One of my friends shared this picture on Facebook—

(which came via mathtricks.org)—and suggested that the teacher who graded this assignment should not be teaching math at all. I suspected that the grading had fallen prey to a heavy teaching load for an elementary school teacher who might not be as comfortable as they’d like to be with mathematical concepts, so I wrote this response (which I’ve edited slightly):The teacher is doing something rather sophisticated—solving a more general problem—which is what makes it easy to trip up on the apparent simplicity of this question. Consider the following similar questions:

“It took Marie 10 minutes to paint two boards. If she works just as fast, how long will it take her to paint three boards of the same size?”

“It took Marie 20 minutes to saw a board into 5 pieces. If she works just as fast, how long will it take her to saw another board into 6 pieces?”

In the case of the first alternative question I’ve proposed, the teacher’s reasoning would be entirely correct: 10 minutes for 2 boards means 15 minutes for three boards. Although this is not the question that’s being asked, sometimes it’s helpful to think of situations where an incorrect sequence of reasoning becomes correct in order to identify where the mistakes are.

In the case of the second alternative question I’ve proposed, think about how you would solve it. Would you divide the 20 minutes into 5 equal periods of time, or 4? Would you blame someone for dividing by 5 the first time they attempted to solve the problem? Once you figure out that what's important is the 4 cuts it takes, rather than the 5 pieces that are produced, then you can solve any such problem. For example, “If Marie takes an hour to cut a board into 6 pieces, then how long will it take to saw another board into 12 pieces?” (The answer, btw, is

not2 hours.)The reasoning the teacher wrote on the paper is clearly of this latter kind. Their mistake is not in

computation, but in choosing what aspect of the problem deserves attention, namely the cuts in the wood and not the resulting pieces. This leads to nothing more than an “off by 1” error, which is easily corrected. I would be happy to see this reasoning written on a student’s paper, because I would know that only a small correction is needed, after which the student could solve the much more general problem, thanks to a demonstrated understanding of proportion.Math teachers have to be prepared to look for this kind of demonstrated understanding in order to hone in on where a student is making mistakes in their reasoning. This particular case is an example of someone who is teaching math, but probably also a lot of other subjects, and may or may not have training in mathematical thinking. So the more sophisticated concept—proportionality—steps in and overrides a simpler formulation of the problem, which just involves counting. This kind of mix-up is common not just in students, but among all people. Which is why I don't think it’s incompetence, but a symptom of the need for more mathematical training for teachers.

I’m curious how other math teachers would have responded to this discussion. There are certainly those in the math community that can give clearer expression to what I was trying to say. Other commenters on Facebook seemed baffled that a teacher could make this mistake in grading, but I think it’s not such a serious error in reasoning (except that the teacher should have been correcting this on students’ papers, rather than making the mistake on their own).

So, what do you think?

## 10 comments:

I agree that this does not confirm incompetence; I am guessing that those who are making such comments either do not teach or are not aware of how many mistakes they make.

I would be curious to know whether the teacher wrote the question. I would be highly sympathetic if the teacher did not write the question. I could imagine a competent teacher whipping out an answer key and rushing to grade this during her only prep period; that would be a very easy mistake to make.

I think that there is a pretty major problem if the teacher wrote the problem himself in order to test the concept that leads to the answer of 20 minutes.

So I think it very much depends on how the question came to be.

Good point, Bret. I think it’s unlikely that the teacher wrote the problem himself/herself. One commenter in the FB discussion suggested that the answer key might have been wrong. Lots of the commenters try to claim that the problem doesn’t have enough information, and maybe there are interpretations for which the teacher is right. I don’t have much patience for explanations that amount to “every math problem must be stated as precisely as possible”, however.

But to get back to your point, this seems to me like a question written specifically to highlight modeling, i.e., taking note of which details are relevant to a solution. That’s a fairly sophisticated process itself, and I can’t imagine that someone who wanted to test proportionality would come up with this.

To me, this question is designed to trick the solver in this exact way, so getting tricked by it is pretty common I'd suspect. Not much shame in that.

The teacher-error of grading the problem incorrectly is a significant one, but it happens. What really matters is how the issue is resolved: it could be a teachable moment ("Everyone makes mistakes!") or it could be swept under the rug.

However, writing things like 10 = 2, 15 = 3 is another significant teacher-error, and one with no real defense.

