. The solution to this is to practice long division to the point that it is more automatic and requires less conscious attention. This way, students can focus more on the context of the question and the reasonableness of their answers.