<br>
<br>
<p>Fractions are a big problem area in elementary-level arithmetic. I have seen students that
have trouble with problems involving fractions because their calculators don’t natively
represent them in any form other than as decimals.</p>
<p>Fractions are quite complicated as presented. Adults usually don’t see it that way as they
are familiar with the implicit framework around fractional operations.</p>
<ul>
<li>We assume the decimal numbering system. I think that most would have difficulty doing fractional operations in base 16 or base 11. Most would probably convert to base 10, do the arithmetic and then convert back to the original base instead of working natively in the other base.</li>
<li>We assume a certain amount of the rules of number theory. When we see 15/40, we mentally simplify because we know that there is a common factor to the numerator and denominator.</li>
<li>Many people, children and adults, have trouble adding fractions with dissimilar denominators. The algebra is fairly simple for those that remember it but sometimes you don’t want to work with large numbers (the product of denominators) so you try some factorizations. There are processes, procedures and algorithms that adults take for granted that kids don’t know.</li>
</ul>
<p>In elementary school, we teach kids facts and procedures and sometimes we toss in properties. In general, I don’t believe that the terms associative, commutative, distributive, existence, identity and inverse are used but the ideas are. They are presented as laws or facts implicitly in the process of learning to do operations. Most adults do operations knowing these things implicitly. Many are aware of them as they are covered in middle-school and high-school but they are not covered in a formal way. The first chapter of Spivak’s Calculus does a pretty good job at introducing the student to a more formal understanding of arithmetic.</p>
<p>It isn’t surprising that a lot of students get confused in fractions as they don’t see the whole picture and framework. They can certainly learn fractions well enough so that they become proficient in operations but I sometimes wonder if the process would be better with exposing young children to some of the formal stuff.</p>