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All of us know, when teaching, the "right" choice of examples is important. Though, making the "right" choice is one of those things that are easier said than done. Here is the story of a series of unfortunate examples I used in my class yesterday. My class was for undergraduates students, but what I observed is quite general. Thus please consider an answer for this question even if you are just involved in teaching school mathematics. The subject of the class was double integrals. Here are the three "simple" examples that I used one after the other to exemplify changing the order of integration in double integrals.

$\int\limits_0^2 {\int\limits_0^1 f(x,y) dxdy }$

$\int\limits_0^1 {\int\limits_0^{y-1} f(x,y) dxdy }$

$\int\limits_0^1 {\int\limits_x^{2x} f(x,y) dydx }$

They look three innocent examples. But let us see what happened. We happily changed the order of integration in the first integral, accompanying it with all the usual explanations. Thus:

$\int\limits_0^2 {\int\limits_0^1 f(x,y) dxdy }$ changed to $\int\limits_0^1 {\int\limits_0^2 f(x,y) dydx }$

As you might guess, what the students got from the whole process was only how the final answer looks. As a result:

$\int\limits_0^1 {\int\limits_0^{y-1} f(x,y) dxdy }$ changed to $\int\limits_0^{y-1} {\int\limits_0^1 f(x,y) dydx }$

We draw the region of integration, discussed the problem again and came to the following correct answer:

$\int\limits_0^1 {\int\limits_0^{y-1} f(x,y) dxdy }$ to $\int\limits_0^1 {\int\limits_0^{x-1} f(x,y) dydx }$

As you might guess again, what the students got from the whole process was only how the final answer looks. As a result, they got the following answer for the next problem.

$\int\limits_0^1 {\int\limits_x^{2x} f(x,y) dydx }$ to $\int\limits_0^1 {\int\limits_y^{2y} f(x,y) dxdy }$

Unlucky me, the function that I had used, for the sake of simplification,was $f(x,y)=1$ which gives the same numerical answer whether you calculate $\int\limits_0^1 {\int\limits_x^{2x} 1 dydx }$ or $\int\limits_0^1 {\int\limits_y^{2y} 1 dxdy }$!!

It wasn't the first time that I observed such a phenomenon, but perhaps it was the most bizarre one so far. In fact, this is a famous phenomenon with some famous examples. All of us have heard of those students who think all the triangles should have a base parallel to the edge of the paper since all the triangles they had seen were as such. I am also sure, every teaching day, we observe one or two of these mis-generalizations based on examples. The puropse of this post is to share these observations.

          What is your series of unfortunate examples?
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Personally, I refer to this phenomenon as students "submarining" a broken understanding on a particular kind of problem.

Example #1: Our in-house elementary algebra textbook, in its first edition, had this problem:

If $1.05x = 22.05$, then $x = ?$

Note that the result is the same whether the student correctly divides both sides by 1.05, or incorrectly thinks that they can subtract 1.05 ($x = 21$).

Example #2: Many textbooks will have an example of reducing rational expressions like this:

Reduce $\frac{x^2 - 9}{x+3}$.

Observe that the numerator is a difference of squares, and the denominator is one of its factors. I went several years before one group of students pointed out an "easier" (incorrect) way of accomplishing the reduction: just divide the numerator and denominator term-wise. That is: $x^2/x = x$, and $-9/3 = -3$, so the answer is $x-3$.

Note that more disturbingly than just this one example, this broken process does actually generate the correct answer for any problem of this form. Any difference of squares on top (any numbers), and any of its factors on the bottom (regardless of which sign appears), will behave the same way.

Conclusion: It really is critically important for instructors to work out the details of any exercise in advance, and pay close attention to any mechanics that may be confusing or obfuscate the initial mechanism. Students being able to generate the correct answer by some broken process will allow the broken understanding to be "submarined" and not visible. As an instructor, I feel like a major part of the job is to be able to function like a forensic specialist and "dredge up where the bodies are buried" with a high-quality series of in-class exercises.

