In 2011 I was very fortunate to be awarded an internship as an elementary school teaching assistant in my hometown. I worked with 3rd, 4th, and 5th graders who were having difficulties with mathematics.
One thing that I noticed while working with these students is that they are still curious enough to be motivated by the fact that something is cool in its own right without having to scrutinize its utility. In some sense, curiosity is the only genuine prerequisite for mathematics. Unfortunately, we squander this resource by offering a curriculum barren of anything not assumed to be helpful on the end of year exams\(^1\).
Given that this constraint is not going away any time soon, what more can we do with the curriculum we already have? Let's consider a few points:
- Starting in 3rd grade, students begin to build and work with multiplication tables, which, aside from closure, are not unlike Cayley tables.
- Also in 3rd grade, students learn to tell time by reading a clock. They are asked questions such as "What time is 5 hours before 3:00?" and "If it is 3:15, where will the minute hand be in 1 hour?". These are precisely the same questions that undergraduate math students are asked in an initial survey of \(\mathbb{Z}_{12}\) and \(\mathbb{Z}_{60}\).
- Fourth graders are asked to divide two natural numbers and consider the remainder
- In order to simplify fractions, fifth grade students are asked to find the greatest common factor of two natural numbers. This is a great opportunity to introduce prime factorization.
- Notions of Reflection, Rotation, and Translation are covered explicitly in 5th grade geometry standards, at least in Tennessee.
So, why not introduce group theory to fifth graders? The only thing they are really lacking is a structured concept of a "Set". Further, given that all of these students are headed towards an algebra curriculum\(^2\), it makes sense to introduce them to concepts like operators, inverses, identities, associativity, and commutativity ahead of time in a structured, possibly even visual way, rather than relying on an intuitional approach later on.
For example, consider the task of explaining how fractions are added. This involves rewriting one or both fractions in terms of a common denominator. There are two challenges here: (a) how does one find a common denominator? and (b) why is it legitimate to rewrite a fraction as some other fraction?
The answer to the second question hinges, mathematically, on the concept of a group identity element. In this case, we know that for any fraction \(\frac{a}{b}\), it is the case that \(1 \cdot \frac{a}{b} = \frac{a}{b}\). However, it is also the case that for any \(n \in \mathbb{Z}\), \(\frac{n}{n} = 1\). Thus, the natural connection to make is that for any \(n \in \mathbb{Z}\), \(\frac{a}{b} = \frac{na}{nb}\). I have seen many students struggle to connect those two facts, and my guess is that it seems to violate their mathematical intuition. I am very suspicious that discussing the identity concept more explicitly would help clarify this.
Another concept that frequently troubles students at this stage is that statements such as "\(\frac{1}{7} \cdot 7 = 1\)" should be true. One way to introduce this concept is by representing "parts of a whole" visually, as in pizza slices. However, presenting this as a necessary repercussion of the fact that group elements have inverses captures the idea without relying on intuition.
For example, consider the task of explaining how fractions are added. This involves rewriting one or both fractions in terms of a common denominator. There are two challenges here: (a) how does one find a common denominator? and (b) why is it legitimate to rewrite a fraction as some other fraction?
The answer to the second question hinges, mathematically, on the concept of a group identity element. In this case, we know that for any fraction \(\frac{a}{b}\), it is the case that \(1 \cdot \frac{a}{b} = \frac{a}{b}\). However, it is also the case that for any \(n \in \mathbb{Z}\), \(\frac{n}{n} = 1\). Thus, the natural connection to make is that for any \(n \in \mathbb{Z}\), \(\frac{a}{b} = \frac{na}{nb}\). I have seen many students struggle to connect those two facts, and my guess is that it seems to violate their mathematical intuition. I am very suspicious that discussing the identity concept more explicitly would help clarify this.
Another concept that frequently troubles students at this stage is that statements such as "\(\frac{1}{7} \cdot 7 = 1\)" should be true. One way to introduce this concept is by representing "parts of a whole" visually, as in pizza slices. However, presenting this as a necessary repercussion of the fact that group elements have inverses captures the idea without relying on intuition.
\(^1\) See any work by Diane Ravitch for a more thorough discussion of this problem.
\(^2\) Ostensibly. See Tavis Smiley's work on the "School to Prison Pipeline".
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