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Math Mistakes (mathmistakes.org)
130 points by DanBC on Sept 28, 2015 | hide | past | web | favorite | 42 comments

This concept of "teaching with keywords" absolutely terrifies me.

I distinctly remember encountering Metric Geometry* in high school. It didn't teach me geometry, it taught me how to think, and I'm forever grateful. Teaching math with keywords is the opposite of this.

[*] Probably not the correct name in English. It was geometry without algebra; proving theorems and finding loci based on some axioms about angles, parallel lines and so on. What is this called? Euclidean Geometry?

It was geometry without algebra; proving theorems and finding loci based on some axioms about angles, parallel lines and so on. What is this called? Euclidean Geometry?

"Euclidean geometry" would do for a name, but to specifically emphasize the point that algebra is not involved, you can say "synthetic geometry." The geometry of the plane can also be taught from an analytic geometry perspective (using ordered pairs of points on the Cartesian plane) as a first high school course, as in the book Vectors and Transformations in Plane Geometry by Tondeur.[1]

[1] http://www.mathpop.com/bookhtms/tondeur.htm

Interesting enough, you can also build your whole geometry on the operation of reflection and a bunch of axioms. See Calculus of Reflections at eg http://rd.springer.com/article/10.1007%2Fs00022-012-0123-5 (surprisingly for Springer, not behind a paywall).

There's algebra and calculating, but with reflection operations here, not with coordinate pairs.

> (surprisingly for Springer, not behind a paywall).

OK, that was only true at work.

Yes, it sounds very familiar. Thanks!

One of the posts illustrates that nicely, I think.


There's a sheet of a bunch of sums. The student has got them all correct. But we don't know if the student actually understands the concepts, or if the student is using their wrong understanding and an algorithm that coincidentally gives the right answer.

The comments give a good take down of that style of teaching.

I didn't find it to be a particularly convincing "takedown". What's being taught in the example is clearly "the ability to execute a simple algorithm" not "an understanding of the concepts" so criticizing it for not doing the latter seems silly. Maybe it is the case as one commenter says that second graders should be doing the latter and not the former until later grades. OTOH maybe not! I'd be curious as to the evidence either way.

> But we don't know if the student actually understands the concepts, or if the student is using their wrong understanding and an algorithm that coincidentally gives the right answer.

Reminds me of https://en.wikipedia.org/wiki/Chinese_room

I remember that class, IIRC it was just called Geometry at my school. It took me about six weeks to figure out that we weren't doing calculations and so I felt like I was eternally weeks behind that semester as I struggled to internally catch up with the theorems and principles being taught. I always felt like the teacher was doing a good job, I had just missed something (I may have missed a couple of crucial days at the beginning of the year).

I do. Thanks :)

Ah, Geometry, the great divider. I was fiddling with game development in late high school and the most natural approach to geometric problems for me was to just think in terms of coordinates and disregard the geometric primitives per se. A friend of mine was the same way - just plop a coordinate system, express every point as an algebraic equation and away we do with all these pesky lines on our pages, just nice algebra is left. Then you show it to the teacher and he makes the observation you filled two pages with calculations that can be replaced with one geometric property and the addition of one line to the diagram. Fun times :) .

Teaching kids to use keywords to help decode word problems has been a standard practice for at least a generation.

Not in the terrible third world education system I went through, thankfully.

[No sarcasm here, it is generally terrible, but I went to a relatively good place]

Where and when I went to high school, it was "Plane Geometry."

People interestd by this may also be interested in 'The Most Common Errors in Undergraduate Mathematics': http://www.math.vanderbilt.edu/~schectex/commerrs/

This particular entry is quite interesting, it gets at a distinction I'm not sure I had explicitly thought about before: http://mathmistakes.org/recursive-and-relational-thinking-an...

I think the author of that post undervalues the strength of the recursive approach. It is, in general, much easier to apply a recursive approach, and once you have done so you can often mechanically convert your solution into a relational one. Even if you cannot, you have still converted the question into a standard notation math problem, which is normally the hardest part of the problem.

Can anyone think of a "writing mistakes" version of this blog?

Making public another's writing may be too far.... But it would be interesting to mine the data and classify non-grammatical writing mistakes, specifically to ease the stress of grading college junior research papers -- some of which are written at a middle-school writing level -- by parsing sentences and auto-applying applicable writing mistake classification, to which I can gently glide my eyes to and convert to something intelligible.

