Let’s imagine students who know that 2 books + 3 books is 5 books, 2 cookies + 3 cookies is 5 cookies, and that 2 chairs + 3 chairs is 5 chairs. These students are taking a geography class on Australia. They learn a lot about that country and are told about an animal called a kangaroo. They are shown a video about kangaroos and learn that it hops on powerful hind legs and that the mother keeps her baby (which they all call Joey) in a pouch on her stomach. Yet, there is one thing that is true of kangaroos that these students are not told. In spite of the fact that Australia is a distant country with many strange creatures, they are not told that 2 kangaroos + 3 kangaroos is 5 kangaroos? Why not?
Now let’s look at three students, student A, B, and C. All three are asked to add 2/7 + 3/7. Student A searches inside his brain for a toolbox where he stores rules for operations on fractions. In it, he identifies a rule that applies to the problem at hand. It says: “To add two fractions with the same denominator, you add the numerators and keep the denominator the same.” He uses that rule to correctly add 2/7 + 3/7. Student B also reaches into his toolbox, but it is a little bit messy in there and he gets the answer wrong because he confuses the rule about adding with the rule about multiplying: he correctly adds the two numerators but also the two denominators. Student C doesn’t use a toolbox. She takes it for granted that 2 sevenths + 3 sevenths is 5 sevenths for the same reason that 2 kangaroos + 3 kangaroos is 5 kangaroos.
Of these three students, which one knows more math, the one who knows the rule and applies it correctly or the one who doesn’t need the rule?
More importantly, if we look at what takes place in classrooms and textbooks, which of these students are we seeking to create by our teaching?
And, whether we like it or not, if we look at the reality of what takes place and has taken place for a very long time, isn’t it obvious that in aiming to create student A we are also unavoidably creating student B?