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## Vi Hart and the teaching of Mathematics

For those of you who are not "in the know" there are a set of Mathematical Doodling videos circulating Facebook and blogs (thus) by a self proclaimed Recreational Mathematician Vi Hart.

now if you watch Vi's video collection you get the distinct impression that she dislikes the way that mathematics is taught. I think that most people who have found a passion for mathematics probably agree that modern math instruction has very little to do with any of the beautiful or particularly interesting aspects of mathematics. There is very little inspirational in a list of properties such as:

a + b = b + a
a * b = b * a
a + ( b + c ) = ( a + b ) + c
a * ( b * c ) = ( a * b ) * c
a ( b * c ) = a * b + a * c
0 + a = a
1 * a = a
0 * a = 0

and learning the names of these properties when you are in grade school really serves no purpose what so ever. I suppose that when your kids are arguing over who sits where in the back seat you could yell that its commutative and they might understand. Or at least be momentarily stunned as the fear of an impromptu math lesson washes over them.

But it has always crushed me that kids will discover the above properties on their own as they struggle to find ways to understand arithmetic and teachers will chide them for not doing things the way they were taught. For example many kids are like me and are not very good at arithmetic and so will look for ways to simplify a problem:

56 * 31 = 56 * 30 + 56 = (56 + 56 + 56) * 10 + 56 = "add 56 three times tack on a zero and then add 56 1 more time"

Sure this seem like a lot of extra work and really it is just the multiplication algorithm broken up, but in order to REALIZE that this works you need to fundamentally understand the properties of numbers and THAT IS THE WHOLE POINT. Kids come up with this stuff on their own. Rather than showing them that they
A. They are right.
B. They *discovered something* that WAS something.

teachers generally chide them for not "doing it right".

I was always told: You can't use shortcuts because shortcuts don't always work. I would be really confused because all of my attempts to find a situation where it didn't work failed. So it seemed to me that it must always work. No teacher ever showed me counter examples to one of my "shortcuts" and no teacher EVER help me prove or disprove my various theories.

Because I continued to have terrible trouble in math and always had to "teach myself" ways of "seeing" why things worked I eventually found that despite my teacher's best efforts I was actually very good at math. I am dyslexic and still HORRIBLE at arithmetic. But to compensate I needed to find ways of breaking problems down into smaller problems and that required me to understand the structure of numbers and that made understanding algebra a little easier and eventually someone pointed out to me that I was rather good at math.

But I refuse to give any of my grade school or middle school teacher's ANY credit for it. Sure, one could say that by torturing me, beating at my self-esteem etc. they forced me to compensate for my "disabilities" -- but I can now only imagine what things might have been like if we had worked WITH my abilities rather than against them.

I don't think it takes much digging to find disdain for the way in which mathematics is taught. A subject that is full of patterns, and discovery, and wonder gets reduced to multiplication tables and rules with big names like "the commutative property of addition".

Anyway if you know a kid who seems to hate math maybe showing them some of Vi Hart's videos will help a little.

### 18 Comments On This Entry

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#### EnvXOwner

18 December 2010 - 10:55 AM
Twas an excellent video. I to suffer at mathematics.
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#### z3r0sh1ft

18 December 2010 - 01:35 PM
Very nice. Now, why can't I think like that?
I was never really happy learning math at school. I always questioned what was that specific problem's purpose and very few did I ever get an answer. It seemed math was just "x + 1 = 2, now solve it" with no real reason as to why.
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#### Munawwar

19 December 2010 - 12:25 AM

Quote

No teacher ever showed me counter examples to one of my "shortcuts"...

I'd like to know what this "shortcut" is

Quote

I don't think it takes much digging to find disdain for the way in which mathematics is taught.

