It Ain't No Repeated Addition
It's Still Not Repeated Addition
Multiplication and Those Pesky British Spellings
Devlin is a good mathematician. However, he keeps on repeating that he is not a K-12 teacher and so can't offer advice as to how to teach at that level. But he does offer lots of advice about what shouldn't be taught. So in that sense he is a tease and a provocateur.
One of his recommended readings, from the third article, is more helpful for practitioners:
How do we teach children to be numerate? (pdf, 23pp) by Mike Askew and Margaret Brown
Calculations can be identified with several different types of interpretations and contextual problems. For example 4 x 5 can be linked to:
Similarly division calculations can be taught in two ways. For example, 20 / 5 can be associated with:
- repeated sets (eg. 4 boxes each with 5 hats)
- multiplicative comparison (scale factor) (eg. 4 hats and 5 times as many scarves)
- rectangular arrays (eg. 4 rows of 5 hats)
- cartesian product (eg. the number of different possibilities of wearing a hat and a scarf from 4 hats and 5 scarves)
Of these possible interpretations, research has shown that multiplication as repeated addition and division as sharing appear to be widely understood by primary aged children. However, as the examples above show, understanding the meaning of multiplication is more complex (Nunes and Bryant, 1996) and difficulties with fully understanding multiplication and division persist into secondary school (Hart, 1981)
- measurement / grouping (quotition) (eg. 20 apples put into bags of 5, how many bags get filled?)
- sharing (partitioning) (eg. 20 apples put into 5 bags, how many apples in each bag?)
There is evidence that such early ideas - multiplication as repeated addition and division as sharing - have an enduring effect and can limit children's later understanding of these operations. For example, understanding multiplication only as repeated addition may lead to misconceptions as "multiplication make bigger" and "division makes smaller" (Hart 1981, Greer 1988) ....
Language is important here as different expressions will greatly influence children's solution methods. For example, interpreting 52 x 3 as "52 times 3" or "52 lots of 3" may lead to a less efficient calculation method then reading the symbols as "52 multiplied by 3" or "3 fifty-twos"
Hart K (1981). Hierarchies in mathematics education. Educational Studies in Mathematics, 12, 205-218.
Nunes T and Bryant P. Children Doing Mathematics (google books, amazon)