I mentioned on Twitter yesterday that one way to relieve the boredom of grading math papers is to explore mistakes. If a statement is wrong, what would it take to make it right? Is it approximately correct? Is there some different context where it is correct? Several people said they’d like to see examples, so this blog post is a sort of response.
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One famous example of this is the so-called Freshman’s Dream theorem:
(a + b)p = ap + bp
This is not true over the real numbers, but it is true, for example, when working with integers mod p.
(More generally, the Freshman’s Dream is true in any ring of characteristic p. This is more than an amusing result; it’s useful in applications of finite fields.)
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A common misunderstanding in calculus is that a series converges if its terms converge to zero. The canonical counterexample is the harmonic series. It’s terms converge to zero, but the sum diverges.
But this can’t happen in the p-adic numbers. There if the terms of a series converge to zero, the series converges (though maybe not absolutely).
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Here’s something sorta along these lines. It looks wrong, and someone might arrive at it via a wrong understanding, but it’s actually correct.
sin(x – y) sin(x + y) = (sin(x) – sin(y)) (sin(x) + sin(y))
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Odd integers end in odd digits, but that might not be true if you’re not working in base 10. See Odd numbers in odd bases.
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You can misunderstand how percentages work, but still get a useful results. See Sales tax included.
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When probabilities are small, you can often get by with adding them together even when strictly speaking they don’t add. See Probability mistake can make a good approximation.
Typo: might NOT be true
A post on mistakes needs at least one typo. :)
YES to this!
When my 3-year attempt to become a full-time independent consultant had hit a huge lull, where I was in that middle space of having no clients but also having no desire to give up, I realized I still needed to productively fill my empty hours.
I spent the first half of 2018 volunteering at a local high school, which gave me lunch and the time after 2:30 PM to build my network and pursue clients. I was primarily tasked with tutoring math with students who needed only their math credits for graduation. The pressure was extreme, as was the students’ emotional frustration, anxiety and occasionally panic. These students were self-paced, the primary goal being to get those math credits. About half were on their second senior year, two on their third.
I foundered badly at the start. Then I started viewing students’ incorrect answers not as something to “fix”, but as their own steps on their own path to learning math. So I instead started Socratic dialogs on those problems, letting students find for themselves what went awry, consulting the text and their own common sense.
This approach did get encouraging results (after the obligatory period of “Just tell me!” pleas of frustration). At first I thought they were simply becoming more familiar with the material, but on their own terms, rather than how the established pedagogy had been feeding them.
That would be wrong, well, certainly not primary: I soon learned what was happening was that these students were learning to “struggle productively”. They were learning to “own their shit” and push through it.
This often meant understanding when taking a break was the most productive path toward the solution. They were beginning to separate a difficult emotional state from the work at hand. They were learning to care for their mental and emotional health, rather than blame it on math.
Their “mistakes” often became the “excuse” for taking a break. I’d send them out for a walk around the quad in our gorgeous weather. I was fortunate the teacher I was working with gave me some leeway in this area, for she didn’t like students to miss any precious time in class. The students generally returned ready to work, to dig through their prior solution to understand what went wrong, then do better.
I was delighted when they started figuring things out without my Socratic prompting. They had converted their “struggle” into a productive personal process. And they had outgrown any need for my help, outside of an initial quick walk-through of new material.
But best of all were those “I got it!” moments, when, on their own, students came to new understandings. There was one particular student who had made minimal progress during the first third of the semester. After only 2 weeks of the Socratic Method, she became able to make progress on her own. One of the best things about tutoring her was the low “OhhhOOOOOoooh” that would escape her when she “got it”. I knew she was doing well if I heard that once in a while.
She then grew firm in her determination to finish the class on time, and to graduate with her peers, despite having only 5 weeks left to do 9 weeks of work. She became good at estimating her “enough to pass” knowledge level, and would attack exams fearlessly, despite knowing there was much she still didn’t know.
This also meant she was still learning *during* the exam. I remember well one particular exam during which she had three “OhhhOOOOOoooh” moments. I rushed to take her exam from her when she was finished. She scored a 95%, her highest math exam score ever.
After her initial burst of joy, she grabbed back her exam, determined to correctly solve the one problem she had missed.
I was struggling to hold back tears. It wasn’t just the math success: She was becoming a woman who would not be beaten by her own limitations or those others placed upon her.
She totally owned her shit.
Math can do that.