Posted by: Dave Richeson | December 27, 2009

Showing two expressions are equal: stop the madness

I know that most you who read my blog teach mathematics at either the high school or college level. I’d like to ask you a question (at the end of the post). It is about how our schools teach students to show that two mathematical quantities are equal. This has bothered me ever since I started teaching college mathematics (more than 10 years ago).

For the sake of an example, suppose I gave a first-year college student the following problem.

Exercise. Show that $2x(x+2)-(2x+1)(x+1)+1=x$.

I expect that nearly all of the students (regardless of their level of mathematical expertise) would give the following answer:

$\begin{array}{rcl} 2x(x+2)-(2x+1)(x+1)+1 & = & x \\ 2x^2+4x-(2x^2+3x+1)+1 & = & x \\ 2x^2+4x-2x^2-3x-1+1 & = & x \\ x & = & x \\ & \checkmark & \end{array}$

Why are students taught this technique? This is a major pet peeve of mine. I do everything I can to remove this technique from their arsenal.

One of the most most important things that I try to teach my students is that the solution to a problem is at least as important as the answer to the problem, and that the way a student presents the solution is also important. Ideally, a solution would be a logical argument containing mathematical expressions and accompanying explanations written in complete sentences. Just as an English or history professor wouldn’t accept a bullet-pointed list of disorganized sentence fragments, I don’t want a scattered collection of mathematical symbols that I have to piece together to see the solution.

Very few students are prepared to write rigorous mathematical arguments when they come to college. That’s fine; from what I can tell, many high school mathematics teachers do not stress this. We slowly train the students to write in this way. The arguments get more sophisticated and our expectations rise as they progress through the curriculum: from calculus, to discrete mathematics, to real analysis. But even in our lowest level classes we want the students’ solutions to be miniature arguments that progress logically from the assumptions to the answer.

My problem with the “high school” approach shown above is that when I’m reading a mathematical argument, and I see an “=” I want to know that at that point in the argument, that the two sides are equal. So, when I see $2x(x+2)-(2x+1)(x+1)+1 = x,''$ in the first line of the argument above it looks to me as if the students are begging the question—they start the argument by asserting the very equality that they are trying to prove.

Some students use $\stackrel{?}{=}$ for all of the equal signs until the last one. This is better, but $\stackrel{?}{=}$isn’t a recognized mathematical symbol (in fact, I don’t think there is a LaTeX symbol for it, I had to “hack” it using \stackrel{?}{=} to get it to appear here).

What the student really means is that a given line is true if and only if the next line is true. Then, because the last line is true ($x=x$), so is the first line. Another way of saying this is that the truth set of one line is the same as the truth set of the next line. Since the truth set of the last line is the set of real numbers, the proposed equality is true for all $x$. If a student wrote “if and only if” or “iff” or “$\Leftrightarrow$” between each line, I’d be much more comfortable with the technique (although I still think it would be inelegant).

So, what would I prefer? I’d like either a vertical argument like this, where each line is clearly equal to the one before it:

$\begin{array}{rcl} 2x(x+2)-(2x+1)(x+1)+1 & = & 2x^2+4x-(2x^2+3x+1)+1 \\ & = & 2x^2+4x-2x^2-3x-1+1 \\ & = & x \end{array}$

or a horizontal sequence such as $2x(x+2)-(2x+1)(x+1)+1=$ $2x^2+4x-(2x^2+3x+1)+1= 2x^2+4x-2x^2-3x-1+1=x$.

So here are my questions.

To high school teachers: when is this technique taught to students, and why? Is this the way it is presented in all the textbooks?

To college professors: do you agree with me about this, or am I being needlessly pedantic? (By the way, my colleagues are well aware of my obsession with this issue. One colleague definitely feels the same way I do, the rest say that they agree, but I can tell it doesn’t bother them as much.)

Responses

1. Yes I agree; no you are neither pedantic nor obsessive.
What such “reasoning” shows is that if
$2x(x+2)-(2x+1)(x+1)=x$ then $x=x$
What students, and perhaps their teachers, do not understand is that we can reach the same conclusion $x=x$ even if $2x(x+2)-(2x+1)(x+1)\neq x$
In fact this is not reasoning at all. It is an accepted practice in school mathematics, and teachers should be ashamed to promote it as mathematical reasoning.

2. Whoops, my apologies for leaving out the “+1″ – makes my point even more strongly, I feel.

3. I don’t do proofs with students often, but I do agree with you. In trig, I ask them to write a string of things they know to be equal, starting with one of LS or RS (for left or right side), and ending with the other.

