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.
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2$\begingroup$ Are you working from an established textbook? $\endgroup$– Daniel R. CollinsCommented Sep 12, 2023 at 23:20
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1$\begingroup$ Cross posted at math.stackexchange.com/questions/4765689/… . That's usually frowned on at SE. If you do cross post, each post should link to the other. $\endgroup$– Ethan BolkerCommented Sep 14, 2023 at 14:27
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$\begingroup$ @DanielR.Collins I worked with a textbook, but it doesn't go into the explanation. Do you know a good book? $\endgroup$– wayneCommented Sep 14, 2023 at 16:15
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1$\begingroup$ @wayne: Got it, answered below. $\endgroup$– Daniel R. CollinsCommented Sep 14, 2023 at 17:56
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2 Answers
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I've experienced positive results by first having students spend some time writing out sums in full (or using ellipsis notation if there are many terms).
That way, it gets annoying to spend so much time writing the sums, and I can present sigma notation as "a shorthand developed by mathematicians who, like you, were tired of spending so much time writing out sums."
Once a student understands on a visceral level what problem an abstraction is solving, they are far more receptive to the abstraction as a solution to that problem. (Otherwise, if they don't really "get" what problem the abstraction is solving, then it just feels like needless complexity, and their eyes glaze over.)
Of course, after the problem of writing out sums is experienced and sigma notation is introduced as a solution, it's still necessary to scaffold the pedagogy well, starting with simple examples (the sum of even numbers from $2$ to $100,$ the sum of squares from $1^2$ to $10^2,$ etc).
And students would still need to have mastered the necessary prerequisites, such as writing a sequence given its formula and vice versa (and a prerequisite for that would be evaluating functions).
But properly motivating the notation should at least get you over the initial hump.
answered Sep 12, 2023 at 14:58
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8$\begingroup$ "a shorthand developed my (sic) mathematicians who, like you, were tired of spending so much time writing out sums." Very neat - they see the benefit. $\endgroup$ Commented Sep 12, 2023 at 17:34
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5$\begingroup$ The sum of odd numbers is so much more interesting than the sum of even numbers $\endgroup$– Flydog57Commented Sep 13, 2023 at 17:14
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5$\begingroup$ My wife often complains about a grade-school math teacher’s failure to explain the problem that estimation was solving, which led her to be highly resistant to the concept well into college. Love that 3rd paragraph for capturing that. $\endgroup$– KRyanCommented Sep 13, 2023 at 17:15
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1$\begingroup$ @KRyan I also remember tedious exercises to "practice approximations" where I quickly decided that it was simpler to perform exact calculations then round the result off, rather than try to understand what it was that the teacher wanted us to do with rounding stuff halfway through to get a purposefully inaccurate result. $\endgroup$– StefCommented Oct 17, 2023 at 16:47
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$\begingroup$ Even when using ellipsis notation, at some point the students might want to start writing $1 + 4 + ... + i^2 + ... + 10000$ instead of $1 + 4 + ... + 10000$ to remove possible ambiguities, and then it's even easier to move on to $\sum_{i=1}^{100} i^2$. $\endgroup$– StefCommented Oct 17, 2023 at 16:51
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For a textbook reference, you'll find that big-sigma summation notation is covered in most textbooks for college algebra and precalculus, complete with scaffolded exercises on the topic. On the one hand, it's likely to be in the latter part of the book and not covered in related courses; but any such book would provide a good resource to fill in a gap like this.
One open-access example would be: OpenStax College Algebra, Sec. 9.4: Series and Their Notations.
I think it's important to note the name and history of the Greek big-sigma symbol as the ancestor of our capital "S", and so it makes sense as a mnemonic for a big summation. An interesting side-question would then be to see if students can intuit a good capital Greek letter to use for a large product.
answered Sep 14, 2023 at 17:53