What implication arrows, if any, should I require in teaching?

Question

Q: Solve $x+5=0$.

A: $x+5=0\implies x=-5$.

This answer would be given full marks.

Isn’t it better to tell students to use $\equiv$ or $\iff$? Because that is what lets them say $-5$ is a solution to the equation. One big upside of this would be that students would never be confused about why extraneous solutions may be introduced when squaring both sides of an equation.

Answer 1

What does and doesn't work in math education is an empirical question. For instance, how math is taught in Singapore means what works for those students may be different than how math works and what is taught in Chicago. So I'd push back on any inclination that education needs to be this or that way when it comes to notation.

I think the best approach might be to explain the difference between arithmetic equality, logical equivalence (either with three lines or bidirectional arrow) all of which are equivalence relations. Thus, to use either of these is to emphasize the equivalence relation, which entails reflexivity, symmetry, and transitivity. To use the material conditional either as an 'if-then' or 'because' construction in natural language emphasizes logical consequence. So, when you're teaching solving for unknowns, I'd start with logical consequence and then move to explaining how the bi-conditional constitutes identity. And, like this post, I'd do in plain, natural language.

Part of the difference in how you approach this is a function of the level of sophistication of the student. A really bright student will hear this once or twice and understand, whereas someone who struggles may be confused by the distinctions in symbols since they've never likely been explained this. Certainly not at the elementary level, probably not at the secondary level, and probably poorly at the higher educational level. The use of the material conditional, for me, is superior because it emphasizes process, rather than entity. An arrow means intuitively, start here and end there, and with each use, one can talk about why we move from the initial to final state. With the equals symbol, it may be a little more mysterious why the statement changes. To really aid a student, you can write a brief reason above the left-to-right arrow. For instance, if you have substituted a constant value into a variable, simply mark "SUB x" over the arrow. If you have have changed the order of an addend and adder, you simply write "COMM ADD" to say you have invoked the commutativity of addition as your justification.

In this way, much like a two-column proof where a rationale is provided, but less tabular in form, you can layout the concepts behind the symbolism. One of the chief problems with poor math education is an over-use of symbols and an under-use of natural language to explain. Thus, there are hordes of students who can manipulate symbols successfully, and have no semantic grounding, and hence no real understanding of why they do what they do.

So, no matter what symbols you use, just ensure students understand why.

Answer 2

I think there's no good reason to insist on $\iff$. First of all the obvious: $p\iff q$ always implies $p\implies q$, but not the other way around. Since beginning students may not have that totally nailed down yet it's better for them to err on the side of $\implies$ rather than mistakenly write $\iff$.

Now about extraneous solutions, I think that technique wouldn't help much anyway because the point is that if you square the equation and get two solutions $x_1$ and $x_2$ you still need to check which of them satisfies the original equation, whether you are aware of the "broken" $\iff$ relation or not. So actually you are more likely to check if you are aware that it is $\implies$ all the way through, since it is a reminder that the equation implies the solutions, but not the other way around.

In other words, I think what's more important is to understand that when we solve an equation, we are interested in the solution, and therefore the direction of $\implies$ is what interests us, that is, we are interested in making a series of logical steps that imply one or more possible solutions. For that exact same reason, we need to check which of our solutions satisfy the equation if we get more than a single one.

Of course, at the more advanced stage the student would know ahead of time how many solutions to expect, but I'm assuming the question is not concerned with that level of math yet. Although even at an early stage, it is quite easy to explain to the student that an equation of integer degree $k$, that is containing $x^k$ as the highest order term in $x$, can have at most $k$ solutions - this can just be given as a simple "axiom" at first, with a promise to be explained more deeply some time later...

Edit: After thinking more about this, I find it important to add that as we know, it's taken for granted that we don't teach propositional logic at an early age. I'm not sure if we should teach this. (At that age, it is certainly in the area of "experimental education" as far as I know). However, if it happens that the young students are taught propositional logic, then it recasts the question in a totally different light. In that case, I would say it would make much more sense to expect the student to be able to distinguish $\iff$ steps from $\implies$ steps, and that it would in general aid them, not only in algebra equations but also in all areas of math. So perhaps this question of whether and how to introduce propositional logic early on, is one that should precede questions about the usage of propositional logic.

Answer 3

Whatever symbol you use probably doesn't matter very much, it's how you explain it that matters.

If you tell your students to write "⟹" between the equations then they will write that because you told them to; the same goes for "⟺" or anything else. But if you want them to understand the conceptual meaning of this symbol then you have to teach them that concept. Teaching them correct use of a symbol and then showing correct examples of how that symbol is used is not sufficient to teach the concept that the symbol represents.

There is a related and widespread misconception among high-school students, that the equals symbol "=" means "the answer is". That's probably because in the examples they are taught, "=" appears before the answer, and when they wrote an answer like "−5" their teacher told them they should have written "x = −5" instead. This leads to students writing things like "12 + 5 = 17 × 3 = 51" because 17 is the answer to 12 + 5, and 51 is the answer to 17 × 3. Such students might also reject or be confused by equations like "17 = 12 + 5", because 17 isn't something that needs to be worked out, and 12 + 5 is not an answer to it.

Likewise, I think that high-school students are unlikely to learn the concept of logical implication from the procedural rule that says they are supposed to write "⟹" in between steps when they are working out a problem. If your students need to understand logical implication, I think it would be better to teach it as its own concept first, and then incorporate it into how they solve algebraic problems later. On the other hand, if they don't need to understand logical implication yet, then you shouldn't insist on them using a symbol to represent it.

In case you read that previous paragraph and thought "but of course students must understand logical implication in order to solve algebraic equations!": no, they don't. They solve algebraic equations using a set of procedures and rules you have taught them. Understanding logical implication may be necessary to justify why those procedures and techniques are correct; but you aren't asking them to justify that. As far as they're concerned, "write ⟹ between the steps" is just another rule, or "write ⟺ between the steps, except write ⟹ when you square both sides" is a more complicated rule. Students can learn these rules and follow them without understanding that they produce justifications for their answers.

Answer 4

The identity symbol would be a poor choice as this is frequently used as a convenient notation for "for all x". Overloading it with a second meaning for "equivalent propositions" would be a poor notation.

While symbolic manipulation is a useful skill, it isn't "teaching algebra". Whether you use implication symbols, therefore triglyphs or words like "so" misses the point that the equation "x+5=0" expresses some property of a number or set of numbers. That you are answering the question "What is the number/are the numbers that when you add five to them gives you the value zero?"

Cover up methods "[finger] + 5 = 0. The thing under my finger must have a value of -5. So x = -5." shift students away from memorising manipulation "rules" to understanding what the meaning of the equation.

These methods are scaffolding, and at some point, efficient methods of symbolic manipulation can be introduced. Or rather, aspects of the symbolic manipulation can be introduced incrementally.

When it finally becomes significant, for example in a solution of |x+2|=2x-1 or 3sin(x)=2cos(x) by "squaring both sides and using the pythagorian identity" students may be ready for careful use of implication symbols, and understanding how this can create addition "solutions". For students at the "x+5=0" level, that would merely be confusing.

Therefore my answer is "no", when teaching algebra, you should encourage understanding of the structure over formal symbolic manipulation.