If they write what you showed, they lose points. If they write it with the ? above the =, I put “ok…”, which I’ve told them means I could deduct points the next time, and I show them the proper way.

4. We’re they really taught this? My hypothesis has been that they don’t see the difference between simplifying an expression and writing a proof.

5. I agree too, and make similar pains (especially in lower-level courses) to distinguish between proving an identity — which is what you’re describing above — and solving an equation.

I do note that the two problems are separate but related: an identity in x holds if and only if the solution to the corresponding (conditional) equation consists of all real numbers for which the two sides of the equality are defined.

As for the “?=” notation, I admit to using it as a quickie shorthand in lower level courses, and make a deliberate point to “phase it out” in higher level courses for which rigor is more central. Paraphrasing Morris Kline, “match the rigor to the age of the students, not to the age of the mathematics.”

6. Don’t know about the US, but where I live, I have never met a maths teacher that teaches their students to write proofs this way. However, students still do it.

One of my students does this (or something equivalent) without fail every paper he sits. I deduct marks for it every time, and every time he gets his paper back he queries the marking. I think Sue is right – fundamentally, many of them don’t know what they’re doing.

(PS, we don’t teach ‘em to do all those weird things that they do to algebraic fractions, either.)

7. I think that your own understanding and level of expertise, and number of years handling and exploring the field of mathematics is affecting how you want students to approach the subject. It is similar to a parent wanting a child to understand life’s lessons. It just isn’t going to happen with a minimal exposure to it.

Since it has been so, so many years since I actually took a math class, I can’t say for certain, but I don’t recall algebra being taught in a “proof” sort of way. I do remember a class called “Theory of Calculus” which, at the time, I found to be a nightmare experience. I also remember the professor teaching it like it should have been the most natural thing in the world to us – after all, we were math majors. I would probably do better with it now, years later. I also remember Abstract Algebra being very different from how I had been used to thinking prior to that point. Again, I would probably enjoy the class much more now.

But I guess if you want to see more reasoning, you would have to start differently from right out of the gate. You would have to ask the student “what is it about three and two that equal five”? or “why can we replace a three, six and four with a thirteen? or can we do that?” or “can we always be sure that four and two will balance six?”. And that is not likely to be happening in any first grade classrooms any time too soon.

But back to the question at hand in the blog. Perhaps the things that bother a true mathematician would not phase me in the least because I have spent a good number of years in corporate and have seen what levels of math realistically get used by 99% of the population. Despite the fact that I have a math degree, I honestly don’t think that a thing above a high school level algebra class ever became necessary for me to know in my work in corporate. Even when I had to do some programming, and when I had to work with data bases and then determine what kind of info and combinations of info had to be pulled from that for reports, nothing above a high school level of algebra was ever required. Years ago when we had to enter our own formulas, etc for our spreadsheets, no higher levels of math were required. So from a practical point of view, no, what you bring up in this post would never bother me.

Truthfully, the only time I had to use any higher level of math was if I happened to teach a math class. So while I remember loving my Differential Equations class and my Non-Euclidean Geometry class, I can’t for the life of me think of one place that those skills were ever required of me.

In my mind, when a student is in a College Algebra class, I am looking at someone who needs a practical level of math for 99% of the time. I suggest to my students that they do their best in College Algebra and also in Statistics because I think those contain a lot of useful and practical math. Also I noticed for students crossing over into computer programming, the 100 level math classes are very useful. While some of the proof type reasoning can be done at that level, I honestly don’t think it is essential. For the rare student who will use levels of mathematics beyond that, I would worry about more detailed thinking and reasoning when they get to those higher level classes.

8. You’re not being pedantic, and this sort of “proof technique” is far too common among the college math students I teach and leads to no good.

My experience is that students who are brought up on proving equalities in this way end up having no “BS detector” when it comes to circular reasoning in more complicated proofs. If they are trained that to prove an equality, it’s OK to start off stating the equality, then they will eventually prove that a group is abelian by assuming xy=yx; they will prove two lines are parallel by assuming they never intersect; they will prove that a sequence converges by letting x be the point at which it converges; and in all those cases they “work backwards” to end up with a true statement. The problem is that this true statement is often conditional (not the result of an if-and-only-if process) and depends on the truth they assumed. Proving equalities by assuming the equality first may look benign at the lower levels but it leads to a serious conceptual flaw in the notions of equality itself (and its cousin, logical equivalence) that creates cracks in the foundation of any further study.

And the thing is, it’s so simple to fix this problem at the root. You can take the proof as you had it initially, and then simply erase the right-hand sides of the “=” signs, and you have a nearly correct proof that never assumes anything uncalled-for and works from one expression to the next via logical and correct steps.