Its not only math. Same applies to chemistry and physics. Actually, in my memory, chemistry was the worst (except the lab experiments).
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#### NickDMax

19 December 2010 - 08:41 AM
yea most of my "shortcuts" probably are not really short. For example to multiply by 9 I often do 10*(n-1)+10-n which actually sounds more complicated than it is. while I don't have my multiplication tables memorized I do have "completions of 10" memorized. That is if you say 6 I know 4 is its 10-completion. (10's compliment?).

So while I have no idea what 9*7 is, I know in an instant that it is 63. Again it is a calculation I do not a memorization. Normally I would do this on my fingers, but since using fingers to do math is basically frowned upon (although you should always let your kids do it as it makes math and numbers more physical for them) I had to find ways of hiding what I was doing.

A teacher might say, but that does not work for 13 but since that breaks up into 90 + 27 the skill still gave me a way to solve the problem.

I should also note that I don't think I really have the completions of 10 memorized... I still catch myself counting them up sometimes.

In chemistry I needed a way to deal with memorizing formulas. Basically what I did was memorize how to "draw the table" -- so I could quickly sketch out the periodic table of elements (I never did get all of the elements but I did get quite a few -- unfortunately I didn't know the names, but the symbols were helpful enough) or I would ditto-copy in my head some table of acids or whatever from the book. Note that I had no way to access the information before I "drew the table".

I did this in math for a little while. For example the quadratic equation was initially just a picture in my head and I had to draw it before I could convert it into a meaningful mathematical formula. However with math things changed and eventually I "understood" the formulas. I do not generally memorize them. Very often I have to derive them from lesser knowledge. Yes this makes me VERY SLOW at math and can make taking tests rather difficult if there is not a "formula table" I can look at.

For calculus I would often do a "table dump" of the rules for derivatives or integrals. Again not really some kind of memorization but a quick run down of operations on simple graphs. IF I didn't know what the various operations did to a graph I would not know how to take derivatives. IF I didn't know how to take the derivatives I probably could not take integrals either as most of those formulas I extract from thinking about how you "reverse" the derivative. (except division -- I have always "drawn" that. I can actually derive it from multiplication but there are some complicated parts in the process and so the shortcut there is just to memorize the formula which for me means "remember how to draw").
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#### smacdav

19 December 2010 - 09:22 AM
I was a mathematics educator for 17 years at the community college level. It drove me nuts that people of all ages, not just small children, were taught that math was about memorizing a bunch of formulas and rules. I tried to teach my students that understanding the problem was the key, but still most of them tried to rely on memorization because it had been drilled into them for years that you couldn't do math unless you memorized things.

One of the real problems with math education in the USA is that the elementary school teachers who teach math generally don't understand it (and don't want to). As far as they are concerned, it is hard and there is only one way to do it; do it a different way and you're doing it wrong. Sadly, even at the level I was teaching, where the instructors all were required to have a masters degree in math or math education, there were still those who discouraged innovative ways of solving a problem. Personally, if a student used an unusual method of reaching an answer, I would (a) make sure that what they did was mathematically correct (in that it would always arrive at the right answer), ( remark on the interesting approach to the problem, and © if the way they did it seemed particularly over-complicated, I would point out an easier way to do it (but that they didn't have to change their approach--it was just something to consider). I learned a lot of interesting ways of approaching problems that way.

Oh, and I have always multiplied by 9 in exactly the way you do; no point in memorizing any more than you have to.
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#### Munawwar

19 December 2010 - 10:02 AM
To multiply by 9 I do 10n-n = 9n

I too had my "shortcuts" for some parts of mathematics/physics. I had derived some *ingenious* formulae. But never used them for any practical purpose.

For certain topics, I would learn/practice in a pure mechanical way rather than logical way.
For chemistry I had around 90 chemical equations to remember, so I wrote all of them without their products (only the LHS), took a scan and made N number of copies. I used to practice everyday. Today, I don't remember even one of them! (2H2+O2->2H2O?).
I once tried to relate all the equations to one another. I ended up with a piece of paper with a mesh of arrows and formulae. Complete mess and an epic fail! I guess there is no other way to remember them.

