Explaining Sigma-Notation

Question

today, I attempted to introduce the summation notation $\Sigma$ to my students. The notation was unfamiliar to the students beforehand. I worked through many examples with them, but for most of them, working with such an abstract notation remains challenging. Do you have any suggestions on how to best teach someone who has very little prior knowledge of math? What is the most intuitive method to introduce this notation? I also tried using a for loop, but it didn't seem to help.

Answer 1

Frame challenge.

Why introduce that notation at all?

Whenever I see a summation I want to understand I unpack it and write the first few terms (and the last if it's a finite sum) with an ellipsis. I (try) never (to) manipulate those expressions, or play with the index of summation.

If you must teach that notation because your students will encounter it in their reading, teach how to read it that way. Don't ask or expect them to write it.

Answer 2

The main difficulty is not the notation $\sum$ itself, but the use of bound variables, especially when combined with other variables. I would simply explain it using a series of simple examples and then ask the students to design their own examples: \begin{align*} \sum_{x = 1}^5 x \quad&\text{is a shortcut for $1 + 2 + 3 + 4 + 5$}.\\ \sum_{i = 1}^5 i \quad&\text{is also a shortcut for } 1 + 2 + 3 + 4 + 5 \\ \sum_{x = -1}^2 x \quad&\text{is a shortcut for } -1 + 0 + 1 + 2 \\ \sum_{x = 1}^5 x^2 \quad&\text{is a shortcut for } 1^2 + 2^2 + 3^2 + 4^2 + 5^2 \\ \sum_{x = -1}^2 x^2 \quad&\text{is a shortcut for } (-1)^2 + 0^2 + 1^2 + 2^2 \\ \sum_{x = 3}^5 (3x-1) \quad&\text{is a shortcut for } (3 * 3-1) + (3*4-1) + (3*5-1) \\ \end{align*}