But if we wait until the college years to fix this, it’s often too far ingrained to be fixed. High school teachers (and parents who check their kids’ homework!) need to be very careful and diligent in making sure this never happens in the first place!

9. As a logician, of course I agree with you. My “solution” is to write both sides of the equation side-by-side *without* the equality symbol. Then, I go through with the expansion and simplification steps independently on each side. If the expressions are the same at the end, I write “equals” at that line.

At my university, the vast majority of students have no conception of mathematics as anything other than a collection of fairly robotic processes that result in “right” or “wrong” answers. Their language is utterly imprecise and reflects this simplistic view. For example, they reliably use the term “solve” to refer to almost any kind of mathematical activity, including simplifying, expanding, even differentiating or integrating. Similarly, many of them use the equality symbol to mean many things, including what most of use would identify as some version of logical implication. (I’m sure I’m not the only one who has seen x^2 = 2x on a calculus exam.)

I’ve thought a lot about how to address these points, but have no real solutions, other than leading by example.

10. I am a graduate student in Mathematics at a large University and admit that I still struggle with this concept from time to time. I wish I had had professors like yourself that would have beat this out of me early in my undergraduate work. Unfortunately that wasn’t the case.

For a college mathematics major it is essential that this mistake is corrected very early on.

However, I would like to make one other point. I also tutor high school math students who are struggling in Algebra, Trig and Pre-calc. These students can’t grasp the error that they are making. I have enough trouble getting them to add and multiply fractions without trying to use a calculator.

I guess what I am saying is that for the vast majority of high school math students who won’t see or use anything beyond Algebra II there are much more fundamental problems that need to be addressed.

11. Wow, thank you all for your thoughtful responses!

I wanted to make a couple of things clear. First of all, I’m certainly not blaming the students. As a few of you pointed out, they are still learning how write mathematics in a rigorous fashion. Even the best students have a long way to go when they get to college.

I’m also not necessarily blaming the high school teachers. I’m giving them the benefit of the doubt that they are teaching the material as presented in the textbooks.

My main point is, why do *so many* students (nearly all of them) use this exact same technique? It is so uniform (start with the equality they want to prove, simplify one side, get x=x, then put a checkmark under it) that I was sure that it must be explicitly taught that way in the textbooks. If it is, then I do blame the textbook writers, for they should know better. If not, then I think it is a remarkable coincidence that there is such consistency in the students’ solutions.

12. The best “cure” to this psuedo-technique that I’ve found is the following:
-2 = 2
(-2)^2 = 2^2
4 = 4

I think the above comment about the similarity between proving an equality and solving an equation is correct. Students first learn to solve algebraic equations, and so they apply the same steps to solving equalities. Their teacher doesn’t correct them (or perhaps even encourages it) and so the behavior becomes ingrained.

13. I teach Alg I and Alg II in HS and to be honest, we don’t do any ‘real’ proofs in my class. Not sure what they do in Geometry or higher levels of math at our school, but it seems like maybe students are used to solving equations (where you would simplify each side as much as possible) and are kind of going through the motions without really thinking. I also know I haven’t done much of ‘horizontal’ work so they may also be used to writing everything vertically. Not sure if you are teaching students who are math majors, but I would also guess that much like HS students, college students who aren’t really interested in math may not care all that much about the proper way to write the math or if they really understand what they are doing. Just my opinion!

But it is interesting to see what bothers college teachers compared to what annoys HS teachers! We just finished teaching complex numbers (VERY SIMPLE VERSION) and I don’t know how many times I saw on a test that sqrt(-144) = sqrt(-12i). what!?!?!? doesn’t even make any sense! So if students aren’t understanding the much more basic math, I am not surprised they aren’t transitioning to the thinking you want in college.

PS- I haven’t looked at a textbook in forever so I have no idea how they present it!

14. I loved math in HS and majored in math in college. I realized that there are two types of math courses: the “solve the problem” courses and the “prove the theorem” courses. Before college the only “proof” course most students see is a part of their geometry class.

“Solving” kind of activities include: find the sum, factor, simplify, find the derivative, etc.
“Proving” uses all those techniques but within a new logical framework.

In your example you are asking your students to prove something, but superficially it looks just like a problem to solve. Maybe that explains their method.

15. My solution to this problem is to force a student to read what she has written out loud. An especially egregious example is when = is used to mean “and then I write on my paper.”

“If and only if” is also an equality. Paul Halmos was wise to introduce “iff” for that, since = and are too confusingly visually similar. (Although so are “if” and “iff.”)

I regularly get this from non-math majors:

f(x) = x^2 – x = f’(x) = 2x -1 = 0 = 1/2.