Integration was something I was good at and I haven't used it for a single practical purpose. Again, I have forgotten it all.

Sad, many portions of my knowledge is gone wasted.
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#### NickDMax

19 December 2010 - 12:17 PM
Here is the thing with my method over memorization. I have not done calculus in years but I do know what "taking an derivative" does to the graph and so I can still quickly start enumerating the derivative formulas. Again from those I can extract integration the corresponds to the various derivative (note that this only works to a point to generate a table of integrals, there are just some that are really too complicated to extract easily -- that is why there is are "tables of integrals" so that you don't HAVE to memorize them).

AS for the 10n-n -- I do that one too if n is large enough. But subtraction is not the easiest for me.
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#### ishkabible

19 December 2010 - 09:04 PM
you now what really ticks me off, when a teacher asks you to right out the steps to a problem then tells you your wrong becuase you didn't do it 'there way'. there was a really simple problem that asked us to write the steps in finding the maximum area of a rectangle forum by a given link of fence. my answer went something " set length equal to the perimeter divided by 4 then set area equal to length squared. f(P)=(P/4)^2" but apparently becuase we where learning 'liner programing' i was supposed to do it that way witch involved writing the equations of 4 lines, finding the 4 intersecting points of these lines, writing a area function and testing each of the intersection points in the area function; needless to say i was baffled at why any one would do it that way.
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#### D.Mulroy

20 December 2010 - 05:54 AM
We use a system at my highschool known as CPM (College Prepatory Math). This curriculum all but gives the math teachers an excuse not to teach. With cpm students are put into groups of four and are encouraged to work as "team" to solve problems. Thats hards to do when you have no instruction what so ever from a teacher. We are expected to combine our critical thinking skills to teach ourselves math. Testing with this program is horrible to and is why my schools math average is the lowest in the county - along side another school whom uses CPM. You have two tests in cpm; a team test (which is worth a very low percentage of your grade) which covers the major topics you had been working on. Students normally do well on this test. Then the individual test. This test covers the smaller topics covered in the previous chapter that you maybe had practice at in three or less homework assignments since cpm thinks "spiralling" is the way to go. Anyways it just frustrates me, and many of the students in my school, that we are getting a cruddy education in math and when we head off to college we are going to be behind in that area. Which especially sucks for students taking math intensive majors.
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#### NickDMax

20 December 2010 - 08:18 AM
@ishkabible -- When learning particular techniques for solving problems often you need to choose simplified examples and then restrict how students are to approach the problem. This is done all of the time in programming. Often there is a pre-built function in a language to do something like sort or search, but there is more value in a student writing a sort/search for themselves then in using the built in function.

You see the question was not: "Find the maximum area with a given rectangular perimeter" it was: "Demonstrate linear programming techniques to find the maximum area with a given rectangular perimeter".

Personally I would probably approached the problem from calculus unless told to use linear programming.

Of course I do feel your pain. As I often was marked off for not using FOIL (first outer inner last) -- as I use what I call the "cannon" method which will work when multiplying any size polynomial (although the battle can get pretty messy for large polynomials). I am terrible at spelling (dyslexic) so acronyms are not particularly helpful to me.

Though phrases are. I will always remember "SOH CAH TOA" == "Some Ornery Hamsters Can Always Handle Trained Oriental Attackers" which was blurted out on the spot by a friend of mine when asked if we could come up with something better than SOHCAHTOA -- it was so sudden and so outlandish that it has stuck with me. I don't actually use it -- but it is always in my head. -- But whenever someone asks what that "Soak your toe-a" thing was, I run though the phrase and reconstruct the acronym.