Lastly, the stars and graphics suggest this is some kind of published "Worksheet", and thus unlikely that the teacher constructed this problem.

I would have totally fallen for the trick and yes, it's designed to trick. I suspect the answer key was wrong, too.

I honestly don't understand the huge fuss some parents and math teachers make about correcting errors. If the parent went to the teacher and said Hey, this is wrong, and the teacher said look, that's what the key said, leave it be--assuming, of course, it's elementary school. Your kids will survive. Yes, absolutely the teacher should talk about it if she understands the mistake. If not, who cares? It's not worth getting bent out of shape about. I teach high school; the kids don't remember a quarter of what they were taught anyway.

"However, writing things like 10 = 2, 15 = 3 is another significant teacher-error"

You didn't see the word "pieces" there? She's applying it to everything on the right hand side.

The teacher-error is modeling this inappropriate use of the equal sign. "10 minutes" does not equal "two pieces".

Modeling sloppy use of fundamental notation creates lots of problems later on.

I was also going to mention the 10 = 2 pieces, 15 = 3 pieces, etc. It's sloppy, to be sure, but more meaningful than the common "equals sign denotes a completed operation" usage. And at some point students have to come to grips with the fact that "=" doesn't necessarily mean "having exactly the same meaning as". My father-in-law claims that his father, who was also a mathematician, never really accepted equations like 3/6=2/4, precisely because his students consistently struggled with equality of fractions. I tell my students repeatedly that, in expressions like lim_{x->infinity} x^2 = infinity, the equals sign means something different than in 2+3=5, because infinity isn't a number; nevertheless, it is standard to write infinite limits this way. And I've seen textbooks that include strings of symbols like 2+3=5+4=9 (which I find appalling). So I'm inclined to accept this use of "=" as meaningful, if non-standard.

I agree that it would really be nice to know more about what came before and after this bit of assessment—particularly whether the error was pointed out to the teacher, and they acknowledged the mistake. It's one thing to be tripped up by a common error, another to assert one's authority over and against proper reason. I have no reason to believe that the teacher in this case responded in one way or another.

The = sign is certainly an overloaded operator, and its use to mean drastically different things often goes unnoticed. (One example I use with Calculus teachers is to ask them what the = sign means in an application of integration by parts.)

But I don't agree with your defense of "10 minutes = 2 pieces". This is not a technical redefinition, like what it means for a limit to equal infinity, or what it means for two ordered pair of integers to be equivalent. This is just a lazy shortcut.

We all take lazy shortcuts sometimes, but teachers must be exemplars of proper mathematical behavior. Taking shortcuts like this encourages students to do the same. This will manifest itself later on in students who declare variables like "Let x = John", or who pay no attention to units when analyzing problems and solutions. And it can be tough to remediate low-level, ingrained behavior like that.

Fair enough. It's a common usage, but not a mathematical one, and I agree that teachers should model good practice.

Do you think that the sloppy use of "=" is correlated with the analytical errors made in the solution? Assuming this grading was done by an elementary school teacher, which of the mistakes most urgently needs correction? What can be done to guide this teacher in math pedagogy?

I know it's problematic to infer too much about a situation from a single example, but I still suspect that this example arose out of a situation where the teacher is only barely comfortable with the material they have to teach, irrespective of how committed they are to good teaching.

There's no error sir at all. Is this a board on a table saw, for which the person saws once, then twice, until two pieces fall off? Or, is this a board that one can hold in his or her own hand, and cut in half for a period of ten minutes? In the first interpretation of the problem, the teacher is accurate. In the second interpretation, the student is accurate. The issue with the problem is one of grammar. All too often, math teachers attack peers when they really should be attacking the problem's grammar. The problem should have stated the context. For example, a piece of wood is suspended across two saw-horses, and then .... blah blah blah ... Without that, these 1950s problems are only obvious when students and teachers have experience with them. oemb1905.livejournal.com ... Algebra and Calculus teacher ...

oemb1905,

I agree that the exercise is problematic, but I don't think you can simply blame it on faulty grammar. The use of the phrases "into two pieces" and "into three pieces" pretty clearly rules out the first interpretation you give. As Patrick mentioned earlier, the question is designed to trick the solver. But I also think you're right that the main trick to solving such problems is having been previously exposed to them.

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