Arghhh!!

Related post to follow.

16. Ut-oh, the symbol < = > showed up as the null symbol in the above. Gotta remember to HTML escape those crazy angle brackets.

Dave, how about a clue as to what WordPress stuff works in your responder’s emails. For example, I added Latex to my own blog, but it doesn’t apply to responders’s posts. And my WordPress installation allows registered posters to delete their posts, yet yours doesn’t have a delete feature.

This wasn’t the related post promised above, so here’s one more to follow.

• I have no idea, Gene. I can edit or delete comments, so if you want me to do either, just let me know. My blog is on WordPress.com, not on a local WordPress installation. So I don’t have much control over anything.

17. I thought about the “symbol adoption” problem as I’ve been preparing to teach History of Math for the first time starting Feb. 1.

Even today we force people to write out in English what their symbols mean. One of my sources says that it’s not just bad handwriting but distrust of symbols going back to the Middle Ages that gives us the following very familiar redundancy:

Pay to the order of Gene Chase \$1,000One million and no/100 ~~~~~dollars.

… which … er … you are welcome to do. Your book royalties must be nearing that much. :-)

18. [...] Third on the list of expectations are the the kinds of mathematical errors I want my students to avoid. Maybe mathematical is the wrong word. These are really logical and notational errors. This section could afford to grow a bit. (I’m teaching Calculus II next semester, hence the items included in this section so far.) I should search on Twitter for the #needaredstamp posts to recall the kinds of errors and omissions everyone was talking about at the end of last semester. When writing this section I also had in mind the discussion going on over at Division By Zero. [...]

19. I am a high school teacher and I have always learned this way throughout high school and college. I also teach this way. My kids come to me knowing this way and I continue to enforce it.

I don’t see a problem with it. The two expressions are equal and we are merely proving it so. If they aren’t equal at the end then we write a not equal sign. And that is that.

• Hi Elissa,
Thank you for expressing a contrary view on this matter. Here are my thoughts on why I don’t think it is not a good technique to teach the students. As you probably remember from being a math major, there are only a few accepted ways of proving a theorem: direct proof (start with the assumptions, show that the theorem can follow from it), proof by contradiction (assume the theorem is false, and show that a contradiction arises), and proof by induction (which is a specific technique for proving something is true for the natural numbers).

The technique I’ve shown above is not one of these. The student starts with the statement the theorem, then the student manipulates it to get a sequence of equivalent statements (although they never say that explicitly), then end with a statement that is clearly true. Thus they can conclude that the original statement was true. As I said, it could be made rigorous by inserting iff between each line, but in general such arguments are very tricky and we don’t encourage students to use this technique.

Thus, at the college level we are having to break students of this habit. I think it is a shame because the “correct” technique is almost identical. As Robert Talbert wrote above, “You can take the proof as [I] had it initially, and then simply erase the right-hand sides of the “=” signs, and you have a nearly correct proof that never assumes anything uncalled-for and works from one expression to the next via logical and correct steps.”

• Elissa,

The problem is that in saying “the two expressions are equal and we are merely proving it so”, students are assuming what they are trying to prove. In Dave’s example, we know (by inspection) that the equation holds and we are just doing a bunch of steps to corroborate our inspection. Or this might appear as an exercise in a textbook and we typically (and perhaps wrongly) simply take the book at its word that the expressions are equal.

But in many problems like this, we do /not/ know that the equation (or inequality, or congruence relationship, or whatever the problem calls for) is true. The relationship we are trying to prove (or perhaps disprove) is not simply an exercise but the result of a mathematical conjecture, and we have no rigorous idea of its truth and therefore no pretext for assuming that it is true. Even if it appears in a book as an exercise or result, we don’t want students to develop the bad mathematical habit of simply accepting statements as true and then doing work — busy work — to provide the steps.

And as I said above, this kind of technique creates many bad notions about logic and proof that have bad consequences further downstream for math students.

Many novelists will often write their stories from the ending to the beginning to ensure that the story is coherent. But it would be bad form, and a bad novel, that actually starts at the last page and continues to the first! Likewise working backwards to develop a draft of a problem solution is perfectly OK and quite common. But a backwards solution is not OK.

20. Thank you for the thoughtful responses. Many of you are arguing that HS students aren’t asked to prove very many things, but they are often asked to solve equations, where they are working with sequences of equalities. That is a good point, and maybe that explains some of it.

However, in that case students are allowed to move things back and forth across the equal sign, and in the example I gave, students don’t do that. They don’t typically move everything to one side and end up with 0=0, for example. Many, many students do what I showed DOWN TO THE CHECKMARK at the end of the sequence of equalities. The consistency is hard to ignore.

