On Chained Relations

In all of the college math courses I teach -- from basic algebra, precalculus, calculus, discrete mathematics, etc. -- there's a particular piece of syntax that perpetually trips up students, and it's this: chained relations

To be clear, chained relations are compound statements in mathematics with more than one relational symbol (including equalities and inequalities). Crack open any math textbook and you're bound to see almost any piece of symbolic expression written in that format. And yet my students are always tripping all over themselves at the difficulty of either reading or writing them. Have you ever noticed this before? Let's consider the several factors contributing to this difficulty:

  1. Even a single equality is hard for people to truly understand. Numerous academic papers have been written on this. More than one person has pointed out that the use of the equals-sign in grade-school problems and the calculator point people in the incorrect direction of a functional understanding, rather than a relational understanding.

  2. There is no explicit instruction in the form in any curriculum. To my knowledge, I've never seen the status of chained relations directly addressed or tested in any math textbook at any level (again, whether in basic algebra, precalculus, etc., etc.). At some point instructors just start using it and we assume students will understand by osmosis.

  3. The compound form is entirely foreign to a natural language like English. Consider something super simple like \(a = b = c\). Translated literally to English, it says, "a is equal to b is equal to c" -- and that's a run-on sentence, disallowed by the rules of English grammar. But here in the algebraic language we have an entirely novel mode of permitted expression.

Considering that last point,we might observe that there is (surprisingly, for such a basic point?) unresolved confusion about how one should even pronounce out loud a simple chained relation. For example:

(Note that while the question is essentially the same, those two queries have entirely different top-voted answers.)

In my opinion, the status of chained relations is one of those classic blindspot/submarined issues that's buried in math education, and winds up troubling students throughout their career. To instructors: it's "obvious" and never rises to consciousness as an issue. To students: it's a quagmire that's never clearly addressed or exercised.

To this end, I've found that I need to start my discrete mathematics classes foremost with direct instruction on this issue; namely a short document that I ask students to read -- and to which I'll be referring them throughout the semester when mistakes are made. You can download it here:

On Chained Relations (PDF)

And then to practice reading them, a timed quiz at the Automatic Algebra site:

Quiz on Chained Relations

Interestingly with that quiz, I've had different math-trained professionals try it and tell me variously that (a) it was entirely trivial and of unclear value, or (b) it was entirely impossible within the span given on the timer. Isn't that interesting? What do you think?