Following are some interesting examples of how the mathematical mind often works in some of the students that I tutor. These are not rare examples, I encounter them on an almost daily basis.
Tutor: "What is the answer to the following?"
\begin{align*} &&1&0 \\ &-&&9 \end{align*}Student: "Well. you can't take nine from zero, so you have to borrow from the \(1\) making it a \(0\) and making the \(0\) in the right-hand column a \(10\). Then, \(10\) minus \(9\) equals \(1\)."
So why didn't \(10\) minus \(9\) equals \(1\) in the first place? Apparently these students do not see the original one-zero as \(10\), but as individual digits. They therefore apply the borrowing rules of subtraction that they have been taught. If they are asked "What is \(10\) minus \(9\)?", they finger-count their way down to \(1\). In this case, the numbers have not been presented in a column-by-column basis, so the borrowing rules are not applied.
Tutor: "What is half a dollar plus half a dollar?"
Student: "One dollar."
Tutor: "So what is \(\frac{1}{2}+\frac{1}{2}\) ?"
Student: "\(\frac{2}{4}\)".
It seems that in the student's mind, the first question was about money, the second about fractions, a totally unrelated topic. This example is one of many that brings to question the value of explaining fraction addition and division using fraction sticks and various geometric shapes. The students that I work with seem to treat these supplemental exercises as additional topics to be learned, not as enhancements to understanding the rules of fraction arithmetic.
Tutor: "What is \(4\) divided by \(2\)?"
Student: "\(2\)".
Tutor: "What does the following reduce to?" \(\frac{4}{2}\)
Student (after some deliberation): \(\frac{2}{1}\).
Apparently the student does not see \(\frac{4}{2}\) as a division problem, but as a problem in fractions. Even \(\frac{2}{1}\) is often not immediately seen as simply \(2\).
John Schwarz