Also, I understand that HS students aren’t asked to prove very many things, but they are asked to simplify algebraic expressions. Some of you refer to my question as a proof, but it is really just asking them to show that $2x(x+2)-(2x+1)(x+1)+1$ simplifies to $x$.

21. [...] } Dave Richeson at Division By Zero wrote recently about a “proof technique” for proving equalities or inequalities that is far too [...]

22. Too many students have seen indirect proofs in geometry and think that the same logic can work in algebra. Silly them.

23. This didn’t start out to be a long-winded rant, but I think it ended up that way. :-(

The hypothetico-deductive method in the sciences alternates between guesses (inductive leaps) and verifications (deductive, data-driven proofs). In mathematical discourse, analysis and synthesis play analogous roles.

“Analysis” is not a synonym for “Calculus” as most undergraduate math majors think, unless they have had a history of math course. Analysis is simply starting out with what you’d like to prove and working backwards. Since you don’t know what the number will be, you call it x. Or, since you don’t know what the root will be (in regula falsi), you guess and then correct. Or, since you don’t know how to trisect this line segment, you trisect that line segment instead, and then figure out how to apply the first solution to the second.

All the best mathematical expositors do not burn their bridges behind them. They show the analysis that leads up to the synthesis that is the eventual proof. Of course not all steps will be reversible! We should teach students to be wary of that at every step.

IMHO, to teach students to burn their analytic bridges behind them to present to their instructors some gem of a proof is inhumane. When Paul Halmos became editor of the American Mathematical Monthly he systematically discarded all papers offered to the MAA for publication which offered (synthetic) proof without (analytic) insight.

One of the reasons that Euler is so revered as a mathematical expositor is that he exposed both the analytic and the synthetic parts of his thinking.

Could we be rushing students into “proving” things in a purely syntactic fashion without more “what if” thinking captured to paper?

Mathematics education proceeds by the telling of little lies to simplify, only later saying, “Well, we overlooked this little thing.” Here are several domains in which we should do this:

* x^2 = 2 has no solution. (Until later.)

* x^2 + 1 = 0 has no solution. (Until later.)

* All topological spaces are metrizable. (Until later.)

* All functions are infinitely differentiable. (Until later.)

* All infinite series of the form a_0 x^0 + a_1 x^1 + a_2 x^2 + … converge. (Until later.)

I especially like this last example because it was a fertile assumption for Euler, and because there are two different directions from which one can diverge “later.” One can provide criteria for convergence–what Imre Lakatos calls “monster barring”–or one can simply say “I don’t care” and get a nice theory of generating functions.

We first want our students to open their minds. Then we want them to polish some gems that they find there. Like Suzuki violin teaching, getting students to play the dots on the music staff should come late in the process.

24. If I understand you correctly, any and all problems you have with “the high school method” are adequately addressed by adding onto the end of the verification the words “and vice versa”. Yes?

Or possibly the reader (and writer) being clear about the mildly subtle distinction between a derivation and a verification in the first place would suffice.

25. >>My main point is, why do *so many* students (nearly all of them) use this exact same technique? It is so uniform (start with the equality they want to prove, simplify one side, get x=x, then put a checkmark under it)

I got it!

I’ve been thinking about why it would look EXACTLY the same, and the answer is that this is how you do the following problem.

“Q: Is x=3 a solution to the equation 3x-4=5x-2 or not? Show your work.”

Now in this specific case, I would not consider it “bad math” to work down, because you are not doing a proof that works for all numbers. You are just working with ONE. And high school students have to do this all. the. time.

So they write
“3x-4 = 5x-2
3(3)-4 ?= 5(3)-2
9-4 ?= 15-2
5 =/= 13
No, x=3 is not a solution to the equation.”

And I have enough problems getting THAT out of them! They like to write it like this

“3*3-4 …… 5 … [doodle of a farting bird]….. 15-2 ….. 13 ….. NO”

Students simply don’t DO algebraic proofs in non-honors high school classes, and when you introduce proofs to them it looks like this typical test question. So they respond with the closest thing they have.

Knowing that, I’d suggest taking the time to explicitly explain the difference between *proving* something for all eternity, and *testing* one given number.

~ Lib

26. Iisn´t it clear to everyone that the first line is true if, and only if, the last line is? While I agree that it’s not very neat, I don’t see how it’s fundamentally wrong, as long as one realizes the fact that the ‘real proof’ should be read backwards. And in any case, that’s also the way most real math problems are solved! What I’m trying to say is that the way it’s presented exactly shows the thought process of the student, which is very natural if you’d ask me.