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    $\begingroup$ Though in a way your second example is more important, and more common, I loved your first example. It is really one of those unfortunate ones that you say "how come?", the ones that you need one of CSI characters as your forensic specialist. $\endgroup$ – Amir Asghari Mar 18 '16 at 22:46
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One mathematical example that has been explored is the somewhat pathological nature of "anomalous fractions" where digit cancellation produces correct simplification. For instance:

$$\frac{16}{64} = \frac{1}{4}$$

Why? Well... just cancel the $6$s!

The Wolfram link above to anomalous cancellation is missing a few important citations. I dug these up while co-writing a paper with Bharath Sriraman (google scholar) which has been sent off:

Sriraman, B. & Dickman, B. (in press). “Mathematical Pathologies as Pathways into Creativity.” ZDM, 49(2).

Sriraman had known of such an example via the somewhat recent paper of Osler:

Osler, T. (2007). Lucky fractions: Where bad arithmetic gives correct results. Journal of Mathematics and Computer Education, 41(2), 162-167

I had been aware of an earlier book, which perhaps fits the OP's question more generally:

Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Greenwood Publishing Group. Google Books.

The author was a doctoral student of Stephen Brown, whose name may be familiar from a wonderful reference: Brown and Walter's The Art of Problem Posing (and its various reprintings and follow-up compendium, Problem Posing: Reflections and Applications).

Consider the excerpt from Reconceiving as follows: (p. 3):

enter image description here

The book looks at a variety of "error case studies" (p. 12) for which I have mentioned F/O (click through for larger resolution):

enter image description here

About the fraction cancellation, Borasi footnotes: "Further inquiry about possible generalizations of the 'error' discussed here was also pursued by a student in one of my teacher education courses and later published (see Johnson, 1985)" (p. 71). That reference goes to:

Johnson, E. (1985). Algebraic and Numerical Explorations Inspired by the Simplification: 16/64 = 1/4. Focus on Learning Problems in Mathematics, 7, 15-28.

So: I recommend this family of citations on anomalous cancellation, in particular, and suggest looking through Borasi's book more generally to see if there are other "unfortunate examples" of interest!

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    $\begingroup$ In fact, I was aware of Borasi's work via a bit earlier publication that might be also of your interest: Borasi, R.(1994), 'Capitalizing on Errors as "Springboards for Inquiry", A Teaching Experiment', Journal for research in mathematics education 25 (2), 166-208. $\endgroup$ – Amir Asghari Mar 19 '16 at 10:21
  • $\begingroup$ @AmirAsghari Yes, the article to which you refer (and others of Borasi and colleagues) were drawn from in compiling the book at hand. $\endgroup$ – Benjamin Dickman Mar 19 '16 at 23:21
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mis-generalizations based on examples

  1. From calculus course (I'm TA)

There is a limit problem on midterm $$\lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}$$

The answer is 0 and the problem is expected to be solved by multiplying $\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}$ but this question makes many people believe they can do $\infty-\infty=0$.

  1. From abstract algebra course. (I'm student)

Example given in lecture: $\mathbb{Z}[x]$ is not a PID because $(2,x)$ cannot be generated by an element.

Midterm: Prove or disprove every ideal in $\mathbb{Z}[x]$ can be generated by 2 elements.

Answer: Consider $(4,2x,x^2)$......not hard to do the rest. Many people include myself believed the statement is true because we only remembered the "standard example" given in lectures.

  1. From the same abstract algebra course.

Example given in lecture: Let $p$ be prime, $f(x)=x^{p-1}+...+x+1$ is irreducible in $\mathbb{Z}[x]$. To prove it, use Eisenstein Criterion on $f(x+1)$.

Midterm: True or False, let $p$ be prime, $f(x)=x^{p-1}+...+x+1$ is irreducible in $(\mathbb{Z}/p\mathbb{Z})[x]$.

Answer: False, $f(1)=0$ in $\mathbb{Z}/p\mathbb{Z}$ therefore...... and before you ask, yes, I got tricked again. :(

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    $\begingroup$ Your last two examples are literally unfortunate :) Thank you for sharing them. I hope your midterm had some other questions as well :) $\endgroup$ – Amir Asghari Mar 19 '16 at 11:47

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