@D.Mulroy -- It sounds like your school is poorly executing a system that probably has potential. The biggest problem with such systems are they the teachers really need to know their mathematics for it to be successful. They need to be able to point the "eager young minds" in the right direction. The thing is that it only takes a slight scratch at the surface of mathematics to uncover unanswered questions.
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#### Munawwar

20 December 2010 - 10:54 AM
I didn't know about "SOH CAH TOA". We were taught it's Indian version, hah.
Long ago, I formed a diagrammatic representation for all needed trigonometric values. It made acronyms unnecessary.
.
For 60 degree, 1 and root 3 would be swaped.
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#### NickDMax

20 December 2010 - 12:57 PM
The reason I don't use SOHCAHTOA is because of computer graphics. In my mind COS is the X-axis, SIN the the Y-axis and TAN is the slop (rise over run or y/x or SIN/COS)

These come from the formulas for converting polar coordinates to rectangular:

x = r * cos(theta)
y = r * sin(theta)

which come up all over the place for example with Phasors: r*e^(i * thata) = r*cos(theta) + i*r*sin(thata) which is plotted with the real-axis as x and the imaginary as y is the same basic formula as used for conversion from polar to rectangular. It also comes up when you want to do rotations and all kinds of other neat things.

So in my head. X is the COS, Y the SIN. So a little bit of algebra:

cos(theta) = x / r
sin(theta) = y / r
tan(theta) = m = y / x = sin(theta)/cos(theta)

of course there is a picture here too of a triangle with r and it hypotenuses extending from the origin.

So for me there is a model that I understand and from the model I can extract information, and if the extraction is simple I don't even have to be aware of the steps because it just kind of "makes sense" and I can "see" formulas without much effort.

This is the part that math teachers don't seem to understand. Mathematics is a language. It "speaks" to you. If you get an understanding of the model being described by a formula then it is much easier to remember the formula because it actually "says" something about the model.
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#### moopet

20 December 2010 - 02:31 PM
I believe that at some point in the history of mathematics in my school, one Silly Old Harry may well have Caught A Herring while Trawling Off America.
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#### Munawwar

20 December 2010 - 11:38 PM
And there was the resistor color code in physics (BBROYGBVGW). All were taught the boring "B B ROY of Great Britain has a Very Good Wife"...but our teacher had the courage to teach us a better one - the first one in this link. lmao and will never forget it.
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#### D.Mulroy

21 December 2010 - 05:42 AM

Quote

@D.Mulroy -- It sounds like your school is poorly executing a system that probably has potential. The biggest problem with such systems are they the teachers really need to know their mathematics for it to be successful. They need to be able to point the "eager young minds" in the right direction. The thing is that it only takes a slight scratch at the surface of mathematics to uncover unanswered questions.

All but maybe two fo the math teachers in my school use this program as an excuse for them to sit in the back of the class and chat on the districts e-mail. Really quite sad and annoying.
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#### NickDMax

21 December 2010 - 08:14 AM

Munawwar, on 21 December 2010 - 01:38 AM, said:

And there was the resistor color code in physics (BBROYGBVGW). All were taught the boring "B B ROY of Great Britain has a Very Good Wife"...but our teacher had the courage to teach us a better one - the first one in this link. lmao and will never forget it.

is it bad that was the one we learned (granted it was the Navy) with no alternatives? I sure didn't learn it in physics class -- why does your physics class cover basic electronics?
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#### Munawwar

21 December 2010 - 09:12 AM
I don't remember where we used the resistor. All I remember that it was for a lab class.
Anyhow I still use it for calculating resistor values
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#### ishkabible

21 December 2010 - 08:47 PM
i use tiger phases to remember things like trig functions. any i time i say (out load) "opposite over adjacent" i automatically remember tangent. same for "opposite over hypotenuse" and "adjacent over hypotenuse". for some reason or another i don't seem to have trouble remembering the reciprocals. "hypotenuse over adjacent" doesn't bring secant to my head, i just seem to remember it's the reciprocal of cos.

edit: by the way @Munawwar, those acronyms are horrible
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