I had a quick look and I didn't see anything on the site about whether dyslexia or dyspraxia have common errors that show up in math.

There was an interesting programme on BBC Radio Four this morning about how dyslexia and dyspraxia can show up in art. http://www.bbc.co.uk/programmes/b06d2fxf

In theory the relatively recent adoption of Common Core math teaching techniques should help with many of these sorts of mistakes. I'm not well versed enough to know whether there's any sign of that already, but certainly the intention of teaching students to actually think about how math works and why a certain technique produces the right answer is laudable.

Maybe Math Mistakes needs a related site "Grammar and Spelling Mistakes":

I notice that the kid didn’t write them as (x,y) but wrote them as x,y. I wonder how come he did that? Or, more precisely, I wonder if he doesn’t see much of a difference between (x,y) and x,y or if three is some other reason for leaving off the parentheses.

It is interesting to see kids invent their own perfectly adequate, if peculiar, notation. In the first example with the quadrilaterals, the answers were all right. The math is not wrong, just written differently.

That gave me mixed feelings. On the one hand, the teacher did an admirable job (for a teacher) of seeing the correct thinking behind the "wrong" notation. On the other hand, the teacher still accused the student of making a mistake.

If nobody else understands your right answer, it may not be worth much. At this level, it doesn't matter at all--anyone can see based on the constraints what they're doing and understand them.

But this may not always be the case. For example, when they get to intervals and suddenly need to properly discriminate between intervals like (1,2], (1,2), [1,2) and [1,2].

I might ding that by just one point--if you use non-standard notation, you must properly define it. You won't always be dealing with cases where your idiosyncrasies have obvious answers and it's better to form good habits that allow you to communicate clearly with others. There may be many good ways to depart from convention and I'd hesitate to stop anyone from doing that, but I think that if defining your departure from convention is too burdensome to be worthwhile, then you do not have a compelling case to depart from it in the first place.

If I were a teacher and I saw a student come up with a peculiar but correct notation, I'd be ecstatic.

Indeed, developing new notation should be not only allowed, not only encouraged, but even required as an assignment at various points during mathematical training, at every age level. Also to be encouraged is conscientious analysis and critique of existing notations, algorithms, and explanations.

Society advances in two ways: in small incremental steps, and in large leaps. When we stick to existing notations and concepts we can make incremental improvements. But the large leaps often come from creating better notations and better conceptual understandings, which can turn substantial amounts of knowledge and logical thinking into obvious consequences of an organized and intuitive system.

Agreed. Feynman diagrams are a great example of this phenomenon. They might be 'pictures', but they are still mathematical notation.

I'd say "the notation x/y means a fraction, and cannot be used for coordinates. Likewise, (x, y) means a coordinate and cannot be used for fractions. They are related but should not be conflated."

That makes it sound way to absolute. You should say something more like "please write coordinates like (x,y)".

Incidentally, in some contexts, (x,y) is how we write fractions (they are just an equivalence class based on an ordered pair of integers).

As long as I am on the subject, (x,y) also means the inner product of x and one. And I had one teacher for whom (x,y) means f(x,y) where f would change more or less every section based on convience.

On second thought, maybe we should be a little stricter in teaching students that math notation is rigid. If we raise a generation of mathematicians believing it, then they might clean up the current mess of notation we have today.

If you want to make a mathematician sweat, just ask them whether ((a, b), c) is the same as (a, b, c).

As I once said after someone, after a discussion on some trivial mathematical argument that he didn't grasp, said "OK, but one plus one equals two, right?": "it depends".

The answer is "What does '(a, b)' mean?"

Yes, up to isomorphism.

It might be that in reading the teachers comment, he parsed the parentheses as part of English grammar, not the mathematical notation. Similar to how, if the comment said to write coordinates as "x,y", I would not include the quotation marks in my answer.

That was my thinking as well.

For the future people: http://mathmistakes.org/naming-coordinates-feedback-and-revi...

(I linked to the site in general, and that's the post currently on that site's front page).

Here's the old go-to for English errors: http://public.wsu.edu/~brians/errors/errors.html

I think they call that "every internet forum or comment section ever".

It would be interesting to see a new teaching methodology develop around teaching mostly just the concepts that people get wrong with the assumption that most people don't need much help with the stuff everyone is getting wrong.

haha i totally remember that "law of sines" example. nothing to do with the "law of sines" over time, i would regularly just exhaust my brain and i would just imaginarily throw random variables to give me solutions, making my homework full of errors lol.

Amazing blog!

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